{"Proposition of Appendix shows that a suf\ufb01cient condition\nfor the identi\ufb01ability in the case of Gaussian and Boltzmann\nlinear policies is that the second moment matrix of the fea-\nture vector Es\u0012d\u0019\u0006\n\u0016\u0010\n\u001epsq\u001epsqT\u0018\nis non\u2013singular along with\nthe fact that the policy \u0019\u0012plays each action with positive\nprobability for the Boltzmann policy.\nConcentration Result We are now ready to present a con-\ncentration result, of independent interest, for the parameters\nand the negative log\u2013likelihood that represents the central\ntool of our analysis (details and derivation in Appendix).\nUnder Assumption and Assumption, let D\u0010\ntpsi;aiqun\ni\u00101be a dataset of n\u00a10independent sam-\nples, where si\u0012d\u0019\u0012\u0006\n\u0016 andai\u0012\u0019\u0012\u0006p\u0004|siq. Let\np\u0012\u0010argmin\u0012P\u0002p`p\u0012qand\u0012\u0006\u0010argmin\u0012P\u0002`p\u0012q. If\nthe empirical FIM:\npFp\u0012q\u00101\nnn\u00b8\ni\u00101Ea\u0012\u0019\u0012p\u0004|sq\u0010\ntps;a;\u0012qtps;a;\u0012qT\u0018\n(1)\nhas a positive minimum eigenvalue p\u0015min\u00a10for all\u0012P\u0002,\nthen, for any \u000ePr0;1s, with probability at least 1\u0001\u000e:\n\u000f\u000f\u000fp\u0012\u0001\u0012\u0006\u000f\u000f\u000f\n2\u00a4\u001b\np\u0015minc\n2d\nnlog2d\n\u000e:\nFurthermore, with probability at least 1\u0001\u000e, individually:\n`pp\u0012q\u0001`p\u0012\u0006q\u00a4d2\u001b4\np\u00152\nminnlog2d\n\u000e\np`p\u0012\u0006q\u0001p`pp\u0012q\u00a4d2\u001b4\np\u00152\nminnlog2d\n\u000e:\nThe theorem shows that the L2\u2013norm of the difference be-\ntween the maximum likelihood parameter p\u0012and the true pa-\nrameter\u0012\u0006concentrates with rate Opn\u00011{2qwhile the like-\nlihoodp`and its expectation `concentrate with faster rate\nOpn\u00011q. Note that the result assumes that the empirical FIM\npFp\u0012qhas a strictly positive eigenvalue p\u0015min\u00a10. This con-\ndition can be enforced as long as the true Fisher matrix Fp\u0012q\nhas a positive minimum eigenvalue \u0015min, i.e. under identi-\n\ufb01ability assumption (Lemma) and given a suf\ufb01ciently large\nnumber of samples. Proposition of Appendix provides the\nminimum number of samples such that with probability at\nleast1\u0001\u000eit holds that p\u0015min\u00a10.\nIdenti\ufb01cation Rule Analysis The goal of the analysis of\nthe identi\ufb01cation rule is to \ufb01nd the critical value cp1qso that\nthe following probabilistic requirement is enforced.\nLet\u000eP r0;1s. An identi\ufb01cation rule producing pIis\u000e\u2013\ncorrect if:Pr\u0000pI\u0018I\u0006\b\n\u00a4\u000e.\nWe denote with \u000b\u00101\nd\u0001d\u0006E\u0010\u0007\u0007 \niRI\u0006:iPpIc(\u0007\u0007\u0018\nthe ex-\npected fraction of parameters that the agent does not control\nselected by the identi\ufb01cation rule and with \f\u00101\nd\u0006E\u0010\u0007\u0007 \niP\nI\u0006:iRpIc(\u0007\u0007\u0018\nthe expected fraction of parameters that the\nagent does control not selected by the identi\ufb01cation rule. Wenow provide a result that bounds \u000band\fand employs them\nto derive\u000e\u2013correctness.\nLetpIcbe the set of parameter indexes selected by the\nIdenti\ufb01cation Rule obtained using n\u00a10i.i.d. samples\ncollected with \u0019\u0012\u0006, with\u0012\u0006P\u0002. Then, under Assump-\ntion and Assumption, let \u0012\u0006\ni\u0010argmin\u0012P\u0002i`p\u0012qfor all\niP t1;:::;duand\u0017\u0010min \n1;\u0015min\n\u001b2(\n. Ifp\u0015min\u00a5\u0015min\n2?\n2and\n`p\u0012\u0006\niq\u0001lp\u0012\u0006q\u00a5cp1q, it holds that:\n\u000b\u00a42dexp\"\n\u0001cp1q\u00152\nminn\n16d2\u001b4*\n\f\u00a42d\u00011\nd\u0006\u00b8\niPI\u0006exp\"\n\u0001plp\u0012\u0006\niq\u0001lp\u0012\u0006q\u0001cp1qq\u0015min\u0017n\n16pd\u00011q2\u001b2*\n:\nFurthermore, the Identi\ufb01cation Rule is ppd\u0001d\u0006q\u000b\u0000d\u0006\fq\u2013\ncorrect.\nSince\u000band\fare functions of cp1q, we could, in prin-\nciple, employ Theorem to enforce a value \u000e, as in De\ufb01ni-\ntion, and derive cp1q. However, Theorem is not very attrac-\ntive in practice as it holds under an assumption regarding the\nminimum eigenvalue of the FIM and the corresponding es-\ntimate, i.e. p\u0015min\u00a5\u0015min\n2?\n2, that cannot be veri\ufb01ed in practice\nsince\u0015minis unknown. Similarly, the constants d\u0006,lp\u0012\u0006\niq\nandlp\u0012\u0006qare typically unknown. We will provide in Section\na heuristic for setting cp1q.\nPolicy Space Identi\ufb01cation in a Con\ufb01gurable\nEnvironment\nThe identi\ufb01cation rules presented so far are unable to dis-\ntinguish between a parameter set to zero because the agent\ncannot control it, or because zero is its optimal value. To\novercome this issue, we employ the Conf\u2013MDP properties\nto select a con\ufb01guration in which the parameters we want to\nexamine have an optimal value other than zero. Intuitively, if\nwe want to test whether the agent can control parameter \u0012i,\nwe should place the agent in an environment !iP\nwhere\n\u0012iis maximally important for the optimal policy. This intu-\nition is justi\ufb01ed by Theorem, since to maximize the power\nof the test ( 1\u0001\f), all other things being equal, we should\nmaximize the log\u2013likelihood gap lp\u0012\u0006\niq\u0001lp\u0012\u0006q, i.e. parame-\nter\u0012ishould be essential to justify the agent\u2019s behavior. Let\nIPt1;:::;dube a set of parameter indexes we want to test,\nour ideal goal is to \ufb01nd the environment !Isuch that:\n!IPargmax!P\ntlp\u0012\u0006\nIp!qq\u0001lp\u0012\u0006p!qqu; (2)\nwhere\u0012\u0006p!q Pargmax\u0012P\u0002JM!p\u0012qand\u0012\u0006\nIp!q P\nargmax\u0012P\u0002IJM!p\u0012qare the parameters of the optimal\npolicies in the environment M!in\u0005\u0002and\u0005\u0002Irespec-\ntively. Clearly, given the samples Dcollected with a sin-\ngle optimal policy \u0019\u0006p!0qin a single environment M!0,\nsolving problem (2) is hard as it requires performing an off\u2013\ndistribution optimization both on the space of policy param-\neters and con\ufb01gurations. For these reasons, we consider a\nsurrogate objective that assumes that the optimal parameter\nin the new con\ufb01guration can be reached by performing a sin-\ngle gradient step\nLetIPt1;:::;duandI\u0010t1;:::;duzI. For a vector v, we\ndenote with vjIthe vector obtained by setting to zero the": "\\documentclass[letterpaper]{article} % DO NOT CHANGE THIS\n\\usepackage{aaai20}  % DO NOT CHANGE THIS\n\\usepackage{times}  % DO NOT CHANGE THIS\n\\usepackage{helvet} % DO NOT CHANGE THIS\n\\usepackage{courier}  % DO NOT CHANGE THIS\n\\usepackage[hyphens]{url}  % DO NOT CHANGE THIS\n\\usepackage{graphicx} % DO NOT CHANGE THIS\n\\urlstyle{rm} % DO NOT CHANGE THIS\n\\def\\UrlFont{\\rm}  % DO NOT CHANGE THIS\n\\usepackage{graphicx}  % DO NOT CHANGE THIS\n\\frenchspacing  % DO NOT CHANGE THIS\n\\setlength{\\pdfpagewidth}{8.5in}  % DO NOT CHANGE THIS\n\\setlength{\\pdfpageheight}{11in}  % DO NOT CHANGE THIS\n\n\n\\usepackage{booktabs}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{dsfont}\n\\usepackage{xspace} \n\\usepackage{amsthm}  \n\\usepackage{thmtools}\n\\usepackage{thm-restate}\n\\usepackage{etoolbox}\n\\usepackage{bm}\n\\usepackage{amsfonts}\n\\usepackage[capitalise, noabbrev]{cleveref}\n\\usepackage{pifont}\n\\usepackage{fixltx2e}\n\\usepackage{mathtools}\n\\usepackage{mathabx}\n\\usepackage{algorithm}\n\\usepackage{algorithmicx,algpseudocode}\n\\usepackage{mathtools}\n\\usepackage{subcaption}\n\\usepackage{mathrsfs}\n\\usepackage[percent]{overpic}\n\n\n\n\n\\begin{document}\n\nProposition of Appendix shows that a sufficient condition for the identifiability in the case of Gaussian and Boltzmann linear policies is that the second moment matrix of the feature vector $\\mathbb{E}_{s \\sim d_{\\mu}^{\\pi^*}} \\left[ \\mathbf{\\phi}(s)\\mathbf{\\phi}(s)^T \\right]$ is non--singular along with the fact that the policy $\\pi_{\\mathbf{\\theta}}$ plays each action with positive probability for the Boltzmann policy.\n\n\\paragraph{Concentration Result} We are now ready to present a concentration result, of independent interest, for the parameters and the negative log--likelihood that represents the central tool of our analysis (details and derivation in Appendix).\n\n\nUnder Assumption and Assumption, let $\\mathcal{D} = \\{(s_i,a_i)\\}_{i=1}^n$ be a dataset of $n>0$ independent samples, where $s_i \\sim d_{\\mu}^{\\pi_{\\mathbf{\\theta}^*}}$ and $a_i \\sim \\pi_{\\mathbf{\\theta}^*}(\\cdot|s_i)$. Let $\\widehat{\\mathbf{\\theta}} = arg\\,min_{\\mathbf{\\theta} \\in \\Theta} \\widehat{\\ell}(\\mathbf{\\theta})$ and $\\mathbf{\\theta}^* = arg\\,min_{\\mathbf{\\theta} \\in \\Theta} {\\ell}(\\mathbf{\\theta})$ . If the empirical FIM:\n\\begin{equation}\n\t\\widehat{\\mathcal{F}}(\\mathbf{\\theta}) = \\frac{1}{n} \\sum_{i=1}^n \\mathbb{E}_{a \\sim \\pi_{\\mathbf{\\theta}}(\\cdot|s)} \\left[\\mathbf{\\overline{t}}(s,a,\\mathbf{\\theta})\\mathbf{\\overline{t}}(s,a,\\mathbf{\\theta})^T\\right]\n\\end{equation}\nhas a positive minimum eigenvalue $\\widehat{\\lambda}_{\\min} > 0$ for all $\\mathbf{\\theta} \\in \\Theta$, then, for any $\\delta \\in [0,1]$, with probability at least $1-\\delta$:\n\t\\begin{equation*}\n\t\t\\left\\| \\widehat{ \\mathbf{\\theta}} - \\mathbf{\\theta}^* \\right\\|_2 \\le \\frac{\\sigma}{\\widehat{\\lambda}_{\\min}} \\sqrt{\\frac{2d}{n} \\log \\frac{2d}{\\delta}}.\n\t\t\\end{equation*}\nFurthermore, with probability at least $1-\\delta$, individually: \n\t\\begin{align*}\n\t\t&\\ell(\\widehat{\\mathbf{\\theta}}) - \\ell({\\mathbf{\\theta}}^*) \\le \\frac{d^2\\sigma^4}{\\widehat{\\lambda}_{\\min}^2 n}  \\log \\frac{2d}{\\delta}\\\\\n\t\t&\\widehat{\\ell}({\\mathbf{\\theta}}^*) - \\widehat{\\ell}(\\widehat{\\mathbf{\\theta}})  \\le \\frac{ d^2\\sigma^4}{\\widehat{\\lambda}_{\\min}^2 n}  \\log \\frac{2d}{\\delta}.\n\t\\end{align*}\n\nThe theorem shows that the $L^2$--norm of the difference between the maximum likelihood parameter $\\widehat{\\mathbf{\\theta}}$ and the true parameter ${\\mathbf{\\theta}^*}$ concentrates with rate $\\mathcal{O}(n^{-1/2})$ while the likelihood $\\widehat{\\ell}$ and its expectation $\\ell$ concentrate with faster rate $\\mathcal{O}(n^{-1})$. \nNote that the result assumes that the empirical FIM $\\widehat{\\mathcal{F}}(\\mathbf{\\theta})$ has a strictly positive eigenvalue $\\widehat{\\lambda}_{\\min} > 0$. This condition can be enforced as long as the true Fisher matrix ${\\mathcal{F}}(\\mathbf{\\theta})$ has a positive minimum eigenvalue $\\lambda_{\\min}$, i.e. under identifiability assumption (Lemma) and given a sufficiently large number of samples. Proposition of Appendix provides the minimum number of samples such that \nwith probability at least $1-\\delta$ it holds that $\\widehat{\\lambda}_{\\min} > 0$.\n\n\\paragraph{Identification Rule Analysis} The goal of the analysis of the identification rule is to find the critical value $c(1)$ so that the following probabilistic requirement is enforced.\n\nLet $\\delta \\in [0,1]$. An identification rule producing $\\widehat{{I}}$ is \\emph{$\\delta$--correct} if: $\\Pr \\big( \\widehat{{I}} \\neq {I}^* \\big)\\le \\delta$. \n\n\n We denote with $\\alpha = \\frac{1}{d-d^*} \\mathbb{E} \\big[ \\big| \\big\\{ i \\notin I^* : i \\in \\widehat{I}_{c} \\big\\} \\big| \\big]$ the expected fraction of parameters that the agent does not control selected by the identification rule and with $\\beta = \\frac{1}{d^*} \\mathbb{E} \\big[ \\big| \\big\\{ i \\in I^* : i \\notin \\widehat{I}_{c} \\big\\} \\big| \\big]$ the expected fraction of parameters that the agent does  control not selected by the identification rule.\nWe now provide a result that bounds $\\alpha$ and $\\beta$ and employs them to derive $\\delta$--correctness.\n\n\tLet $\\widehat{I}_{c}$ be the set of parameter indexes selected by the Identification Rule obtained using $n>0$ i.i.d. samples collected with $\\pi_{\\mathbf{\\theta}^*}$, with $\\mathbf{\\theta}^* \\in \\Theta$. Then, under Assumption and Assumption, let ${\\mathbf{\\theta}}_i^* = arg\\,min_{\\mathbf{\\theta} \\in \\Theta_i} \\ell(\\mathbf{\\theta})$ for all $i \\in \\{1,...,d\\}$ and $\\nu = \\min \\left\\{1, \\frac{\\lambda_{\\min}}{\\sigma^2} \\right\\}$. If $\\widehat{\\lambda}_{\\min} \\ge \\frac{\\lambda_{\\min}}{2\\sqrt{2}}$ and $\\ell({\\mathbf{\\theta}}_i^*) - {l}({\\mathbf{\\theta}^*}) \\ge c(1)$, it holds that:\n\t{\n\t\\begin{align*}\n\t\t&\\alpha  \\le 2d \\exp \\left\\{ -\\frac{c(1) {\\lambda}_{\\min}^2 n}{16d^2 \\sigma^4} \\right\\}\\\\\n\t\t&\\beta \\le \\frac{2d - 1}{d^*} \\sum_{i \\in I^*} \\exp\\left\\{ - \\frac{ \\left( {l}({\\mathbf{\\theta}}_i^*) - {l}({\\mathbf{\\theta}^*}) - c(1) \\right) {\\lambda}_{\\min} \\nu n}{16(d-1)^2 \\sigma^2 } \\right\\}.\n\t\\end{align*}\n\t}\n\tFurthermore, the Identification Rule is $\\left((d-d^*)\\alpha +d^*\\beta\\right)$--correct.\n\n\nSince $\\alpha$ and $\\beta$ are functions of $c(1)$, we could, in principle, employ Theorem to enforce a value $\\delta$, as in Definition, and derive $c(1)$. However, Theorem is not very attractive in practice as it holds under an assumption regarding the minimum eigenvalue of the FIM and the corresponding estimate, i.e. $\\widehat{\\lambda}_{\\min} \\ge \\frac{\\lambda_{\\min}}{2\\sqrt{2}}$, that cannot be verified in practice since $\\lambda_{\\min}$ is unknown. Similarly, the constants $d^*$, ${l}({\\mathbf{\\theta}}_i^*)$ and ${l}({\\mathbf{\\theta}^*})$ are typically unknown. We will provide in Section a heuristic for setting $c(1)$. \n\n\n\\section{Policy Space Identification in a Configurable Environment}\nThe identification rules presented so far are\nunable to distinguish between a parameter set to zero because the agent\ncannot control it, or because zero is its optimal value. To overcome this issue, we employ the Conf--MDP properties to select\na configuration in which the parameters we want to examine have an optimal value other than zero. Intuitively, if we want to test whether the agent can control parameter $\\theta_i$, we should place the agent in an environment $\\mathbf{\\omega}_i \\in \\Omega$ where $\\theta_i$ is maximally important\nfor the optimal policy. This intuition is justified by Theorem, since to maximize the \\emph{power} of the test ($1-\\beta$), all other things being equal, we should maximize the\nlog--likelihood gap ${l}({\\mathbf{\\theta}_i^*}) - {l}({\\mathbf{\\theta}^*})$, i.e. parameter $\\theta_i$ should\nbe essential to justify the agent's behavior. Let $I \\in \\{1,...,d\\}$ be a set of parameter\nindexes we want to test, our ideal goal is to find the environment $\\mathbf{\\omega}_I$ such that:\n\\begin{equation}\n\t\\mathbf{\\omega}_I \\in arg\\,max_{\\mathbf{\\omega} \\in \\Omega} \\left\\{ {l}({\\mathbf{\\theta}_I^*}(\\mathbf{\\omega})) - {l}({\\mathbf{\\theta}^*}(\\mathbf{\\omega})) \\right\\},\n\\end{equation}\nwhere ${\\mathbf{\\theta}^*}(\\mathbf{\\omega}) \\in arg\\,max_{\\mathbf{\\theta} \\in \\Theta} J_{\\mathcal{M}_{\\mathbf{\\omega}}}(\\mathbf{\\theta})$ and ${\\mathbf{\\theta}}_I^*(\\mathbf{\\omega}) \\in arg\\,max_{\\mathbf{\\theta} \\in \\Theta_I} J_{\\mathcal{M}_{\\mathbf{\\omega}}}(\\mathbf{\\theta})$ are the parameters of the optimal policies \nin the environment $\\mathcal{M}_{\\mathbf{\\omega}}$ in $\\Pi_{\\Theta}$ and $\\Pi_{\\Theta_I}$ respectively. Clearly, given the samples $\\mathcal{D}$ collected with a single optimal policy $\\pi^*(\\mathbf{\\omega}_0)$ in a single environment $\\mathcal{M}_{\\mathbf{\\omega}_0}$, solving problem~\\eqref{eq:confProblem} is hard as it requires performing an off--distribution optimization both on the space of policy parameters and configurations. For these reasons, we consider a surrogate objective that assumes that the optimal parameter in the new configuration can be reached by performing a single gradient step\n\n\nLet $I \\in \\{1,...,d\\}$ and $\\overline{I} =\\{1,...,d\\} \\setminus I$. For a vector $\\mathbf{v}$, we denote with $\\mathbf{v} \\rvert_I$ the vector obtained by setting to zero the\n\n\\end{document}\n"}
{"components in I. Let\u0012\u0006p!0q P\u0002the initial parameter.\nLet\u000b\u00a50,\u0012\u0006\nIp!q\u0010\u00120\u0000\u000br\u0012JM!p\u0012\u0006p!0qqjIand\u0012\u0006p!q\u0010\n\u00120\u0000\u000br\u0012JM!p\u0012\u0006p!0qq. Then, under Assumption, we have:\n`p\u0012\u0006\nIp!qq\u0001`p\u0012\u0006p!qq\u00a5\u0015min\u000b2\n2}r\u0012JM!p\u0012\u0006p!0qqjI}2\n2:\nThus, we maximize the L2\u2013norm of the gradient com-\nponents that correspond to the parameters we want to test.\nSince we have at our disposal only samples Dcollected with\nthe current policy \u0019\u0012\u0006p!0qand in the current environment !0,\nwe have to perform an off\u2013distribution optimization over !.\nTo this end, we employ an approach analogous to that of\nwhere we optimize the empirical version of the objective\nwith a penalization that accounts for the distance between\nthe distribution over trajectories:\nCIp!{!0q\u0010\u000f\u000f\u000fpr\u0012JM!{!0p\u0012\u0006p!0qqjI\u000f\u000f\u000f2\n2\u0001\u0010d\npd2p!}!0q\nn;(1)\nwherepr\u0012JM!{!0p\u0012\u0006p!0qqis an off-distribution estimator of\nthe gradient r\u0012JM!p\u0012\u0006p!0qqusing samples collected with\n!0,pd2is the estimated 2-Renyi divergence that works as\na penalization to discourage too large updates and \u0010\u00a50\nis a regularization parameter. The expression of the esti-\nmated gradient, 2-Renyi divergence and the pseudocode are\nreported in Appendix.\nExperimental Evaluation\nIn this section, we present the experimental evaluation of the\nidenti\ufb01cation rules in three RL domains. To set the values of\ncplqwe resort to the Wilk\u2019s asymptotic approximation (The-\norem) to enforce (asymptotic) guarantees on the type I er-\nror. For Identi\ufb01cation Rule we perform 2dstatistical tests by\nusing the same dataset D. Thus, we partition \u000eusing Bon-\nferroni correction and setting cplq \u0010\u001f2\nl;1\u0001\u000e{2d, where\u001f2\nl;\u0018\nis the\u0018\u2013quintile of a chi square distribution with ldegrees\nof freedom. Instead, for Identi\ufb01cation Rule, we perform d\nstatistical test, and thus, we set cp1q\u0010\u001f2\n1;1\u0001\u000e{d.\nDiscrete Grid World\nThe grid world environment is a simple representation of a\ntwo-dimensional world (5 \u00025 cells) in which an agent has to\nreach a target position by moving in the four directions. The\ngoal of this set of experiments is to show the advantages\nof con\ufb01guring the environment when performing the pol-\nicy space identi\ufb01cation using rule. The initial position of the\nagent and the target position are drawn at the beginning of\neach episode from a Boltzmann distribution \u0016!. The agent\nplays a Boltzmann linear policy \u0019\u0012with binary features \u001ein-\ndicating its current row and column and the row and column\nof the goal. For each run, the agent can control a subset I\u0006\nof the parameters \u0012I\u0006associated with those features, which\nis randomly selected. Furthermore, the supervisor can con-\n\ufb01gure the environment by changing the parameters !of the\ninitial state distribution \u0016!. Thus, the supervisor can induce\nthe agent to explore certain regions of the grid world and,\nconsequently, change the relevance of the corresponding pa-\nrameters in the optimal policy.Figure shows the empirical p\u000bandp\f, i.e. the fraction of\nparameters that the agent does not control that are wrongly\nselected and the fraction of those the agent controls that are\nnot selected respectively, as a function of the number nof\nepisodes used to perform the identi\ufb01cation. We compare two\ncases: conf where the identi\ufb01cation is carried out by also\ncon\ufb01guring the environment, i.e. optimizing Equation (1),\nandno-conf in which the identi\ufb01cation is performed in the\noriginal environment only. In both cases, we can see that\np\u000bis almost independent of the number of samples, as it is\ndirectly controlled by the critical value cp1q. Differently, p\f\ndecreases as the number of samples increases, i.e. the power\nof the test 1\u0001p\fincreases with n. Remarkably, we observe\nthat con\ufb01guring the environment gives a signi\ufb01cant advan-\ntage in understanding the parameters controlled by the agent\nw.r.t using a \ufb01xed environment, as p\fdecreases faster in the\nconf case. This phenomenon also justi\ufb01es empirically our\nchoice of objective (Equation (1)) for selecting the new envi-\nronment. Hyperparameters, further experimental results, to-\ngether with experiments on a continuous version of the grid\nworld, are reported in Appendix\u2013.\nMinigolf\nIn the Minigolf environment, an agent hits a ball using a put-\nter with the goal of reaching the hole in the minimum num-\nber of attempts. Surpassing the hole causes the termination\nof the episode and a large penalization. The agent selects the\nforce applied to the putter by playing a Gaussian policy lin-\near in some polynomial features (complying to Lemma) of\nthe distance from the hole ( x) and the friction of the green\n(f). We consider two agents: A1has access to both the xand\nfwhereas A2knows onlyx. Thus, we expect that A1learns\na policy that allows reaching the hole in a smaller number of\nhits, compared to A2, as it can calibrate force according to\nfriction; whereas A2has to be more conservative, being un-\naware off. There is also a supervisor in charge of selecting,\nfor the two agents, the best putter length !, i.e. the con\ufb01g-\nurable parameter of the environment. In this experiment, we\nwant to highlight that knowing the policy space might be of\ncrucial importance when learning in a Conf\u2013MDP.\nFigure-left shows the performance of the optimal policy\nas a function of the putter length !. We can see that for agent\nA1the optimal putter length is !\u0006\nA1\u00105while for agent\nA2is!\u0006\nA2\u001011:5. Figure-right compares the performance\nof the optimal policy of agent A2when the putter length !\nis chosen by the supervisor using four different strategies.\nIn (i) the con\ufb01guration is sampled uniformly in the interval\nr1;15s. In (ii) the supervisor employs the optimal con\ufb01gura-\ntion for agent A1(!\u00105), i.e. assuming the agent is aware\nof the friction. (iii) is obtained by selecting the optimal con-\n\ufb01guration of the policy space produced by using our identi-\n\ufb01cation rule. Finally, (iv) is derived by employing an oracle\nthat knows the true agent\u2019s policy space ( !\u001011:5). We can\nsee that the performance of the identi\ufb01cation procedure (iii)\nis comparable with that of the oracle (iv) and notably higher\nthan the performance when employing an incorrect policy\nspace (ii). Hyperparameters and additional experiments are": "\\documentclass[letterpaper]{article} % DO NOT CHANGE THIS\n\\usepackage{aaai20}  % DO NOT CHANGE THIS\n\\usepackage{times}  % DO NOT CHANGE THIS\n\\usepackage{helvet} % DO NOT CHANGE THIS\n\\usepackage{courier}  % DO NOT CHANGE THIS\n\\usepackage[hyphens]{url}  % DO NOT CHANGE THIS\n\\usepackage{graphicx} % DO NOT CHANGE THIS\n\\urlstyle{rm} % DO NOT CHANGE THIS\n\\def\\UrlFont{\\rm}  % DO NOT CHANGE THIS\n\\usepackage{graphicx}  % DO NOT CHANGE THIS\n\\frenchspacing  % DO NOT CHANGE THIS\n\\setlength{\\pdfpagewidth}{8.5in}  % DO NOT CHANGE THIS\n\\setlength{\\pdfpageheight}{11in}  % DO NOT CHANGE THIS\n\n\n\\usepackage{booktabs}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{dsfont}\n\\usepackage{xspace} \n\\usepackage{amsthm}  \n\\usepackage{thmtools}\n\\usepackage{thm-restate}\n\\usepackage{etoolbox}\n\\usepackage{bm}\n\\usepackage{amsfonts}\n\\usepackage[capitalise, noabbrev]{cleveref}\n\\usepackage{pifont}\n\\usepackage{fixltx2e}\n\\usepackage{mathtools}\n\\usepackage{mathabx}\n\\usepackage{algorithm}\n\\usepackage{algorithmicx,algpseudocode}\n\\usepackage{mathtools}\n\\usepackage{subcaption}\n\\usepackage{mathrsfs}\n\\usepackage[percent]{overpic}\n\n\n\n\n\\begin{document}\n\ncomponents in $I$. Let $\\mathbf{\\theta}^*(\\mathbf{\\omega}_0) \\in \\Theta$ the initial parameter. Let $\\alpha \\ge 0$, $\\mathbf{\\theta}_I^* (\\mathbf{\\omega}) =  \\mathbf{\\theta}_0 + \\alpha \\nabla_{\\mathbf{\\theta}} J_{\\mathcal{M}_{\\mathbf{\\omega}}} (\\mathbf{\\theta}^*(\\mathbf{\\omega}_0)) \\rvert_I$ and $\\mathbf{\\theta}^* (\\mathbf{\\omega}) =  \\mathbf{\\theta}_0 + \\alpha \\nabla_{\\mathbf{\\theta}} J_{\\mathcal{M}_{\\mathbf{\\omega}}} (\\mathbf{\\theta}^*(\\mathbf{\\omega}_0))$. Then, under Assumption, we have:\n\t\\begin{equation*}\n\t\t{\\ell}({\\mathbf{\\theta}_I^*}(\\mathbf{\\omega})) - {\\ell}({\\mathbf{\\theta}^*}(\\mathbf{\\omega})) \\ge \\frac{\\lambda_{\\min} \\alpha^2}{2} \\left\\| \\nabla_{\\mathbf{\\theta}} J_{\\mathcal{M}_{\\mathbf{\\omega}}} (\\mathbf{\\theta}^*(\\mathbf{\\omega}_0)) \\rvert_{\\overline{I}} \\right\\|_2^2.\n\t\\end{equation*}\n\n\nThus, we maximize the $L^2$--norm of the gradient components that correspond to the parameters we want to test. Since we have at our disposal only samples $\\mathcal{D}$ collected with the current policy $\\pi_{\\mathbf{\\theta}^*(\\mathbf{\\omega}_0)}$ and in the current environment $\\mathbf{\\omega}_0$, we have to perform an off--distribution optimization over $\\mathbf{\\omega}$. To this end, we employ an approach analogous to that of where we optimize the empirical version of the objective with a penalization that accounts for the distance between the distribution over trajectories:\n\\begin{equation}\n\\resizebox{0.88\\linewidth}{!}{$\n\\displaystyle \\mathcal{C}_I(\\mathbf{\\omega}/\\mathbf{\\omega}_0) = \\left\\| \\widehat{\\nabla}_{\\mathbf{\\theta}} J_{\\mathcal{M}_{\\mathbf{\\omega}/\\mathbf{\\omega}_0}}(\\mathbf{\\theta}^*(\\mathbf{\\omega}_0)) \\rvert_{\\overline{I}} \\right\\|_2^2 - \\zeta \\sqrt{\\frac{\t\\widehat{d}_2 (\\mathbf{\\omega} \\| \\mathbf{\\omega}_0) }{n}},$}\n\\end{equation}\nwhere $\\widehat{\\nabla}_{\\mathbf{\\theta}} J_{\\mathcal{M}_{\\mathbf{\\omega}/\\mathbf{\\omega}_0}}(\\mathbf{\\theta}^*(\\mathbf{\\omega}_0))$ is an off-distribution estimator of the gradient $\\nabla_{\\mathbf{\\theta}} J_{\\mathcal{M}_{\\mathbf{\\omega}}} (\\mathbf{\\theta}^*(\\mathbf{\\omega}_0))$ using samples collected with $\\mathbf{\\omega}_0$, $\\widehat{d}_2$ is the estimated 2-Renyi divergence that works as a penalization to discourage too large updates and $\\zeta \\ge 0$ is a regularization parameter. The expression of the estimated gradient, 2-Renyi divergence and the pseudocode are reported in Appendix.\n\n\n\n\\section{Experimental Evaluation}\nIn this section, we present the experimental evaluation of the identification rules in three RL domains.\nTo set the values of $c(l)$ we resort to the Wilk's asymptotic approximation (Theorem) to enforce (asymptotic) guarantees on the type I error. For Identification Rule we perform $2^d$ statistical tests by using the same dataset $\\mathcal{D}$. Thus, we partition $\\delta$ using Bonferroni correction and setting $c(l) = \\chi^2_{l,1-{\\delta}/{2^d}}$, where $\\chi^2_{l,\\xi}$ is the $\\xi$--quintile of a chi square distribution with $l$ degrees of freedom. Instead, for Identification Rule, we perform $d$ statistical test, and thus, we set $c(1) = \\chi^2_{1,1-{\\delta}/{d}}$.\n\n\\subsection{Discrete Grid World}\nThe grid world environment is a simple representation of a two-dimensional world (5$\\times$5 cells) in which an agent has to reach a target position by moving in the four directions. The goal of this set of experiments is to show the advantages of configuring the environment when performing the policy space identification using rule. The initial position of the agent and the target position are drawn at the beginning of each episode from a Boltzmann distribution $\\mu_{\\mathbf{\\omega}}$. The agent plays a Boltzmann linear policy $\\pi_{\\mathbf{\\theta}}$ with binary features $\\mathbf{\\phi}$ indicating its current row and column and the row and column of the goal. For each run, the agent can control a subset $I^*$ of the parameters $\\mathbf{\\theta}_{I^*}$ associated with those features, which is randomly selected. Furthermore, the supervisor can configure the environment by changing the parameters $\\mathbf{\\omega}$ of the initial state distribution $\\mu_{\\mathbf{\\omega}}$. Thus, the supervisor can induce the agent to explore certain regions of the grid world and, consequently, change the relevance of the corresponding parameters in the optimal policy.\n\nFigure shows the empirical $\\widehat{\\alpha}$ and $\\widehat{\\beta}$, i.e. the fraction of parameters that the agent does not control that are wrongly selected and the fraction of those the agent controls that are not selected respectively, as a function of the number $n$ of episodes used to perform the identification. We compare two cases: \\emph{conf} where the identification is carried out by also configuring the environment, i.e. optimizing Equation~\\eqref{eq:obj}, and \\emph{no-conf} in which the identification is performed in the original environment only. In both cases, we can see that $\\widehat{\\alpha}$ is almost  independent of the number of samples, as it is directly controlled by the critical value $c(1)$. Differently, $\\widehat{\\beta}$ decreases as the number of samples increases, i.e. the power of the test $1-\\widehat{\\beta}$ increases with $n$. Remarkably, we observe that configuring the environment gives a significant advantage in understanding the parameters controlled by the agent w.r.t using a fixed environment, as $\\widehat{\\beta}$ decreases faster in the \\emph{conf} case. This phenomenon also justifies empirically our choice of objective (Equation~\\eqref{eq:obj}) for selecting the new environment. Hyperparameters, further experimental results, together with experiments on a continuous version of the grid world, are reported in Appendix--.\n\n\\subsection{Minigolf}\nIn the Minigolf environment, an agent hits a ball using a putter with the goal of reaching the hole in the minimum number of attempts. Surpassing the hole causes the termination of the episode and a large penalization. The agent selects the force applied to the putter by playing a Gaussian policy linear in some polynomial features (complying to Lemma) of the distance from the hole ($x$) and the friction of the green ($f$). We consider two agents: $\\mathscr{A}_1$ has access to both the $x$ and $f$ whereas $\\mathscr{A}_2$ knows only $x$. Thus, we expect that $\\mathscr{A}_1$ learns a policy that allows reaching the hole in a smaller number of hits, compared to $\\mathscr{A}_2$, as it can calibrate force according to friction; whereas $\\mathscr{A}_2$ has to be more conservative, being unaware of $f$. There is also a supervisor in charge of selecting, for the two agents, the best putter length $\\omega$, i.e. the configurable parameter of the environment. In this experiment, we want to highlight that knowing the policy space might be of crucial importance when learning in a Conf--MDP.\n\nFigure-left shows the performance of the optimal policy as a function of the putter length $\\omega$. We can see that for agent $\\mathscr{A}_1$ the optimal putter length is $\\omega^*_{\\mathscr{A}_1}=5$ while for agent $\\mathscr{A}_2$ is $\\omega^*_{\\mathscr{A}_2}=11.5$.\nFigure-right compares the performance of the optimal policy of agent $\\mathscr{A}_2$ when the putter length $\\omega$ is chosen by the supervisor using four different strategies. In (i) the configuration is sampled uniformly in the interval $[1,15]$. In (ii) the supervisor employs the optimal configuration for agent $\\mathscr{A}_1$ ($\\omega=5$), i.e. assuming the agent is aware of the friction. (iii) is obtained by selecting the optimal configuration of the policy space produced by using our identification rule. Finally, (iv) is derived by employing an oracle that knows the true agent's policy space ($\\omega=11.5$). We can see that the performance of the identification procedure (iii) is comparable with that of the oracle (iv) and notably higher than the performance when employing an incorrect policy space (ii). Hyperparameters and additional experiments are\n\n\\end{document}\n"}
{"1 Lorentzian matter-coupled F(4)gauged supergravity\nThe theory of matter-coupled F(4) gauged supergravity was \frst studied in , with some\napplications and extensions given in . Below we present a short review of this theory, similar\nto that given in .\n1.1 The bosonic Lagrangian\nWe begin by recalling the \feld content of the 6-dimensional supergravity multiplet,\n(ea\n\u0016;  A\n\u0016; A\u000b\n\u0016; B\u0016\u0017; \u001fA; \u001b) (1)\nThe \feldea\n\u0016is the 6-dimensional frame \feld, with spacetime indices denoted by f\u0016;\u0017g\nand local Lorentz indices denoted by fa;bg. The \feld  A\n\u0016is the gravitino with the index\nA;B = 1;2 denoting the fundamental representation of the gauged SU(2)Rgroup. The\nsupergravity multiplet contains four vectors A\u000b\n\u0016labelled by the index \u000b= 0;:::3. It will\noften prove useful to split \u000b= (0;r) withr= 1;:::; 3 anSU(2)Radjoint index. Finally, the\nremaining \felds consist of a two-form B\u0016\u0017, a spin-1\n2\feld\u001fA, and the dilaton \u001b. The only\nallowable matter in the d= 6,N= 2 theory is the vector multiplet, which has the following\n\feld content\n(A\u0016; \u0015A; \u001e\u000b)I(2)\nwhereI= 1;:::;n labels the distinct matter multiplets included in the theory. The presence\nof thennew vector \felds AI\n\u0016allows for the existence of a further gauge group G+of dimension\ndimG+=n, in addition to the gauged SU(2)RR-symmetry. The presence of this new gauge\ngroup contributes an additional parameter to the theory, in the form of a coupling constant\n\u0015. Throughout this section, we will denote the structure constants of the additional gauge\ngroupG+byCIJK. However, these will play no role in what follows, since we will be\nrestricting to the case of only a single vector multiplet n= 1, in which case G+=U(1).\nIn (half-)maximal supergravity, the dynamics of the 4 nvector multiplet scalars \u001e\u000bIis given\nby a non-linear sigma model with target space G=K ; see e.g. . The group Gis the global\nsymmetry group of the theory, while Kis the maximal compact subgroup of G. As such, in\nthe Lorentzian case the target space is identi\fed with the following coset space,\nM=SO(4;n)\nSO(4)\u0002SO(n)\u0002SO(1;1) (3)\nwhere the second factor corresponds to the scalar \u001bwhich is already present in the gauged\nsupergravity without added matter. In the particular case of n= 1, explored here and in ,\nthe \frst factor is nothing but four-dimensional hyperbolic space H4. When we analytically\ncontinue to the Euclidean case, it will prove very important that we analytically continue\nthe coset space as well, resulting in a dS 4coset space. This will be discussed more in the\nfollowing section.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\n\\begin{document}\n\n1 Lorentzian matter-coupled $F(4)$ gauged supergravity\n\nThe theory of matter-coupled $F(4)$ gauged supergravity was first studied in , with some\n\napplications and extensions given in . Below we present a short review of this theory, similar\n\nto that given in .\n\n1.1 The bosonic Lagrangian\n\nWe begin by recalling the field content of the 6-dimensional supergravity multiplet,\n\n({\\it e}\u00b5{\\it a}, ?\u00b5{\\it A, A}\u00b5a, {\\it B}\u00b5?, ?{\\it A}, s) (1)\n\nThe field {\\it e}\u00b5{\\it a} is the 6-dimensional frame field , with spacetime indices denoted by \\{\u00b5, ?\\}\n\nand local Lorentz indices denoted by $\\{a,\\ b\\}$. The field ?\u00b5{\\it A} is the gravitino with the index\n\n$A, B = 1$, 2 denoting the fundamental representation of the gauged $SU(2)_{R}$ group. The\n\nsupergravity multiplet contains four vectors {\\it A}\u00b5a labelled by the indexa $= 0$, . . .3. It will\n\noften prove useful to split $\\ovalbox{\\tt\\small REJECT}=(0,\\ r)$ with $r=1$, . . . , 3 an $SU(2)_{R}$ adjoint index. Finally, the\n\nremaining fields consist of atwo-form {\\it B}\u00b5?, aspin-$\\displaystyle \\frac{1}{2}$ field?{\\it A}, and the dilatons. The only\n\nallowable matter in the $d=6, \\mathcal{N}=2$ theory is the vector multiplet, which has the following\n\nfield content\n\n({\\it A}\u00b5, ?{\\it A}, fa){\\it I} (2)\n\nwhere $I=1$, . . . , $n$ labels the distinct matter multiplets included in the theory. The presence\n\nof the{\\it n} new vector fields {\\it A}\u00b5{\\it I} allows for the existence of a further gauge group $G_{+}$ of dimension\n\n$\\dim G_{+}=n$, in addition to the gauged $SU(2)_{R}\\mathrm{R}$-symmetry. The presence of this new gauge\n\ngroup contributes an additional parameter to the theory, in the form of a coupling constant\n\n?. Throughout this section, we will denote the structure constants of the additional gauge\n\ngroup $G_{+}$ by $C_{IJK}$. However, these will play no role in what follows, since we will be\n\nrestricting to the case of only a single vector multiplet $n = 1$, in which case $G_{+} = U(1)$ .\n\nIn (half-)maximal supergravity, the dynamics of the $4n$ vector multiplet scalars fa{\\it I} is given\n\nby a non-linear sigma model with target space $G/K$; see e.g. . The group{\\it G} is the global\n\nsymmetry group of the theory, while $K$ is the maximal compact subgroup of $G$. As such, in\n\nthe Lorentzian case the target space is identified with the following coset space,\n\\begin{center}\n$\\displaystyle \\mathcal{M}=\\frac{SO(4,n)}{SO(4)\\times SO(n)}\\ \\times SO(1,1)$   (3)\n\\end{center}\nwhere the second factor corresponds to the scalar s which is already present in the gauged\n\nsupergravity without added matter. In the particular case of{\\it n} $= 1$, explored here and in ,\n\nthe first factor is nothing but four-dimensional hyperbolic space $\\mathrm{H}_{4}$. When we analytically\n\ncontinue to the Euclidean case, it will prove very important that we analytically continue\n\nthe coset space as well, resulting in a $\\mathrm{d}\\mathrm{S}_{4}$ coset space. This will be discussed more in the\n\nfollowing section.\n\n1\n\\end{document}\n"}
{"In both the Lorentzian and Euclidean cases, a convenient way of formulating the coset\nspace non-linear sigma model is to have the scalars \u001e\u000bIparameterize an element LofG.\nThe so-called coset representative Lis an (n+4)\u0002(n+4) matrix with matrix elements L\u0003\n\u0006,\nfor \u0003;\u0006 = 1;:::n + 4. Using this representative, one may construct a left-invariant 1-form,\nL\u00001dL2g (1)\nwhere g= Lie(G). To build a K-invariant kinetic term from the above, we decompose\nL\u00001dL=Q+P (2)\nwhereQ2k= Lie(K) andPlies in the complement of king. Explicitly, the coset vielbein\nforms are given by,\nPI\n\u000b=\u0000\nL\u00001\u0001I\n\u0003\u0000\ndL\u0003\n\u000b+f\u0003\n\u0000\u0005A\u0000L\u0005\n\u000b\u0001\n(3)\nwhere thef\u0000\n\u0003\u0006are structure constants of the gauge algebra, i.e.\n[T\u0003;T\u0006] =f\u0000\n\u0003\u0006T\u0000 (4)\nWe may then use Pto build the kinetic term for the vector multiplet scalars as,\nLcoset=\u00001\n4ePI\u000b\u0016PI\u000b\u0016(5)\nwheree=p\njdetgjand we've de\fned PI\u000b\n\u0016=PI\u000b\ni@\u0016\u001ei;fori= 0;:::; 4n\u00001. With this\nformulation for the coset space non-linear sigma model, we may now write down the full\nbosonic Lagrangian of the theory. We will be interested in the case in which only the metric\nand the scalars are non-vanishing. In this case the Lorentzian theory is given by\ne\u00001L=\u00001\n4R+@\u0016\u001b@\u0016\u001b\u00001\n4PI\u000b\u0016PI\u000b\u0016\u0000V (6)\nwith the scalar potential Vgiven by\nV=\u0000e2\u001b\u00141\n36A2+1\n4BiBi+1\n4(CI\ntCIt+ 4DI\ntDIt)\u0015\n+m2e\u00006\u001bN00\n\u0000me\u00002\u001b\u00142\n3AL00\u00002BiL0i\u0015\n(7)\nThe scalar potential features the following quantities,\nA=\u000frstKrst Br=\u000frstKst0\nCt\nI=\u000ftrsKrIs DIt=K0It (8)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nIn both the Lorentzian and Euclidean cases, a convenient way of formulating the coset space non-linear sigma model is to have the scalars $\\phi^{\\alpha I}$ parameterize an element $L$ of $G$. The so-called coset representative $L$ is an $(n+4)\\times(n+4)$ matrix with matrix elements $L^\\Lambda_{\\,\\,\\,\\,\\Sigma}$, for  $\\Lambda, \\Sigma = 1, \\dots n+4$. Using this representative, one may construct a left-invariant 1-form, \n\\begin{eqnarray}\nL^{-1} d L \\in \\mathfrak{g}\n\\end{eqnarray}\nwhere $\\mathfrak{g} = \\mathrm{Lie}(G)$. To build a $K$-invariant kinetic term from the above, we decompose\n\\begin{eqnarray}\nL^{-1} d L = Q + P\n\\end{eqnarray}\nwhere $Q \\in \\mathfrak{k} = \\mathrm{Lie}(K)$ and $P$ lies in the complement of $\\mathfrak{k}$ in $\\mathfrak{g}$.  Explicitly, the coset vielbein forms are  given by, \n\\begin{eqnarray}\nP^I_{\\,\\,\\,\\,\\alpha} = \\left(L^{-1}\\right)^I_{\\,\\,\\,\\,\\Lambda} \\left(d L^\\Lambda_{\\,\\,\\,\\,\\alpha} + f^\\Lambda_{\\,\\,\\,\\,\\Gamma \\Pi} A^\\Gamma L^\\Pi_{\\,\\,\\,\\,\\alpha} \\right)\n\\end{eqnarray}\nwhere the $f_{\\Lambda \\Sigma}^{\\,\\,\\,\\,\\,\\,\\,\\,\\Gamma}$ are structure constants of the gauge algebra, i.e.\n\\begin{eqnarray}\n[T_\\Lambda, T_\\Sigma] =  f_{\\Lambda \\Sigma}^{\\,\\,\\,\\,\\,\\,\\,\\,\\Gamma} \\, T_\\Gamma\n\\end{eqnarray}\nWe may then use $P$ to build the kinetic term for the vector multiplet scalars as, \n\\begin{eqnarray}\n{\\cal L}_{\\mathrm{coset}} = -{1\\over 4} e P_{I \\alpha \\mu} P^{I \\alpha \\mu}\n\\end{eqnarray}\nwhere $e = \\sqrt{|\\mathrm{det} \\,g|}$ and we've defined $P_\\mu^{I \\alpha} = P_i^{I \\alpha} \\partial_\\mu \\phi^i,$ for $i = 0, \\dots, 4n-1$.  With this formulation for the coset space non-linear sigma model, we may now write down the full bosonic Lagrangian of the theory. We will be interested in the case in which only the metric and the scalars are non-vanishing. In this case the Lorentzian theory is given by\n \\begin{eqnarray}\n e^{-1} {\\cal L} = -{1 \\over 4} R + \\partial_\\mu \\sigma \\partial^\\mu \\sigma - {1\\over 4} P_{I \\alpha \\mu} P^{I \\alpha \\mu} - V\n \\end{eqnarray}\n with the scalar potential $V$ given by \n \\begin{eqnarray}\nV = -e^{2 \\sigma} \\left[ {1 \\over 36} A^2 + {1 \\over 4}B^i B_i + {1 \\over 4} (C^I_t C_{I t} + 4 D^I_t D_{I t}) \\right] + m^2 e^{-6 \\sigma} {\\cal N}_{00} \n\\nonumber\\\\\n\\vphantom{.} \\hspace{0.5in} - m e^{- 2 \\sigma} \\left[{2 \\over 3} A L_{00} - 2 B^i L_{0 i} \\right]\n\\end{eqnarray} \nThe scalar potential features the following quantities, \n\\begin{eqnarray}\nA = \\epsilon^{r s t} K_{r s t} \\hspace{1 in}\\,\\,B^r\\, = \\,\\epsilon^{r s t} K_{s t 0}\n\\nonumber\\\\\nC_I^t = \\epsilon^{t r s} K_{r I s} \\hspace{1 in} D_{I t}\\,\\, = \\,\\, K_{0 I t}\n\\end{eqnarray}\n\n \n\\end{document}\n"}
{"with the so-called \\boosted structure constants\" Kgiven by,\nKrs\u000b=g\u000f`mnL`\nr(L\u00001)m\nsLn\n\u000b+\u0015CIJKLI\nr(L\u00001)J\nsLK\n\u000b\nK\u000bIt=g\u000f`mnL`\n\u000b(L\u00001)m\nILn\nt+\u0015CMJKLM\n\u000b(L\u00001)J\nILK\nt (1)\nWe remind the reader that r;s;t = 1;2;3 are obtained from splitting the index \u000binto a 0\nindex and an SU(2)Radjoint index. Also appearing in the Lagrangian is N00, which is the\n00 component of the matrix\nN\u0003\u0006=L\u000b\n\u0003\u0000\nL\u00001\u0001\n\u000b\u0006\u0000LI\n\u0003\u0000\nL\u00001\u0001\nI\u0006(2)\n0.1 Supersymmetry variations\nWe now review the supersymmetry variations for the fermionic \felds in the Lorentzian\ntheory. In the following section, we will discuss the continuation of this theory to Euclidean\nsignature, which is complicated by the necessary modi\fcation of the symplectic Majorana\ncondition imposed on the spinor \felds. In order to write the fermionic variations, it is \frst\nnecessary to introduce a matrix \n7de\fned as\n\n7=i\n0\n1\n2\n3\n4\n5(3)\nand satisfying ( \n7)2=\u00001. With this, the supersymmetry transformations of the fermions\nin the Lorentzian case can be given as\n\u000e\u001fA=i\n2\n\u0016@\u0016\u001b\"A+NAB\"B\n\u000e A\u0016=D\u0016\"A+SAB\n\u0016\"B\n\u000e\u0015I\nA=i^PI\nri\u001br\nAB@\u0016\u001ei\n\u0016\"B\u0000i^PI\n0i\u000fAB@\u0016\u001ei\n7\n\u0016\"B+MI\nAB\"B(4)\nwhere we have de\fned\nSAB=i\n24[Ae\u001b+6me\u00003\u001b(L\u00001)00]\"AB\u0000i\n8[Bte\u001b\u00002me\u00003\u001b(L\u00001)t0]\n7\u001bt\nAB\nNAB=1\n24[Ae\u001b\u000018me\u00003\u001b(L\u00001)00]\"AB+1\n8[Bte\u001b+6me\u00003\u001b(L\u00001)t0]\n7\u001bt\nAB\nMI\nAB=(\u0000CI\nt+ 2i\n7DI\nt)e\u001b\u001bt\nAB\u00002me\u00003\u001b(L\u00001)I\n0\n7\"AB; (5)\nIn the above, the matrix \u001br\nABde\fned as\u001br\nAB\u0011\u001brC\nB\"CAis symmetric in A;B. For more\ndetails, see our previous paper.\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nwith the so-called ``boosted structure constants\" $K$ given by,\n\\begin{eqnarray}\nK_{r s \\alpha} = g \\,\\epsilon_{\\ell m n} L^{\\ell}_{\\,\\,\\,\\, r} (L^{-1})_s^{\\,\\,\\,\\, m} L^n_{\\,\\,\\,\\,\\alpha} + \\lambda \\,C_{I J K} L^I_{\\,\\,\\,\\,r} (L^{-1})_s^{\\,\\,\\,\\,J}L^K_{\\,\\,\\,\\,\\,\\alpha}\n\\nonumber\\\\\nK_{\\alpha I t} = g\\, \\epsilon_{\\ell m n} L^{\\ell}_{\\,\\,\\,\\, \\alpha} (L^{-1})_I^{\\,\\,\\,\\, m} L^n_{\\,\\,\\,\\,t} + \\lambda \\,C_{M J K} L^M_{\\,\\,\\,\\,\\,\\alpha} (L^{-1})_I^{\\,\\,\\,\\,J}L^K_{\\,\\,\\,\\,\\,t}\n\\end{eqnarray}\nWe remind the reader that $r,s,t = 1, 2, 3$ are obtained from splitting the index $\\alpha$ into a 0 index and an $SU(2)_R$ adjoint index.  Also appearing in the Lagrangian is ${\\cal N}_{00}$, which is the $00$ component of the matrix \n\\begin{eqnarray}\n{\\cal N}_{\\Lambda \\Sigma} = L_\\Lambda^{\\,\\,\\,\\,\\alpha}\\left( L^{-1}\\right)_{\\alpha \\Sigma} -  L_\\Lambda^{\\,\\,\\,\\,I}\\left( L^{-1}\\right)_{I \\Sigma} \n\\end{eqnarray}\n\\subsection{Supersymmetry variations}\nWe now review the supersymmetry variations for the fermionic fields in the Lorentzian theory. In the following section, we will discuss the continuation of this theory to Euclidean signature, which is complicated by the necessary modification of the symplectic Majorana condition imposed on the spinor fields.\nIn order to write the fermionic variations, it is first necessary to introduce a matrix $\\gamma^7$ defined as \n\\begin{eqnarray}\n\\gamma^7=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3\\gamma^4\\gamma^5\n\\end{eqnarray}\nand satisfying $(\\gamma^7)^2=-\\mathds{1}$. With this, the supersymmetry transformations of the fermions in the Lorentzian case can be given as\n\\begin{eqnarray}\n\\delta \\chi_A = {i \\over 2} \\gamma^\\mu \\partial_\\mu \\sigma \\varepsilon_A + N_{AB} \\varepsilon^B \n\\nonumber\\\\\\nonumber\\\\\n\\delta \\psi_{A \\mu} = {\\cal D}_\\mu \\varepsilon_A + S_{AB} \\gamma_\\mu \\varepsilon^B\n\\nonumber\\\\\\nonumber\\\\\n\\delta \\lambda^I_A = i \\hat P^I_{r i} \\sigma^r_{AB} \\partial_\\mu \\phi^i \\gamma^\\mu \\varepsilon^B -i \\hat P^I_{0 i} \\epsilon_{AB} \\partial_\\mu \\phi^i \\gamma^7 \\gamma^\\mu \\varepsilon^B + M^I_{AB} \\varepsilon^B\n\\end{eqnarray}\nwhere we have defined\n\\begin{eqnarray}\nS_{AB}=\\!\\frac{i}{24}[Ae^{\\sigma}\\! +\\!\n6me^{-3\\sigma}(L^{-1})_{00}]\\varepsilon_{AB}\\! -\\!\n\\frac{i}{8}[B_te^{\\sigma}-2me^{-3\\sigma}(L^{-1})_{t0}]\\gamma^7\\sigma^t_{AB}\\nonumber\n\\\\\\nonumber\\\\\nN_{AB}=\\!\\frac{1}{24}[Ae^{\\sigma}\\! -\\!\n18me^{-3\\sigma}(L^{-1})_{00}]\\varepsilon_{AB}\\! +\\!\n\\frac{1}{8}[B_te^{\\sigma}\\! +\\! 6me^{-3\\sigma}(L^{-1})_{t0}]\\gamma^7\\sigma^t_{AB}\\nonumber\\\\\n\\nonumber\\\\\nM^I_{AB}=\\!(-C^I_{~t}+2i\\gamma^7D^I_{~t})e^{\\sigma}\\sigma^t_{AB}-\n2me^{-3\\sigma}(L^{-1})^I_{\\ \\ 0}\\gamma^7\\varepsilon_{AB}, \n\\end{eqnarray}\nIn the above, the matrix $\\sigma^r_{AB}$ defined as $\\sigma^r_{AB}\\equiv\\sigma^{rC}_{~~B}\\varepsilon_{CA}$ is symmetric in $A,B$. For more details, see our previous paper.\n\\end{document}\n"}
{"0.1 Mass deformations\nIn the following, we consider the coset with n= 1, i.e. a single vector multiplet. The coset\nrepresentative is expressed in terms of four scalars \u001ei;i= 0;1;2;3 via\nL=3Y\ni=0e\u001eiKi(1)\nwhereKiare the non compact generators of SO(4;1); see for details. Note that \u001e0is an\nSU(2)Rsinglet, while the other three scalars \u001erform anSU(2)Rtriplet. The scalar potential\nfor this speci\fc case can be obtained from and takes the following form\nV(\u001b;\u001ei) =\u0000g2e2\u001b+1\n8me\u00006\u001b\u0014\n\u000032ge4\u001bcosh\u001e0cosh\u001e1cosh\u001e2cosh\u001e3+ 8mcosh2\u001e0\n+msinh2\u001e0\u0012\n\u00006 + 8 cosh2\u001e1cosh2\u001e2cosh(2\u001e3) + cosh(2(\u001e1\u0000\u001e2))\n+ cosh(2(\u001e1+\u001e2)) + 2 cosh(2 \u001e1) + 2 cosh(2 \u001e2)\u0013\u0015\n(2)\nThe supersymmetric AdS 6vacuum is given by setting g= 3mand setting all scalars to\nvanish. The masses of the linearized scalar \nuctuation around the AdS vacuum determine\nthe dimensions of the dual scalar operators in the SCFT via\nm2l2= \u0001(\u0001\u00005) (3)\nwherelis the curvature radius of the AdS 6vacuum. For the scalars at hand, one \fnds\nm2\n\u001bl2=\u00006 m2\n\u001e0l2=\u00004 m2\n\u001erl2=\u00006; r= 1;2;3 (4)\nHence the dimensions of the dual operators are\n\u0001O\u001b= 3; \u0001O\u001e0= 4; \u0001O\u001er= 3; r= 1;2;3 (5)\nIn these CFT operators were expressed in terms of free hypermultiplets (i.e. the singleton\nsector). The case of n= 1 corresponds to having a single free hypermultiplet, consisting of\nfour real scalars qI\nAand two symplectic Majorana spinors  I. HereI= 1;2 is theSU(2)R\nR-symmetry index and A= 1;2 is theSU(2) \navor symmetry index. The gauge invariant\noperators appearing in are related to these fundamental \felds as follows,\nO\u001b= (q\u0003)A\nIqI\nA;O\u001e0=\u0016 I I;O\u001er= (q\u0003)A\nI(\u001br)B\nAqI\nB; r= 1;2;3 (6)\nNote that the \frst two operators correspond to mass terms for the scalars and fermions,\nrespectively, in the hypermultiplet. The third operator is a triplet with respect to the\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\n\\subsection{Mass deformations}\nIn the following, we consider the coset with $n=1$, i.e. a single vector multiplet. The coset representative is expressed in terms of four scalars $\\phi^i, i=0,1,2,3$ via\n\\begin{eqnarray}\nL=\\prod_{i=0}^3e^{\\phi^i K^i}\n\\end{eqnarray}\nwhere $K^i$ are the non compact generators of $SO(4,1)$; see  for details. Note that  $\\phi^0$ is an $SU(2)_R$ singlet, while the other three scalars $\\phi^r$ form an $SU(2)_R$ triplet. \nThe scalar potential for this specific case can be obtained from and takes the following form\n\\begin{align}\nV(\\sigma,\\phi^i) =- g^2 e^{2 \\sigma }+\\frac{1}{8} m e^{-6 \\sigma } \\bigg[-32 g e^{4 \\sigma } \\cosh \\phi^0 \\cosh \\phi^1 \\cosh \\phi^2 \\cosh \\phi^3+8 m \\cosh ^2\\phi^0\n\\nonumber\\\\\n+m \\sinh ^2 \\phi^0 \\bigg(-6+8 \\cosh ^2\\phi^1 \\cosh ^2 \\phi^2 \\cosh (2 \\phi^3)+\\cosh (2 (\\phi^1-\\phi^2))\\nonumber\\\\\n+\\cosh (2 (\\phi^1+\\phi^2))+2 \\cosh (2 \\phi^1)+2 \\cosh (2 \\phi^2)\\bigg)\\bigg]\n\\end{align}\nThe supersymmetric AdS$_6$ vacuum is given by setting $g=3m$ and setting all scalars to vanish. The masses of the  linearized  scalar fluctuation around the AdS vacuum  determine the dimensions of the dual scalar operators in the SCFT via\n\\begin{eqnarray}\nm^2l^2= \\Delta(\\Delta-5)\n\\end{eqnarray}\nwhere $l$ is the curvature radius of the AdS$_6$ vacuum. For the scalars at hand, one finds\n\\begin{eqnarray}\nm_\\sigma^2 l^2 = -6 \\hspace{0.8 in} m_{\\phi^0}^2 l^2 = -4 \\hspace{0.8 in} m_{\\phi^r}^2 l^2 = -6\\,\\,,\\,\\,\\,\\,r=1,2,3\n\\end{eqnarray}\nHence the dimensions of the dual operators are \n\\begin{eqnarray}\n\\Delta_{{\\cal O}_\\sigma} = 3, \\hspace{0.8 in} \\Delta_{{\\cal O}_{\\phi^0}} = 4, \\hspace{0.8 in} \\Delta_{{\\cal O}_{\\phi^r}} = 3\\,\\,,\\,\\,\\,\\,r=1,2,3\n\\end{eqnarray}\nIn  these CFT operators were expressed in terms of free hypermultiplets (i.e. the singleton sector). The case of $n=1$ corresponds to having a single free hypermultiplet, consisting of four real scalars $q_A^I$ and two symplectic Majorana spinors $\\psi ^I$. Here $I=1,2$ is the $SU(2)_R$ R-symmetry index and $A=1,2$ is the $SU(2)$ flavor symmetry index. The gauge invariant operators appearing in are related to these fundamental fields as follows,\n\\begin{eqnarray}\n{\\cal O}_{\\sigma}=(q^*)^A_{\\;\\;I} q_{\\;\\;A}^I,  \\hspace{0.4 in} {\\cal O}_{\\phi^0}= \\bar \\psi_I \\psi^I,  \\hspace{0.4 in}{\\cal O}_{\\phi^r}=  (q^*) ^A_{\\;\\;I}  (\\sigma^r)_A^{\\;\\; B}  {q^I}_B\\,\\,, \\,\\,\\,\\,\\,r=1,2,3\n\\end{eqnarray}\nNote that the first two operators correspond to mass terms for the scalars and fermions, respectively, in the hypermultiplet. The third operator is a triplet with respect to the\n\\end{document}\n"}
{"R-symmetry. As argued in , the \feld \u001e0is the top component of the global current\nsupermultiplet. Therefore a deformation by O\u001e0will break superconformal symmetry but\npreserve all Poincare supersymmetry . However, deformation by O\u001e0alone is inconsistent.\nPoincare supersymmetry demands that we also turn on the scalar masses O\u001b. Moreover,\nsupersymmetry on S5requires an additional operator in the action that breaks the super-\nconformalSU(2)Rsymmetry to U(1)Rsymmetry . Without loss of generality, we may choose\nthis operator to be O\u001e3.\n1 Euclidean theory and BPS solutions\nIn this section we will obtain the six-dimensional holographic dual of a mass deformation\nof a 5D SCFT on S5. Such a dual is given by S5-sliced domain wall solutions of matter-\ncoupled Euclidean F(4) gauged supergravity. In order to obtain such solutions, we must \frst\ncontinue the Lorentzian signature gauged supergravity outlined above to Euclidean signature,\nwhich has subtleties for both the scalar and fermionic sectors. Once the Euclidean theory\nis obtained, we turn on relevant scalars necessary to support the domain wall. As discussed\nin the previous section, at least three scalars must be turned on to obtain supersymmetric\nsolutions. The ansatz for the domain wall solutions takes the following form\nds2=du2+e2f(u)ds2\nS5; \u001b =\u001b(u); \u001ei=\u001ei(u); i= 0;3 (1)\nwith the remaining \felds set to zero. Next we will obtain a consistent set of BPS equations\non the above ansatz, and then solve them numerically. When solving them, we will demand\nas an initial condition that for some \fnite uthe metric factor e2fvanishes, so that the\ngeometry closes o\u000b smoothly.\n1.1 Euclidean action\nThe Euclidean action may be obtained from the Lorentzian one by \frst performing a simple\nWick rotation of Lorentzian time t!\u0000ix6. This makes the spacetime metric negative de\f-\nnite, since the metric in the Lorentzian theory was taken to be of mostly negative signature.\nHowever, we will choose to work with the Euclidean theory with positive de\fnite metric.\nMaking this modi\fcation involves a change in the sign of the Ricci scalar. Then noting that\nthe Euclidean action is related to the Lorentzian action by exp\u0000\niSLor\u0001\n= exp\u0000\n\u0000SEuc\u0001\n, the\n\fnal result of the Wick rotation is the following Euclidean action,\nS6D=1\n4\u0019G 6Z\nd6xp\nGL;L=\u0012\n\u00001\n4R+@\u0016\u001b@\u0016\u001b+1\n4Gij(\u001e)@\u0016\u001ei@\u0016\u001ej+V(\u001b;\u001ei)\u0013\n(2)\nwhere the spacetime metric Gis positive de\fnite and G6is the six-dimensional Newton's\nconstant. By abuse of notation, Gij(\u001e) with indices refers to the metric on the scalar manifold\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n R-symmetry. As argued in , the field $\\phi^0$ is the top component of the global current supermultiplet. Therefore a deformation by ${\\cal O}_{\\phi^0}$ will break superconformal symmetry but preserve all Poincare supersymmetry . However, deformation by ${\\cal O}_{\\phi^0}$ alone is inconsistent. Poincare supersymmetry demands that we also turn on the scalar masses ${\\cal O}_{\\sigma}$. Moreover, supersymmetry on $S^5$ requires an additional operator in the action that breaks the superconformal $SU(2)_R$ symmetry to $U(1)_R$ symmetry . Without loss of generality, we may choose this operator to be ${\\cal O}_{\\phi^3}$.\n\\section{Euclidean theory and BPS solutions}\n\\setcounter{equation}{0}\nIn this section we will obtain the six-dimensional holographic dual of a mass deformation of a 5D SCFT on $S^5$. Such a dual is given by $S^5$-sliced domain wall solutions of matter-coupled Euclidean $F(4)$ gauged supergravity. In order to obtain such solutions, we must first continue the Lorentzian signature gauged supergravity outlined above to Euclidean signature, which has subtleties for both the scalar and fermionic sectors. Once the Euclidean theory is obtained, we turn on relevant scalars necessary to support the domain wall. As discussed in the previous section, at least three scalars must be turned on to obtain supersymmetric solutions. The ansatz for the domain wall solutions takes the following form\n\\begin{eqnarray}\nds^2 = du^2 + e^{2f(u)} ds_{S^5}^2,\\hspace{0.4in} \\sigma=\\sigma(u),\\hspace{0.4in}  \\phi^i=\\phi^i(u), \\hspace{0.1 in}i=0 ,3\n\\end{eqnarray}\nwith the remaining fields set to zero. Next we will obtain a consistent set of BPS equations on  the above ansatz, and then solve them numerically. When solving them, we will demand as an initial condition that for some finite $u$ the metric factor $e^{2f}$ vanishes, so that the geometry closes off smoothly.\n\\subsection{Euclidean action}\nThe Euclidean action may be obtained from the Lorentzian one by first performing a simple Wick rotation of Lorentzian time $t\\rightarrow-ix^6$. This makes the spacetime metric negative definite, since the metric in the Lorentzian theory was taken to be of mostly negative signature. However, we will choose to work with the Euclidean theory with positive definite metric. Making this modification involves a change in the sign of the Ricci scalar. Then noting that the Euclidean action is related to the Lorentzian action by $\\exp \\left(i S^{Lor}\\right)=\\exp \\left(-S^{Euc}\\right)$, the final result of the Wick rotation is the following Euclidean action,\n\\begin{eqnarray}\nS_{6D}=\\frac{1}{4\\pi G_6}\\int d^6x ~\\sqrt{G} {\\cal L} ~,~~~~ {\\cal L}= \\left(-{1 \\over 4} R +\\partial_\\mu\\sigma\\partial^\\mu\\sigma+{1\\over 4}G_{ij}(\\phi)\\partial_\\mu \\phi^i \\partial^\\mu \\phi^j + V(\\sigma,\\phi^i)\\right)\n\\end{eqnarray}\nwhere the spacetime metric $G$ is positive definite and $G_6$ is the six-dimensional Newton's constant. By abuse of notation, $G_{ij}(\\phi)$ with indices refers to the metric on the scalar manifold\n\\end{document}\n"}
{"which for the coset representative is given by\nGij= diag\u0000\ncosh2\u001e1cosh2\u001e2cosh2\u001e3;cosh2\u001e2cosh2\u001e3;cosh2\u001e3;1\u0001\n(1)\nIn addition to performing the above Wick rotation, we also perform a Wick rotation on the\nsigma model\nSO(4;1)\nSO(4)!SO(4;1)\nSO(3;1)'dS4 (2)\nThe metric on the sigma model is now that of dS 4, as opposed to the H4that we had in the\nLorentzian case . This can be obtained by making the following change to the H4coset,\n\u001er!i\u001err= 1;2;3 (3)\nIt would be interesting to understand this analytic continuation from \frst principles and its\nrelation to Euclidean supersymmetry, possibly along the lines of . For now, we just note that\nsuch a Wick rotated model seems necessary to obtain regular, supersymmetric solutions.\n0.1 Euclidean supersymmetry\nThe next task is to identify the form of the Euclidean supersymmetry variations. Motivation\nfor the form of these variations may be obtained by analysis of the free di\u000berential algebra\n(FDA) of the F(4) gauged supergravity theory with H6vacuum, as discussed in Appendix .\nThe \fnal result for this FDA is given in, and is noted to be of the same form as the FDA\nfor the theory with dS 6background (identi\fed in ), with two di\u000berences. The \frst obvious\ndi\u000berence is that the metrics di\u000ber - the space considered in was dS 6with mostly minus\nsignature, whereas we are currently focused on positive de\fnite H6. However, both of these\nspaces have R\u0016\u0017=\u000020m2g\u0016\u0017. The second di\u000berence is in the de\fnition of Dirac conjugate\nspinors. However, once the di\u000berence in de\fnition of the gamma matrices is accounted for,\nthe only di\u000berence is a factor of i, i.e.\n\u0016 (H6)\nA=i\u0016 (dS6)\nA (4)\nBecause of these similarities, the supersymmetry variations in the current case are expected\nto be of a similar form to that of . In particular, the variations of the fermions are expected\nto be of the form\n\u000e\u001fA=\u00001\n2\n\u0016@\u0016\u001b\"A+NAB\"B+:::\n\u000e A\u0016=D\u0016\"A+iSAB\n\u0016\"B+:::\n\u000e\u0015I\nA=\u0000^PI\nri\u001br\nAB@\u0016\u001ei\n\u0016\"B+^PI\n0i\u000fAB@\u0016\u001ei\n7\n\u0016\"B+MI\nAB\"B+::: (5)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nwhich for the coset representative is given by\n\\begin{eqnarray}\nG_{ij}=\\mathrm{diag}\\left(\\cosh^2 \\phi^1 \\cosh^2 \\phi^2 \\cosh^2 \\phi^3, \\cosh^2 \\phi^2 \\cosh^2 \\phi^3, \\cosh ^2 \\phi^3, 1\\right)\n\\end{eqnarray}\nIn addition to performing the above Wick rotation, we also perform a Wick rotation on the sigma model  \n\\begin{eqnarray}\n{SO(4,1) \\over SO(4)} \\to{SO(4,1) \\over SO(3,1)} \\simeq dS_4\n\\end{eqnarray}\nThe metric on the sigma model is now that of dS$_4$, as opposed to the $\\mathbb{H}_4$ that we had in the Lorentzian case . This can be obtained by making the following change to the $\\mathbb{H}_4$ coset,\n\\begin{eqnarray}\n \\phi_r\\rightarrow i \\phi_r \\hspace{0.5 in}r=1,2,3\n \\end{eqnarray}\nIt would be interesting to understand this analytic continuation from first principles and its relation to Euclidean supersymmetry, possibly along the lines of . For now, we just note that such a Wick rotated model seems necessary to obtain regular, supersymmetric solutions. \n\\subsection{Euclidean supersymmetry}\nThe next task is to identify the form of the Euclidean supersymmetry variations. Motivation for the form of these variations may be obtained by analysis of the free differential algebra (FDA) of the $F(4)$ gauged supergravity theory with $\\mathbb{H}_6$ vacuum, as discussed in Appendix . The final result for this FDA is given in, and is noted to be of the same form as the FDA for the theory with dS$_6$ background (identified in ), with two differences. The first obvious difference is that the metrics differ - the space considered in  was dS$_6$ with mostly minus signature, whereas we are currently focused on positive definite $\\mathbb{H}_6$. However, both of these spaces have $R_{\\mu\\nu} = - 20 m^2 g_{\\mu\\nu}$. The second difference is in the definition of Dirac conjugate spinors. However, once the difference in definition of the gamma matrices is accounted for, the only difference is a factor of $i$, i.e.\n\\begin{eqnarray}\n\\bar \\psi_A^{(\\mathbb{H}_6)} =  i \\bar \\psi_A^{(dS_6)} \n\\end{eqnarray}\nBecause of these similarities, the supersymmetry variations in the current case are expected to be of a similar form to that of . In particular, the variations of the fermions are expected to be of the form\n\\begin{eqnarray}\n\\delta \\chi_A = - {1\\over 2} \\gamma^\\mu \\partial_\\mu \\sigma \\varepsilon_A + N_{AB} \\varepsilon^B + \\dots\n\\nonumber\\\\\\nonumber\\\\\n\\delta \\psi_{A \\mu} = {\\cal D}_\\mu \\varepsilon_A + i S_{AB} \\gamma_\\mu \\varepsilon^B +\\dots\n\\nonumber\\\\\\nonumber\\\\\n\\delta \\lambda^I_A = - \\hat P^I_{r i} \\sigma^r_{AB} \\partial_\\mu \\phi^i \\gamma^\\mu \\varepsilon^B + \\hat P^I_{0 i} \\epsilon_{AB} \\partial_\\mu \\phi^i \\gamma^7 \\gamma^\\mu \\varepsilon^B + M^I_{AB} \\varepsilon^B+\\dots\n\\end{eqnarray}\n\\end{document}\n"}
{"whereNAB,SAB, andMI\nABare again given by, but now with the appropriate rede\fnition\nof the coset representative as per. It should be noted that while the FDA analysis presented\nin Appendix is a strong motivation for the form of the supersymmetry variations presented\nabove, it is not a proof. To actually derive the form of these variations, one must \frst\nintroduce curvature terms representing deviations from zero of each line in the free di\u000berential\nalgebra. An application of the exterior derivative to the resulting expressions then gives rise\nto Bianchi identities, which must be solved before obtaining the explicit form of the fermion\nvariations. This is a rather involved process, and so for the moment we will content ourselves\nwith the motivating comments provided by the FDA. We will take the eventual presence of\nsmooth supersymmetric solutions consistent with the equations of motion as a posteriori\nevidence for the legitimacy of these variations. A nice property of the variations above is\nthe fact that they are consistent with the following SO(6)-invariant symplectic Majorana\ncondition,\n\u0016 A=\u000fAB T\nBC (1)\nThe consistency of such a condition allows us to work with symplectic Majorana spinors just\nas in the Lorentzian case, though the symplectic Majorana condition utilized here is di\u000berent\nthan that of the Lorentzian case. As mentioned before, we will be concerned with only the\nsimplest case of a single non-zero SU(2)R-charged vector multiplet scalar \u001e3, i.e. we take\n\u001e1=\u001e2= 0. It can be easily veri\fed that this is a consistent truncation, and is in fact the\nmost general choice of non-vanishing \felds that can preserve SO(4;2)\u0002U(1)R. With this\nconsistent truncation, the functions NAB,SAB, andMI\nABappearing in the supersymmetry\nvariations reduce to\nSAB=iS0\u000fAB+iS3\n7\u001b3\nAB\nNAB=\u0000N0\u000fAB\u0000N3\n7\u001b3\nAB\nMI\nAB=M0\n7\u000fAB+M3\u001b3\nAB (2)\nwhere we have de\fned\nS0=1\n4\u0000\ngcos\u001e3e\u001b+me\u00003\u001bcosh\u001e0\u0001\nS3=1\n4im e\u00003\u001bsinh\u001e0sin\u001e3\nN0=\u00001\n4\u0000\ngcos\u001e3e\u001b\u00003me\u00003\u001bcosh\u001e0\u0001\nN3=\u00003\n4ime\u00003\u001bsinh\u001e0sin\u001e3\nM0= 2m e\u00003\u001bcos\u001e3sinh\u001e0\nM3=\u00002ig e\u001bsin\u001e3(3)\nImportantly, note that S3,N3, andM3are now purely imaginary, in contrast to the Lorentzian\ncase . In all that follows we will set m=\u00001=2\u0011such that the radius of AdS 6is one.\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nwhere $N_{AB}$, $S_{AB}$, and $M^I_{AB}$ are again given by, but now with the appropriate redefinition of the coset representative as per. It should be noted that while the FDA analysis presented in Appendix  is a strong motivation for the form of the supersymmetry variations presented above, it is not a proof. To actually derive the form of these variations, one must first introduce curvature terms representing deviations from zero of each line in the free differential algebra. An application of the exterior derivative to the resulting expressions then gives rise to Bianchi identities, which must be solved before obtaining the explicit form of the fermion variations. This is a rather involved process, and so for the moment we will content ourselves with the motivating comments provided by the FDA. We will take the eventual presence of smooth supersymmetric solutions consistent with the equations of motion as \\textit{a posteriori} evidence for the legitimacy of these variations.\nA nice property of the variations above is the fact that they are consistent with the following $SO(6)$-invariant symplectic Majorana condition, \n\\begin{eqnarray}\n\\bar \\psi_A = \\epsilon^{AB} \\psi_B^T {\\cal C}\n\\end{eqnarray}\nThe consistency of such a condition allows us to work with symplectic Majorana spinors just as in the Lorentzian case, though the symplectic Majorana condition utilized here is different than that of the Lorentzian case.\nAs mentioned before, we will be concerned with only the simplest case of a single non-zero $SU(2)_R$-charged vector multiplet scalar $\\phi^3$, i.e. we take $\\phi^1=\\phi^2=0$. It can be easily verified that this is a\nconsistent truncation, and is in fact the most general choice of non-vanishing fields that can preserve $SO(4, 2) \\times U(1)_R$. With this consistent truncation, the functions $N_{AB}$, $S_{AB}$, and $M^I_{AB}$ appearing in the supersymmetry variations reduce to \n\\begin{eqnarray}\nS_{AB} = i S_0 \\epsilon_{AB} + i S_3 \\gamma^7 \\sigma^3_{AB}\n\\nonumber\\\\\nN_{AB}= -N_0 \\epsilon_{AB} -N_3 \\gamma^7 \\sigma^3_{AB}\n\\nonumber\\\\\nM^I_{AB}= M_0 \\gamma^7 \\epsilon_{AB} + M_3  \\sigma^3_{AB}\n\\end{eqnarray}\nwhere we have defined \n\\begin{eqnarray}\nS_0=\\frac14 \\left(g\\cos \\phi^3 e^\\sigma+m e^{-3\\sigma}\\cosh \\phi^0\\right)\n\\nonumber\\\\\nS_3=\\frac14 i \\,m ~e^{-3\\sigma}\\sinh \\phi^0 \\sin \\phi^3\n\\nonumber\\\\\nN_0=-\\frac14 \\left(g\\cos \\phi^3 e^\\sigma-3m e^{-3\\sigma}\\cosh \\phi^0\\right)\n\\nonumber\\\\\nN_3=-\\frac34 i\\, m e^{-3\\sigma}\\sinh \\phi^0 \\sin \\phi^3\n\\nonumber\\\\\nM_0=2m ~e^{-3\\sigma}\\cos \\phi^3\\sinh \\phi^0\n\\nonumber\\\\\nM_3=-2 i\\, g ~e^{\\sigma}\\sin \\phi^3\n\\end{eqnarray}\nImportantly, note that $S_3$, $N_3$, and $M_3$ are now purely imaginary, in contrast to the Lorentzian case . In all that follows we will set $m=-1/2\\, \\eta$ such that the radius of AdS$_6$ is one.\n\\end{document}\n"}
