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Proposition 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.

\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).

Under 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:
\begin{equation}
	\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]
\end{equation}
has 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 $1-\delta$:
	\begin{equation*}
		\left\| \widehat{ \mathbf{\theta}} - \mathbf{\theta}^* \right\|_2 \le \frac{\sigma}{\widehat{\lambda}_{\min}} \sqrt{\frac{2d}{n} \log \frac{2d}{\delta}}.
		\end{equation*}
Furthermore, with probability at least $1-\delta$, individually:
	\begin{align*}
		&\ell(\widehat{\mathbf{\theta}}) - \ell({\mathbf{\theta}}^*) \le \frac{d^2\sigma^4}{\widehat{\lambda}_{\min}^2 n}  \log \frac{2d}{\delta}\\
		&\widehat{\ell}({\mathbf{\theta}}^*) - \widehat{\ell}(\widehat{\mathbf{\theta}})  \le \frac{ d^2\sigma^4}{\widehat{\lambda}_{\min}^2 n}  \log \frac{2d}{\delta}.
	\end{align*}

The 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 $\mathcal{O}(n^{-1})$.
Note 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
with probability at least $1-\delta$ it holds that $\widehat{\lambda}_{\min} > 0$.

\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.

Let $\delta \in [0,1]$. An identification rule producing $\widehat{{I}}$ is \emph{$\delta$--correct} if: $\Pr \big( \widehat{{I}} \neq {I}^* \big)\le \delta$.

 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.
We now provide a result that bounds $\alpha$ and $\beta$ and employs them to derive $\delta$--correctness.

	Let $\widehat{I}_{c}$ 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:
	{
	\begin{align*}
		&\alpha  \le 2d \exp \left\{ -\frac{c(1) {\lambda}_{\min}^2 n}{16d^2 \sigma^4} \right\}\\
		&\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\}.
	\end{align*}
	}
	Furthermore, the Identification Rule is $\left((d-d^*)\alpha +d^*\beta\right)$--correct.

Since $\alpha$ 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 $c(1)$.

\section{Policy Space Identification in a Configurable Environment}
The identification rules presented so far are
unable to distinguish between a parameter set to zero because the agent
cannot control it, or because zero is its optimal value. To overcome this issue, we employ the Conf--MDP properties to select
a 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
for 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
log--likelihood gap ${l}({\mathbf{\theta}_i^*}) - {l}({\mathbf{\theta}^*})$, i.e. parameter $\theta_i$ should
be essential to justify the agent's behavior. Let $I \in \{1,...,d\}$ be a set of parameter
indexes we want to test, our ideal goal is to find the environment $\mathbf{\omega}_I$ such that:
\begin{equation}
	\mathbf{\omega}_I \in arg\,max_{\mathbf{\omega} \in \Omega} \left\{ {l}({\mathbf{\theta}_I^*}(\mathbf{\omega})) - {l}({\mathbf{\theta}^*}(\mathbf{\omega})) \right\},
\end{equation}
where ${\mathbf{\theta}^*}(\mathbf{\omega}) \in arg\,max_{\mathbf{\theta} \in \Theta} J_{\mathcal{M}_{\mathbf{\omega}}}(\mathbf{\theta})$ 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
in 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

Let $I \in \{1,...,d\}$ 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

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components 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:
	\begin{equation*}
		{\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.
	\end{equation*}

Thus, 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:
\begin{equation}
\resizebox{0.88\linewidth}{!}{$
\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{	\widehat{d}_2 (\mathbf{\omega} \| \mathbf{\omega}_0) }{n}},$}
\end{equation}
where $\widehat{\nabla}_{\mathbf{\theta}} J_{\mathcal{M}_{\mathbf{\omega}/\mathbf{\omega}_0}}(\mathbf{\theta}^*(\mathbf{\omega}_0))$ 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.

\section{Experimental Evaluation}
In this section, we present the experimental evaluation of the identification rules in three RL domains.
To 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 $c(1) = \chi^2_{1,1-{\delta}/{d}}$.

\subsection{Discrete Grid World}
The 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.

Figure 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--.

\subsection{Minigolf}
In 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.

Figure-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 $\omega^*_{\mathscr{A}_2}=11.5$.
Figure-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

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1 Lorentzian matter-coupled $F(4)$ gauged supergravity

The theory of matter-coupled $F(4)$ gauged supergravity was first studied in , with some

applications and extensions given in . Below we present a short review of this theory, similar

to that given in .