{"0.1 BPS Equations\nWe now use the vanishing of the fermionic variations to obtain BPS equations for the warp\nfactor and the three non-zero scalars.\n0.1.1 Dilatino equation and projector\nWe begin by imposing the vanishing of the dilatino variation, \u000e\u001fA= 0, which implies\n1\n2\n5\u001b0\"A=N0\"A+N3\n7(\u001b3)B\nA\"B (1)\nThis equation can be interpreted as a projection condition on the spinors \"A. Consistency\nof this projection condition then requires that\n\u001b0= 2\u0011q\nN2\n0+N2\n3 (2)\nwhere\u0011=\u00061. Plugging this BPS equation back into then yields a second form of the\nprojection condition,\n\n5\"A=G0\"A\u0000G3\n7(\u001b3)B\nA\"B (3)\nwhich is more useful in the derivation of the other BPS equations. In the above, we have\nde\fned\nG0=\u0011N0p\nN2\n0+N2\n3G3=\u0000\u0011N3p\nN2\n0+N2\n3(4)\n0.1.2 Gravitino equation\nThe analysis of the gravitino equation \u000e A\u0016= 0 proceeds in exactly the same way as for the\nLorentzian case studied in . The procedure gives rise to a \frst-order equation for the warp\nfactorfand an algebraic constraint. To avoid excessive overlap with that paper, we simply\ncite the result,\nf0= 2(G0S0+G3S3)e\u00002f= 4(G0S0+G3S3)2\u00004(S2\n0+S2\n3) (5)\n0.1.3 Gaugino equations\nFinally, we turn toward the gaugino equation \u000e\u0015I\nA= 0. Again the analysis of this equation\nproceeds in an exactly analogous manner to the Lorentzian case . The result is\ncos\u001e3(\u001e0)0=\u0000(G0M0+G3M3) (\u001e3)0=i(G3M0\u0000G0M3) (6)\nThe right-hand sides of both equations are real, and thus give rise to real solutions when\nappropriate initial conditions are imposed.\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\\subsection{BPS Equations}\nWe now use the vanishing of the fermionic variations to obtain BPS equations for the warp factor and the three non-zero scalars. \n\\subsubsection{Dilatino equation and projector}\nWe begin by imposing the vanishing of the dilatino variation, $\\delta\\chi_A = 0$, which implies\n\\begin{eqnarray}\n{1\\over 2} \\gamma^5 \\sigma' \\varepsilon_A = N_0 \\varepsilon_A + N_3 \\gamma^7 {(\\sigma^3)^B}_A \\varepsilon_B\n\\end{eqnarray}\nThis equation can be interpreted as a projection condition on the spinors $\\varepsilon_A$. Consistency of this projection condition then requires that \n\\begin{eqnarray}\n \\sigma' = 2 \\eta \\sqrt{N_0^2 + N_3^2}\n\\end{eqnarray}\nwhere $\\eta = \\pm1$. Plugging this BPS equation back into then yields a second form of the projection condition,\n\\begin{eqnarray}\n\\gamma^5 \\varepsilon_A = G_0 \\varepsilon_A - G_3 \\gamma^7 {(\\sigma^3)^B}_A \\varepsilon_B\n\\end{eqnarray}\nwhich is more useful in the derivation of the other BPS equations. In the above, we have defined \n\\begin{eqnarray}\nG_0 = \\eta {N_0 \\over \\sqrt{N_0^2 + N_3^2}}\\hspace{1 in} G_3 = -\\eta {N_3 \\over \\sqrt{N_0^2 + N_3^2}}\n\\end{eqnarray}\n\\subsubsection{Gravitino equation}\nThe analysis of the gravitino equation  $\\delta \\psi_{A \\mu}=0$ proceeds in exactly the same way as for the Lorentzian case studied in . The procedure gives rise to a first-order equation for the warp factor $f$ and an algebraic constraint. To avoid excessive overlap with that paper, we simply cite the result,\n\\begin{eqnarray}\nf^{\\prime }=2(G_{0}S_{0}+G_{3}S_{3})\\hspace {.5in}e^{-2f}=4(G_{0}S_{0}+G_{3}S_{3})^{2}-4(S_{0}^{2}+S_{3}^{2})\\hspace {.5in}(5)\n\\end{eqnarray}\n\\subsubsection{Gaugino equations}\nFinally, we turn toward the gaugino equation $\\delta \\lambda^I_A=0$. Again the analysis of this equation proceeds in an exactly analogous manner to the Lorentzian case . The result is\n\\begin{eqnarray}\n\\cos \\phi^3 (\\phi^0)' = - (G_0 M_0 + G_3 M_3) \\hspace{0.5 in} (\\phi^3)' = i (G_3 M_0 - G_0 M_3)\n\\end{eqnarray}\nThe right-hand sides of both equations are real, and thus give rise to real solutions when appropriate initial conditions are imposed.\n\\end{document}\n"}
{"0.0.1 Summary of \frst-order equations\nTo summarize, the \frst-order equations for the warp factor fand the scalars \u001b;\u001e0;\u001e3are\nfound to be\nf0= 2 (G0S0+G3S3)\n\u001b0= 2\u0011q\nN2\n0+N2\n3\ncos\u001e3\u0000\n\u001e0\u00010=\u0000(G0M0+G3M3)\n\u0000\n\u001e3\u00010=i(G3M0\u0000G0M3) (1)\nFurthermore, for consistency these were required to satisfy the algebraic constraint\ne\u00002f= 4 (G0S0+G3S3)2\u00004\u0000\nS2\n0+S2\n3\u0001\n(2)\nThe various functions featured in these equations were de\fned in and .\n0.1 Numeric solutions\nIn order to get acceptable numerical solutions from these equations, we must choose ap-\npropriate initial conditions. It is easy to check that the following initial conditions ensure\nsmoothness of all three scalars, as well as the vanishing of e2fat the origin,\n\u001e3\n0= sin\u00001\u00141\n8 tanh\u001e0\n0\u0012\n\u00003 +q\n9 + 16 tanh2\u001e0\n0\u0013\u0015\n\u001b0=1\n4log2\n664cosh\u001e0\n0\u0010\n5 +p\n9 + 16 tanh2\u001e0\n0\u0011\np\n6r\n8 + coth2\u001e0\n0\u0010\n\u00003 +p\n9 + 16 tanh2\u001e0\n0\u00113\n775(3)\nWe have de\fned for notational convenience \u001e\u000b\n0\u0011\u001e\u000b(0) and\u001b0\u0011\u001b(0). For these initial\nconditions to be real, we must ensure that\njf(\u001e0\n0)j\u00141f(\u001e0\n0)\u00111\n8 tanh\u001e0\n0\u0012\n\u00003 +q\n9 + 16 tanh2\u001e0\n0\u0013\n(4)\nNoting that\nlim\n\u001e0\n0!\u00001f(\u001e0\n0) =\u00001\n4lim\n\u001e0\n0!+1f(\u001e0\n0) =1\n4(5)\nand also that f(\u001e0\n0) is monotonically increasing, i.e.\ndf\nd\u001e0\n0>08\u001e0\n02R (6)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\\subsubsection{Summary of first-order equations}\nTo summarize, the first-order equations for the warp factor $f$ and the scalars $\\sigma,\\phi^0,\\phi^3$ are found to be\n\\begin{eqnarray}\nf'=2\\left(G_0S_0+ G_3S_3\\right)\n\\nonumber \\\\%\n \\sigma'=2 \\eta \\sqrt{N_0^2 + N_3^2}\n\\nonumber \\\\%\n\\cos \\phi^3 \\left(\\phi^0\\right)'=-\\left(G_0M_0+G_3M_3\\right)\n\\nonumber\\\\%\n \\left(\\phi^3\\right)'=i \\left(G_3 M_0-G_0M_3\\right)\n\\end{eqnarray}\nFurthermore, for consistency these were required to satisfy the algebraic constraint\n\\begin{eqnarray}\ne^{-2f}=4\\left(G_0S_0+G_3S_3\\right)^2-4 \\left(S_0^2+S_3^2\\right)\n\\end{eqnarray}\nThe various functions featured in these equations were defined in and .\n\\subsection{Numeric solutions}\nIn order to get acceptable numerical solutions from these equations, we must choose appropriate initial conditions. It is easy to check that the following initial conditions ensure smoothness of all three scalars, as well as the vanishing of $e^{2f}$ at the origin,\n\\begin{eqnarray}\n \\phi^3_0 = \\sin^{-1} \\left[{1 \\over 8 \\tanh\\phi^0_0}\\left(-3 + \\sqrt{9 + 16 \\tanh^2 \\phi^0_0}\\right) \\right]\n\\nonumber\\\\\n\\sigma_0 = {1 \\over 4} \\log \\left[{\\cosh \\phi^0_0 \\left(5 + \\sqrt{9 + 16 \\tanh^2 \\phi^0_0} \\right) \\over \\sqrt{6}\\sqrt{8 + \\coth^2 \\phi^0_0 \\left(-3 + \\sqrt{9 + 16 \\tanh^2 \\phi^0_0} \\right)}}\\right]\n\\end{eqnarray}\nWe have defined for notational convenience $\\phi^\\alpha_0 \\equiv \\phi^\\alpha(0)$ and $\\sigma_0 \\equiv \\sigma(0)$. For these initial conditions to be real, we must ensure that\n\\begin{eqnarray}\n|f(\\phi^0_0)| \\leq 1 \\hspace{0.7 in} f(\\phi^0_0) \\equiv {1 \\over 8 \\tanh\\phi^0_0}\\left(-3 + \\sqrt{9 + 16 \\tanh^2 \\phi^0_0}\\right) \n\\end{eqnarray}\nNoting that \n\\begin{eqnarray}\n\\lim_{\\phi^0_0 \\rightarrow - \\infty} f(\\phi^0_0) = - {1 \\over 4} \\hspace{0.8 in} \\lim_{\\phi^0_0 \\rightarrow + \\infty} f(\\phi^0_0) ={1 \\over 4}\n\\end{eqnarray}\nand also that $f(\\phi^0_0)$ is monotonically increasing, i.e.\n\\begin{eqnarray}\n{d f \\over d \\phi^0_0} >0\\hspace{0.3in} \\forall \\phi^0_0 \\in \\mathbb R\n\\end{eqnarray}\n\\end{document}\n"}
{"allows us to conclude that this is always the case for real initial conditions \u001e0\n0. Thus we\nhave a one parameter family of real smooth solutions, labeled by the IR parameter \u001e0\n0. With\nthis in mind, we may choose any value of \u001e0\n0and solve the BPS equations in numerically.\nIn Figure , we plot the solutions obtained for the following choices of initial condition:\n\u001e0\n0=f0:25;0:5;1;1:5;2g. In order to get smooth solutions for u>0, we must take \u0011=\u00001.\nIt is straighforward to verify that the resulting solutions are completely smooth and have\nthe expected vanishing of e2fat the origin, implying that the spacetime smoothly pinches\no\u000b. Furthermore, e2f=e2uis seen to asymptote to a constant, which we denote by e2fk.\n0.1 UV asymptotic expansions\nAs in the holographic Janus solutions in Lorentzian signature , the BPS equations may also\nbe used to obtain the UV asymptotic behavior of the solutions. To do so, we begin by\nde\fning an asymptotic coordinate z=e\u0000u, where the asymptotic S5boundary is reached by\ntakingu!1 . Consequently, an asymptotic expansion is an expansion around z= 0. The\ncoe\u000ecients in the UV expansions of the non-zero \felds may now be solved for order-by-order\nusing the BPS equations. One \fnds explicitly that all coe\u000ecients are determined in terms of\nonly three independent parameters \u000b,\f, andfk, in accord with the fact that there are three\nindependent \frst-order di\u000berential equations. The \frst few terms in the expansions are\nf(z) =\u0000logz+fk\u0000\u00121\n4e\u00002fk+1\n16\u000b2\u0013\nz2+O(z4)\n\u001b(z) =3\n8\u000b2z2+1\n4efk\u000b\fz3+O(z4)\n\u001e0(z) =\u000bz\u0000\u00125\n4\u000be\u00002fk+23\n48\u000b3\u0013\nz3+O(z4)\n\u001e3(z) =e\u0000fk\u000bz2+\fz3+O(z4) (1)\nWe have obtained the expansions up to O(z8), but we display only the \frst few terms here.\n1 Holographic sphere free energy\nThe goal of this section is to obtain the holographic free energy, i.e. the renormalized on-shell\naction. We begin by writing the full action,\nS=S6D+SGH\nS6D=Z\ndu d5xp\nGL SGH=\u00001\n2Z\nd5xp\nK (1)\nwhereS6Dis the six-dimensional Euclidean action given in and SGHis the Gibbons-Hawking\nterm. The\nappearing in SGHis the determinant of the induced metric on the boundary\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nallows us to conclude that this is always the case for real initial conditions $\\phi^0_0$. Thus we have a one parameter family of real smooth solutions, labeled by the IR parameter $\\phi^0_0$.\nWith this in mind, we may choose any value of $\\phi^0_0$ and solve the BPS equations in numerically. In Figure , we plot the solutions obtained for the following choices of initial condition: $\\phi^0_0=\\{0.25,\\,0.5,\\,1,\\,1.5,\\,2 \\}$. In order to get smooth solutions for $u>0$, we must take $\\eta = -1$. It is straighforward to verify that  the resulting solutions are completely smooth and have the expected vanishing of $e^{2f}$ at the origin, implying that the spacetime smoothly pinches off. Furthermore, $e^{2f}/e^{2u}$ is seen to asymptote to a constant, which we denote by $e^{2 f_k}$.\n\\subsection{UV asymptotic expansions}\nAs in the holographic Janus solutions in Lorentzian signature , the BPS equations may also be used to obtain the UV asymptotic behavior of the solutions. To do so, we begin by defining an asymptotic coordinate $z = e^{- u}$, where the asymptotic $S^5$  boundary is reached by taking $u\\to \\infty$. Consequently, an asymptotic expansion is an expansion around $z=0$. The coefficients in the UV expansions of the non-zero fields may now be solved for order-by-order using the BPS equations. One finds explicitly that all coefficients are determined in terms of only three independent parameters $\\alpha$, $\\beta$, and $f_k$, in accord with the fact that there are three independent first-order differential equations. The first few terms in the expansions are\n\\begin{eqnarray}\nf(z) = -\\log z +f_k -  \\left({1\\over 4}e^{-2f_k} + {1\\over 16}\\alpha^2\\right)  \\,z^2 + O(z^4) \\nonumber\\\\\n\\sigma(z) = {3\\over 8} \\alpha^2 \\,z^2 +  {1\\over 4} e^{f_k} \\alpha \\beta \\,z^3 + O(z^4) \n\\nonumber\\\\\n\\phi^0(z) = \\alpha \\, z - \\left({5\\over 4}\\alpha \\,e^{-2f_k} + {23\\over 48} \\alpha^3\\right) \\,z^3 + O(z^4)\n\\nonumber\\\\\n\\phi^3(z) = e^{-f_k} \\alpha  z^2 + \\beta \\, z^3 + O(z^4) \n\\end{eqnarray}\nWe have obtained the expansions up to $O(z^8)$, but we display only the first few terms here.\n\\section{Holographic sphere free energy}\n\\setcounter{equation}{0}\nThe goal of this section is to obtain the holographic free energy, i.e. the renormalized on-shell action. We begin by writing the full action, \n\\begin{eqnarray}\n\\vphantom{.} \\hspace{1.7 in} S = S_{\\mathrm{6D}} + S_{\\mathrm{GH}}\n\\nonumber\\\\\\nonumber\\\\\n\\vphantom{.} S_{\\mathrm{6D}} = \\int du~d^5 x ~\\sqrt{G}\\, {\\cal L} \\hspace{1 in}S_{\\mathrm{GH}}=- {1 \\over 2} \\int d^5 x \\sqrt{\\gamma}\\, {\\cal K}\n\\end{eqnarray}\nwhere $S_{\\mathrm{6D}}$ is the six-dimensional Euclidean action given in and $S_{\\mathrm{GH}}$ is the Gibbons-Hawking term. The $\\gamma$ appearing in $S_{\\mathrm{GH}}$ is the determinant of the induced metric on the boundary \n\\end{document}\n"}
{"(located at some cuto\u000b distance u= \u0003), whileKis the trace of the extrinsic curvature\nKijof the radial S5slices. The latter is de\fned as\nKi=d\ndu\ni (1)\nIn general, the on-shell action is divergent and requires renormalization. The addition of\nin\fnite counterterms is standard in holographic renormalization , but in the current case\nwe must also add \fnite counterterms in order to preserve supersymmetry . We will begin\nour exploration of counterterms in this section by \frst considering the \fnite counterterms in\nthe limit of a \nat domain wall, after which we move onto in\fnite counterterms in the more\ngeneral case of a curved domain wall. Finally, appropriate curved space \fnite counterterms\nwill be \fxed by demanding \fniteness of the one-point functions of the dual operators.\n0.1 Finite counterterms\nIn order to obtain \fnite counterterms, we will make use of the Bogomolnyi trick . To do\nso, we will \frst need to identify a superpotential W. Though we will \fnd that no exact\nsuperpotential can be found for our solutions - in the sense that there is no superpotential\nwhich can recast all of the BPS equations in gradient \now form - we will be able to identify\nanapproximate superpotential. By \\approximate\" here, we mean that it does yield gradient\n\now equations up to terms of order O(z5), where the asymptotic coordinate zwas de\fned\nearlier asz=e\u0000u. This is useful since, as we will see later, we will only need terms up to\nO(z5) to obtain all divergent and \fnite counterterms. Terms of higher order will all vanish\nin the\u000f!0 limit, i.e. when the UV cuto\u000b is removed. Thus the approximate superpotential\nwill yield all \fnite counterterms.\n0.1.1 Approximate superpotential\nIn order to identify a candidate superpotential, we begin by recalling the form of the scalar\npotentialV. With the choice of coset representative and consistent truncation outlined in\nSection , one \fnds that\nV(\u001b;\u001ei) =\u00009m2e2\u001b\u000012m2e\u00002\u001bcosh\u001e0cos\u001e3+m2e\u00006\u001bcosh2\u001e0+m2e\u00006\u001bcos 2\u001e3sinh2\u001e0\nThis scalar potential can in fact be rewritten as\nV= 4(N2\n0+N2\n3) +1\n4(M2\n0+M2\n3)\u000020(S2\n0+S2\n3) (2)\nThen for BPS solutions, implies that\nV= (\u001b0)2+1\n4\u0010\n\u0000(\u001e30)2+ cos2\u001e3(\u001e00)2\u0011\n\u000020(S2\n0+S2\n3) (3)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n(located at some cutoff distance $u= \\Lambda$), while ${\\cal K}$ is the trace of the extrinsic curvature ${\\cal K}_{ij}$ of the radial $S^5$ slices. The latter is defined as\n\\begin{eqnarray}\n{\\cal K}_{i}=\\frac{d}{du} \\gamma_{i}\n\\end{eqnarray}\nIn general, the on-shell action is divergent and requires renormalization. The addition of infinite counterterms is standard in holographic renormalization   , but in the current case we must also add finite counterterms in order to preserve supersymmetry .  We will begin our exploration of counterterms in this section by first considering the finite counterterms in the limit of a flat domain wall, after which we move onto infinite counterterms in the more general case of a curved domain wall. Finally, appropriate curved space finite counterterms will be fixed by demanding finiteness of the one-point functions of the dual operators.\n\\subsection{Finite counterterms}\nIn order to obtain finite counterterms, we will make use of the Bogomolnyi trick . To do so, we will first need to identify a superpotential $W$. Though we will find that no exact superpotential can be found for our solutions - in the sense that there is no superpotential which can recast all of the BPS equations in gradient flow form -  we will be able to identify an \\textit{approximate} superpotential. By ``approximate\" here, we mean that it does yield gradient flow equations up to terms of order $O(z^5)$, where the asymptotic coordinate $z$ was defined earlier as $z=e^{-u}$. This is useful since, as we will see later, we will only need terms up to $O(z^5)$ to obtain all divergent and finite counterterms. Terms of higher order will all vanish in the $\\epsilon \\rightarrow 0$ limit, i.e. when the UV cutoff is removed. Thus the approximate superpotential will yield  all finite counterterms. \n\\subsubsection{Approximate superpotential}\nIn order to identify a candidate superpotential, we begin by recalling the form of the scalar potential $V$. With the choice of coset representative and consistent truncation outlined in Section , one finds that\n\\begin{eqnarray}\nV(\\sigma,\\phi^i) = -9 m^2 e^{2 \\sigma }-12m^2 e^{- 2 \\sigma} \\cosh \\phi^0 \\cos \\phi^3 + m^2 e^{-6 \\sigma} \\cosh^2 \\phi^0+m^2 e^{-6 \\sigma} \\cos 2 \\phi^3 \\sinh^2 \\phi^0\n\\nonumber\n\\end{eqnarray}\n This scalar potential can in fact be rewritten as\n\\begin{eqnarray}\nV = 4 (N_0^2 + N_3^2) + {1 \\over 4} (M_0^2 +M_3^2) - 20 (S_0^2 + S_3^2)\n\\end{eqnarray}\nThen for BPS solutions, implies that \n\\begin{eqnarray}\nV = (\\sigma')^2 + {1 \\over 4}\\left(-({\\phi^3}')^2 + \\cos^2 \\phi^3 ({\\phi^0}')^2\\right)- 20 (S_0^2 + S_3^2)\n\\end{eqnarray}\n\\end{document}\n"}
{"This motivates us to de\fne a superpotential Was\nW=q\nS2\n0+S2\n3 (1)\nUnfortunately, this superpotential does notallow one to write the BPS equations for both\n\u001e0and\u001e3as gradient \now equations. The reason for this failure is that the integrability\ncondition required to convert the BPS equation into a gradient \now form is not satis\fed;\nsee e.g. Appendix C.2.1 of . We thus follow the strategy of to construct an approximate\nsuperpotential. Our model consists of two consistent truncations that admit \nat domain walls\nand an exact superpotential. These are the \u001e3= 0;\u001e06= 0 truncation and the \u001e0= 0;\u001e36= 0\ntruncation. The corresponding \now equations are (we set \u0011=\u00001 henceforth)\n\u001e00=\u00008@\u001e0Wj\u001e3=0 \u001e30= 8@\u001e3Wj\u001e0=0 (2)\nrespectively. In either truncation, the BPS equations for the warp factor and dilaton \u001bcan\nbe put in the following form,\nf0= 2W \u001b0= 2@\u001bW (3)\nAn important fact is that, though the gradient \now equations of do not hold exactly in the\nfull model with \u001e06= 0;\u001e36= 0, they dohold up to and including O(z5). Looking at the form\nof the UV asymptotics of the scalar \felds, one may expand the superpotential of keeping\nonly terms contributing up to this order. This gives\nW=1\n2+3\n4\u001b2+1\n16(\u001e0)2\u00003\n16(\u001e3)2+1\n192(\u001e0)4\u00003\n16(\u001e0)2\u001b+::: (4)\nwhere the dots represent terms of order O(z6). This is the approximate superpotential we\nwill use in what follows.\n0.0.1 Bogomolnyi trick\nWe now use the Bogomolnyi trick to get the \fnite counterterms needed to preserve super-\nsymmetry in the case of a \nat domain wall. The central idea of the Bogomolnyi trick is that\nfor a BPS solution, the renormalized on-shell action must vanish. In order to make use of\nthis fact, we will \frst want to recast the on-shell action in a simpler form. To do so, we\nbegin by inserting. We \fnd that\nL=\u00001\n4R\u000020W2+ 2Lkin (5)\nwhere we've de\fned\nLkin= (\u001b0)2+1\n4h\n\u0000(\u001e30)2+ cos2\u001e3(\u001e00)2i\n(6)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nThis motivates us to define a superpotential $W$ as \n\\begin{eqnarray}\nW = \\sqrt{S_0^2 + S_3^2}\n\\end{eqnarray}\n\\iffalse\nThis can also be motivated from the integrability of the gravitino equation. Namely, one would usually expect integrability to give something proportional to $W^2$, i.e.\n\\begin{eqnarray}\n[D_m , D_n] \\epsilon_A = 2 W^2 \\,\\gamma_{mn} \\epsilon_A\n\\end{eqnarray}\nand indeed one finds.\nRequiring that the warp factor BPS equation may be re-expressed as an equation linear in the superpotential, i.e.\n\\begin{eqnarray}\nf' = 2 (G_0 S_0 + G_3 S_3)\n\\nonumber\\\\\n= 2 \\, \\gamma \\,W\n\\end{eqnarray}\ngives us a factor $\\gamma$ defined as\n\\begin{eqnarray}\n\\gamma = {G_0 S_0 + G_3 S_3 \\over \\sqrt{S_0^2 + S_3^2}}\n\\end{eqnarray}\nWe now make the non-trivial check that, with these definitions of $W$ and $\\gamma$, the BPS equation for $\\sigma$ takes the expected gradient flow form, \n\\begin{eqnarray}\n\\sigma ' =2 \\eta \\, \\sqrt{N_0^2 + N_3^2}\n\\nonumber\\\\\n= -2 \\left({- S_0 N_0 + S_3 N_3 \\over G_0 S_0 + G_3 S_3}\\right)\n\\nonumber\\\\\n= -2\\, \\gamma^{-1} \\,\\partial_\\sigma W\n\\end{eqnarray}\nexactly as expected.\n\\fi\nUnfortunately, this superpotential does \\textit{not} allow one to write the BPS equations for both $\\phi^0$ and $\\phi^3$ as gradient flow equations. The reason for this failure is that the integrability condition required to convert the BPS equation into a gradient flow form is not satisfied; see e.g. Appendix C.2.1 of . We thus follow the strategy of  to construct an approximate superpotential. Our model consists of two consistent truncations that admit flat domain walls and an exact superpotential. These are the $\\phi^3=0,\\phi^0\\ne0$ truncation and the $\\phi^0=0,\\phi^3\\ne0$ truncation. The corresponding flow equations are (we set $\\eta=-1$ henceforth)\n\\begin{eqnarray}\n{\\phi^0}' = -8 \\, \\partial_{\\phi^0} W|_{\\phi^3=0} \\hspace{1 in} {\\phi^3}' =8 \\, \\partial_{\\phi^3} W|_{\\phi^0=0}\n\\end{eqnarray}\nrespectively. In either truncation, the BPS equations for the warp factor and dilaton $\\sigma$ can be put in the following form,\n\\begin{eqnarray}\nf'=2\\,W\\hspace{1 in} \\sigma'=2 \\,\\partial_\\sigma W\n\\end{eqnarray}\nAn important fact is that, though the gradient flow equations of do not hold exactly in the full model with $\\phi^0\\ne0,\\phi^3\\ne0$, they \\textit{do} hold up to and including $O(z^5)$. Looking at the form of the UV asymptotics of the scalar fields, one may expand the superpotential of keeping only terms contributing up to this order. This gives \n\\begin{eqnarray}\nW =  {1\\over 2} + {3 \\over 4} \\sigma^2 + {1 \\over 16} (\\phi^0)^2 -{3 \\over 16} (\\phi^3)^2 + {1 \\over 192} (\\phi^0)^4 -{3 \\over 16} (\\phi^0)^2 \\sigma + \\dots\n\\end{eqnarray}\nwhere the dots represent terms of order $O(z^6)$. This is the approximate superpotential we will use in what follows.\n\\subsubsection{Bogomolnyi trick}\nWe now use the Bogomolnyi trick  to get the finite counterterms needed to preserve supersymmetry in the case of a flat domain wall. The central idea of the Bogomolnyi trick is that for a BPS solution, the renormalized on-shell action must vanish. In order to make use of this fact, we will first want to recast the on-shell action in a simpler form.\nTo do so, we begin by inserting. We find that \n\\begin{eqnarray}\n{\\cal L} =- {1 \\over 4}R -20 W^2 + 2 {\\cal L}_{\\mathrm{kin}}\n\\end{eqnarray}\nwhere we've defined \n\\begin{eqnarray}\n{\\cal L}_{\\mathrm{kin}} = (\\sigma')^2 + {1 \\over 4}\\left[-({\\phi^3}')^2 + \\cos^2 \\phi^3 ({\\phi^0}')^2\\right]\n\\end{eqnarray}\n\\end{document}\n"}
{"The non-zero components of the Ricci tensor are\nRuu=\u00005\u0000\nf00+ (f0)2\u0001\nRmn=\u0000gmn\u0000\nf00+ 5(f0)2\u0001\n(1)\nwhile the Ricci scalar is given by\nR=\u000010f00\u000030(f0)2(2)\nFurthermore, we have thatp\nG=e5fpg, wheregis the determinant of the unit S5metric.\nUpon integration by parts, part of the Einstein-Hilbert term cancels with the Gibbons-\nHawking term to give the following simple expression\nS=Z\nduZ\nd5xpge5f\u0002\n\u00005\u0000\n(f0)2+ 4W2\u0001\n+ 2Lkin\u0003\n(3)\nThe restriction to the \nat case was not strictly necessary so far, but it will be crucial in the\nnext step. The gradient \now equations, together with the chain-rule, allows us to rewrite\nLkin=\u00002W0(4)\nPlugging this into and using the BPS equation of the warp factor, we \fnd\nS=\u00004Z\nd5xpge5fW\f\f\f\u0003\n0(5)\nwhere \u0003 is the UV cuto\u000b. Only the \u0003 part of the action contributes, since e5fWj0vanishes\ndue to the close-o\u000b of the geometry. Removing the UV cuto\u000b \u0003 !1 is equivalent to\nremoving the cuto\u000b \"on our asymptotic coordinate z, i.e.\"!0. From the UV asymptotics\nwe \fnd that in this limit the factor e5fdiverges like\ne5f\u00181\n\"5(6)\nThis is the reason for the previous claims that only the terms up to O(z5) in the superpoten-\ntial are relevant for obtaining counterterms. All the higher-order terms vanish as the cuto\u000b\nis removed. We may thus legitimately insert the approximate superpotential into to get the\ncounterterms,\nS(W)\nct= 4Z\nd5xp\n\u00141\n2+3\n4\u001b2+1\n16(\u001e0)2\u00003\n16(\u001e3)2+1\n192(\u001e0)4\u00003\n16(\u001e0)2\u001b\u0015\n(7)\nwhere\nis the induced metric on the z=\"boundary. All \felds are evaluated at z=\". This\ngives all \fnite and in\fnite counterterms for the \nat domain wall solutions.\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nThe non-zero components of the Ricci tensor are\n\\begin{eqnarray}\nR_{uu} = - 5 \\left(f'' + (f')^2 \\right) \\hspace{1 in} R_{mn} =  -g_{mn} \\left(f'' + 5(f')^2 \\right)\n\\end{eqnarray}\nwhile the Ricci scalar is given by\n\\begin{eqnarray}\nR = - 10 f'' - 30(f')^2 \n\\end{eqnarray}\nFurthermore, we have that $\\sqrt{G} = e^{5f} \\sqrt{g}$, where $g$ is the determinant of the unit $S^5$ metric. Upon integration by parts, part of the Einstein-Hilbert term cancels with the Gibbons-Hawking term to give the following simple expression\n\\begin{eqnarray}\nS = \\int du \\int d^5 x \\sqrt{g}\\, e^{5f}\\left[-5 \\left((f')^2 +4 W^2 \\right)+2 {\\cal L}_{\\mathrm{kin}}\\right]\n\\end{eqnarray}\nThe restriction to the flat case was not strictly necessary so far, but it will be crucial in the next step.\nThe gradient flow equations, together with the chain-rule, allows us to rewrite\n\\begin{eqnarray}\n{\\cal L}_{\\mathrm{kin}} =-2\\,  W'\n\\end{eqnarray}\nPlugging this into and using the BPS equation of the warp factor, we find\n\\begin{eqnarray}\nS = -4  \\int d^5 x \\sqrt{g}\\,e^{5f} W \\Big|_0^\\Lambda\n\\end{eqnarray}\nwhere $\\Lambda$ is the UV cutoff. Only the $\\Lambda$ part of the action contributes, since $e^{5f} W|_0$ vanishes due to the close-off of the geometry. \nRemoving the UV cutoff $\\Lambda\\rightarrow \\infty$ is equivalent to removing the cutoff $\\varepsilon$ on our asymptotic coordinate $z$, i.e. $\\varepsilon\\rightarrow 0$. From the UV asymptotics we find that in this limit the factor $e^{5f}$ diverges like \n\\begin{eqnarray}\ne^{5f} \\sim {1 \\over \\varepsilon^5}\n\\end{eqnarray}\nThis is the reason for the previous claims that only the terms up to $O(z^5)$ in the superpotential are relevant for obtaining counterterms. All the higher-order terms vanish as the cutoff is removed. \nWe may thus legitimately insert the approximate superpotential into to get the counterterms, \n\\begin{eqnarray}\nS^{(W)}_{\\mathrm{ct}} = 4 \\int d^5x \\sqrt{\\gamma} \\left[  {1\\over 2} + {3 \\over 4} \\sigma^2 + {1 \\over 16} (\\phi^0)^2 -{3 \\over 16} (\\phi^3)^2 + {1 \\over 192} (\\phi^0)^4 -{3 \\over 16} (\\phi^0)^2 \\sigma \\right]\n\\end{eqnarray}\nwhere $\\gamma$ is the induced metric on the $z = \\varepsilon$ boundary. All fields are evaluated at $z = \\varepsilon$. This gives all finite and infinite counterterms for the flat domain wall solutions.\n\\end{document}\n"}
{"0.1 In\fnite counterterms\nWe now turn towards the identi\fcation of the in\fnite counterterms in the more general\ncurved domain wall case. We may \frst solve for all of the in\fnite counterterms via the usual\nholographic renormalization procedure. Once we have these, we will\n1. Check that in the \nat limit, they reduce to the divergent pieces of the \nat counterterms\nfound above.\n2. Add to them the \fnite pieces found in but missing in the holographic renormalization\nprocedure.\nFor simplicity, we will perform holographic renormalization on supersymmetric solutions\nonly, and thus the in\fnite counterterms we obtain are universal for supersymmetric solutions\nonly. We begin by using the expression for the on-shell Ricci scalar,\nR= 4(\u001b0)2+h\n\u0000(\u001e30)2+ cos2\u001e3(\u001e00)2i\n+ 6V (1)\nto rewrite the action as\nS6D=\u00001\n2Z\ndud5xpge5fV (2)\nWe have not included the Gibbons-Hawking term yet, but will do so later. The \frst step of\nholographic renormalization is to isolate the divergent terms. We may do so by expanding\nall \felds using their UV asymptotics, then integrating over small zand evaluating on the\ncuto\u000b\u000f. Doing so, we \fnd\nS6D=\u00001\n2Z\nd5xpge5fk\u00141\n\u000f5+1\n3\u000f3\u0010\n25f2+\u0000\n\u001e0\n1\u00012\u0011\n+1\n24\u000f\u0010\n1500f2\n2+ 600f4+ 120f2\u0000\n\u001e0\n1\u00012\u0000\u0000\n\u001e0\n1\u00014\n+48\u001e0\n1\u001e0\n3+ 36\u0010\n\u0000\u0000\n\u001e3\n2\u00012+ 4\u001b2\n2\u0011\u0011i\n(3)\nwhere we've thrown out all non-divergent contributions. Note that the integration would\nnaively give a log \u000f, but this vanishes on the BPS equations since they constrain the UV\nasymptotic expansion coe\u000ecients in the following way,\n25f5+ 2\u001e0\n1\u001e0\n4\u00003\u001e3\n2\u001e3\n3+ 12\u001b2\u001b3= 0 (4)\nThe absence of the logarithmic term is to be expected, since any dual \fve-dimensional \feld\ntheory is anomaly-free. The Gibbons-Hawking term is\nSGH=\u00005\n2Z\nd5xpge5ff0(5)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\n\\subsection{Infinite counterterms}\nWe now turn towards the identification of the infinite counterterms in the more general curved domain wall case. We may first solve for all of the infinite counterterms via the usual holographic renormalization procedure. Once we have these, we will\n\\begin{enumerate}\n\\item Check that in the flat limit, they reduce to the divergent pieces of the flat counterterms found above.\n\\item Add to them the finite pieces found in but missing in the holographic renormalization procedure. \n\\end{enumerate}\nFor simplicity, we will perform holographic renormalization on supersymmetric solutions only, and thus the infinite counterterms we obtain are universal for supersymmetric solutions only. \nWe begin by using the expression for the on-shell Ricci scalar, \n\\begin{eqnarray}\nR = 4 (\\sigma')^2 + \\left[-({\\phi^3}')^2 + \\cos^2 \\phi^3 ({\\phi^0}')^2 \\right] + 6 V\n\\end{eqnarray}\nto rewrite the action as \n\\begin{eqnarray}\nS_{\\mathrm{6D}} = -{1 \\over 2} \\int du\\, d^5 x \\sqrt{g} \\,e^{5f}  V\n\\end{eqnarray}\nWe have not included the Gibbons-Hawking term yet, but will do so later. The first step of holographic renormalization is to isolate the divergent terms. We may do so by expanding all fields using their UV asymptotics, then integrating over small $z$ and evaluating on the cutoff $\\epsilon$. Doing so, we find \n\\begin{eqnarray}\nS_{\\mathrm{6D}} = -{1\\over 2} \\int d^5 x \\sqrt{g} e^{5 f_k} \\left[{1 \\over \\epsilon^5} + {1 \\over 3 \\epsilon^3}\\left(25 f_2 + \\left(\\phi^0_1\\right)^2 \\right) \\right.\n\\nonumber\\\\\n\\hspace{1.3 in} + {1 \\over 24\\epsilon}\\left(1500 f_2^2 + 600 f_4 + 120 f_2 \\left(\\phi^0_1\\right)^2  - \\left(\\phi^0_1\\right)^4 \\right.\\nonumber\\\\\n\\vphantom{.} \\hspace{1.1 in}\\left. \\left.  \\hspace{1.3 in}+ 48\\, \\phi^0_1 \\phi^0_3 + 36\\left(-\\left(\\phi^3_2\\right)^2 + 4 \\sigma_2^2\\right)  \\right)\\right]\n\\end{eqnarray}\nwhere we've thrown out all non-divergent contributions. Note that the integration would naively give a $\\log \\epsilon$, but this vanishes on the BPS equations since they constrain the UV asymptotic expansion coefficients in the following way,\n\\begin{eqnarray}\n25 f_5 + 2 \\,\\phi^0_1 \\phi^0_4 - 3 \\,\\phi^3_2 \\phi^3_3 + 12 \\sigma_2 \\sigma_3 = 0\n\\end{eqnarray}\nThe absence of the logarithmic term is to be expected, since any dual five-dimensional field theory is anomaly-free. The Gibbons-Hawking term is \n\\begin{eqnarray}\nS_{\\mathrm{GH}} = -{5 \\over 2} \\int d^5 x \\sqrt{g}\\, e^{5f} f'\n\\end{eqnarray}\n\\end{document}\n"}
{"We again use the asymptotic expansions to write\nSGH=\u00005\n2Z\nd5xpge5fk\u00141\n\u000f5+3f2\n\u000f3+1\n2\u000f\u0000\n5f2\n2+ 2f4\u0001\u0015\n(1)\nAdding the two together, we \fnd in total that\nS6D+SGH=\u0000Z\nd5xpge5fk\u00142\n\u000f5+1\n6\u000f3\u0010\n20f2\u0000\u0000\n\u001e0\n1\u00012\u0011\n\u00001\n48\u000f\u0000\n1200f2\n2+ 480f4\n+120f2\u0000\n\u001e0\n1\u00012\u0000\u0000\n\u001e0\n1\u00014+ 48\u001e0\n1\u001e0\n3\u000036(\u001e3\n2)2+ 144\u001b2\n2\u0011i\n(2)\nWe must now undergo the task of inverting all of the UV modes to rewrite the action in\nterms of induced \felds at the cut-o\u000b surface (since it is the latter which transform nicely\nunder bulk di\u000beomorphism). Before quoting the result, we note that at the cut-o\u000b z=\u000f,\nthe induced metric \nijis given by\n\nij=e2f\f\f\nz=\u000fg(S5)\nij (3)\nThe Ricci tensor and Ricci scalar are given by\nRij[\n] = 4e\u00002f\nij\f\f\nz=\u000fR[\n] = 20e\u00002f\f\f\nz=\u000f(4)\nIn terms of these quantities, we \fnd that the inverted form of the divergent part of the\non-shell action is\nS=\u0000Z\nd5xp\n\u0014\n2 +1\n4\u0000\n\u001e0\u00012+3\n4\u0000\n\u001e3\u00012\u00003\u001b2+7\n12\u0000\n\u001e0\u00014\n+1\n12R[\n]\u00001\n320R[\n]2\u00003\n32R[\n]\u0000\n\u001e0\u00012\u0015\n(5)\nWe may now address the two points mentioned at the start of this subsection. To begin,\nwe check that in the \nat limit, we reproduce the divergent terms obtained in. In particular,\nwe expect that the \frst line of should be equal to \u0000S(W)\nctup to and including order O(z4).\nThough the expressions look di\u000berent at \frst sight, it can be checked via the relationships\nbetween expansion coe\u000ecients in (along with their higher order counterparts) that in the\nlimite\u00002f!0 the two expressions indeed areequivalent up to O(z4). Thus all of their\ndivergent contributions are the same in the \nat limit. However, even in this limit the two\ndi\u000ber at order O(z5), which means that they have di\u000berent \fnite contributions. As mentioned\nearlier, the \fnite terms we must work with are those coming from . An action which has\nboth the required \fnite and in\fnite counterterms is\nSct=Z\nd5xp\n\u0014\n2 +1\n4\u0000\n\u001e0\u00012+3\n4\u0000\n\u001e3\u00012+ 3\u001b2+1\n48\u0000\n\u001e0\u00014\u00003\n4\u0000\n\u001e0\u00012\u001b\n+1\n12R[\n]\u00001\n320R[\n]2\u00003\n32R[\n]\u0000\n\u001e0\u00012\u0015\n(6)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\n\nWe again use the asymptotic expansions to write \n\\begin{eqnarray}\nS_{\\mathrm{GH}} = -{5 \\over 2} \\int d^5 x \\sqrt{g} e^{5 f_k} \\left[{1 \\over \\epsilon^5} +{3 f_2\\over \\epsilon^3} +{1 \\over2 \\epsilon} \\left(5 f_2^2 + 2 f_4\\right) \\right]\n\\end{eqnarray}\nAdding the two together, we find in total that \n\\begin{eqnarray}\nS_{\\mathrm{6D}}+S_{\\mathrm{GH}} = -\\int d^5 x \\sqrt{g} e^{5 f_k} \\left[{2 \\over \\epsilon^5} +{1 \\over 6 \\epsilon^3}\\left(20 f_2 - \\left(\\phi^0_1\\right)^2 \\right) \\right.- {1 \\over 48\\epsilon}\\left(1200 f_2^2 + 480 f_4  \\right.  \\nonumber\\\\\n\\vphantom{.}\\hspace{0.2 in}\\left. \\left. \\hspace{.5in} + 120 f_2 \\left(\\phi^0_1\\right)^2 - \\left(\\phi^0_1\\right)^4 + 48\\, \\phi^0_1 \\phi^0_3 -36(\\phi^3_2)^2 + 144 \\sigma_2^2  \\right)\\right]\n\\end{eqnarray}\nWe must now undergo the task of inverting all of the UV modes to rewrite the action in terms of induced fields at the cut-off surface (since it is the latter which transform nicely under bulk diffeomorphism). Before quoting the result, we note that at the cut-off $z = \\epsilon$, the induced metric $\\gamma_{i j}$ is given by \n\\begin{eqnarray}\n\\gamma_{i j}=  e^{2 f}\\big|_{z = \\epsilon} \\,g^{(S^5)}_{i j}\n\\end{eqnarray}  \nThe Ricci tensor and Ricci scalar are given by\n\\begin{eqnarray}\nR_{ij}[\\gamma] = 4e^{-2f}\\gamma_{ij}\\big|_{z = \\epsilon}~~~~~~~~~~~~~~R[\\gamma] = 20 \\,e^{- 2 f}\\big|_{z = \\epsilon}\n\\end{eqnarray}\nIn terms of these quantities, we find that the inverted form of the divergent part of the on-shell action is \n\\begin{eqnarray}\nS = - \\int d^5 x \\sqrt{\\gamma} \\left[2 + {1 \\over 4} \\left(\\phi^0\\right)^2 + {3 \\over 4} \\left(\\phi^3\\right)^2 - 3 \\sigma^2 +{7 \\over 12 }\\left(\\phi^0\\right)^4\\right.\n\\nonumber\\\\\n\\vphantom{.}\\hspace{1.4 in}\\left. + {1 \\over 12} R[\\gamma] - {1 \\over 320} R[\\gamma]^2 - {3 \\over 32} R[\\gamma] \\left(\\phi^0\\right)^2 \\right]\n\\end{eqnarray}\nWe may now address the two points mentioned at the start of this subsection. To begin, we check that in the flat limit, we reproduce the divergent terms obtained in. In particular, we expect that the first line of should be equal to $-S_{ct}^{(W)}$ up to and including order $O(z^4)$. Though the expressions look different at first sight, it can be checked via the relationships between expansion coefficients in  (along with their higher order counterparts) that in the limit $e^{-2f} \\rightarrow 0$ the two expressions indeed \\textit{are} equivalent up to $O(z^4)$. Thus all of their divergent contributions are the same in the flat limit. However, even in this limit the two differ at order $O(z^5)$, which means that they have different finite contributions. As mentioned earlier, the finite terms we must work with are those coming from . An action which has both the required finite and infinite counterterms is\n\\begin{eqnarray}\nS_{\\mathrm{ct}} = \\int d^5 x \\sqrt{\\gamma} \\left[2 + {1 \\over 4} \\left(\\phi^0\\right)^2 + {3 \\over 4} \\left(\\phi^3\\right)^2 + 3 \\sigma^2 +{1 \\over 48 }\\left(\\phi^0\\right)^4 - {3 \\over 4} \\left(\\phi^0\\right)^2 \\sigma\\right.\n\\nonumber\\\\\n\\vphantom{.}\\hspace{1.4 in}\\left. + {1 \\over 12} R[\\gamma] - {1 \\over 320} R[\\gamma]^2 - {3 \\over 32} R[\\gamma] \\left(\\phi^0\\right)^2 \\right]\n\\end{eqnarray}\n\\end{document}\n"}
{"The three gravitational counterterms 2 ;R[\n];andR[\n]2match with the ones obtained\nin . On our S5domain-wall ansatz, the term proportional to the square of the Ricci tensor\nsimpli\fes in terms of the square of the Ricci scalar Rij[\n]R[\n]ij=1\n5R[\n]2. Note that there is\nstill a question of curved space \fnite counterterms, which we have not yet \fxed. If we insist\non including only terms even under\n'0!\u0000'0and '3!\u0000'3(1)\n(which is a symmetry of the action) it can be shown that the only way to add terms which\nchange the curved space \fnite counterterms but leave the other counterterms unchanged is\nto add a combination of the form\n(\u001e3)2\u00001\n20R[\n](\u001e0)2= 2e\u0000fk\f alphaz5+O(z6) (2)\nThis freedom is \fxed by demanding that the vevs of the dual operators stay \fnite. We will\nsimply quote the result here,\nSct=Z\nd5xp\n\u0014\n2 +1\n4\u0000\n\u001e0\u00012\u00001\n2\u0000\n\u001e3\u00012+ 3\u001b2+1\n48\u0000\n\u001e0\u00014\u00003\n4\u0000\n\u001e0\u00012\u001b\n+1\n12R[\n]\u00001\n320R[\n]2\u00001\n32R[\n]\u0000\n\u001e0\u00012\u0015\n(3)\nand postpone showing that this gives \fnite vacuum expectation values to the next subsection.\nAt this level, everything has seemed unique. However, when thinking in terms of the induced\n\felds instead of the modes appearing in asymptotic expansions, the counterterms of are just\none of many possible sets of counterterms that can be written down. In particular, since\non-shell we have the relationship\nI0\u00115\u001b2+45\n64('0)4\u000015\n4('0)2\u001b=O(z6) (4)\nwe may add I0freely to without changing either \fnite or in\fnite contributions. However,\nthe inclusion of this term will have an impact on some of the one-point functions, which we\ncalculate next.\n0.1 Vevs and free energy\nThe renormalized on-shell action is given by\nSren=S6D+SGH+Sct+ \nZ\nd5xp\nI0 (5)\nwhere the counterterm action Sctis given by , \n is a constant parameterizing choice of\nscheme, and I0is given in . Note that the free energy is independent of the choice of \n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nThe three gravitational counterterms $2, R[\\gamma],$ and $R[\\gamma]^2$ match with the ones obtained in . On our $S^5$ domain-wall ansatz, the term proportional to the square of the Ricci tensor simplifies in terms of the square of the Ricci scalar $R_{ij}[\\gamma]R[\\gamma]^{ij}=\\frac15 R[\\gamma]^2$. \nNote that there is still a question of curved space finite counterterms, which we have not yet fixed. If we insist on including only terms even under \n\\begin{eqnarray}\n\\varphi^0 \\rightarrow -\\varphi^0 \\hspace{0.7in}\\mathrm{and} \\hspace{0.7in} \\varphi^3 \\rightarrow -\\varphi^3\n\\end{eqnarray} \n(which is a symmetry of the action) it can be shown that the only way to add terms which change the curved space finite counterterms but leave the other counterterms unchanged is to add a combination of the form \n\\begin{eqnarray}\n(\\phi^3)^2 - {1 \\over 20} R[\\gamma] (\\phi^0)^2 = 2\\, e^{-f_k} \\beta\\ alpha \\,z^5 + O(z^6)\n\\end{eqnarray}\nThis freedom is fixed by demanding that the vevs of the dual operators stay finite. We will simply quote the result here, \\begin{eqnarray}\nS_{\\mathrm{ct}} = \\int d^5 x \\sqrt{\\gamma} \\left[2 + {1 \\over 4} \\left(\\phi^0\\right)^2 - {1 \\over 2} \\left(\\phi^3\\right)^2 + 3 \\sigma^2 +{1 \\over 48 }\\left(\\phi^0\\right)^4 - {3 \\over 4} \\left(\\phi^0\\right)^2 \\sigma\\right.\n\\nonumber\\\\\n\\vphantom{.}\\hspace{1.4 in}\\left. + {1 \\over 12} R[\\gamma] - {1 \\over 320} R[\\gamma]^2 - {1 \\over 32} R[\\gamma] \\left(\\phi^0\\right)^2 \\right]\n\\end{eqnarray}\nand postpone showing that this gives finite vacuum expectation values to the next subsection.\nAt this level, everything has seemed unique. However, when thinking in terms of the induced fields instead of the modes appearing in asymptotic expansions, the counterterms of  are just one of many possible sets of counterterms that can be written down. In particular, since on-shell we have the relationship \n\\begin{eqnarray}\nI_0 \\equiv 5 \\sigma^2 + {45 \\over 64} (\\varphi^0)^4 - {15 \\over 4} (\\varphi^0)^2 \\sigma =  O(z^6)\n\\end{eqnarray}\nwe may add $I_0$ freely to  without changing either finite or infinite contributions. However, the inclusion of this term will have an impact on some of the one-point functions, which we calculate next.\n\\subsection{Vevs and free energy}\nThe renormalized on-shell action is given by \n\\begin{eqnarray}\nS_{\\mathrm{ren}} = S_{\\mathrm{6D}} + S_{\\mathrm{GH}} + S_{\\mathrm{ct}} + \\Omega \\int d^5 x~  \\sqrt{\\gamma}\\, I_0\n\\end{eqnarray}\nwhere the counterterm action $S_{ct}$ is given by , $\\Omega$ is a constant parameterizing choice of scheme, and $I_0$ is given in . Note that the free energy is independent of the choice of $\\Omega$ \n\\end{document}\n"}
{"sinceI0isO(z6) and hence vanishes in the \u000f!0 limit. However, some of the one-point\nfunctions willdepend on \n. It may be the case that only certain choices of \n correspond\nto supersymmetric schemes, but since the \fnal free energy will be independent of \n we will\nnot worry about this choice. While in principle gives us the free energy, its evaluation on\nour numerical solutions is complicated by the integration over uinS6D. As such, we will\ntake a slightly roundabout approach to the calculation of the free energy, \frst calculating\nits derivative dF=d\u000b and then integrating over the UV parameter \u000b. This will allow us\nto circumvent the integration over u. In order to get dF=d\u000b , it will \frst be necessary to\ncalculate the one-point functions of the dual \feld theory operators. This is the topic of the\nfollowing subsection.\n0.0.1 One-point functions\nBy the usual AdS/CFT dictionary, the one-point functions of the operators dual to the three\nscalar \felds and the metric are given by\nhO\u001bi= lim\n\u000f!01\n\u000f31p\n\u000eSren\n\u000e\u001bhO\u001e0i= lim\n\u000f!01\n\u000f41p\n\u000eSren\n\u000e\u001e0\nhO\u001e3i= lim\n\u000f!01\n\u000f31p\n\u000eSren\n\u000e\u001e3hTi\nji= lim\n\u000f!01\n\u000f51p\n\njk\u000eSren\n\u000e\nik(1)\nWe may obtain the explicit values of these vacuum expectation values by varying the on-shell\naction . The variation of the counterterm action Sctis straightforward. The variation of S6D\ngives rise to one piece which vanishes on the equations of motion, as well as a boundary term\nwhich must be accounted for. We \fnd\nhO\u001bi= lim\n\u000f!01\n\u000f3\u0014\n\u00002z@z\u001b+ 6\u001b\u00003\n4('0)2+ \n\u0012\n10\u001b\u000015\n4\u0000\n\u001e0\u00012\u0013\u0015\nhO\u001e0i= lim\n\u000f!01\n\u000f4\u0014\n\u00001\n2cos2\u001e3z@z\u001e0+1\n2\u001e0+1\n12\u0000\n\u001e0\u00013\u00003\n2\u001e0\u001b\u00001\n16R\u001e0\n+ \n\u001245\n16\u0000\n\u001e0\u00013\u000015\n2\u001e0\u001b\u0013 \u0015\nhO\u001e3i= lim\n\u000f!01\n\u000f3\u00141\n2z@z\u001e3\u0000\u001e3\u0015\nhTi\nji= lim\n\u000f!01\n\u000f5\u00141\n2\u0000\nK\nij\u0000Kij\u0001\n+2p\n\u000eSct\n\u000e\nij\u0015\n(2)\nEvaluating the limits, we get the following one-point functions\nhO\u001bi=5\n2efk\u000b\f\nhO\u001e0i=3\n2e\u0000fk\f\u000015\n8efk\u000b2\f\nhO\u001e3i=1\n2\fhTi\nii=\u00005\n2e\u0000fk\u000b\f (3)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nsince $I_0$ is $O(z^6)$ and hence vanishes in the $\\epsilon \\rightarrow 0$ limit. However, some of the one-point functions \\textit{will} depend on $\\Omega$. It may be the case that only certain choices of $\\Omega$ correspond to supersymmetric schemes, but since the final free energy will be independent of $\\Omega$ we will not worry about this choice.\nWhile in principle  gives us the free energy, its evaluation on our numerical solutions is complicated by the integration over $u$ in $S_{6D}$. As such, we will take a slightly roundabout approach to the calculation of the free energy, first calculating its derivative $dF/d\\alpha$ and then integrating over the UV parameter $\\alpha$. This will allow us to circumvent the integration over $u$. In order to get $dF/d\\alpha$, it will first be necessary to calculate the one-point functions of the dual field theory operators. This is the topic of the following subsection. \n\\subsubsection{One-point functions}\nBy the usual AdS/CFT dictionary, the one-point functions of the operators dual to the three scalar fields and the metric are given by \n\\begin{align}\n\\langle {\\cal O}_{\\sigma} \\rangle = \\lim_{\\epsilon \\rightarrow 0}  \\frac{1}{\\epsilon^3} {1 \\over \\sqrt{\\gamma}} {\\delta S_{ren} \\over \\delta \\sigma}~~~~~~~~~~\n\\langle {\\cal O}_{\\phi^0} \\rangle = \\lim_{\\epsilon \\rightarrow 0}  \\frac{1}{\\epsilon^4} {1 \\over \\sqrt{\\gamma}} {\\delta S_{ren} \\over \\delta \\phi^0} \n\\nonumber\\\\\n \\langle {\\cal O}_{\\phi^3} \\rangle = \\lim_{\\epsilon \\rightarrow 0}  \\frac{1}{\\epsilon^3} {1 \\over \\sqrt{\\gamma}} {\\delta S_{ren} \\over \\delta \\phi^3}~~~~~~~~~~\n \\langle {T^i}_j \\rangle  = \\lim_{\\epsilon \\rightarrow 0}  \\frac{1}{\\epsilon^5} {1 \\over \\sqrt{\\gamma}} \\gamma_{jk}{\\delta S_{ren} \\over \\delta\\gamma_{ik}}\n\\end{align}\nWe may obtain the explicit values of these vacuum expectation values by varying the on-shell action . The variation of the counterterm action $S_{ct}$ is straightforward. The variation of $S_{6D}$ gives rise to one piece which vanishes on the equations of motion, as well as a boundary term which must be accounted for. We find\n\\begin{align}\n\\langle {\\cal O}_{\\sigma} \\rangle = \\lim_{\\epsilon \\rightarrow 0} {1 \\over \\epsilon^3}  \\left[- 2 z \\partial_z \\sigma + 6\\sigma - {3 \\over 4}( \\varphi^0)^2 +\\Omega\\left(10\\sigma-\\frac{15}{4} \\left(\\phi^0\\right)^2\\right)  \\right]\\nonumber \\\\\n\\langle {\\cal O}_{\\phi^0} \\rangle = \\lim_{\\epsilon \\rightarrow 0} {1 \\over \\epsilon^4}  \\bigg[- \\frac12 \\cos^2\\phi^3 z \\partial_z \\phi^0 +\\frac12 \\phi^0 +\\frac{1}{12} \\left(\\phi^0\\right)^3-\\frac32 \\phi^0\\sigma-\\frac{1}{16}R\\phi^0 \\nonumber\\\\\n~~~~~~~~~~~~~~~~~~~~+\\Omega\\left(\\frac{45}{16}\\left(\\phi^0\\right)^3-\\frac{15}{2}\\phi^0\\sigma\\right)\\bigg]\\nonumber \\\\\n\\langle {\\cal O}_{\\phi^3} \\rangle = \\lim_{\\epsilon \\rightarrow 0} {1 \\over \\epsilon^3}  \\left[ \\frac12 z \\partial_z \\phi^3-\\phi^3\\right]\\nonumber  \\\\\n\\langle {T^i}_j\\rangle = \\lim_{\\epsilon \\rightarrow 0} {1 \\over \\epsilon^5}\\left[\\frac12 \\left(\\cal K \\gamma^{ij}-\\cal K^{ij}\\right)+\\frac{2}{\\sqrt{\\gamma}}\\frac{\\delta S_{ct}}{\\delta \\gamma_{ij}}\\right]\n\\end{align}\nEvaluating the limits, we get the following one-point functions\n\\begin{align}\n\\langle {\\cal O}_{\\sigma} \\rangle = \\frac52 e^{f_k}\\alpha\\beta\\,\\Omega ~~~~~~~~~~~\n\\langle {\\cal O}_{\\phi^0} \\rangle = \\frac32 e^{-f_k} \\beta -\\frac{15}{8} e^{f_k}\\alpha^2\\beta\\, \\Omega\n\\nonumber\\\\\n \\langle {\\cal O}_{\\phi^3} \\rangle = \\frac12 \\beta~~~~~~~~~~~~~~~~~~~ \\langle {T^i}_i \\rangle  =  -\\frac52 e^{-f_k} \\alpha\\beta\n\\end{align}\n\\end{document}\n"}
{"The expectation values of the operator O\u001e3and the trace of the energy-momentum tensor\nare independent of \n. As a check, we note that the four one-point functions satisfy the\nfollowing operator relation, which is associated to the violation of conformal invariance by\nnon-zero classical beta functions,\nhTi\nii=\u0000X\nO(d\u0000\u0001O)\u001eOhOi (1)\n0.0.1 Derivative of the free energy\nFollowing , we may now compute the derivative of Fwith respect to \u000bas follows. First we\nnote that\ndF\nd\u000b=dSren\nd\u000b= lim\n\u000f!0Z\nd5xX\n\felds \b\u000e\u0000p\nLren\u0001\n\u000e\bd\b\nd\u000b\f\f\f\f\nz=\u000f(2)\nIn our case, the terms appearing in the sum over \felds are\n\u000e\u0000p\nLren\u0001\n\u000e\u001b=p\nhO\u001bi\u000f3+:::\u000e\u0000p\nLren\u0001\n\u000e\u001e0=p\nhO0\n\u001ei\u000f4+:::\n\u000e\u0000p\nLren\u0001\n\u000e\u001e3=p\nhO3\n\u001ei\u000f3+:::\u000e\u0000p\nLren\u0001\n\u000e\nij=1\n2p\nhTiji\u000f5+::: (3)\nThe dots represent terms of strictly lower order in \u000f. Furthermore, from the form of the UV\nasymptotic expansions , we have\nd\u001b\nd\u000b=3\n4\u000b\u000f2+O(\u000f3)d\u001e0\nd\u000b=\u000f+O(\u000f3)\nd\u001e3\nd\u000b=\u0012\n1\u0000\u000bdfk\nd\u000b\u0013\ne\u0000fk\u000f2+O(\u000f3)d\nij\nd\u000b=\u00002dfk\nd\u000be\u00002fk\u000f2+O(\u000f2) (4)\nCombining the pieces , with the results for the one-point functions in , we \fnd that the\ncontribution of the metric in is suppressed by \u000f2compared to other terms. The derivative of\nthe free energy is then\ndF\nd\u000b= lim\n\u000f!0Z\nd5xp\n\u000f5\u00143\n2\fe\u0000fk+1\n2\fe\u0000fk\u0012\n1\u0000\u000bdfk\nd\u000b\u0013\n+O(\u000f)\u0015\n= vol 0\u0000\nS5\u00011\n2\fe4fk\u0012\n4\u0000\u000bdfk\nd\u000b\u0013\n(5)\nwhere vol 0(S5) =\u00193is the volume of a unit S5. The \n dependence in the one-point functions\ncancels out, consistent with the fact that Fitself is independent of \n. We thus obtain the\n\fnal result\ndF\nd\u000b=\u00192\n8G6\fe4fk\u0012\n4\u0000\u000bdfk\nd\u000b\u0013\n(6)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nThe expectation values of the operator ${\\cal O}_{\\phi^3}$ and the trace of the energy-momentum tensor are independent of $\\Omega$. As a check, we note that the four one-point functions satisfy the following operator relation, which is associated to the violation of conformal invariance by non-zero classical beta functions,\n\\begin{eqnarray}\n\\langle{T^i}_i\\rangle = -\\sum_{\\cal O}(d-\\Delta_{\\cal O})\\,\\phi_{\\cal O}\\,\\langle{\\cal O}\\rangle \n\\end{eqnarray}\n\\subsubsection{Derivative of the free energy}\nFollowing , we may now compute the derivative of $F$ with respect to $\\alpha$ as follows. First we note that\n\\begin{eqnarray}\n{d F \\over d \\alpha} = {d S_{\\mathrm{ren}} \\over d\\alpha} =\\lim_{\\epsilon \\rightarrow 0} \\int d^5x \\sum_{\\mathrm{fields}\\,\\, \\Phi} {\\delta \\left(\\sqrt{\\gamma} {\\cal L}_{\\mathrm{ren}}\\right) \\over \\delta \\Phi }{d \\Phi \\over d \\alpha}\\,\\bigg|_{z = \\epsilon}\n\\end{eqnarray}\nIn our case, the terms appearing in the sum over fields are\n\\begin{align}\n{\\delta \\left(\\sqrt{\\gamma} {\\cal L}_{\\mathrm{ren}}\\right) \\over \\delta \\sigma } =  \\sqrt{\\gamma}\\,  \\langle O_\\sigma \\rangle \\epsilon^3 + \\dots~~~~~~~~~\n{\\delta \\left(\\sqrt{\\gamma} {\\cal L}_{\\mathrm{ren}}\\right) \\over \\delta \\phi^0 }  = \\sqrt{\\gamma} \\, \\langle O_\\phi^0 \\rangle \\epsilon^4 + \\dots\n\\nonumber\\\\\n{\\delta \\left(\\sqrt{\\gamma} {\\cal L}_{\\mathrm{ren}}\\right) \\over \\delta \\phi^3 } = \\sqrt{\\gamma}\\,  \\langle O_\\phi^3 \\rangle \\epsilon^3+ \\dots~~~~~~~~~\n{\\delta \\left(\\sqrt{\\gamma} {\\cal L}_{\\mathrm{ren}}\\right) \\over \\delta \\gamma^{ij} } = {1\\over 2}  \\sqrt{\\gamma}\\, \\langle T_{ij} \\rangle  \\epsilon^5+ \\dots\n\\end{align}\nThe dots represent terms of strictly lower order in $\\epsilon$. Furthermore, from the form of the UV asymptotic expansions , we have  \n\\begin{align}\n{d\\sigma \\over d \\alpha} =  {3 \\over 4}\\alpha \\epsilon^2 + O(\\epsilon^3)~~~~~~~~~~~~~~~~~~~~~~~~~~~\n{d\\phi^0 \\over d \\alpha} = \\epsilon + O(\\epsilon^3)~~~~~~~~\\nonumber\\\\\n{d\\phi^3 \\over d \\alpha} = \\left(1- \\alpha {d f_k \\over d \\alpha}\\right)e^{-f_k}\\epsilon^2 + O(\\epsilon^3)~~~~~~~~\n\\frac{d\\gamma^{ij}}{d\\alpha} = -2\\frac{df_k}{d\\alpha}e^{-2f_k}\\epsilon^2 + O(\\epsilon^2)\n\\end{align}\nCombining the pieces , with the results for the one-point functions in , we find that the contribution of the metric in is suppressed by $\\epsilon^2$ compared to other terms. The derivative of the free energy is then\n\\begin{eqnarray}\n{d F \\over d \\alpha} = \\lim_{\\epsilon \\rightarrow 0} \\int d^5 x \\sqrt{\\gamma} \\, \\epsilon^5 \\left[{3 \\over 2} \\beta e^{- f_k} + {1\\over 2} \\beta e^{- f_k}\\left(1 -\\alpha {d f_k \\over d \\alpha} \\right) + O(\\epsilon) \\right]\n\\nonumber\\\\\n= \\mathrm{vol}_0\\left(S^5 \\right) \\, {1\\over 2} \\beta \\,e^{4 f_k} \\left(4 - \\alpha {d f_k \\over d \\alpha} \\right)\n\\end{eqnarray}\nwhere $\\mathrm{vol}_0(S^5)=\\pi^3$ is the volume of a unit $S^5$. The $\\Omega$ dependence in the one-point functions cancels out, consistent with the fact that  $F$ itself is independent of $\\Omega$.\nWe thus obtain the final result\n\\begin{eqnarray}\n{d F \\over d \\alpha} =  {\\pi^2 \\over 8\\, G_6}  \\beta \\,e^{4 f_k} \\left(4-\\alpha {d f_k \\over d \\alpha} \\right)\n\\end{eqnarray}\n\\end{document}\n"}
{"Note that we've reintroduced the six-dimensional Newton's constant G6, which had been\npreviously set to 4 \u0019G 6= 1. This factor is important for the identi\fcation with the free\nenergy on the \feld theory side. Treating (\n\u0016\u000b) andfk(\u000b) as functions of \u000b, this gives us an\nexpression which may be numerically integrated to obtain the free energy F(\u000b)\u0000F(0) of\nthe domain wall. The functional forms of (\n\u0016\u000b);fk(\u000b) are obtained by \ftting curves to the\nnumerical data, as shown in Figure . Integrating to obtain F(\u000b)\u0000F(0) gives the result\nshown in Figure .\n1 Field theory calculation\nLocalization is a powerful tool used to obtain exact results in supersymmetric quantum \feld\ntheories. In the large Nlimit, results obtained via localization calculations can be compared\nwith results obtained via holography. The goal of this section is to calculate the sphere free\nenergy for a \fve-dimensional mass-deformed SCFT using localization, and then to compare\nit to the holographic result obtained in the previous section. A potential complication\nis that the \fve-dimensional \feld theory dual to the matter-coupled six-dimensional gauged\nsupergravity described in section has not been fully identi\fed. This is because the full gauged\nsupergravity has not been shown to arise as a consistent truncation of any ten-dimensional\ntheory. In the following, the tentative \feld theory dual we will use for the localization\ncalculation in the IR is a USp(2N) gauge theory coupled to Nffundamental representation\nhypermultiplets, and a single hypermultiplet in the anti-symmetric representation. As we\nwill review below, this theory is believed to be obtained from the D4-D8 system in type I'\nstring theory/massive type IIA supergravity. One fundamental limitation in our comparison\nbetween \feld theory and holographic results is that our holographic RG \now is completely\nnumerical, and there is no analytic formula for the free energy that can be derived from it.\nNevertheless, we will \fnd qualitative similarities between the holographic free energy and\nthe localization result for the free energy of the aforementioned USp(2N) gauge theory with\nmass deformation. For completeness, we will review the origin of the \feld theory from the\nbrane system before presenting the localization calculation.\n1.1 The D4-D8 system\nThe original D4-D8 system is a brane con\fguration in type I' string theory involving ND4\nbranes on R1;8\u0002S1=Z2. The D4 branes have their worldvolume along R1;8and sit at points\nalong the interval S1=Z2. There is an O8\u0000plane living at each of the two ends of the interval.\nThese orientifold planes carry \u000016 units of D8 brane charge, and thus require the inclusion\nof 16 D8 branes at points along the interval for tadpole cancellation. The usual construction\nis to stackNfD8 branes atop one of the O8\u0000planes and to stack the remaining (16 \u0000Nf)\nD8 branes atop the other O8\u0000plane. One then considers the case in which the ND4 branes\nare very near to the former stack, in which case the second boundary may be neglected. We\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nNote that we've reintroduced the six-dimensional Newton's constant $G_6$, which had been previously set to $4 \\pi G_{6} = 1$. This factor is important for the identification with the free energy on the field theory side.\nTreating $\\b(\\alpha)$ and $f_k(\\alpha)$ as functions of $\\alpha$, this gives us an expression which may be numerically integrated to obtain the free energy $F(\\alpha)-F(0)$ of the domain wall. The functional forms of $\\b(\\alpha),f_k(\\alpha)$ are obtained by fitting curves to the numerical data, as shown in Figure . Integrating to obtain $F(\\alpha) - F(0)$ gives the result shown in Figure .\n\\section{Field theory calculation}\n\\setcounter{equation}{0}\nLocalization  is a powerful tool used to obtain exact results in supersymmetric quantum field theories. In the large $N$ limit, results obtained via localization calculations  can be compared with  results obtained via holography. The goal of this section is to calculate the sphere free energy for a five-dimensional mass-deformed SCFT using localization, and then to compare it to the holographic result obtained in the previous section.\nA potential complication is  that the five-dimensional field theory dual to the matter-coupled six-dimensional gauged supergravity described in section  has not been fully identified. This is because the full gauged supergravity has not been shown to arise as a consistent truncation of any ten-dimensional theory. In the following, the tentative field theory dual we will use for the localization calculation in the IR is a $USp(2N)$ gauge theory coupled to $N_f$ fundamental representation hypermultiplets, and a single hypermultiplet in the anti-symmetric representation. As we will review below, this theory is believed to be obtained from the D4-D8 system  in type I' string theory/massive type IIA supergravity.\nOne fundamental limitation in our comparison between field theory and holographic results is that our holographic RG flow is completely numerical, and there is no analytic formula for the free energy that can be derived from it. Nevertheless, we will find qualitative similarities between the holographic free energy and the localization result for the free energy of the aforementioned $USp(2N)$ gauge theory with mass deformation. For completeness, we will review the origin of the field theory from the brane system before presenting the localization calculation.\n\\subsection{The D4-D8 system}\nThe original D4-D8 system  is a brane configuration in type I' string theory involving $N$ D4 branes on $\\mathbb R^{1,8} \\times S^1/\\mathbb Z^2$. The D4 branes have their worldvolume along $\\mathbb R^{1,8}$ and sit at points along the interval $S^1/\\mathbb Z^2$. There is an O8$^-$ plane living at each of the two ends of the interval. These orientifold planes carry $-16$ units of D8 brane charge, and thus require the inclusion of 16 D8 branes at points along the interval for tadpole cancellation. The usual construction is to stack $N_f$ D8 branes atop one of the O8$^-$ planes and to stack the remaining $(16-N_f)$ D8 branes atop the other O8$^-$ plane. One then considers the case in which the $N$ D4 branes are very near to the former stack, in which case the second boundary may be neglected. We\n\\end{document}\n"}
{"the solutions of the F(4) gauged supergravity theory being studied here can be uplifted to\nan AdS 6\u0002S4background of massive type IIA, our solutions should be captured by the D4-D8\nbrane framework. To identify the details of the relevant brane con\fguration, we \frst recall\nfrom section that the group which is gauged in the supergravity theory is SU(2)R\u0002G+, where\nG+is the additional gauge group arising from the presence of vector multiplets. Indeed, the\npresence of nvector \felds AI\n\u0016allows for the existence of a gauge group G+of dimension\ndimG+=n. The gauge group G+in the bulk corresponds to a \navor symmetry group\nENf+1of the boundary SCFT . The RG-\now triggered by the gauge coupling breaks this\nsymmetry group to SO(2Nf)\u0002U(1) in the IR. Deformation by the relevant mass parameters\nwill generically break SO(2Nf) further. For the solution studied in this paper, an SO(2)\nsymmetry survives, which suggests that a minimal choice for the dual \feld theory would be\none with Nf= 1 (i.e. a single D8 brane). However, even in this minimal case the enhanced\ngauge group E2\u0018=SU(2)\u0002U(1) of the conformal \fxed point is found to have dimension\ndimE2= 4, which suggests that the holographic dual to such a theory should contain at\nleast four bulk vector multiplets. Fortunately, it is possible to embed our n= 1 solution\nin a theory with n= 4, which can accommodate the extended \navor symmetry in the UV.\nSetting the \felds of the three additional vector multiplets to vanish then reproduces exactly\nthe solutions explored in this paper. In fact, such an embedding is possible for any value\nofn > 1. This suggests that our holographic solutions are generic enough to capture the\nbehavior of all single-mass deformations of ENf+1theories for any Nf. As such, we will carry\nout the localization calculation in section for generic Nf. We will \fnd that for every choice\nof 1\u0014Nf\u00147, one obtains a good match between the analytic \feld theory expression and\nour previous numerical results. Having addressed the identi\fcation of \navor symmetries, it\nis natural to interpret the holographic solutions of this paper as dual to RG \nows emanating\nfrom the same UV SCFTs that were found to be the duals of pure Roman's supergravity.\nThe \now is driven by three relevant operators of dimension \u0001 = 3 ;4;3, in addition to the\ngauge coupling deformation which brings the non-Lagrangian UV SCFT to an IR Yang-\nMills-matter theory. In the IR, the three relevant deformations are interpreted respectively\nas a mass term for the hypermultiplet scalars, a mass term for the hypermultiplet fermions,\nand a dimension three operator needed to preserve supersymmetry on the \fve-sphere . The\nexplicit form of these deformations is shown in . To support this interpretation, we now\ncalculate the free energy of the mass-deformed USp(2N) gauge theory and compare it to the\nholographic result displayed in Figure . For the unfamiliar reader, we will \frst reproduce\nthe results of , where the USp(2N) theory without mass deformation was studied. The\ntechniques used for the mass-deformed theory will be the same, and the new calculation is\npresented in section .\n0.1 Undeformed USp(2N)gauge theory\nIn , localization techniques were used to \fnd the perturbative partition function of N= 1\n\fve-dimensional Yang-Mills theory with matter in a representation RonS5, with the result\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nthe solutions of the $F(4)$ gauged supergravity theory being studied here can be uplifted to an AdS$_6 \\times S^4$ background of massive type IIA, our solutions should be captured by the D4-D8 brane framework. To identify the details of the relevant brane configuration, we first recall from section  that the group which is gauged in the supergravity theory is $SU(2)_R \\times G_+$, where $G_+$ is the additional gauge group arising from the presence of vector multiplets. Indeed, the presence of $n$ vector fields $A_\\mu^I$ allows for the existence of a gauge group $G_+$ of dimension $\\mathrm{dim} \\,G_+ = n$. The gauge group $G_+$ in the bulk corresponds to a flavor symmetry group $E_{N_f+1}$ of the boundary SCFT . The RG-flow triggered by the gauge coupling breaks this symmetry group to $SO(2N_f)\\times U(1)$ in the IR. Deformation by the relevant mass parameters will generically break $SO(2N_f)$ further. For the solution  studied in this paper, an $SO(2)$ symmetry survives, which suggests that a minimal choice for the dual field theory would be one with $N_f=1$ (i.e. a single D8 brane).\nHowever, even in this minimal case the enhanced gauge group $E_2 \\cong SU(2) \\times U(1)$ of the conformal fixed point is found to have dimension $\\mathrm{dim}\\, E_2=4$, which suggests that the holographic dual to such a theory should contain at least four bulk vector multiplets. Fortunately, it is possible to embed our $n=1$ solution in a theory with $n=4$, which can accommodate the extended flavor symmetry in the UV. Setting the fields of the three additional vector multiplets to vanish then reproduces exactly the solutions explored in this paper. In fact, such an embedding is possible for any value of $n>1$. This suggests that our holographic solutions are generic enough to capture the behavior of all single-mass deformations of $E_{N_f+1}$ theories for any $N_f$. As such, we will carry out the localization calculation in section  for generic $N_f$. We will find that for every choice of $1 \\leq N_f \\leq7$, one obtains a good match between the analytic field theory expression and our previous numerical results.\nHaving addressed the identification of flavor symmetries, it is natural to interpret the holographic solutions of this paper as dual to RG flows emanating from the same UV SCFTs that were found to be the duals of pure Roman's supergravity.  The flow is driven by three relevant operators of dimension $\\Delta = 3,4,3$, in addition to the gauge coupling deformation which brings the non-Lagrangian UV SCFT to an IR Yang-Mills-matter theory. In the IR, the three relevant deformations are interpreted respectively as a mass term for the hypermultiplet scalars, a mass term for the hypermultiplet fermions, and a dimension three operator needed to preserve supersymmetry on the five-sphere . The explicit form of these deformations is shown in .  \nTo support this interpretation, we now calculate the free energy of the mass-deformed $USp(2N)$ gauge theory and compare it to the holographic result displayed in Figure . For the unfamiliar reader, we will first reproduce the results of , where the $USp(2N)$ theory without mass deformation was studied. The techniques used for the mass-deformed theory will be the same, and the new calculation is presented in section .\n\\subsection{Undeformed $USp(2N)$ gauge theory}\nIn , localization techniques were used to find the perturbative partition function of ${\\cal N} = 1$ five-dimensional Yang-Mills theory with matter in a representation $R$ on $S^5$, with the result\n\\end{document}\n"}
{"result given by\nZ=1\njWjZ\nCartan[d\u001b]e\u00008\u00193r\ng2\nYMTr(\u001b2)det Ad\u0010\nsin(i\u0019\u001b)e1\n2f(i\u001b)\u0011\n\u0002Y\nIdet RI\u0010\n(cos(i\u0019\u001b))1\n4e\u00001\n4f(1\n2\u0000i\u001b)\u00001\n4f(1\n2+i\u001b)\u0011\n+ O\u0012\ne\u000016\u00193r\ng2\nYM\u0013\n(1)\nwhereris the radius of S5,\u001bis a dimensionless matrix, and fis de\fned as\nf(y) =i\u0019y3\n3+y2log\u0000\n1\u0000e\u00002\u0019iy\u0001\n+iy\n\u0019Li2\u0000\ne\u00002\u0019iy\u0001\n+1\n2\u00192Li3\u0000\ne\u00002\u0019iy\u0001\n\u0000\u0010(3)\n2\u00192(2)\nThe quotient by the Weyl group in amounts to division by a simple numerical factor jWj=\n2NN!. The integral over \u001bis not restricted to a Weyl chamber. Though this localization\nresult was obtained in the IR theory, it is expected to hold in the UV due to the assumed\nQ-exactness of the irrelevant UV completion terms. One may rewrite the partition function\nin terms of the free energy as\nZ=1\njWjZ\nCartan[d\u001b]e\u0000F(\u001b)+O\u0012\ne\u000016\u00193r\ng2\nYM\u0013\nF(\u001b) =4\u00193r\ng2\nYMTr\u001b2+ Tr AdFV(\u001b) +X\nITrRIFH(\u001b) (3)\nThe de\fnitions of FV(\u001b) andFH(\u001b) follow simply from , and using one may obtain the\nfollowing large argument expansions\nFV(\u001b)\u0019\u0019\n6j\u001bj3\u0000\u0019j\u001bjFH(\u001b)\u0019\u0000\u0019\n6j\u001bj3\u0000\u0019\n8j\u001bj (4)\nIt was argued in that in the large Nlimit, the perturbative Yang-Mills term - i.e. the \frst\nterm in the expression for F(\u001b) in - can be neglected, as can be the instanton contributions.\nThus in our evaluation of the free energy, we will only concern ourselves with the contri-\nbutions coming from FV(\u001b) andFH(\u001b). The \frst step in the evaluation of is recasting the\nmatrix integral in a simpler form. The integral over \u001bin is an integration over the Coulomb\nbranch, which is parameterized by the non-zero vevs of \u001b. One may write\n\u001b= diagf\u00151;:::;\u0015N;\u0000\u00151;:::;\u0000\u0015Ng (5)\nsinceUSp(2N) hasNelements in its Cartan. The integration variables are these N \u0015i.\nNormalizing the weights of the fundamental representation of USp(2N) to be\u0006eiwithei\nforming a basis of unit vectors for RN, it follows that the adjoint representation has weights\n\u00062eiandei\u0006ejfor alli6=j, whereas the anti-symmetric representation has only weights\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nresult given by\n\\begin{eqnarray}\nZ = {1 \\over |{\\cal W}|} \\int_{\\mathrm{Cartan}}[d\\sigma] \\,\\,e^{- {8 \\pi^3 r \\over g_{YM}^2} \\mathrm{Tr}(\\sigma^2)}\\rm det \\,_{\\mathrm{Ad}}\\left(\\sin(i \\pi \\sigma) e^{{1\\over 2} f(i \\sigma)} \\right)\n\\nonumber\\\\\n\\vphantom{.}\\hspace{0.6 in} \\times \\prod_I \\rm det \\,_{R_I}\\left((\\cos(i \\pi \\sigma))^{1 \\over 4} e^{-{1 \\over 4} f({1\\over 2} - i \\sigma) - {1 \\over 4} f({1\\over 2} + i \\sigma)} \\right) + O\\left(e^{- 16 \\pi^3 r \\over g_{YM}^2}\\right)\n\\end{eqnarray}\nwhere $r$ is the radius of $S^5$, $\\sigma$ is a dimensionless matrix, and $f$ is defined as \n\\begin{eqnarray}\nf(y) = {i \\pi y^3 \\over 3} + y^2 \\log\\left(1-e^{-2 \\pi i y} \\right) + {i y \\over \\pi} \\mathrm{Li}_2\\left(e^{-2 \\pi i y} \\right) + {1 \\over 2 \\pi^2} \\mathrm{Li}_3\\left(e^{-2 \\pi i y }\\right) - {\\zeta(3) \\over 2 \\pi^2}\n\\end{eqnarray}\nThe quotient by the Weyl group in  amounts to division by a simple numerical factor $|{\\cal W}| = 2^N N!$. The integral over $\\sigma$ is not restricted to a Weyl chamber. Though this localization result was obtained in the IR theory, it is expected to hold in the UV due to the assumed $Q$-exactness of the irrelevant UV completion terms. \n One may rewrite the partition function in terms of the free energy as \n\\begin{eqnarray}\nZ = {1 \\over |{\\cal W}|} \\int_{\\mathrm{Cartan}}[d\\sigma] \\,e^{-F({\\sigma})}+ O\\left(e^{- 16 \\pi^3 r \\over g_{YM}^2}\\right)\n\\nonumber\\\\\nF(\\sigma) = {4 \\pi^3 r \\over g_{YM}^2} \\mathrm{Tr}\\,\\sigma^2 + \\mathrm{Tr}_{\\mathrm{Ad}} F_V(\\sigma) + \\sum_I \\mathrm{Tr}_{R_I} F_H(\\sigma)\n\\end{eqnarray}\nThe definitions of $F_V(\\sigma)$ and $F_H(\\sigma)$ follow simply from , and using  one may obtain the following large argument expansions\n\\begin{eqnarray}\nF_V(\\sigma) \\approx {\\pi \\over 6} |\\sigma|^3 - \\pi |\\sigma| \\hspace{0.7 in} F_H(\\sigma) \\approx - {\\pi \\over 6} |\\sigma|^3 - {\\pi \\over 8} | \\sigma|\n\\end{eqnarray}\nIt was argued in  that in the large $N$ limit, the perturbative Yang-Mills term - i.e. the first term in the expression for $F(\\sigma)$ in  -  can be neglected, as can be the instanton contributions. Thus in our evaluation of the free energy, we will only concern ourselves with the contributions coming from $F_V(\\sigma)$ and $F_H(\\sigma)$. \nThe first step in the evaluation of  is recasting the matrix integral in a simpler form. The integral over $\\sigma$ in  is an integration over the Coulomb branch, which is parameterized by the non-zero vevs of $\\sigma$. One may write \n\\begin{eqnarray}\n\\sigma = \\mathrm{diag}\\{\\lambda_1,\\dots, \\lambda_N, - \\lambda_1, \\dots, -\\lambda_N \\}\n\\end{eqnarray}\nsince $USp(2N)$ has $N$ elements in its Cartan. The integration variables are these $N$ $\\lambda_i$. Normalizing the weights of the fundamental representation of $USp(2N)$ to be $\\pm e_i$ with $e_i$ forming a basis of unit vectors for $\\mathbb R^N$, it follows that the adjoint representation has weights $\\pm 2 e_i$ and $e_i \\pm e_j$ for all $i \\neq j$, whereas the anti-symmetric representation has only weights\n\\end{document}\n"}
{"ei\u0006ejfor alli6=j. The free energy in the speci\fc case of a vector multiplet in the\nadjoint, a single antisymmetric hypermultiplet, and Nffundamental hypermultiplets then is\nF(\u0015i) =X\ni6=j[FV(\u0015i\u0000\u0015j) +FV(\u0015i+\u0015j) +FH(\u0015i\u0000\u0015j) +FH(\u0015i+\u0015j)]\n+X\ni[FV(2\u0015i) +FV(\u00002\u0015i) +NfFH(\u0015i) +NfFH(\u0000\u0015i)] (1)\nThe next step is to look for extrema of this function in the speci\fc case of \u0015i\u00150 for all\ni. Extrema in the case of non-positive \u0015ican be obtained from these through action of the\nWeyl group. To calculate the extrema, one \frst assumes that as N!1 , the vevs scale as\n\u0015i=N\u000bxifor\u000b>0 andxiof orderO(N0). One then introduces a density function\n\u001a(x) =1\nNNX\ni=1\u000e(x\u0000xi) (2)\nwhich in the continuum limit should approach an L1function normalized as\nZ\ndx\u001a(x) = 1 (3)\nIn terms of this density function, one \fnds that\nF\u0019\u00009\u0019\n8N2+\u000bZ\ndxdy\u001a (x)\u001a(y) (jx\u0000yj+jx+yj) +\u0019(8\u0000Nf)\n3N1+3\u000bZ\ndx\u001a(x)jxj3(4)\nwhere the large argument expansions have been used, and terms subleading in Nhave been\ndropped. This only has non-trivial saddle points when both terms scale the same with N,\nwhich demands that \u000b= 1=2 and gives the famous result that F/N5=2. Extremizing the\nfree energy over normalized density functions then gives\nF\u0019\u00009p\n2\u0019N5=2\n5p\n8\u0000Nf(5)\nThis value of the free energy is to be identi\fed with the renormalized on-shell action of the\nsupersymmetric AdS 6solution. This identi\fcation yields the following relation between the\nsix-dimensional Newton's constant G6and the parameters NandNfof the dual SCFT,\nG6=5\u0019p\n8\u0000Nf\n27p\n2N\u00005=2(6)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n $e_i \\pm e_j$ for all $i \\neq j$. The free energy in the specific case of a vector multiplet in the adjoint, a single antisymmetric hypermultiplet, and $N_f$ fundamental hypermultiplets then is\n\\begin{eqnarray}\nF(\\lambda_i) = \\sum_{i \\neq j} \\left[F_V(\\lambda_i - \\lambda_j) + F_V(\\lambda_i + \\lambda_j) + F_H(\\lambda_i - \\lambda_j) + F_H(\\lambda_i + \\lambda_j) \\right] \n\\nonumber\\\\\n\\vphantom{.}\\hspace{0.3in}+ \\sum_i \\left[F_V(2 \\lambda_i) + F_V(-2 \\lambda_i) + N_f F_H(\\lambda_i) + N_f F_H(-\\lambda_i) \\right]\n\\end{eqnarray}\nThe next step is to look for extrema of this function in the specific case of $\\lambda_i \\geq 0$ for all $i$. Extrema in the case of non-positive $\\lambda_i$ can be obtained from these through action of the Weyl group.\nTo calculate the extrema, one first assumes that as $N \\rightarrow \\infty$, the vevs scale as $\\lambda_i = N^\\alpha x_i$ for $\\alpha>0$ and $x_i$ of order $O(N^0)$. One then introduces a density function\n\\begin{eqnarray}\n\\rho(x) = {1 \\over N} \\sum_{i=1}^N \\delta(x-x_i) \n\\end{eqnarray}\nwhich in the continuum limit should approach an $L^1$ function normalized as \n\\begin{eqnarray}\n\\int dx\\, \\rho(x) = 1\n\\end{eqnarray}\nIn terms of this density function, one finds that \n\\begin{eqnarray}\nF \\approx -{9 \\pi \\over 8} N^{2 + \\alpha} \\int dx dy \\,\\rho(x) \\rho(y) \\left(|x-y| + |x+y| \\right) + {\\pi (8-N_f) \\over 3} N^{1 + 3 \\alpha} \\int dx\\, \\rho(x)\\, |x|^3\n\\end{eqnarray}\nwhere the large argument expansions  have been used, and terms subleading in $N$ have been dropped. This only has non-trivial saddle points when both terms scale the same with $N$, which demands that $\\alpha=1/2$ and gives the famous result that $F\\propto N^{5/2}$. Extremizing the free energy over normalized density functions then gives \n\\begin{eqnarray}\nF \\approx - {9 \\sqrt{2} \\pi N^{5/2} \\over 5 \\sqrt{8-N_f}}\n\\end{eqnarray}\nThis value of the free energy is to be identified with the renormalized on-shell action of the supersymmetric AdS$_6$ solution. This identification yields the following relation between the six-dimensional Newton's constant $G_6$ and the parameters $N$ and $N_f$ of the dual SCFT,\n\\begin{eqnarray}\nG_6= \\frac{5\\pi\\sqrt{8-N_f}}{27\\sqrt{2}} ~N^{-5/2}\n\\end{eqnarray}\n\\end{document}\n"}
{"0.1 Mass-deformed USp(2N)gauge theory\nAs discussed previously, we now give a mass to a single hypermultiplet in the fundamental\nrepresentation. This amounts to making a shift \u001b!\u001b+min the relevant functional\ndeterminant. The result of this shift may be accounted for in by writing\nF(\u0015i;m) =X\ni6=j[FV(\u0015i\u0000\u0015j) +FV(\u0015i+\u0015j) +FH(\u0015i\u0000\u0015j) +FH(\u0015i+\u0015j)]\n+X\ni[FV(2\u0015i) +FV(\u00002\u0015i) +FH(\u0015i+m) +FH(\u0000\u0015i+m)\n+(Nf\u00001)FH(\u0015i) + (Nf\u00001)FH(\u0000\u0015i)] (1)\nAs before, we assume that \u0015i=N\u000bxifor\u000b > 0 and introduce a density \u001a(x) satisfying .\nUsing the expansions , we \fnd the analog of to be\nF(\u0016)\u0019\u00009\u0019\n8N2+\u000bZ\ndxdy\u001a (x)\u001a(y) (jx\u0000yj+jx+yj) +\u0019\n3(9\u0000Nf)N1+3\u000bZ\ndx\u001a(x)jxj3\n\u0000\u0019\n6N1+3\u000bZ\ndx\u001a(x)\u0002\njx+\u0016j3+jx\u0000\u0016j3\u0003\n(2)\nwhere for convenience we have de\fned \u0016\u0011m=N\u000b. As in the undeformed case, there is a\nnon-trivial saddle point only when \u000b= 1=2. A normalized density function which extremizes\nthe free energy is\n\u001a(x) =1\n(8\u0000Nf)x2\n\u0003\u0000\u00162( 2(9\u0000Nf)jxj\u0000jx+\u0016j\u0000jx\u0000\u0016j)x\u0003=s\n9 + 2\u00162\n2(8\u0000Nf)(3)\nwith\u001a(x) having support only on the interval x2[0;x\u0003]. Inserting this result back into then\ngives our \fnal result,\nF(\u0016) =\u0019\n135 \n(Nf\u00001)j\u0016j5\u0000s\n2\n8\u0000Nf(9 + 2\u00162)5=2!\nN5=2(4)\nWe may check that when \u0016= 0, we reobtain the result of the undeformed case . With this\nresult and G6given by we may now try to compare G6(F(\u0016)\u0000F(0)) to the same result\ncalculated holographically in Figure . Importantly, since \u0016scales asN\u00001=2, we see that in\nthe largeNlimit the \frst term of is subleading and may be neglected. Thus to leading\norder inN, the combination G6F(\u0016) is in fact independent of Nf. Since comparison with\nthe holographic result requires taking the large Nlimit, our supergravity solutions will be\nunable to capture information about the precise \navor content of the SCFT dual. This agrees\nwith the previous comments that, from the point of view of six-dimensional supergravity,\nthen= 1 solutions we are considering can be consistently embedded into theories with any\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\\subsection{Mass-deformed $USp(2N)$ gauge theory}\nAs discussed previously, we now give a mass to a single hypermultiplet in the fundamental representation. This amounts to making a shift $\\sigma \\rightarrow \\sigma + m $ in the relevant functional determinant. The result of this shift may be accounted for in  by writing\n\\begin{eqnarray}\nF(\\lambda_i,m) = \\sum_{i \\neq j} \\left[F_V(\\lambda_i - \\lambda_j) + F_V(\\lambda_i + \\lambda_j) + F_H(\\lambda_i - \\lambda_j) + F_H(\\lambda_i + \\lambda_j) \\right] \n\\nonumber\\\\\n\\vphantom{.}\\hspace{0.25in}+ \\sum_i \\left[F_V(2 \\lambda_i)+F_V(-2 \\lambda_i) +F_H(\\lambda_i+m) + F_H(-\\lambda_i+m) \\right.\n\\nonumber\\\\\n\\vphantom{.}\\hspace{1in}\\left. + (N_f-1) F_H(\\lambda_i) + (N_f-1) F_H(-\\lambda_i)  \\right]\\,\\,\\,\\,\n\\end{eqnarray}\nAs before, we assume that $\\lambda_i = N^\\alpha x_i$ for $\\alpha>0$ and introduce a density $\\rho(x)$ satisfying . Using the expansions , we find the analog of  to be\n\\begin{eqnarray}\nF(\\mu) \\approx - {9 \\pi \\over 8}N^{2 + \\alpha} \\int dx dy\\, \\rho(x) \\rho(y) \\left(|x-y| + |x+y| \\right)  + { \\pi \\over 3} (9-N_f)N^{1 + 3 \\alpha} \\int dx\\, \\rho(x) \\,|x|^3\n\\nonumber\\\\\n\\vphantom{.} \\hspace{0.3 in} - {\\pi \\over 6} N^{1 + 3 \\alpha} \\int dx \\,\\rho(x) \\left[ |x + \\mu|^3 + |x-\\mu|^3 \\right]\n\\end{eqnarray}\nwhere for convenience we have defined $\\mu \\equiv m/N^\\alpha$. As in the undeformed case, there is a non-trivial saddle point only when $\\alpha=1/2$.  A normalized density function which extremizes the free energy is\n\\begin{eqnarray}\n\\rho(x) = {1 \\over (8-N_f) x_*^2 - \\mu^2}\\left(\\,2(9-N_f) |x| - |x+ \\mu| - |x-\\mu|\\, \\right) \\hspace{0.3 in} x_* = \\sqrt{9 + 2 \\mu^2 \\over 2(8-N_f)}\\,\\,\\,\n\\end{eqnarray}\nwith $\\rho(x)$ having support only on the interval $x \\in [0,x_*]$. Inserting this result back into  then gives our final result,\n\\begin{eqnarray}\nF(\\mu) = {\\pi \\over 135}  \\left( (N_f-1) |\\mu|^5-\\sqrt{2\\over 8-N_f}\\,(9+2 \\mu^2)^{5/2} \\right)N^{5/2}\n\\end{eqnarray}\nWe may check that when $\\mu=0$, we reobtain the result of the undeformed case .\nWith this result and $G_6$ given by we may now try to compare $G_6(F(\\mu)-F(0))$ to the same result calculated holographically in Figure . Importantly, since $\\mu$ scales as $N^{-1/2}$, we see that in the large $N$ limit the first term of  is subleading and may be neglected. Thus to leading order in $N$, the combination $G_6 F(\\mu)$ is in fact independent of $N_f$. Since comparison with the holographic result requires taking the large $N$ limit, our supergravity solutions will be unable to capture information about the precise flavor content of the SCFT dual. This agrees with the previous comments that, from the point of view of six-dimensional supergravity, the $n=1$ solutions we are considering can be consistently embedded into theories with any\n\\end{document}\n"}
{"1 Gamma matrix and spinor conventions\nFor concreteness, we take the following basis of gamma matrices\n\n1=\u001b2\n12\n\u001b3\n\n2=\u001b2\n12\n\u001b1\n\n3=12\n\u001b1\n\u001b2\n\n4=12\n\u001b3\n\u001b2\n\n5=\u001b1\n\u001b2\n12\n\n6=\u001b3\n\u001b2\n12 (1)\nThese gamma matrices satisfy the Cli\u000bord algebra\nf\n\u0016;\n\u0017g= 2\u000e\u0016\u0017 (2)\nas appropriate for a positive de\fnite Euclidean spacetime. All matrices are purely imaginary\nand satisfy\n(\n\u0016)y=\n\u0016 (\n\u0016)2=1 (3)\nWe will now be interested in a seven-dimensional Cli\u000bord algebra, which will require the\nintroduction of a new matrix \n7. The reason we are interested in this is that we would like\nto represent hyperbolic space H6as a hypersurface in a seven-dimensional ambient space.\nThis allows us to determine properties of the Dirac spinors in the Euclidean-continued F(4)\ngauged supergravity theory with H6background by \frst considering Dirac spinors in seven\ndimensions and then performing a timelike reduction. In particular, we will choose a 7D\nmetric of signature (+ ;+;+;+;+;+;\u0000) for the ambient space. Then hyperbolic space H6\nis given by the following quadratic form\nx2\n1+\u0001\u0001\u0001+x2\n6\u0000x2\n7=\u0000L2(4)\nThe seven-dimensional Cli\u000bord algebra is made up of the set of matrices f\n1;:::;\n 6;\n7g,\nwith\n7satisfying\n(\n7)2=\u00001f\n\u0016;\n7g= 08\u00166= 7 (5)\nAs usual, we use the notation \n7= (\n7)\u00001, so that by the above we have \n7=\u0000\n7. We now\ndiscuss Dirac spinors in d= 7. We de\fne the Dirac conjugate of  Ato be\n\u0016 A= y\nAG\u00001(6)\nfor some matrix G. There are two possible choices for G, which in the particular case of the\nambient space above are\nG1=\n7G2=\n1:::\n6(7)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\\section{Gamma matrix and spinor conventions }\n\\setcounter{equation}{0}\nFor concreteness, we take the following basis of gamma matrices\n\\begin{eqnarray}\n\\gamma_1 =\\sigma_2 \\otimes \\mathds{1}_2 \\otimes \\sigma_3\n\\nonumber\\\\\n\\gamma_2 =\\sigma_2 \\otimes \\mathds{1}_2 \\otimes \\sigma_1\n\\nonumber\\\\\n\\gamma_3 = \\mathds{1}_2  \\otimes \\sigma_1  \\otimes \\sigma_2\n\\nonumber\\\\\n\\gamma_4 = \\mathds{1}_2 \\otimes \\sigma_3  \\otimes \\sigma_2\n\\nonumber\\\\\n\\gamma_5 =\\sigma_1 \\otimes \\sigma_2 \\otimes \\mathds{1}_2\n\\nonumber\\\\\n\\gamma_6 = \\sigma_3 \\otimes \\sigma_2 \\otimes \\mathds{1}_2\n\\end{eqnarray}\nThese gamma matrices satisfy the Clifford algebra\n\\begin{eqnarray}\n\\{ \\gamma_\\mu, \\gamma_\\nu\\} = 2 \\delta_{\\mu \\nu} \n\\end{eqnarray}\nas appropriate for a positive definite Euclidean spacetime. All matrices are purely imaginary and satisfy \n\\begin{eqnarray}\n(\\gamma_\\mu)^\\dagger = \\gamma_\\mu \\hspace{1 in} \\left(\\gamma_\\mu \\right)^2 =  \\mathds{1}\n\\end{eqnarray}\nWe will now be interested in a seven-dimensional Clifford algebra, which will require the introduction of a new matrix $\\gamma_7$. The reason we are interested in this is that we would like to represent hyperbolic space $\\mathbb{H}_6$ as a hypersurface in a seven-dimensional ambient space. This allows us to determine properties of the Dirac spinors in the Euclidean-continued $F(4)$ gauged supergravity theory with $\\mathbb{H}_6$ background by first considering Dirac spinors in seven dimensions and then performing a timelike reduction. In particular, we will choose a 7D metric of signature $(+,+,+,+,+,+,-)$ for the ambient space. Then hyperbolic space $\\mathbb{H}_6$ is given by the following quadratic form\n\\begin{eqnarray}\nx_1^2 + \\dots +x_6^2 - x_7^2 = - L^2 \n\\end{eqnarray} \nThe seven-dimensional Clifford algebra is made up of the set of matrices $\\{ \\gamma_1, \\dots, \\gamma_6, \\gamma_7\\}$, with $\\gamma_7$ satisfying \n\\begin{eqnarray}\n(\\gamma_7)^2 = - \\mathds{1} \\hspace{1 in} \\{ \\gamma_\\mu, \\gamma_7\\} = 0 \\,\\,\\,\\forall \\mu \\neq 7\n\\end{eqnarray}\nAs usual, we use the notation $\\gamma^7 = (\\gamma_7)^{-1}$, so that by the above we have $\\gamma^7 = - \\gamma_7$. \nWe now discuss Dirac spinors in $d=7$. We define the Dirac conjugate of $\\psi_A$ to be \n\\begin{eqnarray}\n\\bar \\psi_A = \\psi_A^\\dagger G^{-1} \n\\end{eqnarray}\nfor some matrix $G$. There are two possible choices for $G$ , which in the particular case of the ambient space above are \n\\begin{eqnarray}\nG_1 = \\gamma^7 \\hspace{1 in} G_2 = \\gamma^1 \\dots \\gamma^6\n\\end{eqnarray}\n\\end{document}\n"}
{"These will turn out to be the same, so we just work with the former. Thus we have that\n\u0016 A= y\nA\n7 (1)\nIf we choose \n7such that\n(\n7)y=\u0000\n7 (2)\nwe can express the Hermitian conjugates of our gamma matrices as\n\ny\n\u0016=\u0011G\u00001\n\u0016G (3)\nImportantly, with G=G1in , we have\n\u0011=\u00001 (4)\nThis will be important in Appendix when the consistency of the symplectic Majorana condi-\ntion is analyzed. For now, we just recall that the symplectic Majorana condition must take\nthe form\n\u0016 A=\u000fAB T\nBC (5)\nwhere\nC2= 1CT=C \nT\n\u0016=\u0000C\u00001\n\u0016C (6)\nWe now want to reduce from d= 7 tod= 6. In particular, we reduce on the time-like\ndirectionx7. This entails \fnding a Euclidean induced metric on the six-dimensional surface\n. From the point of view of the Cli\u000bord algebra, we must remove the matrix \n7to get a\nsix-dimensional Cli\u000bord algebra. However, the properties of the matrix \n7remain the same.\nIn fact, we may choose\n\n7=\n0\n1\n2\n3\n4\n5 (7)\nwhich satis\fes all of the properties ,.\n1 Free di\u000berential algebra\nIn this Appendix, we will construct the free di\u000berential algebra (FDA) of a supergravity the-\nory with H6background in order to motivate the form of the supersymmetry variations given\nin . The \frst step of constructing the FDA is to write down the Maurer-Cartan equations\n(MCEs), which may be thought of as the geometrization of the (anti-)commutation relations\nof the superalgebra. In short, instead of de\fning the algebra via the (anti-)commutators of\nits generators, the MCEs encode the algebraic structure in integrability conditions. In\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n\nThese will turn out to be the same, so we just work with the former. Thus we have that \n\\begin{eqnarray}\n\\bar \\psi_A = \\psi_A^\\dagger \\gamma_7\n\\end{eqnarray}\nIf we choose $\\gamma_7$ such that \n\\begin{eqnarray}\n(\\gamma_7)^\\dagger = - \\gamma_7\n\\end{eqnarray}\nwe can express the Hermitian conjugates of our gamma matrices as\n\\begin{eqnarray}\n\\gamma_\\mu^\\dagger = \\eta\\, G^{-1} \\gamma_\\mu G\n\\end{eqnarray}\nImportantly, with $G=G_1$ in , we have \n\\begin{eqnarray}\n\\eta = -1\n\\end{eqnarray}\nThis will be important in Appendix  when the consistency of the symplectic Majorana condition is analyzed. \nFor now, we just recall that the symplectic Majorana condition must take the form \n\\begin{eqnarray}\n\\bar \\psi_A = \\epsilon^{AB} \\psi_B^T\\, {\\cal C}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n{\\cal C}^2 = 1 \\hspace{0.7 in} {\\cal C}^T = {\\cal C} \\hspace{0.7in} \\gamma_\\mu^T = - {\\cal C}^{-1} \\gamma_\\mu {\\cal C}\n\\end{eqnarray}\nWe now want to reduce from $d=7$ to $d=6$. In particular, we reduce on the time-like direction $x_7$. This entails finding a Euclidean induced metric on the six-dimensional surface . From the point of view of the Clifford algebra, we must remove the matrix $\\gamma_7$ to get a six-dimensional Clifford algebra. However, the properties of the matrix $\\gamma^7$ remain the same. In fact, we may choose\n\\begin{eqnarray}\n\\gamma_7 = \\gamma_0\\gamma_1\\gamma_2\\gamma_3\\gamma_4\\gamma_5\n\\end{eqnarray}\nwhich satisfies all of the properties ,.\n\\section{Free differential algebra}\n\\setcounter{equation}{0}\nIn this Appendix, we will construct the free differential algebra (FDA) of a supergravity theory with $\\mathbb{H}_6$ background in order to motivate the form of the supersymmetry variations given in .\nThe first step of constructing the FDA is to write down the Maurer-Cartan equations (MCEs), which may be thought of as the geometrization of the (anti-)commutation relations of the superalgebra. In short, instead of defining the algebra via the (anti-)commutators of its generators, the MCEs encode the algebraic structure in integrability conditions. In \n\\end{document}\n"}
{"the supergravity context, a nice introduction to the MCEs, as well as to the free di\u000ber-\nential algebras to be introduced shortly, may be found in . In the current case, the MCEs\nare\n0 =DVa+1\n2\u0016 A\na\n7 A\n0 =Rab\u00004m2VaVb+m\u0016 A\nab A\n0 =dAr\u00001\n2g\u000frstAsAt\u0000i\u0016 A B\u001br AB\n0 =D a+m\na\n7 AVa(1)\nHerea= 1;:::; 6 andVaare the six-dimensional frame \felds, given in terms of the seven-\ndimensional spin-connection as Va=1\n2m!a7. These may be compared to the analogous\nexpressions in the dS/AdS cases of . As a simple check, the second equation of tells us that\nwhen A= 0,\nR\u0016\u0017=\u000020m2g\u0016\u0017 (2)\nwhich is precisely as expected for an H6background. The next step is to enlarge the MCEs\nto a free di\u000berential algebra (FDA) by adding the following equations for the additional\nvector and 2-form \felds of the full d= 6F(4) supergravity theory,\ndA\u0000mB+\u000b\u0016 A\n7 A= 0 dB+\f\u0016 A\na AVa= 0 (3)\nAbove,\u000band\fare two coe\u000ecients, which can be shown to satisfy\n\f=\u00002\u000b (4)\nfor our metric conventions. For the ambient space signature ( t;s) = (1;6), it is furthermore\nfound that \f= 2i, and thus we have \u000b=\u0000i. We would now like to compare the FDA above\nto the results of . To do so, we must \frst shift our notations by shifting\n\na!\n7\na\na!\u0000\n7\na (5)\nThis preserves the square of the gamma matrices, and hence the signature of the metric.\nThe de\fnition of the Dirac conjugate spinor remains the same under this change. So the\nFDA for the H6theory in these conventions is,\n0 =DVa+1\n2\u0016 A\na A\n0 =Rab\u00004m2VaVb+m\u0016 A\nab A\n0 =dAr\u00001\n2g\u000frstAsAt\u0000i\u0016 A B\u001br AB\n0 =D a\u0000m\na AVa\n0 =dA\u0000mB\u0000i\u0016 A\n7 A\n0 =dB\u00002i\u0016 A\n7\na AVa(6)\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\n the supergravity context, a nice introduction to the MCEs, as well as to the free differential algebras to be introduced shortly, may be found in . In the current case, the MCEs are \n\\begin{eqnarray}\n0 ={\\cal D} V^a + {1\\over 2} \\bar \\psi_A \\gamma^a \\gamma^7 \\psi^A \n\\nonumber\\\\\n0= R^{ab} - 4 m^2 V^a V^b + m \\bar \\psi_A \\gamma^{a b} \\psi^A \n\\nonumber\\\\\n0= d A^r - {1\\over 2} g \\epsilon^{rst} A_s A_t - i \\bar \\psi_A \\psi_B \\sigma^{r\\,\\,AB}\n\\nonumber\\\\\n0= D \\psi_a + m \\gamma_a \\gamma_7 \\psi_A V^a\n\\end{eqnarray}\nHere $a = 1,\\dots,6$ and $V^a$ are the six-dimensional frame fields, given in terms of the seven-dimensional spin-connection as $V^a= {1 \\over 2m}\\omega^{a7}$. These may be compared to the analogous expressions in the dS/AdS cases of . \nAs a simple check, the second equation of  tells us that when $\\psi^A =0$, \n\\begin{eqnarray}\nR_{\\mu \\nu} = - 20 m^2 g_{\\mu \\nu}\n\\end{eqnarray}\nwhich is precisely as expected for an $\\mathbb{H}_6$ background.\nThe next step is to enlarge the MCEs to a free differential algebra (FDA) by adding the following equations for the additional vector and 2-form fields of the full $d=6$ $F(4)$ supergravity theory, \n\\begin{eqnarray}\ndA - m B + \\alpha \\bar \\psi_A \\gamma_7 \\psi^A = 0 \\hspace{1 in} d B + \\beta \\bar \\psi_A \\gamma_a \\psi^A V^a = 0\n\\end{eqnarray}\nAbove, $\\alpha$ and $\\beta$ are two coefficients, which can be shown  to satisfy \n\\begin{eqnarray}\n\\beta = - 2 \\alpha\n\\end{eqnarray}\nfor our metric conventions. For the ambient space signature $(t,s) = (1,6)$, it is furthermore found that $\\beta = 2i$, and thus we have $\\alpha = -i$. \nWe would now like to compare the FDA above to the results of . To do so, we must first shift our notations by shifting \n\\begin{eqnarray}\n\\gamma^a \\rightarrow \\gamma^7 \\gamma^a \\hspace{0.7 in} \\gamma_a \\rightarrow - \\gamma_7 \\gamma_a\n\\end{eqnarray}\nThis preserves the square of the gamma matrices, and hence the signature of the metric. The definition of the Dirac conjugate spinor  remains the same under this change. So the FDA for the $\\mathbb{H}_6$ theory in these conventions is, \n\\begin{eqnarray}\n0 ={\\cal D} V^a + {1\\over 2} \\bar \\psi_A \\gamma^a \\psi^A \n\\nonumber\\\\\n0= R^{ab} - 4 m^2 V^a V^b + m \\bar \\psi_A \\gamma^{a b} \\psi^A \n\\nonumber\\\\\n0= d A^r - {1\\over 2} g \\epsilon^{rst} A_s A_t - i \\bar \\psi_A \\psi_B \\sigma^{r\\,\\,AB}\n\\nonumber\\\\\n0= D \\psi_a - m \\gamma_a \\psi_A V^a\n\\nonumber\\\\\n0=dA - m B - i \\bar \\psi_A \\gamma_7 \\psi^A \n\\nonumber\\\\\n0=d B -2 i \\bar \\psi_A \\gamma_7 \\gamma_a \\psi^A V^a\n\\end{eqnarray}\n\\end{document}\n"}
{"We may now compare the FDA written above to that obtained in the AdS 6case, which\nfor convenience we reproduce below,\n0 =DVa\u0000i\n2\u0016 A\na A\n0 =Rab+ 4m2VaVb+m\u0016 A\nab A\n0 =dAr\u00001\n2g\u000frstAsAt\u0000i\u0016 A B\u001br AB\n0 =D a\u0000im\n a AVa\n0 =dA\u0000mB\u0000i\u0016 A\n7 A\n0 =dB+ 2\u0016 A\n7\na AVa(1)\nWe see that formally, we may obtain the H6FDA from the AdS 6FDA by exchanging\nm!\u0000im  A! A\u0016 A!i\u0016 AAr!iArg!\u0000ig B!\u0000B A!iA\nThese exchanges are compatible with the relation g= 3m. Finally, we will check that the\nH6FDA is compatible with the symplectic Majorana condition. This is a statement about\nthe fourth equation of . We begin by de\fning\nr A\u0011D A\u0000q\na AVa(2)\nwhereq=mforH6andq=imfor AdS 6. We then \fnd that\nr A=D y\nAG\u00001\u0000q\u0003 y\nAG\u00001G\ny\naG\u00001Va=D\u0016 A\u0000q\u0003\u0011\u0016 A\naVa\n\u000fABr T\nBC=\u000fABD T\nBC\u0000q\u000fAB T\nBCC\u00001\nT\naCVa=D\u0016 A+q\u0016 A\naVa(3)\nwhere\u0011is de\fned implicitly in . We thus \fnd that the symplectic Majorana condition is\nconsistent only when\n\u0000q\u0003\u0011=q (4)\nForH6, the consistency of the symplectic Majorana condition thus requires \u0011=\u00001, which\nwe have already seen to be the case in . On the other hand, in the AdS 6case, one would\ninstead have required \u0011= 1. Checking the results of con\frms that this was so.\n1": "\\documentclass[12pt]{article}\n\\setlength{\\topmargin}{-.3in}\n\\setlength{\\oddsidemargin}{0in}\n\\setlength{\\textheight}{8.2in}\n\\setlength{\\textwidth}{6.5in}\n\\setlength{\\footnotesep}{\\baselinestretch\\baselineskip}\n\\newlength{\\abstractwidth}\n\\setlength{\\abstractwidth}{\\textwidth}\n\\addtolength{\\abstractwidth}{-6pc}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{latexsym}\n\\usepackage{epsf}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{dsfont}\n\\usepackage{subfigure}\n\\usepackage{hyperref}\n\\pagestyle{plain}\n\\begin{document}\n\nWe may now compare the FDA written above to that obtained in the AdS$_6$ case, which for convenience we reproduce below, \n\\begin{eqnarray}\n0 ={\\cal D} V^a - {i \\over 2} \\bar \\psi_A \\gamma^a \\psi^A \n\\nonumber\\\\\n0= R^{ab} + 4 m^2 V^a V^b + m \\bar \\psi_A \\gamma^{a b} \\psi^A \n\\nonumber\\\\\n0= d A^r - {1\\over 2} g \\epsilon^{rst} A_s A_t - i \\bar \\psi_A \\psi_B \\sigma^{r\\,\\,AB}\n\\nonumber\\\\\n0= D \\psi_a - i m \\gamma_a \\psi_A V^a\n\\nonumber\\\\\n0=dA - m B - i \\bar \\psi_A \\gamma_7 \\psi^A \n\\nonumber\\\\\n0=d B +2  \\bar \\psi_A \\gamma_7 \\gamma_a \\psi^A V^a\n\\end{eqnarray}\nWe see that formally, we may obtain the $\\mathbb{H}_6$ FDA from the AdS$_6$ FDA by exchanging\n\\begin{align}\nm \\rightarrow - i m \\qquad \\psi_A \\rightarrow \\psi_A \\qquad \\bar \\psi_A \\rightarrow i \\bar \\psi_A \\qquad A^r \\rightarrow i A^r \\qquad g \\rightarrow - i g \\qquad B \\rightarrow - B \\qquad A \\rightarrow i A\\nonumber\n\\end{align}\nThese exchanges are compatible with the relation $g = 3m$. \nFinally, we will check that the $\\mathbb{H}_6$ FDA is compatible with the symplectic Majorana condition. This is a statement about the fourth equation of . We begin by defining \n\\begin{eqnarray}\n\\nabla \\psi_A \\equiv D \\psi_A - q \\gamma_a \\psi_A V^a\n\\end{eqnarray}\nwhere $q = m$ for $\\mathbb{H}_6$ and $q=im$ for AdS$_6$. We then find that \n\\begin{eqnarray}\n\\overline{\\nabla \\psi_A} = D \\psi_A^\\dagger G^{-1} - q^* \\psi_A^\\dagger G^{-1} G \\gamma_a^\\dagger G^{-1} V^a = D \\bar \\psi_A - q^* \\eta \\,\\bar \\psi_A \\gamma_a V^a\n\\nonumber\\\\\n\\epsilon^{AB} \\nabla \\psi_B^T {\\cal C} = \\epsilon^{AB} D\\psi_B^T {\\cal C} - q \\epsilon^{AB} \\psi_B^T {\\cal C} {\\cal C}^{-1} \\gamma_a^T {\\cal C} V^a = D \\bar \\psi_A+q \\bar \\psi_A \\gamma_a V^a\n\\end{eqnarray}\nwhere $\\eta$ is defined implicitly in . We thus find that the symplectic Majorana condition is consistent only when \n\\begin{eqnarray}\n-q^* \\eta = q\n\\end{eqnarray}\nFor $\\mathbb{H}_6$, the consistency of the symplectic Majorana condition thus requires $\\eta=-1$, which we have already seen to be the case in . On the other hand, in the AdS$_6$ case, one would instead have required $\\eta=1$. Checking the results of  confirms that this was so. \n\\end{document}\n"}
{"1\nHS theory can also be considered in the sense of\nAdS/CFT correspondence, where the HS theory of mass-\nless HS \felds corresponds to a limiting case of the string\ntheory for the string tension going to zero. Large N su-\nperconformal \feld theories were studied as holographic\nduals for higher spin gauge theories in perturbative ex-\npansion around AdS spacetime . Klebanov and Polyakov\nhave proposed a duality between the singlet sector of\nthe critical 3-d O(N) vector model with ( \u001ea\u001ea)2interac-\ntion and minimal bosonic theory in AdS4 which contains\nmassless gauge \felds with even spin. The analog of AdS4\nconjecture by Klebanov and Polyakov appeared in AdS3\nconjecturing a duality between a complex scalar coupled\nto higher-spin \felds in Vasiliev's gravity in 3 dimensions\nand WN minimal model CFT in t'Hooft limit denoted\nby coset representation\nSU(N)k\bSU(N)1\nSU(N)k+1; (1)\nwhere we de\fne the t'Hooft limit with N;k! 1 for\n\u0015\u0011N\nk+N. This duality has been veri\fed by the number\nof studies, correspondence of global symmetries in bulk\nand at the boundary , correspondence of the 1-loop par-\ntition function in the bulk and at the large NCFT , and\npartition function of the HS black hole at high temper-\nature in the bulk and at the boundary CFT, as well as\nfor the 3-point functions when \u0015=1\n2ands= 2;3;4 for\nthe scalar-scalar-HS \feld (00s) correlator in the t'Hooft\nlimit. Via three-point functions, tests of the conjecture\nhave been done in (for 00s correlator with general \u0015),\nand in . In this work, we extract the coupling of the\n00s three-point correlator by considering the linearised\nVasiliev equations of motion, and we verify it by choos-\ning the spin to be three and comparing with result in\n. The result corresponds to coupling of the three-point\ncorrelation function up to selected normalisation. While\nwe consider general \u0015, the similar work has been done\nfor the \fxed \u0015in . The structure of the work is as fol-\nlows: In the section two we consider linearised equations\nof motion in the Vasiliev's theory, in the section three we\nconsider the higher spin \feld in the metric formulation,\nand in section four we conclude.\nI. LINEARIZED EQUATIONS OF MOTION\nLet us \frst consider the coe\u000ecient coming from the\nVasiliev linearised equations. Vasiliev's theory contains\n\fve equations for the master \felds W which is spacetime\n1-form, B and S\u000bwhich are spacetime 0-forms. The gen-\nerating functions are dependent on the coordinates of the\nspacetime, auxiliary bosonic twistor variables (referred\nto as \"oscillators\") and Cli\u000bord element pairs, where in\nde\fnitions we follow conventions from . The oscillators\nand various other ingredients are used to de\fne the \"de-\nformed\" oscillator star-commutation relations which give\nrise tohs[\u0015] higher spin algebra. Two of the above men-\ntioned equations that will be of the interest here are\ndW=W^?W (2)\ndB=W ?B\u0000B?W (3)We can rewrite W with projector operators\nP\u0006=1\u0006 \n2(4)\nfor elements of the Cli\u000bord pairs such that W=\n\u0000P +A\u0000P\u0000Afor\nP\u0006 1= 1P\u0006=\u0006P\u0006P\u0006 2= 2P\u0007 (5)\nwhere A are Chern-Simons gauge \felds which take value\nin the Lie algebra hs[ \u0015]. In this formulation the equation\ndW=W^?W (6)\ngives\ndA+A^?A= 0 (7)\nDA+A^?A= 0 (8)\nwhere A and Aare positive polynomials of the positive\ndegree in products of deformed oscillators. () and () are\nin that case equal to \feld equations hs[\u0015]\nhs[\u0015] Chern-\nSimons theory.\nThe generators of hs[ \u0015] are de\fned with spin index sand\nmode index masVs\nmfors\u00152 whilejmj< sand obey\nstar product\nVs\nm?Vt\nn=s+t\u0000js\u0000tj\u00001X\nu=1;2;3gst\nu(m;n;\u0015)Vs+t\u0000u\nm\u0000n (9)\nwhere\ngst\nu(m;n;\u0015) = (\u00001)u+1gts\nu(m;n;\u0015) (10)\nare speci\fc coe\u000ecients dependent on \u0015and de\fned ac-\ncording to conventions . The equations describe interac-\ntion of arbitrary higher spin background with lienarized\nscalars. The coupling that we are interested in can be ex-\ntracted from rewriting the master \feld B as a linearized\n\nuctuation around vacuum value \u0017\nB=\u0017+P+ 2C(x;~y\u000b) +P\u0000 2~C(x;~y\u000b) (11)\nand expanding the master \feld Cin the deformed oscil-\nlators ~y\u000bin the equation\ndC+A?C\u0000C?A= 0: (12)\nThat allows us determining the generalised Klein-Gordon\n(KG) equation in the background of HS \felds. While the\nexpansion of the master \feld C in formalism of bosonic\nVasiliev's theory is given by\nC=C1\n0+C\u000b\f~y\u000b~y\f+C\u000b\f\u001b\u0015~y\u000b~y\f~y\u001b~y\u0015+::: (13)": "\\documentclass[prd,superscriptaddress,twocolumn,10pt]{revtex4}\n\\usepackage{amsmath,amssymb}\n\\usepackage{verbatim}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{color} \n\\DeclareFontFamily{OT1}{rsfs}{}\n\\DeclareFontShape{OT1}{rsfs}{m}{n}{ <-7> rsfs5 <7-10> rsfs7 <10->rsfs10}{} \n\\DeclareMathAlphabet{\\mycal}{OT1}{rsfs}{m}{n} \n\\begin{document}\nHS theory can also be considered in the sense of AdS/CFT correspondence, where the HS theory of massless HS fields corresponds to a limiting case of the string theory for the string tension going to zero. Large N superconformal field theories were studied as holographic duals for higher spin gauge theories in perturbative expansion around AdS spacetime .\nKlebanov and Polyakov  have proposed a duality  between the singlet sector of the critical 3-d O(N) vector model with $(\\phi^a\\phi^a)^2$ interaction and minimal bosonic theory in AdS4 which contains massless gauge fields with even spin. The analog of AdS4 conjecture by Klebanov and Polyakov appeared in AdS3  conjecturing a duality between a complex scalar coupled to higher-spin fields in Vasiliev's gravity in 3 dimensions and WN minimal model CFT in t'Hooft limit denoted by coset representation\n \\begin{equation}\n\\frac{SU(N)_k\\oplus SU(N)_1}{SU(N)_{k+1}},\n\\end{equation}\nwhere we define the t'Hooft limit with $N,k\\rightarrow\\infty$ for $\\lambda\\equiv\\frac{N}{k+N}$. \nThis duality has been verified by the number of studies, correspondence of global symmetries in bulk and at the boundary , correspondence of the 1-loop partition function in the bulk and at the large NCFT  , and partition function of the HS black hole at high temperature in the bulk and at the boundary CFT, as well as for the 3-point functions when $\\lambda=\\frac{1}{2}$ and $s=2,3,4$ for the scalar-scalar-HS field (00s) correlator in the t'Hooft limit. \nVia three-point functions, tests of the conjecture have been done in  (for 00s correlator with general $\\lambda$), and in .\nIn this work, we extract the coupling of the 00s  three-point correlator  by considering the linearised Vasiliev equations of motion, and we verify it by choosing the spin to be three and comparing with result in .\nThe result corresponds to coupling of the three-point correlation function up to selected normalisation. \nWhile we consider general $\\lambda$, the similar work has been done for the fixed $\\lambda$ in .\nThe structure of the work is as follows: In the section two we consider linearised equations of motion in the Vasiliev's theory, in the section three we consider the higher spin field in the metric formulation, and in section four we conclude.\n  \\section{Linearized equations of motion}\nLet us first consider the coefficient coming from the Vasiliev linearised equations. Vasiliev's theory contains five equations for the master fields W which is spacetime 1-form, B and $S_{\\alpha}$ which are spacetime 0-forms. The generating functions are dependent on the coordinates of the spacetime, auxiliary bosonic twistor variables (referred to as \"oscillators\") and Clifford element pairs, where in definitions we follow conventions from .\nThe oscillators  and various other ingredients are used to define the \"deformed\" oscillator star-commutation relations which give rise to $hs[\\lambda]$ higher spin algebra. \nTwo of the above mentioned equations that will be of the interest here are\n\\begin{align}\ndW&=W\\wedge\\star W \\\\\ndB&=W\\star B-B\\star W \n\\end{align}\nWe can rewrite W with projector operators \n\\begin{equation}\n\\mathcal{P}_{\\pm}=\\frac{1\\pm\\psi}{2} \n\\end{equation}\nfor $\\psi$ elements of the Clifford pairs such that $W=-\\mathcal{P}_{+}A-\\mathcal{P}_{-}\\overline{A}$\nfor \n\\begin{align}\n\\mathcal{P}_{\\pm}\\psi_{1}&=\\psi_1\\mathcal{P}_{\\pm}=\\pm\\mathcal{P}_{\\pm} &\\mathcal{P}_{\\pm}\\psi_2=\\psi_2\\mathcal{P}_{\\mp}\n\\end{align}\nwhere A are Chern-Simons gauge fields which take value in the Lie algebra hs$[\\lambda]$. In this formulation the equation \n\\begin{equation}\ndW=W\\wedge\\star W\n \\end{equation}\ngives\n\\begin{align}\ndA+A\\wedge\\star A =0 \\\\ \nD\\overline{A}+\\overline{A}\\wedge \\star \\overline{A}=0 \n\\end{align}\nwhere A and $\\overline{A}$ are positive polynomials of the positive degree in products of deformed oscillators.\n() and () are in that case equal to field equations $hs[\\lambda]\\otimes hs[\\lambda]$  Chern-Simons theory. \\\\ \nThe generators of hs$[\\lambda]$ are defined with spin index \\textit{s} and mode index \\textit{m}  as\n$V^s_{m}$ for $s\\geq2 $\nwhile $|m|<s$ and obey\n star product\n\\begin{equation}\nV_m^s\\star V_n^t=\\sum_{u=1,2,3}^{s+t-|s-t|-1}g_u^{st}(m,n;\\lambda)V^{s+t-u}_{m-n}\n\\end{equation}\nwhere \n\\begin{equation}g_u^{st}(m,n;\\lambda)=(-1)^{u+1}g_u^{ts}(m,n;\\lambda)\\end{equation}\nare specific coefficients dependent on $\\lambda$ and defined according to conventions .\nThe equations describe interaction of arbitrary higher spin background with lienarized scalars.\nThe coupling that we are interested in can be extracted from rewriting the master field B as a linearized fluctuation around vacuum value $\\mathcal{\\nu}$\n\\begin{align}\nB=\\mathcal{\\nu}+\\mathcal{P}_{+}\\psi_2C(x,\\tilde{y}_{\\alpha})+\\mathcal{P}_-\\psi_2\\tilde{C}(x,\\tilde{y}_{\\alpha})\n\\end{align}\nand expanding the master field $C$ in the deformed oscillators $\\tilde{y}_{\\alpha}$ in the equation\n\\begin{equation}\ndC+A\\star C- C \\star \\overline{A}=0.\n\\end{equation}\n That allows us determining the generalised Klein-Gordon (KG) equation in the background of HS fields.\nWhile the expansion of the master field C in formalism of bosonic Vasiliev's theory is given by\n \\begin{equation}\n C=C_0^1+C^{\\alpha\\beta}\\tilde{y}_{\\alpha}\\tilde{y}_{\\beta}+C^{\\alpha\\beta\\sigma\\lambda}\\tilde{y}_{\\alpha}\\tilde{y}_{\\beta}\\tilde{y}_{\\sigma}\\tilde{y}_{\\lambda}+...\n \\end{equation}\n\\end{document}\n"}
{"1\nwith implied star product and symmetric C compo-\nnents. Master \feld components are now separated in\nphysical scalar \feld C1\n0and higher ones, related to it on-\nshell by derivatives. The expansion of C is given by\nC=1X\ns=1X\njmj<sCs\nmVs\nm (1)\nforCs\nm\u0018C\u000b1\u000b2::\u000b2s\u00001where m and number of oscillators\n~y1versus ~y2are related with 2 m=N1\u0000N2andCs\nmare\nfunctions of spacetime coordinates. Auxiliary tensors are\nabsorbed within a \feld. The \felds Aand \u0016Aare expanded\nanalogously\nA=1X\ns=2X\njmj<sAs\nmVs\nmA=1X\ns=2X\njmj<sAs\nmVs\nm:(2)\nThe standard procedure of \fnding the generalised KG\nequation consists of inserting the expressions for A,\u0016A\nandCin () and determining the smallest possible set\nof equations needed to \fnd the scalar equation in arbi-\ntrary background. Standard procedure can be described\nconsidering equation () in AdS background since it is a\nfoundation for the following computations. The vacuum\nC1\n0equation without AdS \felds is ordinary KG equation\nwhile one can determine the higher components in the\nterms ofC1\n0. The AdS connection consists of the spin-2\ngenerators that form SL(2), subalgebra of hs[\u0015]\nA=e\u001aV2\n1dz+V2\n0d\u001a (3)\n\u0016A=e\u001aV2\n\u00001d\u0016z\u0000V2\n0d\u001a (4)\nwith AdS metric\nds2=d\u001a2+e2\u001adzd\u0016z: (5)\nThe higher spins \felds vanish, and we are working in Eu-\nclidean metric and Fe\u000berman-Graham gauge. The gen-\neral form of the Cequation () in the AdS background\nis\n@\u001aCs\nm+ 2Cs+1\nm+Cs+1\nmg(s+1)2\n3 (m;0) = 0 (6)\n@Cs\nm+e\u001a(Cs\u00001\nm\u00001+1\n2g2s\n2(1;m\u00001)Cs\nm (7)\n+1\n2g2(s+1)\n3 (1;m\u00001)Cs+1\nm\u00001) = 0\n@Cs\nm\u0000e\u001a(Cs\u00001\nm+1\u00001\n2g2s\n2(\u00001;m+ 1)Cs\nm+1 (8)\n+1\n2g2(s+1)\n3 (\u00001;m+ 1)Cs+1\nm+1) = 0\nforjmj< s,@=@z;@=@zand the\u0015-dependence in\nthe structure constants suppressed. In the simplest case\nchoosings= 1;s= 2 one can solve for the higher com-\nponents in C and obtain the Klein-Gordon KG equation\n\u0002\n@2\n\u001a+ 2@\u001a+ 4e\u00002\u001a@\u0016@\u0000(\u00152\u00001)\u0003\nC1\n0= 0: (9)Consistency condition on equations is that all the com-\nponents of C have smooth solution when expressed using\nC1\n0. The strategy for determining the minimal set of\nequations for C1\n0is to select components of C that are of\nthe formCm+1\n\u0006mand therefore the smallest spin for \fxed\nm (e.g.C1\n0;C2\n\u00061;::). That are minimal components. One\nneedsVs\nm;\u001aequations for \fxed m, solve for non-minimal\ncomponents in terms of minimal ones and \u001aderivatives,\nforA\u001a=\u0000A\u001a=V2\n0. After solving for minimal ones,\none needs to solve Vs\nm;zandVs\nm;zequations in terms of\nC1\n0and its derivatives. Once that we have expressed the\nhigher components of C in terms of C1\n0we can determine\nthe part that de\fnes the KG equation and the generalised\npart that appears due to the HS background. To obtain\nthe equation of motion for the scalar \feld up to linear\norder we consider the variation of the gauge \feld and\napply the KG equation on it. This and the standard pro-\ncedure for obtaining the linearised equation of motion for\nthe scalar \feld described above should be equal once the\ngauge parameter is chosen conveniently. That approach\ncan be written in the following way. First we express the\nhigher components of the C \feld in terms of the combi-\nnation of the derivatives on C1\n0in the background AdS.\nFocusing on the master \feld C, the equation () is invari-\nant under the hs[\u0015]\bhs[\u0015] gauge invariance when\nC!C+C?\u0016\u0003\u0000\u0003?C (10)\nfor\n\u0003(\u001a;z;\u0016z) =2s\u00001X\nn=11\n(n\u00001)!(\u0000@)n\u00001\u0015(s)(z;\u0016z)e(s\u0000n)\u001aVs\ns\u0000n:\n(11)\nWhere we take \u0003 to be chiral, so \u0016\u0003 = 0. The \feld in the\nhigher spin background is obtained by transformation\n~Cs\nm=Cs\nm\u0000(\u0003?C)s\nm: (12)\nThe \feldCs\nmwe express in terms of the C1\n0. To do that\nwe focus on the set of equations (),(20),(). The product\nof the C \feld with \u0003 gives combination of higher com-\nponents of C in AdS background which can, as we will\nshow, be expressed in terms of C1\n0. On the \feld C1\n0we\ncan use the transformation () and obtain\n~C1\n0=C1\n0\u0000(\u0003?C)1\n0: (13)\nSince we are at the linear order, once we have C1\n0we\ncan rewrite it as ~C1\n0which is de\fned on the higher spin\nbackground. From the expression for the gauge \feld \u0003 ()\nand the relation for the star product () we can determine\nthe variation of the scalar \feld C1\n0\n(\u000eC)1\n0=\u00002s\u00001X\nn=11\n(n\u00001)!(\u0000@)n\u00001\u0003(s)(14)\n\u00021\n2gss\n2s\u00001(s\u0000n;n\u0000s)Cs\n\u0000(s\u0000n)e(s\u0000n)\u001a;(15)": "\\documentclass[prd,superscriptaddress,twocolumn,10pt]{revtex4}\n\\usepackage{amsmath,amssymb}\n\\usepackage{verbatim}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{color} \n\\DeclareFontFamily{OT1}{rsfs}{}\n\\DeclareFontShape{OT1}{rsfs}{m}{n}{ <-7> rsfs5 <7-10> rsfs7 <10->rsfs10}{} \n\\DeclareMathAlphabet{\\mycal}{OT1}{rsfs}{m}{n} \n\\begin{document}\n with implied star product and symmetric C components.\n  Master field components are now separated in physical scalar field $C_0^1$  and higher ones, related to it on-shell by derivatives. \nThe expansion of C is given by\n\\begin{equation}\nC=\\sum_{s=1}^{\\infty}\\sum_{|m|<s}C_m^sV_m^s\n\\end{equation}\nfor $C_m^s\\sim C^{\\alpha_1\\alpha_2..\\alpha_{2s-1}}$\nwhere m and number of oscillators $\\tilde{y}_{1}$ versus $\\tilde{y}_2$ are related with $2m=N_1-N_2$ and $C_m^s$ are functions of spacetime coordinates. Auxiliary tensors are absorbed within a field.\nThe fields $A$ and $\\bar{A}$ are expanded analogously\n\\begin{align}\nA&=\\sum_{s=2}^{\\infty}\\sum_{|m|<s}A_m^sV_m^s & \\overline{A}=\\sum_{s=2}^{\\infty}\\sum_{|m|<s}\\overline{A}_m^sV_m^s.\n\\end{align}\nThe standard procedure of finding the generalised KG equation consists of inserting the expressions for $A$, $\\bar{A}$ and $C$ in  () and determining the smallest possible set of equations needed to find the scalar equation in arbitrary background.\nStandard procedure can be described considering equation () in AdS background since it is a foundation for the following computations. The vacuum $C_0^1$ equation without AdS fields is ordinary KG  equation while one can determine the higher components in the terms of $C_0^1$. \nThe AdS connection consists of the spin-2 generators that form SL(2), subalgebra of $hs[\\lambda]$ \n\\begin{align}\nA&=e^{\\rho}V_1^2dz+V_0^2d\\rho\\\\\n\\bar{A}&=e^{\\rho}V_{-1}^2d\\bar{z}-V_0^2d\\rho\n\\end{align}\nwith AdS metric\n\\begin{equation}\nds^2=d\\rho^2+e^{2\\rho}dzd\\bar{z}. \n\\end{equation}\nThe higher spins fields vanish, and we are working in Euclidean metric and  Fefferman-Graham gauge. The general form of the $C$ equation () in the AdS background is \n\\begin{align}\n&\\partial_{\\rho}C_m^s+2C_m^{s+1}+C_m^{s+1}g_3^{(s+1)2}(m,0)=0  \\end{align}\n\\begin{align}\n \\partial C_m^s+e^{\\rho}(C_{m-1}^{s-1}+\\frac{1}{2}g_2^{2s}(1,m-1)C^s_m\\\\ \\nonumber+\\frac{1}{2}g_3^{2(s+1)}(1,m-1)C_{m-1}^{s+1})=0 \\end{align}\n\\begin{align}\n\\overline{\\partial}C^s_m-e^{\\rho}( C_{m+1}^{s-1}-\\frac{1}{2}g_2^{2s}(-1,m+1)C^s_{m+1}\\\\+\\frac{1}{2}g_3^{2(s+1)}(-1,m+1)C^{s+1}_{m+1})=0 \\nonumber\n\\end{align}\nfor $|m|<s$, $\\partial=\\partial_z,\\overline{\\partial}=\\partial_{\\overline{z}}$ and the $\\lambda$-dependence in the structure constants suppressed. \nIn the simplest case choosing $s=1,s=2$ one can solve for the higher components in C and obtain the Klein-Gordon KG equation \n\\begin{equation}\n\\left[ \\partial_{\\rho}^2+2\\partial_{\\rho}+4e^{-2\\rho}\\partial\\bar{\\partial}-(\\lambda^2-1) \\right]C_0^1=0.\n\\end{equation}\n Consistency condition on equations is that all the components of C have smooth solution when expressed using $C_0^1$. The strategy for determining the minimal set of equations for $C_0^1$ is to select components of C that are of the form $C_{\\pm m}^{m+1}$ and therefore the smallest spin for fixed m (e.g. $C_0^1, C_{\\pm1}^2,..$). That are minimal components. \n One needs $V_{m,\\rho}^s$ equations for fixed m, solve for non-minimal components in terms of minimal ones and $\\rho$ derivatives, for $A_{\\rho}=-\\overline{A}_{\\rho}=V_{0}^2$. After solving for minimal ones, one needs to solve $V_{m,z}^s$ and $V_{m,\\overline{z}}^s$ equations in terms of $C_0^1$ and its derivatives. \nOnce that we have expressed the higher components of C in terms of $C_0^1$ we can determine the part that defines the KG equation and the\ngeneralised part that appears due to the HS background.\nTo obtain the equation of motion for the scalar field up to linear order we consider the variation of the gauge field and apply the KG equation on it. This and the standard procedure for obtaining the linearised equation of motion for the scalar field described above\nshould be equal once the gauge parameter is chosen conveniently.\nThat approach can be written in the following way.\nFirst we express the higher components of the C field in terms of the combination of the derivatives on $C_0^1$ in the background AdS. \nFocusing on the master field $C$, the equation () is invariant under the $hs[\\lambda]\\oplus hs[\\lambda]$  gauge invariance when \n\\begin{align}\nC\\rightarrow C+C\\star \\bar{\\Lambda}-\\Lambda\\star C \n\\end{align}\nfor \n\\begin{equation}\n\\Lambda(\\rho,z,\\bar{z})=\\sum_{n=1}^{2s-1}\\frac{1}{(n-1)!}(-\\partial)^{n-1}\\lambda^{(s)}(z,\\bar{z})e^{(s-n)\\rho}V^{s}_{s-n}. \n\\end{equation}\nWhere we take $\\Lambda$ to be chiral, so $\\bar{\\Lambda}=0$.\nThe field in the higher spin background is obtained by transformation\n\\begin{align}\n\\tilde{C}_m^s=C_m^s-(\\Lambda\\star C)^s_m.\n\\end{align}\nThe field $C_m^s$ we express in terms of the $C_0^1$.\nTo do that we focus on the set of equations (),(20),().\nThe product of the C field with $\\Lambda$ gives combination of higher components of C in AdS background which can, as we will show, be expressed in terms of $C_0^1$. On the field $C_0^1$ we can use the transformation () and obtain \n\\begin{align}\n\\tilde{C}_0^1=C_0^1-(\\Lambda\\star C)^1_0.\n\\end{align}\nSince we are at the linear order, once we have $C_0^1$ we can rewrite it as $\\tilde{C}_0^1$ which is defined on the higher spin background. \nFrom the expression for the gauge field $\\Lambda$ () and the relation for the star product () we can determine the variation of the scalar field $C^1_0$\n\\begin{align}\n(\\delta C)^1_0 &=-\\sum_{n=1}^{2s-1}\\frac{1}{(n-1)!}(-\\partial)^{n-1}\\Lambda^{(s)} \\\\ &\\times\\frac{1}{2}g_{2s-1}^{ss}(s-n,n-s)C^s_{-(s-n)}e^{(s-n)\\rho}, \n\\end{align}\n\\end{document}\n"}
{"1\nforCs\n\u0000(s\u0000n)an arbitrary component of the master \feld\nC. Takingm!\u0000min the set of equations (), (20), ()\nands=m+ 1 in (), we can iteratively determine the\ndependence of the Cm+1\nm on theC1\n0. From the equation\n() we obtain\n@zCm+1\n\u0000m+e\u001a\n2g2(m+2)\n3 (1;\u0000m\u00001)Cm+2\n\u0000m\u00001= 0 (1)\ntaking into consideration that for certain components Cs\nn\nit is requiredjnj\u0014s\u00001 this iteratively leads to relation\nofCm+1\nm andC1\n0, and from the () analogously for Cm+1\n\u0000m\nandC1\n0. The general form of the Cs\n\u0006is then given in\nterms ofCm+1\n\u0006mand coe\u000ecients gts\nu(m;n). Knowing Cs\n\u0006\nandCm+1\n\u0006mallows to obtain\n(\u000eC)1\n0=sX\nn=1fs;n\n\u0006(\u0015)@n\u00001\nz\u0003(s)@s\u0000n\nz\u001e (2)\nfor\u001e\u0011C1\n0andfs;n\n\u0006(\u0015) expressed in terms of coe\u000ecients\ngst\nu(m;n) Using the replacement @\u001a!\u0000(1\u0006\u0015) and writ-\ning explicitly \frst few n values for fs;n\n\u0006(\u0015), allows to de-\ntermine its general expression\nfs;n\n\u0006(\u0015) = (\u00001)s \u0000(s+\u0015)\n\u0000(s\u0000n+ 1\u0006\u0015)1\n2n\u00001(2(n\n2\u00001))!!\u0000n\u00001\n2\u0001\n!\n\u0002n\u00001\n2Y\nj=1s+ 1\u0000n\n2s\u00002j\u00001: (3)\nSubstituting () in () one obtains the variation of the\nscalar \feld\n(\u000eC)1\n0=sX\nn=1(\u00001)s \u0000(s\u0006\u0015)\n\u0000(s\u0000n+ 1\u0006\u0015)1\n2n\u00001\u0000\n2\u0000n\n2\u0001\n\u00001\u0001\n!!\u0000n\u00001\n2\u0001\n!\n\u0002(n\u00001\n2)Y\nj=1s+j\u0000n\n2s\u00002j\u00001@n\u00001\nz\u0003(s)@s\u0000n\nzC1\n0: (4)\nTo consider the coe\u000ecient in front, we focus on the term\nwith the lowest number of @zderivatives on the gauge\n\feld \u0003(s), obtained for n=1. Then, () becomes\n(\u000eC)1\n0jn=1= (\u00001)s\u0003(s)@s\u00001C1\n0: (5)\nTo obtain the linearised equation of motion for the scalar\n\feld we act on () with KG operator (). This can be\nwritten as\n\u0003KG~C1\n0=\u0003KGC1\n0+\u0003KG\u000eC1\n0: (6)\nTaking@\u001a!(1\u0006\u0015) infs;n\n\u0006(\u0015) we have taken and con-\nsidering the term with highest number of derivatives on\nC1\n0leads to\n\u0003KGjhighest number of derivatives (\u000eC)1\n0= (7)\n= (\u00001)s4e\u00002\u001a@(\u0016@\u0003(s)@(s\u00001)C1\n0) (8)\n= (\u00001)s4e\u00002\u001a[@\u0016@\u0016\u0003(s)@(s\u00001)C1\n0+\u0016@\u0003(s)@sC1\n0\n+@\u0003(s)\u0016@@(s\u00001)C1\n0+ \u0003(s)\u0016@@sC1\n0]: (9)The term in () that is of further interest is the one mul-\ntiplying 4e\u00002\u001a@\u0016@acting on\u000eC1\n0which is convenient to\ncompute in the metric formulation.\nI. METRIC FORMULATION\nIn the metric formulation we can express the higher\nspin \feld of arbitrary spin swith\n\u001e\u00161:::::\u0016 s=tr\u0010\n~e(\u00161:::~e\u0016s\u00001~E\u0016s)\u0011\n(10)\nwhere ~E\u0016s=~A\u0016\u0000~\u0016A\u0016and ~A\u0016and~\u0016A\u0016we de\fne below.\nThe dreibein is determined from the background AdS\nmetric ()\nez=1\n2e\u001a(L1+L\u00001) =1\n2e\u001a(V2\n1+V2\n\u00001) (11)\ne\u0016z=1\n2e\u001a(L1\u0000L\u00001) =1\n2e\u001a(V2\n1\u0000V2\n\u00001) (12)\ne\u001a=L0=V2\n0: (13)\nThe invariance of the equation () under the gauge trans-\nformation for hs[\u0015]\bhs[\u0015] for the \felds A means\nA!A+d\u0003 + [A;\u0003]?\u0011~A (14)\n\u0016A!\u0016A+d\u0016\u0003 +\u0002\u0016A;\u0016\u0003\u0003\n?\u0011~\u0016A: (15)\nSince \u0003 parameter is chiral it means \u0016\u0003 = 0 and the \feld\n~\u0016Ais essentially unchanged. The \feld ~A\u0016is then\n~A=AAdS+d\u0003 + [AAdS;\u0003]?: (16)\nd\u0003 reads\nd\u0003 =2s\u00001X\nn=11\n(n\u00001)!Vs\ns\u0000ne(s\u0000n)\u001a[(\u0000@)n\u00001@\u0003(s)(z;\u0016z)dz(17)\n+ (\u0000@)n\u00001\u0016@\u0003(s)(z;\u0016z)d\u0016z+ (\u0000@)n\u00001\u0003(s)(z;\u0016z)(s\u0000n)d\u001a]\n(18)\nand\n[AAdS;\u0003]?= [e\u001aV2\n1dz+V2\n0d\u001a;\n2s\u00001X\nn=11\n(n\u00001)!(\u0000@)n\u00001\u0003(s)(z;\u0016z)e(s\u0000n)\u001aVs\ns\u0000n]\n(19)\nTo read out the coupling we focus on \u0016 z::::\u0016zcomponent of\nthe \feldC1\n0with lowest number of derivatives on gauge\n\feld \u0003(s). The?multiplication of the dreibeins in ()\nin that case contributes only with \frst gst\nu(m;n;\u0015) co-\ne\u000ecient with the each following dreibein that is being\nmultiplied. More explicitly\ne\u0016z?e\u0016z=1\n22e2\u001a\u0000\nV2\n1\u0000V2\n\u00001\u0001\n?(V2\n1\u0000V2\n\u00001) (20)\nFrom () we notice that the lowest number of derivatives\non \u0003 will appear for lowest n, i.e. for n= 1 in summation\n(). Knowing the relation for the trace of higher spin\ngenerators, the required generator Vs\ns\u0000nwill than": "\\documentclass[prd,superscriptaddress,twocolumn,10pt]{revtex4}\n\\usepackage{amsmath,amssymb}\n\\usepackage{verbatim}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{color} \n\\DeclareFontFamily{OT1}{rsfs}{}\n\\DeclareFontShape{OT1}{rsfs}{m}{n}{ <-7> rsfs5 <7-10> rsfs7 <10->rsfs10}{} \n\\DeclareMathAlphabet{\\mycal}{OT1}{rsfs}{m}{n} \n\\begin{document}\nfor $C^s_{-(s-n)}$ an arbitrary component of the master field $C$.\nTaking $m\\rightarrow -m$ in the set of equations (), (20), () and $s=m+1$ in (), we can iteratively determine the dependence of the $C^{m+1}_m$ on the $C_0^1$. From the equation () we obtain\n\\begin{equation}\n\\partial_z C_{-m}^{m+1}+\\frac{e^{\\rho}}{2}g_3^{2(m+2)}(1,-m-1)C^{m+2}_{-m-1}=0\n\\end{equation}\ntaking into consideration that for certain components $C_n^s$ it is required $|n|\\leq s-1$ this iteratively leads to relation of $C_m^{m+1}$ and $C_0^1$, and from  \nthe () analogously for $C^{m+1}_{-m}$ and $C_0^1$.\nThe general form of the $C^s_{\\pm}$ is\nthen given in terms of $C^{m+1}_{\\pm m}$ and coefficients $g_u^{ts}(m,n)$.\nKnowing $C^s_{\\pm}$ and $C^{m+1}_{\\pm m}$ allows to obtain  \n\\begin{align} \n(\\delta C)^1_0&=\\sum_{n=1}^sf_{\\pm}^{s,n}(\\lambda)\\partial_z^{n-1}\\Lambda^{(s)}\\partial_z^{s-n}\\phi \n\\end{align}\nfor $\\phi\\equiv C_0^1$ and $f^{s,n}_{\\pm}(\\lambda)$ expressed in terms  of coefficients $g_u^{st}(m,n)$\nUsing the replacement $\\partial_{\\rho}\\rightarrow-(1\\pm\\lambda)$ \nand \n writing explicitly first few n values for $f_{\\pm}^{s,n}(\\lambda)$, allows to determine its general expression \n\\begin{align}\nf_{\\pm}^{s,n}(\\lambda)&=(-1)^s\\frac{\\Gamma(s+\\lambda)}{\\Gamma(s-n+1\\pm\\lambda)}\\frac{1}{2^{n-1}(2(\\frac{n}{2}-1))!!\\left(\\frac{n-1}{2}\\right)!}\\nonumber\\\\\n&\\times\\prod_{j=1}^{\\frac{n-1}{2}}\\frac{s+1-n}{2s-2j-1} .\n\\end{align}\n\\noindent Substituting () in () one obtains the variation of the scalar field \n\\small\n\\begin{align}\n\\displaystyle{(\\delta C)_0^1}&\\displaystyle{=\\sum_{n=1}^{s}(-1)^s\\frac{\\Gamma(s\\pm\\lambda)}{\\Gamma(s-n+1\\pm\\lambda)}\\frac{1}{2^{n-1}\\left(2\\left(\\frac{n}{2}\\right)-1\\right)!!\\left(\\frac{n-1}{2}\\right)!} \\nonumber}\\\\ & \\times\\prod_{j=1}^{\\left( \\frac{n-1}{2} \\right)}\\frac{s+j-n}{2s-2j-1}\\partial_z^{n-1}\\Lambda^{(s)}\\partial_z^{s-n}C_0^1. \n\\end{align}\n\\normalsize\nTo consider the coefficient in front, we focus on the term with the lowest number of $\\partial_z$ derivatives on the gauge field $\\Lambda^{(s)}$,  obtained for n=1. Then, () becomes\n\\begin{equation}\n(\\delta C)_0^1|_{n=1}=(-1)^s\\Lambda^{(s)}\\partial^{s-1}C_0^1 .\n\\end{equation}\nTo obtain the linearised equation of motion for the scalar field we act on () with KG operator ().\nThis can be written as\n\\begin{equation}\n\\Box_{KG}\\tilde{C}^1_0=\\Box_{KG}C_0^1+\\Box_{KG}\\delta C_0^1 .\n\\end{equation}\nTaking   $\\partial_{\\rho}\\rightarrow(1\\pm\\lambda)$ in $f_{\\pm}^{s,n}(\\lambda)$ we have taken and considering the term with highest number of derivatives on $C_0^1$  leads to\n\\begin{align}\n&\\Box_{KG}|_{\\text{highest number of derivatives}}(\\delta C)_0^1=\\\\& = (-1)^s4e^{-2\\rho}\\partial(\\bar{\\partial}\\Lambda^{(s)}\\partial^{(s-1)}C_0^1)\\\\&=(-1)^s4e^{-2\\rho}[ \\partial\\bar{\\partial}\\bar{\\Lambda}^{(s)}\\partial^{(s-1)}C_0^1+\\bar{\\partial}\\Lambda^{(s)}\\partial^{s}C_0^1\\nonumber\\\\&+\\partial\\Lambda^{(s)}\\bar{\\partial}\\partial^{(s-1)}C_0^1+\\Lambda^{(s)}\\bar{\\partial}\\partial^sC_0^1 ].\n\\end{align}\nThe term in () that is of further interest is the one multiplying $4e^{-2\\rho}\\partial\\bar{\\partial}$ acting on $\\delta C_0^1$ which is convenient to compute in the metric formulation.\n\\section{Metric formulation}\nIn the metric formulation we can express the higher spin field of arbitrary spin $s$ with\n\\begin{equation}\n\\phi_{\\mu_1.....\\mu_s}=tr\\left( \\tilde{e}_{(\\mu_1}...\\tilde{e}_{\\mu_{s-1}}\\tilde{E}_{\\mu_s)} \\right)\n\\end{equation}\nwhere $\\tilde{E}_{\\mu s}=\\tilde{A}_{\\mu}-\\tilde{\\bar{A}}_{\\mu}$ and $\\tilde{A}_{\\mu}$ and $\\tilde{\\bar{A}}_{\\mu}$ we define below. \nThe  dreibein is determined from the background AdS metric ()\n\\begin{align}\ne_{z}&=\\frac{1}{2}e^{\\rho}(L_1+L_{-1})=\\frac{1}{2}e^{\\rho}(V_1^2+V_{-1}^{2}) \\\\\ne_{\\bar{z}}&=\\frac{1}{2}e^{\\rho}(L_1-L_{-1})=\\frac{1}{2}e^{\\rho}(V_1^2-V_{-1}^2) \\\\\ne_{\\rho}&=L_0=V_0^2.\n\\end{align}\nThe invariance of the equation () under the gauge transformation for $hs[\\lambda]\\oplus hs[\\lambda]$ for the fields A means \n\\begin{align}\nA&\\rightarrow A+d \\Lambda +\\left[A,\\Lambda \\right]_{\\star}\\equiv \\tilde{A}\\\\\n\\bar{A}&\\rightarrow \\bar{A}+d \\bar{\\Lambda} +\\left[\\bar{A},\\bar{\\Lambda} \\right]_{\\star}\\equiv \\tilde{\\bar{A}}.\n\\end{align}\nSince $\\Lambda$ parameter is chiral it means $\\bar{\\Lambda}=0$ and the field $\\tilde{\\bar{A}}$ is essentially unchanged. The field $\\tilde{A}_{\\mu}$ is then  \n\\begin{equation}\n\\tilde{A}=A_{AdS}+d\\Lambda+\\left[A_{AdS},\\Lambda\\right]_{\\star}.\n\\end{equation}\n $d\\Lambda$  reads\n\\small\n\\begin{align}\nd\\Lambda&=\\sum_{n=1}^{2s-1}\\frac{1}{(n-1)!}V_{s-n}^se^{(s-n)\\rho}[ (-\\partial)^{n-1}\\partial\\Lambda^{(s)}(z,\\bar{z}) dz\\\\&+(-\\partial)^{n-1}\\bar{\\partial}\\Lambda^{(s)}(z,\\bar{z})d\\bar{z}+(-\\partial)^{n-1}\\Lambda^{(s)}(z,\\bar{z})(s-n)d\\rho ] \n\\end{align}\n\\normalsize\nand \n\\begin{align}\n\\left[A_{AdS},\\Lambda\\right]_{\\star}&= [ e^{\\rho}V_1^2dz+V_0^2d\\rho,\\nonumber\\\\ &\\sum_{n=1}^{2s-1}\\frac{1}{(n-1)!}(-\\partial)^{n-1}\\Lambda^{(s)}(z,\\bar{z})e^{(s-n)\\rho}V^s_{s-n} ]\n\\end{align}\nTo read out the coupling we focus on $\\bar{z}....\\bar{z}$ component of the field $C_0^1$ with lowest number of derivatives on gauge field $\\Lambda^{(s)}$. The $\\star$ multiplication of the dreibeins in () in that case contributes only with first $g_{u}^{st}(m,n;\\lambda)$  coefficient with the each following dreibein that is being multiplied. More explicitly \n\\begin{align}\ne_{\\bar{z}}\\star e_{\\bar{z}}&=\\frac{1}{2^2}e^{2\\rho}\\left(V_1^2-V_{-1}^2\\right)\\star(V_{1}^2-V_{-1}^2)\n\\end{align}\nFrom () we notice that the lowest number of derivatives on $\\Lambda$ will appear for lowest n, i.e.  for $n=1$ in summation (). Knowing the relation for the trace of higher spin generators, the required generator $V^s_{s-n}$ will than\n\\end{document}\n"}
{"1\nbe of the form Vs\ns\u00001, as we see below, which means that\nmultiplication of HS generators we have to consider is\nV2\n\u00001?V2\n\u00001?::::?V2\n\u00001: (1)\nThen\nV2\n\u00001?V2\n\u00001=1\n2(g22\n1(\u00001;\u00001)V3\n\u00002+g22\n2(\u00001;\u00001)V2\n\u00002\n+g22\n3(\u00001;\u00001)V1\n\u00002) (2)\nwhere theg22\n2(\u00001;\u00001) =g22\n3(\u00001;\u00001) = 0. Multiplying\nwith following V2\n\u00001, etc. on e can conclude\nV2\n\u00001?V2\n\u00001?::::?V2\n\u00001| {z }\ns\u00001=1\n2s\u00001g2(s\u00001)\n1 (\u00001;\u0000(s\u00002))Vs\n\u0000(s\u00001)\n(3)\nwhile\ng2(s\u00001)\n2 (\u00001;\u0000(s\u00002)) =g2(s\u00001)\n3 (\u00001;\u0000(s\u00002)) = 0:(4)\nThat means we have found the contribution to the \u0016 z:::\u0016z\ncomponent multiplied with lowest derivative on \u0003(s)due\nto de\fnition of trace for generators Vs\nn\ntr\u0000\nVs\nmVt\nn\u0001\n=Ns(\u00001)s\u0000m\u00001\n(2s\u00002)!\u0000(s+m)\u0000(s\u0000m)\u000est\u000em;\u0000n:(5)\nfor\nNs\u00113\u00014s\u00003p\u0019q2s\u00004\u0000(s)\n(\u00152\u00001)\u0000(s+1\n2)(1\u0000\u0015)s\u00001(1 +\u0015)s\u00001(6)\nand (a)n=\u0000(a+n)\n\u0000(a)ascending Pochhammer symbol. The\noverall constant is set to\ntr(V2\n1V2\n\u00001) =\u00001: (7)\nLet us go back to \u001e\u0016z::::\u0016zcomponent. The star product\ne\u0016z?:::?e \u0016zwill contribute with1\n2s\u00001e(s\u00001)\u001aVs\n\u0000(s\u00001)if we\nconsider as explained above the lowest derivative on \u0003(s).\nWe can denote this as\ne\u0016z?::::?e \u0016z(V2\n\u00001?::::?V2\n\u00001) =1\n2s\u00001e(s\u00001)\u001aVs\n\u0000(s\u00001):(8)\nThe ~E\u0016zs=~A\u0016zs\u0000~\u0016A\u0016zsneeds to be able to satisfy the\nconditions of the trace () in star multiplication with e\u0016z?\n:::?e \u0016z, the only HS generator that contributes is Vs\ns\u00001\ngenerator. When we gauge the \feld A\u0016\u0016s,d\u0016zcomponent\nappears in d\u0003 whileAAdSand [AAdS;\u0003]?do not have\nd\u0016zcomponent. The~\u0016A\u0016zshasd\u0016zcomponent that comes\nfrom \u0016AAdSpart and it is e\u001aV2\n\u00001d\u0016z. This however will\nnot appear with the right number of derivatives on \u0003.\nSince we have chosen \u0003 to be chiral and \u0016\u0003 = 0, that was\nthe only contribution from~\u0016A\u0016z. Altogether, we can write\n\u001e\u0016z:::\u0016zcomponent for the \u0016@\u0003(s)derivative as\n\u001e\u0016z::::\u0016zj\u0016@\u0003(s)=tr\u00141\n2s\u00001e(s\u00001)Vs\n\u0000(s\u00001)?Vs\ns\u00001e(s\u00001)\u001a\u0016@\u0003(s)(z;\u0016z)\u0015\n(9)\n=1\n2s\u00001e2(s\u00001)\u001a\u0016@\u0003(s)Ns: (10)Inserting the normalisation Nswe obtain\n\u001e\u0016z:::\u0016zj\u0016@\u0003(s)=1\n2s\u00001e2(s\u00001)\u001a\u0016@\u0003(s)\n\u00023\u00014p\u001944\u00002s\u0000(s)\u0000(s+\u0015)\u0000(s\u0000\u0015)\n(\u00152\u00001)\u0000(s+1\n2)\u0000(1\u0000\u0015)\u0000(1 +\u0015):(11)\nThe expression \u001e\u0016z:::\u0016zwe want to compare with expres-\nsion () for highest derivative on C1\n0and \u0016@\u0003(s). In the\ncomputation of the vertex this would be a term\n\u001ez:::z\u001erz:::rz\u001e (12)\nfor\u001ez:::zhigher spin \feld with sindices and \u001escalar \feld.\nRaising indices contributes with a factor 2se\u00002s\u001a, so that\nthe \feld\u001ez:::zbecomes\n\u001ez:::z=1\n2e\u00002\u001a\u0016@\u0003(s)3\u000144\u00002s\n\u0002\u0000(s)\u0000(s+\u0015)\u0000(s\u0000\u0015)\n(\u00152\u00001)\u0000(s+1\n2)\u0000(1\u0000\u0015)\u0000(1 +\u0015): (13)\nWhen we take the ratio with\n\u0003KGjhighest number of derivatives( \u000eC1\n0)j\u0016@\u0003=\n(\u00001)s4e\u0000s\u001a\u0016@\u0003(s)@sC1\n0we get (schematically written)\n\u001ez:::zj\u0016@\u0003(s)\n\u0003KGjhighest number of derivatives( \u000eC)1\n0j\u0016@\u0003(s)\n= (\u00001)s1\n23p\u001944\u00002s\u0000(s)\u0000(s+\u0015)\u0000(s\u0000\u0015)\n(\u00152\u00001)\u0000(s+1\n2)\u0000(1\u0000\u0015)\u0000(1 +\u0015):(14)\nwhich taking into account the normalisation gives the\ncoupling for the 00s three point function.\nI. CONCLUSION AND OUTLINE\nWe have considered the three-point coupling using\nmetric-like formation to express the higher spin \feld and\nusing the linearised Vasiliev's equations of motion. The\nobtained result can also be veri\fed using the alterna-\ntive methods, for example following the procedure by .\nThe generalisation of the result to higher point functions\nwould be non-trivial since in order to compute higher or-\nder vertices, one would have to consider perturbations\naround the background AdS \feld with higher spin \felds\nup to that required higher order.": "\\documentclass[prd,superscriptaddress,twocolumn,10pt]{revtex4}\n\\usepackage{amsmath,amssymb}\n\\usepackage{verbatim}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{color} \n\\DeclareFontFamily{OT1}{rsfs}{}\n\\DeclareFontShape{OT1}{rsfs}{m}{n}{ <-7> rsfs5 <7-10> rsfs7 <10->rsfs10}{} \n\\DeclareMathAlphabet{\\mycal}{OT1}{rsfs}{m}{n} \n\\begin{document}\n be of the form $V^s_{s-1}$, as we see below, which means that multiplication of HS generators we have to  consider is \n\\begin{equation}\nV_{-1}^2\\star V_{-1}^2\\star....\\star V_{-1}^2.\n\\end{equation}\nThen \n\\begin{align}\nV_{-1}^2\\star V_{-1}^2 &=\\frac{1}{2}( g_1^{22}(-1,-1)V_{-2}^3+g_{2}^{22}(-1,-1)V_{-2}^2\\nonumber\\\\&+g_{3}^{22}(-1,-1)V_{-2}^1 )\n\\end{align}\nwhere the $g_2^{22}(-1,-1)=g_3^{22}(-1,-1)=0$. Multiplying with following $V_{-1}^2$, etc. on e can conclude\n\\begin{equation}\n\\underbrace{V_{-1}^{2}\\star V_{-1}^2\\star ....\\star V_{-1}^2}_{s-1}=\\frac{1}{2^{s-1}}g_1^{2(s-1)}(-1,-(s-2))V_{-(s-1)}^s\n\\end{equation}\nwhile \n\\begin{equation}\ng_{2}^{2(s-1)}(-1,-(s-2))=g_{3}^{2(s-1)}(-1,-(s-2))=0.\n\\end{equation}\nThat  means we have found the contribution to the $\\bar{z}...\\bar{z}$ component multiplied with lowest derivative on $\\Lambda^{(s)}$ due to definition of trace for generators $V_n^s$ \n\\small\n\\begin{equation}\ntr\\left( V_m^s V_n^t \\right)=N_s\\frac{(-1)^{s-m-1}}{(2s-2)!}\\Gamma(s+m)\\Gamma(s-m)\\delta^{st}\\delta_{m,-n}. \n\\end{equation}\n\\normalsize\nfor \n\\begin{equation}\nN_s\\equiv \\frac{3\\cdot 4^{s-3}\\sqrt{\\pi}q^{2s-4}\\Gamma(s)}{(\\lambda^2-1)\\Gamma(s+\\frac{1}{2})}(1-\\lambda)_{s-1}(1+\\lambda)_{s-1}\n\\end{equation}\nand $(a)_n=\\frac{\\Gamma(a+n)}{\\Gamma(a)}$ ascending Pochhammer symbol. The overall constant is set to \\begin{equation}tr(V_1^2V_{-1}^2)=-1.\n\\end{equation}\nLet us go back to $\\phi_{\\bar{z}....\\bar{z}}$ component. The star product $e_{\\bar{z}}\\star...\\star e_{\\bar{z}}$ will contribute with $\\frac{1}{2^{s-1}}e^{(s-1)\\rho}V^{s}_{-(s-1)}$ if we consider as explained above the lowest derivative on $\\Lambda^{(s)}$. We can denote this as \n\\begin{align}\ne_{\\bar{z}}\\star....\\star e_{\\bar{z}}(V_{-1}^2\\star ....\\star V_{-1}^2)=\\frac{1}{2^{s-1}}e^{(s-1)\\rho}V^{s}_{-(s-1)}.\\end{align} The $\\tilde{E}_{\\bar{z}_s}=\\tilde{A}_{\\bar{z}_s}-\\tilde{\\bar{A}}_{\\bar{z}_s}$ needs to be able to satisfy the conditions of the trace () in star multiplication with $e_{\\bar{z}}\\star...\\star e_{\\bar{z}}$, the only HS generator that  contributes is $V^s_{s-1}$ generator. When we gauge the field $A_{\\bar{\\mu}_s}$, $d\\bar{z}$ component appears in $d\\Lambda$ while $A_{AdS}$ and $[A_{AdS},\\Lambda]_{\\star}$ do not have $d\\bar{z}$ component. The $\\tilde{\\bar{A}}_{\\bar{z}_s}$ has $d\\bar{z}$ component that comes from $\\bar{A}_{AdS}$ part and it is $e^{\\rho}V_{-1}^2d\\bar{z}$. This however will not appear with the right number of derivatives on $\\Lambda$. Since we have chosen $\\Lambda$ to be chiral and $\\bar{\\Lambda}=0$, that was the only contribution from $\\tilde{\\bar{A}}_{\\bar{z}}$. \nAltogether, we can write $\\phi_{\\bar{z}...\\bar{z}}$ component for the $\\bar{\\partial}\\Lambda^{(s)}$ derivative as\n\\small\n\\begin{align}\n\\phi_{\\bar{z}....\\bar{z}}|_{\\bar{\\partial}\\Lambda^{(s)}}&=tr\\left[\\frac{1}{2^{s-1}}e^{(s-1)}V^s_{-(s-1)}\\star V^s_{s-1}e^{(s-1)\\rho}\\bar{\\partial}\\Lambda^{(s)}(z,\\bar{z})\\right]\\\\\n&=\\frac{1}{2^{s-1}}e^{2(s-1)\\rho}\\bar{\\partial}\\Lambda^{(s)} N_s.\n\\end{align}\n\\normalsize\nInserting the normalisation $N_s$ we obtain\n\\begin{align}\\nonumber\n\\phi_{\\bar{z}...\\bar{z}}|_{\\bar{\\partial}\\Lambda^{(s)}}&=\\frac{1}{2^{s-1}}e^{2(s-1)\\rho}\\bar{\\partial}\\Lambda^{(s)}\\\\&\\times\n\\frac{3\\cdot 4 \\sqrt{\\pi}4^{4-2s}\\Gamma(s)\\Gamma(s+\\lambda)\\Gamma(s-\\lambda)}{(\\lambda^2-1)\\Gamma(s+\\frac{1}{2})\\Gamma(1-\\lambda)\\Gamma(1+\\lambda)}.\n\\end{align}\nThe expression $\\phi_{\\bar{z}...\\bar{z}}$ we want to compare with expression () for highest derivative on $C_0^1$ and $\\bar{\\partial}\\Lambda^{(s)}$. \nIn the computation of the vertex this would be a term \n\\begin{equation}\n\\phi^{z...z}\\mathcal{\\phi}\\nabla_z...\\nabla_{z}\\mathcal{\\phi}\n\\end{equation}\nfor $\\phi^{z...z}$ higher spin field with $s$ indices and $\\mathcal{\\phi}$ scalar field. Raising indices\n contributes with a factor $2^se^{-2s\\rho}$, so that the field $\\phi^{z...z}$ becomes\n\\begin{align}\n\\phi^{z...z}&=\\frac{1}{2}e^{-2\\rho}\\bar{\\partial}{\\Lambda}^{(s)}3\\cdot4^{4-2s}\\nonumber \\\\ &\\times\n\\frac{\\Gamma(s)\\Gamma(s+\\lambda)\\Gamma(s-\\lambda)}{(\\lambda^2-1)\\Gamma(s+\\frac{1}{2})\\Gamma(1-\\lambda)\\Gamma(1+\\lambda)}.\n\\end{align}\nWhen  we take the ratio with $\\Box_{KG}|_{\\text{highest number of derivatives}(\\delta C_0^1)|_{\\bar{\\partial}\\Lambda}}=(-1)^s4e^{-s\\rho}\\bar{\\partial}\\Lambda^{(s)}\\partial^sC_0^1$ we get (schematically written)\n\\begin{align}\n&\\frac{\\phi^{z...z}|_{\\bar{\\partial}\\Lambda^{(s)}}}{\\Box_{KG}|_{\\text{highest number of derivatives}(\\delta C)_0^1|_{\\bar{\\partial}\\Lambda^{(s)}}}}\\nonumber\\\\&=(-1)^s\\frac{1}{2}3\\sqrt{\\pi}\\frac{4^{4-2s}\\Gamma(s)\\Gamma(s+\\lambda)\\Gamma(s-\\lambda)}{(\\lambda^2-1)\\Gamma(s+\\frac{1}{2})\\Gamma(1-\\lambda)\\Gamma(1+\\lambda)}.\n\\end{align}\nwhich taking into account the normalisation gives the coupling for the 00s three point function.\n\\section{Conclusion and Outline}\nWe have considered the three-point coupling using metric-like formation to express the higher spin field and using the linearised Vasiliev's equations of motion.  The obtained result can also be verified using the alternative methods, for example following the procedure by . \nThe generalisation of the result to higher point functions would be non-trivial since in order to compute higher order vertices, one would have to consider perturbations around the background AdS field with higher spin fields up to that required higher order. \n\\end{document}\n"}
{"Let \u0001\u001c\n1=E[Y1(1;M(1))\u0000Y1(0;M(0))j\u001c] denote the ATE conditional on \u001c2\nfa;c;de;ng;\u0012\u001c\n1(d) and\u000e\u001c\n1(d) denote the corresponding direct and indirect e\u000bects.\nBecauseM(1) =M(0) = 0 for any never-taker, the indirect e\u000bect for this group is\nby de\fnition zero ( \u000en\n1(d) =E[Y1(d;0)\u0000Y1(d;0)j\u001c=n] = 0) and \u0001n\n1=E[Y1(1;0)\u0000\nY1(0;0)j\u001c=n] =\u0012n\n1(1) =\u0012n\n1(0) =\u0012n\n1equals the direct e\u000bect for never-takers. Corre-\nspondingly, because M(1) =M(0) = 1 for any always-taker, the indirect e\u000bect\nfor this group is by de\fnition zero ( \u000ea\n1(d) =E[Y1(d;1)\u0000Y1(d;1)j\u001c=a] = 0)\nand \u0001a\n1=E[Y1(1;1)\u0000Y1(0;1)j\u001c=a] =\u0012a\n1(1) =\u0012a\n1(0) =\u0012a\n1equals the di-\nrect e\u000bect for always-takers. For the compliers, both direct and indirect e\u000bects\nmay exist. Note that M(d) =ddue to the de\fnition of compliers. Accord-\ningly,\u0012c\n1(d) =E[Y1(1;d)\u0000Y1(0;d)j\u001c=c] equals the direct e\u000bect for compliers,\n\u000ec\n1(d) =E[Y1(d;1)\u0000Y1(d;0)j\u001c=c] equals the indirect e\u000bect for compliers, and\n\u0001c\n1=E[Y1(1;1)\u0000Y1(0;0)j\u001c=c] equals the total e\u000bect for compliers. In the ab-\nsence of any direct e\u000bect, the indirect e\u000bects on the compliers are homogeneous,\n\u000ec\n1(1) =\u000ec\n1(0) =\u000ec\n1, and correspond to the local average treatment e\u000bect. Analogous\nresults hold for the de\fers. As already mentioned, we will also consider direct e\u000bects\nconditional on speci\fc values D=dand mediator states M=M(d) =m, which are\ndenoted by \u0012d;m\n1(d) =E[Y1(1;m)\u0000Y1(0;m)jD=d;M(d) =m]. These parameters\nare identi\fed under weaker assumptions than strata-speci\fc e\u000bects, but are also less\nstraightforward to interpret, as they refer to mixtures of two strata. For instance,\n\u00121;0\n1(1) =E[Y1(1;0)\u0000Y1(0;0)jD= 1;M(1) = 0] is the e\u000bect on a mixture of never-\ntakers and de\fers, as these two groups satisfy M(1) = 0. Likewise, \u00120;0\n1(0) refers to\nnever-takers and compliers satisfying M(0) = 0,\u00120;1\n1(0) to always-takers and de\fers\nsatisfyingM(0) = 1, and \u00121;1\n1(1) to always-takers and compliers satisfying M(1) = 1.\n0.1 Quantile e\u000bects\nWe denote by FYt(d;m )(y) = Pr(Yt(d;m)\u0014y) the cumulative distribution function\nofYt(d;m) at outcome level y. Its inverse, F\u00001\nYt(d;m )(q) = inffy:FYt(d;m )(y)\u0015qg, is\nthe quantile function of Yt(d;m) at rankq. The total QTE are denoted by\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nLet $\\Delta_1^{\\tau} = E[Y_1(1,M(1))-Y_1(0,M(0))|\\tau]$ denote the ATE conditional on $\\tau \\in \\{a,c,de,n\\}$; $\\theta_1^{\\tau}(d)$ and $\\delta_1^{\\tau}(d)$ denote the corresponding direct and indirect effects. Because $M(1)=M(0)=0$ for any never-taker, the indirect effect for this group is by definition zero $(\\delta_1^{n}(d)=E[Y_1(d,0) -Y_1(d,0)|\\tau=n]=0)$ and $\\Delta_1^{n} = E[Y_1(1,0)-Y_1(0,0)|\\tau=n]=\\theta_1^{n}(1)=\\theta_1^{n}(0)=\\theta_1^{n}$ equals the direct effect for never-takers. Correspondingly, because $M(1)=M(0)=1$ for any always-taker, the indirect effect for this group is by definition zero $(\\delta_1^{a}(d)=E[Y_1(d,1) -Y_1(d,1)|\\tau=a]=0)$ and $\\Delta_1^{a} = E[Y_1(1,1)-Y_1(0,1)|\\tau=a]=\\theta_1^{a}(1)=\\theta_1^{a}(0)=\\theta_1^{a}$ equals the direct effect for always-takers. For the compliers, both direct and indirect effects may exist. Note that $M(d)=d$ due to the definition of compliers. Accordingly, $\\theta_1^{c}(d) = E[Y_1(1,d)-Y_1(0,d)|\\tau=c]$ equals the direct effect for compliers, $\\delta_1^{c}(d)= E[Y_1(d,1) -Y_1(d,0)|\\tau=c]$ equals the indirect effect for compliers, and $\\Delta_1^{c}= E[Y_1(1,1) -Y_1(0,0)|\\tau=c]$ equals the total effect for compliers. In the absence of any direct effect, the indirect effects on the compliers are homogeneous, $\\delta_1^{c}(1)=\\delta_1^{c}(0)=\\delta_1^{c}$, and correspond to the local average treatment effect. Analogous results hold for the defiers.\nAs already mentioned, we will also consider direct effects conditional on specific values $D=d$ and mediator states $M=M(d)=m$, which are denoted by $\\theta_1^{d,m}(d)=E[Y_1(1,m)-Y_1(0,m)|D=d,M(d)=m]$. These parameters are identified under weaker assumptions than strata-specific effects, but are also less straightforward to interpret, as they refer to mixtures of two strata. For instance, $\\theta_1^{1,0}(1)=E[Y_1(1,0)-Y_1(0,0)|D=1,M(1)=0]$ is the effect on a mixture of  never-takers and defiers, as these two groups satisfy $M(1)=0$. Likewise, $\\theta_1^{0,0}(0)$ refers to never-takers and compliers satisfying $M(0)=0$,  $\\theta_1^{0,1}(0)$ to always-takers and defiers satisfying $M(0)=1$, and $\\theta_1^{1,1}(1)$ to always-takers and compliers satisfying $M(1)=1$.\n\\subsection{Quantile effects}\nWe denote by $F_{Y_{t}(d,m)}(y) = \\Pr(Y_t(d,m) \\leq y)$ the cumulative distribution function of $Y_t(d,m)$ at outcome level $y$. Its inverse, $F_{Y_{t}(d,m)}^{-1}(q) = \\inf \\{y : F_{Y_t(d,m)}(y) \\geq q \\}$, is the quantile function of $Y_t(d,m)$ at rank $q$. The total QTE are denoted by \n\\end{document}\n"}
{"\u00011(q) =F\u00001\nY1(1;M(1))(q)\u0000F\u00001\nY1(0;M(0))(q). The QTE can be disentangled into the\ndirect quantile e\u000bects, denoted by \u00121(q;d) =F\u00001\nY1(1;M(d))(q)\u0000F\u00001\nY1(0;M(d))(q), and\nthe indirect quantile e\u000bects, denoted by \u000e1(q;d) =F\u00001\nY1(d;M (1))(q)\u0000F\u00001\nY1(d;M (0))(q).\nThe conditional distribution function in stratum \u001cisFYt(d;m)j\u001c(y) = Pr(Yt(d;m)\u0014\nyj\u001c) and the corresponding conditional quantile function is F\u00001\nYt(d;m)j\u001c(q) = inffy:\nFYt(d;m)j\u001c(y)\u0015qgfor\u001c2 fa;c;d;ng. Using the previously described strati\fca-\ntion framework, we de\fne the QTE conditional on \u001c2 fa;c;de;ng: \u0001\u001c\n1(q) =\nF\u00001\nY1(1;M(1))j\u001c(q)\u0000F\u00001\nY1(0;M(0))j\u001c(q). The direct quantile treatment e\u000bect among never-\ntakers equals \u0001n\n1(q) =F\u00001\nY1(1;0)jn(q)\u0000F\u00001\nY1(0;0)jn(q) =\u0012n\n1(q). The direct quantile e\u000bect\namong always-takers equals \u0001a\n1(q) =F\u00001\nY1(1;1)ja(q)\u0000F\u00001\nY1(0;1)ja(q) =\u0012a\n1(q). The total\nQTE among compliers equals \u0001c\n1(q) =F\u00001\nY1(1;1)jc(q)\u0000F\u00001\nY1(0;0)jc(q), the direct quantile\ne\u000bect among compliers equals \u0012c\n1(q;d) =F\u00001\nY1(1;d)jc(q)\u0000F\u00001\nY1(0;d)jc(q), and the indirect\nquantile e\u000bect among compliers equals \u000ec\n1(q;d) =F\u00001\nY1(d;1)jc(q)\u0000F\u00001\nY1(d;0)jc(q). Finally,\nwe de\fne the direct quantile treatment e\u000bects conditional on speci\fc values D=d\nand mediator states M=M(d) =m,\n\u0012d;m\n1(q;1) =F\u00001\nY1(1;m)jD=d;M (1)=m(q)\u0000F\u00001\nY1(0;m)jD=d;M (1)=m(q) and\n\u0012d;m\n1(q;0) =F\u00001\nY1(1;m)jD=d;M (0)=m(q)\u0000F\u00001\nY1(0;m)jD=d;M (0)=m(q);\nwith the quantile function F\u00001\nYt(d;m)jD=d;M (d)=m(q) = inffy:FYt(d;m)jD=d;M (d)=m(y)\u0015\nqgand the distribution function FYt(d;m)jD=d;M (d)=m(y) = Pr(Yt(d;m)\u0014yjD=\nd;M(d) =m).\n0.1 Observed distribution and quantile transformations\nWe subsequently de\fne various functions of the observed data required for the iden-\nti\fcation results. The conditional distribution function of the observed outcome Yt\nconditional on treatment value dand mediator state m, is given by FYtjD=d;M =m(y) =\nPr(Yt\u0014yjD=d;M =m) ford;m2f0;1g. The corresponding conditional quantile\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n$\\Delta_1(q) = F_{Y_1(1,M(1))}^{-1}(q) -F_{Y_1(0,M(0))}^{-1}(q)$.\nThe QTE can be disentangled into the direct quantile effects, denoted by $\\theta_1(q,d) = F_{Y_1(1,M(d))}^{-1}(q) -F_{Y_1(0,M(d))}^{-1}(q)$, and the indirect quantile effects, denoted by $\\delta_1(q,d) = F_{Y_1(d,M(1))}^{-1}(q) -F_{Y_1(d,M(0))}^{-1}(q)$.\nThe conditional distribution function in stratum $\\tau$ is $F_{Y_{t}(d,m)|\\tau}(y) = \\Pr(Y_t(d,m) \\leq y |\\tau)$ and the corresponding conditional quantile function is $F_{Y_t(d,m)|\\tau}^{-1}(q) = \\inf \\{y : F_{Y_{t}(d,m)|\\tau}(y) \\geq q \\}$ for $\\tau \\in \\{a,c,d,n\\}$. Using the previously described stratification framework, we define the QTE conditional on $\\tau \\in \\{a,c,de,n\\}$: $\\Delta_1^{\\tau}(q) = F_{Y_1(1,M(1))|\\tau}^{-1}(q)-F_{Y_1(0,M(0))|\\tau}^{-1}(q)$. The direct quantile treatment effect among never-takers equals $\\Delta_1^{n} (q)= F_{Y_1(1,0)|n}^{-1}(q)-F_{Y_1(0,0)|n}^{-1}(q) =\\theta_1^{n}(q)$. The direct quantile effect among always-takers equals $\\Delta_1^{a} (q)= F_{Y_1(1,1)|a}^{-1}(q)-F_{Y_1(0,1)|a}^{-1}(q) =\\theta_1^{a}(q)$. The total QTE among compliers equals $\\Delta_1^{c}(q) = F_{Y_1(1,1)|c}^{-1}(q)-F_{Y_1(0,0)|c}^{-1}(q)$, the direct quantile effect among compliers equals $\\theta_1^{c}(q,d) = F_{Y_1(1,d)|c}^{-1}(q)-F_{Y_1(0,d)|c}^{-1}(q)$, and the indirect quantile effect among compliers equals $\\delta_1^{c}(q,d) = F_{Y_1(d,1)|c}^{-1}(q)-F_{Y_1(d,0)|c}^{-1}(q)$. Finally, we define the direct quantile treatment effects conditional on specific values $D=d$ and mediator states $M=M(d)=m$,\n\\begin{align*}\n\\theta_1^{d,m}(q,1)=F_{Y_1(1,m)|D=d,M(1)=m}^{-1}(q)-F_{Y_1(0,m)|D=d,M(1)=m}^{-1}(q) \\mbox{ and} \\\\\n\\theta_1^{d,m}(q,0)=F_{Y_1(1,m)|D=d,M(0)=m}^{-1}(q)-F_{Y_1(0,m)|D=d,M(0)=m}^{-1}(q),\n\\end{align*}\nwith the quantile function $F_{Y_t(d,m)|D=d,M(d)=m}^{-1}(q) = \\inf \\{y : F_{Y_{t}(d,m)|D=d,M(d)=m}(y) \\geq q \\}$ and the distribution function $F_{Y_{t}(d,m)|D=d,M(d)=m}(y) = \\Pr(Y_t(d,m) \\leq y |D=d,M(d)=m)$.\n\\subsection{Observed distribution and quantile transformations}\nWe subsequently define various functions of the observed data required for the identification results. The conditional distribution function of the observed outcome $Y_t$ conditional on treatment value $d$ and mediator state $m$, is given by $F_{Y_{t}|D=d,M=m}(y) = \\Pr(Y_t \\leq y |D=d,M=m)$ for $d,m \\in \\{0,1\\}$. The corresponding conditional quantile \n\\end{document}\n"}
{"function is F\u00001\nYtjD=d;M =m(q) = inffy:FYtjD=d;M =m(y)\u0015qg. Furthermore,\nQdm(y) :=F\u00001\nY1jD=d;M =m\u000eFY0jD=d;M =m(y) =F\u00001\nY1jD=d;M =m(FY0jD=d;M =m(y))\nis the quantile-quantile transform of the conditional outcome from period 0 to 1\ngiven treatment dand mediator status m. This transform maps yat rank qin\nperiod 0 ( q=FY0jD=d;M =m(y)) into the corresponding y0at rank qin period 1\n(y0=F\u00001\nY1jD=d;M =m(q)).\n1 Identi\fcation and Estimation\n1.1 Identi\fcation\nThis sections discusses the identifying assumptions along with the identi\fcation\nresults for the various direct and indirect e\u000bects. We note that our assumptions\ncould be adjusted to only hold conditional on a vector of observed covariates. In\nthis case, the identi\fcation results would hold within cells de\fned upon covari-\nate values. In our main discussion, however, covariates are not considered for the\nsake of ease of notation. For notational convenience, we maintain throughout that\nPr(T=t; D =d; M =m)>0 for t; d; m2 f1;0g, implying that all possible\ntreatment-mediator combinations exist in the population in both time periods. Our\n\frst assumption implies that potential outcomes are characterized by a continuous\nnonparametric function, denoted by h, that is strictly monotonic in a scalar Uthat\nre\nects unobserved heterogeneity.\nAssumption 1: Strict monotonicity of continuous potential outcomes in unob-\nserved heterogeneity.\nThe potential outcomes satisfy the following model: Yt(d; m) =h(d; m; t; U ), with\nthe general function hbeing continuous and strictly increasing in the scalar unob-\nservable U2Rfor all d; m; t2f0;1g.\nAssumption 1 requires the potential outcomes to be continuous implying that there\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nfunction is $F_{Y_{t}|D=d,M=m}^{-1}(q) = \\inf \\{y : F_{Y_t|D=d,M=m}(y) \\geq q \\}$. Furthermore,\n\\begin{equation*}\nQ_{dm}(y) := F_{Y_{1}|D=d,M=m}^{-1} \\circ F_{Y_{0}|D=d,M=m}(y) = F_{Y_{1}|D=d,M=m}^{-1}(F_{Y_{0}|D=d,M=m}(y))\n\\end{equation*}\nis the quantile-quantile transform of the conditional outcome from period 0 to 1 given treatment $d$ and mediator status $m$. This transform maps $y$ at rank $q$ in period 0 ($q = F_{Y_{0}|D=d,M=m}(y)$) into the corresponding $y'$ at rank $q$ in period 1 ($y'= F_{Y_{1}|D=d,M=m}^{-1}(q)$).\n\\section{Identification and Estimation}\n\\subsection{Identification}\nThis sections discusses the identifying assumptions along with the identification results for the various direct and indirect effects. We note that our assumptions could be adjusted to only hold conditional on a vector of observed covariates. In this case, the identification results would hold within cells defined upon covariate values. In our main discussion, however, covariates are not considered for the sake of ease of notation. For notational convenience, we maintain throughout that $\\Pr(T=t, D=d, M=m)>0$ for $t,d,m$ $\\in\\{1,0\\}$, implying that all possible treatment-mediator combinations exist in the population in both time periods. Our first assumption implies that potential outcomes are characterized by a continuous nonparametric function, denoted by $h$, that is strictly monotonic in a scalar $U$ that reflects unobserved heterogeneity.\\vspace{5 pt}\\\\\n\\textbf{Assumption 1:} Strict monotonicity of continuous potential outcomes in unobserved heterogeneity.\\\\\nThe potential outcomes satisfy the following model: $Y_t(d,m)= h(d,m, t, U)$, with the general function $h$ being continuous and strictly increasing in the scalar unobservable $U \\in \\mathbb{R}$ for all $d,m,t \\in \\{0,1\\}$.\\vspace{5 pt}\\\\\nAssumption 1 requires the potential outcomes to be continuous implying that there\n\\end{document}\n"}
{"is a one-to-one correspondence between a potential outcome's distribution and\nquantile functions, which is a condition for point identi\fcation. For discrete po-\ntential outcomes, only bounds on the e\u000bects could be identi\fed, in analogy to the\ndiscussion in for total (rather than direct and indirect) e\u000bects. Assumption 1 also\nimplies that individuals with identical unobserved characteristics Uhave the same\npotential outcomes Yt(d; m), while higher values of Ucorrespond to strictly higher\npotential outcomes Yt(d; m). Strict monotonicity is automatically satis\fed in addi-\ntively separable models, but Assumption 1 also allows for more \nexible non-additive\nstructures that arise in nonparametric models. The next assumption rules out an-\nticipation e\u000bects of the treatment or the mediator on the outcome in the baseline\nperiod. This assumption is plausible if assignment to the treatment or the mediator\ncannot be foreseen in the baseline period, such that behavioral changes a\u000becting the\npre-treatment outcome are ruled out.\nAssumption 2: No anticipation e\u000bect of MandDin the baseline period.\nY0(d; m)\u0000Y0(d0; m0) = 0, for d; d0; m; m0f1;0g:\nSimilarly, and assume the assignment to the treatment group does not a\u000bect the\npotential outcomes as long as the treatment is not yet realized. Furthermore, we as-\nsume conditional independence between unobserved heterogeneity and time periods\ngiven the treatment and no mediation.\nAssumption 3: Conditional independence of UandTgiven D= 1; M= 0 or\nD= 0; M= 0.\n(a)UTjD= 1; M= 0,\n(b)UTjD= 0; M= 0.\nUnder Assumption 3a, the distribution of Uis allowed to vary across groups de-\n\fned upon treatment and mediator state, but not over time within the group\nwith D= 1; M= 0. Assumption 3b imposes the same restriction conditional on\nD= 0; M= 0. Assumption 3 thus imposes stationarity of Uwithin groups de\fned\nonDandM. This assumption is weaker than (and thus implied by) requiring that\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nis a one-to-one correspondence between a potential outcome's distribution and quantile functions, which is a condition for point identification. For discrete potential outcomes, only bounds on the effects could be identified, in analogy to the discussion in  for total (rather than direct and indirect) effects. Assumption 1 also implies that individuals with identical unobserved characteristics $U$ have the same potential outcomes $Y_t(d,m)$, while higher values of $U$ correspond to strictly higher potential outcomes $Y_t(d,m)$. Strict monotonicity is automatically satisfied in additively separable models, but Assumption 1 also allows for more flexible non-additive structures that arise in nonparametric models. \nThe next assumption rules out anticipation effects of the treatment or the mediator on the outcome in the baseline period. This assumption is plausible if assignment to the treatment or the mediator cannot be foreseen in the baseline period, such that behavioral changes affecting the pre-treatment outcome are ruled out.\\vspace{5 pt}\\\\\n\\textbf{Assumption 2:} No anticipation effect of $M$ and $D$ in the baseline period.\\\\\n$Y_0(d,m) - Y_0(d',m') = 0\\mbox{, for } d, d', m, m' \\{1,0\\}.$\n\\vspace{5 pt}\\\\\nSimilarly,  and  assume the assignment to the treatment group does not affect the potential outcomes as long as the treatment is not yet realized. %This is implied by Assumption 1 and 2.\nFurthermore, we assume conditional independence between unobserved heterogeneity and time periods given the treatment and no mediation. \\vspace{5 pt}\\\\\n\\textbf{Assumption 3:} Conditional independence of $U$ and  $T$ given $D=1,M=0$ or $D=0,M=0$.\\\\\n(a) $U T|D=1,M=0$,\\\\\n(b) $U T|D=0,M=0$.\\vspace{5 pt}\\\\\nUnder Assumption 3a, the distribution of $U$ is allowed to vary across groups defined upon treatment and mediator state, but not over time within the group with $D=1,M=0$. Assumption 3b imposes the same restriction conditional on $D=0,M=0$. Assumption 3 thus imposes stationarity of $U$ within groups defined on $D$ and $M$. This assumption is weaker than (and thus implied by) requiring that\n\\end{document}\n"}
{"Uis constant across Tfor each individual i. For example, Assumption 3 is sat-\nis\fed in the \fxed e\u000bect model U=\u0011+vt, with\u0011being a time-invariant individual-\nspeci\fc unobservable (\fxed e\u000bect) and vtan idiosyncratic time-varying unobservable\nwith the same distribution in both time periods. and impose time invariance condi-\ntional on the treatment status, UTjD=d, to identify the average treatment e\u000bect\non the treated, '1=E[Y1(1;M(1))\u0000Y1(0;M(0))jD= 1] or local average treatment\ne\u000bect,'1=E[Y1(1;M(1))\u0000Y1(0;M(0))j\u001c=c], respectively. We additionally con-\ndition on the mediator status to identify direct and indirect e\u000bects. For our next\nassumption, we introduce some further notation. Let FUjd;m(u)) = Pr(U\u0014ujD=\nd;M =m) be the conditional distribution of Uwith support Udm.\nAssumption 4: Common support given M= 0.\n(a)U10\u0012U00,\n(b)U00\u0012U10.\nAssumption 4a is a common support assumption, implying that any possible value\nofUin the population with D= 1;M= 0 is also contained in the population\nwithD= 0;M= 0. Assumption 4b imposes that any value of Uconditional on\nD= 0;M= 0 also exists conditional on D= 1;M= 0. Both assumptions together\nimply that the support of Uis the same in both populations, albeit the distribu-\ntions may generally di\u000ber. Assumptions 1 to 3 permit identifying direct e\u000bects on\nmixed populations of never-takers and de\fers as well as never-takers and compliers,\nrespectively, as formally stated in Theorem 1.\nTheorem 1: Under Assumptions 1{3,\n(a) and Assumption 4a, the average and quantile direct e\u000bects under d= 1 con-\nditional on D= 1 andM(1) = 0 are identi\fed:\n\u00121;0\n1(1) =E[Y1\u0000Q00(Y0)jD= 1;M= 0];\n\u00121;0\n1(q;1) =F\u00001\nY1jD=1;M=0(q)\u0000F\u00001\nQ00(Y0)jD=1;M=0(q):\n(b) and Assumption 4b, the average and quantile direct e\u000bects under d= 0\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n$U$ is constant across $T$ for each individual $i$.  For example, Assumption 3 is satisfied in the fixed effect model $U = \\eta + v_t$, with $\\eta$ being a time-invariant individual-specific unobservable (fixed effect) and $v_t$ an idiosyncratic time-varying unobservable with the same distribution in both time periods.\n and  impose time invariance conditional on the treatment status, $U T|D=d$, to identify the average treatment effect on the treated, $\\varphi_1=E[Y_1(1,M(1))-Y_1(0,M(0))|D=1]$ or local average treatment effect, $\\varphi_1=E[Y_1(1,M(1))-Y_1(0,M(0))|\\tau=c]$, respectively. We additionally condition on the mediator status to identify direct and indirect effects. %Furthermore, we can identify the ATE and indirect effects under additional assumptions about the assignment of $D$, which we discuss in Section .\nFor our next assumption, we introduce some further notation. Let $F_{U|d,m}(u) )= \\Pr(U \\leq u |D=d,M=m)$ be the conditional distribution of $U$ with support $\\mathbb{U}_{dm}$. \\\\ %\\vspace{5 pt}\\\\\n\\textbf{Assumption 4:} Common support given $M=0$.\\\\\n(a) $\\mathbb{U}_{10}\\subseteq \\mathbb{U}_{00}$,\\\\\n(b) $\\mathbb{U}_{00}\\subseteq \\mathbb{U}_{10}$.\\vspace{5 pt}\\\\\nAssumption 4a is a common support assumption, implying that any possible value of $U$ in the population with $D=1,M=0$ is also contained in the population with $D=0,M=0$. Assumption 4b imposes that any value of $U$ conditional on $D=0,M=0$ also exists conditional on $D=1,M=0$. Both assumptions together imply that the support of $U$ is the same in both populations, albeit the distributions may generally differ.\nAssumptions 1 to 3 permit identifying direct effects on mixed populations of never-takers and defiers as well as never-takers and compliers, respectively, as formally stated in Theorem 1.\\\\\n\\noindent \\textbf{Theorem 1:} Under Assumptions 1\u20133,\n\\begin{itemize}\n\\item[(a)] and Assumption 4a, the average and quantile direct effects under $d=1$ conditional on $D=1$ and $M(1)=0$ are identified:\n\\begin{align*}\n\\theta_1^{1,0}(1)= E[Y_1-Q_{00}(Y_0)|D=1,M=0], \\\\\n\\theta_1^{1,0}(q,1)= F_{Y_1|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_0)|D=1,M=0}^{-1}(q).\n\\end{align*}\n\\item[(b)] and Assumption 4b, the average and quantile direct effects under $d=0$\n\\end{itemize}\n\\end{document}\n"}
{"\u00120;0\n1(0) =E[Q10(Y0)\u0000Y1jD= 0;M= 0];\n\u00120;0\n1(q;0) =F\u00001\nQ10(Y0)jD=0;M=0(q)\u0000F\u00001\nY1jD=0;M=0(q):\n(a) and Assumption 6a, the average and quantile direct e\u000bects under d= 0 con-\nditional on D= 0 andM(0) = 1 are identi\fed:\n\u00120;1\n1(0) =E[Y1\u0000Q11(Y0)jD= 0;M= 1];\n\u00120;1\n1(q;0) =F\u00001\nY1jD=0;M=1(q)\u0000F\u00001\nQ11(Y0)jD=0;M=1(q):\n(b) and Assumption 6b, the average and quantile direct e\u000bects under d= 1 is\nidenti\fed conditional on D= 1 andM(1) = 1 are identi\fed:\n\u00121;1\n1(1) =E[Q01(Y0)\u0000Y1jD= 1;M= 1];\n\u00121;1\n1(q;1) =F\u00001\nQ01(Y0)jD=1;M=1(q)\u0000F\u00001\nY1jD=1;M=1(q):\nProof. See Appendix . In the instrumental variable framework, any direct e\u000bects\nof the instrument are typically ruled out by imposing the exclusion restriction, in\norder to identify the causal e\u000bect of an endogenous regressor on the outcome, see\nfor instance . By considering Das instrument and Mas endogenous regressor,\n\u00121;0\n1(1) =\u00120;0\n1(0) =\u00120;1\n1(0) =\u00121;1\n1(1) = 0 yield testable implications of the exclusion\nrestriction under Assumptions 1-6. So far, we did not impose exogeneity of the\ntreatment or mediator. In the following, we assume treatment exogeneity by invoking\nindependence between the treatment and the potential post-treatment variables.\nAssumption 7: Independence of the treatment and potential mediators/outcomes.\nfYt(d;m);M(d)gD, for alld;m;t;2f0;1g:\nAssumption 7 implies that there are no confounders jointly a\u000becting the treatment\non the one hand and the mediator and/or outcome on the other hand. It is\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\begin{align*}\n\\theta_1^{0,0}(0)= E[Q_{10}(Y_0)-Y_1|D=0,M=0], \\\\\n\\theta_1^{0,0}(q,0)= F_{Q_{10}(Y_0)|D=0,M=0}^{-1}(q)-F_{Y_1|D=0,M=0}^{-1}(q).\n\\end{align*}\n\\begin{itemize}\n\\item[(a)] and Assumption 6a, the average and quantile direct effects under $d=0$ conditional on $D=0$ and $M(0)=1$ are identified:\n\\begin{align*}\n\\theta_1^{0,1}(0)= E[Y_1-Q_{11}(Y_0)|D=0,M=1],\\\\\n\\theta_1^{0,1}(q,0)= F_{Y_1|D=0,M=1}^{-1}(q)-F_{Q_{11}(Y_0)|D=0,M=1}^{-1}(q).\n\\end{align*}\n\\item[(b)] and Assumption 6b, the average and quantile direct effects under $d=1$ is identified conditional on $D=1$ and $M(1)=1$ are identified:\n\\begin{align*}\n\\theta_1^{1,1}(1)= E[Q_{01}(Y_0)-Y_1|D=1,M=1],\\\\\n\\theta_1^{1,1}(q,1)= F_{Q_{01}(Y_0)|D=1,M=1}^{-1}(q)-F_{Y_1|D=1,M=1}^{-1}(q).\n\\end{align*}\n\\end{itemize}\n\\textbf{Proof.} See Appendix .\nIn the instrumental variable framework, any direct effects of the instrument are typically ruled out by imposing the exclusion restriction, in order to identify the causal effect of an endogenous regressor on the outcome, see for instance . By considering $D$ as instrument and $M$ as endogenous regressor, $\\theta_1^{1,0}(1)=\\theta_1^{0,0}(0)=\\theta_1^{0,1}(0)=\\theta_1^{1,1}(1)=0$ yield testable implications of the exclusion restriction under Assumptions 1-6.\nSo far, we did not impose exogeneity of the treatment or mediator. In the following, we assume treatment exogeneity by invoking independence between the treatment and the potential post-treatment variables.\\vspace{5 pt}\\\\\n\\textbf{Assumption 7:} Independence of the treatment and potential mediators/outcomes.\\\\\n$ \\{Y_t(d,m),M(d)\\} D \\mbox{, for all } d,m,t, \\in \\{0,1\\}.$\\vspace{5 pt}\\\\\nAssumption 7 implies that there are no confounders jointly affecting the treatment on the one hand and the mediator and/or outcome on the other hand. It is\n\\end{document}\n"}
{"under treatment randomization as in successfully conducted experiments. This\nallows identifying the ATE: \u0001 1=E[Y1jD= 1]\u0000E[Y1jD= 0]. Furthermore, we\nassume the mediator to be weakly monotonic in the treatment.\nAssumption 8: Weak monotonicity of the mediator in the treatment.\nPr(M(1)\u0015M(0)) = 1:\nAssumption 8 is standard in the instrumental variable literature on local aver-\nage treatment e\u000bects when denoting by Dthe instrument and by Mthe endoge-\nnous regressor, see and . It rules out the existence of de\fers. As discussed in\nthe Appendix , the total ATE \u0001 1=E[Y1jD= 1]\u0000E[Y1jD= 0] and QTE\n\u00011(q) =F\u00001\nY1jD=1(q)\u0000F\u00001\nY1jD=0(q) for the entire population are identi\fed under As-\nsumption 7. Furthermore, Assumptions 7 and 8 yield the strata proportions, de-\nnoted byp\u001c= Pr(\u001c), as functions of the conditional mediator probabilities given\nthe treatment, which we denote by p(mjd)= Pr(M=mjD=d) ford;m2f0;1g\n(see Appendix ):\npa=p1j0;pc=p1j1\u0000p1j0=p0j0\u0000p0j1;pn=p0j1: (1)\nFurthermore, Assumptions 2, 7, and 8 imply that (see Appendix )\n\u00010;c=E[Y0(1;1)\u0000Y0(0;0)jc] =E[Y0jD= 1]\u0000E[Y0jD= 0]\np1j1\u0000p1j0= 0: (2)\nTherefore, a rejection of the testable implication E[Y0jD= 1]\u0000E[Y0jD= 0] = 0\nin the data would point to a violation of these assumptions. Assumptions 7 and\n8 permit identifying additional parameters, namely the total, direct, and indirect\ne\u000bects on compliers, and the direct e\u000bects on never- and always-takers, as shown in\nTheorems 3 to 5. This follows from the fact that de\fers are ruled out and that the\nproportions and potential outcome distributions of the various principal strata are\nnot selective w.r.t. the treatment.\nTheorem 3: Under Assumptions 1{3, 7-8,\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nunder treatment randomization as in successfully conducted experiments. This allows identifying the ATE: $\\Delta_1 = E[Y_1|D=1] -E[Y_1|D=0]$.\nFurthermore, we assume the mediator to be weakly monotonic in the treatment.\\vspace{5 pt}\\\\\n\\textbf{Assumption 8:} Weak monotonicity of the mediator in the treatment.\\\\\n$\\Pr(M(1) \\geq M(0)) =1.$\\vspace{5 pt}\\\\\nAssumption 8 is standard in the instrumental variable literature on local average treatment effects when denoting by $D$ the instrument and by $M$ the endogenous regressor, see  and . It rules out the existence of defiers.\nAs discussed in the Appendix , the total ATE $\\Delta_1= E[Y_1 |D=1]- E[Y_1 |D=0]$ and QTE $\\Delta_1(q) = F_{Y_{1} |D=1}^{-1}(q)- F_{Y_{1} |D=0}^{-1}(q)$ for the entire population are identified under Assumption 7. Furthermore, Assumptions 7 and 8 yield the strata proportions, denoted by $p_{\\tau}= \\Pr(\\tau)$, as functions of the conditional mediator probabilities given the treatment, which we denote by $p_{(m|d)}=\\Pr(M=m|D=d)$ for $d, m \\in \\{0,1\\}$ (see Appendix ):\n\\begin{equation} \np_a =p_{1|0}, p_c = p_{1|1}-p_{1|0} = p_{0|0}-p_{0|1}, p_n = p_{0|1}.\n\\end{equation}\nFurthermore, Assumptions 2, 7, and 8 imply that (see Appendix )\n\\begin{equation} \n\\Delta_{0,c} =E[Y_0(1,1) - Y_0(0,0)|c] = \\frac{E[Y_0|D=1] -E[Y_0|D=0] }{p_{1|1} - p_{1|0}} =  0.\n\\end{equation}\nTherefore, a rejection of the testable implication $E[Y_0|D=1] -E[Y_0|D=0]=0$ in the data would point to a violation of these assumptions.\nAssumptions 7 and 8 permit identifying additional parameters, namely the total, direct, and indirect effects on compliers, and the direct effects on never- and always-takers, as shown in Theorems 3 to 5. This follows from the fact that defiers are ruled out and that the proportions and potential outcome distributions of the various principal strata are not selective w.r.t.\\ the treatment.\\\\\n\\noindent \\textbf{Theorem 3:} Under Assumptions 1\u20133, 7-8,\n\\end{document}\n"}
{"a) and Assumption 4a, the average and quantile direct e\u000bects on never-takers are\nidenti\fed:\n\u0012n\n1=\u00121;0\n1(1) and\u0012n\n1(q) =\u00121;0\n1(q;1):\nb) and Assumption 4, the average direct e\u000bect under d= 0 on compliers is\nidenti\fed:\n\u0012c\n1(0) =p0j0\np0j0\u0000p0j1\u00120;0\n1(0)\u0000p0j1\np0j0\u0000p0j1\u00121;0\n1(1):\nFurthermore, the potential outcome distributions under d= 0 on compliers\nare identi\fed:\nFY1(1;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FQ10(Y0)jD=0;M=0(y)\n\u0000p0j1\np0j0\u0000p0j1cFY1jD=1;M=0(y);(1)\nFY1(0;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FY1jD=0;M=0(y)\n\u0000p0j1\np0j0\u0000p0j1FQ00(Y0)jD=1;M=0(y):(2)\nTherefore, the direct quantile e\u000bect under d= 0 on compliers, \u0012c\n1(q;0) =\nF\u00001\nY1(1;0)jc(q)\u0000F\u00001\nY1(0;0)jc(q), is identi\fed.\nProof. See Appendix . Theorem 4: Under Assumptions 1{2, 5, 7-8,\na) and Assumption 6a, the average and quantile direct e\u000bects on always-takers\nare identi\fed:\n\u0012a\n1=\u00120;1\n1(0) and\u0012a\n1(q) =\u00120;1\n1(q;0):\nb) and Assumption 6, the average direct e\u000bect under d= 1 on compliers is\nidenti\fed:\n\u0012c\n1(1) =p1j1\np1j1\u0000p1j0\u00121;1\n1(1)\u0000p1j0\np1j1\u0000p1j0\u00120;1\n1(0):\nFurthermore, the potential outcome distributions under d= 1 for compliers\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\begin{itemize}\n\\item[a)] and Assumption 4a, the average and quantile direct effects on never-takers are identified:\n\\begin{equation*}\n\\theta_1^n= \\theta_1^{1,0}(1) \\mbox{ and } \\theta_1^n(q)= \\theta_1^{1,0}(q,1).\n\\end{equation*}\n\\item[b)] and Assumption 4, the average direct effect under $d = 0$ on compliers is identified:\n\\begin{align*}\n\\displaystyle \\theta_1^{c}(0) = \\frac{p_{0|0}}{p_{0|0} - p_{0|1}} \\theta_1^{0,0}(0)   - \\frac{p_{0|1}}{p_{0|0} - p_{0|1}} \\theta_1^{1,0}(1) .\n\\end{align*}\nFurthermore, the potential outcome distributions under $d = 0$ on compliers are identified:\n\\begin{equation} \\begin{array}{rl} \\displaystyle\n F_{Y_{1}(1,0)|\\tau=c}(y) = \\displaystyle \\frac{p_{0|0}}{p_{0|0} - p_{0|1}}  F_{Q_{10}(Y_{0})|D=0,M=0}(y) \\\\ \\displaystyle \\qquad - \\frac{p_{0|1}}{ p_{0|0} - p_{0|1}c}  F_{Y_1|D=1,M=0}(y), \\end{array}\n \\end{equation}\n \\begin{equation}\n \\begin{array}{rl}\\displaystyle\nF_{Y_{1}(0,0)|\\tau=c}(y) = \\displaystyle \\frac{p_{0|0}}{p_{0|0} - p_{0|1}} F_{Y_{1}|D=0,M=0}(y) \\\\ \\displaystyle \\qquad - \\frac{p_{0|1} }{p_{0|0} - p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y)  . \n\\end{array}\n\\end{equation}\nTherefore, the direct quantile effect under $d = 0$ on compliers, $\\theta_1^{c}(q,0) = F_{Y_{1}(1,0)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$, is identified.\n\\end{itemize}\n\\textbf{Proof.} See Appendix .\n\\noindent \\textbf{Theorem 4:} Under Assumptions 1\u20132, 5, 7-8,\n\\begin{itemize}\n\\item[a)] and Assumption 6a, the average and quantile direct effects on\nalways-takers are identified:\n\\begin{equation*}\n\\theta_1^a= \\theta_1^{0,1}(0) \\mbox{ and } \\theta_1^a(q)= \\theta_1^{0,1}(q,0).\n\\end{equation*}\n\\item[b)] and Assumption 6, the average direct effect under $d = 1$ on compliers is identified:\n\\begin{align*}\n\\theta_{1}^{c}(1) = \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} \\theta_1^{1,1}(1) -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}\\theta_1^{0,1}(0).\n\\end{align*}\nFurthermore, the potential outcome distributions under $d = 1$ for compliers\n\\end{itemize}\n\\end{document}\n"}
{"are identi\fed:\nFY1(1;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FQ10(Y0)jD=0;M=0(y)\n\u0000p0j1\np0j0\u0000p0j1cFY1jD=1;M=0(y);(1)\nFY1(0;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FY1jD=0;M=0(y)\n\u0000p0j1\np0j0\u0000p0j1FQ00(Y0)jD=1;M=0(y):(2)\nTherefore, the direct quantile e\u000bect under d= 0 on compliers, \u0012c\n1(q;0) =F\u00001\nY1(1;0)jc(q)\u0000\nF\u00001\nY1(0;0)jc(q), is identi\fed. Proof. See Appendix . Theorem 4: Under Assumptions\n1{2, 5, 7-8,\na) and Assumption 6a, the average and quantile direct e\u000bects on always-takers\nare identi\fed:\n\u0012a\n1=\u00120;1\n1(0) and\u0012a\n1(q) =\u00120;1\n1(q;0):\nb) and Assumption 6, the average direct e\u000bect under d= 1 on compliers is\nidenti\fed:\n\u0012c\n1(1) =p1j1\np1j1\u0000p1j0\u00121;1\n1(1)\u0000p1j0\np1j1\u0000p1j0\u00120;1\n1(0):\nFurthermore, the potential outcome distributions under d= 1 for compliers\nare identi\fed:\nFY1(1;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FY1jD=1;M=1(y)\n\u0000p1j0\np1j1\u0000p1j0FQ11(Y0)jD=0;M=1(y);(3)\nFY1(0;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FQ01(Y0)jD=1;M=1(y)\n\u0000p1j0\np1j1\u0000p1j0FY1jD=0;M=1(y):(4)\nTherefore, the direct quantile e\u000bect under d= 1 on compliers \u0012c\n1(q;1) =\nF\u00001\nY1(1;1)jc(q)\u0000F\u00001\nY1(0;1)jc(q) is identi\fed.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nare identified:\n\\begin{equation} \\begin{array}{rl} \\displaystyle\n F_{Y_{1}(1,0)|\\tau=c}(y) = \\displaystyle \\frac{p_{0|0}}{p_{0|0} - p_{0|1}}  F_{Q_{10}(Y_{0})|D=0,M=0}(y) \\\\ \\displaystyle \\qquad - \\frac{p_{0|1}}{ p_{0|0} - p_{0|1}c}  F_{Y_1|D=1,M=0}(y), \\end{array}\n \\end{equation}\n \\begin{equation}\n \\begin{array}{rl}\\displaystyle\nF_{Y_{1}(0,0)|\\tau=c}(y) = \\displaystyle \\frac{p_{0|0}}{p_{0|0} - p_{0|1}} F_{Y_{1}|D=0,M=0}(y) \\\\ \\displaystyle \\qquad - \\frac{p_{0|1} }{p_{0|0} - p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y)  . \n\\end{array}\n\\end{equation}\nTherefore, the direct quantile effect under $d = 0$ on compliers, $\\theta_1^{c}(q,0) = F_{Y_{1}(1,0)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$, is identified.\n\\textbf{Proof.} See Appendix .\n\\noindent \\textbf{Theorem 4:} Under Assumptions 1\u20132, 5, 7-8,\n\\begin{itemize}\n\\item[a)] and Assumption 6a, the average and quantile direct effects on\nalways-takers are identified:\n\\begin{equation*}\n\\theta_1^a= \\theta_1^{0,1}(0) \\mbox{ and } \\theta_1^a(q)= \\theta_1^{0,1}(q,0).\n\\end{equation*}\n\\item[b)] and Assumption 6, the average direct effect under $d = 1$ on compliers is identified:\n\\begin{align*}\n\\theta_{1}^{c}(1) = \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} \\theta_1^{1,1}(1) -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}\\theta_1^{0,1}(0).\n\\end{align*}\nFurthermore, the potential outcome distributions under $d = 1$ for compliers are identified:\n\\begin{equation} \\begin{array}{rl} \\displaystyle\n F_{Y_{1}(1,1)|\\tau=c}(y) = \\displaystyle \\frac{p_{1|1}}{p_{1|1} - p_{1|0}} F_{Y_{1}|D=1,M=1}(y) \\\\ \\displaystyle \\qquad - \\frac{p_{1|0} }{p_{1|1} - p_{1|0}}F_{Q_{11}(Y_0)|D=0,M=1}(y),  \\end{array}\n \\end{equation}\n \\begin{equation}\n \\begin{array}{rl}\\displaystyle\nF_{Y_{1}(0,1)|\\tau=c}(y) =\\displaystyle \\frac{p_{1|1}}{p_{1|1} - p_{1|0}}  F_{Q_{01}(Y_{0})|D=1,M=1}(y)\\\\  \\displaystyle \\qquad - \\frac{p_{1|0}}{ p_{1|1} - p_{1|0}}  F_{Y_1|D=0,M=1}(y).  \\end{array}\n\\end{equation}\nTherefore, the direct quantile effect under $d = 1$ on compliers $\\theta_1^{c}(q,1) = F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,1)|c}^{-1}(q)$ is identified.\n\\end{itemize}\n\\end{document}\n"}
{"Proof. See Appendix . Theorem 5: Under Assumptions 1-3, 5, 7-8,\na) and Assumptions 4a, 6a, the total average treatment e\u000bect on compliers is\nidenti\fed:\n\u0001c\n1=p1j1\np1j1\u0000p1j0E[Y1jD= 1;M= 1]\u0000p1j0\np1j1\u0000p1j0E[Q11(Y0)jD= 0;M= 1]\n\u0000p0j0\np1j1\u0000p1j0E[Y1jD= 0;M= 0] +p0j1\np1j1\u0000p1j0E[Q00(Y0)jD= 1;M= 0]:\nFurthermore, the total quantile treatment e\u000bect on compliers \u0001c\n1(q) =F\u00001\nY1(1;1)jc(q)\u0000\nF\u00001\nY1(0;0)jc(q) is identi\fed using the inverse of () and ().\nb) and Assumptions 4a, 6b, the average indirect e\u000bect under d= 0 on compliers\nis identi\fed:\n\u000ec\n1(0) =p1j1\np1j1\u0000p1j0E[Q11(Y0)jD= 1;M= 1]\u0000p1j0\np1j1\u0000p1j0E[Y1jD= 0;M= 1]\n\u0000p0j0\np1j1\u0000p1j0E[Y1jD= 0;M= 0] +p0j1\np1j1\u0000p1j0E[Q00(Y0)jD= 1;M= 0]:\nFurthermore, the quantile indirect e\u000bect under d= 0 on compliers \u000ec\n1(q;0) =\nF\u00001\nY1(0;1)jc(q)\u0000F\u00001\nY1(0;0)jc(q) is identi\fed using the inverse of () and ().\nc) and Assumptions 4b, 6a, the average indirect e\u000bect under d= 1 on compliers\nis identi\fed:\n\u000ec\n1(1) =p1j1\np1j1\u0000p1j0E[Y1jD= 1;M= 1]\u0000p1j0\np1j1\u0000p1j0E[Q11(Y0)jD= 0;M= 1]\n\u0000p0j0\np1j1\u0000p1j0E[Q00(Y0)jD= 0;M= 0] +p0j1\np1j1\u0000p1j0E[Y1jD= 1;M= 0]:\nFurthermore, the quantile indirect e\u000bect under d= 1 on compliers \u000ec\n1(q;1) =\nF\u00001\nY1(1;1)jc(q)\u0000F\u00001\nY1(1;0)jc(q) is identi\fed using the inverse of () and ().\nProof. See Appendix .\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document}\n\\textbf{Proof.} See Appendix .\n\\noindent \\textbf{Theorem 5:} Under Assumptions 1-3, 5, 7-8,\n\\begin{itemize}\n\\item[a)] and Assumptions 4a, 6a, the total average treatment effect on compliers is identified:\n\\begin{align*}\n\\Delta_1^c= \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1|D=1,M=1] -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)|D=0,M=1] \\\\\n \u2212 \\frac{p_{0|0}}{ p_{1|1} - p_{1|0}} E[Y_1|D=0,M=0] +\\frac{p_{0|1}}{p_{1|1} - p_{1|0}}E[Q_{00}(Y_0)|D=1,M=0].\n\\end{align*}\nFurthermore, the total quantile treatment effect on compliers $\\Delta_1^{c}(q) = F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is identified using the inverse of () and ().\n\\item[b)] and Assumptions 4a, 6b, the average indirect effect\nunder $d = 0$ on compliers is identified:\n\\begin{align*}\n\\delta_1^c(0) = \\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_0)|D=1,M=1] - \\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_1|D=0,M=1]\\\\\n\\hspace{-0.15cm}- \\frac{p_{0|0}}{p_{1|1} - p_{1|0}} E[Y_1|D=0,M=0] +\\frac{p_{0|1}}{p_{1|1} - p_{1|0}}E[Q_{00}(Y_0)|D=1,M=0].\n\\end{align*}\nFurthermore, the quantile indirect effect\nunder $d = 0$ on compliers $\\delta_1^{c}(q,0) = F_{Y_{1}(0,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is identified using the inverse of () and ().\n\\item[c)] and Assumptions 4b, 6a, the average indirect effect\nunder $d = 1$ on compliers is identified:\n\\begin{align*}\n\\delta_1^{c}(1) = \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1|D=1,M=1] -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)|D=0,M=1]\\\\\n\\hspace{-0.15cm}- \\frac{p_{0|0}}{p_{1|1} - p_{1|0}}E[Q_{00}(Y_0)|D=0,M=0] + \\frac{p_{0|1}}{p_{1|1} - p_{1|0}}E[Y_1|D=1,M=0].\n\\end{align*}\nFurthermore, the quantile indirect effect\nunder $d = 1$ on compliers $\\delta_1^{c}(q,1) = F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(1,0)|c}^{-1}(q)$ is identified using the inverse of () and ().\n\\end{itemize}\n\\textbf{Proof.} See Appendix .\n\\end{document}\n"}
{"0.1 Estimation\nAs in Assumption 5.1 of , we assume standard regularity conditions, namely that\nconditional on T=t,D=d, and M=m,Yis a random draw from that sub-\npopulation de\fned in terms of t; d; m2f1;0g. Furthermore, the outcome in the\nsubpopulations required for the identi\fcation results of interest must have compact\nsupport and a density that is bounded from above and below as well as continu-\nously di\u000berentiable. Denote by Nthe total sample size across both periods and all\ntreatment-mediator combinations and by i2f1; :::; Ngan index for the sampled\nsubject, such that ( Yi; Di; Mi; Ti) correspond to sample realizations of the random\nvariables ( Y; D; M; T ). The total, direct, and indirect e\u000bects may be estimated\nusing the sample analogy principle, which replaces population moments with sam-\nple moments. For instance, any conditional mediator probability given the treat-\nment, Pr( M=mjD=d), is to be replaced by an estimate thereof in the sample,\nPN\ni=1IfMi=m;D i=dgPN\ni=1IfDi=dg. A crucial step is the estimation of the quantile-quantile trans-\nforms. The application of such quantile transformations dates at least back to ,\nsee also , , and for recent applications. First, it requires estimating the condi-\ntional outcome distribution, FYtjD=d;M =m(y), by the conditional empirical distribu-\ntion ^FYtjD=d;M =m(y) =1Pn\ni=1IfDi=d;Mi=m;Ti=tgP\ni:Di=d;Mi=m;Ti=tIfYi\u0014yg. Second,\ninverting the latter yields the empirical quantile function ^F\u00001\nYtjD=d;M =m(q). The em-\npirical quantile-quantile transform is then obtained by\n^Qdm(y) =^F\u00001\nY1jD=d;M =m(^FY0jD=d;M =m(y)):\nThis permits estimating the average and quantile e\u000bects of interest. Average e\u000bects\nare estimated by replacing any (conditional) expectations with the corresponding\nsample averages in which the estimated quantile-quantile transforms enter as plug-\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\subsection{Estimation}\n As in Assumption 5.1 of , we assume standard regularity conditions, namely that conditional on $T=t$, $D=d$, and $M=m$, $Y$ is a random draw from that subpopulation defined in terms of $t,d,m$ $\\in$ $\\{1,0\\}$. Furthermore, the outcome in the subpopulations required for the identification results of interest must have compact support and a density that is bounded from above and below as well as continuously differentiable. Denote by $N$ the total sample size across both periods and all treatment-mediator combinations and by $i$ $\\in$ $\\{1,...,N\\}$ an index for the sampled subject, such that $(Y_i,D_i,M_i,T_i$) correspond to sample realizations of the random variables $(Y,D,M,T$).\n The total, direct, and indirect effects may be estimated using the sample analogy principle, which replaces population moments with sample moments. For instance, any conditional mediator probability given the treatment, $\\Pr(M=m|D=d)$, is to be replaced by an estimate thereof in the sample, $ \\frac{\\sum_{i=1}^{N} I\\{M_i=m,D_i=d\\}}{\\sum_{i=1}^{N} I\\{D_i=d\\}}$. A crucial step is the estimation of the quantile-quantile transforms. The application of such quantile transformations dates at least back to , see also , , and  for recent applications. First, it requires estimating the conditional outcome distribution, $F_{Y_t|D=d,M=m}(y)$, by the conditional empirical distribution $\\hat{F}_{Y_t|D=d,M=m}(y)=\\frac{1}{\\sum_{i=1}^{n}I\\{D_i=d, M_i=m, T_i=t\\}}\\sum_{i:D_i=d,M_i=m,T_i=t}I\\{Y_i\\leq y\\}$. Second, inverting the latter yields the empirical quantile function $\\hat{F}_{Y_t|D=d,M=m}^{-1}(q)$. The empirical quantile-quantile transform is then obtained by\n  \\begin{eqnarray*}\n \\hat{Q}_{dm}(y) = \\hat{F}_{Y_{1}|D=d,M=m}^{-1}(\\hat{F}_{Y_{0}|D=d,M=m}(y)).\n  \\end{eqnarray*}\n This permits estimating the average and quantile effects of interest. Average effects are estimated by replacing any (conditional) expectations with the corresponding sample averages in which the estimated quantile-quantile transforms enter as plug-\n\\end{document}\n"}
{"0.1 Estimation\n-in estimates. Taking \u00121;0\n1(see Theorem 1) as an example, an estimate thereof is\n^\u00121;0\n1(1) =1Pn\ni=1IfDi= 1;Mi= 0;Ti= 1gX\ni:Di=1;M i=0;Ti=1Yi\n\u00001Pn\ni=1IfDi= 1;Mi= 0;Ti= 0gX\ni:Di=1;M i=0;Ti=0^Q00(Yi):\nLikewhise, quantile e\u000bects are estimated based on the empirical quantiles. For the\nestimation of total ATE and QTE, show that the resulting estimators arep\nN-\nconsistent and asymptotically normal, see their Theorems 5.1 and 5.3. These prop-\nerties also apply to our context when splitting the sample into subgroups based on\nthe values of a binary treatment and mediator (rather than the treatment only).\nFor instance, the implications of Theorem 1 in when considering subsamples with\nD= 1 andD= 0 carry over to considering subsamples with D= 1;M= 0 and\nD= 0;M= 0 for estimating the average direct e\u000bect on never-takers. In contrast to\n, however, some of our identi\fcation results include the conditional mediator prob-\nabilities Pr( M=mjD=d). As the latter are estimated withp\nN-consistency, too,\nit follows that the resulting e\u000bect estimators are againp\nN-consistent and asymp-\ntotically normal. We use a non-parametric bootstrap approach to calculate the\nstandard errors. show the validity of the bootstrap approach for such kind of esti-\nmators, which follows from their asymptotic normality. For the case that identifying\nassumptions to only hold conditional on observed covariates, denoted by X, estima-\ntion must be adapted to allow for control variables. Following a suggestion by in\ntheir Section 5.1, basing estimation on outcome residuals in which the association of\nXandYhas been purged by means of a regression is consistent under the additional\nassumption that the e\u000bects of DandMare homogeneous across covariates. As an\nalternative, propose a \nexible semiparametric estimator that does not impose such\na homogeneity-in-covariates assumption and showp\nN-consistency and asymptotic\nnormality.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\subsection{Estimation}\n-in estimates. Taking $\\theta_1^{1,0}$ (see Theorem 1) as an example, an estimate thereof is\n \\begin{eqnarray*}\n\\hat{\\theta}_1^{1,0}(1)=\\frac{1}{\\sum_{i=1}^{n}I\\{D_i=1,M_i=0,T_i=1\\}}\\sum_{i:D_i=1,M_i=0,T_i=1}Y_i\\\\ -\\frac{1}{\\sum_{i=1}^{n}I\\{D_i=1,M_i=0,T_i=0\\}}\\sum_{i:D_i=1,M_i=0,T_i=0}\\hat{Q}_{00}(Y_i).\n \\end{eqnarray*}\nLikewhise, quantile effects are estimated based on the empirical quantiles.\nFor the estimation of total ATE and QTE,  show that the resulting estimators are $\\sqrt{N}$-consistent and asymptotically normal, see their Theorems 5.1 and 5.3. These properties also apply to our context when splitting the sample into subgroups based on the values of a binary treatment and mediator (rather than the treatment only). For instance, the implications of Theorem 1 in  when considering subsamples with $D=1$ and $D=0$ carry over to considering subsamples with $D=1, M=0$ and $D=0, M=0$ for estimating the average direct effect on never-takers. In contrast to , however, some of our identification results include the conditional mediator probabilities $\\Pr(M=m|D=d)$. As the latter are estimated with $\\sqrt{N}$-consistency, too, it follows that the resulting effect estimators are again $\\sqrt{N}$-consistent and asymptotically normal. %which can be shown in a two step GMM framework in the spirit of .\nWe use a non-parametric bootstrap approach to calculate the standard errors.  show the validity of the bootstrap approach for such kind of estimators, which follows from their asymptotic normality.\nFor the case that identifying assumptions to only hold conditional on observed covariates, denoted by $X$, estimation must be adapted to allow for control variables. Following a suggestion by  in their Section 5.1, basing estimation on outcome residuals in which the association of $X$ and $Y$ has been purged by means of a regression is consistent under the additional assumption that the effects of $D$ and $M$ are homogeneous across covariates. As an alternative,  propose a flexible semiparametric estimator that does not impose such a homogeneity-in-covariates assumption and show $\\sqrt{N}$-consistency and asymptotic normality.\n\\end{document}\n"}
{"1 Simulations\nTo shape the intuition for our identi\fcation results, this section presents a brief\nsimulation based on the following data generating process (DGP):\nT\u0018Binom (0:5); D\u0018Binom (0:5); U\u0018Unif (\u00001;1); V\u0018N(0;1)\nindependent of each other, and\nM=IfD+U+V > 0g; Y T= \u0003((1 + D+M+D\u0001M)\u0001T+U):\nTreatment Das well as the observed time period Tare randomized, while the\nmediator-outcome association is confounded due to the unobserved time constant\nheterogeneity U. The potential outcome in period 1 is given by Y1(d; M(d0)) =\n\u0003((1 + d+M(d0) +d\u0001M(d0)) +U), where \u0003 denotes a link function. If the latter\ncorresponds to the identity function, our model is linear and implies a homogeneous\ntime trend Tequal to 1. If \u0003 is nonlinear, the time trend is heterogeneous, which\ninvalidates the common trend assumption of di\u000berence-in-di\u000berences models. Mis\nnot only a function of DandU, but also of the unobserved random term V, which\nguarantees common support w.r.t. U, see Assumptions 4 and 6. Compliers, always-\ntakers, and never-takers satisfy, respectively: c=IfU+V\u00140;1 +U+V > 0g,\na=IfU+V > 0g, and n=If1 +U+V\u00140g. In the simulations with 1,000\nreplications, we consider two sample sizes ( N= 1;000;4;000) and investigate the\nbehaviour of our change-in-changes methods as well as the di\u000berence-in-di\u000berences\napproach of in both a linear (\u0003 equal to identity function) and nonlinear outcome\nmodel where \u0003 equals the exponential function. To implement the change-in-changes\nestimators in the simulations as well as the application in Section , we make use of\nthe `cic' command in the qteR-package by with its default values. Table reports the\nbias, standard deviation (`sd'), root mean squared error (`rmse'), true e\u000bect (`true'),\nand the relative root mean squared error in percent of the true e\u000bect (`relr')\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\section{Simulations}\nTo shape the intuition for our identification results, this section presents a brief simulation based on the following data generating process (DGP):\n\\begin{equation*}\nT \\sim Binom(0.5),\\textrm{ }D \\sim Binom(0.5),\\textrm{ }U\\sim Unif(-1,1),\\textrm{ } V\\sim N(0,1)\n\\end{equation*}\nindependent of each other, and\n\\begin{equation*}\nM=I\\{D+U+V>0\\},\\qquad Y_T=\\Lambda((1+D+M+D\\cdot M)\\cdot T+U).\n\\end{equation*}\nTreatment $D$ as well as the observed time period $T$ are randomized, while the mediator-outcome association is confounded due to the unobserved time constant heterogeneity $U$. The potential outcome in period $1$ is given by $Y_1(d,M(d'))=\\Lambda((1+d+M(d')+d\\cdot M(d'))+U)$, where $\\Lambda$ denotes a link function. If the latter corresponds to the identity function,  our model is linear and implies a homogeneous time trend $T$ equal to 1. If $\\Lambda$ is nonlinear, the time trend is heterogeneous, which invalidates the common trend assumption of difference-in-differences models. $M$ is not only a function of $D$ and $U$, but also of the unobserved random term $V$, which guarantees common support w.r.t.\\ $U$, see Assumptions 4 and 6. Compliers, always-takers, and never-takers satisfy, respectively: $c=I\\{U+V\\leq 0, 1+U+V>0\\}$, $a=I\\{U+V>0\\}$, and $n=I\\{1+U+V\\leq0\\}$.\nIn the simulations with 1,000 replications, we consider two sample sizes ($N=1,000, 4,000$) and investigate the behaviour of our change-in-changes methods as well as the difference-in-differences approach of  in both a linear ($\\Lambda$ equal to identity function) and nonlinear outcome model where $\\Lambda$ equals the exponential function. To implement the change-in-changes estimators in the simulations as well as the application in Section , we make use of the `cic' command in the \\texttt{qte} R-package by  with its default values.\nTable  reports the bias, standard deviation (`sd'), root mean squared error (`rmse'), true effect (`true'), and the relative root mean squared error in percent of the true effect (`relr') \n\\end{document}\n"}
{"of the respective estimators of \u0012n\n1,\u0012a\n1, \u0001c,\u0012c\n1(1),\u0012c\n1(0),\u000ec\n1(1), and\u000ec\n1(0) for the lin-\near model. In this case, the identifying assumptions underlying both the change-in-\nchanges (Panel A.) and di\u000berence-in-di\u000berences (Panel B.) estimators are satis\fed.\nSpeci\fcally, the homogeneous time trend on the individual level satis\fes any of the\ncommon trend assumptions in , while the monotonicity of YinUand the indepen-\ndence ofTandUsatis\fes the key assumptions of this paper. For this reason any of\nthe estimates in Table are close to being unbiased and appear to converge to the true\ne\u000bect at the parametric rate when comparing the results for the two di\u000berent sample\nsizes. Table provides the results for the exponential outcome model, in which the\ntime trend is heterogeneous and interacts with Uthrough the nonlinear link func-\ntion. While the change-in-changes assumptions hold (Panel A.), average time trends\nare heterogeneous across complier types such that the di\u000berence-in-di\u000berences ap-\nproach (Panel B.) of is inconsistent. Accordingly, the biases of the change-in-changes\nestimates generally approach zero as the sample size increases, while this is not the\ncase for the di\u000berence-in-di\u000berences estimates. Change-in-changes yields a lower\nroot mean squared error than the respective di\u000berence-in-di\u000berences estimator in all\nbut one case (namely ^\u000ec\n1(0) withN= 1;000) and its relative attractiveness increases\nin the sample size due to its lower bias. In our \fnal simulation design, we maintain\nthe exponential outcome model but assume Dto be selective w.r.t. Urather than\nrandom. To this end, the treatment model in is replaced by D=IfU+Q > 0g,\nwith the independent variable Q\u0018N(0;1) being an unobserved term. Under this\nviolation of Assumption 7, complier shares and e\u000bects are no longer identi\fed, which\nis con\frmed by the simulation results presented in Table . The bias in the change-\nin-changes based total, direct, and indirect e\u000bects on compliers do not vanish as\nthe sample size increases. Furthermore, under non-random assignment of D(while\nmaintaining monotonicity of MinD), the never-takers' and always-takers' respec-\ntive distributions of Udi\u000ber across treatment. Therefore, average direct e\u000bects\namong the total of never or always-takers, respectively, are not identi\fed. Yet, \u00121;0\n1,\nwhich is still identi\fed by the same estimator as before, yields the direct e\u000bect\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nof the respective estimators of $\\theta_1^n$, $\\theta_1^a$, $\\Delta_c$,  $\\theta_1^c(1)$, $\\theta_1^c(0)$, $\\delta_1^c(1)$, and $\\delta_1^c(0)$ for the linear model. In this case, the identifying assumptions underlying both the change-in-changes (Panel A.) and difference-in-differences (Panel B.) estimators are satisfied. Specifically, the homogeneous time trend on the individual level satisfies any of the common trend assumptions in , while the monotonicity of $Y$ in $U$ and the independence of $T$ and $U$ satisfies the key assumptions of this paper. For this reason any of the estimates in Table  are close to being unbiased and appear to converge to the true effect at the parametric rate when comparing the results for the two different sample sizes.\nTable  provides the results for the exponential outcome model, in which the time trend is heterogeneous and interacts with $U$ through the nonlinear link function. While the change-in-changes assumptions hold (Panel A.), average time trends are heterogeneous across complier types such that the difference-in-differences approach (Panel B.) of  is inconsistent. Accordingly, the biases of the change-in-changes estimates generally approach zero as the sample size increases, while this is not the case for the difference-in-differences estimates. Change-in-changes yields a lower root mean squared error than the respective difference-in-differences estimator in all but one case (namely $\\hat{\\delta}_1^c(0)$ with $N=1,000$) and its relative attractiveness increases in the sample size due to its lower bias.\nIn our final simulation design, we maintain the exponential outcome model but assume $D$ to be selective w.r.t.\\ $U$ rather than random. To this end, the treatment model in is replaced by $D=I\\{U+Q>0\\}$, with the independent variable $Q\\sim N(0,1)$ being an unobserved term. Under this violation of Assumption 7, complier shares and effects are no longer identified, which is confirmed by the simulation results presented in Table . The bias in the change-in-changes based total, direct, and indirect effects on compliers do not vanish as the sample size increases. Furthermore, under non-random assignment of $D$ (while maintaining monotonicity of $M$ in $D$), the never-takers' and always-takers' respective distributions of $U$ differ across treatment. Therefore, average direct effects among the total of never or always-takers, respectively, are not identified. Yet, $\\theta_1^{1,0}$, which is still identified by the same estimator as before, yields the direct effect\n\\end{document}\n"}
{"among never-takers with D= 1 (as de\fers do not exist). Likewise, \u00120;1\n1corre-\nsponds to the direct e\u000bect on always-takers with D= 0. Indeed, the results in Table\nsuggest that both parameters are consistently estimated with the change-in-changes\nmodel (Panel A.).\n1 Proof of Theorem 1\n1.1 Average direct e\u000bect under d =1 conditional on D =1\nand M (1) =0\nIn the following, we prove that \u00121;0\n1(1) =E[Y1(1;0)\u0000Y1(0;0)jD= 1;Mi(1) =\n0] =E[Y1\u0000Q00(Y0)jD= 1;M= 0]. Using the observational rule, we obtain\nE[Y1(1;0)jD= 1;M(1) = 0] = E[Y1jD= 1;M= 0]. Accordingly, we have to\nshow thatE[Y1(0;0)jD= 1;M(1) = 0] = E[Q00(Y0)jD= 1;M= 0] to \fnish the\nproof. Denote the inverse of h(d;m;t;u ) byh\u00001(d;m;t ;y), which exists because of\nthe strict monotonicity required in Assumption 1. Under Assumptions 1 and 3a, the\nconditional potential outcome distribution function equals\nFYt(d;0)jD=1;M=0(y)A1= Pr(h(d;m;t;U )\u0014yjD= 1;M= 0;T=t);\n= Pr(U\u0014h\u00001(d;m;t ;y)jD= 1;M= 0;T=t);\nA3a= Pr(U\u0014h\u00001(d;m;t ;y)jD= 1;M= 0);\n=FUj10(h\u00001(d;m;t ;y));(1.1)\nford;d02 f0;1g. We use these quantities in the following. First, evaluating\nFY1(0;0)jD=1;M=0(y) ath(0;0;1;u) gives\nFY1(0;0)jD=1;M=0(h(0;0;1;u)) =FUj10(h\u00001(0;0;1;h(0;0;1;u))) =FUj10(u):\nApplyingF\u00001\nY1(0;0)jD=1;M=0(q) to both sides, we have\nh(0;0;1;u) =F\u00001\nY1(0;0)jD=1;M=0(FUj10(u)): (1.2)\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\namong never-takers with $D=1$ (as defiers do not exist). Likewise, $\\theta_1^{0,1}$ corresponds to the direct effect on always-takers with $D=0$. Indeed, the results in Table  suggest that both parameters are consistently estimated with the change-in-changes model (Panel A.).\n\\numberwithin{equation}{section}\n\\section{Proof of Theorem 1 }\n\\subsection{Average direct effect under $\\mathbf{d=1}$ conditional on $\\mathbf{D=1}$ and $\\mathbf{M(1)=0}$}\nIn the following, we prove that $\\theta_1^{1,0}(1)= E[Y_1(1,0)-Y_1(0,0)|D=1,M_i (1)=0]=  E[Y_1-Q_{00}(Y_0)|D=1,M=0]$. Using the observational rule, we obtain $E[Y_1(1,0)|D=1,M(1)=0]=E[Y_1|D=1,M=0]$. Accordingly, we have to show that $E[Y_1(0,0)|D=1,M(1)=0]=E[Q_{00}(Y_0)|D=1,M=0]$ to finish the proof.\nDenote the inverse of $h(d,m,t,u)$ by $h^{-1}(d,m,t;y)$, which exists because of the strict monotonicity required in Assumption 1. Under Assumptions 1 and 3a, the conditional potential outcome distribution function equals\n\\begin{equation} \n\\begin{array}{rl}\n F_{Y_t(d,0)|D=1,M=0}(y)  \\stackrel{A1}{=} \\Pr(h(d,m,t,U) \\leq y|D=1,M=0,T=t) ,\\\\\n= \\Pr(U \\leq h^{-1}(d,m,t;y)|D=1,M=0,T=t) ,\\\\\n\\stackrel{A3a}{=} \\Pr(U \\leq h^{-1}(d,m,t;y)|D=1,M=0) ,\\\\\n= F_{U|10} ( h^{-1}(d,m,t;y)),\n\\end{array}\n\\end{equation}\nfor $d,d' \\in \\{0,1\\}$. We use these quantities in the following.\nFirst, evaluating $F_{Y_1(0,0)|D=1,M=0}(y)$ at $h(0,0,1,u)$ gives\n\\begin{equation*}\nF_{Y_1(0,0)|D=1,M=0}(h(0,0,1,u)) = F_{U|10} ( h^{-1}(0,0,1;h(0,0,1,u)))  =F_{U|10} ( u).\n\\end{equation*}\nApplying $F_{Y_1(0,0)|D=1,M=0}^{-1}(q)$ to both sides, we have\n\\begin{equation} \nh(0,0,1,u)  =F_{Y_1(0,0)|D=1,M=0}^{-1}(F_{U|10} ( u)).\n\\end{equation}\n\\end{document}\n"}
{"Second, for FY0(0;0)jD=1;M=0(y) we have\nF\u00001\nUjD=1;M=0(FY0(0;0)jD=1;M=0(y)) =h\u00001(0;0;0;y): (1)\nCombining () and () yields,\nh(0;0;1; h\u00001(0;0;0;y)) =F\u00001\nY1(0;0)jD=1;M=0\u000eFY0(0;0)jD=1;M=0(y): (2)\nNote that h(0;0;1; h\u00001(0;0;0;y)) maps the period 1 (potential) outcome of an in-\ndividual with the outcome yin period 0 under non-treatment without the me-\ndiator. Accordingly, E[F\u00001\nY1(0;0)jD=1;M=0\u000eFY0(0;0)jD=1;M=0(Y0)jD= 1; M = 0] =\nE[Y1(0;0)jD= 1; M= 0]. We can identify FY0(0;0)jD=1;M=0(y) under Assumption\n2, but we cannot identify FY1(0;0)jD=1;M=0(y). However, we show in the follow-\ning that we can identify the overall quantile-quantile transform F\u00001\nY1(0;0)jD=1;M=0\u000e\nFY0(0;0)jD=1;M=0(y) under the additional Assumption 3b. Under Assumptions 1 and\n3b, the conditional potential outcome distribution function equals\nFYt(d;0)jD=0;M=0(y)A1= Pr( h(d; m; t; U )\u0014yjD= 0; M= 0; T=t);\n= Pr( U\u0014h\u00001(d; m; t ;y)jD= 0; M= 0; T=t);\nA3b= Pr( U\u0014h\u00001(d; m; t ;y)jD= 0; M= 0);\n=FUj00(h\u00001(d; m; t ;y));(3)\nford; d02f0;1g. We repeat similar steps as above. First, evaluating FY1(0;0)jD=0;M=0(y)\nath(0;0;1; u) gives\nFY1(0;0)D=0;M=0(h(0;0;1; u)) =FUj00(h\u00001(0;0;1;h(0;0;1; u))) = FUj00(u):\nApplying F\u00001\nY1(0;0)jD=0;M=0(q) to both sides, we have\nh(0;0;1; u) =F\u00001\nY1(0;0)jD=0;M=0(FUj00(u)): (4)\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nSecond, for $F_{Y_0(0,0)|D=1,M=0}(y)$ we have\n\\begin{equation}\nF_{U|D=1,M=0}^{-1} ( F_{Y_0(0,0)|D=1,M=0}(y)) =   h^{-1}(0,0,0;y).\n\\end{equation}\nCombining () and () yields,\n\\begin{equation} \nh(0,0,1,h^{-1}(0,0,0;y))  =F_{Y_1(0,0)|D=1,M=0}^{-1} \\circ F_{Y_0(0,0)|D=1,M=0}(y) .\n\\end{equation}\nNote that $h(0,0,1,h^{-1}(0,0,0;y))$ maps the period 1 (potential) outcome of an individual with the outcome $y$ in period 0 under non-treatment without the mediator. Accordingly, $E[F_{Y_1(0,0)|D=1,M=0}^{-1} \\circ F_{Y_0(0,0)|D=1,M=0}(Y_0)|D=1,M=0]= E[Y_1(0,0)|D=1,M=0]$. We can identify $F_{Y_0(0,0)|D=1,M=0}(y)$ under Assumption 2, but we cannot identify $F_{Y_1(0,0)|D=1,M=0}(y)$. However, we show in the following that we can identify the overall quantile-quantile transform $F_{Y_1(0,0)|D=1,M=0}^{-1} \\circ F_{Y_0(0,0)|D=1,M=0}(y)$ under the additional Assumption 3b.\nUnder Assumptions 1 and 3b, the conditional potential outcome distribution function equals\n\\begin{equation} \n\\begin{array}{rl}\n F_{Y_t(d,0)|D=0,M=0}(y)  \\stackrel{A1}{=} \\Pr(h(d,m,t,U) \\leq y|D=0,M=0,T=t) ,\\\\\n= \\Pr(U \\leq h^{-1}(d,m,t;y)|D=0,M=0,T=t) ,\\\\\n\\stackrel{A3b}{=} \\Pr(U \\leq h^{-1}(d,m,t;y)|D=0,M=0) ,\\\\\n= F_{U|00} ( h^{-1}(d,m,t;y)),\n\\end{array}\n\\end{equation}\nfor $d,d' \\in \\{0,1\\}$. We repeat similar steps as above. First, evaluating $F_{Y_1(0,0)|D=0,M=0}(y)$ at $h(0,0,1,u)$ gives\n\\begin{equation*}\nF_{Y_1(0,0)D=0,M=0}(h(0,0,1,u)) = F_{U|00} ( h^{-1}(0,0,1;h(0,0,1,u)))  =F_{U|00} ( u).\n\\end{equation*}\nApplying $F_{Y_1(0,0)|D=0,M=0}^{-1}(q)$ to both sides, we have\n\\begin{equation} \nh(0,0,1,u)  =F_{Y_1(0,0)|D=0,M=0}^{-1}(F_{U|00} ( u)).\n\\end{equation}\n\\end{document}\n"}
{"Second, for FY0(0;0)jD=0;M=0(y) we have\nF\u00001\nUj00(FY0(0;0)jD=0;M=0(y)) =h\u00001(0;0;0;y): (1)\nCombining () and () yields,\nh(0;0;1; h\u00001(0;0;0;y)) =F\u00001\nY1(0;0)jD=0;M=0\u000eFY0(0;0)jD=0;M=0(y): (2)\nThe left sides of () and () are equal. In contrast to (), () contains only distributions\nthat can be identi\fed from observable data. In particular, FYt(0;0)jD=0;M=0(y) =\nPr(Yt(0;0)\u0014yjD= 0; M = 0) = Pr( Yt\u0014yjD= 0; M = 0). Accordingly,\nwe can identify F\u00001\nY1(0;0)jD=1;M=0\u000eFY0(0;0)jD=1;M=0(y) by Q00(y)\u0011F\u00001\nY1jD=0;M=0\u000e\nFY0jD=0;M=0(y). Parsing Y0through Q00(\u0001) in the treated group without mediator\ngives\nE[Q00(Y0)jD= 1; M= 0]\n=E[F\u00001\nY1jD=0;M=0\u000eFY0jD=0;M=0(Y0)jD= 1; M= 0];\n=E[F\u00001\nY1(0;0)jD=0;M=0\u000eFY0(0;0)jD=0;M=0(Y0(1;0))jD= 1; M= 0];\nA1;A3b=E[h(0;0;1; h\u00001(0;0;0;Y0(1;0)))jD= 1; M= 0];\nA2=E[h(0;0;1; h\u00001(0;0;0;Y0(0;0)))jD= 1; M= 0];\nA1;A3a=E[F\u00001\nY1(0;0)jD=1;M=0\u000eFY0(0;0)jD=1;M=0(Y0(0;0))jD= 1; M= 0];\n=E[Y1(0;0)jD= 1; M= 0] = E[Y1(0;0)jD= 1; M(1) = 0] ;(3)\nwhich has data support because of Assumption 4a.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nSecond, for $F_{Y_0(0,0)|D=0,M=0}(y)$ we have\n\\begin{equation} \nF_{U|00}^{-1} ( F_{Y_0(0,0)|D=0,M=0}(y)) =   h^{-1}(0,0,0;y).\n\\end{equation}\nCombining () and () yields,\n\\begin{equation} \nh(0,0,1,h^{-1}(0,0,0;y))  =F_{Y_1(0,0)|D=0,M=0}^{-1} \\circ F_{Y_0(0,0)|D=0,M=0}(y) .\n\\end{equation}\nThe left sides of () and () are equal. In contrast to (), () contains only distributions that can be identified from observable data. In particular, $F_{Y_t(0,0)|D=0,M=0}(y)  =\\Pr(Y_t(0,0) \\leq y|D=0,M=0) =  \\Pr(Y_t \\leq y|D=0,M=0)$. Accordingly, we can identify $F_{Y_1(0,0)|D=1,M=0}^{-1} \\circ F_{Y_0(0,0)|D=1,M=0}(y)$ by $Q_{00} (y) \\equiv F_{Y_1|D=0,M=0}^{-1} \\circ F_{Y_0|D=0,M=0}(y) $.\nParsing $Y_0$ through $Q_{00}(\\cdot)$ in the treated group without mediator gives\n\\begin{equation} \\begin{array}{rl}\n E[Q_{00}(Y_0)|D=1,M=0] \\\\ \\qquad =  E[F_{Y_1|D=0,M=0}^{-1} \\circ F_{Y_0|D=0,M=0}(Y_0 )|D=1,M=0], \\\\\n \\qquad =  E[F_{Y_1(0,0)|D=0,M=0}^{-1} \\circ F_{Y_0(0,0)|D=0,M=0}(Y_0(1,0))|D=1,M=0], \\\\\n\\qquad \\stackrel{A1,A3b}{=}  E[h(0,0,1,h^{-1}(0,0,0; Y_0(1,0)))|D=1,M=0], \\\\\n\\qquad \\stackrel{A2}{=} E[h(0,0,1,h^{-1}(0,0,0; Y_0(0,0)))|D=1,M=0], \\\\\n\\qquad \\stackrel{A1,A3a}{=}E[F_{Y_1(0,0)|D=1,M=0}^{-1} \\circ F_{Y_0(0,0)|D=1,M=0}(Y_0 (0,0))|D=1,M=0],\\\\\n\\qquad =E[Y_1(0,0)|D=1,M=0]=E[Y_1(0,0)|D=1,M(1)=0], \\end{array}\n\\end{equation}\nwhich has data support because of Assumption 4a.\n\\end{document}\n"}
{"0.1 Quantile direct e\u000bect under d =1 conditional on D =1\nand M (1) =0\nIn the following, we prove that\n\u00121;0\n1(q;1) =F\u00001\nY1(1;0)jD=1;M(1)=0(q)\u0000F\u00001\nY1(0;0)jD=1;M(1)=0(q);\n=F\u00001\nY1jD=1;M=0(q)\u0000F\u00001\nQ00(Y0)jD=1;M=0(q):\nFor this purpose, we have to show that\nFY1(1;0)jD=1;M(1)=0(y) =FY1jD=1;M=0(y) and (1)\nFY1(0;0)jD=1;M(1)=0(y) =FQ00(Y0)jD=1;M=0(y); (2)\nwhich is su\u000ecient to show that the quantiles are also identi\fed. We can show () using\nthe observational rule FY1(1;0)jD=1;M(1)=0(y) =FY1jD=1;M=0(y) =E[1fY1\u0014ygjD=\n1;M= 0], with 1f\u0001gbeing the indicator function. Using (), we obtain\nFQ00(Y0)jD=1;M=0(y)\n=E[1fQ00(Y0)\u0014ygjD= 1;M= 0];\n=E[1fF\u00001\nY1jD=0;M=0\u000eFY0jD=0;M=0(Y0)\u0014ygjD= 1;M= 0];\n=E[1fY1(0;0)\u0014ygjD= 1;M= 0];\n=FY1(0;0)jD=1;M(1)=0(y);(3)\nwhich proves ().\n0.2 Average direct e\u000bect under d =0 conditional on D =0\nand M (0) =0\nIn the following, we show that \u00120;0\n1(0) =E[Y1(1;0)\u0000Y1(0;0)jD= 0;M(0) =\n0] =E[Q10(Y0)\u0000Y1jD= 0;M= 0]. Using the observational rule, we obtain\nE[Y1(0;0)jD= 0;M(0) = 0] =E[Y1jD= 0;M= 0]. Accordingly, we have to show\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\subsection{Quantile direct effect under $\\mathbf{d=1}$ conditional on $\\mathbf{D=1}$ and $\\mathbf{M(1)=0}$}\nIn the following, we prove that\n\\begin{align*}\\theta_1^{1,0}(q,1) = F_{Y_{1}(1,0)|D=1,M(1)=0}^{-1}(q)-F_{Y_{1}(0,0)|D=1,M(1)=0}^{-1}(q),\\\\= F_{Y_{1}|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).\n\\end{align*} For this purpose, we have to show that\n\\begin{align}\nF_{Y_{1}(1,0)|D=1,M(1)=0}(y)   = F_{Y_{1}|D=1,M=0}(y) \\mbox{ and}  \\\\\nF_{Y_{1}(0,0)|D=1,M(1)=0} (y)  =F_{Q_{00}(Y_{0})|D=1,M=0}(y) ,\n\\end{align}\nwhich is sufficient to show that the quantiles are also identified. We can show () using the observational rule $F_{Y_{1}(1,0)|D=1,M(1)=0}(y)= F_{Y_{1}|D=1,M=0}(y)= E[1\\{Y_1 \\leq y\\} |D=1,M=0]$, with $1\\{\\cdot\\}$ being the indicator function.\nUsing (), we obtain\n\\begin{equation}  \\begin{array}{rl}\nF_{Q_{00}(Y_{0})|D=1,M=0}(y)\\\\  \\qquad = E[1\\{Q_{00}(Y_0) \\leq y \\}|D=1,M=0],\\\\\n\\qquad =E[1\\{ F_{Y_1|D=0,M=0}^{-1} \\circ F_{Y_0|D=0,M=0}(Y_0 ) \\leq y \\} |D=1,M=0],\\\\\n\\qquad =E[1\\{ Y_1(0,0)\\leq y \\}|D=1,M=0],\\\\ \\qquad =F_{Y_{1}(0,0)|D=1,M(1)=0} (y), \\end{array}\n\\end{equation}\nwhich proves ().\n\\subsection{Average direct effect under $\\mathbf{d=0}$ conditional on $\\mathbf{D=0}$ and $\\mathbf{M(0)=0}$}\nIn the following, we show that $\\theta_1^{0,0}(0)= E[Y_1(1,0)-Y_1(0,0)|D=0,M(0)=0]=  E[Q_{10}(Y_0)-Y_1|D=0,M=0]$. Using the observational rule, we obtain $E[Y_1(0,0)|D=0,M(0)=0]=E[Y_1|D=0,M=0]$. Accordingly, we have to show\n\\end{document}\n"}
{"that E[Y1(1;0)jD= 0; M(0) = 0] = E[Q10(Y0)jD= 0; M= 0] to \fnish the\nproof. First, we use () to evaluate FY1(1;0)jD=0;M=0(y) ath(1;0;1; u)\nFY1(1;0)jD=0;M=0(h(1;0;1; u)) =FUj10(h\u00001(1;0;1;h(1;0;1; u))) = FUj10(u):\nApplying F\u00001\nY1(1;0)jD=0;M=0(q) to both sides, we have\nh(1;0;1; u) =F\u00001\nY1(1;0)jD=0;M=0(FUj10(u)): (1)\nSecond, for FY0(1;0)jD=0;M=0(y) we have\nF\u00001\nUj10(FY0(1;0)jD=0;M=0(y)) =h\u00001(1;0;0;y); (2)\nusing (). Combining () and () yields,\nh(1;0;1; h\u00001(1;0;0;y)) =F\u00001\nY1(1;0)jD=0;M=0\u000eFY0(1;0)jD=0;M=0(y): (3)\nNote that h(1;0;1; h\u00001(1;0;0;y)) maps the period 1 (potential) outcome of an indi-\nvidual with the outcome yin period 0 under treatment without the mediator. Ac-\ncordingly, E[F\u00001\nY1(1;0)jD=0;M=0\u000eFY0(1;0)jD=0;M=0(Y0)jD= 0; M= 0] = E[Y1(1;0)jD=\n1; M= 0]. We can identify FY0(1;0)jD=0;M=0(y) under Assumption 2, but we cannot\nidentify FY1(1;0)jD=0;M=0(y). However, we show in the following that we can identify\nthe overall quantile-quantile transform F\u00001\nY1(1;0)jD=0;M=0\u000eFY0(1;0)jD=0;M=0(y) under\nthe additional Assumption 3a. First, we use () to evaluate FY1(1;0)jD=1;M=0(y) at\nh(1;0;1; u)\nFY1(1;0)jD=10;M=0(h(1;0;1; u)) =FUj10(h\u00001(1;0;1;h(1;0;1; u))) = FUj10(u):\nApplying F\u00001\nY1(1;0)jD=1;M=0(q) to both sides, we have\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nthat $E[Y_1(1,0)|D=0,M(0)=0]=E[Q_{10}(Y_0)|D=0,M=0]$ to finish the proof.\nFirst, we use () to evaluate $F_{Y_1(1,0)|D=0,M=0}(y)$ at $h(1,0,1,u)$\n\\begin{equation*}\nF_{Y_1(1,0)|D=0,M=0}(h(1,0,1,u)) = F_{U|10} ( h^{-1}(1,0,1;h(1,0,1,u)))  =F_{U|10} ( u).\n\\end{equation*}\nApplying $F_{Y_1(1,0)|D=0,M=0}^{-1}(q)$ to both sides, we have\n\\begin{equation} \nh(1,0,1,u)  =F_{Y_1(1,0)|D=0,M=0}^{-1}(F_{U|10} ( u)).\n\\end{equation}\nSecond, for $F_{Y_0(1,0)|D=0,M=0}(y)$ we have\n\\begin{equation}\nF_{U|10}^{-1} ( F_{Y_0(1,0)|D=0,M=0}(y)) =   h^{-1}(1,0,0;y),\n\\end{equation}\nusing (). Combining () and () yields,\n\\begin{equation} \nh(1,0,1,h^{-1}(1,0,0;y))  =F_{Y_1(1,0)|D=0,M=0}^{-1} \\circ F_{Y_0(1,0)|D=0,M=0}(y) .\n\\end{equation}\nNote that $h(1,0,1,h^{-1}(1,0,0;y))$ maps the period 1 (potential) outcome of an individual with the outcome $y$ in period 0 under treatment without the mediator. Accordingly, $E[F_{Y_1(1,0)|D=0,M=0}^{-1} \\circ F_{Y_0(1,0)|D=0,M=0}(Y_0)|D=0,M=0]= E[Y_1(1,0)|D=1,M=0]$. We can identify $F_{Y_0(1,0)|D=0,M=0}(y)$ under Assumption 2, but we cannot identify $F_{Y_1(1,0)|D=0,M=0}(y)$. However, we show in the following that we can identify the overall quantile-quantile transform $F_{Y_1(1,0)|D=0,M=0}^{-1} \\circ F_{Y_0(1,0)|D=0,M=0}(y)$ under the additional Assumption 3a.\nFirst, we use () to evaluate $F_{Y_1(1,0)|D=1,M=0}(y)$ at $h(1,0,1,u)$\n\\begin{equation*}\nF_{Y_1(1,0)|D=10,M=0}(h(1,0,1,u)) = F_{U|10} ( h^{-1}(1,0,1;h(1,0,1,u)))  =F_{U|10} ( u).\n\\end{equation*}\nApplying $F_{Y_1(1,0)|D=1,M=0}^{-1}(q)$ to both sides, we have\n\\end{document}\n"}
{"h(1;0;1; u) =F\u00001\nY1(1;0)jD=1;M=0(FUj10(u)): (1)\nSecond, for FY0(1;0)jD=0;M=0(y) we have\nF\u00001\nUj10(FY0(1;0)jD=1;M=0(y)) =h\u00001(1;0;0;y); (2)\nusing (). Combining () and () yields,\nh(1;0;1; h\u00001(1;0;0;y)) =F\u00001\nY1(1;0)jD=1;M=0\u000eFY0(1;0)jD=1;M=0(y): (3)\nThe left sides of () and () are equal. In contrast to (), () contains only distributions\nthat can be identi\fed from observable data. In particular, FYt(1;0)jD=1;M=0(y) =\nPr(Yt(1;0)\u0014yjD= 1; M = 0) = Pr( Yt\u0014yjD= 1; M = 0). Accordingly,\nwe can identify F\u00001\nY1(1;0)jD=0;M=0\u000eFY0(1;0)jD=0;M=0(y) by Q10(y)\u0011F\u00001\nY1jD=1;M=0\u000e\nFY0jD=1;M=0(y). Parsing Y0through Q10(\u0001) in the non-treated group without me-\ndiator gives\nE[Q10(Y0)jD= 0; M= 0]\n=E[F\u00001\nY1jD=1;M=0\u000eFY0jD=1;M=0(Y0)jD= 0; M= 0];\n=E[F\u00001\nY1(1;0)jD=1;M=0\u000eFY0(1;0)jD=1;M=0(Y0(0;0))jD= 0; M= 0];\nA1;A3a=E[h(1;0;1; h\u00001(1;0;0;Y0(0;0)))jD= 0; M= 0];\nA2=E[h(1;0;1; h\u00001(1;0;0;Y0(1;0)))jD= 1; M= 0];\nA1;A3b=E[F\u00001\nY1(1;0)jD=0;M=0\u000eFY0(1;0)jD=0;M=0(Y0(1;0))jD= 0; M= 0];\n=E[Y1(1;0)jD= 0; M= 0] = E[Y1(1;0)jD= 0; M(0) = 0] ;(4)\nwhich has data support because of Assumption 4b.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\n\\begin{equation} \nh(1,0,1,u)  =F_{Y_1(1,0)|D=1,M=0}^{-1}(F_{U|10} ( u)).\n\\end{equation}\nSecond, for $F_{Y_0(1,0)|D=0,M=0}(y)$ we have\n\\begin{equation} \nF_{U|10}^{-1} ( F_{Y_0(1,0)|D=1,M=0}(y)) =   h^{-1}(1,0,0;y),\n\\end{equation}\nusing (). Combining () and () yields,\n\\begin{equation} \nh(1,0,1,h^{-1}(1,0,0;y))  =F_{Y_1(1,0)|D=1,M=0}^{-1} \\circ F_{Y_0(1,0)|D=1,M=0}(y) .\n\\end{equation}\nThe left sides of () and () are equal. In contrast to (), () contains only distributions that can be identified from observable data. In particular, $F_{Y_t(1,0)|D=1,M=0}(y)  =\\Pr(Y_t(1,0) \\leq y|D=1,M=0) =  \\Pr(Y_t \\leq y|D=1,M=0)$. Accordingly, we can identify $F_{Y_1(1,0)|D=0,M=0}^{-1} \\circ F_{Y_0(1,0)|D=0,M=0}(y)$ by $Q_{10} (y) \\equiv F_{Y_1|D=1,M=0}^{-1} \\circ F_{Y_0|D=1,M=0}(y) $.\nParsing $Y_0$ through $Q_{10}(\\cdot)$ in the non-treated group without mediator gives\n\\begin{equation}  \\begin{array}{rl}\nE[Q_{10}(Y_0)|D=0,M=0] \\\\  \\qquad =  E[F_{Y_1|D=1,M=0}^{-1} \\circ F_{Y_0|D=1,M=0}(Y_0 )|D=0,M=0], \\\\\n \\qquad =  E[F_{Y_1(1,0)|D=1,M=0}^{-1} \\circ F_{Y_0(1,0)|D=1,M=0}(Y_0(0,0))|D=0,M=0], \\\\\n\\qquad \\stackrel{A1,A3a}{=}  E[h(1,0,1,h^{-1}(1,0,0; Y_0(0,0)))|D=0,M=0], \\\\\n\\qquad \\stackrel{A2}{=} E[h(1,0,1,h^{-1}(1,0,0; Y_0(1,0)))|D=1,M=0] ,\\\\\n\\qquad \\stackrel{A1,A3b}{=}E[F_{Y_1(1,0)|D=0,M=0}^{-1} \\circ F_{Y_0(1,0)|D=0,M=0}(Y_0 (1,0))|D=0,M=0],\\\\\n\\qquad =E[Y_1(1,0)|D=0,M=0]=E[Y_1(1,0)|D=0,M(0)=0],\n\\end{array} \\end{equation}\nwhich has data support because of Assumption 4b.\n\\end{document}\n"}
{"0.1 Quantile direct e\u000bect under d =0 conditional on D =0\nand M (0) =0\nIn the following, we prove that\n\u00120;0\n1(q;0) =F\u00001\nY1(1;0)jD=0;M(0)=0(q)\u0000F\u00001\nY1(0;0)jD=0;M(0)=0(q);\n=F\u00001\nQ10(Y0)jD=0;M=0(q)\u0000F\u00001\nY1jD=0;M=0(q):\nFor this purpose, we have to show that\nFY1(1;0)jD=0;M(0)=0(y) =FQ10(Y0)jD=0;M=0(y) and (1)\nFY1(0;0)jD=0;M(0)=0(y) =FY1jD=0;M=0(y); (2)\nwhich is su\u000ecient to show that the quantiles are also identi\fed. We can show () using\nthe observational rule FY1(0;0)jD=0;M(0)=0(y) =FY1jD=0;M=0(y) =E[1fY1\u0014ygjD=\n0;M= 0]. Using (), we obtain\nFQ10(Y0)jD=0;M=0(y)\n=E[1fQ10(Y0)\u0014ygjD= 0;M= 0];\n=E[1fF\u00001\nY1jD=1;M=0\u000eFY0jD=1;M=0(Y0)\u0014ygjD= 0;M= 0];\n=E[1fY1(1;0)\u0014ygjD= 0;M= 0];\n=FY1(1;0)jD=0;M(0)=0(y);\nwhich proves ().\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\subsection{Quantile direct effect under $\\mathbf{d=0}$ conditional on $\\mathbf{D=0}$ and $\\mathbf{M(0)=0}$}\nIn the following, we prove that\n\\begin{align*}\n\\theta_1^{0,0}(q,0) = F_{Y_{1}(1,0)|D=0,M(0)=0}^{-1}(q)-F_{Y_{1}(0,0)|D=0,M(0)=0}^{-1}(q),\\\\= F_{Q_{10}(Y_{0})|D=0,M=0}^{-1}(q)-F_{Y_{1}|D=0,M=0}^{-1}(q).\n\\end{align*}\nFor this purpose, we have to show that\n\\begin{align}\nF_{Y_{1}(1,0)|D=0,M(0)=0}(y)   = F_{Q_{10}(Y_{0})|D=0,M=0}(y) \\mbox{ and}  \\\\\nF_{Y_{1}(0,0)|D=0,M(0)=0}(y)  =F_{Y_{1}|D=0,M=0}(y) ,\n\\end{align}\nwhich is sufficient to show that the quantiles are also identified. We can show () using the observational rule $F_{Y_{1}(0,0)|D=0,M(0)=0}(y)= F_{Y_{1}|D=0,M=0}(y)= E[1\\{Y_1 \\leq y\\} |D=0,M=0]$.\nUsing (), we obtain\n\\begin{equation*} \\begin{array}{rl}\nF_{Q_{10}(Y_{0})|D=0,M=0}(y) \\\\  \\qquad = E[1\\{Q_{10}(Y_0) \\leq y \\}|D=0,M=0],\\\\\n\\qquad =E[1\\{ F_{Y_1|D=1,M=0}^{-1} \\circ F_{Y_0|D=1,M=0}(Y_0 ) \\leq y \\} |D=0,M=0],\\\\\n\\qquad =E[1\\{ Y_1(1,0)\\leq y \\}|D=0,M=0],\\\\\n\\qquad =F_{Y_{1}(1,0)|D=0,M(0)=0}(y), \\end{array}\n\\end{equation*}\nwhich proves ().\n\\end{document}\n"}
{"1 Proof of Theorem 2\n1.1 Average direct e\u000bect under d =0 conditional on D =0\nand M (0) =1\nIn the following, we show that \u00120;1\n1(0) =E[Y1(1;1)\u0000Y1(0;1)jD= 0;M(0) =\n1] =E[Q11(Y0)\u0000Y1jD= 0;M= 1]. Using the observational rule, we obtain\nE[Y1(0;1)jD= 0;M(0) = 1] = E[Y1jD= 0;M= 1]. Accordingly, we have to\nshow thatE[Y1(1;1)jD= 0;M(0) = 1] = E[Q11(Y0)jD= 0;M= 1] to \fnish the\nproof. Under Assumptions 1 and 5a, the conditional potential outcome distribution\nfunction equals\nFYt(d;0)jD=1;M=0(y)A1= Pr(h(d;m;t;U )\u0014yjD= 0;M= 1;T=t);\n= Pr(U\u0014h\u00001(d;m;t ;y)jD= 0;M= 1;T=t);\nA5a= Pr(U\u0014h\u00001(d;m;t ;y)jD= 0;M= 1);\n=FUj01(h\u00001(d;m;t ;y));(1)\nford;d02 f0;1g. We use these quantities in the following. First, evaluating\nFY1(1;1)jD=0;M=1(y) ath(1;1;1;u) gives\nFY1(1;1)jD=0;M=1(h(1;1;1;u)) =FUj01(h\u00001(1;1;1;h(1;1;1;u))) =FUj01(u):\nApplyingF\u00001\nY1(1;1)jD=0;M=1(q) to both sides, we have\nh(1;1;1;u) =F\u00001\nY1(1;1)jD=0;M=1(FUj01(u)): (2)\nSecond, for FY0(1;1)jD=0;M=1(y) we have\nF\u00001\nUj01(FY0(1;1)jD=0;M=1(y)) =h\u00001(1;1;0;y): (3)\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\section{Proof of Theorem 2 }\n\\subsection{Average direct effect under $\\mathbf{d=0}$ conditional on $\\mathbf{D=0}$ and $\\mathbf{M(0)=1}$}\nIn the following, we show that $\\theta_1^{0,1}(0)= E[Y_1(1,1)-Y_1(0,1)|D=0,M(0)=1]=  E[Q_{11}(Y_0)-Y_1|D=0,M=1]$. Using the observational rule, we obtain $E[Y_1(0,1)|D=0,M(0)=1]=E[Y_1|D=0,M=1]$. Accordingly, we have to show that $E[Y_1(1,1)|D=0,M(0)=1]=E[Q_{11}(Y_0)|D=0,M=1]$ to finish the proof.\nUnder Assumptions 1 and 5a, the conditional potential outcome distribution function equals\n\\begin{equation} \n\\begin{array}{rl}\n F_{Y_t(d,0)|D=1,M=0}(y)  \\stackrel{A1}{=} \\Pr(h(d,m,t,U) \\leq y|D=0,M=1,T=t) ,\\\\\n= \\Pr(U \\leq h^{-1}(d,m,t;y)|D=0,M=1,T=t) ,\\\\\n\\stackrel{A5a}{=} \\Pr(U \\leq h^{-1}(d,m,t;y)|D=0,M=1) ,\\\\\n= F_{U|01} ( h^{-1}(d,m,t;y)),\n\\end{array}\n\\end{equation}\nfor $d,d' \\in \\{0,1\\}$. We use these quantities in the following.\nFirst, evaluating $F_{Y_1(1,1)|D=0,M=1}(y)$ at $h(1,1,1,u)$ gives\n\\begin{equation*}\nF_{Y_1(1,1)|D=0,M=1}(h(1,1,1,u)) = F_{U|01} ( h^{-1}(1,1,1;h(1,1,1,u)))  =F_{U|01} ( u).\n\\end{equation*}\nApplying $F_{Y_1(1,1)|D=0,M=1}^{-1}(q)$ to both sides, we have\n\\begin{equation} \nh(1,1,1,u)  =F_{Y_1(1,1)|D=0,M=1}^{-1}(F_{U|01} ( u)).\n\\end{equation}\nSecond, for $F_{Y_0(1,1)|D=0,M=1}(y)$ we have\n\\begin{equation}\nF_{U|01}^{-1} ( F_{Y_0(1,1)|D=0,M=1}(y)) =   h^{-1}(1,1,0;y).\n\\end{equation}\n\\end{document}\n"}
{"Combining () and () yields,\nh(1;1;1; h\u00001(1;1;0;y)) =F\u00001\nY1(1;1)jD=0;M=1\u000eFY0(1;1)jD=0;M=1(y): (1)\nNote that h(1;1;1; h\u00001(1;1;0;y)) maps the period 1 (potential) outcome of an in-\ndividual with the outcome yin period 0 under treatment with the mediator. Ac-\ncordingly, E[F\u00001\nY1(1;1)jD=0;M=1\u000eFY0(1;1)jD=0;M=1(Y0)jD= 0; M= 1] = E[Y1(1;1)jD=\n0; M= 1]. We can identify FY0(1;1)jD=0;M=1(y) =FY0jD=0;M=1(y) under Assump-\ntion 2, but we cannot identify FY1(1;1)jD=0;M=1(y). However, we show in the follow-\ning that we can identify the overall quantile-quantile transform F\u00001\nY1(1;1)jD=0;M=1\u000e\nFY0(1;1)jD=0;M=1(y) under the additional Assumption 5b. Under Assumptions 1 and\n5b, the conditional potential outcome distribution function equals\nFYt(d;1)jD=1;M=1(y)A1= Pr( h(d; m; t; U )\u0014yjD= 1; M= 1; T=t);\n= Pr( U\u0014h\u00001(d; m; t ;y)jD= 1; M= 1; T=t);\nA5b= Pr( U\u0014h\u00001(d; m; t ;y)jD= 1; M= 1);\n=FUj11(h\u00001(d; m; t ;y));(2)\nford; d02f0;1g. We repeat similar steps as above. First, evaluating FY1(1;1)jD=1;M=1(y)\nath(1;1;1; u) gives\nFY1(1;1)jD=1;M=1(h(1;1;1; u)) =FUj11(h\u00001(1;1;1;h(1;1;1; u))) = FUj11(u):\nApplying F\u00001\nY1(1;1)jD=1;M=1(q) to both sides, we have\nh(1;1;1; u) =F\u00001\nY1(1;1)jD=1;M=1(FUj11(u)): (3)\nSecond, for FY0(1;1)jD=1;M=1(y) we have\nF\u00001\nUj11(FY0(1;1)jD=1;M=1(y)) =h\u00001(1;1;1;y): (4)\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nCombining () and () yields,\n\\begin{equation} \nh(1,1,1,h^{-1}(1,1,0;y))  =F_{Y_1(1,1)|D=0,M=1}^{-1} \\circ F_{Y_0(1,1)|D=0,M=1}(y) .\n\\end{equation}\nNote that $h(1,1,1,h^{-1}(1,1,0;y))$ maps the period 1 (potential) outcome of an individual with the outcome $y$ in period 0 under treatment with the mediator. Accordingly, $E[F_{Y_1(1,1)|D=0,M=1}^{-1} \\circ F_{Y_0(1,1)|D=0,M=1}(Y_0)|D=0,M=1]= E[Y_1(1,1)|D=0,M=1]$. We can identify $F_{Y_0(1,1)|D=0,M=1}(y)= F_{Y_0|D=0,M=1}(y)$ under Assumption 2, but we cannot identify $F_{Y_1(1,1)|D=0,M=1}(y)$. However, we show in the following that we can identify the overall quantile-quantile transform $F_{Y_1(1,1)|D=0,M=1}^{-1} \\circ F_{Y_0(1,1)|D=0,M=1}(y)$ under the additional Assumption 5b.\nUnder Assumptions 1 and 5b, the conditional potential outcome distribution function equals\n\\begin{equation} \n\\begin{array}{rl}\n F_{Y_t(d,1)|D=1,M=1}(y)  \\stackrel{A1}{=} \\Pr(h(d,m,t,U) \\leq y|D=1,M=1,T=t) ,\\\\\n= \\Pr(U \\leq h^{-1}(d,m,t;y)|D=1,M=1,T=t) ,\\\\\n\\stackrel{A5b}{=} \\Pr(U \\leq h^{-1}(d,m,t;y)|D=1,M=1) ,\\\\\n= F_{U|11} ( h^{-1}(d,m,t;y)),\n\\end{array}\n\\end{equation}\nfor $d,d' \\in \\{0,1\\}$. We repeat similar steps as above. First, evaluating $F_{Y_1(1,1)|D=1,M=1}(y)$ at $h(1,1,1,u)$ gives\n\\begin{equation*}\nF_{Y_1(1,1)|D=1,M=1}(h(1,1,1,u)) = F_{U|11} ( h^{-1}(1,1,1;h(1,1,1,u)))  =F_{U|11} ( u).\n\\end{equation*}\nApplying $F_{Y_1(1,1)|D=1,M=1}^{-1}(q)$ to both sides, we have\n\\begin{equation} \nh(1,1,1,u)  =F_{Y_1(1,1)|D=1,M=1}^{-1}(F_{U|11} ( u)).\n\\end{equation}\nSecond, for $F_{Y_0(1,1)|D=1,M=1}(y)$ we have\n\\begin{equation} \nF_{U|11}^{-1} ( F_{Y_0(1,1)|D=1,M=1}(y)) =   h^{-1}(1,1,1;y).\n\\end{equation}\n\\end{document}\n"}
{"1 Proof of Theorem 3\n1.1 Average direct e\u000bect on the never-takers\nIn the following, we show that \u0012n\n1=E[Y1(1;0)\u0000Y1(0;0)j\u001c=n] =E[Y1\u0000Q00(Y0)jD=\n1;M= 0]. From (), we obtain the \frst ingredient E[Y1(1;0)j\u001c=n] =E[Y1jD=\n1;M= 0]. Furthermore, from () we have E[Q00(Y0)jD= 1;M= 0] =E[Y1(0;0)jD=\n1;M(1) = 0]. Under Assumption 7 and 8,\nE[Y1(0;0)jD= 1;M(1) = 0]A7=E[Y1(0;0)jD= 1;\u001c=n]\nA8=E[Y1(0;0)j\u001c=n]:(1)\n1.2 Quantile direct e\u000bect on the never-takers\nWe prove that\n\u0012n\n1(q) =F\u00001\nY1(1;0)j\u001c=n(q)\u0000F\u00001\nY1(0;0)j\u001c=n(q);\n=F\u00001\nY1jD=1;M=0(q)\u0000F\u00001\nQ00(Y0)jD=1;M=0(q):\nThis requires showing that\nFY1(1;0)j\u001c=n(y) =FY1jD=1;M=0(y) and (2)\nFY1(0;0)j\u001c=n(y) =FQ00(Y0)jD=1;M=0(y): (3)\nUnder Assumptions 7 and 8,\nFYtjD=1;M=0(y) =E[1fYt\u0014ygjD= 1;M= 0]\nA7;A8=E[1fYt(1;0)\u0014ygj\u001c=n]\n=FYt(1;0)j\u001c=n(y);(4)\nwhich proves (). From (), we have\nFQ00(Y0)jD=1;M=0(y) =FY1(0;0)jD=1;M(1)=0(y) =E[1fY1(0;0)\u0014ygjD= 1;M(1) = 0]:\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\section{Proof of Theorem 3 }\n\\subsection{Average direct effect on the never-takers}\nIn the following, we show that $\\theta_1^n= E[Y_1(1,0)-Y_1(0,0)|\\tau=n]= E[Y_1-Q_{00}(Y_0)|D=1,M=0]$. From (), we obtain the first ingredient $E[Y_1(1,0)|\\tau=n]=E[Y_1|D=1,M=0]$. Furthermore, from () we have $E[Q_{00}(Y_0)|D=1,M=0] = E[Y_1(0,0)|D=1,M(1)=0]$. Under Assumption 7 and 8,\n\\begin{equation}  \\begin{array}{rl} E[Y_1(0,0)|D=1,M(1)=0]\\stackrel{A7}{=}E[Y_1(0,0)|D=1,\\tau=n]\\\\\\stackrel{A8}{=}E[Y_1(0,0)|\\tau=n].\n\\end{array} \\end{equation}\n\\subsection{Quantile direct effect on the never-takers}\nWe prove that\n\\begin{align*}\n\\theta_1^n (q)= F_{Y_{1}(1,0)|\\tau=n}^{-1}(q)- F_{Y_{1}(0,0)|\\tau=n}^{-1}(q), \\\\\n= F_{Y_1|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).\n\\end{align*}\n This requires showing that\n\\begin{align}\nF_{Y_{1}(1,0)|\\tau=n}(y) =F_{Y_1|D=1,M=0}(y) \\mbox{ and} \\\\\nF_{Y_{1}(0,0)|\\tau=n}(y) = F_{Q_{00}(Y_{0})|D=1,M=0}(y). \n\\end{align}\nUnder Assumptions 7 and 8,\n\\begin{equation} \n\\begin{array}{rl}\nF_{Y_t|D=1,M=0} (y) = E[1\\{Y_t\\leq y\\}|D=1,M=0] \\\\ \\stackrel{A7,A8}{=}E[1\\{Y_t(1,0)\\leq y\\}|\\tau=n] \\\\ = F_{Y_{t}(1,0)|\\tau=n} (y), \\end{array}\n\\end{equation}\nwhich proves (). From (), we have\n\\begin{equation*}\nF_{Q_{00}(Y_{0})|D=1,M=0}(y) = F_{Y_{1}(0,0)|D=1,M(1)=0} (y) = E[1\\{Y_1(0,0) \\leq y\\}|D=1,M(1)=0].\n\\end{equation*}\n\\end{document}\n"}
{"Under Assumption 7 and 8,\nE[1fY1(0;0)\u0014ygjD= 1;M(1) = 0]A7;A8=E[1fY1(0;0)\u0014ygj\u001c=n]\n=FY1(0;0)j\u001c=n(y);(1)\nwhich proves.\n0.1 Average direct e\u000bect under d =0 on compliers\nIn the following, we show that\n\u0012c\n1(0) =E[Y1(1;0)\u0000Y1(0;0)j\u001c=c];\n=p0j0\np0j0\u0000p0j1E[Q10(Y0)\u0000Y1jD= 0;M= 0]\n\u0000p0j1\np0j0\u0000p0j1E[Y1\u0000Q00(Y0)jD= 1;M= 0]:\nPlugging () in () under T= 1, we obtain\nE[Y1jD= 0;M= 0] =pn\npn+pcE[Q00(Y0)jD= 1;M= 0]\n+pc\npn+pcE[Y1(0;0)j\u001c=c]:\nThis allows identifying\nE[Y1(0;0)j\u001c=c] =p0j0\np0j0\u0000p0j1E[Y1jD= 0;M= 0]\n\u0000p0j1\np0j0\u0000p0j1E[Q00(Y0)jD= 1;M= 0]:(2)\nAccordingly, we have to show the identi\fcation of E[Y1(1;0)jc] to \fnish the proof.\nFrom () we have E[Y1(1;0)jD= 0;M= 0] =E[Q10(Y0)jD= 0;M= 0]. Applying\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nUnder Assumption 7 and 8,\n\\begin{equation}  \\begin{array}{rl} E[1\\{Y_1(0,0) \\leq y\\}|D=1,M(1)=0]\\stackrel{A7,A8}{=}E[1\\{Y_1(0,0) \\leq y\\}|\\tau=n]\\\\  = F_{Y_{1}(0,0)|\\tau=n}(y),\n\\end{array} \\end{equation}\nwhich proves.\n\\subsection{Average direct effect under $\\mathbf{d = 0}$ on compliers}\nIn the following, we show that\n\\begin{align*}\n\\displaystyle \\theta_1^{c}(0) = E[Y_1(1,0)-Y_1(0,0)|\\tau=c], \\\\ =\n\\frac{p_{0|0}}{p_{0|0} - p_{0|1}}E[Q_{10}(Y_0)- Y_1|D=0,M=0] \\\\\n  - \\frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Y_1 - Q_{00}(Y_0)|D=1,M=0].\n\\end{align*}\nPlugging () in () under $T=1$, we obtain\n\\begin{equation*}\n\\begin{array}{rl}\nE[Y_1|D=0,M=0]  = \\displaystyle  \\frac{p_n}{p_n + p_c}E[Q_{00}(Y_0)|D=1,M=0] \\\\\n\\displaystyle \\quad +   \\frac{p_c}{p_n + p_c}E[Y_1(0,0)|\\tau=c].\n\\end{array}\n\\end{equation*}\nThis allows identifying\n\\begin{equation} \n\\begin{array}{rl}\nE[Y_1(0,0)|\\tau=c] = \\displaystyle \\frac{p_{0|0}}{ p_{0|0} - p_{0|1}} E[Y_1|D=0,M=0] \\\\\n \\displaystyle \\quad -\\frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Q_{00}(Y_0)|D=1,M=0].\n\\end{array}\n\\end{equation}\nAccordingly, we have to show the identification of $E[Y_1(1,0)|c]$ to finish the proof. From () we have $E[Y_1(1,0)|D=0,M=0] = E[Q_{10}(Y_0)|D=0,M=0]$. Applying\n\\end{document}\n"}
{"the law of iterative expectations, gives\nE[Y1(1;0)jD= 0;M= 0] =pn\npn+pcE[Y1(1;0)jD= 0;M= 0;\u001c=n]\n+pc\npn+pcE[Y1(1;0)jD= 0;M= 0;\u001c=c];\nA7=pn\npn+pcE[Y1(1;0)j\u001c=n] +pc\npn+pcE[Y1(1;0)j\u001c=c]:\nAfter some rearrangements and using (), we obtain\nE[Y1(1;0)j\u001c=c] =pn+pc\npcE[Q10(Y0)jD= 0;M= 0]\u0000pn\npcE[Y1jD= 1;M= 0]:\nThis gives\nE[Y1(1;0)j\u001c=c] =p0j0\np0j0\u0000p0j1E[Q10(Y0)jD= 0;M= 0]\n\u0000p0j1\np0j0\u0000p0j1E[Y1jD= 1;M= 0];(1)\nusingpn=p0j1, andpc+pn=p0j0.\n0.1 Quantile direct e\u000bect under d =0 on compliers\nWe show that\nFY1(1;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FQ10(Y0)jD=0;M=0(y)\u0000p0j1\np0j0\u0000p0j1cFY1jD=1;M=0(y) and\nFY1(0;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FY1jD=0;M=0(y)\u0000p0j1\np0j0\u0000p0j1FQ00(Y0)jD=1;M=0(y);\nwhich proves that \u0012c\n1(q;0) =F\u00001\nY1(1;0)jc(q)\u0000F\u00001\nY1(0;0)jc(q) is identi\fed. From (), we have\nFY1(1;0)jD=0;M(0)=0(y) =FQ10(Y0)jD=0;M=0(y). Applying the law of iterative gives\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nthe law of iterative expectations, gives\n\\begin{align*}\nE[Y_1(1,0)|D=0,M=0] = \\frac{p_n}{p_n+p_c} E[Y_1(1,0)|D=0,M=0, \\tau=n] \\\\+ \\frac{p_c}{p_n+p_c}E[Y_1(1,0)|D=0,M=0,\\tau= c], \\\\\n\\stackrel{A7}{=} \\frac{p_n}{p_n+ p_c} E[Y_1(1,0)|\\tau=n] + \\frac{p_c}{p_n+p_c}E[Y_1(1,0)|\\tau= c].\n\\end{align*}\nAfter some rearrangements and using (), we obtain\n\\begin{equation*}\nE[Y_1(1,0)|\\tau= c]  = \\frac{p_n+p_c}{p_c} E[Q_{10}(Y_0)|D=0,M=0] -   \\frac{p_n}{p_c} E[Y_1|D=1,M=0].\n\\end{equation*}\nThis gives\n\\begin{equation} \n\\begin{array}{rl}\nE[Y_1(1,0)|\\tau=c] =  \\displaystyle \\frac{p_{0|0}}{p_{0|0} - p_{0|1}}E[Q_{10}(Y_0)|D=0,M=0] \\\\  \\displaystyle \\quad  - \\frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Y_1|D=1,M=0],\n\\end{array}\n\\end{equation}\nusing $p_n = p_{0|1}$, and $p_c +p_n = p_{0|0}$.\n\\subsection{Quantile direct effect under $\\mathbf{d = 0}$ on compliers}\nWe show that\n\\begin{align*}\n F_{Y_{1}(1,0)|\\tau=c}(y) = \\frac{p_{0|0}}{p_{0|0} - p_{0|1}}  F_{Q_{10}(Y_{0})|D=0,M=0}(y) - \\frac{p_{0|1}}{ p_{0|0} - p_{0|1}c}  F_{Y_1|D=1,M=0}(y) \\mbox{ and} \\\\\nF_{Y_{1}(0,0)|\\tau=c}(y) = \\frac{p_{0|0}}{p_{0|0} - p_{0|1}} F_{Y_{1}|D=0,M=0}(y) - \\frac{p_{0|1} }{p_{0|0} - p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y)  ,\n\\end{align*}\nwhich proves that $\\theta_1^{c}(q,0) = F_{Y_{1}(1,0)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is identified.\nFrom (), we have $F_{Y_{1}(1,0)|D=0,M(0)=0}(y)   = F_{Q_{10}(Y_{0})|D=0,M=0}(y)$.\nApplying the law of iterative gives\n\\end{document}\n"}
{"expectations gives\nFY1(1;0)jD=0;M(0)=0(y) =pn\npn+pcFY1(1;0)jD=0;M(0)=0;\u001c=n(y)\n+pc\npn+pcFY1(1;0)jD=0;M(0)=0;\u001c=c(y);\nA7=pn\npn+pcFY1(1;0)j\u001c=n(y) +pc\npn+pcFY1(1;0)j\u001c=c(y):\nUsing () and rearranging the equation gives,\nFY1(1;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FQ10(Y0)jD=0;M=0(y)\u0000p0j1\np0j0\u0000p0j1FY1jD=1;M=0(y):(1)\nIn analogy to (), the outcome distribution under D= 0 andM= 0 equals:\nFY1jD=0;M=0(y) =pn\npn+pcFY1(0;0)j\u001c=n(y) +pc\npn+pcFY1(0;0)j\u001c=c(y):\nUsing () and rearranging the equation gives\nFY1(0;0)j\u001c=c(y) =p0j0\np0j0\u0000p0j1FY1jD=0;M=0(y)\u0000p0j1\np0j0\u0000p0j1FQ00(Y0)jD=1;M=0(y):(2)\n1 Proof of Theorem 4\n1.1 Average direct e\u000bect on the always-takers\nIn the following, we show that \u0012a\n1=E[Y1(1;1)\u0000Y1(0;1)j\u001c=a] =E[Q11(Y0)\u0000Y1jD=\n0;M= 1]. From (), we obtain the \frst ingredient E[Y1(0;1)ja] =E[Y1jD= 0;M=\n1]. Furthermore, from () we have E[Q11(Y0)jD= 0;M= 1] =E[Y1(1;1)jD=\n0;M(0) = 1]. Under Assumption 7 and 8,\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}expectations gives\n\\begin{align*}\nF_{Y_{1}(1,0)|D=0,M(0)=0}(y)   = \\frac{p_n}{p_n + p_c} F_{Y_{1}(1,0)|D=0,M(0)=0,\\tau=n}(y)\\\\  +\\frac{p_c}{p_n + p_c} F_{Y_{1}(1,0)|D=0,M(0)=0,\\tau=c}(y),\\\\\n \\stackrel{A7}{=} \\frac{p_n}{p_n + p_c} F_{Y_{1}(1,0)|\\tau=n}(y) +\\frac{p_c}{p_n + p_c} F_{Y_{1}(1,0)|\\tau=c}(y).\n\\end{align*}\nUsing () and rearranging the equation gives,\n\\begin{equation} \nF_{Y_{1}(1,0)|\\tau=c}(y) = \\frac{p_{0|0}}{p_{0|0} - p_{0|1}}  F_{Q_{10}(Y_{0})|D=0,M=0}(y) - \\frac{p_{0|1}}{ p_{0|0} - p_{0|1}}  F_{Y_1|D=1,M=0}(y).\n\\end{equation}\nIn analogy to (), the outcome distribution under $D=0$ and $M=0$ equals:\n\\begin{equation*}\nF_{Y_{1}|D=0,M=0}(y) =\\frac{p_n}{p_n + p_c}F_{Y_{1}(0,0)|\\tau=n}(y)  + \\frac{p_c}{p_n + p_c}F_{Y_{1}(0,0)|\\tau=c}(y).\n\\end{equation*}\nUsing () and rearranging the equation gives\n\\begin{equation} \nF_{Y_{1}(0,0)|\\tau=c}(y)= \\frac{p_{0|0}}{p_{0|0} - p_{0|1}} F_{Y_{1}|D=0,M=0}(y) - \\frac{p_{0|1} }{p_{0|0} - p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y) .\n\\end{equation}\n\\section{Proof of Theorem 4 }\n \\subsection{Average direct effect on the\nalways-takers}\nIn the following, we show that $\\theta_1^a= E[Y_1(1,1)-Y_1(0,1)|\\tau=a]= E[Q_{11}(Y_0)-Y_1|D=0,M=1]$. From (), we obtain the first ingredient $E[Y_1(0,1)|a]=E[Y_1|D=0,M=1]$. Furthermore, from () we have $E[Q_{11}(Y_0)|D=0,M=1] = E[Y_1(1,1)|D=0,M(0)=1]$. Under Assumption 7 and 8,\n\\end{document}\n"}
{"0.1 Quantile direct e\u000bect on the always-takers\nWe prove that\n\u0012a\n1(q) =F\u00001\nY1(1;1)j\u001c=a(q)\u0000F\u00001\nY1(0;1)j\u001c=a(q);\n=F\u00001\nQ11(Y0)jD=0;M=1(q)\u0000F\u00001\nY1jD=0;M=1(q):\nThis requires showing that\nFY1(1;1)j\u001c=a(y) =FQ11(Y0)jD=0;M=1(y) and (1)\nFY1(0;1)j\u001c=a(y) =FY1jD=0;M=1(y): (2)\nUnder Assumptions 7 and 8,\nFYtjD=0;M=1(y) =E[1fYt\u0014ygjD= 0;M= 1]\nA7;A8=E[1fYt(0;1)\u0014ygj\u001c=a]\n=FYt(0;1)j\u001c=a;(y):(3)\nwhich proves (). From (), we have\nFQ11(Y0)jD=0;M=1(y) =FY1(1;1)jD=0;M(0)=1(y) =E[1fY1(1;1)\u0014ygjD= 0;M(0) = 1]:\nUnder Assumption 7 and 8,\nE[1fY1(1;1)\u0014ygjD= 0;M(0) = 1]A7;A8=E[1fY1(1;1)\u0014ygj\u001c=a]\n=FY1(1;1)j\u001c=a(y);(4)\nwhich proves.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n \\subsection{Quantile direct effect on the\nalways-takers}\nWe prove that\n\\begin{align*}\n\\theta_1^a (q) = F_{Y_1(1,1)|\\tau=a}^{-1}(q)- F_{Y_1(0,1)|\\tau=a}^{-1}(q), \\\\\n= F_{Q_{11}(Y_0)|D=0,M=1}^{-1}(q)-F_{Y_1|D=0,M=1}^{-1}(q).\n\\end{align*}\nThis requires showing that\n\\begin{align}\nF_{Y_1(1,1)|\\tau=a}(y) =F_{Q_{11}(Y_0)|D=0,M=1}(y) \\mbox{ and} \\\\\nF_{Y_1(0,1)|\\tau=a}(y)  = F_{Y_1|D=0,M=1}(y). \n\\end{align}\nUnder Assumptions 7 and 8,\n\\begin{equation} \n\\begin{array}{rl}\nF_{Y_t|D=0,M=1} (y) = E[1\\{Y_t\\leq y\\}|D=0,M=1] \\\\ \\stackrel{A7,A8}{=}E[1\\{Y_t(0,1)\\leq y\\}|\\tau=a] \\\\ = F_{Y_{t}(0,1)|\\tau=a}, (y). \\end{array}\n\\end{equation}\nwhich proves (). From (), we have\n\\begin{equation*}\nF_{Q_{11}(Y_{0})|D=0,M=1}(y) = F_{Y_{1}(1,1)|D=0,M(0)=1}(y)=E[1\\{Y_1(1,1) \\leq y\\}|D=0,M(0)=1].\n\\end{equation*}\nUnder Assumption 7 and 8,\n\\begin{equation}  \\begin{array}{rl} E[1\\{Y_1(1,1) \\leq y\\}|D=0,M(0)=1]\\stackrel{A7,A8}{=}E[1\\{Y_1(1,1) \\leq y\\}|\\tau=a]\\\\  = F_{Y_1(1,1)|\\tau=a}(y),\n\\end{array} \\end{equation}\nwhich proves.\n\\end{document}\n"}
{"0.1 Average direct e\u000bect under d =1 on compliers\nIn the following, we show that\n\u0012c\n1(1) =E[Y1(1;1)\u0000Y1(0;1)j\u001c=c];\n=p1j1\np1j1\u0000p1j0E[Y1\u0000Q01(Y0)jD= 1;M= 1]\n\u0000p1j0\np1j1\u0000p1j0E[Q11(Y0)\u0000Y1jD= 0;M= 1]:\nPlugging () in (), we obtain\nE[Y1jD= 1;M= 1] =pa\npa+pcE[Q11(Y0)jD= 0;M= 1]\n+pc\npa+pcE[Y1(1;1)j\u001c=c]:\nThis allows identifying\nE[Y1(1;1)j\u001c=c] =p1j1\np1j1\u0000p1j0E[Y1jD= 1;M= 1]\n\u0000p1j0\np1j1\u0000p1j0E[Q11(Y0)jD= 0;M= 1]:(1)\nFrom () we have E[Y1(0;1)jD= 1;M= 1] =E[Q01(Y0)jD= 1;M= 1]. Applying\nthe law of iterative expectations, gives\nE[Y1(0;1)jD= 1;M= 1] =pa\npa+pcE[Y1(0;1)jD= 1;M= 1;\u001c=a]\n+pc\npa+pcE[Y1(0;1)jD= 1;M= 1;\u001c=c];\nA7=pa\npa+pcE[Y1(0;1)j\u001c=a] +pc\npa+pcE[Y1(0;1)j\u001c=c]:\nAfter some rearrangements and using (), we obtain\nE[Y1(0;1)j\u001c=c] =pa+pc\npcE[Q01(Y0)jD= 1;M= 1]\u0000pa\npcE[Y1jD= 0;M= 1]:\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\subsection{Average direct effect\nunder $\\mathbf{d = 1}$ on compliers}\nIn the following, we show that\n\\begin{align*}\n\\theta_{1}^{c}(1) = E[Y_1(1,1)-Y_1(0,1)|\\tau=c], \\\\ = \n\\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1-Q_{01}(Y_0) |D=1,M=1] \\\\ -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)-Y_1|D=0,M=1].\n\\end{align*}\nPlugging () in (), we obtain\n\\begin{equation*}\n\\begin{array}{rl}\nE[Y_1|D=1,M=1] = \\displaystyle \\frac{p_a}{p_a + p_c}E[Q_{11}(Y_0)|D=0,M=1] \\\\\n \\displaystyle \\quad + \\frac{p_c}{p_a + p_c} E[Y_1(1,1)|\\tau=c].\n\\end{array}\n\\end{equation*}\nThis allows identifying\n\\begin{equation} \n\\begin{array}{rl}\nE[Y_1(1,1)|\\tau=c] = \\displaystyle \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1|D=1,M=1] \\\\\n \\displaystyle \\quad -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)|D=0,M=1].\n\\end{array}\n\\end{equation}\nFrom () we have $E[Y_1(0,1)|D=1,M=1] = E[Q_{01}(Y_0)|D=1,M=1]$. Applying the law of iterative expectations, gives\n\\begin{align*}\nE[Y_1(0,1)|D=1,M=1] = \\frac{p_a}{p_a+p_c} E[Y_1(0,1)|D=1,M=1, \\tau=a] \\\\+ \\frac{p_c}{p_a+p_c}E[Y_1(0,1)|D=1,M=1,\\tau= c], \\\\\n\\stackrel{A7}{=} \\frac{p_a}{p_a+ p_c} E[Y_1(0,1)|\\tau=a] + \\frac{p_c}{p_a+p_c}E[Y_1(0,1)|\\tau= c].\n\\end{align*}\nAfter some rearrangements and using (), we obtain\n\\begin{equation*}\nE[Y_1(0,1)|\\tau= c]  = \\frac{p_a+p_c}{p_c} E[Q_{01}(Y_0)|D=1,M=1] -   \\frac{p_a}{p_c} E[Y_1|D=0,M=1].\n\\end{equation*}\n\\end{document}\n"}
{"This gives\nE[Y1(0;1)j\u001c=c] =p1j1\np1j1\u0000p1j0E[Q01(Y0)jD= 1;M= 1]\n\u0000p1j0\np1j1\u0000p1j0E[Y1jD= 0;M= 1];(1)\nwithpa=p1j0, andpc+pa=p1j1.\n0.1 Quantile direct e\u000bect under d =1 on compliers\nWe show that\nFY1(1;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FY1jD=1;M=1(y)\u0000p1j0\np1j1\u0000p1j0FQ11(Y0)jD=0;M=1(y) and\nFY1(0;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FQ01(Y0)jD=1;M=1(y)\u0000p1j0\np1j1\u0000p1j0FY1jD=0;M=1(y);\nwhich proves that \u0012c\n1(q;1) =F\u00001\nY1(1;1)jc(q)\u0000F\u00001\nY1(0;1)jc(q) is identi\fed. In analogy to (),\nthe outcome distribution under D= 0 andM= 0 equals:\nFY1jD=1;M=1(y) =pa\npa+pcFY1(1;1)j\u001c=a(y) +pc\npa+pcFY1(1;1)j\u001c=c(y):\nUsing () and rearranging the equation gives\nFY1(1;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FY1jD=1;M=1(y)\u0000p1j0\np1j1\u0000p1j0FQ11(Y0)jD=0;M=1(y):(2)\nFrom (), we have FY1(0;1)jD=1;M(1)=1(y) =FQ01(Y0)jD=1;M=1(y). Applying the law of\niterative expectations gives\nFY1(0;1)jD=1;M(1)=1(y) =pa\npa+pcFY1(0;1)jD=1;M(1)=1;\u001c=a(y)\n+pc\npa+pcFY1(0;1)jD=1;M(1)=1;\u001c=c(y);\nA7=pa\npa+pcFY1(0;1)j\u001c=a(y) +pc\npa+pcFY1(0;1)j\u001c=c(y):\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nThis gives\n\\begin{equation} \n\\begin{array}{rl}\nE[Y_1(0,1)|\\tau=c] =  \\displaystyle \\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{01}(Y_0)|D=1,M=1] \\\\\n \\displaystyle \\quad - \\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_1|D=0,M=1],\n\\end{array}\n\\end{equation}\nwith $p_a = p_{1|0}$, and $p_c +p_a =p_{1|1}$.\n\\subsection{Quantile direct effect\nunder $\\mathbf{d = 1}$ on compliers}\nWe show that\n\\begin{align*}\n F_{Y_{1}(1,1)|\\tau=c}(y) =  \\frac{p_{1|1}}{p_{1|1} - p_{1|0}} F_{Y_{1}|D=1,M=1}(y) - \\frac{p_{1|0} }{p_{1|1} - p_{1|0}}F_{Q_{11}(Y_0)|D=0,M=1}(y) \\mbox{ and} \\\\\nF_{Y_{1}(0,1)|\\tau=c}(y) = \\frac{p_{1|1}}{p_{1|1} - p_{1|0}}  F_{Q_{01}(Y_{0})|D=1,M=1}(y) - \\frac{p_{1|0}}{ p_{1|1} - p_{1|0}}  F_{Y_1|D=0,M=1}(y)  ,\n\\end{align*}\nwhich proves that $\\theta_1^{c}(q,1) = F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,1)|c}^{-1}(q)$ is identified.\nIn analogy to (), the outcome distribution under $D=0$ and $M=0$ equals:\n\\begin{equation*}\nF_{Y_{1}|D=1,M=1}(y)=\\frac{p_a}{p_a + p_c}F_{Y_{1}(1,1)|\\tau=a}(y)  + \\frac{p_c}{p_a + p_c}F_{Y_{1}(1,1)|\\tau=c}(y).\n\\end{equation*}\nUsing () and rearranging the equation gives\n\\begin{equation}\nF_{Y_{1}(1,1)|\\tau=c}(y)= \\frac{p_{1|1}}{p_{1|1} - p_{1|0}} F_{Y_{1}|D=1,M=1}(y) - \\frac{p_{1|0} }{p_{1|1} - p_{1|0}}F_{Q_{11}(Y_0)|D=0,M=1}(y) . \n\\end{equation}\nFrom (), we have $F_{Y_{1}(0,1)|D=1,M(1)=1}(y)  =F_{Q_{01}(Y_{0})|D=1,M=1}(y)$. Applying the law of iterative expectations gives\n\\begin{align*}\nF_{Y_{1}(0,1)|D=1,M(1)=1}(y)   =\\frac{p_a}{p_a + p_c} F_{Y_{1}(0,1)|D=1,M(1)=1,\\tau=a}(y)\n\\\\  +\\frac{p_c}{p_a + p_c} F_{Y_{1}(0,1)|D=1,M(1)=1,\\tau=c}(y),\\\\\n \\stackrel{A7}{=} \\frac{p_a}{p_a + p_c} F_{Y_{1}(0,1)|\\tau=a}(y) +\\frac{p_c}{p_a + p_c} F_{Y_{1}(0,1)|\\tau=c}(y).\n\\end{align*}\n\\end{document}\n"}
{"Using () and rearranging the equation gives\nFY1(1;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FY1jD=1;M=1(y)\u0000p1j0\np1j1\u0000p1j0FQ11(Y0)jD=0;M=1(y):(1)\nFrom (), we have FY1(0;1)jD=1;M(1)=1(y) =FQ01(Y0)jD=1;M=1(y). Applying the law of\niterative expectations gives\nFY1(0;1)jD=1;M(1)=1(y) =pa\npa+pcFY1(0;1)jD=1;M(1)=1;\u001c=a(y)\n+pc\npa+pcFY1(0;1)jD=1;M(1)=1;\u001c=c(y);\nA7=pa\npa+pcFY1(0;1)j\u001c=a(y) +pc\npa+pcFY1(0;1)j\u001c=c(y):\nUsing () and rearranging the equation gives,\nFY1(0;1)j\u001c=c(y) =p1j1\np1j1\u0000p1j0FQ01(Y0)jD=1;M=1(y)\u0000p1j0\np1j1\u0000p1j0FY1jD=0;M=1(y):(2)\n1 Proof of Theorem 5\n1.1 Average treatment e\u000bect on the compliers\nIn () and (), we show that\n\u0012c\n1=E[Y1(1;1)\u0000Y1(0;0)j\u001c=c];\n=p1j1\np1j1\u0000p1j0E[Y1jD= 1;M= 1]\u0000p1j0\np1j1\u0000p1j0E[Q11(Y0)jD= 0;M= 1]\n\u0000p0j0\np0j0\u0000p0j1E[Y1jD= 0;M= 0] +p0j1\np0j0\u0000p0j1E[Q00(Y0)jD= 1;M= 0]:\n1.2 Quantile treatment e\u000bect on the compliers\nIn () and (), we show that FY1(1;1)jc(y) andFY1(0;0)jc(y) are identi\fed. Accordingly,\n\u0001c\n1(q) =F\u00001\nY1(1;1)jc(q)\u0000F\u00001\nY1(0;0)jc(q) is identi\fed.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\nUsing () and rearranging the equation gives\n\\begin{equation}\nF_{Y_{1}(1,1)|\\tau=c}(y)= \\frac{p_{1|1}}{p_{1|1} - p_{1|0}} F_{Y_{1}|D=1,M=1}(y) - \\frac{p_{1|0} }{p_{1|1} - p_{1|0}}F_{Q_{11}(Y_0)|D=0,M=1}(y) . \n\\end{equation}\nFrom (), we have $F_{Y_{1}(0,1)|D=1,M(1)=1}(y)  =F_{Q_{01}(Y_{0})|D=1,M=1}(y)$. Applying the law of iterative expectations gives\n\\begin{align*}\nF_{Y_{1}(0,1)|D=1,M(1)=1}(y)   =\\frac{p_a}{p_a + p_c} F_{Y_{1}(0,1)|D=1,M(1)=1,\\tau=a}(y)\n\\\\  +\\frac{p_c}{p_a + p_c} F_{Y_{1}(0,1)|D=1,M(1)=1,\\tau=c}(y),\\\\\n \\stackrel{A7}{=} \\frac{p_a}{p_a + p_c} F_{Y_{1}(0,1)|\\tau=a}(y) +\\frac{p_c}{p_a + p_c} F_{Y_{1}(0,1)|\\tau=c}(y).\n\\end{align*}\nUsing () and rearranging the equation gives,\n\\begin{equation} \nF_{Y_{1}(0,1)|\\tau=c}(y) = \\frac{p_{1|1}}{p_{1|1} - p_{1|0}}  F_{Q_{01}(Y_{0})|D=1,M=1}(y) - \\frac{p_{1|0}}{ p_{1|1} - p_{1|0}}  F_{Y_1|D=0,M=1}(y).\n\\end{equation}\n\\section{Proof of Theorem 5 }\n\\subsection{Average treatment effect on the compliers}\nIn () and (), we show that\n\\begin{align*}\n\\theta_1^c=  E[Y_1(1,1) -Y_1(0,0)|\\tau=c], \\\\ = \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1|D=1,M=1] -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)|D=0,M=1] \\\\\n \u2212 \\frac{p_{0|0}}{ p_{0|0} - p_{0|1}} E[Y_1|D=0,M=0] +\\frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Q_{00}(Y_0)|D=1,M=0] .\n\\end{align*}\n\\subsection{Quantile treatment effect on the compliers}\nIn () and (), we show that $F_{Y_{1}(1,1)|c}(y)$ and $F_{Y_{1}(0,0)|c}(y)$ are identified. Accordingly, $\\Delta_1^{c}(q) = F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is identified.\n\\end{document}\n"}
{"0.1 Average indirect e\u000bect under d= 0on compliers\nIn () and (), we show that\n\u000ec\n1(0) =E[Y1(0;1)\u0000Y1(0;0)j\u001c=c];\n=p1j1\np1j1\u0000p1j0E[Q11(Y0)jD= 1;M= 1]\u0000p1j0\np1j1\u0000p1j0E[Y1jD= 0;M= 1]\n\u0000p0j0\np0j0\u0000p0j1E[Y1jD= 0;M= 0] +p0j1\np0j0\u0000p0j1E[Q00(Y0)jD= 1;M= 0]:\n0.2 Quantile indirect e\u000bect under d= 0on compliers\nIn () and (), we show that FY1(0;1)jc(y) andFY1(0;0)jc(y) are identi\fed. Accordingly,\n\u000ec\n1(q;0) =F\u00001\nY1(0;1)jc(q)\u0000F\u00001\nY1(0;0)jc(q) is identi\fed.\n0.3 Average indirect e\u000bect under d= 1on compliers\nIn () and (), we show that\n\u000ec\n1(1) =E[Y1(1;1)\u0000Y1(1;0)j\u001c=c];\n=p1j1\np1j1\u0000p1j0E[Y1jD= 1;M= 1]\u0000p1j0\np1j1\u0000p1j0E[Q11(Y0)jD= 0;M= 1]\n\u0000p0j0\np0j0\u0000p0j1E[Q00(Y0)jD= 0;M= 0] +p0j1\np0j0\u0000p0j1E[Y1jD= 1;M= 0]:\n0.4 Quantile indirect e\u000bect under d= 1on compliers\nIn () and (), we show that FY1(1;1)jc(y) andFY1(1;0)jc(y) are identi\fed. Accordingly,\n\u000ec\n1(q;1) =F\u00001\nY1(1;1)jc(q)\u0000F\u00001\nY1(1;0)jc(q) is identi\fed.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[flushmargin]{footmisc}\n\\usepackage{setspace}\n\\usepackage[comma]{natbib}\n\\usepackage{float}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{ae}\n\\usepackage{caption}\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}\n\\usepackage{graphicx}\n\\usepackage[colorinlistoftodos]{todonotes}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n\\begin{document} \\doublespacing \\pagestyle{plain}\n\\subsection{Average indirect effect\nunder $d = 0$ on compliers}\nIn () and (), we show that\n\\begin{align*}\n\\delta_{1}^{c}(0) = E[Y_1(0,1) -Y_1(0,0)|\\tau=c],\\\\\n= \\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_0)|D=1,M=1] - \\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_1|D=0,M=1]\\\\\n- \\frac{p_{0|0}}{ p_{0|0} - p_{0|1}} E[Y_1|D=0,M=0] +\\frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Q_{00}(Y_0)|D=1,M=0].\n\\end{align*}\n\\subsection{Quantile indirect effect\nunder $d = 0$ on compliers}\nIn () and (), we show that $F_{Y_{1}(0,1)|c}(y)$ and $F_{Y_{1}(0,0)|c}(y)$ are identified. Accordingly, $\\delta_1^{c}(q,0) = F_{Y_{1}(0,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is identified.\n\\subsection{Average indirect effect\nunder $d = 1$ on compliers}\nIn () and (), we show that\n\\begin{align*}\n\\delta_1^{c}(1) = E[Y_1(1,1) -Y_1(1,0)|\\tau= c],\\\\\n= \\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1|D=1,M=1] -\\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)|D=0,M=1]\\\\\n- \\frac{p_{0|0}}{p_{0|0} - p_{0|1}}E[Q_{00}(Y_0)|D=0,M=0] + \\frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Y_1|D=1,M=0].\n\\end{align*}\n\\subsection{Quantile indirect effect\nunder $d = 1$ on compliers}\nIn () and (), we show that $F_{Y_{1}(1,1)|c}(y)$ and $F_{Y_{1}(1,0)|c}(y)$ are identified. Accordingly,  $\\delta_1^{c}(q,1) = F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(1,0)|c}^{-1}(q)$ is identified.\n\\end{document}\n"}
{"At each simulation step tand for each \u03b9, the path length re\ufb02ected by\nthe\u03b9-th scatter l(\u03b9)\nt=|(x\u03b9\nt,y\u03b9\nt)\u2212(0,0)|+|(x\u2217\nt,y\u2217\nt)\u2212(x\u03b9\nt,y\u03b9\nt)|and its deriva-\ntive with respect to timed\ndtl(\u03b9)\ntare computed. The corresponding transmis-\nsion delay time \u03c3(\u03b9)\nt, the constant phase o\ufb00set \u03b8(\u03b9)\nt, and the Doppler frequency\nf(\u03b9)\nDtcaused by the \u03b9-th scatterer following the rules \u03c3(\u03b9)\nt=l(\u03b9)\nt/c0,\u03b8(\u03b9)\nt=\n((\u2212l(\u03b9)\ntfcarrier/c0) mod 1)\u00b72\u03c0, andf(\u03b9)\nDt=\u2212d\ndtl(\u03b9)\ntfcarrier/c0, respectively, as well\nasthereceivedsignalamplitude a(\u03b9)\ntcomputedusingthefree-spacepropagation\nmodela(\u03b9)\nt=c0/(4\u03c0fcarrierl(\u03b9)\nt)(cf.) are recorded for each scatterer \u03b9. (Here,\nc0refers to the speed of light in vacuum.) In a setting without line of sight,\nusing linearisation of the phase o\ufb00set with respect to the Doppler frequency,\nthe time-variant channel impulse response evaluated at time t+\u03c4for each sim-\nulation step tand small\u03c4resulting from the multipath transmission simulated\nusing the above parameters can be approximated by\nh(\u00b7,t+\u03c4) =1/radicalBig/summationtext255\n\u03b9=0(a(\u03b9)\nt)2255/summationdisplay\n\u03b9=0a(\u03b9)\ntexp(i\u03b8(\u03b9)\nt+i2\u03c0f(\u03b9)\nDt\u03c4)\u03b4\u03c3(\u03b9)\nt(\u00b7).\nFor any signal{S\u03c4}0\u2264\u03c4<Tbeing transmitted in the block beginning at time\nsteptthrough the simulated channel, this consideration leads to a received\nsignal{R\u03c4}0\u2264\u03c4<Tin the form of\nR\u03c4= (h(\u00b7,t+\u03c4)\u2217S\u00b7)(\u03c4)\n=1/radicalBig/summationtext255\n\u03b9=0(a(\u03b9)\nt)2255/summationdisplay\n\u03b9=0a(\u03b9)\ntexp(i\u03b8(\u03b9)\nt+i2\u03c0f(\u03b9)\nDt\u03c4)(\u03b4\u03c3(\u03b9)\nt(\u00b7)\u2217S\u00b7)(\u03c4).(1)\nThis parametrisation is used in and delivers a realistic approximation of real-\nworld scenarios for numbers of summands greater than 100. In order to allow\ncontinuous time delays to be applied to discrete time signals, the impulse func-\ntion\u03b4\u03c3(\u03b9)\nt(\u00b7)in () is convolved with a windowed sinc (\u00b7)function scaled with\na given bandwidth. Overall, the channel transmission including pulse shaping\nwith bandwidth restricted to half the sample rate\nand additive noise is approximated by replacing the \u03b4\u03c3(\u03b9)\nt(\u00b7)in () by\nsin(\u03c0(\u00b7/2))/(\u03c0(\u00b7/2))1[\u22128,8]and adding independent and identically distributed\nGaussian white noise \u223cN (0,\u03c32)to the transmitted signal with power \u03c32re-\nsulting in a signal-to-noise ratio of 12dB.\n1 Channel Estimation\nThe time-variant channel transfer functions Fh(\u00b7,t+\u03c4)fort= 0,..., 4095T\nand 0\u2264\u03c4 < Tsimulated in Section are approximated by a time series of\nblock wise time-invariant transfer functions {Fht}t=0,...,4095based on which\nthe estimation and prediction of the channel transmission are conducted. For\neach transmission block beginning at time step t, in order to estimate the\ncorresponding channel transfer function Fht, a complex-valued (white noise)\ntest signal{\u02dcSt\n\u03c4}\u03c4=0,...,N\u22121whose Fourier transform has constant amplitude\nand random phases \u223cU(\u2212\u03c0,\u03c0)is generated and then transmitted through the\nchannel simulated in Section resulting in a received signal {Rt\n\u03c4}\u03c4=0,...,N\u22121.\n1": "\\documentclass[a4paper,extrafontsizes,article]{memoir}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{tabularx}\n\\usepackage{caption}\n\\usepackage{siunitx}\n\\usepackage{hyperref}\n\\counterwithout{section}{chapter}\n\\makeatletter\n\\let\\@fnsymbol\\@arabic\n\\makeatother\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\\begin{document}\nAt each simulation step $t$ and for each $\\iota$, the path length reflected by the $\\iota$-th scatter $l^{(\\iota)}_t =\\left\\vert(x_t^\\iota,y_t^\\iota)-(0,0)\\right\\vert + \\left\\vert(x_t^*,y_t^*)-(x_t^\\iota,y_t^\\iota)\\right\\vert$ and its derivative with respect to time $\\frac{d}{dt}l_t^{(\\iota)}$ are computed. The corresponding transmission delay time $\\sigma_t^{(\\iota)}$, the constant phase offset $\\theta_t^{(\\iota)}$,  and the Doppler frequency $f_{D_t}^{(\\iota)}$ caused by the $\\iota$-th scatterer following the rules $\\sigma_t^{(\\iota)}=l_t^{(\\iota)}/c_0$, $\\theta_t^{(\\iota)}=((-l_t^{(\\iota)}f_{\\text{carrier}}/c_0) \\bmod 1)\\cdot 2\\pi$, and $f_{D_t}^{(\\iota)}=-\\frac{d}{dt}l_t^{(\\iota)}f_{\\text{carrier}}/{c_0}$, respectively, as well as the received signal amplitude $a_t^{(\\iota)}$ computed using the free-space propagation model   $a_t^{(\\iota)}=c_0/(4\\pi f_{\\text{carrier}} l_t^{(\\iota)})$ (cf.) are recorded for each scatterer $\\iota$. (Here, $c_0$ refers to the speed of light in vacuum.)\nIn a setting without line of sight, using linearisation of the phase offset with respect to the Doppler frequency, the time-variant channel impulse response evaluated at time $t+\\tau$ for each simulation step $t$ and small $\\tau$ resulting from the multipath transmission simulated using the above parameters can be approximated by\n\\[\nh(\\cdot,t+\\tau)=\\frac{1}{\\sqrt{\\sum_{\\iota=0}^{255} (a_t^{(\\iota)}) ^2}}\\sum_{\\iota=0}^{255} a_t^{(\\iota)}\\exp(i\\theta_t^{(\\iota)}+i2\\pi f_{D_t}^{(\\iota)}\\tau)\\delta_{\\sigma_t^{(\\iota)}}(\\cdot).\n\\]\nFor any signal $\\{S_\\tau\\}_{0\\leq\\tau< T}$ being transmitted in the block beginning at time step $t$ through the simulated channel, this consideration leads to a received signal $\\{R_\\tau\\}_{0\\leq\\tau< T}$ in the form of\n\\begin{align}\n\\nonumber R_\\tau &= (h(\\cdot,t+\\tau) *S_\\cdot)(\\tau)\\\\\n& = \\frac{1}{\\sqrt{\\sum_{\\iota=0}^{255} (a_t^{(\\iota)}) ^2}}\\sum_{\\iota=0}^{255} a_t^{(\\iota)}\\exp(i\\theta_t^{(\\iota)}+i2\\pi f_{D_t}^{(\\iota)}\\tau)(\\delta_{\\sigma_t^{(\\iota)}}(\\cdot)*S_\\cdot)(\\tau).\n\\end{align}\nThis parametrisation is used in  and delivers a realistic approximation of real-world scenarios for numbers of summands greater than $100$ . In order to allow continuous time delays to be applied to discrete time signals, the impulse function $\\delta_{\\sigma_t^{(\\iota)}}(\\cdot)$ in () is convolved with a windowed $\\text{sinc}(\\cdot)$ function scaled with a given bandwidth.  Overall, the channel transmission including pulse shaping with bandwidth restricted to half the sample rate\\linebreak and additive noise is approximated by replacing the $\\delta_{\\sigma_t^{(\\iota)}}(\\cdot)$ in () by\\linebreak $\\sin(\\pi(\\cdot/2))/(\\pi(\\cdot/2))\\mathbf{1}_{[-8,8]}$ and adding independent and identically distributed Gaussian white noise $\\sim\\mathcal{N}(0,\\sigma^2)$ to the transmitted signal with power $\\sigma^2$ resulting in a signal-to-noise ratio of $12\\text{dB}$.\n\\section{Channel Estimation}\nThe time-variant channel transfer functions $\\mathcal{F}h(\\cdot,t+\\tau)$ for $t=0,\\dots, 4095T$ and $0\\leq \\tau<T$ simulated in Section are approximated by a time series of block wise time-invariant transfer functions $\\{\\mathcal{F}h^t\\}_{t=0,\\dots, 4095}$ based on which the estimation and prediction of the channel transmission are conducted.\nFor each transmission block beginning at time step $t$, in order to estimate the corresponding channel transfer function $\\mathcal{F}h^{t}$, a complex-valued (white noise) test signal $\\{\\tilde{S}_\\tau^t\\}_{\\tau=0,\\dots,N-1}$ \nwhose Fourier transform has constant amplitude and random phases $\\sim\\mathcal{U}(-\\pi,\\pi)$ is generated\nand then transmitted through the channel simulated in Section resulting in a received signal $\\{R_\\tau^t\\}_{\\tau=0,\\dots,N-1}$.\n\\end{document}\n"}
{"ZENIT (Zero Emission Neighborhood Investment Tool) is\na tool for minimizing the cost of investing and operating the\nenergy system of a Zero Emission Neighborhood (ZEN). It\nuses a MILP model to \ufb01nd the optimal type and size of\ntechnology needed to provide heat and electricity to a ZEN.\nThe concept contains much more than only the energy system\n(materials and architecture to name two examples) but this\ntool\u2019s focus is energy systems. The idea behind ZEN is to limit\nemissions and that it is possible to compensate for the various\nemissions of CO 2in the neighborhood by exporting electricity\nto the grid. Indeed, by exporting electricity produced from\non-site renewable to the grid, we assume that the production\nin the system is reduced by the corresponding amount and\nthus reduces the emission of the total system. The model\nused in this paper is based on the model presented in . In\nthis section, the main elements of the model will be repeated\nand the differences with the model from presented. For the\ndetails on the model not repeated, one can refer to that paper.\nThen the implementation and case chosen will be presented.\nThe optimization is written in Python and uses Gurobi as a\nsolver. In this paper, we interpret the de\ufb01nition of a ZEN as\na neighborhood that has 0 emissions over its lifetime, which\nis set to 60 years. However due to practical reasons and to\nreduce the computational time, different periods of the lifetime\ncan be de\ufb01ned using one representative year for each. In this\nstudy we use a single period. The different decision variables\nare the amount of investment in each technology for heat,\npower and energy storage as well as the operation related\nvariables de\ufb01ned for each hour (e.g. amount of electricity\nproduced, amount of fuel consumed). Multiple constraints are\nused, to enforce the CO 2limitations necessary in the ZEN\ncontext and to represent the operation of the neighborhood\nand in particular of each technology. It is important to note\nthat part load limitations and start up/shutdown constraints are\nnot implemented. The objective function of the optimization\nis the following: Minimize :\nX\niCdisc\ni\u0001xi+bhg\u0001Chg+1\n\"tot\nr;DX\niCmaint\ni\u0001xi\n+1\n\"tot\nr;D\u0012X\nt\u0010X\nfff;t\u0001Pfuel\nf+ (Pspot\nt +Pgrid\n+Pret)\u0001(yimp\nt +X\nestygbimp\nt;est )\u0000Pspot\nt\u0001yexp\nt\u0011\u0013\n(1)\nWhereCdisc\ni is the discounted investment cost in technology\niincluding re-investments and salvage value, xithe capacity\nof technology i,bhgthe binary variable for investment in\na heating grid, Chgthe cost of the heating grid, \"tot\nr;Dthe\ndiscount factor for the whole lifetime of the neighborhood\nwith discount rate r, Cmaint\ni the annual maintenance cost,\nff;tthe fuel consumption, Pfuel\nfthe fuel price, Pspot\nt the\nelectricity spot price, PgridandPretthe grid and retailer\ntariffs,yimp\ntandyexp\ntthe import and export of electricity from\nthe neighborhood and ygbimp\nt;est the import of electricity to the\nstorage. The subscript trepresent timesteps, ithe technologieswith in particular ffor the technologies using other fuel than\nelectricity and estfor energy storages. It minimizes the cost\nof investing in the different technologies and the operating\ncosts, fuels, electricity and O&M costs and contains the costs\nof the heating grid and a binary associated with it that also\ngives access to technologies at a neighborhood level. The\nmost important constraint in the case of ZENs is the CO 2\nbalance ( ??), whose principle was explained earlier. In ( ??),\nygbimp\nt;est ,ygbexp\nt;est andypbexp\nt;est are respectively the import and\nexport on the grid side battery and the export from the on-\nsite technologies producing electricity, 'CO2e and'CO2\nfthe\nCO 2factors for electricity and other fuels, \u0011estthe ef\ufb01ciency\nof battery and yexp\nt;gthe export of electricity from the on-site\ntechnologies.\nX\nt((yimp\nt +X\nestygbimp\nt;est )\u0001'CO2\ne )\n+X\ntX\nf('CO2\nf\u0001ff;t)\u0014X\nt(X\nest(ygbexp\nt;est\n+ypbexp\nt;est )\u0001\u0011est+X\ngyexp\nt;g)\u0001'CO2\ne (2)\nFig. presents graphically the electricity balance and the differ-\nent equations associated. Different technologies are included\nin the study; some of them are only available at the building\nlevel and others at the neighborhood level in a centralized pro-\nduction plant. The different technologies are: at the building\nlevel : Solar Panels (PV), Solar Thermal (ST), Heat Pumps\n(HP), Biomass Boiler (BB), Electric Boiler (EB), Gas Boiler\n(GB); and at the neighborhood levels: CHP (nCHP), Gas\nBoiler (nGB), Electric Boiler (nEB), Heat Pumps (nHP). In\naddition, Batteries (Bat) and Heat Storage (HS) are available\nat both levels. Different subcategories can be available to\nchoose from within each category, for instance air-water or\nwater-water heat pumps. In parenthesis is the notation used\nfor the rest of the study for each technology. Unlike the\nmodel in , the model used for this paper uses a disaggregrated\nheat load. The buildings\u2019 load are not summed, but types\nof buildings are identi\ufb01ed and the loads are aggregated per\nbuilding type. It is possible to use a completely disaggregated\nheat load but the lack of available data motivated not doing\nit. The input data necessary to run ZENIT are the electric and\nheating loads (ideally separated between domestic hot water\nand space heating), the outside and ground temperatures, the\nsolar insolation and the electricity prices. Hourly timeseries\nfor each representative years are necessary. A description of\nthe neighborhood and its buildings with the \ufb02oor and the roof\narea, and the layout of the neighborhoods is also needed. In\nthis study we assume the heating grid is there (and set the\ncorresponding binary to 1) because there is one in the location\nthat inspired this case. Its characteristics (layout, losses and\ncost) are then necessary but a module can be used to provide\nan estimate of the losses and cost based on the layout of the\nneighborhood. The CO 2factors used were 17 g CO 2/kWh for\nelectricity, 277 g CO 2/kWh for gas and 7 g CO 2/kWh for wood\nchips . The electricity produced via solar panels or": "\\documentclass[conference]{IEEEtran}\n\\usepackage{cite}\n\\usepackage[pdftex]{graphicx}\n\\usepackage{amsmath,amssymb,amsfonts}\n\\usepackage{algorithmic}\n  \\usepackage[caption=false,font=footnotesize]{subfig}\n\\usepackage{url}\n\\usepackage{xcolor}\n\\usepackage{eurosym}\n\n\\begin{document}\nZENIT (Zero Emission Neighborhood Investment Tool) is a tool for minimizing the cost of investing and operating the energy system of a Zero Emission Neighborhood (ZEN). It uses a MILP model to find the optimal type and size of technology needed to provide heat and electricity to a ZEN.\nThe concept contains much more than only the energy system (materials and architecture to name two examples) but this tool's focus is energy systems.\nThe idea behind ZEN is to limit emissions and that it is possible to compensate for the various emissions of $CO_2$ in the neighborhood by exporting electricity to the grid. Indeed, by exporting electricity produced from on-site renewable to the grid, we assume that the production in the system is reduced by the corresponding amount and thus reduces the emission of the total system.\nThe model used in this paper is based on the model presented in . In this section, the main elements of the model will be repeated and the differences with the model from  presented. For the details on the model not repeated, one can refer to that paper. Then the implementation and case chosen will be presented.\nThe optimization is written in Python and uses Gurobi as a solver. In this paper, we interpret the definition of a ZEN as a neighborhood that has 0 emissions over its lifetime, which is set to 60 years. However due to practical reasons and to reduce the computational time, different periods of the lifetime can be defined using one representative year for each. In this study we use a single period.\nThe different decision variables are the amount of investment in each technology for heat, power and energy storage as well as the operation related variables defined for each hour (e.g. amount of electricity produced, amount of fuel consumed).\nMultiple constraints are used, to enforce the $CO_2$ limitations necessary in the ZEN context and to represent the operation of the neighborhood and in particular of each technology. It is important to note that part load limitations and start up/shutdown constraints are not implemented.\nThe objective function of the optimization is the following:\n\\textit{Minimize}:\n\\begin{multline}\n    \\sum_{i} C_i^{disc}\\cdot x_i + b_{hg} \\cdot C_{hg} + \\frac{1}{\\varepsilon^{tot}_{r,D}} \\sum_{i} C_i^{maint} \\cdot x_i\\\\+\\frac{1}{\\varepsilon^{tot}_{r,D}}\\bigg(\\sum_{t}\\Big(\\sum_{f}f_{f,t}\\cdot P_{f}^{fuel} + (P_{t}^{spot}+P^{grid}\\\\+P^{ret})\\cdot (y_{t}^{imp}+\\sum_{est}y_{t,est}^{gb\\_imp})-P_{t}^{spot}\\cdot y_{t}^{exp}\\Big)\\bigg)\n\\end{multline}\nWhere $C_i^{disc}$ is the discounted investment cost in technology $i$ including re-investments and salvage value, $x_i$ the capacity of technology $i$, $b_{hg}$ the binary variable for investment in a heating grid, $C_{hg}$ the cost of the heating grid, $\\varepsilon^{tot}_{r,D}$ the discount factor for the whole lifetime of the neighborhood with discount rate r, $C_i^{maint}$ the annual maintenance cost, $f_{f,t}$ the fuel consumption, $P_{f}^{fuel}$ the fuel price, $P_{t}^{spot}$ the electricity spot price, $P^{grid}$ and $P^{ret}$ the grid and retailer tariffs, $y_{t}^{imp}$ and $y_{t}^{exp}$ the import and export of electricity from the neighborhood and $y_{t,est}^{gb\\_imp}$ the import of electricity to the storage. The subscript $t$ represent timesteps, $i$ the technologies with in particular $f$ for the technologies using other fuel than electricity and $est$ for energy storages.\nIt minimizes the cost of investing in the different technologies and the operating costs, fuels, electricity and O\\&M costs and contains the costs of the heating grid and a binary associated with it that also gives access to technologies at a neighborhood level.\nThe most important constraint in the case of ZENs is the $CO_2$ balance \\eqref{co2bal}, whose principle was explained earlier. In \\eqref{co2bal}, $y_{t,est}^{gb\\_imp}$, $y_{t,est}^{gb\\_exp}$ and $y_{t,est}^{pb\\_exp}$ are respectively the import and export on the grid side battery and the export from the on-site technologies producing electricity, $\\varphi^{CO_2}_{e}$ and $\\varphi^{CO_2}_{f}$ the $CO_2$ factors for electricity and other fuels, $\\eta_{est}$ the efficiency of battery and $y_{t,g}^{exp}$ the export of electricity from the on-site technologies.\n\\begin{multline}\n    \\sum_{t}((y_{t}^{imp}+\\sum_{est}y_{t,est}^{gb\\_imp}) \\cdot \\varphi^{CO_2}_{e}) \\\\+ \\sum_{t}\\sum_{f} (\\varphi^{CO_2}_{f} \\cdot f_{f,t}) \\leq \\sum_{t}(\\sum_{est}(y_{t,est}^{gb\\_exp}\\\\+y_{t,est}^{pb\\_exp})\\cdot \\eta_{est} +\\sum_{g}y_{t,g}^{exp}) \\cdot \\varphi^{CO_2}_{e}\n\\end{multline}\nFig.  presents graphically the electricity balance and the different equations associated.\nDifferent technologies are included in the study; some of them are only available at the building level and others at the neighborhood level in a centralized production plant. The different technologies are: at the building level : Solar Panels (PV), Solar Thermal (ST), Heat Pumps (HP), Biomass Boiler (BB), Electric Boiler (EB), Gas Boiler (GB); and at the neighborhood levels: CHP (nCHP), Gas Boiler (nGB), Electric Boiler (nEB), Heat Pumps (nHP). In addition, Batteries (Bat) and Heat Storage (HS) are available at both levels. Different subcategories can be available to choose from within each category, for instance air-water or water-water heat pumps. In parenthesis is the notation used for the rest of the study for each technology.\nUnlike the model in , the model used for this paper uses a disaggregrated heat load. The buildings' load are not summed, but types of buildings are identified and the loads are aggregated per building type. It is possible to use a completely disaggregated heat load but the lack of available data motivated not doing it.\nThe input data necessary to run ZENIT are the electric and heating loads (ideally separated between domestic hot water and space heating), the outside and ground temperatures, the solar insolation and the electricity prices. Hourly timeseries for each representative years are necessary. A description of the neighborhood and its buildings with the floor and the roof area, and the layout of the neighborhoods is also needed.\nIn this study we assume the heating grid is there (and set the corresponding binary to 1) because there is one in the location that inspired this case. Its characteristics (layout, losses and cost) are then necessary but a module can be used to provide an estimate of the losses and cost based on the layout of the neighborhood. \nThe $CO_2$ factors used were 17 g$CO_2$/kWh for electricity, 277 g$CO_2$/kWh for gas  and 7 g$CO_2$/kWh for wood chips . The electricity produced via solar panels or\n\\end{document}\n"}
{"solar thermal on-site do not have CO 2associated. Embed-\nded emissions were not included. For additional details on the\nmodel and the references of the input data used in the model,\nrefer to .\nI. G RIDTARIFFS DESCRIPTION\nThe Norwegian electricity consumption\u2019s recent trend is a\nconsumption where peak demand increases relatively more\nthan annual demand. This trend must be met by new incentives\nto shave peak load in order to avoid costly distribution grid\ninvestments. Grid tariffs are one effective way to solve this\nissue. In this paper we suggest three new grid tariffs and\ncompare the results with the current grid tariff. The \ufb01rst\nanalyzed grid tariff is energy based and is the current tariff\nin Norway. It consists of an annual \ufb01xed price and a grid\nenergy cost per kWh consumed. As this rate is \ufb02at, it does\nnot incentivize \ufb02exible resources nor consumption patterns\nwhich results in lower peak demand. The annual cost can be\ncalculated using ( ??).\nCtot= 137 + 0;0225 \u0001X\ntyimp tot\nt (1)\nThe second grid tariff is a time-of-use based tariff which\npenalizes import when there is typically scarcity in the grid.\nThe tariff has a basic cost, which is double during peak load\nhours (7-10am and 6-9pm) and reduced to half during low load\nhours (11pm-5am). The effect of increasing electric vehicle\nand demand response penetration on the peak hours is ignored.\nThe total costs are given by ( ??).\nCtot=X\nt\u0000\n0;0123 \u0001yimp low\nt + 0;0246 \u0001yimp med\nt\n+ 0;0492 \u0001yimp peak\nt\u0001\n(2)\nThe third tariff was originally described in , and is called\ncapacity subscription. It contains a \ufb01xed annual cost ( e/year),\na capacity cost ( e/kW), an energy cost ( e/kWh) and an excess\ndemand charge ( e/kWh). The energy cost is signi\ufb01cantly\nhigher when the imports are above the subscription. The main\nadvantage of this tariff is that it incentivizes peak shaving\nand creates a market for capacity where consumers pay for\nthe resource which in fact is scarce in the distribution grid:\ncapacity. Disadvantages are complexity and the uncertainty\nin consumer behaviour. In addition, the optimal subscribed\ncapacity is unknown in advance. Finding its value is further\ndiscussed in . In this paper, the subscribed capacity is a\nvariable in the optimization. In reality, the consumer would\nhave to choose it and it would most likely not be the optimal\nvalue. The costs are calculated with ( ??).\nCtot= 108 \u0001csub+X\nt\u0000\n0;005\u0001yimp below\nt + 0;1\u0001yimp above\nt\u0001\n(3)\nThe fourth tariff is a dynamic tariff where grid scarcity is\ntaken into account. As an extra incentive to reduce impacts\non the grid, a penalization Cscis given for consumption in\nhours with grid scarcity. Scarcity \u000esc\ntin the system is de\ufb01nedas the 5% of hours in the region (NO1) when the load is the\nhighest. The percentage chosen is arbitrary and could be tuned\nor changed into a threshold by the regulator. The total costs\nare given by ( ??). In addition, as an added incentive to help the\ngrid, a bonus for exporting in those hours is added, at the same\ncost as the scarcity tariff. In ( ??),\u000esc\ntis a binary parameter\nde\ufb01ning for each hour if there is scarcity in the grid.\nCtot=X\nt\u0010\u0000\n0;0225 \u0001(1\u0000\u000esc\nt) +\u000esc\nt\u00010;1\u0001\n\u0001yimp tot\nt\n\u00000;1\u0001\u000esc\nt\u0001yexp tot\nt\u0011\n(4)\nII. R ESULTS\nIn Norway, the legislation regarding prosumers is changing,\nmoving from a situation where exports are limited to 100kW\nto a situation of unrestrained export. For this reason, both\ncases are investigated to explore the consequences on the\ndesign of ZENs of the different grid tariffs in these cases.\nThe investment in the energy system can be seen in Table\nand in Table , respectively for the case without and with\nlimitation on exports. The results are presented in the format\nProd Plant/ Student Housing/ Normal Of\ufb01ces/ Passive Of\ufb01ces.\nThe investments stay similar, no new technology is introduced\nor replaced. However, small variations in the amount of each\ntechnology appear, in particular heat storage. The difference\nbetween the energy system with and without export limit\nis greater, namely due to storages. A large battery pack is\nnecessary in order to store the PV production while it waits to\nbe exported, i.e. to accommodate the bottleneck. In addition,\nlarge investments in heat storages and electric boilers are done.\nThe subscribed capacity resulting of the optimization is of\n134,5kW for the case with no export limit, and of 124kW\nin the case with export limits. Fig. presents the total cost of\nthe neighborhood\u2019s energy system (investment and operation)\nand the total revenue for the DSO, both over the lifetime and\ndiscounted to the start of the study. There are small variations\nin the cost in all cases. Subscribed capacity and dynamic\npricing cause an increase in the total cost for the ZEN between\n3 and 5% compared with the energy case. On the other hand,\nthe time of use scheme allows for a cost reduction of around\n12% in the case without export limit and 5% with export\nlimit. The DSO revenue from the ZEN are higher when using\nthe other pricing schemes than with the energy scheme when\nthere is no export limit. When there is export limits, the DSO\nrevenue stays the same because the battery allows to self-\nconsume more and \u201danticipates\u201d the higher price periods and\nbuys electricity when the price is lower. The revenue in the\ncase of export limits are about half of the revenue of the case of\nno export limit except in the case of subscribed capacity where\nthe subscription tariff allows to maintain the revenue. The cost\nincrease in the ZEN is of the same order of magnitude as the\nincrease in revenue for the DSO except for ToU where the cost\nof the ZEN decreases while the revenue for the DSO increases.\nToU has a bene\ufb01cial effect from both points of view in this\naspect. The duration curves Fig. , in the case of no export\nlimit, are not affected much by the tariff scheme in place.": "\\documentclass[conference]{IEEEtran}\n\\usepackage{cite}\n\\usepackage[pdftex]{graphicx}\n\\usepackage{amsmath,amssymb,amsfonts}\n\\usepackage{algorithmic}\n  \\usepackage[caption=false,font=footnotesize]{subfig}\n\\usepackage{url}\n\\usepackage{xcolor}\n\\usepackage{eurosym}\n\n\\begin{document}\n solar thermal on-site do not have $CO_2$ associated. Embedded emissions were not included.\nFor additional details on the model and the references of the input data used in the model, refer to .\n\\section{Grid Tariffs Description}\nThe Norwegian electricity consumption's recent trend is a consumption where peak demand increases relatively more than annual demand. This trend must be met by new incentives to shave peak load in order to avoid costly distribution grid investments. Grid tariffs are one effective way to solve this issue. In this paper we suggest three new grid tariffs and compare the results with the current grid tariff.\nThe first analyzed grid tariff is energy based and is the current tariff in Norway. It consists of an annual fixed price and a grid energy cost per kWh consumed. As this rate is flat, it does not incentivize flexible resources nor consumption patterns which results in lower peak demand. The annual cost can be calculated using \\eqref{energytarifftotal}.\n\\begin{equation}\n    C^{tot} = 137 + 0,0225\\cdot \\sum_t y_{t}^{imp\\_tot}\n\\end{equation}\nThe second grid tariff is a time-of-use based tariff which penalizes import when there is typically scarcity in the grid. The tariff has a basic cost, which is double during peak load hours (7-10am and 6-9pm) and reduced to half during low load hours (11pm-5am). The effect of increasing electric vehicle and demand response penetration on the peak hours is ignored. The total costs are given by \\eqref{TOUtotal}.\n\\begin{multline}\n    C^{tot} = \\sum_t \\big( 0,0123\\cdot y_{t}^{imp\\_low} + 0,0246\\cdot y_{t}^{imp\\_med} \\\\ \n    + 0,0492\\cdot y_{t}^{imp\\_peak} \\big)\n\\end{multline}\nThe third tariff was originally described in , and is called capacity subscription. It contains a fixed annual cost (\\euro/year), a capacity cost (\\euro/kW), an energy cost (\\euro/kWh) and an excess demand charge (\\euro/kWh). The energy cost is significantly higher when the imports are above the subscription. The main advantage of this tariff is that it incentivizes peak shaving and creates a market for capacity where consumers pay for the resource which in fact is scarce in the distribution grid: capacity. Disadvantages are complexity and the uncertainty in consumer behaviour. In addition, the optimal subscribed capacity is unknown in advance. Finding its value is further discussed in . In this paper, the subscribed capacity is a variable in the optimization. In reality, the consumer would have to choose it and it would most likely not be the optimal value.\nThe costs are calculated with \\eqref{subcaptotal}.\n\\begin{equation}\n   C^{tot} = 108 \\cdot c^{sub} + \\sum_t \\big( 0,005\\cdot y_{t}^{imp\\_below} + 0,1\\cdot y_{t}^{imp\\_above} \\big)\n\\end{equation}\nThe fourth tariff is a dynamic tariff where grid scarcity is taken into account. As an extra incentive to reduce impacts on the grid, a penalization $C^{sc}$ is given for consumption in hours with grid scarcity. Scarcity $\\delta _{t}^{sc}$ in the system is defined as the 5\\% of hours in the region (NO1) when the load is the highest. The percentage chosen is arbitrary and could be tuned or changed into a threshold by the regulator. The total costs are given by \\eqref{dynamictotal}. In addition, as an added incentive to help the grid, a bonus for exporting in those hours is added, at the same cost as the scarcity tariff. In \\eqref{dynamictotal}, $\\delta _{t}^{sc}$ is a binary parameter defining for each hour if there is scarcity in the grid.\n\\begin{multline}\n    C^{tot} = \\sum_t \\Big( \\big( 0,0225\\cdot(1-\\delta _{t}^{sc}) + \\delta _{t}^{sc} \\cdot 0,1 \\big) \\cdot y_{t}^{imp\\_tot} \\\\ - 0,1 \\cdot  \\delta _{t}^{sc} \\cdot y_{t}^{exp\\_tot} \\Big)\n\\end{multline}\n\\section{Results}\nIn Norway, the legislation regarding prosumers is changing, moving from a situation where exports are limited to 100kW to a situation of unrestrained export. For this reason, both cases are investigated to explore the consequences on the design of ZENs of the different grid tariffs in these cases.\nThe investment in the energy system can be seen in Table  and in Table , respectively for the case without and with limitation on exports. The results are presented in the format Prod Plant/ Student Housing/ Normal Offices/ Passive Offices.\nThe investments stay similar, no new technology is introduced or replaced. However, small variations in the amount of each technology appear, in particular heat storage. The difference between the energy system with and without export limit is greater, namely due to storages. A large battery pack is necessary in order to store the PV production while it waits to be exported, i.e. to accommodate the bottleneck. In addition, large investments in heat storages and electric boilers are done.\nThe subscribed capacity resulting of the optimization is of 134,5kW for the case with no export limit, and of 124kW in the case with export limits.\nFig.  presents the total cost of the neighborhood's energy system (investment and operation) and the total revenue for the DSO, both over the lifetime and discounted to the start of the study. \nThere are small variations in the cost in all cases. Subscribed capacity and dynamic pricing cause an increase in the total cost for the ZEN between 3 and 5\\% compared with the energy case. On the other hand, the time of use scheme allows for a cost reduction of around 12\\% in the case without export limit and 5\\% with export limit.\nThe DSO revenue from the ZEN are higher when using the other pricing schemes than with the energy scheme when there is no export limit. When there is export limits, the DSO revenue stays the same because the battery allows to self-consume more and \"anticipates\" the higher price periods and buys electricity when the price is lower. The revenue in the case of export limits are about half of the revenue of the case of no export limit except in the case of subscribed capacity where the subscription tariff allows to maintain the revenue. \nThe cost increase in the ZEN is of the same order of magnitude as the increase in revenue for the DSO except for ToU where the cost of the ZEN decreases while the revenue for the DSO increases. ToU has a beneficial effect from both points  of view in this aspect.\nThe duration curves Fig. , in the case of no export limit, are not affected much by the tariff scheme in place. \n\\end{document}\n"}
{"LetXbe aRp-valued random variable with distribution function F. For\na point x2Rp, we consider the statistical data depth of xwith respect to\nFbed(x;F) such that dsatis\fes the four properties given in ?and?and\nreported in Appendix of the Supplementary Material. Given an independent\nand identically distributed sample X1;:::;Xnof sizen, we denote ^Fn(\u0001) its\nempirical distribution function and by d(x;^Fn) the sample depth. We assume\nthat,d(x;^Fn) is a uniform consistent estimator of d(x;F), that is,\nsup\nxjd(x;^Fn)\u0000d(x;F)ja:s:!0 n!1;\na property enjoined by many statistical data depth functions, e.g., among\nothers simplicial depth ( ?), half-space depth ( ?). One important feature of\nthe depth functions is the \u000b-depth trimmed region given by R\u000b(F) =fx2\nRp:d(x;F)\u0015\u000bg; for any\f2[0;1], we will denote R\f(F) the smallest\nregionR\u000b(F) that has probability larger that or equal to \faccording to F.\nThroughout, subscripts and superscripts for depth regions are used for depth\nlevels and probability contents, respectively. Let C\f(F) be the complement\ninRpof the setR\f(F). Letm= max xd(x;F), be the maximum of the\ndepth, for simplicial depth m\u00142\u0000p, for half-space depth m\u00141=2. Given a\nhigh order quantile \f, we de\fne a \flter of dimension pbased on\ndn= sup\nx2C\f(F)fd(x;^Fn)\u0000d(x;F)g+; (1)\nwherefag+represents the positive part of a, and we mark as outliers all\nthebndn=mcobservations with the smallest population depth (where bac\nis the largest integer less then or equal to a). This de\fne a \flter in the\ngeneral dimension p. We have the following result, with obvious proof. If\nsupxjd(x;^Fn)\u0000d(x;F)j=o(n) (a.s.) then ndn!0 asn!1 . If the above\nresult holds, then the \flter would be consistent. In the next subsection we\nare going to illustrate this approach using the half-space depth.\n0.1 Filters based on Half-space Depth\nLetXbe aRp-valued random variable with distribution function F. For a\npoint x2Rp, the half-space depth of xwith respect to Fis de\fned as the\nminimum probability of all closed half-spaces including x:\ndHS(x;F) = min\nH2H(x)PF(X2H):\nwhereH(x) indicates the set of all half-spaces in Rpcontaining x2Rp. A\nrandom vector X2Rpis said elliptically symmetric distributed, denoted by\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\nLet $\\bf{X}$ be a $\\mathbb{R}^p$-valued random variable with distribution function $F$. For a point $\\bf{x} \\in \\mathbb{R}^p$, we consider the statistical data depth of $\\bf{x}$ with respect to $F$ be $d(\\bf{x};F)$ such that $d$ satisfies the four properties given in \\citet{Liu1990} and \\citet{Zuo2000a} and reported in Appendix  of the Supplementary Material. Given an independent and identically distributed sample $\\bf{X}_1, \\ldots, \\bf{X}_n$ of size $n$, we denote $\\hat{F}_n(\\cdot)$ its empirical distribution function and by $d(\\bf{x}; \\hat{F}_n)$ the sample depth. We assume that, $d(\\bf{x}; \\hat{F}_n)$ is a uniform consistent estimator of $d(\\bf{x}; F)$, that is, \n\\begin{equation*}\n\\sup_{\\bf{x}}| d(\\bf{x}; \\hat{F}_n) - d(\\bf{x}; F) | \\stackrel{a.s.}{\\rightarrow} 0 \\qquad n \\rightarrow \\infty ,\n\\end{equation*}\na property enjoined by many statistical data depth functions, e.g., among others simplicial depth \\citep{Liu1990}, half-space depth \\citep{Tukey1975}. One important feature of the depth functions is the $\\alpha$-depth trimmed region given by $R_\\alpha(F) = \\{ \\bf{x} \\in \\mathbb{R}^p: d(\\bf{x}; F) \\ge \\alpha\\}$; for any $\\beta \\in [0,1]$, we will denote $R^\\beta(F)$ the smallest region $R_\\alpha(F)$ that has probability larger that or equal to $\\beta$ according to $F$. Throughout, subscripts and superscripts for depth regions are used for depth levels and probability contents, respectively. Let $C^\\beta(F)$ be the complement in $\\mathbb{R}^p$ of the set $R^\\beta(F)$. Let $m = \\max_{\\bf{x}} d(\\bf{x};F)$, be the maximum of the depth, for simplicial depth $m \\le 2^{-p}$, for half-space depth $m \\le 1/2$.\nGiven a high order quantile $\\beta$, we define a filter of dimension $p$ based on\n\\begin{equation}\nd_n = \\sup_{\\bf{x} \\in C^\\beta(F)} \\{ d(\\bf{x}; \\hat{F}_n) - d(\\bf{x}; F) \\}^+ ,\n\\end{equation}\nwhere $\\{a\\}^+$ represents the positive part of $a$, and we mark as outliers all the $\\lfloor n d_n/m \\rfloor$ observations with the smallest population depth (where $\\lfloor a \\rfloor$ is the largest integer less then or equal to $a$). This define a filter in the general dimension $p$.\nWe have the following result, with obvious proof.\nIf $\\sup_{\\bf{x}}| d(\\bf{x}; \\hat{F}_n) - d(\\bf{x}; F) | = o(n)$ (a.s.) then $n d_n \\rightarrow 0$ as $n \\rightarrow \\infty$.\nIf the above result holds, then the filter would be consistent. In the next subsection we are going to illustrate this approach using the half-space depth.\n\\subsection{Filters based on Half-space Depth}\nLet $\\bf{X}$ be a $\\mathbb{R}^p$-valued random variable with distribution function $F$. For a point $\\bf{x} \\in \\mathbb{R}^p$, the half-space depth of $\\bf{x}$ with respect to $F$ is defined as the minimum probability of all closed half-spaces including $\\bf{x}$:\n\\begin{equation*}\nd_{HS} (\\bf{x};F) = \\min_{H \\in \\mathcal{H}(\\bf{x})} P_F(\\bf{X} \\in H).\n\\end{equation*}\nwhere $\\mathcal{H}(\\bf{x})$ indicates the set of all half-spaces in $\\mathbb{R}^p$ containing $\\bf{x} \\in \\mathbb{R}^p$.\nA random vector $\\bf{X} \\in \\mathbb{R}^p$ is said elliptically symmetric distributed, denoted by\n\\end{document}\n"}
{"X\u0018Ep(h;\u0016;\u0006), if it has a density function given by\nf0(x)/j\u0006\u00001=2jh((x\u0000\u0016)>\u0006\u00001(x\u0000\u0016)):\nwherehis a non-negative scalar function, \u0016is the location parameter and\n\u0006is ap\u0002ppositive de\fnite matrix. Denote by F0the corresponding distri-\nbution function and by \u0001 x= (x\u0000\u0016)>\u0006\u00001(x\u0000\u0016) the squared Mahalanobis\ndistance of a p-dimensional point x. By Theorem 3.3 of ?if a depth is a\u000ene\nequivariant () and has maximum at \u0016() (see Appendix ) then a depth is such\nthatd(x;F0) =g(\u0001x) for some non increasing function gand we can restrict\nourselves without loss of generality, to the case \u0016=0and\u0006=Iwhere Iis the\nidentity matrix of dimension p. Under this setting, it is easy to see that the\nhalf-space depth of a given point xis given by dHS(x;F0) =1\u0000F0;1(p\u0001x),\nwhereF0;1is a marginal distribution of X. If the function his such that\nexp(\u00001\n2\u0001)\nh(\u0001)!0; \u0001!1;\nthen, there exists a \u0001\u0003such that for all xso that \u0001 x>\u0001\u0003,dHS(x;F0)\u0015\ndHS(x;\b), where \b is the distribution function of the standard normal.\nHence,\nsup\nfx:\u0001x>\u0001\u0003g[dHS(x;\b)\u0000dHS(x;F0)]<0\nand therefore, for all \f >1\u00002F0;1(\u0000p\n(\u0001\u0003)),\nsup\nC\f(F0)[dHS(x;\b)\u0000dHS(x;F0)]<0:\nGiven an independent and identically distributed sample X1;:::;Xn, we\nde\fne the \flter in general dimension pintroduced previously, where here we\nuse the half-space depth\ndn= sup\nx2C\f(F)fdHS(x;^Fn)\u0000dHS(x;F(T0n;C0n))g+;\nwhere\fis a high order quantile, ^Fn(\u0001) is the empirical distribution function\nandF(T0n;C0n) is a chosen reference distribution which depends on a pair\nof initial location and dispersion estimators, T0nandC0n. Hereafter, we\nare going to use the normal distribution F=N(T0n;C0n). For T0nand\nC0none might use, e.g., the coordinate-wise median and the coordinate-wise\nMAD for a univariate \flter as in ?. In order to compute the value dn, we\nhave to identify the set C\f(F) =fx2RpjdHS(x;F)\u0014dHS(\u0011\f;F)gwhere\n\u0011\fis a large quantile of F. By Corollary 4.3 in and denoting with\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\n$\\bf{X} \\sim E_p(h, \\bf{\\mu}, \\bf{\\Sigma})$, if it has a density function given by\n\\begin{equation*}\nf_0(\\bf{x}) \\propto |\\bf{\\Sigma}^{-1/2}| h((\\bf{x}-\\bf{\\mu})^\\top\\bf{\\Sigma}^{-1}(\\bf{x}-\\bf{\\mu})) .\n\\end{equation*}\nwhere $h$ is a non-negative scalar function, $\\bf{\\mu}$ is the location parameter and $\\bf{\\Sigma}$ is a $p \\times p$ positive definite matrix.\nDenote by $F_0$ the corresponding distribution function and by $\\Delta_{\\bf{x}} = (\\bf{x} - \\bf{\\mu})^\\top \\bf{\\Sigma}^{-1} (\\bf{x} - \\bf{\\mu})$ the squared Mahalanobis distance of a $p$-dimensional point $\\bf{x}$. By Theorem 3.3 of \\citet{Zuo2000b} if a depth is affine equivariant () and has maximum at $\\bf{\\mu}$ () (see Appendix ) then a depth is such that $d(\\bf{x}; F_0) = g(\\Delta_{\\bf{x}})$ for some non increasing function $g$ and we can restrict ourselves without loss of generality, to the case $\\bf{\\mu} = \\bf{0}$ and $\\bf{\\Sigma} = \\bf{I}$ where $\\bf{I}$ is the identity matrix of dimension $p$. Under this setting, it is easy to see that the half-space depth of a given point $\\bf{x}$ is given by $d_{HS}(\\bf{x}; F_0) = 1 - F_{0,1}(\\sqrt{\\Delta_{\\bf{x}}})$, where $F_{0,1}$ is a marginal distribution of $\\bf{X}$.\nIf the function $h$ is such that\n\\begin{equation*}\n\\frac{\\exp(-\\frac{1}{2} \\Delta)}{h(\\Delta)} \\rightarrow 0 , \\qquad \\Delta \\rightarrow \\infty ,\n\\end{equation*}\nthen, there exists a $\\Delta^\\ast$ such that for all $\\bf{x}$ so that $\\Delta_{\\bf{x}} > \\Delta^\\ast$, $d_{HS}(\\bf{x}; F_0) \\ge d_{HS}(\\bf{x}; \\Phi)$, where $\\Phi$ is the distribution function of the standard normal. Hence,\n\\begin{equation*}\n\\sup_{\\{\\bf{x}: \\Delta_{\\bf{x}} > \\Delta^\\ast\\}} [d_{HS}(\\bf{x}; \\Phi) - d_{HS}(\\bf{x}; F_0)] < 0\n\\end{equation*}\nand therefore, for all $\\beta > 1 - 2 F_{0,1}(-\\sqrt(\\Delta^{\\ast}))$,  \n\\begin{equation*}\n\\sup_{C^\\beta(F_0)} [d_{HS}(\\bf{x}; \\Phi) - d_{HS}(\\bf{x}; F_0)] < 0 \\ .\n\\end{equation*}\nGiven an independent and identically distributed sample $\\bf{X}_1, \\ldots, \\bf{X}_n$, we define the filter in general dimension $p$ introduced previously, where here we use the half-space depth\n\\begin{equation*}\nd_n = \\sup_{\\bf{x} \\in C^\\beta(F)} \\{ d_{HS}(\\bf{x}; \\hat{F}_n) - d_{HS}(\\bf{x}; F(\\bf{T}_{0n}, \\bf{C}_{0n})) \\}^+ ,\n\\end{equation*}\nwhere $\\beta$ is a high order quantile, $\\hat{F}_n(\\cdot)$ is the empirical distribution function and $F(\\bf{T}_{0n}, \\bf{C}_{0n})$ is a chosen reference distribution which depends on a pair of initial location and dispersion estimators, $\\bf{T}_{0n}$ and $\\bf{C}_{0n}$. Hereafter, we are going to use the normal distribution $F = N(\\bf{T}_{0n}, \\bf{C}_{0n})$. For $\\bf{T}_{0n}$ and $\\bf{C}_{0n}$ one might use, e.g., the coordinate-wise median and the coordinate-wise MAD for a univariate filter as in \\citet{Zamar2017}. In order to compute the value $d_n$, we have to identify the set $C^\\beta(F) = \\{ \\bf{x} \\in \\mathbb{R}^p | d_{HS}(\\bf{x},F) \\le d_{HS}(\\eta_\\beta,F) \\}$ where $\\eta_\\beta$ is a large quantile of $F$. By Corollary 4.3 in  and denoting with\n\\end{document}\n"}
{"\u0001x= (x\u0000T0n)>C\u00001\n0n(x\u0000T0n) the squared Mahalanobis distance of x\nusing the initial location and dispersion estimates, the set can be rewritten\nasC\f(F) =fx2Rpj\u0001x>(\u001f2\np)\u00001(\f)g, where (\u001f2\np)\u00001(\f) is a large quan-\ntile of a chi-squared distribution with pdegrees of freedom. Now we want\nto show that the result given by Proposition holds for this particular case.\nConsider a random vector ( X1;:::;Xn)\u0018F0(\u00160;\u00060) and suppose that F0\nis an elliptically symmetric distribution. Also consider a pair of location and\ndispersion estimators T0nandC0nsuch that T0n!\u00160andC0n!\u00060a.s..\nLetFbe a chosen reference distribution and ^Fnthe empirical distribution\nfunction. If the reference distribution satis\fes\nsup\nx2C\f(F0)[dHS(x;F)\u0000dHS(x;F0)]<0\nwhere\fis some large quantile of F0, then\nndn!0 asn!1\nProof. In?, it is proved that for i.i.d. X1;X2;:::;Xnwith distribution F0,\nasn!1\nsup\nt2RdjdHS(t;F0)\u0000dHS(t;^Fn)j!0a.s.\nNote that, by the continuity of F,F(T0n;C0n)!F(\u00160;\u00060) a.s.. Hence, for\neach\">0 there exists n0such that for all n>n 0we have\nsup\nx2C\f(F0)fdHS(x;^Fn)\u0000dHS(x;F(T0n;C0n))g\u0014\nsup\nx2C\f(F0)fdHS(x;^Fn)\u0000dHS(x;F0(\u00160;\u00060))g+\nsup\nx2C\f(F0)fdHS(x;F0(\u00160;\u00060))\u0000dHS(x;F(\u00160;\u00060))g+\nsup\nx2C\f(F0)fdHS(x;F(\u00160;\u00060))\u0000dHS(x;F(T0n;C0n))g\n\u0014\"\n2+ 0 +\"\n2=\"\nIn the next example, we illustrate a univariate \flter based on half-space\ndepth that controls independently the left and the right tail of the distribu-\ntion. In the univariate case, given a point xthere exist only two halfspaces\nincluding it, hence the half-space depth assumes the explicit form\ndHS(x;F) = min(PF((\u00001;x]);PF([x;1)))\n= min(F(x);1\u0000F(x) +PF(X=x));\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\n$\\Delta_x = (\\bf{x} - \\bf{T}_{0n})^\\top \\bf{C}_{0n}^{-1}(\\bf{x} - \\bf{T}_{0n})$ the squared Mahalanobis distance of $\\bf{x}$ using the initial location and dispersion estimates, the set can be rewritten as $C^\\beta(F) = \\{ \\bf{x} \\in \\mathbb{R}^p | \\Delta_x > (\\chi^2_p)^{-1}(\\beta) \\}$, where $(\\chi^2_p)^{-1}(\\beta)$ is a large quantile of a chi-squared distribution with $p$ degrees of freedom.  \nNow we want to show that the result given by Proposition  holds for this particular case. \nConsider a random vector $(\\bf{X}_1, \\ldots , \\bf{X}_n) \\sim F_0(\\bf{\\mu}_0, \\bf{\\Sigma}_0)$ and suppose that $F_0$ is an elliptically symmetric distribution. Also consider a pair of location and dispersion estimators $\\bf{T}_{0n}$ and $\\bf{C}_{0n}$ such that $\\bf{T}_{0n} \\rightarrow \\bf{\\mu}_0$ and $\\bf{C}_{0n} \\rightarrow \\bf{\\Sigma}_0$ a.s.. Let $F$ be a chosen reference distribution and $\\hat{F}_n$ the empirical distribution function. If the reference distribution satisfies\n\\begin{equation*}\n\\sup_{\\bf{x} \\in C^\\beta(F_0)} [d_{HS}(\\bf{x}; F) - d_{HS}(\\bf{x}; F_0)] < 0 \n\\end{equation*}\nwhere $\\beta$ is some large quantile of $F_0$, then   \n\\begin{equation*}\nn d_n \\rightarrow 0 \\mbox{ as } n \\rightarrow \\infty\n\\end{equation*}\n\\begin{proof}\nIn \\citet{Donoho1992}, it is proved that for i.i.d. $\\bf{X}_1, \\bf{X}_2, ... , \\bf{X}_n$ with distribution $F_0$, as $n \\rightarrow \\infty$\n\\begin{equation*}\n\\sup_{\\bf{t} \\in \\mathbb{R}^d} |d_{HS}(\\bf{t},F_0) - d_{HS}(\\bf{t},\\hat{F}_n)| \\rightarrow 0   \\mbox{   a.s.}\n\\end{equation*}\nNote that, by the continuity of $F$, $F(\\bf{T}_{0n}, \\bf{C}_{0n}) \\rightarrow F(\\bf{\\mu}_0, \\bf{\\Sigma}_0)$ a.s..\nHence, for each $\\varepsilon > 0$ there exists $n_0$ such that for all $n > n_0$ we have\n\\begin{align*}\n  \\sup_{\\bf{x} \\in C^\\beta(F_0)} \\{ d_{HS}(\\bf{x};  \\hat{F}_n) - d_{HS}(\\bf{x}; F(\\bf{T}_{0n}, \\bf{C}_{0n})) \\} \\le \\\\\n   \\sup_{\\bf{x} \\in C^\\beta(F_0)} \\{ d_{HS}(\\bf{x}; \\hat{F}_n) - d_{HS}(\\bf{x}; F_0(\\bf{\\mu}_0, \\bf{\\Sigma}_0)) \\} + \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\sup_{\\bf{x} \\in C^\\beta(F_0)} \\{ d_{HS}(\\bf{x}; F_0(\\bf{\\mu}_0, \\bf{\\Sigma}_0)) - d_{HS}(\\bf{x}; F(\\bf{\\mu}_0, \\bf{\\Sigma}_0)) \\} + \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\sup_{\\bf{x} \\in C^\\beta(F_0)} \\{ d_{HS}(\\bf{x}; F(\\bf{\\mu}_0, \\bf{\\Sigma}_0)) - d_{HS}(\\bf{x}; F(\\bf{T}_{0n}, \\bf{C}_{0n})) \\}  \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\le     \\frac{\\varepsilon}{2} + 0 + \\frac{\\varepsilon}{2} = \\varepsilon\n\\end{align*}\n\\end{proof}\nIn the next example, we illustrate a univariate filter based on half-space depth that controls independently the left and the right tail of the distribution.\nIn the univariate case, given a point $x$ there exist only two halfspaces including it, hence the half-space depth assumes the explicit form\n\\begin{align*}\nd_{HS}(x;F)  =  \\min (P_F((-\\infty,x]),P_F([x,\\infty))) \\\\\n\t\t    = \\min (F(x) , 1 - F(x) + P_F(X = x)),\n\\end{align*}\n\\end{document}\n"}
{"and considering the empirical distribution function ^Fn(\u0001), the halfspace\ndepth will be\ndHS(x;^Fn) = min\u00121\nnnX\ni=1I(Xi\u0014x);1\nnnX\ni=1I(Xi\u0015x)\u0013\n:\nConsider T0n= (T0n;1;:::;T0n;p) and S0n= (S0n;1;:::;S0n;p), a pair of\ninitial location and dispersion estimators. Here we choose for T0nandS0n\nrespectively the coordinate-wise median and the median absolute deviation\n(MAD). For each variable ( X1j;X 2j;:::;Xnj) (j= 1;:::;p ), we denote the\nstandardized version of XijbyZij=Xij\u0000T0n;j\nS0n;j. LetFja chosen reference\ndistribution for Zij; here we use the standard normal distribution, i.e., Fj=\n\b. Let ^Fn;jbe the empirical distribution for the standardized values, that is\n^Fn;j(t) =1\nnnX\ni=1I(Zij\u0014t)j= 1;:::;p:\nWe de\fne the proportion of \nagged outliers by\ndn;j= max\u0012\nsup\nt\u0014\u0000\u0011\f;jfdHS(t;^Fn;j)\u0000dHS(t; Fj)g+; sup\nt\u0015\u0011\f;jfdHS(t;^Fn;j)\u0000dHS(t; Fj)g+\u0013\n;\nwhere\u0011\f;j=F\u00001\nj(\f) is a large quantile of Fj. Note that, according to (),\nwe are considering the set C\f(Fj) =fx2R:dHS(x;Fj)< dHS(\u0011\f;j)g,\nwhich results in the simpler form written above considering the de\fnition of\nthe half-space depth in the univariate case. Here, if we consider the order\nstatisticsZ(i);j, de\fnei\u0000= minfi:Z(i);j>\u0000\u0011\f;jgandi+= maxfi:Z(i);j<\n\u0011\f;jg. Using the de\fnition of half-space depth function in the univariate case,\npresented above, the previous expression can be written as\ndn;j= max\u0012\nsup\ni<i\u0000fi\nn\u0000Fj(Z(i);j)g+;sup\ni>i +fFj(Z(i);j)\u0000i\u00001\nng+\u0013\n: (1)\nThen, we \nagbndn;jcobservations with the smallest depth value as cell-wise\noutliers and replace them by NA's.\n0.1 A consistent univariate, bivariate and p-variate \fl-\nter\nGiven a sample X1;:::;Xnwhere Xi2Rp;i=1;:::;n, we \frst apply the\nunivariate \flter described in the previous example to each variable separately.\nFiltered data are indicated through an auxiliary matrix Uof zeros and ones,\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\nand considering the empirical distribution function $\\hat{F}_n(\\cdot)$, the halfspace depth will be \n\\begin{equation*}\nd_{HS}(x,\\hat{F}_n) = \\min \\biggl( \\frac{1}{n} \\sum_{i=1}^n I(X_i \\leq x) , \\frac{1}{n} \\sum_{i=1}^n I(X_i \\geq x) \\biggr). \n\\end{equation*}\nConsider $\\bf{T}_{0n} = (T_{0n,1}, \\ldots, T_{0n,p})$ and $\\bf{S}_{0n} = (S_{0n,1}, \\ldots, S_{0n,p})$, a pair of initial location and dispersion estimators. Here we choose for  $\\bf{T}_{0n}$ and $\\bf{S}_{0n}$ respectively the coordinate-wise median and the median absolute deviation (MAD).\nFor each variable $(X_{1j}, X_{2j}, \\ldots, X_{nj})$ ($j=1, \\ldots, p$), we denote the standardized version of $X_{ij}$ by $Z_{ij}= \\frac{X_{ij} - T_{0n,j}}{S_{0n,j}}$. Let $F_j$ a chosen reference distribution for $Z_{ij}$; here we use the standard normal distribution, i.e., $F_j = \\Phi$.\nLet $\\hat{F}_{n,j}$ be the empirical distribution for the standardized values, that is\n\\begin{equation*}\n\\hat{F}_{n,j}(t) = \\frac{1}{n} \\sum_{i=1}^{n} I(Z_{ij} \\leq t) \\qquad j = 1, \\ldots, p.\n\\end{equation*}\nWe define the proportion of flagged outliers by\n\\small\n\\begin{equation*}\nd_{n,j} = \\max \\biggl( \\sup_{t \\leq -\\eta_{\\beta,j}} \\{d_{HS}(t,\\hat{F}_{n,j}) - d_{HS}(t, F_j)\\}^+ ; \\sup_{t \\geq \\eta_{\\beta,j}} \\{d_{HS}(t,\\hat{F}_{n,j}) - d_{HS}(t, F_j)\\}^+ \\biggr),\n\\end{equation*}\n\\normalsize\nwhere $\\eta_{\\beta,j} = F_j^{-1}(\\beta)$ is a large quantile of $F_j$. Note that, according to (), we are considering the set $C^\\beta(F_j) = \\{x \\in \\mathbb{R} : d_{HS}(x,F_j) < d_{HS}(\\eta_{\\beta,j})\\}$, which results in the simpler form written above considering the definition of the half-space depth in the univariate case.\nHere, if we consider the order statistics $Z_{(i),j}$, define $i_- = \\min \\{ i : Z_{(i),j} > -\\eta_{\\beta,j} \\}$  and $i_+ = \\max \\{ i : Z_{(i),j} < \\eta_{\\beta,j} \\}$. Using the definition of half-space depth function in the univariate case, presented above, the previous expression can be written as \n\\begin{equation}\nd_{n,j} = \\max \\biggl( \\sup_{i < i_-} \\{\\frac{i}{n} -  F_j(Z_{(i),j})\\}^+ , \\sup_{i > i_+} \\{ F_j(Z_{(i),j}) - \\frac{i-1}{n}\\}^+ \\biggr).\n\\end{equation}\nThen, we flag $\\lfloor nd_{n,j} \\rfloor$ observations with the smallest depth value as cell-wise outliers and replace them by NA's.\n\\subsection{A consistent univariate, bivariate and $p$-variate filter}\nGiven a sample $\\bf{X}_1, \\ldots, \\bf{X}_n$ where $\\bf{X}_i \\in \\mathbb{R}^p, i = 1, \\ldots, n$, we first apply the univariate filter described in the previous example to each variable separately. Filtered data are indicated through an auxiliary matrix $\\bf{U}$ of zeros and ones,\n\\end{document}\n"}
{"with zero corresponding to a NA value. Next we want to identify the\nbivariate outliers by iterating the \flter over all possible pairs of variables.\nConsider a pair of variables X(jk)=f(Xij;Xik)g;i=1;:::;n. The ini-\ntial location and dispersion estimators are, respectively, the coordinate-wise\nmedian and the 2 \u00022 sub-matrix S(jk)of the estimate Scomputed by the\ngeneralized S-estimator on non-\fltered data. Note that, this ensure the pos-\nitive de\fniteness property for Sand eachd\u0002dsub-matrix corresponding to\na subset of dvariables. For bivariate points with no \nagged components by\nthe univariate \flter we compute the squared Mahalanobis distance \u0001(jk)\niand\nhence apply the bivariate \flter, for all 1 < j < k < p . At the end we want\nto identify the cells ( i;j) which have to be \nagged as cell-wise outliers. The\nprocedure used for this purpose is described in ?and reported here. Let\nJ=f(i;j;k ) : \u0001(jk)\niis \nagged as bivariate outlier g\nbe the set of triplets which identi\fes the pairs of cells \nagged by the bivariate\n\flter in rows i= 1;:::;n . For each cell ( i;j) in the data, we count the number\nof \nagged pairs in the i-th row in which the considered cell is involved:\nmij= #fk: (i;j;k )2Jg:\nIn absence of contamination, mijfollows approximately a binomial distribu-\ntionBin(P\nk 6=jUjk;\u000e) where\u000erepresents the overall proportion of cell-wise\noutliers undetected by the univariate \flter. Hence, we \nag the cell ( i;j) if\nmij> c ij, wherecijis the 0:99-quantile of Bin(P\nk 6=jUjk;0:1). Finally, we\nperform the p-variate \flter as described in subsection to the full data matrix.\nDetected observations (rows) are directly \nagged as p-variate (case-wise) out-\nliers. We denote the procedure based on univariate, bivariate and p-variate\n\flters, HS-UBPF.\n0.1 A sequencing \fltering procedure\nSuppose we would like to apply a sequence of k\flters with di\u000berent dimension\n1\u0014d1\u0014d2\u0014:::\u0014dk\u0014p. For each di,i= 1;:::;k , the \flter updates\nthe data matrix adding NA values to the di-tuples identi\fed as di-variate\noutliers. In this way, each \flter applies only those di-tuples that have not\nbeen \nagged as outliers by the \flters with lower dimension. Initial values for\neach procedures rather than d1would be obtained by applying the GSE to\nthe actual \fltered values. This procedure aims to be a valid alternative to\nthat used in the presented HS-UBPF \flter to perform a sequence of \flters\nwith di\u000berent dimensions. However, this is a preliminary idea, indeed it has\nnot been implemented yet.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\nwith zero corresponding to a NA value. Next we want to identify the bivariate outliers by iterating the filter over all possible pairs of variables. Consider a pair of variables $\\bf{X}^{(jk)} = \\{ (X_{ij},X_{ik}) \\}, i = 1, \\ldots, n$. The initial location and dispersion estimators are, respectively, the coordinate-wise median and the $2 \\times 2$ sub-matrix $S^{(jk)}$ of the estimate $S$ computed by the generalized S-estimator on non-filtered data. Note that, this ensure the positive definiteness property for $S$ and each $d \\times d$ sub-matrix corresponding to a subset of $d$ variables. For bivariate points with no flagged components by the univariate filter we compute the squared Mahalanobis distance $\\Delta^{(jk)}_i$ and hence apply the bivariate filter, for all $1 < j < k < p$. At the end we want to identify the cells $(i,j)$ which have to be flagged as cell-wise outliers. The procedure used for this purpose is described in \\citet{Zamar2017} and reported here. Let\n\\begin{equation*}\nJ = \\{ (i,j,k) : \\Delta_i^{(jk)} \\mbox{ is flagged as bivariate outlier} \\}\n\\end{equation*}\nbe the set of triplets which identifies the pairs of cells flagged by the bivariate filter in rows $i = 1, \\ldots, n$. For each cell $(i,j)$ in the data, we count the number of flagged pairs in the $i$-th row in which the considered cell is involved:\n\\begin{equation*}\nm_{ij} = \\#\\{ k : (i,j,k) \\in J\\}.\n\\end{equation*}\nIn absence of contamination, $m_{ij}$ follows approximately a binomial distribution $Bin(\\sum_{k \\not = j}\\bf{U}_{jk},\\delta)$ where $\\delta$ represents the overall proportion of cell-wise outliers undetected by the univariate filter. Hence, we flag the cell $(i,j)$ if $m_{ij} > c_{ij}$, where $c_{ij}$ is the $0.99$-quantile of $Bin(\\sum_{k \\not = j}\\bf{U}_{jk},0.1)$.\nFinally, we perform the $p$-variate filter as described in subsection  to the full data matrix. Detected observations (rows) are directly flagged as $p$-variate (case-wise) outliers. We denote the procedure based on univariate, bivariate and $p$-variate filters, HS-UBPF.\n\\subsection{A  sequencing filtering procedure}\nSuppose we would like to apply a sequence of $k$ filters with different dimension $1 \\le d_1 \\le d_2 \\le \\ldots \\le d_k \\le p$. For each $d_i$, $i = 1, \\ldots, k$, the filter updates the data matrix adding NA values to the $d_i$-tuples identified as $d_i$-variate outliers. In this way, each filter applies only those $d_i$-tuples that have not been flagged as outliers by the filters with lower dimension.\nInitial values for each procedures rather than $d_1$ would be obtained by applying the GSE to the actual filtered values.\nThis procedure aims to be a valid alternative to that used in the presented HS-UBPF filter to perform a sequence of filters with different dimensions. However, this is a preliminary idea, indeed it has not been implemented yet.\n\\end{document}\n"}
{"1 Gervini-Yohai d-variate \flter\nIn this Section we are going to show that the \flters introduced in are a special\ncase of our approach, using the following Gervini-Yohai depth\ndGY(t;F;G) =1\u0000G(\u0001(t;\u0016(F);\u0006(F)));\nwhereGis a continuous distribution function, \u0016(F) and \u0006(F) are the lo-\ncation and scatter matrix functionals and \u0001( t;F) = \u0001( t;\u0016(F);\u0006(F)) =\n(t\u0000\u0016(F))>\u0006(F)\u00001(t\u0000\u0016(F)) is the squared Mahalanobis distance. Ap-\npendix shows that this is a statistical data depth function. Let fGng1\nn=1be\na sequence of discrete distribution functions that might depends on ^Fnand\nsuch that suptjGn(t)\u0000G(t)ja:s:!0, we might de\fne the \fnite sample version\nof the Gervini-Yohai depth as\ndGY(t;^Fn;Gn) =1\u0000Gn(\u0001(t;\u0016(^Fn);\u0006(^Fn)));\nhowever for \fltering purpose we will use two alternative de\fnitions later on.\nThe use of Gn, that might depend on the data, instead of Gmakes this\nsample depth semiparametric. We notice that the Mahalanobis depth, which\nis completely parametric, cannot be used for the purpose of de\fning a \flter\nin a similar fashion. Let 1 \u0014d\u0014p,j1;:::;j dbe and-tuple of the integer\nnumbers 1;:::;p and, for easy of presentation, let Yi= (Xij1;:::;Xijd) be a\nsubvector of dimension dofXi. Consider a pair of initial location and scatter\nestimators\nT(d)\n0n=0\n@T0n;j1\n:::\nT0n;jd1\nA and C(d)\n0n=0\n@C0n;j1j1:::C 0n;j1jd\n:::::::::\nC0n;jdj1:::C 0n;jdjd1\nA:\nNow, de\fne the squared Mahalanobis distance for a data point Yiby \u0001 i=\n\u0001(Yi;^Fn) =\u0001(Yi;T(d)\n0n;C(d)\n0n). Consider Gthe distribution function of a\n\u001f2\nd,Hthe distribution function of \u0001 = \u0001( \u0001;F) and let ^Hnbe the empirical\ndistribution function of \u0001 i(1\u0014i\u0014n). We consider two \fnite sample\nversion of the Gervini-Yohai depth, i.e.,\ndGY(t;^Fn;G) =1\u0000G(\u0001(t;^Fn));\nand\ndGY(t;^Fn;^Hn) =1\u0000^Hn(\u0001(t;^Fn)):\nThe proportion of \nagged d-variate outliers is de\fned by\ndn= sup\nt2AfdGY(t;^Fn;^Hn)\u0000dGY(t;^Fn;G)g+:\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\n\\section{Gervini-Yohai $d$-variate filter}\nIn this Section we are going to show that the filters introduced in  are a special case of our approach, using the following Gervini-Yohai depth\n\\begin{equation*}\nd_{GY}(\\bf{t}, F, G) = 1 - G(\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F))) ,\n\\end{equation*}\nwhere $G$ is a continuous distribution function, $\\bf{\\mu}(F)$ and $\\bf{\\Sigma}(F)$ are the location and scatter matrix functionals and $\\Delta(t, F) = \\Delta(\\bf{t}, \\bf{\\mu}(F), \\bf{\\Sigma}(F)) = (\\bf{t} - \\bf{\\mu}(F))^\\top \\bf{\\Sigma}(F)^{-1} (\\bf{t} - \\bf{\\mu}(F))$ is the squared Mahalanobis distance. Appendix  shows that this is a statistical data depth function. Let $\\{ G_n \\}_{n=1}^\\infty$ be a sequence of discrete distribution functions that might depends on $\\hat{F}_n$ and such that $\\sup_{t} |G_n(t) - G(t)| \\stackrel{a.s.}{\\rightarrow} 0$, we might define the finite sample version of the Gervini-Yohai depth as\n\\begin{equation*}\nd_{GY}(\\bf{t}, \\hat{F}_n, G_n) = 1 - G_n(\\Delta(\\bf{t},\\bf{\\mu}(\\hat{F}_n),\\bf{\\Sigma}(\\hat{F}_n))) \\ ,\n\\end{equation*}\nhowever for filtering purpose we will use two alternative definitions later on.\nThe use of $G_n$, that might depend on the data, instead of $G$ makes this sample depth semiparametric. We notice that the Mahalanobis depth, which is completely parametric, cannot be used for the purpose of defining a filter in a similar fashion.\nLet $1 \\le d \\le p$, $j_1, \\ldots, j_d$ be an $d$-tuple of the integer numbers $1, \\ldots, p$ and, for easy of presentation, let $\\bf{Y}_i = (X_{ij_1}, \\ldots , X_{ij_d})$ be a subvector of dimension $d$ of $\\bf{X}_i$. Consider a pair of initial location and scatter estimators\n\\begin{equation*}\n \\bf{T}_{0n}^{(d)} =  \\left (\n\t\t\t\t\\begin{array}{ll}\n\t\t\t\tT_{0n,j_1} \\\\\n \t\t\t\t\\ldots \\\\\n\t\t\t\tT_{0n,j_d}\n\t\t\t\t\\end{array}\n\t\t\t\\right )\n\t\t\t\\quad \\mbox{ and } \\quad \n        \\bf{C}_{0n}^{(d)} =  \\left (\n\t\t\t\t\\begin{array}{lll}\n\t\t\t\tC_{0n,j_1j_1}  \\ldots    C_{0n,j_1j_d} \\\\\n \t\t\t\t\\ldots             \\ldots       \\ldots            \\\\\n\t\t\t\tC_{0n,j_dj_1} \\ldots     C_{0n,j_dj_d}\n\t\t\t\t\\end{array}\n\t\t\t\\right ) \\ .\n\\end{equation*}\nNow, define the squared Mahalanobis distance for a data point $\\bf{Y}_i$ by $\\Delta_i = \\Delta(\\bf{Y}_i, \\hat{F}_n) = \\Delta(\\bf{Y}_i, \\bf{T}_{0n}^{(d)}, \\bf{C}_{0n}^{(d)})$. Consider $G$ the distribution function of a $\\chi_d^2$, $H$ the distribution function of $\\Delta = \\Delta(\\cdot, F)$ and let $\\hat{H}_n$ be the empirical distribution function of $\\Delta_i$ ($1 \\le i \\le n$). We consider two finite sample version of the Gervini-Yohai depth, i.e., \n\\begin{equation*}\nd_{GY}(\\bf{t}, \\hat{F}_n, G) = 1 - G(\\Delta(\\bf{t}, \\hat{F}_n)) ,\n\\end{equation*}\nand\n\\begin{equation*}\nd_{GY}(\\bf{t}, \\hat{F}_n, \\hat{H}_n) = 1 - \\hat{H}_n(\\Delta(\\bf{t}, \\hat{F}_n)) .\n\\end{equation*}\nThe proportion of flagged $d$-variate outliers is defined by\n\\begin{equation*}\nd_n = \\sup_{\\bf{t} \\in A} \\{ d_{GY}(\\bf{t}, \\hat{F}_n, \\hat{H}_n) - d_{GY}(\\bf{t}, \\hat{F}_n, G) \\}^+ .\n\\end{equation*}\n\\end{document}\n"}
{"HereA=ft2Rd:dGY(t;F;G)\u0014dGY(\u0010;F;G)g, where\u0010is any point\ninRdsuch that \u0001( \u0010;F) =\u0011and\u0011=G\u00001(\u000b) is a large quantile of G. Then,\nwe \nagbndncobservations. It is easy to see that,\ndn= sup\nt2Af[1\u0000^Hn(\u0001(t;^Fn))]\u0000[1\u0000G(\u0001(t;^Fn))]g+\n= sup\nt2AfG(\u0001(t;^Fn))\u0000^Hn(\u0001(t;^Fn))g+\n= sup\n\u0001\u0015\u0011fG(\u0001)\u0000^Hn(\u0001)g+\nsincedGYis a non increasing function of the squared Mahalanobis distance\nof the point t. We can rephrase Proposition 2. in ?, that states the con-\nsistency property of the \flter as follows. Consider a random vector Y=\n(X1;:::;Xd)\u0018F0and a pair of location and scatter estimators T0nand\nC0nsuch that T0n!\u00160=\u0016(F0)2RdandC0n!\u00060=\u0006(F0) a.s..\nConsider any continuous distribution function Gand let ^Hnbe the empirical\ndistribution function of \u0001 iandH0(t) = Pr(( Y\u0000\u00160)t\u0006\u00001\n0(Y\u0000\u00160)\u0014t). If\nthe distribution Gsatis\fes:\nmax\nt2AfdGY(t;F0;H0)\u0000dGY(t;F0;G)g\u00140; (1)\nwhereA=ft2Rd:dGY(t;F0;G)\u0014dGY(\u0010;F0;G)g, where\u0010is any point\ninRdsuch that \u0001( \u0010;F0) =\u0011and\u0011=G\u00001(\u000b) is a large quantile of G, then\nn0\nn!0 a.s.\nwhere\nn0=bndnc:\nProof. Note that\ndGY(t;^Fn;^Hn)\u0000dGY(t;^Fn;G) =G(\u0001(t;T0n;C0n))\u0000^Hn(\u0001(t;T0n;C0n))\nand condition in equation () is equivalent to\nmax\n\u0001\u0015\u0011fG(\u0001)\u0000H0(\u0001)g\u00140;\nThe rest of the proof is the same as in Proposition 2. of ?.\nFigure shows the bivariate scatter plot of WTS versus HTLD, HTLD\nversus WSBC and WSBC versus SUR where the GY-UBF and HS-UBF\n\flters are applied, respectively. The bivariate observations with at least\none component \nagged as outlier are in blue, and outliers detected by the\nbivariate \flter are in orange. We see that the HS-UBF identi\fes less outliers\nwith respect to the GY-UBF.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\nHere $A = \\{ \\bf{t} \\in \\mathbb{R}^d: d_{GY}(\\bf{t}, F, G) \\leq d_{GY}(\\bf{\\zeta}, F, G) \\}$, where $\\bf{\\zeta}$ is any point in $\\mathbb{R}^d$ such that $\\Delta(\\bf{\\zeta}, F)  = \\eta$ and $\\eta = G^{-1}(\\alpha)$ is a large quantile of $G$. Then, we flag $\\lfloor nd_{n} \\rfloor$ observations. It is easy to see that,\n\\begin{align*}\nd_n  = \\sup_{\\bf{t} \\in A} \\{ [1 - \\hat{H}_n(\\Delta(\\bf{t}, \\hat{F}_n))] - [1 - G(\\Delta(\\bf{t}, \\hat{F}_n))] \\}^+ \\\\\n =  \\sup_{\\bf{t} \\in A} \\{ G(\\Delta(\\bf{t}, \\hat{F}_n)) - \\hat{H}_n(\\Delta(\\bf{t}, \\hat{F}_n)) \\}^+ \\\\\n = \\sup_{\\Delta \\ge \\eta} \\{ G(\\Delta) - \\hat{H}_n(\\Delta) \\}^+\n\\end{align*}\nsince $d_{GY}$ is a non increasing function of the squared Mahalanobis distance of the point $\\bf{t}$.\nWe can rephrase Proposition 2. in \\citet{Zamar2017}, that states the consistency property of the filter as follows.\nConsider a random vector $\\bf{Y} = (X_{1}, \\ldots, X_{d}) \\sim F_0$ and a pair of location and scatter estimators $\\bf{T}_{0n}$ and $\\bf{C}_{0n}$ such that $\\bf{T}_{0n} \\rightarrow \\bf{\\mu}_0 = \\bf{\\mu}(F_0) \\in \\mathbb{R}^d$ and $\\bf{C}_{0n} \\rightarrow \\bf{\\Sigma}_0 = \\bf{\\Sigma}(F_0)$ a.s.. Consider any continuous distribution function $G$ and let $\\hat{H}_n$ be the empirical distribution function of $\\Delta_i$ and $H_0(t) = \\Pr ((\\bf{Y} - \\bf{\\mu}_0)^t \\bf{\\Sigma}_0^{-1}(\\bf{Y} - \\bf{\\mu}_0) \\le t )$. If the distribution $G$ satisfies: \n\\begin{equation}\n\\max_{\\bf{t} \\in A} \\{ d_{GY}(\\bf{t},F_0,H_0) - d_{GY}(\\bf{t},F_0,G) \\} \\le 0 ,\n\\end{equation}\nwhere $A = \\{ \\bf{t} \\in \\mathbb{R}^d: d_{GY}(\\bf{t}, F_0, G) \\leq d_{GY}(\\bf{\\zeta}, F_0, G) \\}$, where $\\bf{\\zeta}$ is any point in $\\mathbb{R}^d$ such that $\\Delta(\\bf{\\zeta}, F_0)  = \\eta$ and $\\eta = G^{-1}(\\alpha)$ is a large quantile of $G$, then\n\\begin{equation*}\n\\frac{n_0}{n} \\rightarrow 0 \\qquad \\text{a.s.}\n\\end{equation*}\nwhere\n\\begin{equation*}\nn_0 = \\lfloor nd_{n} \\rfloor .\n\\end{equation*}\n\\begin{proof}\nNote that\n\\begin{equation*}\nd_{GY}(\\bf{t},\\hat{F}_n,\\hat{H}_n) - d_{GY}(\\bf{t},\\hat{F}_n, G) = G(\\Delta(\\bf{t},\\bf{T}_{0n},\\bf{C}_{0n})) - \\hat{H}_n(\\Delta(\\bf{t},\\bf{T}_{0n},\\bf{C}_{0n})) \n\\end{equation*}\t\t\nand condition in equation () is equivalent to \n\\begin{equation*}\n\\max_{\\Delta \\ge \\eta} \\{G(\\Delta) - H_0(\\Delta) \\} \\le 0 ,\n\\end{equation*}\nThe rest of the proof is the same as in Proposition 2. of \\citet{Zamar2017}.\n\\end{proof}\nFigure  shows the bivariate scatter plot of WTS versus HTLD, HTLD versus WSBC and WSBC versus SUR where the GY-UBF and HS-UBF filters are applied, respectively. The bivariate observations with at least one component flagged as outlier are in blue, and outliers detected by the bivariate filter are in orange. We see that the HS-UBF identifies less outliers with respect to the GY-UBF.\n\\end{document}\n"}
{"1 Monte Carlo results\nWe performed a Monte Carlo simulation to assess the performance of the\nproposed \flter based on halfspace depth. After the \flter \nags the outlying\nobservations, the generalized S-estimator is applied to the data with added\nmissing values. Our simulation study is based on the same setup described\nin?to compare signi\fcantly the performance of our \flter with respect to\nthe \flter introduced in their work. We considered samples from a Np(0;\u00060),\nwhere all values in diag(\u00060) are equal to 1, p= 10;20;30;40;50 and the\nsample size is n= 10p. We consider the following scenarios:\n\u000fClean data: data without changes.\n\u000fCell-Wise contamination: a proportion \u000fof cells in the data is replaced\nbyXij\u0018N(k;0:12), wherek= 1;:::; 10.\n\u000fCase-Wise contamination: a proportion \u000fof cases in the data matrix\nis replaced by Xi\u00180:5N(cv;0:12I) +0:5N(\u0000cv;0:12I), wherec=q\nk(\u001f2\np)\u00001(0:99),k= 1;2;:::; 20 and vis the eigenvector corresponding\nto the smallest eigenvalue of \u00060with length such that ( v\u0000\u00160)>\u0006\u00001\n0(v\u0000\n\u00160) =1.\nThe proportions of contaminated rows chosen for case-wise contamination are\n\u000f= 0:1;0:2, and\u000f= 0:02;0:05 for cell-wise contamination. The number of\nreplicates in our simulation study is N= 200. We measure the performance\nof a given pair of location and scatter estimators ^ \u0016and ^\u0006using the mean\nsquared error (MSE) and the likelihood ratio test distance (LRT), as in:\nMSE =1\nNNX\ni=1(^\u0016i\u0000\u00160)>(^\u0016i\u0000\u00160)\nLRT (^\u0006;\u00060) =1\nNNX\ni=1D(^\u0006i;\u00060)\n2 Statistical data depth properties\nAdepth function d(\u0001;F) measures the centrality of a point w.r.t. a prob-\nability distribution F.\nd=Rp!R+[f0g; x!d(x;F)\nA statistical depth function should satisfy the following Properties\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\n\\section{Monte Carlo results}\nWe performed a Monte Carlo simulation to assess the performance of the proposed filter based on halfspace depth. After the filter flags the outlying observations, the generalized S-estimator is applied to the data with added missing values. Our simulation study is based on the same setup described in \\citet{Zamar2017} to compare significantly the performance of our filter with respect to the filter introduced in their work.\nWe considered samples from a $N_p(\\bf{0}, \\bf{\\Sigma}_0)$, where all values in  $diag(\\bf{\\Sigma}_0)$ are equal to $1$, $p = 10, 20, 30, 40, 50$ and the sample size is $n = 10p$. We consider the following scenarios:\n\\begin{itemize}\n\\item Clean data: data without changes.\n\\item Cell-Wise contamination: a proportion $\\epsilon$ of cells in the data is replaced by $X_{ij} \\sim N(k,0.1^2)$, where $k = 1, \\ldots, 10$.\n\\item Case-Wise contamination: a proportion $\\epsilon$ of cases in the data matrix is replaced by $\\bf{X}_i \\sim 0.5N(c\\bf{v},0.1^2\\bf{I}) + 0.5N(-c\\bf{v},0.1^2\\bf{I})$, where $c = \\sqrt{k(\\chi^2_p)^{-1}(0.99)}$, $k = 1, 2, \\ldots,20$ and $\\bf{v}$ is the eigenvector corresponding to the smallest eigenvalue of $\\bf{\\Sigma}_0$ with length such that $(\\bf{v}-\\bf{\\mu}_0)^\\top\\bf{\\Sigma}_0^{-1}(\\bf{v}-\\bf{\\mu}_0) = 1$.\n\\end{itemize}\nThe proportions of contaminated rows chosen for case-wise contamination are $\\epsilon = 0.1, 0.2$, and $\\epsilon = 0.02,0.05$ for cell-wise contamination. The number of replicates in our simulation study is $N=200$.\nWe measure the performance of a given pair of location and scatter estimators $\\hat{\\bf{\\mu}}$ and $\\hat{\\bf{\\Sigma}}$ using the mean squared error (MSE) and the likelihood ratio test distance (LRT), as in:\n\\begin{align*}\n MSE = \\frac{1}{N}\\sum_{i=1}^N (\\hat{\\bf{\\mu}}_i - \\bf{\\mu}_0)^\\top (\\hat{\\bf{\\mu}}_i - \\bf{\\mu}_0) \\\\  \n LRT(\\hat{\\bf{\\Sigma}},\\bf{\\Sigma}_0) = \\frac{1}{N}\\sum_{i=1}^N D(\\hat{\\bf{\\Sigma}}_i,\\bf{\\Sigma}_0)\n\\end{align*}\n\\section{Statistical data depth properties}\nA \\textbf{depth function} $d(\\cdot; F)$ measures the centrality of a point w.r.t. a probability distribution $F$. \n\\begin{equation*}\nd = \\mathbb{R}^{p} \\rightarrow \\mathbb{R}^+ \\cup \\{ 0 \\}, \\qquad \\bf{x} \\rightarrow d(\\bf{x}; F)\n\\end{equation*}\nA statistical depth function should satisfy the following Properties\n\\end{document}\n"}
{"P1A\u000ene invariance: d(x;F) =d(Ax+b;FA;b);\nP2Maximality at center: if Fis \\symmetric\" around \u0016thend(x;F)\u0014\nd(\u0016;F) for all x; for a more detailed discussion on symmetry see ?.\nP3Monotonicity: if () holds, then\nd(x;F)\u0014d(\u0016+\u000b(x\u0000\u0016);F)\u000b2[0;1] ;\nP4Approaching zero: kxk!1) d(x;F)!0.\n1 Gervini-Yohai depth\nHere we want to show that the Gervini-Yohai depth, de\fned as dGY(t;F;G) =\n1\u0000G(\u0001(t;\u0016(F);\u0006(F))), is a proper statistical depth function, i.e., it satis-\n\fes the four properties introduced above.\n1. A\u000ene invariance: it follows directly from the a\u000ene invariance property\nof the Mahalanobis distance;\n2. Maximality at center: if Fis elliptically symmetric around \u0016(F),\ndGY(\u0016(F);F;G) =1\u0000G(\u0001(\u0016(F);\u0016(F);\u0006(F))) = 1\u0000G(0):\nFor any t6=\u0016(F) we have\n\u0001(t;\u0016(F);\u0006(F))>0\nG(\u0001(t;\u0016(F);\u0006(F)))\u0015G(0)\n1\u0000G(\u0001(t;\u0016(F);\u0006(F)))\u00141\u0000G(0)\ndGY(t;F;G)\u0014dGY(\u0016(F);F;G);\nwhenGis strictly monotone then strict inequality holds, and \u0016(F) is\nthe unique maximizer of the Gervini-Yohai depth.\n3. Monotonicity:\n\u0001(\u0016(F) +\u000b(t\u0000\u0016(F));\u0016(F);\u0006(F)) = (\u000b(t\u0000\u0016(F)))>\u0006(F)\u00001(\u000b(t\u0000\u0016(F)))\n=\u000b2(t\u0000\u0016(F))>\u0006(F)\u00001(t\u0000\u0016(F))\n=\u000b2\u0001(t;\u0016(F);\u0006(F))\n\u0014\u0001(t;\u0016(F);\u0006(F))\nThendGY(\u0016(F) +\u000b(t\u0000\u0016(F));F;G)\u0015dGY(t;F;G).\n4. Approaching zero: if ktk!1 we have that \u0001( t;\u0016(F);\u0006(F))!1\nand consequently G(\u0001(t;\u0016(F);\u0006(F)))!1. Then\ndGY(t;F;G) =1\u0000G(\u0001(t;\u0016(F);\u0006(F)))!0\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{setspace}\n\\usepackage{graphicx}\n\\usepackage{authblk}\n\\usepackage{amsfonts}\n\\usepackage{natbib}\n\\bibliographystyle{plainnat}\n\\usepackage{xr}\n\\usepackage{hyperref}\n\\usepackage[toc,page]{appendix}\n\\usepackage{enumitem}\n\n\\begin{document}\n\\begin{enumerate}[label=\\textbf{P\\arabic*}]\n\\item  Affine invariance: $d(\\bf{x}; F)=d(\\bf{A} \\bf{x}+ \\bf{b}; F_{\\bf{A},\\bf{b}})$;\n\\item  Maximality at center: if $F$ is ``symmetric'' around $\\bf{\\mu}$ then $d(\\bf{x}; F) \\leq d(\\bf{\\mu}; F)$ for all $\\bf{x}$; for a more detailed discussion on symmetry see \\citet{Serfling2006}. \n\\item  Monotonicity: if () holds, then \n\\begin{equation*}\nd(\\bf{x}; F) \\le d(\\bf{\\mu} + \\alpha (\\bf{x} - \\bf{\\mu}); F) \\qquad \\alpha \\in [0,1] \\ ;\n\\end{equation*}\n\\item  Approaching zero: $\\parallel \\bf{x} \\parallel \\rightarrow \\infty \\Rightarrow d(\\bf{x}; F) \\rightarrow 0$.\n\\end{enumerate}\n\\section{Gervini-Yohai depth}\nHere we want to show that the Gervini-Yohai depth, defined as $d_{GY}(\\bf{t}, F, G) = 1 - G(\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F)))$, is a proper statistical depth function, i.e., it satisfies the four properties introduced above.\n\\begin{enumerate}\n\\item Affine invariance: it follows directly from the affine invariance property of the Mahalanobis distance;\n\\item  Maximality at center: if $F$ is elliptically symmetric around $\\bf{\\mu}(F)$, \n\\begin{equation*}\nd_{GY}(\\bf{\\mu}(F), F, G) = 1 - G(\\Delta(\\bf{\\mu}(F),\\bf{\\mu}(F),\\bf{\\Sigma}(F))) = 1 - G(0) . \n\\end{equation*}\nFor any $\\bf{t} \\not= \\bf{\\mu}(F)$ we have\n\\begin{align*}\n\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F))  > 0 \\\\\nG(\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F)))  \\ge G(0) \\\\\n1-G(\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F)))  \\le 1- G(0) \\\\\nd_{GY}(\\bf{t}, F, G)   \\le d_{GY}(\\bf{\\mu}(F), F, G) , \n\\end{align*}\nwhen $G$ is strictly monotone then strict inequality holds, and $\\bf{\\mu}(F)$ is the unique maximizer of the Gervini-Yohai depth. \n\\item  Monotonicity: \n\\begin{align*}\n\\Delta(\\bf{\\mu}(F) + \\alpha(\\bf{t}-\\bf{\\mu}(F)),\\bf{\\mu}(F),\\bf{\\Sigma}(F))  = (\\alpha(\\bf{t} - \\bf{\\mu}(F)))^\\top \\bf{\\Sigma}(F)^{-1}(\\alpha(\\bf{t} - \\bf{\\mu}(F))) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t = \\alpha^2 (\\bf{t} - \\bf{\\mu}(F))^\\top \\bf{\\Sigma}(F)^{-1}(\\bf{t} - \\bf{\\mu}(F)) \\\\ \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t = \\alpha^2 \\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F)) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\le \\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F))\n\\end{align*}\nThen  $d_{GY}(\\bf{\\mu}(F) + \\alpha(\\bf{t}-\\bf{\\mu}(F)), F, G) \\ge d_{GY}(\\bf{t}, F, G)$.\n\\item Approaching zero: if $\\parallel \\bf{t} \\parallel \\rightarrow \\infty $ we have that $\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F)) \\rightarrow \\infty$ and consequently  $G(\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F))) \\rightarrow 1$.\nThen \n\\begin{equation*}\nd_{GY}(\\bf{t}, F, G) = 1 - G(\\Delta(\\bf{t},\\bf{\\mu}(F),\\bf{\\Sigma}(F))) \\rightarrow 0\n\\end{equation*}\n\\end{enumerate}\n\\end{document}\n"}
{"The models have a subgroup of E8as the symmetry. The subgroup is\ndetermined by the choice of the \nux and punctures, that is it is the subgroup\nofE8commuting with the choice of \nux in the case of closed Riemann surface.\nFor properly quantized \nux this has rank eight, for fractional values of \nux\nthe rank might be smaller. Every puncture is associated with additional\nfactor of SU(2) symmetry. The punctures come in di\u000berent types which we\nrefer to as di\u000berent colors. Models corresponding to di\u000berent surfaces can\nbe glued together by gauging a symmetry corresponding to punctures of the\nsame color. The color of the punctures determines what are the details of the\ngluing. The punctures break the E8symmetry of the six dimensional model\nto U(1)\u0002SU(8) sub group. The \nux might break the symmetry further.\nIn particular the color is determined by the U(1) \u0002SU(8) subgroup of E8\nwhich the puncture keeps. The subgroup preserved by given puncture is\nparametrized by fugacity tfor U(1) and fugacities aifor SU(8) (i= 1;:::; 8\nand8Q\ni=1ai= 1). For di\u000berent colors of punctures the fugacities of one are\nexpressible in terms of monomial products of the other. When we glue two\npunctures together the index of the theory is\nTcombined =TA\nJ(u)\u0002uTB\nJ(u)\u0011(q;q)(p;p)Idu\n4\u0019iu8Q\nj=1\u0000e\u0000\n(qp)1\n21\ntJ\u0000\naJ\nj\u0001\u00001u\u00061\u0001\n\u0000(u\u00062):\nHere the indices AandBstand for Theory A andTheory B . The gamma\nfunctions appearing in the denominator correspond to N= 1 vector \felds\nand the gamma functions in the numerator to a collection of eight chiral\n\felds in fundamental representation of the gauged symmetry. This collection\nof chiral \felds couples to certain chiral operators of the two glued copies\nwhich generalize the moment map operators of the class Scase. We will use\nthe shorthand notation \u0002uto indicate the gluing. Here TJ(u) is an index\nof a theory corresponding to some Riemann surface with puncture of color\nJwith associated symmetry SU(2) u. The parameters tJandaJlabel the\nU(1)\u0002SU(8) symmetry preserved by the puncture. Let us de\fne the basic\nbuilding blocks of our construction. We de\fne the tube TJ;J(z;u) to be\nTJ;J(u;z) = \u0000 e\u0000\nqpt4\u0001 8Y\nj=1\u0000e\u0000\n(qp)1\n2tajz\u00061\u0001\n\u0000e\u0000\n(qp)1\n2ta\u00001\nju\u00061\u0001!\n\u0000e\u00121\nt2u\u00061z\u00061\u0013\n:\nThis tube is the model obtained as compacti\fcation on sphere with two\npunctures and \nux \u00001=2 for U(1) tand zero \nux for other symmetries. The\nmodel is an IR free Wess{Zumino theory. From this we can construct cap\ntheories\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\n\\allowdisplaybreaks\nThe models have a subgroup of $E_8$ as the symmetry. The subgroup is determined by the choice of the flux and punctures, that is it is the subgroup of $E_8$ commuting with the choice of flux in the case of closed Riemann surface. For properly quantized flux this has rank eight, for fractional values of flux the rank might be smaller. Every puncture is associated with additional factor of ${\\rm SU}(2)$ symmetry. The punctures come in different types which we refer to as different colors.\nModels corresponding to different surfaces can be glued together by gauging a symmetry corresponding to punctures of the same color. The color of the punctures determines what are the details of the gluing. The punctures break the $E_8$ symmetry of the six dimensional model to ${\\rm U}(1)\\times {\\rm SU}(8)$ sub group. The flux might break the symmetry further. In particular the color is determined by the ${\\rm U}(1)\\times {\\rm SU}(8)$ subgroup of $E_8$ which the puncture keeps. The subgroup preserved by given puncture is parametrized by fugacity $t$ for ${\\rm U}(1)$ and fugacities $a_i$ for ${\\rm SU}(8)$ ($i=1,\\dots, 8$ and $\\prod\\limits_{i=1}^8 a_i=1$). For different colors of punctures the fugacities of one are expressible in terms of monomial products of the other. When we glue two punctures together the index of the theory is\n\\begin{gather*}\nT_{\\rm combined}=T_{\\mathfrak J}^{A}(u)\\times_u T_{\\mathfrak J}^{B}(u)\\equiv (q;q)(p;p)\\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\prod\\limits_{j=1}^8\\Gamma_e\\big((q p)^{\\frac12 }\\frac1{t^{\\mathfrak J}} \\big(a^{\\mathfrak J}_j\\big)^{-1} u^{\\pm1}\\big)}{\\Gamma(u^{\\pm2})} .\n\\end{gather*}\nHere the indices $A$ and $B$ stand for {\\it Theory A} and {\\it Theory B}. The gamma functions appearing in the denominator correspond to ${\\mathcal N}=1$ vector fields and the gamma functions in the numerator to a collection of eight chiral fields in fundamental representation of the gauged symmetry. This collection of chiral fields couples to certain chiral operators of the two glued copies which generalize the moment map operators of the class~${\\mathcal S}$ case.\nWe will use the shorthand notation $\\times_u$ to indicate the gluing. Here $T_{\\mathfrak J}(u)$ is an index of a theory corresponding to some Riemann surface with puncture of color ${\\mathfrak J}$ with associated symmetry ${\\rm SU}(2)_u$. The parameters $t^{\\mathfrak J}$ and $a^{\\mathfrak J}$ label the ${\\rm U}(1)\\times {\\rm SU}(8)$ symmetry preserved by the puncture.\nLet us define the basic building blocks of our construction. We define the tube $T_{{\\mathfrak J},\\overline{\\mathfrak J}}(z,u)$ to be\n\\begin{gather*}\nT_{{\\mathfrak J},\\overline{\\mathfrak J}}(u,z)=\\Gamma_e\\big(q p t^4\\big)\\left(\\prod_{j=1}^8\\Gamma_e\\big((q p)^{\\frac12}t a_j z^{\\pm1}\\big)\\Gamma_e\\big(( q p)^{\\frac12} t a_j^{-1} u^{\\pm1}\\big)\\right)\\Gamma_e\\left(\\frac1{t^2}u^{\\pm1}z^{\\pm1}\\right) .\n\\end{gather*}\nThis tube is the model obtained as compactification on sphere with two punctures and flux $-1/2$ for ${\\rm U}(1)_t$ and zero flux for other symmetries. The model is an IR free Wess--Zumino theory. From this we can construct cap theories\n\\end{document} \n"}
{"C(M;L ;i)\nJ (z), corresponding to a sphere with single puncture, by computing\nresidues. We de\fne these to be\nC(M;L ;i)\nJ (z) =1\u0010Q\nj6=i\u0000e(ai=aj)\u0011\n\u0000e\u0000\npqt21\na2\ni\u0001\n(q;q)(p;p)Resu!1\n(qp)1\n2qMpLtai1\nuTJ;J(u;z):\nThe cap theory for zero values of MandLis a model corresponding to\nsphere with one puncture and \nux \u00003\n4for U(1) t,7\n8for U(1) i, and\u00001\n8for\nU(1) j. See for details of the derivation of the \nux. The index can be thought\nas partition function on S1\u0002S3, and for non vanishing values of MandLthe\ntheory also has surface defects wrapping the S1and one of the two equators\nofS3. Finally we have a three punctured sphere TJB;JC;JD(w;u;v )\nTJB;JC;JD(w;u;v ) = \u0000 e\u0000\n(qp)1\n2t\u0000\nB\u00001A\u0001\u00061w\u00061\u0001\n\u0000e\u0010qp\nt2\u0011\n(q;q)(p;p)\n\u0002Idh\n4\u0019ih\u0000e\u0000(pq)1\n2\nt2\u0000\nAB\u00001\u0001\u00061h\u00061\u0001\n\u0000e\u0000\nh\u00062\u0001 \u0000e\u0000\nth\u00061w\u00061\u0001\nH\u0000\nu;D;v;C;p\nhB;p\nh\u00001B;A\u0001\n;\nwhere we have de\fned\nH(z1;z2;v1;v2;a;b;A) = (q;q)2(p;p)2Idw1\n4\u0019iw 1Idw2\n4\u0019iw 2\u0000e\u0000(pq)1\n2\nt2w\u00061\n1w\u00061\n2\u0001\n\u0000e\u0000\nw\u00062\n2\u0001\n\u0000e\u0000\nw\u00062\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA1\n2b\u00001w\u00061\n1z\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n4A1\n2bw\u00061\n1z\u00061\n2\u0001\n\u0000e\u0000\n(qp)1\n4tA\u00001\n2bw\u00061\n2z\u00061\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A\u00001\n2b\u00001z\u00061\n2w\u00061\n2\u0001\n\u0000e\u0000\n(qp)1\n4tA\u00001\n2aw\u00061\n1v\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n4A\u00001\n2a\u00001v\u00061\n2w\u00061\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA1\n2a\u00001w\u00061\n2v\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n4A1\n2aw\u00061\n2v\u00061\n2\u0001\n: (1)\nThe above expressions are non trivial to derive. The theory corresponding\nto three punctured spheres is constructed by starting from a gauge theory,\nindex of which is roughly speaking H, and arguing that at some point on\nthe conformal manifold the U(1) symmetry corresponding to fugacityp\na=b\nenhances to SU(2). This is a non trivial fact which follows from dualities.\nThis SU(2) is then taken to be dynamical with addition of some chiral \felds.\nThe resulting index is given above. The statement that this theory corre-\nsponds to three punctured sphere is made by performing a variety of physical\nconsistency checks. Note that the construction also gives a theory having only\nrank \fve symmetry as opposed to rank eight. For the three punctured sphere\nwe have \nux 3 =4 for U(1) tand vanishing \nux for the Cartan generators of\nSU(8). The three punctured sphere depends on four parameters ( A;B;C;D )\nwhich parametrize SO(8) inside SU(8). That is,\n(a1;a2;a3;a4) =A\u00061B\u00061; (a5;a6;a7;a8) =C\u00061D\u00061: (2)\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\n\\allowdisplaybreaks\n$C^{(M,L; i)}_{{\\mathfrak J}}(z)$, corresponding to a sphere with single puncture, by computing residues. We define these to be\n\\begin{gather*}\nC^{(M,L;i)}_{{\\mathfrak J}}(z)=\\frac1{\\Big(\\prod\\limits_{j\\neq i}\\Gamma_e(a_i/a_j)\\Big)\\Gamma_e\\big( pq t^2\\frac1{a_i^2}\\big)(q;q)(p;p)}\\operatorname{Res}_{u\\to \\frac1{(q p)^{\\frac12} q^M p^L t} a_i} \\frac1u T_{{\\mathfrak J},\\overline{\\mathfrak J}}(u,z) .\n\\end{gather*}\n The cap theory for zero values of $M$ and $L$ is a model corresponding to sphere with one puncture and flux $-\\frac34$ for ${\\rm U}(1)_t$, $\\frac78$ for ${\\rm U}(1)_i$, and $-\\frac18$ for ${\\rm U}(1)_j$. See for details of the derivation of the flux.\nThe index can be thought as partition function on ${\\mathbb S}^1\\times {\\mathbb S}^3$, and for non vanishing values of~$M$ and~$L$ the theory\nalso has surface defects wrapping the~${\\mathbb S}^1$ and one of the two equators of~${\\mathbb S}^3$. Finally we have a three punctured sphere $T_{{\\mathfrak J}_B,{\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)$\n\\begin{gather*}\nT_{{\\mathfrak J}_B,{\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v) = \\Gamma_e\\big((q p)^{\\frac12} t\\big(B^{-1}A\\big)^{\\pm1} w^{\\pm1}\\big)\\Gamma_e\\left(\\frac{q p}{t^2}\\right) (q;q)(p;p)\\\\\n\\quad{} \\times \\oint \\frac{{\\rm d} h}{4\\pi i h} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}\\big(A B^{-1}\\big)^{\\pm1}h^{\\pm1}\\big)}{\\Gamma_e\\big(h^{\\pm2}\\big)}\\Gamma_e\\big(t h^{\\pm1} w^{\\pm1}\\big) H\\big(u,D,v,C,\\sqrt{h B},\\sqrt{h^{-1}B}; A\\big),\n\\end{gather*} where we have defined\n\\begin{gather}\nH(z_1,z_2,v_1,v_2, a , b; A ) = (q;q)^2(p;p)^2\\oint\\frac{{\\rm d}w_1}{4\\pi i w_1}\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}w_1^{\\pm1}w_2^{\\pm1}\\big)}{\\Gamma_e\\big(w_2^{\\pm2}\\big)\\Gamma_e\\big(w_1^{\\pm2}\\big)} \\nonumber\\\\\n \\quad{} \\times \\Gamma_e\\big( (q p)^{\\frac14}t A^{\\frac12}b^{-1} w_1^{\\pm1}z_1^{\\pm1}\\big) \\Gamma_e\\big((qp)^{\\frac14} A^{\\frac12} bw_1^{\\pm1}z_2^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} b w_2^{\\pm1} z_1^{\\pm1}\\big)\\nonumber\\\\\n\\quad{} \\times \\Gamma_e \\big((q p)^{\\frac14} A^{-\\frac12} b^{-1} z_2^{\\pm1}w_2^{\\pm1}\\big) \\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} a w_1^{\\pm1} v_1^{\\pm1}\\big)\\Gamma_e\\big( ( q p)^{\\frac14} A^{-\\frac12} a^{-1} v_2^{\\pm1} w_1^{\\pm1}\\big)\\nonumber\\\\\n\\quad{} \\times \\Gamma_e\\big( ( q p )^{\\frac14} t A^{\\frac12} a^{-1} w_2^{\\pm1} v_1^{\\pm1}\\big) \\Gamma_e\\big( ( q p )^{\\frac14} A^{\\frac12} a w_2^{\\pm1} v_2^{\\pm1}\\big) . \n \\end{gather}\nThe above expressions are non trivial to derive. The theory corresponding to three punctured spheres is constructed by starting from a gauge theory, index of which is roughly speaking $H$, and arguing that at some point on the conformal manifold the ${\\rm U}(1)$ symmetry corresponding to fugacity $\\sqrt{a/b}$ enhances to ${\\rm SU}(2)$. This is a non trivial fact which follows from dualities. This~${\\rm SU}(2)$ is then taken to be dynamical with addition of some chiral fields. The resulting index is given above. The statement that this theory corresponds to three punctured sphere is made by performing a variety of physical consistency checks. Note that the construction also gives a theory having only rank five symmetry as opposed to rank eight.\nFor the three punctured sphere we have flux $3/4$ for ${\\rm U}(1)_t$ and vanishing flux for the Cartan generators of ${\\rm SU}(8)$. The three punctured sphere depends on four parameters $(A,B,C,D)$ which parametrize ${\\rm SO}(8)$ inside ${\\rm SU}(8)$. That is,\n\\begin{gather}\n (a_1,a_2,a_3,a_4) =A^{\\pm1} B^{\\pm1} ,\\qquad (a_5,a_6,a_7,a_8) =C^{\\pm1} D^{\\pm1} .\n\\end{gather}\n\\end{document} \n"}
{"In principle there should be three punctured spheres depending on all\neight parameters but the particular construction of gives us a three punctured\nsphere only depending on \fve with the map to eight parameters written\nabove. The three punctures are of di\u000berent color\nw:JB=\u0000\nt;A\u00061B\u00061; C\u00061D\u00061\u0001\n;\nu:JC=\u0000\nt;A\u00061D\u00061; B\u00061C\u00061\u0001\n;\nv:JD=\u0000\nt;A\u00061C\u00061; B\u00061D\u00061\u0001\n:\nWithout loss of any generality let us assume that we will compute residues\nwith respect to a1=AB\u00001. Then as we have only a subgroup of SU(8) we\nneed to specify the \nux for this. We obtain that the \nux for the cap is\n\u0000\nU(1) A;U(1) B;U(1) C;U(1) D\u0001\n=\u00001\n4;\u00001\n4;0;0\u0001\n:\n1 Defect operators\nUsing the building blocks of the previous section we can introduce surface\ndefects into the index computation. Given a model of some \nux and cor-\nresponding to some surface we introduce a defect operator by gluing to the\nsurface \frst two three punctured spheres and then closing two of the punc-\ntures with caps. In case one closes the two punctures with cap de\fned by\nresidues (0 ;0;i) and (0 ;0;i), where by iwe mean ajsuch that ai= 1=aj, one\nadds tube with zero \nux, which gives us the original model without defect.\nWe can indeed check, see Appendix, that the index satis\fes such property\nTJC(u) =TJC(z)\u0002z\u0000\u0000\nTJB;JC;JD(h; z; g )\u0002hC(0;0;i)\nJB(h)\u0001\n\u0002g\u0000\nTJB;JC;JD(v; u; g )\u0002vC(0;0;i)\nJB(v)\u0001\u0001\n: (1)\nHowever, when we close one of the punctures with ( M; L ;i) and other with\n(0;0;i) we introduce a surface defect. Performing the computation with\nM= 1 and L= 0, see Appendix, we can see that the index is given by\nacting on the one with no defect by a di\u000berence operator\nDJB;(1;0;i)\nJDTJD(u) =TJD(g)\u0002g\u0000\u0000\nTJB;JC;JD(h; z; g )\u0002hC(0;0;i)\nJB(h)\u0001\n\u0002z\u0000\nTJB;JC;JD(v; z; u )\u0002vC(1;0;i)\nJB(v)\u0001\u0001\n:\nThe di\u000berence operator is given by\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nIn principle there should be three punctured spheres depending on all eight parameters but the particular construction of gives us a three punctured sphere only depending on five with the map to eight parameters written above.\nThe three punctures are of different color\n\\begin{gather*}\nw\\colon \\ {\\mathfrak J}_B=\\big(t; A^{\\pm1}B^{\\pm1},C^{\\pm1}D^{\\pm1}\\big) , \\\\\nu\\colon \\ {\\mathfrak J}_C=\\big(t;A^{\\pm1}D^{\\pm1},B^{\\pm1}C^{\\pm1}\\big) , \\\\\nv\\colon \\ {\\mathfrak J}_D=\\big(t; A^{\\pm1}C^{\\pm1}, B^{\\pm1}D^{\\pm1}\\big) .\n\\end{gather*}\nWithout loss of any generality let us assume that we will compute residues with respect to $a_1=A B^{-1}$. Then as we have only a subgroup of ${\\rm SU}(8)$ we need to specify the flux for this. We obtain that the flux for the cap is\n\\begin{gather*} \\big({\\rm U}(1)_A,{\\rm U}(1)_B,{\\rm U}(1)_C,{\\rm U}(1)_D\\big) = \\big(\\tfrac14,-\\tfrac14,0,0\\big) .\n\\end{gather*}\n\\section{Defect operators}\nUsing the building blocks of the previous section we can introduce surface defects into the index computation. Given a model of some flux and corresponding to some surface we introduce a~defect operator by gluing to the surface first two three punctured spheres and then closing two of the punctures with caps. In case one closes the two punctures with cap defined by residues $(0,0;i)$ and $(0,0;\\overline i)$, where by $\\overline i$ we mean $a_j$ such that $a_i=1/a_j$, one adds tube with zero flux, which gives us the original model without defect. We can indeed check, see Appendix, that the index satisfies such property\n\\begin{gather}\nT_{{\\mathfrak J}_C}(u)= T_{{\\mathfrak J}_C}(z)\\times_z \\big(\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,z,g)\\times_h C^{(0,0;i)}_{{\\mathfrak J}_B}(h)\\big)\\nonumber\\\\\n\\hphantom{T_{{\\mathfrak J}_C}(u)=}{}\n\\times_g\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(v,u,g)\\times_v C^{(0,0;\\overline i)}_{{\\mathfrak J}_B}(v)\\big)\\big) .\n\\end{gather}\nHowever, when we close one of the punctures with $(M,L;i)$ and other with $(0,0;\\overline i)$ we introduce a surface defect. Performing the computation with $M=1$ and $L=0$, see Appendix, we can see that the index is given by acting on the one with no defect by a~difference operator\n\\begin{gather*}\n{\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;i)}T_{{\\mathfrak J}_D}(u)=\nT_{{\\mathfrak J}_D}(g)\\times_g \\big(\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,z,g)\\times_h C^{(0,0;i)}_{{\\mathfrak J}_B}(h)\\big)\\\\\n\\hphantom{{\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;i)}T_{{\\mathfrak J}_D}(u)=}{}\n\\times_z\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(v,z,u)\\times_v C^{(1,0;\\overline i)}_{{\\mathfrak J}_B}(v)\\big)\\big) .\\nonumber\n\\end{gather*}\nThe difference operator is given by\n\\end{document} \n"}
{"DJB;(1;0;AB\u00001)\nJDTJD(z)\u0018\u0012p\u0000\n(pq)1\n2t\u00001A\u00061C\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00001B\u00061D\u00061z\u0001\n\u0012p\u0000\nqz2\u0001\n\u0012p\u0000\nz2\u0001 TJD(qz)\n+\u0012p\u0000\n(pq)1\n2t\u00001A\u00061C\u00061z\u00001\u0001\n\u0012p((pq)1\n2t\u00001B\u00061D\u00061z\u00001)\n\u0012p\u0000\nqz\u00002\u0001\n\u0012p\u0000\nz\u00002\u0001 TJD\u0000\nqz\u00001\u0001\n+WJB\nJD;\u0000\n1;0;AB\u00001\u0001(z)TJD(z); (1)\nwhere\u0018means equal up to an overall factor which is independent of z. We\nhave denoted\nWJB\nJD;(1;0;AB\u00001)(z)\n=\u0012p\u0000\nq\u00001t\u00004\u0001\n\u0012p\u0000\nq\u00001t\u00004A2B\u00002z2\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061AC\u00061(qz)\u00001\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061B\u00001D\u00061(qz)\u00001\u0001\n\u0012p\u0000\nq\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nq\u00001z\u00002\u0001\n\u0012p\u0000\nt\u00004z2\u0001\n+\u0012p\u0000\nq\u00001t\u00004\u0001\n\u0012p\u0000\nq\u00001t\u00004A2B\u00002z\u00002\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061A\u00001C\u00061z\u00001\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061BD\u00061z\u00001\u0001\n\u0012p\u0000\nq\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nz\u00002\u0001\n\u0012p\u0000\nq\u00001z2\u0001\n\u0012p\u0000\nt\u00004z\u00002\u0001\n+\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0012p\u0000\n(pq)1\n2t2BD\u00061\u0000\nt\u00001z\u0001\u00061\u0001\n\u0012p\u0000\n(pq)1\n2t2A\u00001C\u00061\u0000\nt\u00001z\u0001\u00061\u0001\n\u0012p\u0000\nq\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nt4z\u00002\u0001\n+\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0012p\u0000\n(pq)1\n2t2BD\u00061\u0000\nt\u00001z\u00001\u0001\u00061\u0001\n\u0012p\u0000\n(pq)1\n2t2A\u00001C\u00061\u0000\nt\u00001z\u00001\u0001\u00061\u0001\n\u0012p\u0000\nq\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nz\u00002\u0001\n\u0012p\u0000\nt4z2\u0001\n+\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0012p\u0000\np\u00001q\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nq\u00002t\u00002A2B\u00002\u0001 :\nIn Appendix we give details of the computation leading to this operator.\nOne could consider more general residues by gluing the cap C(L;M ;i)\nJ (v) and\ngeneralLandM. We leave this as an exercise to the interested reader.\n1 Relation to van Diejen model\nThe di\u000berence operator of the previous section is the van Diejen di\u000berence\noperator. Using the notations of and the de\fnitions of Appendix the van\nDiejen operator is given as\nAD(h;z)T(z)\u0011V(h;z)T(qz) +V\u0000\nh;z\u00001\u0001\nT\u0000\nq\u00001z\u0001\n+Vb(h;z);\nwhere\nV(h;z)\u00118Q\nn=1\u0012\u0000\n(pq)1\n2hnz\u0001\n\u0012(z2)\u0012\u0000\nqz2\u0001; V b(h;z)\u00113P\nk=0pk(h)[Ek(\u0018;z)\u0000Ek(\u0018;!k)]\n2\u0012(\u0018)\u0012\u0000\nq\u00001\u0018\u0001;\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\n\\begin{gather}\n{\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;AB^{-1})} T_{{\\mathfrak J}_D}(z) \\sim\n\\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}A^{\\pm1}C^{\\pm1}z\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}B^{\\pm1}D^{\\pm1}z\\big)}{\\theta_p\\big(q z^2\\big)\\theta_p\\big(z^2\\big)} T_{{\\mathfrak J}_D}(qz)\\nonumber\\\\\n\\hphantom{{\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;AB^{-1})} T_{{\\mathfrak J}_D}(z) \\sim}{}\n +\\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}A^{\\pm1}C^{\\pm1}z^{-1}\\big)\\theta_p((pq)^\\frac{1}{2}t^{-1}B^{\\pm1}D^{\\pm1}z^{-1})}{\\theta_p\\big(q z^{-2}\\big)\\theta_p\\big(z^{-2}\\big)} T_{{\\mathfrak J}_D}\\big(qz^{-1}\\big)\\nonumber\\\\\n\\hphantom{{\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;AB^{-1})} T_{{\\mathfrak J}_D}(z) \\sim}{}\n+ W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D,\\big(1,0;AB^{-1}\\big)}(z) T_{{\\mathfrak J}_D}(z) ,\n\\end{gather}\nwhere $\\sim$ means equal up to an overall factor which is independent of~$z$. We have denoted\n\\begin{gather*}\nW^{{\\mathfrak J}_B}_{{\\mathfrak J}_D, (1,0; AB^{-1})}(z)\n\\\\=\\frac{\\theta_p\\big(q^{-1}t^{-4}\\big)\\theta_p\\big(q^{-1}t^{-4}A^2B^{-2}z^2\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}A C^{\\pm1}(qz)^{-1}\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}B^{-1}D^{\\pm1}(qz)^{-1}\\big)}\n{\\theta_p\\big(q^{-2}t^{-4}A^2B^{-2}\\big)\\theta_p\\big(z^2\\big)\\theta_p\\big(q^{-1}z^{-2}\\big)\\theta_p\\big(t^{-4}z^2\\big)}\\\\\n\\quad{} +\\frac{\\theta_p\\big(q^{-1}t^{-4}\\big)\\theta_p\\big(q^{-1}t^{-4}A^2B^{-2}z^{-2}\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}A^{-1} C^{\\pm1}z^{-1}\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}BD^{\\pm1}z^{-1}\\big)}\n{\\theta_p\\big(q^{-2}t^{-4}A^2B^{-2}\\big)\\theta_p\\big(z^{-2}\\big)\\theta_p\\big(q^{-1}z^{2}\\big)\\theta_p\\big(t^{-4}z^{-2}\\big)}\\\\\n\\quad{} +\\frac{\\theta_p\\big(q^{-1}A^2B^{-2}\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{2}B D^{\\pm1}\\big(t^{-1}z\\big)^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{2}A^{-1} C^{\\pm1} \\big(t^{-1} z\\big)^{\\pm1}\\big)}\n{\\theta_p\\big(q^{-2}t^{-4}A^2B^{-2}\\big)\\theta_p\\big(z^2\\big)\\theta_p\\big(t^4z^{-2}\\big)} \\\\\n\\quad{}+\\frac{\\theta_p\\big(q^{-1}A^2B^{-2}\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^{2}B D^{\\pm1}\\big(t^{-1}z^{-1}\\big)^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{2}A^{-1} C^{\\pm1} \\big(t^{-1} z^{-1}\\big)^{\\pm1}\\big)}\n{\\theta_p\\big(q^{-2}t^{-4}A^2B^{-2}\\big)\\theta_p\\big(z^{-2}\\big)\\theta_p\\big(t^4z^2\\big)}\\\\\n\\quad{} +\\frac{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(q^{-1}A^2B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big) \\theta_p\\big(q^{-1}t^{-2}A B^{-1} C^{\\pm1} D^{\\pm1}\\big)}{\\theta_p\\big(p^{-1}q^{-2}t^{-4}A^2 B^{-2}\\big)\\theta_p\\big(q^{-2}t^{-2}A^2B^{-2}\\big)}.\n\\end{gather*}\nIn Appendix we give details of the computation leading to this operator. One could consider more general residues by gluing the cap $C^{(L,M;\\overline i)}_{{\\mathfrak J}}(v)$ and general~$L$ and~$M$. We leave this as an exercise to the interested reader.\n\\section{Relation to van Diejen model}\nThe difference operator of the previous section is the van Diejen difference operator. Using the notations of  and the definitions of Appendix the van Diejen operator is given as\n\\begin{gather*}\nA_D(h;z)T(z)\\equiv V(h;z)T(qz)+V\\big(h;z^{-1}\\big)T\\big(q^{-1}z\\big)+V_b(h;z) ,\n\\end{gather*}\nwhere\n\\begin{gather*}\n V(h;z) \\equiv \\frac{\\prod\\limits_{n=1}^8\\theta\\big((pq)^{\\frac12}h_n z\\big)}{\\theta(z^2)\\theta\\big(qz^2\\big)},\\qquad\n V_b(h;z) \\equiv \\frac{\\sum\\limits_{k=0}^3p_k(h)[\\mathcal{E}_k(\\xi;z)-\\mathcal{E}_k(\\xi;\\omega_k)]}{2\\theta(\\xi)\\theta\\big(q^{-1}\\xi\\big)} ,\n\\end{gather*}\n\\end{document} \n"}
{"where!kare\n!0= 1; ! 1=\u00001; ! 2=p1\n2; ! 3=\u0000p\u00001\n2:\nThe functions pk(h) are\np0(h)\u0011Y\nn\u0012(p1\n2hn); p 1(h)\u0011Y\nn\u0012\u0000\n\u0000p1\n2hn\u0001\n;\np2(h)\u0011pY\nnh\u00001\n2n\u0012(hn); p 3(h)\u0011pY\nnh1\n2n\u0012\u0000\n\u0000h\u00001\nn\u0001\n;\nandEkis\nEk(\u0018;z)\u0011\u0012\u0000\nq\u00001\n2\u0018!\u00001\nkz\u0001\n\u0012\u0000\nq\u00001\n2\u0018!kz\u00001\u0001\n\u0012\u0000\nq\u00001\n2!\u00001\nkz\u0001\n\u0012\u0000\nq\u00001\n2!kz\u00001\u0001:\nThe van Diejen operator and the operator () are the same up to a constant\nfunction (independent of z). It's clear that V(h;z) coincides with the corre-\nsponding term in ( ??) if we make the identi\fcations\nh1;2;3;4=t\u00001A\u00061C\u00061; h 5;6;7;8=t\u00001B\u00061D\u00061:\nSinceVb(h;z) is elliptic in zwith periods 1 and pand it is easy to check\nthatWJB\nJD;(1;0;AB\u00001)(z) is also elliptic with the same period, it is enough to\nshow that the two functions have the same poles and residues to prove that\nthey can di\u000ber only by a function independent of z. In the fundamental\nparallelogram Vbhas poles at (we assume with no loss of generality that\njpj<jqj\u001cjtj<1 and the rest of the variables are on unit circle)\nz=\u0006q\u00001\n2p; z =\u0006q1\n2; z =\u0006p1\n2q\u00061\n2:\nIn addition to such poles the operator ( ??) seems to have poles at z=\n\u0006t\u00002p;\u0006t2;\u0006p1\n2t\u00062andz=\u00061;\u0006p1\n2, but computation of the residue at these\npoles yields zero. The computation of the residue at the poles is straightfor-\nward, the result is ( his either 1 or\u00001)\nResz!hq1\n2WJB\nJD;(1;0;AB\u00001)(z) =\u0000h(p;p)\u00002\u0012p\u0000\nhp1\n2t\u00061AC\u00061\u0001\n\u0012p\u0000\nhp1\n2t\u00061B\u00001D\u00061\u0001\n2q\u00001\n2\u0012p\u0000\nq\u00001\u0001 ;\nResz!hq\u00001\n2WJB\nJD;(1;0;AB\u00001)(z) =h(p;p)\u00002\u0012p\u0000\nhp1\n2t\u00061AC\u00061\u0001\n\u0012p\u0000\nhp1\n2t\u00061B\u00001D\u00061\u0001\n2q1\n2\u0012p\u0000\nq\u00001\u0001 ;\nResz!hp1\n2q1\n2WJB\nJD;(1;0;AB\u00001)(z) =\u0000h(p;p)\u00002A\u00002B2\u0012p\u0000\nht\u00061AC\u00061\u0001\n\u0012p\u0000\nht\u00061B\u00001D\u00061\u0001\n2p\u00003\n2q\u00001\n2\u0012p\u0000\nq\u00001\u0001 ;\nResz!hp1\n2q\u00001\n2WJB\nJD;(1;0;AB\u00001)(z) =h(p;p)\u00002A\u00002B2\u0012p\u0000\nht\u00061AC\u00061\u0001\n\u0012p\u0000\nht\u00061B\u00001D\u00061\u0001\n2p\u00003\n2q1\n2\u0012p\u0000\nq\u00001\u0001 :\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nwhere $\\omega_k$ are\n\\begin{gather*}\n \\omega_0=1 , \\qquad \\omega_1=-1 , \\qquad \\omega_2=p^\\frac12 ,\\qquad \\omega_3=-p^{-\\frac12} .\n\\end{gather*}\nThe functions $p_k(h)$ are\n\\begin{alignat*}{3}\n  p_0(h)\\equiv \\prod_n \\theta(p^\\frac12 h_n) , \\qquad  p_1(h)\\equiv \\prod_n \\theta\\big({-}p^\\frac12 h_n\\big) , \\\\\n  p_2(h)\\equiv p\\prod_n h_n^{-\\frac12}\\theta(h_n) ,\\qquad  p_3(h)\\equiv p\\prod_n h_n^{\\frac12}\\theta\\big({-}h_n^{-1}\\big) ,\n\\end{alignat*}\nand $\\mathcal{E}_k$ is\n\\begin{gather*}\n \\mathcal{E}_k(\\xi;z)\\equiv\\frac{\\theta\\big(q^{-\\frac12} \\xi \\omega_k^{-1} z\\big)\\theta\\big(q^{-\\frac12} \\xi \\omega_k z^{-1}\\big)}{\\theta\\big(q^{-\\frac12}\\omega_k^{-1} z\\big)\\theta\\big(q^{-\\frac12} \\omega_k z^{-1}\\big)} .\n\\end{gather*}\nThe van Diejen operator and the operator () are the same up to a constant function (independent of~$z$). It's clear that $V(h;z)$ coincides with the corresponding term in \\eqref{diuer} if we make the identifications\n\\begin{gather*}\n h_{1,2,3,4}=t^{-1}A^{\\pm1}C^{\\pm1} , \\qquad h_{5,6,7,8}=t^{-1}B^{\\pm1}D^{\\pm1} .\n\\end{gather*}\nSince $V_b(h;z)$ is elliptic in $z$ with periods~$1$ and $p$ and it is easy to check that $W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D, (1,0; AB^{-1})}(z)$ is also elliptic with the same period, it is enough to show that the two functions have the same poles and residues to prove that they can differ only by a function independent of~$z$. In the fundamental parallelogram $V_b$ has poles at (we assume with no loss of generality that $|p|<|q|\\ll |t|<1$ and the rest of the variables are on unit circle)\n\\begin{gather*}\n z=\\pm q^{-\\frac12}p , \\qquad z= \\pm q^{\\frac12}, \\qquad z=\\pm p^\\frac12 q^{\\pm\\frac12}.\n\\end{gather*}\nIn addition to such poles the operator \\eqref{diuer} seems to have poles at $z=\\pm t^{-2}p , \\pm t^2 , \t\\pm p^\\frac12 t^{\\pm2}$ and $z=\\pm1,\\pm p^\\frac12$, but computation of the residue at these poles yields zero. The computation of the residue at the poles is straightforward, the result is ($h$ is either $1$ or $-1$)\n\\begin{gather*}\n\\operatorname{Res}_{z\\to h q^\\frac12} W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D, (1,0; AB^{-1})}(z)=\n-h(p;p)^{-2}\\frac{\\theta_p\\big(h p^\\frac{1}{2}t^{\\pm1}A C^{\\pm1}\\big)\\theta_p\\big(h p^\\frac{1}{2}t^{\\pm1}B^{-1}D^{\\pm1}\\big)}\n{2q^{-\\frac12}\\theta_p\\big(q^{-1}\\big)},\\\\\n\\operatorname{Res}_{z\\to h q^{-\\frac12}} W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D, (1,0; AB^{-1})}(z)=\nh(p;p)^{-2}\\frac{\\theta_p\\big(hp^\\frac{1}{2}t^{\\pm1}A C^{\\pm1}\\big)\\theta_p\\big(h p^\\frac{1}{2}t^{\\pm1}B^{-1}D^{\\pm1}\\big)}\n{2q^{\\frac12}\\theta_p\\big(q^{-1}\\big)},\\\\\n\\operatorname{Res}_{z\\to h p^\\frac12 q^{\\frac12}} W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D, (1,0; AB^{-1})}(z)=\n-h (p;p)^{-2}\\frac{A^{-2}B^{2}\\theta_p\\big(h t^{\\pm1}A C^{\\pm1}\\big)\\theta_p\\big(h t^{\\pm1}B^{-1}D^{\\pm1}\\big)}\n{2p^{-\\frac32}q^{-\\frac12}\\theta_p\\big(q^{-1}\\big)},\\\\\n\\operatorname{Res}_{z\\to h p^\\frac12 q^{-\\frac12}} W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D, (1,0; AB^{-1})}(z)=\nh (p;p)^{-2}\\frac{A^{-2}B^{2}\\theta_p\\big(h t^{\\pm1}A C^{\\pm1}\\big)\\theta_p\\big(h t^{\\pm1}B^{-1}D^{\\pm1}\\big)}\n{2p^{-\\frac32}q^{\\frac12}\\theta_p\\big(q^{-1}\\big)}.\n\\end{gather*}\n\\end{document} \n"}
{"Using the ellipticity of Vband some basic properties of the theta function\nthis is exactly what we get from computing the residues of Vb. Thus, we can\nconclude that the operators are the same up to a constant function. The\nvan Diejen operator does not depend on the type of residue we took, that\nis what type of defect was introduced, but only on the color of puncture\nthrough the choice of the eight parameters. The choice of the defect enters\nthrough the additive constant by which the operator derived from the index\ndi\u000bers from the van Diejen operator. In particular, operators introducing\ndi\u000berent defects commute with each other as they di\u000ber by a constant. We\ncan compute operators which correspond to residues with qexchanged by p,\nthese will correspond to defects wrapping the other equator of S3. All the\noperators should commute and indeed they do as this is true for the van\nDiejen operators. Our derivation of the operator had only \fve parameters\nbut the relation discussed here suggests generalization to eight parameters,\nagain up to the additive constant function.\n1 Koornwinder limit\nWe consider the following limit of the parameters. De\fne\np1\n2eA=AC; p1\n2eC=BDeB=A=C;eD=B=D: (1)\nWe take pto zero keeping the new variables \fxed. In the limit the di\u000berence\noperator is\nDJB;\u0000\n1;0;(eAeB=eCeD)1\n2\u0001\nJDTJD(u)\n\u0018\u0000\n1\u0000q\u00001\n2teA\u00001u\u00001\u0001\u0000\n1\u0000q\u00001\n2teC\u00001u\u00001\u0001\u0000\n1\u0000q1\n2t\u00001eA\u00001u\u0001\u0000\n1\u0000q1\n2t\u00001eC\u00001u\u0001\n\u0000\n1\u0000u2\u0001\u0000\n1\u0000qu2\u0001 TJD(qu)\n+\u0000\n1\u0000q\u00001\n2teA\u00001u\u0001\u0000\n1\u0000q\u00001\n2teC\u00001u\u0001\u0000\n1\u0000q1\n2t\u00001eA\u00001u\u00001\u0001\u0000\n1\u0000q1\n2t\u00001eC\u00001u\u00001\u0001\n\u0000\n1\u0000u\u00002\u0001\u0000\n1\u0000qu\u00002\u0001 TJD(qu\u00001)\n+WJB\nJD;\u0000\n1;0;(eAeB=eCeD)1\n2\u0001TJD(u):\nWe note that conjugating the operator to be\nOJB;\u0000\n1;0;(eAeB=eCeD)1\n2\u0001\nJD= \u0000e\u0000\nq1\n2teA\u00001u\u00061\u0001\u00001\u0000e\u0000\nq1\n2teC\u00001u\u00061\u0001\u00001\n\u0002DJB;\u0000\n1;0;(eAeB=eDeC)1\n2\u0001\nJD\u0000e\u0000\nq1\n2teA\u00001u\u00061\u0001\n\u0000e\u0000\nq1\n2teC\u00001u\u00061\u0001\n:\nWe can write\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nUsing the ellipticity of $V_b$ and some basic properties of the theta function this is exactly what we get from computing the residues of $V_b$. Thus, we can conclude that the operators are the same up to a constant function. The van Diejen operator does not depend on the type of residue we took, that is what type of defect was introduced, but only on the color of puncture through the choice of the eight parameters. The choice of the defect enters through the additive constant by which the operator derived from the index differs from the van Diejen operator. In particular, operators introducing different defects commute with each other as they differ by a constant. We can compute operators which correspond to residues with $q$ exchanged by $p$, these will correspond to defects wrapping the other equator of ${\\mathbb S}^3$. All the operators should commute and indeed they do as this is true for the van Diejen operators. Our derivation of the operator had only five parameters but the relation discussed here suggests generalization to eight parameters, again up to the additive constant function.\n\\section{Koornwinder limit}\nWe consider the following limit of the parameters. Define\n\\begin{gather}\np^{\\frac12} \\widetilde A =A C , \\qquad p^{\\frac12} \\widetilde C= B D \\qquad \\widetilde B=A/C ,\\qquad \\widetilde D=B/D .\n\\end{gather}\nWe take $p$ to zero keeping the new variables fixed. In the limit the difference operator is\n\\begin{gather*}\n{\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,\\big(1,0;(\\widetilde A \\widetilde B/ \\widetilde C \\widetilde D)^{\\frac12}\\big)} T_{{\\mathfrak J}_D}(u)\\\\\n{} \\sim \\frac{\\big(1-q^{-\\frac12}t \\widetilde A^{-1}u^{-1}\\big)\\big(1-q^{-\\frac12}t \\widetilde C^{-1}u^{-1}\\big)\\big(1-q^{\\frac12} t^{-1} \\widetilde A^{-1} u\\big)\\big(1-q^{\\frac12} t^{-1} \\widetilde C^{-1} u\\big)}{\\big(1-u^2\\big)\\big(1-q u^2\\big)} T_{{\\mathfrak J}_D}(q u) \\\\\n \\quad{}+\\frac{\\big(1-q^{-\\frac12}t \\widetilde A^{-1}u\\big)\\big(1-q^{-\\frac12}t \\widetilde C^{-1}u\\big)\\big(1-q^{\\frac12} t^{-1} \\widetilde A^{-1} u^{-1}\\big)\\big(1-q^{\\frac12} t^{-1} \\widetilde C^{-1} u^{-1}\\big)}{\\big(1-u^{-2}\\big)\\big(1-q u^{-2}\\big)} T_{{\\mathfrak J}_D}(q u^{-1}) \\\\\n\\quad{} +W^{{\\mathfrak J}_B}_{{\\mathfrak J}_D,\\big(1,0;(\\widetilde A \\widetilde B/\\widetilde C\\widetilde D )^{\\frac12}\\big)} T_{{\\mathfrak J}_D}(u).\\end{gather*}\n We note that conjugating the operator to be\n\\begin{gather*}\n{\\mathfrak O}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,\\big(1,0;(\\widetilde A \\widetilde B / \\widetilde C \\widetilde D)^{\\frac12}\\big)}=\n\\Gamma_e\\big(q^{\\frac12}t {\\widetilde A}^{-1}u^{\\pm1}\\big)^{-1} \\Gamma_e\\big(q^{\\frac12}t {\\widetilde C}^{-1}u^{\\pm1}\\big)^{-1}\\\\\n \\hphantom{{\\mathfrak O}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,\\big(1,0;(\\widetilde A \\widetilde B / \\widetilde C \\widetilde D)^{\\frac12}\\big)}= }{}\n\\times {\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,\\big(1,0;(\\widetilde A \\widetilde B / \\widetilde D\n\\widetilde C)^{\\frac12}\\big)} \\Gamma_e\\big(q^{\\frac12}t {\\widetilde A}^{-1}u^{\\pm1}\\big) \\Gamma_e\\big(q^{\\frac12}t {\\widetilde C}^{-1}u^{\\pm1}\\big) . \\end{gather*}\nWe can write\n\\end{document} \n"}
{"We can write\nOJB;\u0000\n1;0;(eAeB=eDeC)1\n2\u0001\nJDF(u) =4Q\nl=1(1\u0000alu)\n\u0000\n1\u0000u2\u0001\u0000\n1\u0000qu2\u0001(F(qu)\u0000F(u))\n+4Q\nj=1\u0000\n1\u0000aju\u00001\u0001\n\u0000\n1\u0000u\u00002\u0001\u0000\n1\u0000qu\u00002\u0001\u0000\nF\u0000\nqu\u00001\u0001\n\u0000F(u)\u0001\n+EJB;\u0000\n1;0;(eAeB=eDeC)1\n2\u0001\nJDF(u);\na1=q1\n2teA\u00001; a 2=q1\n2teC\u00001; a 3=q1\n2t\u00001eA\u00001; a 4=q1\n2t\u00001eC\u00001:\nThe \frst two terms de\fne the rank one Koornwinder operator and we have\nan additional constant term. The eigenfunctions are the Askey{Wilson poly-\nnomials. In general these polynomials have four independent parameters al\nbut in our case they are restricted to a1a4=a2a3. The constant term is\nEJB;(1;0;\u0000\neAeB=eDeC)1\n2\u0001\nJD=\"\n\u0000q3t2\n(eAeC)2\u0000q2t2 \n1\u0000eB\neCeDeA\u00001\neC2\u00001 +t\u00002\neAeC!\n\u0000q \n\u0000eB\neC3eAeD+eB\neC2eD+eB\neAeDeC+1\neC2+eBt2\neC2eD!\n+eBeA\neDeC#\n:\nLet us take the limit of the three punctured sphere. We give details of the\ncomputation in Appendix with the \fnal result given here (taking lim\np!0of the\nright-hand side as in ( ??))\nTJB;JC;JD(w;u;v ) =1\u0000\nqp1\nABCDt\u00002;q\u0001\u0012\nqpt21\nABCD;q\u00132\u0012\nqp1\nBACD;q\u0013\n(1)\n\u00021\u0000pqpt\nACv\u00061;pqpt\nDBv\u00061;q\u00011\u0000pqpt\nABw\u00061;pqpt\nDCw\u00061;q\u00011\u0000pqpt\nBCu\u00061;pqpt\nADu\u00061;q\u0001:\nThe index factorizes. In particular on general grounds we expect the index\nof the three punctured sphere to be\n1\u0000pqpt\nACv\u00061;pqpt\nDBv\u00061;q\u00011\u0000pqpt\nABw\u00061;pqpt\nDCw\u00061;q\u0001\n\u00021\u0000pqpt\nBCu\u00061;pqpt\nADu\u00061;q\u0001X\n\u0015C\u0015 \u0015(u) \u0015(w) \u0015(v);\nwhere the sum is over all the eigenvalues of the Koornwinder operator and\n \u0015(z) are Askey{Wilson polynomials. What we have shown above is that\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nWe can write\n\\begin{gather*}\n{\\mathfrak O}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,\\big(1,0;(\\widetilde A \\widetilde B/ \\widetilde D \\widetilde C)^{\\frac12}\\big)} {\\mathfrak F}(u)=\n\\frac{\\prod\\limits_{l=1}^4(1- a_l u)}{\\big(1-u^2\\big)\\big(1-q u^2\\big)}({\\mathfrak F}(q u)-{\\mathfrak F}(u))\\\\\n\\qquad{}\n+\\frac{\\prod\\limits_{j=1}^4\\big(1- a_j u^{-1}\\big)}{\\big(1-u^{-2}\\big)\\big(1-q u^{-2}\\big)}\\big({\\mathfrak F}\\big(q u^{-1}\\big)-{\\mathfrak F}(u)\\big)\n +E_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,\\big(1,0;(\\widetilde A \\widetilde B / \\widetilde D \\widetilde C)^{\\frac12}\\big)} {\\mathfrak F}(u) ,\\\\\na_1=q^{\\frac12}t \\widetilde A^{-1} ,\\qquad\na_2=q^{\\frac12}t \\widetilde C^{-1} ,\\qquad\na_3=q^{\\frac12}t^{-1} \\widetilde A^{-1} ,\\qquad\na_4=q^{\\frac12}t^{-1} \\widetilde C^{-1}.\n\\end{gather*} The first two terms define the rank one Koornwinder operator  and we have an additional constant term. The eigenfunctions are the Askey--Wilson polynomials. In general these polynomials have four independent parameters $a_l$ but in our case they are restricted to $a_1 a_4 =a_2 a_3$. The constant term is\n\\begin{gather*}\nE_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;\\big(\\widetilde A \\widetilde B / \\widetilde D\n\\widetilde C)^{\\frac12}\\big)}=\\left[-\\frac{q^3t^2}{ (\\widetilde A \\widetilde C)^2}-q^2t^2\\left(1-\\frac{\\widetilde B}{\\widetilde C\\widetilde D\\widetilde A}-\\frac1{\\widetilde C^2}-\\frac{1+t^{-2}}{\\widetilde A\\widetilde C}\\right)\\right. \\\\\n\\left.\\hphantom{E_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;\\big(\\widetilde A \\widetilde B / \\widetilde D \\widetilde C)^{\\frac12}\\big)}=}{}\n - q\\left(-\\frac{\\widetilde B}{\\widetilde C^3\\widetilde A\\widetilde D}+\\frac{\\widetilde B}{\\widetilde C^2\\widetilde D}+\\frac{\\widetilde B}{\\widetilde A\\widetilde D\\widetilde C}+\\frac1{\\widetilde C^2}+\\frac{\\widetilde B t^2}{\\widetilde C^2 \\widetilde D}\\right)+\\frac{\\widetilde B\\widetilde A}{\\widetilde D \\widetilde C}\\right].\n\\end{gather*}\nLet us take the limit of the three punctured sphere. We give details of the computation in Appendix with the final result given here (taking $\\lim\\limits_{p\\to 0}$ of the right-hand side as in~\\eqref{limpfuj})\n\\begin{gather}\n T_{{\\mathfrak J}_B,{\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)= \\frac1{\\big(q p\\frac1{ABC D}t^{-2};q\\big)} \\left(q p t^2 \\frac1{A BC D};q\\right)^2\\left(q p \\frac1{B A C D};q\\right) \\\\\n{} \\times \\frac1{\\big(\\sqrt{ q p} \\frac{t}{AC} v^{\\pm1},\\sqrt{q p}\\frac{t}{D B}v^{\\pm1};q\\big)}\n\\frac1{\\big(\\sqrt{q p} \\frac{t}{AB}w^{\\pm1},\\sqrt{ q p} \\frac{t}{D C} w^{\\pm1};q\\big)}\\frac1{\n\\big(\\sqrt{q p } \\frac{t}{B C} u^{\\pm1},\\sqrt {q p } \\frac{ t}{ A D } u^{\\pm1};q\\big)} . \\nonumber\n\\end{gather} The index factorizes. In particular on general grounds  we expect the index of the three punctured sphere to be\n\\begin{gather*}\n\\frac1{\\big(\\sqrt{ q p} \\frac{t}{AC} v^{\\pm1},\\sqrt{q p}\\frac{t}{D B}v^{\\pm1};q\\big)}\n\\frac1{\\big(\\sqrt{q p} \\frac{t}{AB}w^{\\pm1},\\sqrt{ q p} \\frac{t}{D C} w^{\\pm1};q\\big)}\\\\\n\\qquad{}\\times \\frac1{\\big(\\sqrt{q p } \\frac{t}{B C} u^{\\pm1},\\sqrt {q p }\\frac{ t}{ A D } u^{\\pm1};q\\big)}\n \\sum_{\\lambda} C_\\lambda \\psi_\\lambda(u)\\psi_\\lambda(w)\\psi_\\lambda(v) ,\n\\end{gather*} where the sum is over all the eigenvalues of the Koornwinder operator and $\\psi_\\lambda(z)$ are Askey--Wilson polynomials. What we have shown above is that\n\\end{document} \n"}
{"in the limit the three punctured sphere we have de\fned has all C\u0015vanish-\ning but the one corresponding to the constant polynomial. The Koornwinder\npolynomials have higher rank generalizations which should be relevant for\nhigher rank E string theories. In those cases we do not know the three punc-\ntured spheres and the relation to Koornwinder polynomials can provide a\nuseful tool to study the indices of these models. The limit we considered\ndoes not have a special physical meaning a priori, however the fact that\nthe expressions become simple and the fact that one might generalize the\ndiscussion to the higher rank case, make the limit of potential interest.\nA Index de\fnitions\nWe compute the supersymmetric index using the standard de\fnitions of .\nThe index of chiral \feld charged under \navor U(1) symmetry with charge S\nand having R-charge Ris\n\u0000e\u0000\n(qp)R\n2uS\u0001\n:\nThe parameter uis fugacity for the \navor symmetry. We de\fne here\n\u0000e(u) =1Y\ni;j=01\u00001\nuqi+1pj+1\n1\u0000upiqj:\nWe will use the following de\fnitions\n(s;q) =1Y\ni=1\u0000\n1\u0000sqi\u00001\u0001\n; \u0012r(u) =1Y\nj=1\u0000\n1\u0000urj\u00001\u0001\u0000\n1\u0000rj=u\u0001\n:\nFinally we use the condensed conventions\nf\u0000\ny\u00061\u0001\n=f(1=y)f(y); (s1;:::;sk;q) = (s1;q)\u0001 \u0001 \u0001(sk;q):\nContour integrals in the paper are around the unit circle unless we state\notherwise.\nB Computation of the sphere with two punc-\ntures\nWe give here the derivation of equation. The computation involves calculat-\ning several contour integrals over products of elliptic gamma functions\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nin the limit the three punctured sphere we have defined has all~$C_\\lambda$ vanishing but the one corresponding to the constant polynomial. The Koornwinder polynomials have higher rank generalizations which should be relevant for higher rank E string theories. In those cases we do not know the three punctured spheres and the relation to Koornwinder polynomials can provide a useful tool to study the indices of these models. The limit we considered does not have a special physical meaning a priori, however the fact that the expressions become simple and the fact that one might generalize the discussion to the higher rank case, make the limit of potential interest.\n\\appendix\n\\section{Index definitions}\nWe compute the supersymmetric index  using the standard definitions of . The index of chiral field charged under flavor ${\\rm U}(1)$ symmetry with charge $S$ and having R-charge ${\\mathfrak R}$ is\n\\begin{gather*}\n \\Gamma_e\\big((q p)^{\\frac{ \\mathfrak R}2} u^S\\big) .\n\\end{gather*}\nThe parameter $u$ is fugacity for the flavor symmetry. We define here\n\\begin{gather*} \\Gamma_e(u) = \\prod_{i,j=0}^\\infty \\frac{1-\\frac1u q^{i+1}p^{j+1}}{1- u p^i q^j} .\n\\end{gather*} We will use the following definitions\n\\begin{gather*}\n (s;q)= \\prod_{i=1}^\\infty \\big(1-s q^{i-1}\\big) ,\\qquad \\theta_r(u)=\\prod_{j=1}^\\infty \\big(1- u r^{j-1}\\big)\\big(1-r^j/u\\big) .\n\\end{gather*} Finally we use the condensed conventions\n\\begin{gather*}\nf\\big(y^{\\pm1}\\big) =f(1/y) f(y) ,\t\\qquad (s_1,\\dots,s_k;q) =(s_1;q)\\cdots(s_k;q) .\n\\end{gather*}\nContour integrals in the paper are around the unit circle unless we state otherwise.\n\\section{Computation of the sphere with two punctures}\nWe give here the derivation of equation. The computation involves calculating several contour integrals over products of elliptic gamma functions\n\\end{document} \n"}
{"and taking\nC(0;0;AB\u00001)\nJB(w)\u0002wTJB;JC;JD(w;u;v )\n= (q;q)(p;p)Idw\n4\u0019iw8Q\nj=1\u0000e\u0000\n(qp)1\n21\nta\u00001\njw\u00061\u0001\n\u0000\u0000\nw\u00062\u0001C(0;0;AB\u00001)\nJB(w)TJB;JC;JD(w;u;v )\n= (q;q)4(p;p)4Idw\n4\u0019iw8Q\nj=1\u0000e\u0000\n(qp)1\n21\nta\u00001\njw\u00061\u0001\n\u0000\u0000\nw\u00062\u0001 \u0000e\u0000\npqt4\u0001Y\nj6=i\u0000e\u0012pqt2\naiaj\u0013\n08Y\nj=1\u0000e\u0000\n(pq)1\n2tajw\u00061\u0001\n\u0002\u0000e \n(pq)1\n2w\u00061\ntai!\n\u0000e \naiw\u00061\n(pq)1\n2t3!\n\u0000e\u0000\n(qp)1\n2t\u0000\nB\u00001A\u0001\u00061w\u00061\u0001\n\u0000e\u0010qp\nt2\u0011\n\u0002Idy\n4\u0019iy\u0000e\u0000(pq)1\n2\nt2\u0000\nAB\u00001\u0001\u00061y\u00061\u0001\n\u0000e(y\u00062)\u0000e\u0000\nty\u00061w\u00061\u0001Idw1\n4\u0019iw 1Idw2\n4\u0019iw 2\u0000e\u0000(pq)1\n2\nt2w\u00061\n1w\u00061\n2\u0001\n\u0000e\u0000\nw\u00062\n2\u0001\n\u0000e\u0000\nw\u00062\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y1\n2w\u00061\n1u\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y\u00001\n2w\u00061\n1D\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y\u00001\n2w\u00061\n2u\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y1\n2D\u00061w\u00061\n2\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y1\n2w\u00061\n1v\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y\u00001\n2C\u00061w\u00061\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y\u00001\n2w\u00061\n2v\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y1\n2w\u00061\n2C\u00061\u0001\n:\nPlugging in the values of ajfrom where ai=AB\u00001and using the identity\n\u0000e\u0000pq\nz\u0001\n\u0000e(z) = 1 we get\n(q;q)4(p;p)4Idw\n4\u0019iw1\n\u0000\u0000\nw\u00062\u0001\u0000e\u0000\npqt4\u0001\n\u0000e\u0000\npqt2\u0001\n\u0000e\u0000\npqt2B2\u0001\n\u0000e\u0000\npqt2A\u00002\u0001\n\u0002\u0000e\u0000\npqt2A\u00001BC\u00061D\u00061\u0001\n\u0000e \nAB\u00001w\u00061\n(pq)1\n2t3!\n\u0000e\u0000\n(qp)1\n2tA\u00001Bw\u00061\u0001\n\u0000e\u0010qp\nt2\u0011\n\u0002Idy\n4\u0019iy\u0000e\u0000(pq)1\n2\nt2\u0000\nAB\u00001\u0001\u00061y\u00061\u0001\n\u0000e\u0000\ny\u00062\u0001 \u0000e\u0000\nty\u00061w\u00061\u0001\n\u0002Idw1\n4\u0019iw 1Idw2\n4\u0019iw 2\u0000e\u0000(pq)1\n2\nt2w\u00061\n1w\u00061\n2\u0001\n\u0000e\u0000\nw\u00062\n2\u0001\n\u0000e\u0000\nw\u00062\n1\u0001\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y1\n2w\u00061\n1u\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y\u00001\n2w\u00061\n1D\u00061\u0001\n\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y\u00001\n2w\u00061\n2u\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y1\n2D\u00061w\u00061\n2\u0001\n\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y1\n2w\u00061\n1v\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y\u00001\n2C\u00061w\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y\u00001\n2w\u00061\n2v\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y1\n2w\u00061\n2C\u00061\u0001\n:\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\n and taking\n\\begin{gather*}\nC^{(0,0;AB^{-1})}_{{\\mathfrak J}_B}(w)\\times_w T_{{\\mathfrak J}_B,{\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\n\\\\\n=(q;q)(p;p)\\oint\\frac{{\\rm d}w}{4\\pi i w} \\frac{\\prod\\limits_{j=1}^8\\Gamma_e\\big((q p)^{\\frac12 }\\frac1t a_j^{-1} w^{\\pm1}\\big)}{\\Gamma\\big(w^{\\pm2}\\big)} C^{(0,0;AB^{-1})}_{{\\mathfrak J}_B}(w)T_{{\\mathfrak J}_B,{\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\n\\\\\n=(q;q)^4(p;p)^4\\!\\oint\\!\\frac{{\\rm d}w}{4\\pi i w} \\frac{\\prod\\limits_{j=1}^8\\Gamma_e\\big((q p)^{\\frac12 }\\frac1t a_j^{-1} w^{\\pm1}\\big)}{\\Gamma\\big(w^{\\pm2}\\big)}\n\\Gamma_e\\big(pqt^4\\big)\\prod_{j\\neq i}\\Gamma_e\\left(\\frac{pqt^2}{a_ia_j}\\right)0\n\\prod_{j=1}^8\\Gamma_e\\big((pq)^{\\frac{1}{2}}ta_jw^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\left(\\frac{(pq)^{\\frac{1}{2}}w^{\\pm1}}{ta_i}\\right)\\Gamma_e\\left(\\frac{a_iw^{\\pm1}}{(pq)^{\\frac{1}{2}}t^3}\\right)\n \\Gamma_e\\big((q p)^{\\frac12} t\\big(B^{-1}A\\big)^{\\pm1} w^{\\pm1}\\big)\\Gamma_e\\left(\\frac{q p}{t^2}\\right) \\\\\n\\quad{}\\times\\oint \\frac{{\\rm d}y}{4\\pi i y} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}\\big(A B^{-1}\\big)^{\\pm1}y^{\\pm1}\\big)}{\\Gamma_e(y^{\\pm2})}\n\\Gamma_e\\big(t y^{\\pm1} w^{\\pm1}\\big)\\oint\\frac{{\\rm d}w_1}{4\\pi i w_1}\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}w_1^{\\pm1}w_2^{\\pm1}\\big)}{\\Gamma_e\\big(w_2^{\\pm2}\\big)\\Gamma_e\\big(w_1^{\\pm2}\\big)} \\\\\n\\quad{}\\times\\Gamma_e\\big( (q p)^{\\frac14}t A^{\\frac12}B^{-\\frac12} y^{\\frac12} w_1^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((qp)^{\\frac14} A^{\\frac12} B^{\\frac12} y^{-\\frac12} w_1^{\\pm1}D^{\\pm1}\\big)\\\\\n\\quad{}\\times \\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^{\\frac12} y^{-\\frac12} w_2^{\\pm1} u^{\\pm1}\\big)\n\\Gamma_e \\big((q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{\\frac12} D^{\\pm1}w_2^{\\pm1}\\big)\\\\\n\\quad{}\\times\n\\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^\\frac12 y^\\frac12 w_1^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big( ( q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{-\\frac12} C^{\\pm1} w_1^{\\pm1}\\big) \\\\\n\\quad{} \\times\\Gamma_e\\big( ( q p )^{\\frac14} t A^{\\frac12} B^{-\\frac12} y^{-\\frac12} w_2^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big( ( q p )^{\\frac14} A^{\\frac12} B^\\frac12 y^\\frac12 w_2^{\\pm1} C^{\\pm1}\\big) .\n\\end{gather*}\nPlugging in the values of $a_j$ from where $a_i=AB^{-1}$ and using the identity $\\Gamma_e\\big(\\frac{pq}{z}\\big)\\Gamma_e(z)=1$ we get\n\\begin{gather*}\n(q;q)^4(p;p)^4\\oint\\frac{{\\rm d}w}{4\\pi i w} \\frac{1}{\\Gamma\\big(w^{\\pm2}\\big)}\n\\Gamma_e\\big(pqt^4\\big)\\Gamma_e\\big(pqt^2\\big)\\Gamma_e\\big(pqt^2B^2\\big)\\Gamma_e\\big(pqt^2A^{-2}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big(pqt^2A^{-1}BC^{\\pm1}D^{\\pm1}\\big) \\Gamma_e\\left(\\frac{AB^{-1}w^{\\pm1}}{(pq)^{\\frac{1}{2}}t^3}\\right)\n \\Gamma_e\\big((q p)^{\\frac12} tA^{-1}B w^{\\pm1}\\big)\\Gamma_e\\left(\\frac{q p}{t^2}\\right)\\\\\n \\quad{}\\times \\oint \\frac{{\\rm d}y}{4\\pi i y} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}\\big(A B^{-1}\\big)^{\\pm1}y^{\\pm1}\\big)}{\\Gamma_e\\big(y^{\\pm2}\\big)}\\Gamma_e\\big(t y^{\\pm1} w^{\\pm1}\\big)\\\\\n\\quad{} \\times\\oint\\frac{{\\rm d}w_1}{4\\pi i w_1}\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}w_1^{\\pm1}w_2^{\\pm1}\\big)}{\\Gamma_e\\big(w_2^{\\pm2}\\big)\\Gamma_e\\big(w_1^{\\pm2}\\big)} \\Gamma_e\\big( (q p)^{\\frac14}t A^{\\frac12}B^{-\\frac12} y^{\\frac12} w_1^{\\pm1}u^{\\pm1}\\big)\\\\\n\\quad{} \\times \\Gamma_e\\big((qp)^{\\frac14} A^{\\frac12} B^{\\frac12} y^{-\\frac12} w_1^{\\pm1}D^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^{\\frac12} y^{-\\frac12} w_2^{\\pm1} u^{\\pm1}\\big) \\\\\n\\quad{} \\times \\Gamma_e \\big((q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{\\frac12} D^{\\pm1}w_2^{\\pm1}\\big)\n \\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^\\frac12 y^\\frac12 w_1^{\\pm1} v^{\\pm1}\\big)\\\\\n\\quad{} \\times \\Gamma_e\\big( ( q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{-\\frac12} C^{\\pm1} w_1^{\\pm1}\\big) \\Gamma_e\\big( ( q p )^{\\frac14} t A^{\\frac12} B^{-\\frac12} y^{-\\frac12} w_2^{\\pm1} v^{\\pm1}\\big) \\\\\n\\quad{} \\times \\Gamma_e\\big( ( q p )^{\\frac14} A^{\\frac12} B^\\frac12 y^\\frac12 w_2^{\\pm1} C^{\\pm1}\\big) .\n\\end{gather*}\n\\end{document} \n"}
{"We can perform the integral over wusing the inversion formula which\nsetsy=AB\u00001\n(pq)1\n2t2, and we get\n(q;q)2(p;p)2\u0000e\u0000\npqt2B2\u0001\n\u0000e\u0000\npqt2A\u00002\u0001\n\u0000e\u0000\npqt2A\u00001BC\u00061D\u00061\u0001\n\u0000e(pq)\u0000e\u0000\nt\u00004A2B\u00002\u0001\n\u0002\u0000e\u0000\npqA\u00002B2\u0001Idw1\n4\u0019iw 1Idw2\n4\u0019iw 2\u0000e\u0000(pq)1\n2\nt2w\u00061\n1w\u00061\n2\u0001\n\u0000e\u0000\nw\u00062\n2\u0001\n\u0000e\u0000\nw\u00062\n1\u0001\u0000e\u0000\nAB\u00001u\u00061w\u00061\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n2tBD\u00061w\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n2t2A\u00001Bu\u00061w\u00061\n2\u0001\n\u0000e\u0000\nt\u00001B\u00001D\u00061w\u00061\n2\u0001\n\u0000e\u0000\nv\u00061w\u00061\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n2tA\u00001C\u00061w\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n2t2v\u00061w\u00061\n2\u0001\n\u0000e\u0000\nt\u00001AC\u00061w\u00061\n2\u0001\n:\n\u0000e(pq) is zero but the integral over w1is pinched at w1=v\u00061due to the\nterm \u0000 e(v\u00061w\u00061\n1) such that the multiplication is \fnite and we get\n(q;q)(p;p)\u0000e\u0000\npqt2B2\u0001\n\u0000e\u0000\npqt2A\u00002\u0001\n\u0000e\u0000\npqt2A\u00001BC\u00061D\u00061\u0001\n\u0000e\u0000\nt\u00004A2B\u00002\u0001\n\u0000e\u0000\npqA\u00002B2\u0001\n\u0002\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tBD\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tA\u00001C\u00061v\u00061\u0001\n\u0002Idw2\n4\u0019iw 21\n\u0000e\u0000\nw\u00062\n2\u0001\u0000e\u0000\n(qp)1\n2t2A\u00001Bu\u00061w\u00061\n2\u0001\n\u0000e\u0000\nt\u00001B\u00001D\u00061w\u00061\n2\u0001\n\u0000e\u0000\nt\u00001AC\u00061w\u00061\n2\u0001\n:\nIntegral over w2can be evaluated using the elliptic beta integral formula,\nand the \fnal result is\n\u0000e\u0000\npqA\u00002B2\u0001\n\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tBD\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tA\u00001C\u00061v\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tBC\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061u\u00061\u0001\n:\nWe glue this to another three punctured sphere closed with ai=A\u00001Band\nthe claim is that gluing this to a given model we get the same model. Indeed\nwe have\nTJC(u)\u0002u\u0000\u0000\nTJB;JC;JD(w;u;v )\u0002wC(0;0;AB\u00001)\nJB(w)\u0001\n\u0002v\u0000\nTJB;JC;JD(h;z;v )\u0002hC(0;0;A\u00001B)\nJB(h)\u0001\u0001\n= (q;q)2(p;p)2\u0000e\u0000\npq\u0000\nA\u00002B2\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tB\u00001C\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061z\u00061\u0001\n\u0002Idu\n4\u0019iu\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061u\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061u\u00061\u0001\n\u0000\u0000\nu\u00062\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tBC\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061u\u00061\u0001\nTJC(u)\n\u0002Idv\n4\u0019iv\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061v\u00061\u0001\n\u0000\u0000\nv\u00062\u0001\n\u0002\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tBD\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tA\u00001C\u00061v\u00061\u0001\n\u0002\u0000e\u0000\nA\u00001Bz\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061v\u00061\u0001\n:\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nWe can perform the integral over $w$ using the inversion formula which sets $y=\\frac{AB^{-1}}{(pq)^\\frac12t^2}$, and we get\n\\begin{gather*}\n(q;q)^2(p;p)^2\\Gamma_e\\big(pqt^2B^2\\big)\\Gamma_e\\big(pqt^2A^{-2}\\big)\\Gamma_e\\big(pqt^2A^{-1}BC^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e(pq)\\Gamma_e\\big(t^{-4}A^2B^{-2}\\big)\\\\\n\\quad{}\\times \\Gamma_e\\big(pqA^{-2}B^2\\big)\n\\oint\\frac{{\\rm d}w_1}{4\\pi i w_1}\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}w_1^{\\pm1}w_2^{\\pm1}\\big)}{\\Gamma_e\\big(w_2^{\\pm2}\\big)\\Gamma_e\\big(w_1^{\\pm2}\\big)} \\Gamma_e\\big( AB^{-1} u^{\\pm1} w_1^{\\pm1}\\big)\\\\\n\\quad {} \\times \\Gamma_e\\big((qp)^{\\frac12}t BD^{\\pm1} w_1^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac12} t^2 A^{-1} B u^{\\pm1} w_2^{\\pm1}\\big) \\Gamma_e \\big(t^{-1} B^{-1} D^{\\pm1}w_2^{\\pm1}\\big) \\Gamma_e\\big( v^{\\pm1} w_1^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big((q p)^{\\frac12}t A^{-1} C^{\\pm1} w_1^{\\pm1}\\big) \\Gamma_e\\big( (q p)^{\\frac12} t^2 v^{\\pm1} w_2^{\\pm1}\\big) \\Gamma_e\\big( t^{-1} A C^{\\pm1} w_2^{\\pm1}\\big) .\n\\end{gather*}\n$\\Gamma_e(pq)$ is zero but the integral over $w_1$ is pinched at $w_1=v^{\\pm1}$ due to the term $\\Gamma_e(v^{\\pm1}w_1^{\\pm1})$ such that the multiplication is finite and we get\n\\begin{gather*}\n(q;q)(p;p) \\Gamma_e\\big(pqt^2B^2\\big)\\Gamma_e\\big(pqt^2A^{-2}\\big)\\Gamma_e\\big(pqt^2A^{-1}BC^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big(t^{-4}A^2B^{-2}\\big)\\Gamma_e\\big(pqA^{-2}B^2\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big( AB^{-1} u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t BD^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A^{-1} C^{\\pm1} v^{\\pm1}\\big)\\\\\n\\quad{} \\times\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{1}{\\Gamma_e\\big(w_2^{\\pm2}\\big)}\n\\Gamma_e\\big((q p)^{\\frac12} t^2 A^{-1} B u^{\\pm1} w_2^{\\pm1}\\big) \\Gamma_e \\big(t^{-1} B^{-1} D^{\\pm1}w_2^{\\pm1}\\big)\n\\Gamma_e\\big( t^{-1} A C^{\\pm1} w_2^{\\pm1}\\big) .\n\\end{gather*}\nIntegral over $w_2$ can be evaluated using the elliptic beta integral formula, and the final result is\n\\begin{gather*}\n\\Gamma_e\\big(pqA^{-2}B^2\\big) \\Gamma_e\\big( AB^{-1} u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t BD^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A^{-1} C^{\\pm1} v^{\\pm1}\\big)\\\\\n\\quad{}\\times \\Gamma_e\\big((pq)^\\frac12 t B C^{\\pm1}u^{\\pm1}\\big) \\Gamma_e\\big((pq)^\\frac12 t A^{-1}D^{\\pm1}u^{\\pm1}\\big) .\n\\end{gather*}\nWe glue this to another three punctured sphere closed with $a_i=A^{-1}B$ and the claim is that gluing this to a given model we get the same model. Indeed we have\n\\begin{gather*}\n T_{{\\mathfrak J}_C}(u)\\times_u \\big(\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\\times_w C^{(0,0;AB^{-1})}_{{\\mathfrak J}_B}(w)\\big)\\\\\n \\quad{} \\times_v\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,z,v)\\times_h C^{(0,0;A^{-1}B)}_{{\\mathfrak J}_B}(h)\\big)\\big)\\\\\n=(q;q)^2(p;p)^2\\Gamma_e\\big(pq\\big(A^{-2}B^2\\big)^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t AD^{\\pm1}z^{\\pm1}\\big)\\\\\n\\quad{} \\times\\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}u^{\\pm1}\\big)}{\\Gamma\\big(u^{\\pm2}\\big)}\\\\\n\\quad{} \\times\\Gamma_e\\big((pq)^\\frac12 t B C^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t A^{-1}D^{\\pm1}u^{\\pm1}\\big)T_{{\\mathfrak J}_C}(u)\\\\\n\\quad{} \\times\\oint\\frac{{\\rm d}v}{4\\pi i v} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}v^{\\pm1}\\big)}{\\Gamma\\big(v^{\\pm2}\\big)}\\\\\n\\quad{} \\times\\Gamma_e\\big( AB^{-1} u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t BD^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A^{-1} C^{\\pm1} v^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big( A^{-1}B z^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} v^{\\pm1}\\big) .\n\\end{gather*}\n\\end{document} \n"}
{"Terms cancel in the second integral and we get\n= (q;q)2(p;p)2\u0000e\u0000\npq\u0000\nA\u00002B2\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tB\u00001C\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061z\u00061\u0001\n\u0002Idu\n4\u0019iu\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061u\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061u\u00061\u0001\n\u0000\u0000\nu\u00062\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tBC\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061u\u00061\u0001\nTJC(u)\n\u0002Idv\n4\u0019iv1\n\u0000\u0000\nv\u00062\u0001\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\nA\u00001Bz\u00061v\u00061\u0001\n:\nThe integrals can be evaluated using the inversion formula which sets u=z:\n= \u0000 e\u0000\n(pq)1\n2tB\u00001C\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061z\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061z\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBC\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061z\u00061\u0001\nTJC(z):\nWe see that all terms cancel so we get\nTJC(u)\u0002u\u0000\u0000\nTJB;JC;JD(w;u;v )\u0002wC(0;0;AB\u00001)\nJB(w)\u0001\n\u0002v\u0000\nTJB;JC;JD(h;z;v )\u0002hC(0;0;A\u00001B)\nJB(h)\u0001\u0001\n=TJC(z):\n1 Computation of the sphere with two punc-\ntures and a defect\nHere we compute the di\u000berence operator. The computation is a small twist\non the one of the previous section, however it is less straightforward and\nthus for which we perform another duality operation. After the duality the p\nvanishing limit is well de\fned for all \felds. Let us write the \felds surviving\nafter scaling\n(pq)1\n6A1\n3b\u00002\n3t1\n3a\u00002\n3z\u00001\n3\n2z\u00061\n1wj\n2;(pq)1\n6A\u00002\n3b1\n3t1\n3a1\n3z\u00001\n3\n2v\u00061\n1wj\n2;\n(qp)t\u00002;(qp)1\n3A\u00001\n3t\u00004\n3b\u00001\n3a\u00001\n3z1\n3\n2v\u00001\n2\u0000\nwj\n2\u0001\u00001;(qp)1\n3A\u00001\n3t2\n3b\u00001\n3a\u00001\n3z1\n3\n2v\u00001\n2\u0000\nwj\n2\u0001\u00001; ab\u00001tw\u00061\n1;\n(qp)1\n2ab\u00001v\u00001\n2z\u00001\n2;(qp)1\n2b\u00001t\u00001A\u00001w\u00061\n1a\u00001;\n(qp)1\n2a\u00001bz2v2;(qp)1\n2z\u00001\n2t\u00001w\u00061\n1v\u00001\n2; bta\u00001w\u00061\n1;(qp)1\n2tv2Av\u00061\n1;(qp)1\n2abtv 2z\u00061\n1;\nWe would like to thank Hee-Cheol Kim, S. Ruijsenaars, Cumrun Vafa, and\nGabi Zafrir for relevant discussions. The research was supported by Israel\nScience Foundation under grant no. 1696/15 and by I-CORE Program of the\nPlanning and Budgeting Committee.\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nTerms cancel in the second integral and we get\n\\begin{gather*}\n=(q;q)^2(p;p)^2\\Gamma_e\\big(pq\\big(A^{-2}B^2\\big)^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t AD^{\\pm1}z^{\\pm1}\\big)\\\\\n\\quad{}\\times\\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}u^{\\pm1}\\big)}{\\Gamma\\big(u^{\\pm2}\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac12 t B C^{\\pm1}u^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t A^{-1}D^{\\pm1}u^{\\pm1}\\big)T_{{\\mathfrak J}_C}(u)\\\\\n\\quad{}\\times\\oint\\frac{{\\rm d}v}{4\\pi i v} \\frac{1}{\\Gamma\\big(v^{\\pm2}\\big)}\n\\Gamma_e\\big( AB^{-1} u^{\\pm1} v^{\\pm1}\\big) \\Gamma_e\\big( A^{-1}B z^{\\pm1} v^{\\pm1}\\big) .\n\\end{gather*}\nThe integrals can be evaluated using the inversion formula which sets $u=z$:\n\\begin{gather*}\n=\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t AD^{\\pm1}z^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}z^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}z^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t B C^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t A^{-1}D^{\\pm1}z^{\\pm1}\\big)T_{{\\mathfrak J}_C}(z) .\n\\end{gather*}\nWe see that all terms cancel so we get\n\\begin{gather*}\n T_{{\\mathfrak J}_C}(u)\\times_u \\big(\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\\times_w C^{(0,0;AB^{-1})}_{{\\mathfrak J}_B}(w)\\big)\\\\\n\\qquad{} \\times_v\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,z,v)\\times_h C^{(0,0;A^{-1}B)}_{{\\mathfrak J}_B}(h)\\big)\\big)= T_{{\\mathfrak J}_C}(z).\n\\end{gather*}\n\\section{Computation of the sphere with two punctures and a defect}\nHere we compute the difference operator. The computation is a small twist on the one of the previous section, however it is less straightforward and thus  for which we perform another duality operation. After the duality the $p$ vanishing limit is well defined for all fields. Let us write the fields surviving after scaling\n\\begin{gather*}\n(p q)^{\\frac16}A^{\\frac13}b^{-\\frac23} t^{\\frac13}a^{-\\frac23} z_2^{-\\frac13} z_1^{\\pm1}w_2^j , \\ (p q)^{\\frac16}A^{-\\frac23}b^{\\frac13} t^{\\frac13}a^{\\frac13} z_2^{-\\frac13} v_1^{\\pm1} w_2^j , \\\\\n(q p)t^{-2} , \\ (q p)^{\\frac13} A^{-\\frac13} t^{-\\frac43} b ^{-\\frac13} a^{-\\frac13} z_2^{\\frac13} v_2^{-1} \\big(w^j_2\\big)^{-1} , \\\n (q p)^{\\frac13} A^{-\\frac13} t^{\\frac23} b ^{-\\frac13} a^{-\\frac13} z_2^{\\frac13} v_2^{-1} \\big(w^j_2\\big)^{-1} , \\\na b^{-1} t w_1^{\\pm1} ,\\\\\n( q p)^{ \\frac1 2 }a b^{-1} v_2^ {-1} z_2^{-1} , \\ ( q p)^{\\frac12} b^{-1} t^{-1} A^{-1} w_1^{\\pm1} a^{-1} ,\\\\\n(q p)^{\\frac12} a^{-1} b z_2 v_2 , \\ (q p)^{\\frac12} z_2^{-1} t^{-1} w_1^{\\pm1} v_2^{-1} , \\\nb t a^{-1} w_1^{\\pm1} , \\ (q p)^{\\frac12} t v_2 A v_1^{\\pm1} , \\ (q p)^{\\frac12} a b t v_2 z_1^{\\pm1} ,\n\\end{gather*}\nWe would like to thank Hee-Cheol Kim, S.~Ruijsenaars, Cumrun Vafa, and Gabi Zafrir for relevant discussions. The research was supported by Israel Science Foundation under grant no. 1696/15 and by I-CORE Program of the Planning and Budgeting Committee.\n\\end{document} \n"}
{"we discuss it in detail. We compute\nTJB;JC;JD(w;u;v )\u0002wC(1;0;AB\u00001)\nJB(w)\n= (q;q)(p;p)Idw\n4\u0019iw8Q\nj=1\u0000e\u0000\n(qp)1\n21\nta\u00001\njw\u00061\u0001\n\u0000\u0000\nw\u00062\u0001C(1;0;AB\u00001)\nJB(w)TJB;JC;JD(w;u;v )\n\u0018Idw\n4\u0019iw8Q\nj=1\u0000e\u0000\n(qp)1\n21\nta\u00001\njw\u00061\u0001\n\u0000\u0000\nw\u00062\u00018Y\nj=1\u0000e\u0000\n(pq)1\n2tajw\u00061\u0001\n\u0000e \n(pq)1\n2qw\u00061\ntAB\u00001!\n\u0002\u0000e \nAB\u00001w\u00061\n(pq)1\n2qt3!\n\u0000e\u0000\n(qp)1\n2t\u0000\nB\u00001A\u0001\u00061w\u00061\u0001\n\u0000e\u0010qp\nt2\u0011Idy\n4\u0019iy\u0000e\u0000(pq)1\n2\nt2\u0000\nAB\u00001\u0001\u00061y\u00061\u0001\n\u0000e\u0000\ny\u00062\u0001\n\u0002\u0000e\u0000\nty\u00061w\u00061\u0001Idw1\n4\u0019iw 1Idw2\n4\u0019iw 2\u0000e\u0000(pq)1\n2\nt2w\u00061\n1w\u00061\n2\u0001\n\u0000e\u0000\nw\u00062\n2\u0001\n\u0000e\u0000\nw\u00062\n1\u0001\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y1\n2w\u00061\n1u\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y\u00001\n2w\u00061\n1D\u00061\u0001\n\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y\u00001\n2w\u00061\n2u\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y1\n2D\u00061w\u00061\n2\u0001\n\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y1\n2w\u00061\n1v\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y\u00001\n2C\u00061w\u00061\n1\u0001\n\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y\u00001\n2w\u00061\n2v\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y1\n2w\u00061\n2C\u00061\u0001\n;\nwhere \u0018means equality up to overall factors independent of w,u,v. Using\nthe identity \u0000 e\u0000pq\nz\u0001\n\u0000e(z) = 1 and the elliptic beta integral formula we evaluate\nthe integral over wand up to overall factors we get\n\u0000e\u0000\npq2\u0001Idy\n4\u0019iy\u0000e\u0000\n(pq)1\n2qA\u00001By\u00061\u0001\n\u0000e\u0000\n(pq)\u00001\n2q\u00001t\u00002AB\u00001y\u00061\u0001\n\u0000e\u0000\ny\u00062\u0001Idw1\n4\u0019iw 1Idw2\n4\u0019iw 2\n\u0002\u0000e\u0000(pq)1\n2\nt2w\u00061\n1w\u00061\n2\u0001\n\u0000e\u0000\nw\u00062\n2\u0001\n\u0000e\u0000\nw\u00062\n1\u0001\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y1\n2w\u00061\n1u\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y\u00001\n2w\u00061\n1D\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y\u00001\n2w\u00061\n2u\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y1\n2D\u00061w\u00061\n2\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y1\n2w\u00061\n1v\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A\u00001\n2B\u00001\n2y\u00001\n2C\u00061w\u00061\n1\u0001\n\u0002\u0000e\u0000\n(qp)1\n4tA1\n2B\u00001\n2y\u00001\n2w\u00061\n2v\u00061\u0001\n\u0000e\u0000\n(qp)1\n4A1\n2B1\n2y1\n2w\u00061\n2C\u00061\u0001\n: (1)\nWe got a zero multiplying the integral, but we will see next that some of the\nintegrals are pinched giving \fnite result. We start by evaluating the integral\noveryusing the residue theorem and we get that the integral over w1is\npinched due to \u0000 e\u0000\n(qp)1\n4tA\u00001\n2B1\n2y1\n2w\u00061\n1v\u00061\u0001\nterm for the poles\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nwe discuss it in detail. We compute\n\\begin{gather*}\nT_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\\times_w C^{(1,0;AB^{-1})}_{{\\mathfrak J}_B}(w)\\\\\n=(q;q)(p;p)\\oint\\frac{{\\rm d}w}{4\\pi i w} \\frac{\\prod\\limits_{j=1}^8\\Gamma_e\\big((q p)^{\\frac12 }\\frac1t a_j^{-1} w^{\\pm1}\\big)}{\\Gamma\\big(w^{\\pm2}\\big)} C^{(1,0;AB^{-1})}_{{\\mathfrak J}_B}(w)T_{{\\mathfrak J}_B,{\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v) \\\\\n\\sim\\oint\\frac{{\\rm d}w}{4\\pi i w} \\frac{\\prod\\limits_{j=1}^8\\Gamma_e\\big((q p)^{\\frac12 }\\frac1t a_j^{-1} w^{\\pm1}\\big)}{\\Gamma\\big(w^{\\pm2}\\big)}\n\\prod_{j=1}^8\\Gamma_e\\big((pq)^{\\frac{1}{2}}ta_jw^{\\pm1}\\big)\n\\Gamma_e\\left(\\frac{(pq)^\\frac12 q w^{\\pm1}}{tAB^{-1}}\\right)\\\\\n\\quad{} \\times\\Gamma_e\\left(\\frac{AB^{-1}w^{\\pm1}}{(pq)^{\\frac{1}{2}} q t^3}\\right)\n \\Gamma_e\\big((q p)^{\\frac12} t\\big(B^{-1}A\\big)^{\\pm1} w^{\\pm1}\\big)\\Gamma_e\\left(\\frac{q p}{t^2}\\right)\n\\oint \\frac{{\\rm d}y}{4\\pi i y} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}\\big(A B^{-1}\\big)^{\\pm1}y^{\\pm1}\\big)}{\\Gamma_e\\big(y^{\\pm2}\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big(t y^{\\pm1} w^{\\pm1}\\big)\\oint\\frac{{\\rm d}w_1}{4\\pi i w_1}\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}w_1^{\\pm1}w_2^{\\pm1}\\big)}{\\Gamma_e\\big(w_2^{\\pm2}\\big)\\Gamma_e\\big(w_1^{\\pm2}\\big)} \\Gamma_e\\big( (q p)^{\\frac14}t A^{\\frac12}B^{-\\frac12}y^{\\frac12} w_1^{\\pm1}u^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\big((qp)^{\\frac14} A^{\\frac12} B^{\\frac12}y^{-\\frac12} w_1^{\\pm1}D^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^{\\frac12} y^{-\\frac12} w_2^{\\pm1} u^{\\pm1}\\big)\\\\\n\\quad{} \\times \\Gamma_e \\big((q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{\\frac12} D^{\\pm1}w_2^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^\\frac12 y^\\frac12 w_1^{\\pm1} v^{\\pm1}\\big)\\\\\n\\quad{} \\times \\Gamma_e\\big( ( q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{-\\frac12} C^{\\pm1} w_1^{\\pm1}\\big) \\Gamma_e\\big( ( q p )^{\\frac14} t A^{\\frac12} B^{-\\frac12} y^{-\\frac12} w_2^{\\pm1} v^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\big( ( q p )^{\\frac14} A^{\\frac12} B^\\frac12 y^\\frac12 w_2^{\\pm1} C^{\\pm1}\\big),\n\\end{gather*}\nwhere $\\sim$ means equality up to overall factors independent of $w$, $u$, $v$. Using the identity $\\Gamma_e\\big(\\frac{pq}{z}\\big)\\Gamma_e(z)=1$ and the elliptic beta integral formula we evaluate the integral over~$w$ and up to overall factors we get\n\\begin{gather}\n\\Gamma_e\\big(pq^2\\big)\\oint \\frac{{\\rm d}y}{4\\pi i y} \\frac{\\Gamma_e\\big((pq)^\\frac12 q A^{-1} B y^{\\pm1}\\big)\\Gamma_e\\big((pq)^{-\\frac12}q^{-1}t^{-2}AB^{-1}y^{\\pm1}\\big)}{\\Gamma_e\\big(y^{\\pm2}\\big)}\\oint\\frac{{\\rm d}w_1}{4\\pi i w_1}\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2}\\nonumber\\\\\n\\quad{} \\times \\frac{\\Gamma_e\\big(\\frac{(p q)^{\\frac12}}{t^2}w_1^{\\pm1}w_2^{\\pm1}\\big)}{\\Gamma_e\\big(w_2^{\\pm2}\\big)\\Gamma_e\\big(w_1^{\\pm2}\\big)}\\Gamma_e\\big( (q p)^{\\frac14}t A^{\\frac12}B^{-\\frac12} y^{\\frac12} w_1^{\\pm1}u^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac14} A^{\\frac12} B^{\\frac12} y^{-\\frac12} w_1^{\\pm1}D^{\\pm1}\\big)\n\\nonumber\\\\\n \\quad{} \\times \\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^{\\frac12} y^{-\\frac12} w_2^{\\pm1} u^{\\pm1}\\big) \\Gamma_e \\big((q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{\\frac12} D^{\\pm1}w_2^{\\pm1}\\big)\\nonumber\\\\\n \\quad{}\\times \\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^\\frac12 y^\\frac12 w_1^{\\pm1} v^{\\pm1}\\big)\n \\Gamma_e\\big( ( q p)^{\\frac14} A^{-\\frac12} B^{-\\frac12} y^{-\\frac12} C^{\\pm1} w_1^{\\pm1}\\big)\\nonumber\\\\\n\\quad{}\\times \\Gamma_e\\big( ( q p )^{\\frac14} t A^{\\frac12} B^{-\\frac12} y^{-\\frac12} w_2^{\\pm1} v^{\\pm1}\\big) \\Gamma_e\\big( ( q p )^{\\frac14} A^{\\frac12} B^\\frac12 y^\\frac12 w_2^{\\pm1} C^{\\pm1}\\big) .\n\\end{gather}\nWe got a zero multiplying the integral, but we will see next that some of the integrals are pinched giving finite result. We start by evaluating the integral over $y$ using the residue theorem and we get that the integral over $w_1$ is pinched due to $\\Gamma_e\\big((q p)^{\\frac14} t A^{-\\frac12} B^\\frac12 y^\\frac12 w_1^{\\pm1} v^{\\pm1}\\big)$ term for the poles\n\\end{document} \n"}
{"y1= (pq)\u00001\n2t\u00002AB\u00001andy2= (pq)\u00001\n2q\u00001t\u00002AB\u00001. Fory1the integral\noverw1is pinched at w1=v\u00061and we proceed exactly as in appendix B to\nget the same result up to some overall factors\n1\n2\u0000e\u0000\nqt\u00002\u0001\n\u0000e\u0000\npq2t2A\u00002B2\u0001\n\u0000e\u0000\nt\u00002B\u00002\u0001\n\u0000e\u0000\nt\u00002A2\u0001\n\u0000e\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0000e\u0000\n(pq)\u00001t\u00004A2B\u00002\u0001\n\u0002\u0000e\u0000\nt\u00002AB\u00001C\u00061D\u00061\u0001\n\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tBD\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tA\u00001C\u00061v\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tBC\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061u\u00061\u0001\n: (1)\nNow we look at the pole y2. the integral over w1is pinched at w1=q\u00061\n2v\u00061.\nSubstituting these values ( ??) now becomes\n1\n2\u0000e\u0000\nt\u00002\u0001\n\u0000e\u0000\npq3t2A\u00002B2\u0001\n\u0000e\u0000\npq3t4A\u00002B2\u0001\u0000e\u0000\nv2\u0001\n\u0000e\u0000\nqv2\u0001\u0000e\u0000\nq\u00001\n2AB\u00001u\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(qp)1\n2q1\n2tBD\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2v\u0001\u00061\u0001Idw2\n4\u0019iw 21\n\u0000e\u0000\nw\u00062\n2\u0001\u0000e \n(pq)1\n2\nt2q1\n2vw\u00061\n2!\n\u0002\u0000e\u0000\n(qp)1\n2q1\n2t2A\u00001Bw\u00061\n2u\u00061\u0001\n\u0000e\u0000\nq\u00001\n2t\u00001B\u00001D\u00061w\u00061\n2\u0001\n\u0002\u0000e\u0000\n(qp)1\n2q1\n2t2w\u00061\n2v\u00001\u0001\n\u0000e\u0000\nq\u00001\n2t\u00001Aw\u00061\n2C\u00061\u0001\n+\b\nv$v\u00001\t\n:\nThe integral here can be interpreted as index of the SU(2) gauge theory with\nfour \navors. Using the Intriligator{Pouliot duality transformation (that is\nV(s) =Q\n1\u0014j<k\u00148\u0000e(sjsk)V(ppq=s) in the notations of) the expression be-\ncomes\n1\n2\u0000e\u0000\nt\u00002\u0001\n\u0000e\u0000\npq3t2A\u00002B2\u0001\n\u0000e\u0000\npq3t4A\u00002B2\u0001\u0000e\u0000\nv2\u0001\n\u0000e\u0000\nqv2\u0001\u0000e\u0000\nq\u00001\n2AB\u00001u\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(qp)1\n2q1\n2tBD\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0002\u0000e\u0000\n(qp)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\npq2A\u00001Bvu\u00061\u0001\n\u0000e\u0000\n(pq)1\n2t\u00003B\u00001vD\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2t\u00003AvC\u00061\u0001\n\u0000e\u0000\npq2t4A\u00002B2\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n\u0002\u0000e\u0000\npq2t4A\u00001Bv\u00001u\u00061\u0001\n\u0000e\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0000e\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tB\u00001v\u00001D\u00061\u0001\n\u0000e\u0000\nq\u00001t\u00002A2\u0001\n\u0000e\u0000\n(pq)1\n2tAv\u00001C\u00061\u0001\n\u0000e\u0000\npq2\u0001\n\u0002Idw2\n4\u0019iw 21\n\u0000e\u0000\nw\u00062\n2\u0001\u0000e\u0000\nq\u00001\n2t2v\u00001w\u00061\n2\u0001\n\u0000e\u0000\nq\u00001\n2t\u00002AB\u00001w\u00061\n2u\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061w\u00061\n2\u0001\n\u0000e\u0000\n(qp)1\n2q1\n2tA\u00001w\u00061\n2C\u00001\u0001\n\u0000e\u0000\nq\u00001\n2t\u00002vw\u00061\n2\u0001\n+\b\nv$v\u00001\t\n:\nWe have zero multiplying the integral but the integral is pinched at w2=\u0000\nq\u00061\n2t\u00002v\u0001\u00061due to the colliding of poles of the \frst and last elliptic gamma\nfunctions in the last integral. We get di\u000berent contribution for each choice\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\n $y_1=(pq)^{-\\frac12}t^{-2}AB^{-1}$ and $y_2=(pq)^{-\\frac12}q^{-1}t^{-2}AB^{-1}$. For $y_1$ the integral over $w_1$ is pinched at $w_1=v^{\\pm1}$ and we proceed exactly as in appendix B to get the same result up to some overall factors\n\\begin{gather}\n\\frac12 \\frac{\\Gamma_e\\big(qt^{-2}\\big)\\Gamma_e\\big(pq^2t^2A^{-2}B^2\\big)\\Gamma_e\\big(t^{-2}B^{-2}\\big)\\Gamma_e\\big(t^{-2}A^2\\big) \\Gamma_e\\big((pq)^{-1}q^{-1}t^{-4}A^2 B^{-2}\\big)} {\\Gamma_e\\big((pq)^{-1}t^{-4}A^2 B^{-2}\\big)}\\nonumber\\\\\n\\quad{}\\times\\Gamma_e\\big(t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big( AB^{-1} u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t BD^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A^{-1} C^{\\pm1} v^{\\pm1}\\big)\\nonumber\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac12 t B C^{\\pm1}u^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t A^{-1}D^{\\pm1}u^{\\pm1}\\big).\n\\end{gather}\nNow we look at the pole $y_2$. the integral over $w_1$ is pinched at $w_1=q^{\\pm\\frac12}v^{\\pm1}$. Substituting these values \\eqref{c2} now becomes\n\\begin{gather*}\n\\frac12\\frac{\\Gamma_e\\big(t^{-2}\\big)\\Gamma_e\\big(pq^3t^2A^{-2}B^2\\big)}\n{\\Gamma_e\\big(pq^3 t^4 A^{-2}B^2\\big)}\\frac{\\Gamma_e\\big(v^2\\big)}{\\Gamma_e\\big(qv^2\\big)}\n\\Gamma_e\\big( q^{-\\frac12}AB^{-1}u^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}q^\\frac12 t B D^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big( ( q p)^{\\frac12} q^\\frac12 t A^{-1} C^{\\pm1} \\big(q^\\frac12 v\\big)^{\\pm1}\\big)\n\\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{1}{\\Gamma_e\\big(w_2^{\\pm2}\\big)} \\Gamma_e\\left(\\frac{(p q)^{\\frac12}}{t^2}q^\\frac12 v w_2^{\\pm1}\\right)\\\\\n\\quad{}\\times \\Gamma_e\\big((q p)^{\\frac12}q^\\frac12 t^2 A^{-1} B w_2^{\\pm1} u^{\\pm1}\\big) \\Gamma_e \\big(q^{-\\frac12} t^{-1} B^{-1} D^{\\pm1}w_2^{\\pm1}\\big)\\\\\n\\quad{}\\times \\Gamma_e\\big( ( q p )^{\\frac12}q^\\frac12 t^2 w_2^{\\pm1} v^{-1}\\big) \\Gamma_e\\big( q^{-\\frac12}t^{-1} A w_2^{\\pm1} C^{\\pm1}\\big)\n+\\big\\{v \\leftrightarrow v^{-1} \\big\\} .\n\\end{gather*}\nThe integral here can be interpreted as index of the ${\\rm SU}(2)$ gauge theory with four flavors. Using the Intriligator--Pouliot duality transformation (that is $V(\\underline{s})={\\prod\\limits_{1\\leq j < k\\leq 8} \\Gamma_e(s_j s_k)V(\\sqrt{pq}/\\underline{s})}$ in the notations of) the expression becomes\n\\begin{gather*}\n\\frac12\\frac{\\Gamma_e\\big(t^{-2}\\big)\\Gamma_e\\big(pq^3t^2A^{-2}B^2\\big)}\n{\\Gamma_e\\big(pq^3 t^4 A^{-2}B^2\\big)}\\frac{\\Gamma_e\\big(v^2\\big)}{\\Gamma_e\\big(qv^2\\big)}\n\\Gamma_e\\big( q^{-\\frac12}AB^{-1}u^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}q^\\frac12 t B D^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big( ( q p)^{\\frac12} q^\\frac12 t A^{-1} C^{\\pm1} \\big(q^\\frac12 v\\big)^{\\pm1}\\big) \\Gamma_e\\big(pq^2A^{-1}Bvu^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t^{-3}B^{-1}vD^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big((pq)^\\frac12 t^{-3}AvC^{\\pm1}\\big)\\Gamma_e\\big(pq^2t^4A^{-2}B^2\\big)\\Gamma_e\\big((pq)^\\frac12 t A^{-1}u^{\\pm1}D^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 tBu^{\\pm1}C^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e\\big(pq^2t^4A^{-1}B v^{-1}u^{\\pm1}\\big)\\Gamma_e\\big(q^{-1}t^{-2}B^{-2}\\big)\\Gamma_e\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac12 t B^{-1}v^{-1}D^{\\pm1}\\big) \\Gamma_e\\big(q^{-1}t^{-2}A^2\\big)\\Gamma_e\\big((pq)^\\frac12 t Av^{-1}C^{\\pm1}\\big)\\Gamma_e\\big(pq^2\\big)\\\\\n\\quad{}\\times \\oint \\frac{{\\rm d}w_2}{4\\pi i w_2} \\frac{1}{\\Gamma_e\\big(w_2^{\\pm2}\\big)} \\Gamma_e\\big(q^{-\\frac12}t^2 v^{-1} w_2^{\\pm1}\\big) \\Gamma_e\\big(q^{-\\frac12} t^{-2} A B^{-1} w_2^{\\pm1} u^{\\pm1}\\big)\\\\\n\\quad{} \\times\\Gamma_e \\big((pq)^{\\frac12}q^\\frac12 t B D^{\\pm1}w_2^{\\pm1}\\big)\n\\Gamma_e\\big( ( q p )^{\\frac12}q^\\frac12 t A^{-1} w_2^{\\pm1} C^{-1}\\big) \\Gamma_e\\big( q^{-\\frac12}t^{-2} v w_2^{\\pm1} \\big)\n+\\big\\{v \\leftrightarrow v^{-1} \\big\\} .\n\\end{gather*}\nWe have zero multiplying the integral but the integral is pinched at $w_2=\\big(q^{\\pm\\frac12}t^{-2}v\\big)^{\\pm1}$ due to the colliding of poles of the first and last elliptic gamma functions in the last integral. We get different contribution for each choice\n\\end{document} \n"}
{"sign ofq. Substituting each one of the values of w2and using the identities\n\u0000e(qz) =\u0012p(z)\u0000e(z) and \u0000 e\u0000\npq2z\u00001\u0001\n\u0000e(z) =\u0012p\u0000\nq\u00001z\u0001\nwe get\n1\n2\u0000e(t\u00002)\u0000e\u0000\npq3t2A\u00002B2\u0001\n\u0000e\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0000e\u0000\nq\u00001t\u00002B2\u0001\n\u0012p(pq2t4A\u00002B2)\n\u0002\u0000e\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBD\u00061u\u00061\u0001\n\u0002\u0000e\u0000\nq\u00001\n2AB\u00001u\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0002\u0012p\u0000\nq\u00001t\u00004AB\u00001vu\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001tB\u00001v\u00001D\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001tAv\u00001C\u00061\u0001\n\u0012p\u0000\nt\u00004v2\u0001\n\u0012p\u0000\nv2\u0001\n+1\n2\u0000e(t\u00002)\u0000e\u0000\npq3t2A\u00002B2\u0001\n\u0000e\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0000e\u0000\nq\u00001t\u00002B2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0002\u0000e\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001D\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBD\u00061u\u00061\u0001\n\u0002\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2q\u00001t\u00003B\u00001vD\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001t\u00003AvC\u00061\u0001\n\u0012p\u0000\nt4v\u00002\u0001\n\u0012p\u0000\nv2\u0001 +\b\nv$v\u00001\t\n:\nAdding the contribution of ( ??) and taking away overall factors the \fnal\nresult is\nTJB;JC;JD(w;u;v )\u0002wC(1;0;AB\u00001)\nJB(w)\n\u0018\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p(t\u00002)\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\nq\u00001t\u00004AB\u00001vu\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001tB\u00001v\u00001D\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001tAv\u00001C\u00061\u0001\n\u0012p\u0000\nt\u00004v2\u0001\n\u0012\u0000\nv2\u0001\n\u0002\u0000e\u0000\nq\u00001\n2AB\u00001u\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061(q1\n2v)\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n+\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n\u0002\u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2v\u0001\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2q\u00001t\u00003B\u00001vD\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001t\u00003AvC\u00061\u0001\n\u0012p\u0000\nv2\u0001\n\u0012p\u0000\nt4v\u00002\u0001 +\b\nv$v\u00001\t\n+ \u0000e\u0000\nAB\u00001u\u00061v\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBD\u00061v\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001C\u00061v\u00061\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n: (1)\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nsign of $q$. Substituting each one of the values of $w_2$ and using the identities $\\Gamma_e(qz)=\\theta_p(z)\\Gamma_e(z)$ and $\\Gamma_e\\big(pq^2z^{-1}\\big)\\Gamma_e(z)=\\theta_p\\big(q^{-1}z\\big)$ we get\n\\begin{gather*}\n\\frac12 \\frac{\\Gamma_e(t^{-2})\\Gamma_e\\big(pq^3t^2A^{-2}B^2\\big)\\Gamma_e\\big(q^{-1}t^{-2}B^{-2}\\big)\\Gamma_e\\big(q^{-1}t^{-2}B^2\\big)}\n{\\theta_p(pq^2t^4A^{-2}B^2)}\\\\\n\\quad{}\\times\\Gamma_e\\big(q^{-1}t^{-2}A B^{-1}C^{\\pm1}D^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 tA^{-1} D^{\\pm1}u^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t B D^{\\pm1}u^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\big(q^{-\\frac12}A B^{-1}u^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 q^\\frac12 t B D^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 q^\\frac12 t A^{-1} C^{\\pm1} \\big(q^\\frac12 v\\big)^{\\pm1}\\big) \\\\\n\\quad{}\\times\\frac{\\theta_p\\big(q^{-1}t^{-4}AB^{-1}vu^{\\pm1}\\big)\\theta_p\\big((pq)^\\frac12 q^{-1} t B^{-1}v^{-1}D^{\\pm1}\\big)\\theta_p\\big((pq)^\\frac12 q^{-1}t A v^{-1} C^{\\pm1}\\big)}{\\theta_p\\big(t^{-4}v^2\\big)\\theta_p\\big(v^2\\big)}\\\\\n+ \\frac12 \\frac{\\Gamma_e(t^{-2})\\Gamma_e\\big(pq^3t^2A^{-2}B^2\\big)\\Gamma_e\\big(q^{-1}t^{-2}B^{-2}\\big)\\Gamma_e\\big(q^{-1}t^{-2}B^2\\big)}\n{\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big(q^{-1}t^{-2}A B^{-1}C^{\\pm1}D^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 tA^{-1} D^{\\pm1}u^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 t B D^{\\pm1}u^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\big(A B^{-1}u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 q^\\frac12 t B D^{\\pm1}\\big(q^\\frac12 v\\big)^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac12 q^\\frac12 t A^{-1} C^{\\pm1} \\big(q^\\frac12 v\\big)^{\\pm1}\\big) \\\\\n\\quad{}\\times\\frac{\\theta_p\\big((pq)^\\frac12 q^{-1} t^{-3} B^{-1}vD^{\\pm1}\\big)\\theta_p\\big((pq)^\\frac12 q^{-1}t^{-3} A v C^{\\pm1}\\big)}\n{\\theta_p\\big(t^{4}v^{-2}\\big)\\theta_p\\big(v^2\\big)} + \\big\\{v \\leftrightarrow v^{-1} \\big\\} .\n\\end{gather*}\nAdding the contribution of \\eqref{c3} and taking away overall factors the final result is\n\\begin{gather}\nT_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\\times_w C^{(1,0;AB^{-1})}_{{\\mathfrak J}_B}(w)\\nonumber\\\\\n\\sim\\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p(t^{-2})\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\nonumber\\\\\n\\quad{}\\times\\frac{\\theta_p\\big(q^{-1}t^{-4}AB^{-1}vu^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tB^{-1}v^{-1}D^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tAv^{-1}C^{\\pm1}\\big)}\n{\\theta_p\\big(t^{-4}v^2\\big)\\theta\\big(v^2\\big)}\\nonumber\\\\\n\\quad{}\\times\\Gamma_e\\big(q^{-\\frac{1}{2}}A B^{-1}u^{\\pm1}\\big(q^\\frac{1}{2}v\\big)^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}q^\\frac{1}{2}t B D^{\\pm1}\\big(q^\\frac{1}{2}v\\big)^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}q^\\frac{1}{2}tA^{-1}C^{\\pm1}(q^\\frac{1}{2}v)^{\\pm1}\\big)\n\\nonumber\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac{1}{2}tA^{-1}u^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}t B u^{\\pm1}C^{\\pm1}\\big)\\nonumber\\\\\n\\quad{}+\\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\nonumber\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac{1}{2}t A^{-1}u^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}t B u^{\\pm1}C^{\\pm1}\\big)\n\\nonumber\\\\\n\\quad{}\\times\\Gamma_e\\big(AB^{-1}u^{\\pm1}v^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac{1}{2}q^\\frac{1}{2}t B D^{\\pm1}\\big(q^\\frac{1}{2}v\\big)^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}q^\\frac{1}{2}tA^{-1}C^{\\pm1}\\big(q^\\frac{1}{2}v\\big)^{\\pm1}\\big)\n\\nonumber\\\\\n\\quad{}\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}t^{-3}B^{-1}v D^{\\pm1}\\big)\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}t^{-3}A v C^{\\pm1}\\big)}\n{\\theta_p\\big(v^2\\big)\\theta_p\\big(t^4v^{-2}\\big)}+\\big\\{v\\leftrightarrow v^{-1}\\big\\}\\nonumber\\\\\n \\quad{}+\\Gamma_e\\big(A B^{-1}u^{\\pm1}v^{\\pm1}\\big)\\Gamma_e\\big((pq)^{\\frac{1}{2}}t B D^{\\pm1}v^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}t A^{-1}C^{\\pm1}v^{\\pm1}\\big)\\nonumber\\\\\n\\quad{}\\times\n\\Gamma_e\\big((pq)^\\frac{1}{2}tA^{-1}u^{\\pm1}D^{\\pm1}\\big)\\Gamma_e\\big((pq)^\\frac{1}{2}t B u^{\\pm1}C^{\\pm1}\\big).\n\\end{gather}\n\\end{document} \n"}
{"Now we glue this as indicated in Fig.\nTJD(v)\u0002v\u0000\u0000\nTJB;JC;JD(w;u;v )\u0002wC(0;0;A\u00001B)\nJB(w)\u0001\n\u0002u\u0000\nTJB;JC;JD(h;u;z )\u0002hC(1;0;AB\u00001)\nJB(h)\u0001\u0001\n\u0018Idu\n4\u0019iu\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061u\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061u\u00061\u0001\n\u0000\u0000\nu\u00062\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tB\u00001C\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061u\u00061\u0001\n\u0002Idv\n4\u0019iv\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061v\u00061\u0001\n\u0000(v\u00062)\n\u0002\u0000e\u0000\nA\u00001Bu\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061v\u00061\u0001\nTJD(v)\n\u0002TJB;JC;JD(h;u;z )\u0002hC(1;0;AB\u00001)\nJB(h):\nLet's perform the computation for each term in ( ??) separately. From the\n\frst term we get\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002Idu\n4\u0019iu\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061u\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061u\u00061\u0001\n\u0000\u0000\nu\u00062\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tB\u00001C\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061u\u00061\u0001\n\u0002Idv\n4\u0019iv\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061v\u00061\u0001\n\u0000\u0000\nv\u00062\u0001\n\u0002\u0000e\u0000\nA\u00001Bu\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061v\u00061\u0001\nTJD(v)\n\u0002\u0012p\u0000\n(pq)1\n2q\u00001tB\u00001z\u00001D\u00061\u0001\n\u0012p((pq)1\n2q\u00001tAz\u00001C\u00061)\n\u0012p\u0000\nt\u00004z2\u0001\n\u0012\u0000\nz2\u0001\n\u0002\u0000e\u0000\nq\u00001\n2AB\u00001u\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0002\u0000e\u0000\nt\u00004AB\u00001zu\u00061\u0001\n\u0000e\u0000\npq2t4A\u00001Bz\u00001u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n+\b\nz$z\u00001\t\n:\nTerms cancel and the integral over ureduces to an integral over 6 gamma\nfunctions which can be evaluated as before using the elliptic beta integral\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nNow we glue this as indicated in Fig.\n\\begin{gather*}\nT_{{\\mathfrak J}_D}(v)\\times_v \\big(\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\\times_w C^{(0,0;A^{-1}B)}_{{\\mathfrak J}_B}(w)\\big)\\\\\n\\quad{}\\times_u\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,u,z)\\times_h C^{(1,0;AB^{-1})}_{{\\mathfrak J}_B}(h)\\big)\\big)\\\\\n\\sim \\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}u^{\\pm1}\\big)}{\\Gamma\\big(u^{\\pm2}\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t A D^{\\pm1}u^{\\pm1}\\big) \\\\\n\\quad{}\\times\\oint\\frac{{\\rm d}v}{4\\pi i v} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}v^{\\pm1}\\big)}{\\Gamma(v^{\\pm2})}\\\\\n\\quad{}\\times\\Gamma_e\\big( A^{-1}B u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} v^{\\pm1}\\big)T_{{\\mathfrak J}_D}(v)\\\\\n\\quad{}\\times T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,u,z)\\times_h C^{(1,0;AB^{-1})}_{{\\mathfrak J}_B}(h) .\n\\end{gather*}\nLet's perform the computation for each term in \\eqref{c7} separately. From the first term we get\n\\begin{gather*}\n\\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n\\quad{}\\times\\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}u^{\\pm1}\\big)}{\\Gamma\\big(u^{\\pm2}\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t A D^{\\pm1}u^{\\pm1}\\big)\\nonumber\\\\\n\\quad{}\\times\\oint\\frac{{\\rm d}v}{4\\pi i v} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}v^{\\pm1}\\big)}{\\Gamma\\big(v^{\\pm2}\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big( A^{-1}B u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} v^{\\pm1}\\big)T_{{\\mathfrak J}_D}(v)\\\\\n\\quad{}\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tB^{-1}z^{-1}D^{\\pm1}\\big)\n\\theta_p((pq)^\\frac{1}{2}q^{-1}tAz^{-1}C^{\\pm1})}\n{\\theta_p\\big(t^{-4}z^2\\big)\\theta\\big(z^2\\big)}\\\\\n\\quad{}\\times\\Gamma_e\\big(q^{-\\frac{1}{2}}A B^{-1}u^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}q^\\frac{1}{2}t B D^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}q^\\frac{1}{2}tA^{-1}C^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\\\\\n\\quad{}\\times\\Gamma_e\\big(t^{-4}AB^{-1}zu^{\\pm1}\\big)\\Gamma_e\\big(pq^2 t^4 A^{-1}B z^{-1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}tA^{-1}u^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}t B u^{\\pm1}C^{\\pm1}\\big)\\\\\n\\quad{}+\\big\\{z\\leftrightarrow z^{-1}\\big\\} .\n\\end{gather*}\nTerms cancel and the integral over $u$ reduces to an integral over 6 gamma functions which can be evaluated as before using the elliptic beta integral\n\\end{document} \n"}
{"\u0012p(pq2t2A\u00002B2)\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2q\u00001tB\u00001z\u00001D\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001tAz\u00001C\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0012p\u0000\nt\u00004z2\u0001\n\u0012\u0000\nz2\u0001\n\u0002\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061z\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061z\u00061\u0001\n\u0000\u0000\nz\u00062\u0001\n\u0002\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061z\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061z\u00061\u0001\n\u0000e\u0000\nq\u00001A2B\u00002\u0001\n\u0000e\u0000\npq2t4\u0001\n\u0000e\u0000\nt\u00004A2B\u00002z2\u0001\n\u0002\u0000e\u0000\nzz\u00061\u0001\n\u0000e\u0000\npqt4z\u00002\u0001\n\u0000e\u0000\nq\u00001t\u00004A2B\u00002\u0001\n\u0000e\u0000\nq\u00001z\u00001z\u00061\u0001\n\u0002\u0000e\u0000\npq2\u0001\n\u0000e\u0000\npq2t4A\u00002B2z\u00001z\u00061\u0001\n\u0000e\u0000\nt\u00004zz\u00061\u0001\n\u0000e\u0000\nA\u00002B2\u0001\nTJD(z)\n+\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2q\u00001tB\u00001z\u00001D\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001tAz\u00001C\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0012p\u0000\nt\u00004z2\u0001\n\u0012\u0000\nz2\u0001\n\u0002\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061(qz)\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061(qz)\u00061\u0001\n\u0000\u0000\n(qz)\u00062\u0001\n\u0002\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061(qz)\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061(qz)\u00061\u0001\n\u0000e\u0000\nq\u00001A2B\u00002\u0001\n\u0002\u0000e\u0000\npq2t4\u0001\n\u0000e\u0000\nt\u00004A2B\u00002z2\u0001\n\u0000e\u0000\nz(qz)\u00061\u0001\n\u0000e\u0000\npqt4z\u00002\u0001\n\u0000e\u0000\nq\u00001t\u00004A2B\u00002\u0001\n\u0000e\u0000\n(qz)\u00001(qz)\u00061\u0001\n\u0002\u0000e\u0000\npq2\u0001\n\u0000e\u0000\npq2t4A\u00002B2z\u00001(qz)\u00061\u0001\n\u0000e\u0000\nt\u00004z(qz)\u00061\u0001\n\u0000e\u0000\nA\u00002B2\u0001\nTJD(qz) +\b\nz$z\u00001\t\n;\nwhich simpli\fes to\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0000e\u0000\nq\u00001A2B\u00002\u0001\n\u0000e\u0000\nA\u00002B2\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2t\u00061BD\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061A\u00001C\u00061z\u0001\n\u0012p\u0000\nq\u00001t\u00004\u0001\n\u0012p\u0000\nq\u00001t\u00004A2B\u00002z2\u0001\n\u0012p\u0000\nt\u00004z2\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nq\u00001z2\u0001 TJD(z)\n+\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0000e\u0000\nq\u00001A2B\u00002\u0001\n\u0000e\u0000\nA\u00002B2\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2t\u00001B\u00061D\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00001A\u00061C\u00061z\u0001\n\u0012p\u0000\nqz2\u0001\n\u0012\u0000\nz2\u0001 TJD(qz) +\b\nz$z\u00001\t\n:\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\n\\begin{gather*}\n\\frac{\\theta_p(pq^2t^2A^{-2}B^2)\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tB^{-1}z^{-1}D^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tAz^{-1}C^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}q^\\frac{1}{2}t B D^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)}\n{\\theta_p\\big(t^{-4}z^2\\big)\\theta\\big(z^2\\big)}\\\\\n\\times \\frac{\\Gamma_e\\big((pq)^{\\frac{1}{2}}q^\\frac{1}{2}tA^{-1}C^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}z^{\\pm1}\\big)}{\\Gamma\\big(z^{\\pm2}\\big)}\\\\\n\\times\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} z^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} z^{\\pm1}\\big)\n\\Gamma_e\\big(q^{-1}A^2B^{-2}\\big)\\Gamma_e\\big(pq^2t^4\\big)\\Gamma_e\\big(t^{-4}A^2B^{-2}z^2\\big)\\\\\n\\times\\Gamma_e\\big(zz^{\\pm1}\\big)\\Gamma_e\\big(pqt^4z^{-2}\\big)\\Gamma_e\\big(q^{-1}t^{-4}A^2B{-2}\\big)\\Gamma_e\\big(q^{-1}z^{-1}z^{\\pm1}\\big)\\\\\n\\times\\Gamma_e\\big(pq^2\\big)\\Gamma_e\\big(pq^2t^4A^{-2}B^2z^{-1}z^{\\pm1}\\big)\\Gamma_e\\big(t^{-4}zz^{\\pm1}\\big)\n\\Gamma_e\\big(A^{-2}B^2\\big)T_{{\\mathfrak J}_D}(z)\\\\\n+ \\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tB^{-1}z^{-1}D^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}tAz^{-1}C^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}q^\\frac{1}{2}t B D^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)}\n{\\theta_p\\big(t^{-4}z^2\\big)\\theta\\big(z^2\\big)}\\\\\n\\times\\frac{\\Gamma_e\\big((pq)^{\\frac{1}{2}}q^\\frac{1}{2}tA^{-1}C^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}(qz)^{\\pm1}\\big)\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}(qz)^{\\pm1}\\big)}{\\Gamma\\big((qz)^{\\pm2}\\big)}\\\\\n\\times\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} (qz)^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} (qz)^{\\pm1}\\big)\\Gamma_e\\big(q^{-1}A^2B^{-2}\\big)\\\\\n\\times\\Gamma_e\\big(pq^2t^4\\big)\\Gamma_e\\big(t^{-4}A^2B^{-2}z^2\\big)\\Gamma_e\\big(z(qz)^{\\pm1}\\big)\n\\Gamma_e\\big(pqt^4z^{-2}\\big)\\Gamma_e\\big(q^{-1}t^{-4}A^2B^{-2}\\big)\\Gamma_e\\big((qz)^{-1}(qz)^{\\pm1}\\big)\\\\\n\\times\\Gamma_e\\big(pq^2\\big)\\Gamma_e\\big(pq^2t^4A^{-2}B^2z^{-1}(qz)^{\\pm1}\\big)\\Gamma_e\\big(t^{-4}z(qz)^{\\pm1}\\big)\n\\Gamma_e\\big(A^{-2}B^2\\big)T_{{\\mathfrak J}_D}(qz) + \\big\\{z\\leftrightarrow z^{-1}\\big\\},\n\\end{gather*}\nwhich simplifies to\n\\begin{gather*}\n\\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)\\Gamma_e\\big(q^{-1}A^2B^{-2}\\big)\\Gamma_e\\big(A^{-2}B^2\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n\\quad{}\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}BD^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}A^{-1}C^{\\pm1}z\\big)\n\\theta_p\\big(q^{-1}t^{-4}\\big)\\theta_p\\big(q^{-1}t^{-4}A^2B^{-2}z^2\\big)}\n{\\theta_p\\big(t^{-4}z^2\\big)\\theta_p\\big(z^2\\big)\\theta_p\\big(q^{-1}z^2\\big)}T_{{\\mathfrak J}_D}(z)\\\\\n\\quad{}+ \\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)\\Gamma_e\\big(q^{-1}A^2B^{-2}\\big)\\Gamma_e\\big(A^{-2}B^2\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n\\quad{}\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}B^{\\pm1}D^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}A^{\\pm1}C^{\\pm1}z\\big)}\n{\\theta_p\\big(qz^2\\big)\\theta\\big(z^2\\big)}T_{{\\mathfrak J}_D}(qz) + \\big\\{z\\leftrightarrow z^{-1}\\big\\} .\n\\end{gather*}\n\\end{document} \n"}
{"From the second term of we get\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002Idu\n4\u0019iu\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061u\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061u\u00061\u0001\n\u0000\u0000\nu\u00062\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tB\u00001C\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n\u0002\u0000e\u0000\nAB\u00001u\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tBD\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0000e\u0000\n(pq)1\n2q1\n2tA\u00001C\u00061\u0000\nq1\n2z\u0001\u00061\u0001\n\u0002Idv\n4\u0019iv\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061v\u00061\u0001\n\u0000\u0000\nv\u00062\u0001\n\u0002\u0012p\u0000\n(pq)1\n2q\u00001t\u00003B\u00001zD\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001t\u00003AzC\u00061\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nt4z\u00002\u0001\n\u0002\u0000e\u0000\nA\u00001Bu\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061v\u00061\u0001\nTJD(v) +\b\nz$z\u00001\t\n:\nTerms cancel in the integral over uand what is left can be evaluated using\nthe inversion formula as before which sets v=z, after some cancelations we\nget\n\u0000e\u0000\u0000\nAB\u00001\u00012\u0001\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(qp)1\n2tA\u00001C\u00061z\u0001\n\u0012p\u0000\n(qp)1\n2tBD\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001t\u00003B\u00001zD\u00061\u0001\n\u0012p\u0000\n(pq)1\n2q\u00001t\u00003AzC\u00061\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nt4z\u00002\u0001\n\u0002TJD(z) +\b\nz$z\u00001\t\n:\nWe compute the contribution from the last term in ( ??)\nIdu\n4\u0019iu\u0000e\u0000\n(qp)1\n2t\u00001A\u00061D\u00061u\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061C\u00061u\u00061\u0001\n\u0000\u0000\nu\u00062\u0001\n\u0002\u0000e\u0000\n(pq)1\n2tB\u00001C\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tAD\u00061u\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001u\u00061D\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBu\u00061C\u00061\u0001\n\u0002\u0000e\u0000\nAB\u00001u\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tBD\u00061z\u00061\u0001\n\u0000e\u0000\n(pq)1\n2tA\u00001C\u00061z\u00061\u0001\n\u0002Idv\n4\u0019iv\u0000e\u0000\n(qp)1\n2t\u00001A\u00061C\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2t\u00001B\u00061D\u00061v\u00061\u0001\n\u0000\u0000\nv\u00062\u0001\n\u0002\u0000e\u0000\nA\u00001Bu\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tB\u00001D\u00061v\u00061\u0001\n\u0000e\u0000\n(qp)1\n2tAC\u00061v\u00061\u0001\nTJD(v):\nIntegrals can be evaluated using the inversion formula which sets v=z\nalmost everything cancel and we get\n\u0000e\u0000\u0000\nAB\u00001\u00012\u0001\nTJD(z):\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nFrom the second term of we get\n\\begin{gather*}\n\\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n{}\\times\\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}u^{\\pm1}\\big)}{\\Gamma\\big(u^{\\pm2}\\big)}\\\\\n{}\\times\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}u^{\\pm1}\\big) \\Gamma_e\\big((pq)^\\frac12 t A D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}t A^{-1}u^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}t B u^{\\pm1}C^{\\pm1}\\big)\\\\\n{}\\times\\Gamma_e\\big(AB^{-1}u^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}q^\\frac{1}{2}t B D^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}q^\\frac{1}{2}tA^{-1}C^{\\pm1}\\big(q^\\frac{1}{2}z\\big)^{\\pm1}\\big)\\\\\n{}\\times\\oint\\frac{{\\rm d}v}{4\\pi i v} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}v^{\\pm1}\\big)}{\\Gamma\\big(v^{\\pm2}\\big)}\\\\\n{}\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}t^{-3}B^{-1}z D^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}t^{-3}A z C^{\\pm1}\\big)}\n{\\theta_p\\big(z^2\\big)\\theta_p\\big(t^4z^{-2}\\big)}\\\\\n{}\\times\\Gamma_e\\big( A^{-1}B u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} v^{\\pm1}\\big)T_{{\\mathfrak J}_D}(v) + \\big\\{z\\leftrightarrow z^{-1}\\big\\} .\n\\end{gather*}\nTerms cancel in the integral over $u$ and what is left can be evaluated using the inversion formula as before which sets $v=z$, after some cancelations we get\n\\begin{gather*}\n\\frac{\\Gamma_e\\big(\\big(AB^{-1}\\big)^2\\big)\t\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n{}\\!\\times\\!\\frac{\\theta_p\\big((q p)^{\\frac12 }t A^{-1} C^{\\pm1}z\\big)\n\\theta_p\\big((q p)^{\\frac12 }t B D^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}t^{-3}B^{-1}z D^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}q^{-1}t^{-3}A z C^{\\pm1}\\big)}{\\theta_p\\big(z^2\\big)\\theta_p\\big(t^4z^{-2}\\big)}\\\\\n{}\\!\\times\\! T_{{\\mathfrak J}_D}(z)+ \\big\\{z\\leftrightarrow z^{-1}\\big\\} .\n\\end{gather*}\nWe compute the contribution from the last term in \\eqref{c7}\n\\begin{gather*}\n\\oint\\frac{{\\rm d}u}{4\\pi i u} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} C^{\\pm1}u^{\\pm1}\\big)}{\\Gamma\\big(u^{\\pm2}\\big)}\\\\\n{}\\times\\Gamma_e\\big((pq)^\\frac12 t B^{-1} C^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac12 t A D^{\\pm1}u^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}tA^{-1}u^{\\pm1}D^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^\\frac{1}{2}t B u^{\\pm1}C^{\\pm1}\\big)\\\\\n{}\\times\\Gamma_e\\big(A B^{-1}u^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}t B D^{\\pm1}z^{\\pm1}\\big)\n\\Gamma_e\\big((pq)^{\\frac{1}{2}}t A^{-1}C^{\\pm1}z^{\\pm1}\\big)\\\\\n{}\\times\\oint\\frac{{\\rm d}v}{4\\pi i v} \\frac{\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} A^{\\pm1} C^{\\pm1}v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12 }t^{-1} B^{\\pm1} D^{\\pm1}v^{\\pm1}\\big)}{\\Gamma\\big(v^{\\pm2}\\big)}\\\\\n{}\\times\\Gamma_e\\big( A^{-1}B u^{\\pm1} v^{\\pm1}\\big)\\Gamma_e\\big((qp)^{\\frac12}t B^{-1}D^{\\pm1} v^{\\pm1}\\big)\n\\Gamma_e\\big((q p)^{\\frac12}t A C^{\\pm1} v^{\\pm1}\\big)T_{{\\mathfrak J}_D}(v) .\n\\end{gather*}\nIntegrals can be evaluated using the inversion formula which sets $v=z$ almost everything cancel and we get\n\\begin{gather*}\n\\Gamma_e\\big(\\big(AB^{-1}\\big)^2\\big)T_{{\\mathfrak J}_D}(z) .\n\\end{gather*}\n\\end{document} \n"}
{"Dividing by this factor \u0000 e\u0000\u0000\nAB\u00001\u00012\u0001\nand adding all contributions we get\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0002\u0012p\u0000\n(pq)1\n2t\u00001B\u00061D\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00001A\u00061C\u00061z\u0001\n\u0012p\u0000\nqz2\u0001\n\u0012\u0000\nz2\u0001 TJD(qz)\n+\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(pq)1\n2t\u00061BD\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061A\u00001C\u00061z\u0001\n\u0012p\u0000\nq\u00001t\u00004\u0001\n\u0012p\u0000\nq\u00001t\u00004A2B\u00002z2\u0001\n\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0012p\u0000\nt\u00004z2\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nq\u00001z2\u0001 TJD(z)\n+\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\npq2t4A\u00002B2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0002\u0012p\u0000\n(qp)1\n2tA\u00001C\u00061z\u0001\n\u0012p\u0000\n(qp)1\n2tBD\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t3BD\u00061z\u00001\u0001\n\u0012p\u0000\n(pq)1\n2t3A\u00001C\u00061z\u00001\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nt4z\u00002\u0001\n\u0002TJD(z) +\b\nz$z\u00001\t\n+TJD(z):\nTaking away overall factor of\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0012p\u0000\nq\u00001A2B\u00002\u0001;\nwe get\n\u0012p\u0000\n(pq)1\n2t\u00001B\u00061D\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00001A\u00061C\u00061z\u0001\n\u0012p\u0000\nqz2\u0001\n\u0012\u0000\nz2\u0001 TJD(qz)\n+\u0012p\u0000\nq\u00001t\u00004\u0001\n\u0012p\u0000\nq\u00001t\u00004A2B\u00002z2\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061BD\u00061z\u0001\n\u0012p\u0000\n(pq)1\n2t\u00061A\u00001C\u00061z\u0001\n\u0012p\u0000\nq\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nt\u00004z2\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nq\u00001z2\u0001 TJD(z)\n+\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0012p\u0000\n(pq)1\n2t2BD\u00061(t\u00001z)\u00061\u0001\n\u0012p\u0000\n(pq)1\n2t2A\u00001C\u00061(t\u00001z)\u00061\u0001\n\u0012p\u0000\nq\u00002t\u00004A2B\u00002\u0001\n\u0012p\u0000\nz2\u0001\n\u0012p\u0000\nt4z\u00002\u0001 TJD(z)\n+\b\nz$z\u00001\t\n+\u0012p\u0000\nt\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002B\u00002\u0001\n\u0012p\u0000\nq\u00001t\u00002A2\u0001\n\u0012p\u0000\nq\u00001t\u00002AB\u00001C\u00061D\u00061\u0001\n\u0012p\u0000\nq\u00001A2B\u00002\u0001\n\u0012p\u0000\npq2t2A\u00002B2\u0001\n\u0012p\u0000\n(pq)\u00001q\u00001t\u00004A2B\u00002\u0001 TJD(z):\nTo summarize, we have shown that\nTJD(v)\u0002v\u0000\u0000\nTJB;JC;JD(w;u;v )\u0002wC(0;0;A\u00001B)\nJB(w)\u0001\n\u0002u\u0000\nTJB;JC;JD(h;u;z )\u0002hC(1;0;AB\u00001)\nJB(h)\u0001\u0001\n;\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nDividing by this factor $\\Gamma_e\\big(\\big(A B^{-1}\\big)^2\\big)$ and adding all contributions we get\n\\begin{gather*}\n\\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)\n\\theta_p\\big(q^{-1}A^2B^{-2}\\big)}\\\\\n{}\\times \\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}B^{\\pm1}D^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}A^{\\pm1}C^{\\pm1}z\\big)}{\\theta_p\\big(qz^2\\big)\\theta\\big(z^2\\big)}T_{{\\mathfrak J}_D}(qz)\\\\\n{}+ \\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big) \\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n{}\\times\\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}BD^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}A^{-1}C^{\\pm1}z\\big)\n\\theta_p\\big(q^{-1}t^{-4}\\big)\\theta_p\\big(q^{-1}t^{-4}A^2B^{-2}z^2\\big)}\n{\\theta_p\\big(q^{-1}A^2B^{-2}\\big)\\theta_p\\big(t^{-4}z^2\\big)\\theta_p\\big(z^2\\big)\\theta_p\\big(q^{-1}z^2\\big)}T_{{\\mathfrak J}_D}(z)\\\\\n{}+ \\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\\theta_p\\big(pq^2t^4A^{-2}B^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)}\\\\\n{}\\times\\frac{\\theta_p\\big((q p)^{\\frac12 }t A^{-1} C^{\\pm1}z\\big)\n\\theta_p\\big((q p)^{\\frac12 }t B D^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^3B D^{\\pm1} z^{-1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^3 A^{-1} C^{\\pm1} z^{-1}\\big)}\n{\\theta_p\\big(z^2\\big)\\theta_p\\big(t^4z^{-2}\\big)}\\\\\n{}\\times T_{{\\mathfrak J}_D}(z)\n+ \\big\\{z\\leftrightarrow z^{-1}\\big\\} + T_{{\\mathfrak J}_D}(z) .\n\\end{gather*}\nTaking away overall factor of\n\\begin{gather*}\n \\frac{\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)}\n{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big) \\theta_p\\big(q^{-1}A^2B^{-2}\\big)} ,\n\\end{gather*}\n we get\n\\begin{gather*}\n\\frac{\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}B^{\\pm1}D^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{-1}A^{\\pm1}C^{\\pm1}z\\big)}\n{\\theta_p\\big(qz^2\\big)\\theta\\big(z^2\\big)}T_{{\\mathfrak J}_D}(qz)\\\\\n{}+ \\frac{\\theta_p\\big(q^{-1}t^{-4}\\big)\\theta_p\\big(q^{-1}t^{-4}A^2B^{-2}z^2\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}BD^{\\pm1}z\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^{\\pm1}A^{-1}C^{\\pm1}z\\big)}{\\theta_p\\big(q^{-2}t^{-4}A^{2}B^{-2}\\big)\n\\theta_p\\big(t^{-4}z^2\\big)\\theta_p\\big(z^2\\big)\\theta_p\\big(q^{-1}z^2\\big)}T_{{\\mathfrak J}_D}(z)\\\\\n{}+ \\frac{\\theta_p\\big(q^{-1}A^2B^{-2}\\big)\\theta_p\\big((pq)^\\frac{1}{2}t^2B D^{\\pm1} (t^{-1}z)^{\\pm1}\\big)\n\\theta_p\\big((pq)^\\frac{1}{2}t^2 A^{-1} C^{\\pm1} (t^{-1}z)^{\\pm1}\\big)}\n{\\theta_p\\big(q^{-2}t^{-4}A^{2}B^{-2}\\big)\\theta_p\\big(z^2\\big)\\theta_p\\big(t^4z^{-2}\\big)}T_{{\\mathfrak J}_D}(z)\\\\\n{}+ \\big\\{z\\leftrightarrow z^{-1}\\big\\}\\nonumber\\\\\n{}+ \\frac{\\theta_p\\big(t^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}B^{-2}\\big)\\theta_p\\big(q^{-1}t^{-2}A^2\\big)\n\\theta_p\\big(q^{-1}t^{-2}AB^{-1}C^{\\pm1}D^{\\pm1}\\big)\n\\theta_p\\big(q^{-1}A^2B^{-2}\\big)} {\\theta_p\\big(pq^2t^2A^{-2}B^2\\big)\n\\theta_p\\big((pq)^{-1}q^{-1}t^{-4}A^2B^{-2}\\big)} T_{{\\mathfrak J}_D}(z) .\n\\end{gather*}\nTo summarize, we have shown that\n\\begin{gather*}\nT_{{\\mathfrak J}_D}(v)\\times_v \\big(\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(w,u,v)\\times_w C^{(0,0;A^{-1}B)}_{{\\mathfrak J}_B}(w)\\big)\\\\\n\\qquad{} \\times_u\\big(T_{{\\mathfrak J}_B, {\\mathfrak J}_C,{\\mathfrak J}_D}(h,u,z)\\times_h C^{(1,0;AB^{-1})}_{{\\mathfrak J}_B}(h)\\big)\\big) ,\n\\end{gather*}\n\\end{document} \n"}
{"is proportional to\nDJB;(1;0;AB\u00001)\nJDTJD(z);\nwhere DJB;(1;0;AB\u00001)\nJDis given by ().\n1 Koornwinder limit of the trinion\nWe compute here the index of the three punctured sphere in the Koorn-\nwinder limit. The only subtle issue is the evaluation of the integrals in the\nfunction H(??). The integral can be interpreted as index of two coupled\nSU(2) gauge theories with \fve \navors. The integrand does not have a good\nlimit however by manipulating it using Seiberg dualities we can show that\nthe limit is well de\fned and evaluate all the integrals. We will discuss the\nevaluation of this function here. To evaluate Hwe \frst perform Seiberg du-\nality on w2splitting to \fve \navors in particular way, leading to SU(3) theory\nwith \fve \navors\n\u0000\n(qp)1\n4A\u00001\n2tbz\u00061\n1;(qp)1\n4A1\n2a\u00001tv\u00061\n1;(qp)1\n4A\u00001\n2b\u00001z\u00001\n2\u0001\n;\n\u0000\n(qp)1\n21\nt2w\u00061\n1;(qp)1\n4A1\n2av\u00061\n2;(qp)1\n4A\u00001\n2b\u00001z2\u0001\n;\nwhich leads to no w2\navors having negative powers of p. We have the mesons\n(qp)3\n4A\u00001\n2t\u00001bz\u00061\n1w\u00061\n1;(qp)3\n4A1\n2a\u00001t\u00001w\u00061\n1v\u00061\n1;(qp)3\n4t\u00002A\u00001\n2b\u00001z\u00001\n2w\u00061\n1;\n(qp)1\n2batz\u00061\n1v\u00061\n2;(qp)1\n2tAv\u00061\n2v\u00061\n1;(qp)1\n2A\u00001b\u00002;\n(qp)1\n2A\u00001z2tz\u00061\n1;(qp)1\n2ab\u00001v\u00061\n2z\u00001\n2;(qp)1\n2tb\u00001a\u00001z2v\u00061\n1;\nand the new charged \felds are\n\u0003SU(3) w2: (pq)1\n6A1\n3b\u00002\n3t1\n3a\u00002\n3z\u00001\n3\n2z\u00061\n1;(pq)1\n6A\u00002\n3b1\n3t1\n3a1\n3z\u00001\n3\n2v\u00061\n1;(qp)1\n6A1\n3b4\n3t4\n3a\u00002\n3z2\n3\n2;\n\u0003SU(3) w2: (qp)1\n12t2\n3A1\n6a2\n3b\u00001\n3z1\n3\n2w\u00061\n1;(qp)1\n3A\u00001\n3b\u00001\n3t\u00004\n3a\u00001\n3z1\n3\n2v\u00061\n2;(qp)1\n3A2\n3b2\n3t\u00004\n3a2\n3z\u00002\n3\n2;\nand all of these \felds have a good limit when pgoes to zero and the fugacities\nare scaled. We note that some of the mesons charged under w1form mass\nterms with some of the quarks and after the \frst Seiberg duality we have\nfour \navors of w1\n\u0000\n(qp)1\n4A1\n2bz2;(qp)1\n12A1\n6t2\n3a2\n3b\u00001\n3z1\n3\n2\u0000\nwj\n2\u0001\u00001\u0001\n;\n\u0000\n(qp)1\n4A1\n2bz\u00001\n2;(qp)1\n4A\u00001\n2a\u00001v\u00061\n2;(qp)3\n4A\u00001\n2t\u00002b\u00001z\u00001\n2\u0001\n;\n1": "\\documentclass[a4paper,12pt]{article}\n\\usepackage{latexsym}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\pagestyle{plain}\n\\usepackage{fancybox}\n\\usepackage{bm}\n\\begin{document}\nis proportional to\n\\begin{gather*}\n {\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;AB^{-1})} T_{{\\mathfrak J}_D}(z) ,\n\\end{gather*}\nwhere ${\\mathfrak D}_{{\\mathfrak J}_D}^{{\\mathfrak J}_B,(1,0;AB^{-1})}$ is given by~().\n\\section{Koornwinder limit of the trinion}\nWe compute here the index of the three punctured sphere in the Koornwinder limit. The only subtle issue is the evaluation of the integrals in the function~$H$~\\eqref{gff}. The integral can be interpreted as index of two coupled ${\\rm SU}(2)$ gauge theories with five flavors. The integrand does not have a good limit however by manipulating it using Seiberg dualities we can show that the limit is well defined and evaluate all the integrals. We will discuss the evaluation of this function here.\nTo evaluate $H$ we first perform Seiberg duality on $w_2$ splitting to five flavors in particular way, leading to ${\\rm SU}(3)$ theory with five flavors\n\\begin{gather*}\n\\big((qp)^{\\frac14}A^{-\\frac12} t b z_1^{\\pm1} , (qp)^{\\frac14} A^{\\frac12} a^{-1} t v_1^{\\pm1} , (q p)^{\\frac14} A^{-\\frac12} b^{-1} z_2^{-1}\\big) , \\\\\n\\big((q p)^{\\frac12}\\frac1{t^2} w_1^{\\pm1} , (q p)^{\\frac14} A^{\\frac12}a v_2^{\\pm1} , (q p)^{\\frac14}A^{-\\frac12}b^{-1} z_2\\big) ,\n\\end{gather*} which leads to no $w_2$ flavors having negative powers of $p$. We have the mesons\n\\begin{gather*}\n(q p)^{\\frac34}A^{-\\frac12}t^{-1} b z_1^{\\pm1}w_1^{\\pm1} , \\ (q p)^{\\frac34} A^{\\frac12} a^{-1} t^{-1} w_1^{\\pm1}v_1^{\\pm1} , \\ (q p)^{\\frac34} t^{-2} A^{-\\frac12} b^{-1} z_2^{-1} w_1^{\\pm1} , \\\\\n (q p)^{\\frac12} b a t z_1^{\\pm1} v_2^{\\pm1} , \\ (q p)^{\\frac12} t A v_2^{\\pm1} v_1^{\\pm1} , \\ (q p)^{\\frac12} A^{-1} b^{-2} , \\\\\n (q p) ^{\\frac12} A^{-1} z_2 t z_1^{\\pm1} , \\ (q p)^{\\frac12} a b^{-1} v_2^{\\pm1} z_2 ^{-1} , \\ (q p)^{\\frac12} t b^{-1} a^{-1} z_2 v_1^{\\pm1} , \\end{gather*}\nand the new charged fields are\n\\begin{gather*}\n\\square_{{\\rm SU}(3)_{w_2}} \\colon \\ (p q)^{\\frac16}A^{\\frac13}b^{-\\frac23} t^{\\frac13}a^{-\\frac23} z_2^{-\\frac13} z_1^{\\pm1} , \\ (p q)^{\\frac16}A^{-\\frac23}b^{\\frac13} t^{\\frac13}a^{\\frac13} z_2^{-\\frac13} v_1^{\\pm1} , \\\n( q p)^{\\frac16} A^{\\frac13} b^{\\frac43} t^{\\frac43} a^{-\\frac23} z_2^{\\frac23} , \\\\\n\\overline \\square _{{\\rm SU}(3)_{w_2 } } \\colon \\ (q p)^{\\frac1{ 1 2} } t^{\\frac23}A^{\\frac16}a^{\\frac23} b^{-\\frac13}z_2^{\\frac13}w_1^{\\pm1} , \\\n(q p) ^{\\frac13} A^{-\\frac13} b^{-\\frac13} t^{-\\frac43} a^{-\\frac13} z_2^{\\frac13} v_2^{\\pm1} , \\ ( q p )^{\\frac13} A^{\\frac23} b^{\\frac23} t^{-\\frac43} a^{\\frac23} z_2^{-\\frac23} ,\n\\end{gather*}\n and all of these fields have a good limit when $p$ goes to zero and the fugacities are scaled. We note that some of the mesons charged under $w_1$ form mass terms with some of the quarks and after the first Seiberg duality we have four flavors of $w_1$\n\\begin{gather*}\n \\big( (q p )^{\\frac14} A^{\\frac12}b z_2 , (q p) ^{\\frac1{ 1 2 }} A^{\\frac16} t^{\\frac23} a^{\\frac23} b^{-\\frac13} z_2^{\\frac13} \\big(w_2^j\\big)^{-1} \\big) , \\\\\n \\big((q p)^{\\frac14} A^{\\frac12} b z_2^{-1} , (q p)^{\\frac14} A^{-\\frac12} a^{-1} v_2^{\\pm1} , (q p)^{\\frac34} A^{-\\frac12} t^{-2} b^{-1} z_2^{-1} \\big) ,\n\\end{gather*}\n\\end{document} \n"}
{"questioning the presence of ( i;j) in each structure. By construction, the\ndistribution of Xi;jis given by the base pairing probability P(Xi;j(s) = 1) =pi;j.\nAny base pair, ( i;j), has an entropy , de\fned by the information entropy of Xi;j,\ni.e.\nH(Xi;j) =\u0000pi;jlog2pi;j\u0000(1\u0000pi;j) log2(1\u0000pi;j);\nwhere the units of Hare in bits. The entropy H(Xi;j) measures the uncertainty\nof the base pair ( i;j) in \n . When a base pair ( i;j) is certain to either exist\nor not, its entropy H(Xi;j) is 0. However, in case pi;jis closer to 1 =2,H(Xi;j)\nbecomes larger. The r.v. Xi;jpartitions the space \n into two disjoint sub-\nspaces \n 0and \n 1, where \n k=fs2\n :Xi;j(s) =kg(k= 0;1), and the\ninduced distributions are given by\np0(s) =p(s)\n1\u0000pi;jfors2\n0; p 1(s) =p(s)\npi;jfors2\n1:\nIntuitively, H(Xi;j) quanti\fes the average bits of information we would expect\nto gain about the ensemble when querying a base pair ( i;j) This motivates\nus to consider the maximum entropy base pairs , the base pair ( i0;j0) having\nmaximum entropy among all base pairs in \n, i.e.\n(i0;j0) = argmax\n(i;j)H(Xi;j):\nAs we shall prove in Section, Xi0;j0produces maximally balanced splits.\n0.1. The ensemble tree\nEquipped with the notion of ensemble and bit query (i.e. the respective max-\nimum entropy base pairs), we proceed by describing our strategy to identify the\ntarget structure as speci\fed in Problem. The \frst step consists in having a\ncloser look at the space of ensemble reductions. Each split obtained by parti-\ntioning the ensemble \n using r.v. Xi;j, can in turn be bipartitioned itself via\nany of its maximum entropy base pairs. This recursive splitting induces the\nensemble tree ,T(\n), whose vertices are sub-samples and in which its k-th layer\nrepresents a partition of the original ensemble into 2kblocks.T(\n), is a rooted\nbinary tree, in which\n\u000fi;j((x1;:::;x j);(y1;:::;y m)) = (y1;:::;y i\u00001;x1;:::;x j;yi+1;:::y m)\n\u0018i;j(x1;:::;x n) = ((xi;:::;x j);(x1;:::;x i\u00001;xj+1;:::;x n)):\nTo quantify to what extent modularity can discriminate base pairs, we perform\ncomputational experiments on random sequences via splittings. For each se-\nquence, we consider its MFE structure scomputed via ViennaRNA (?). Given\ntwo positions iandj, we cut the entire sequence xinto two fragments, xi;jand\nthe remainder \u0016xi;j, i.e.,\u0018i;j(x) = (xi;j;\u0016xi;j). Subsequently, the two fragments\nxi;jand\u0016xi;jrefold into their MFE structures si;jand \u0016si;j, respectively, which\nare combined into a structure \u000fi;j(si;j;\u0016si;j). If basesiandjare paired in s\n1": "\\documentclass[preprint,authoryear]{elsarticle}\n\\usepackage{amssymb}\n\\journal{}%Mathematical Biosciences}\n\\usepackage{graphicx}\n\\usepackage{amsmath,amssymb,amsfonts,latexsym}\n\\usepackage{color}\n\\usepackage{eucal}%    caligraphic-euler fonts: \\mathcal{ }\n\\usepackage{xspace}\n\\usepackage{hyperref}\n\\usepackage{bm}\n\\usepackage[title]{appendix}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{proof}{Proof}\n\\usepackage[round]{natbib}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\\DeclareMathOperator*{\\argmax}{argmax} % no space, limits underneath in displays\n\\DeclareMathOperator*{\\argmin}{argmin} % no space, limits underneath in displays\n\\begin{document}\nquestioning the presence of  $(i,j)$ in each structure.\nBy construction, the distribution of $X_{i,j}$ is given by the base pairing probability \n$\\mathbb{P}(X_{i,j}(s)=1)=p_{i,j}$.\nAny base pair, $(i,j)$, has an \\emph{entropy}, defined by the information entropy of\n$X_{i,j}$, i.e. \n\\[\nH(X_{i,j})=-p_{i,j}  \\log_2 p_{i,j}  -(1-p_{i,j} )  \\log_2 (1-p_{i,j} ),\n\\]\nwhere the units of $H$ are in bits. The entropy $H(X_{i,j}) $ measures the uncertainty of  the base\npair $(i,j)$ in $\\Omega$ . When a base pair $(i,j)$  is certain to either exist or not, its entropy\n$H(X_{i,j})$ is $0$. However, in case $p_{i,j} $ is closer to $1/2$, $H(X_{i,j})$ becomes larger.\nThe r.v. $X_{i,j}$ partitions the space $\\Omega$ into two disjoint sub-spaces $\\Omega_0$ and\n$\\Omega_1$, where  $ \\Omega_k=\\{s\\in  \\Omega :X_{i,j}(s)=k\\}$ ($k=0,1$), \nand the induced distributions are given by\n\\[\np_0(s)=\\frac{p(s)}{1-p_{i,j}} \\quad \\text{ for } s\\in  \\Omega_0,\\qquad\np_1(s)=\\frac{p(s)}{p_{i,j}} \\quad \\text{ for } s\\in  \\Omega_1.\n\\]\nIntuitively, $H(X_{i,j}) $ quantifies the average bits of information\nwe would expect to gain about the ensemble when querying a   base pair $(i,j)$  \nThis motivates  us to consider the \\emph{maximum entropy base pairs}, the base pair $(i_0,j_0)$\nhaving  maximum entropy among all base pairs in $\\Omega$, i.e. \n\\[\n(i_0,j_0)= \\argmax_{(i,j)}  H(X_{i,j}).\n\\]\nAs we shall prove in Section, $X_{i_0,j_0}$ produces maximally balanced\nsplits.\n        \n\\subsection{The ensemble tree}\nEquipped with the notion of ensemble and bit query (i.e.~the respective maximum entropy base pairs),\nwe proceed by describing our strategy to identify the target structure as specified in Problem.\nThe first step consists in having a closer look at the space of ensemble reductions.\nEach split obtained by partitioning the ensemble $\\Omega$ using r.v. $X_{i,j}$, can in turn be\nbipartitioned itself via any of its maximum entropy base pairs. This recursive splitting induces the\n{\\it ensemble  tree}, $T(\\Omega)$, whose vertices are sub-samples and in which its $k$-th layer represents\na partition of the original ensemble into $2^k$ blocks. $T(\\Omega)$, is a rooted binary tree, in which\n\\begin{align*}\n\\epsilon_{i,j}((x_1,\\dots, x_j),(y_1,\\dots,y_m)) & =  (y_1,\\dots,y_{i-1},x_1,\\dots,x_j,y_{i+1},\\dots y_m)\\\\\n\\xi_{i,j}(x_1,\\dots,x_{n}) & =  ((x_i,\\dots,x_{j}),(x_1,\\dots,x_{i-1},x_{j+1},\\dots,x_{n})).\n\\end{align*}\nTo quantify to what extent modularity can discriminate base pairs, % hypothesis\nwe perform computational experiments on random sequences via splittings. For each sequence, we consider\nits MFE structure $s$ computed via \\textsf{ViennaRNA}~\\citep{Lorenz:11}.\nGiven two positions $i$ and $j$, we cut the entire sequence $\\mathbf{x}$  into two fragments,\n$\\mathbf{x}_{i,j}$ and the remainder  $\\bar{\\mathbf{x}}_{i,j}$, \n\ti.e., $ \\xi_{i,j}(\\mathbf{x})=(\\mathbf{x}_{i,j},\\bar{\\mathbf{x}}_{i,j})$.\nSubsequently, the two fragments $\\mathbf{x}_{i,j}$ and  $\\bar{\\mathbf{x}}_{i,j}$ refold\ninto their MFE structures $s_{i,j}$  and   $\\bar{s}_{i,j}$, respectively,\nwhich are combined into a structure $\\epsilon_{i,j}(s_{i,j}, \\bar{s}_{i,j})$.\nIf bases $i$ and $j$ are paired in $s$\n\\end{document}\n"}
{"such a splitting is referred to as modular In our R\u0013 enyi-Ulam game variation,\nthe expected values of FDR and FOR are the error rates e1ande0in case the\ntruthful answer being yes and no, respectively. Fig. displays the error rates e0\nande1as functions of \u0012. For\u0012= 31, we compute e0\u00190:052 ande1\u00190:007,\ni.e. we have an error rate of 0 :052 for rejecting and an error rate of 0 :007 for\ncon\frming a base pair.\n0.1. Entropy\nTo quantify the uncertainty of an ensemble, we de\fne the structural entropy\nof an ensemble, \n, of an RNA sequence, x, as the Shannon entropy\nH(\n) =\u0000X\ns2\np(s) log2p(s);\nProposition implies that a sample with small structural entropy contains a dis-\ntinguished structure and a proof is given in . We refer to a sample having a\ndistinguished structure of probability at least \u0015as being\u0015-distinguished .\nNext we quantify the reduction of a bit query on an ensemble. Recall that the\nassociated r.v. Xi;jof a base pair ( i;j) partitions the sample \n into two disjoint\nsub-samples \n 0and \n 1, where \n k=fs2\n :Xi;j(s) =kg(k= 0;1). The\nconditional entropy ,H(\njXi;j), represents the expected value of the entropies\nof the conditional distributions on \n, averaged over the conditioning r.v. Xi;j\nand can be computed by\nH(\njXi;j) = (1\u0000pi;j)H(\n0) +pi;jH(\n1):\nThen the entropy reduction R(\n;Xi;j) ofXi;jon \n is the di\u000berence between\nthea priori Shannon entropy H(\n) and the conditional entropy H(\njXi;j), i.e.\nR(\n;Xi;j) =H(\n)\u0000H(\njXi;j):\nP(s2\n\u0003) = (1\u0000e0)l0(1\u0000e1)l1, wherel0andl1denote the number of No-/Yes-\nanswers to queried base pairs along the path, respectively. Fig. displays the\ndistribution of l1. We observe that l1has a mean around 5, i.e., the probabilities\nof queried base pairs being con\frmed and being rejected are roughly equal.\nForl0=l1= 5, we have a theoretical estimate P(s2\n\u0003)\u00190:736. In Fig.\nwe present that P(s2\n\u0003) decreases as the error rate e0increases, for \fxed\ne1= 0:01.\n1. Information theory\nAs the Boltzmann ensemble is a particular type of discrete probability spaces,\nthe information-theoretic results on the ensemble trees will be stated in the more\ngeneral setup. Let \n = ( S;P(S);p) be a discrete probability space consisting\nof the sample space S, its power set P(S) as the\u001b-algebra and the probability\nmeasurep. The Shannon entropy of \n is given by\nH(\n) =\u0000X\ns2Sp(s) log2p(s);\n1": "\\documentclass[preprint,authoryear]{elsarticle}\n\\usepackage{amssymb}\n\\journal{}%Mathematical Biosciences}\n\\usepackage{graphicx}\n\\usepackage{amsmath,amssymb,amsfonts,latexsym}\n\\usepackage{color}\n\\usepackage{eucal}%    caligraphic-euler fonts: \\mathcal{ }\n\\usepackage{xspace}\n\\usepackage{hyperref}\n\\usepackage{bm}\n\\usepackage[title]{appendix}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{proof}{Proof}\n\\usepackage[round]{natbib}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\\DeclareMathOperator*{\\argmax}{argmax} % no space, limits underneath in displays\n\\DeclareMathOperator*{\\argmin}{argmin} % no space, limits underneath in displays\n\\begin{document}\nsuch a splitting is referred to as \\emph{modular}\nIn our R\\'{e}nyi-Ulam game variation, the expected values of FDR and FOR are the error rates $e_1$\nand $e_0$ in case the truthful answer being yes and no, respectively.\nFig. displays the error rates $e_0$ and $e_1$ as functions of $\\theta$.\nFor $\\theta=31$, we compute $e_0\\approx 0.052$ and $e_1 \\approx 0.007$, i.e. ~we have an error rate\nof $ 0.052$ for rejecting and an error rate of $0.007$ for confirming a base pair.\n\\subsection{Entropy}\nTo quantify the uncertainty of an ensemble, we define \nthe \\emph{structural entropy} of an ensemble, $\\Omega$, of an RNA sequence, $\\mathbf{x}$,\nas the Shannon entropy\n\\begin{equation*}\nH(\\Omega) = -\\sum_{s\\in \\Omega} p(s) \\log_2 p(s),\n\\end{equation*}\nProposition implies that a sample with small structural entropy contains a\ndistinguished structure and a proof is given in .\nWe refer to a sample having a distinguished structure of probability at least $\\lambda$ as\nbeing \\emph{$\\lambda$-distinguished}.\n   \nNext we quantify the reduction of a bit query on an ensemble.\nRecall that the associated r.v. $X_{i,j}$  of a base pair $(i,j)$ \npartitions the sample $\\Omega$ into two disjoint sub-samples $\\Omega_0$ and\n$\\Omega_1$, where  $ \\Omega_k=\\{s\\in  \\Omega :X_{i,j}(s)=k\\}$ ($k=0,1$).\nThe \\emph{conditional entropy}, $H(\\Omega|X_{i,j})$,  \nrepresents \nthe expected value of the entropies of the conditional distributions on $\\Omega$,\naveraged over the conditioning r.v. $X_{i,j}$ and can be\ncomputed by \n\\begin{equation*}\nH(\\Omega|X_{i,j})= (1-p_{i,j}) H(\\Omega_0)+ p_{i,j} H(\\Omega_1).\n\\end{equation*}\nThen the \\emph{entropy reduction} $R(\\Omega,X_{i,j})$ of  $X_{i,j}$ on $\\Omega$ \nis the difference between the \\textit{a priori} Shannon entropy $H(\\Omega)$ and the conditional\nentropy $H(\\Omega|X_{i,j}) $, i.e. \n\\begin{equation*}\nR(\\Omega,X_{i,j})= H(\\Omega)-H(\\Omega|X_{i,j}). \n\\end{equation*}\n$\\mathbb{P}(s\\in \\Omega^* )=(1-e_0)^{l_0} (1-e_1)^{l_1}$,\nwhere $l_0$ and  $l_1$ denote the number of No-/Yes-answers to queried base pairs along the path,\nrespectively.  \nFig. displays the distribution of $l_1$. We observe that $l_1$ has\na mean around $5$,\ni.e., the probabilities of queried base pairs being confirmed and being rejected\nare roughly equal.\nFor $l_0=l_1=5$,  we have a theoretical estimate  $\\mathbb{P}(s\\in \\Omega^* )\\approx 0.736$.\nIn Fig.  we present  that $\\mathbb{P}(s\\in \\Omega^* )$ decreases \nas  the error rate $e_0$ increases, for fixed $e_1=0.01$.\n\\section{Information theory}\nAs the Boltzmann ensemble is a particular type of discrete probability spaces,\nthe information-theoretic results on the ensemble trees will be stated in the more general setup.\nLet  $\\Omega=(\\mathcal{S},\\mathcal{P}(\\mathcal{S}),p)$ be a discrete probability space\nconsisting of the sample space $\\mathcal{S}$, its power set $\\mathcal{P}(\\mathcal{S})$ as the $\\sigma$-algebra\nand the probability measure $p$.\nThe \\emph{Shannon entropy} of $\\Omega$ is given by \n\\begin{equation*}\nH(\\Omega) = -\\sum_{s\\in \\mathcal{S}} p(s) \\log_2 p(s),\n\\end{equation*}\n\\end{document}\n"}
{"where the units of Hare in bits. A feature Xis a discrete random variable\nde\fned on \n. Assume that Xhas a \fnite number of values x1; x2; : : : ; x k. Set\nqi=P(X=xi). The Shannon entropy H(X) of the feature Xis given by\nH(X) =\u0000X\niqilog2qi:\nIn particular, the values of Xde\fne a partition of Sinto disjoint subsets Si=\nfs2S:X(s) =xig, for 1\u0014i\u0014k. This further induces kspaces \n i=\n(Si;P(Si); pi), where the induced distribution is given by\npi(s) =p(s)\nqifors2Si;\nandqidenotes the probability of Xhaving value xiand is given by\nqi=P(X=xi) =X\ns2Sip(s):\nLetH(\njX) denote the conditional entropy of \n given the value of feature X.\nThe entropy H(\njX) gives the expected value of the entropies of the condi-\ntional distributions on \n, averaged over the conditioning feature Xand can be\ncomputed by\nH(\njX) =X\niqiH(\ni):\nThen the entropy reduction R(\n; X) of \n for feature Xis the di\u000berence between\nthea priori Shannon entropy H(\n) and the conditional entropy H(\njX), i.e.\nR(\n; X) =H(\n)\u0000H(\njX):\nThe entropy reduction indicates the change on average in information entropy\nfrom a prior state to a state that takes some information as given. Now we\nprove Propositions and. Given two features X1andX2, we can partition \neither \frst by X1and subsequently by X2, or \frst by X2and then by X1, or\njust by a pair of features ( X1; X2). In the following, we will show that all three\napproaches provide the same entropy reduction of \n. Before the proof, we de\fne\nsome notations. The joint probability distribution of a pair of features ( X1; X2)\nis given by qi1;i2=P(X1=x(1)\ni1; X2=x(2)\ni2), and the marginal probability\ndistributions are given by q(1)\ni1=P(X1=x(1)\ni1) and q(2)\ni2=P(X2=x(2)\ni2). Clearly,P\ni1qi1;i2=q(2)\ni2andP\ni2qi1;i2=q(1)\ni1. The joint entropy H(X1; X2) of a pair\n(X1; X2) is de\fned as\nH(X1; X2) =\u0000X\ni1X\ni2qi1;i2log2qi1;i2:\nThe conditional entropy H(X2jX1) of a feature X2given X1is de\fned as the\nexpected value of the entropies of the conditional distributions X2, averaged\n1": "\\documentclass[preprint,authoryear]{elsarticle}\n\\usepackage{amssymb}\n\\journal{}%Mathematical Biosciences}\n\\usepackage{graphicx}\n\\usepackage{amsmath,amssymb,amsfonts,latexsym}\n\\usepackage{color}\n\\usepackage{eucal}%    caligraphic-euler fonts: \\mathcal{ }\n\\usepackage{xspace}\n\\usepackage{hyperref}\n\\usepackage{bm}\n\\usepackage[title]{appendix}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{proof}{Proof}\n\\usepackage[round]{natbib}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\\DeclareMathOperator*{\\argmax}{argmax} % no space, limits underneath in displays\n\\DeclareMathOperator*{\\argmin}{argmin} % no space, limits underneath in displays\n\\begin{document}\nwhere the units of $H$ are\nin bits.\nA \\emph{feature} $X$ is a discrete random variable defined on $\\Omega$. \nAssume that $X$ has a finite number of values $x_1,x_2,\\ldots,x_k$. Set $q_i=\\mathbb{P}(X=x_i)$. \nThe \\emph{Shannon entropy} $H(X)$  of the feature $X$ is given by\n\\begin{equation*}\nH(X) = -\\sum_{i} q_i \\log_2 q_i.\n\\end{equation*}\nIn particular, the values of $X$ define a partition of  $ \\mathcal{S}$ %training set data  \ninto disjoint subsets $ \\mathcal{S}_i=\\{s\\in  \\mathcal{S}:X(s)=x_i\\}$, for $1\\leq i \\leq k$.\nThis further induces $k$ spaces $\\Omega_i=(\\mathcal{S}_i,\\mathcal{P}(\\mathcal{S}_i),p_i)$,\nwhere the induced distribution is given by\n\\[\np_i(s)=\\frac{p(s)}{q_i} \\quad \\text{ for } s\\in  \\mathcal{S}_i,\n\\]\nand $q_i$ denotes the probability of $X$ having value $x_i$ and is given by\n\\begin{equation*}\nq_i =\\mathbb{P}(X=x_i)= \\sum_{s\\in  \\mathcal{S}_i} p(s).\n\\end{equation*}\nLet $H(\\Omega|X) $ denote the \\emph{conditional entropy} of $\\Omega$ given the value of feature $X$. \nThe entropy $H(\\Omega|X) $ gives the expected value of the entropies of the conditional distributions on $\\Omega$, \naveraged over the conditioning feature $X$ and  can be computed by \n\\begin{equation*}\nH(\\Omega|X)= \\sum_i q_i H(\\Omega_i).\n\\end{equation*}\nThen the \\emph{entropy reduction} %(or information gain) \n$R(\\Omega,X)$ of $\\Omega$ for feature $X$ \nis the difference between the \\textit{a priori} Shannon entropy $H(\\Omega)$ and the conditional entropy $H(\\Omega|X) $, i.e. \n\\begin{equation*}\nR(\\Omega,X)= H(\\Omega)-H(\\Omega|X). \n\\end{equation*}\nThe entropy reduction indicates the change on average in information entropy from a prior state to a state that takes some information as given.\nNow we prove Propositions and. % i.e.  $R(\\Omega,X)=H(X)$.\nGiven two features $X_1$ and $X_2$, we can  partition $\\Omega$ either   first by $X_1$ and  subsequently  by  $X_2$,\nor  first by $X_2$ and then  by  $X_1$, or just by  a pair of features $(X_1,X_2)$.  \nIn the following, we will show that all three approaches provide the same entropy reduction of $\\Omega$.\nBefore the proof, we define some notations. \nThe joint probability distribution of  a pair of features $(X_1,X_2)$ is given by \n$q_{i_1,i_2}= \\mathbb{P}(X_1=x^{(1)}_{i_1},X_2=x^{(2)}_{i_2})$, \nand the marginal probability distributions are given by\n$q^{(1)}_{i_1}=\\mathbb{P}(X_1=x^{(1)}_{i_1})$ and $q^{(2)}_{i_2}=\\mathbb{P}(X_2=x^{(2)}_{i_2})$.\nClearly, $\\sum_{i_1}q_{i_1,i_2}=q^{(2)}_{i_2} $ and $\\sum_{i_2}q_{i_1,i_2}=q^{(1)}_{i_1} $.\nThe \\emph{joint entropy} $H (X_1,X_2) $ of a pair  $(X_1,X_2)$ is defined as\n\\begin{equation*}\nH (X_1,X_2)= -\\sum_{i_1} \\sum_{i_2}  q_{i_1,i_2}  \\log_2 q_{i_1,i_2}.\n\\end{equation*}\nThe \\emph{conditional entropy} $H(X_2|X_1)$ of a feature $X_2$ given $X_1$ is \ndefined as the expected value of  the entropies of  the conditional distributions $X_2$, \naveraged\n\\end{document}\n"}
{"over the conditioning feature X1, i.e.\nH(X2jX1) =X\ni1P(X1=x(1)\ni1)H(X2jX1=x(1)\ni1):\nProposition 1 (Chain rule,).\nH(X1; X2) =H(X1) +H(X2jX1): (1)\nProposition 2. LetR(\n; X1; X2)denote the entropy reduction of\n\frst by the\nfeature X1and then by the feature X2, and R(\n;(X1; X2))denote the entropy\nreduction of\nby a pair of features (X1; X2). Then\nR(\n; X1; X2) =R(\n;(X1; X2)): (2)\nThe maximum entropy of an arbitrary feature is achieved when all its outcomes\noccur with equal probability, and this maximum value is proportional to the\nlogarithm of the number of possible outcomes to the base 2. Thus Proposition\nimplies that the more possible outcomes a feature has, the higher entropy reduc-\ntion it could possibly lead to. Meanwhile, a feature with an arbitrary number of\noutcomes can be viewed as a combination of binary features , the ones with two\npossible outcomes. Even though the entropy of the combination of two features\nis greater than each of them, Proposition shows that partitioning the space sub-\nsequently by two features has the same entropy reduction as partitioning by\ntheir combination. Therefore, instead of considering features with outcomes as\nmany as possible, we focus on binary features.\n1. Query repeats\nHere we assess the improvement of the error rates by repeating the same\nquery twice. Let Y(orN) denote the event of the queried base pair existing\n(or not) in the target structure. Let y(orn) denote the event of the experi-\nment con\frming (or rejecting) the base pair. Let nndenote the event of two\nindependent experiments both rejecting the base pair. Similarly, we have yy\nandyn. Utilizing the same sequences and structures as described in Fig., we\nestimate the conditional probabilities P(njN)\u00190:993 and P(njY)\u00190:055. The\nprior probability P(Y) can be computed via the expected number l1of con\frmed\nqueried base pairs on the path, divided by the number of queries in each sam-\nple. Fig. displays the distribution of l1having mean around 5. Thus we adopt\nP(Y) =P(N) = 0:5. By Bayes' theorem, we calculate the posterior\nP(Njnn) =P(nnjN)P(N)\nP(nn)=P(njN)2P(N)\nP(njN)2P(N) +P(njY)2P(Y);\nwhere P(nn) =P(nnjN)P(N) +P(nnjY)P(Y). we have P(nnjN) =P(njN)2\nandP(nnjY) =P(njY)2. Similarly, we compute P(Yjnn),P(Yjyy) and P(Yjyn)\n1": "\\documentclass[preprint,authoryear]{elsarticle}\n\\usepackage{amssymb}\n\\journal{}%Mathematical Biosciences}\n\\usepackage{graphicx}\n\\usepackage{amsmath,amssymb,amsfonts,latexsym}\n\\usepackage{color}\n\\usepackage{eucal}%    caligraphic-euler fonts: \\mathcal{ }\n\\usepackage{xspace}\n\\usepackage{hyperref}\n\\usepackage{bm}\n\\usepackage[title]{appendix}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{proof}{Proof}\n\\usepackage[round]{natbib}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\\DeclareMathOperator*{\\argmax}{argmax} % no space, limits underneath in displays\n\\DeclareMathOperator*{\\argmin}{argmin} % no space, limits underneath in displays\n\\begin{document}\nover the conditioning feature $X_1$, i.e. \n\\begin{equation*}\nH(X_2|X_1)= \\sum_{i_1}  \\mathbb{P}(X_1=x^{(1)}_{i_1})  H(X_2|X_1=x^{(1)}_{i_1}).\n\\end{equation*}\n\\begin{proposition}[Chain rule,]\n\t\\begin{equation}\n\tH(X_1,X_2)=H(X_1)+ H(X_2|X_1).\n\t\\end{equation}\n\\end{proposition}\n\\begin{proposition}\n\tLet  $R(\\Omega,X_1,X_2)$ denote   the \\emph{entropy reduction} of $\\Omega$ first by the feature $X_1$ and then by the feature $X_2$, \n\tand $R(\\Omega,(X_1,X_2))$ denote   the \\emph{entropy reduction} of $\\Omega$  by a pair of features $(X_1,X_2)$. Then\n\t\\begin{equation}\n\tR(\\Omega,X_1,X_2) = R(\\Omega,(X_1,X_2)).\n\t\\end{equation}\n\\end{proposition}\nThe maximum entropy of an arbitrary feature is achieved when all  its outcomes   occur with equal probability, \nand this maximum value is proportional to the logarithm of the number of  possible outcomes  to the base $2$.\nThus Proposition  implies that the more possible outcomes  a feature has, \nthe higher  entropy reduction it could possibly lead to.\nMeanwhile,  a feature with an arbitrary number of outcomes can be viewed as a combination of  \\emph{binary features}, \nthe ones with two possible outcomes.\nEven though the entropy of the combination of two features is greater than  each of them,\nProposition shows that partitioning the space  subsequently  by two features has the same entropy reduction\nas partitioning by their combination. \nTherefore, instead of considering features with outcomes as many as possible, we \nfocus on binary features.\n\\section{Query repeats}\nHere we assess the improvement of the error rates  by repeating the same query  twice.\nLet $Y$ (or $N$) denote the event of the queried base pair  existing (or not) in the target structure. \nLet $y$ (or $n$) denote the event of the experiment confirming (or rejecting) the base pair.\nLet $nn$ denote the event of two independent experiments both rejecting the base pair. \nSimilarly, we have $yy$ and $y n$.\nUtilizing the same sequences and structures as described in  Fig.,\nwe estimate the conditional probabilities $\t\\mathbb{P}(n|N)\\approx 0.993$ and $\t\\mathbb{P}(n|Y)\\approx 0.055$.\nThe \\emph{prior probability} $\\mathbb{P}(Y)$ can be computed via the expected number  $l_1$ \nof confirmed queried base pairs on the path,\ndivided by the number of queries in each sample. \nFig. displays the distribution of $l_1$ having mean around $5$.\nThus we adopt \n$\\mathbb{P}(Y)= \\mathbb{P}(N)=0.5$.\nBy Bayes' theorem, we calculate the \\emph{posterior} \n\\[\n\\mathbb{P}(N|nn)=\\frac{\\mathbb{P}(nn|N) \\mathbb{P}(N)}{\\mathbb{P}(nn)}=\\frac{\\mathbb{P}(n|N)^2 \\mathbb{P}(N)}{\\mathbb{P}(n|N)^2 \\mathbb{P}(N) +\\mathbb{P}(n|Y)^2\\mathbb{P}(Y)},\n\\]\nwhere $\\mathbb{P}(nn)=\\mathbb{P}(nn|N) \\mathbb{P}(N) + \\mathbb{P}(nn|Y) \\mathbb{P}(Y)$.\nwe have $\\mathbb{P}(nn|N) =\\mathbb{P}(n|N)^2$ and $\\mathbb{P}(nn|Y) =\\mathbb{P}(n|Y)^2$.\nSimilarly, we compute $\\mathbb{P}(Y|nn)$, $\\mathbb{P}(Y|yy)$ and $\\mathbb{P}(Y|yn)$\n\\end{document}\n"}