1.1 The bosonic Lagrangian

We begin by recalling the field content of the 6-dimensional supergravity multiplet,

({\it e}µ{\it a}, ?µ{\it A, A}µa, {\it B}µ?, ?{\it A}, s) (1)

The field {\it e}µ{\it a} is the 6-dimensional frame field , with spacetime indices denoted by \{µ, ?\}

and local Lorentz indices denoted by $\{a,\ b\}$. The field ?µ{\it A} is the gravitino with the index

$A, B = 1$, 2 denoting the fundamental representation of the gauged $SU(2)_{R}$ group. The

supergravity multiplet contains four vectors {\it A}µa labelled by the indexa $= 0$, . . .3. It will

often prove useful to split $\ovalbox{\tt\small REJECT}=(0,\ r)$ with $r=1$, . . . , 3 an $SU(2)_{R}$ adjoint index. Finally, the

remaining fields consist of atwo-form {\it B}µ?, aspin-$\displaystyle \frac{1}{2}$ field?{\it A}, and the dilatons. The only

allowable matter in the $d=6, \mathcal{N}=2$ theory is the vector multiplet, which has the following

field content

({\it A}µ, ?{\it A}, fa){\it I} (2)

where $I=1$, . . . , $n$ labels the distinct matter multiplets included in the theory. The presence

of the{\it n} new vector fields {\it A}µ{\it I} allows for the existence of a further gauge group $G_{+}$ of dimension

$\dim G_{+}=n$, in addition to the gauged $SU(2)_{R}\mathrm{R}$-symmetry. The presence of this new gauge

group contributes an additional parameter to the theory, in the form of a coupling constant

?. Throughout this section, we will denote the structure constants of the additional gauge

group $G_{+}$ by $C_{IJK}$. However, these will play no role in what follows, since we will be

restricting to the case of only a single vector multiplet $n = 1$, in which case $G_{+} = U(1)$ .

In (half-)maximal supergravity, the dynamics of the $4n$ vector multiplet scalars fa{\it I} is given

by a non-linear sigma model with target space $G/K$; see e.g. . The group{\it G} is the global

symmetry group of the theory, while $K$ is the maximal compact subgroup of $G$. As such, in

the Lorentzian case the target space is identified with the following coset space,
\begin{center}
$\displaystyle \mathcal{M}=\frac{SO(4,n)}{SO(4)\times SO(n)}\ \times SO(1,1)$   (3)
\end{center}
where the second factor corresponds to the scalar s which is already present in the gauged

supergravity without added matter. In the particular case of{\it n} $= 1$, explored here and in ,

the first factor is nothing but four-dimensional hyperbolic space $\mathrm{H}_{4}$. When we analytically

continue to the Euclidean case, it will prove very important that we analytically continue

the coset space as well, resulting in a $\mathrm{d}\mathrm{S}_{4}$ coset space. This will be discussed more in the

following section.

1
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In 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,
\begin{eqnarray}
L^{-1} d L \in \mathfrak{g}
\end{eqnarray}
where $\mathfrak{g} = \mathrm{Lie}(G)$. To build a $K$-invariant kinetic term from the above, we decompose
\begin{eqnarray}
L^{-1} d L = Q + P
\end{eqnarray}
where $Q \in \mathfrak{k} = \mathrm{Lie}(K)$ and $P$ lies in the complement of $\mathfrak{k}$ in $\mathfrak{g}$.  Explicitly, the coset vielbein forms are  given by,
\begin{eqnarray}
P^I_{\,\,\,\,\alpha} = \left(L^{-1}\right)^I_{\,\,\,\,\Lambda} \left(d L^\Lambda_{\,\,\,\,\alpha} + f^\Lambda_{\,\,\,\,\Gamma \Pi} A^\Gamma L^\Pi_{\,\,\,\,\alpha} \right)
\end{eqnarray}
where the $f_{\Lambda \Sigma}^{\,\,\,\,\,\,\,\,\Gamma}$ are structure constants of the gauge algebra, i.e.
\begin{eqnarray}
[T_\Lambda, T_\Sigma] =  f_{\Lambda \Sigma}^{\,\,\,\,\,\,\,\,\Gamma} \, T_\Gamma
\end{eqnarray}
We may then use $P$ to build the kinetic term for the vector multiplet scalars as,
\begin{eqnarray}
{\cal L}_{\mathrm{coset}} = -{1\over 4} e P_{I \alpha \mu} P^{I \alpha \mu}
\end{eqnarray}
where $e = \sqrt{|\mathrm{det} \,g|}$ 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
 \begin{eqnarray}
 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
 \end{eqnarray}
 with the scalar potential $V$ given by
 \begin{eqnarray}
V = -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}
\nonumber\\
\vphantom{.} \hspace{0.5in} - m e^{- 2 \sigma} \left[{2 \over 3} A L_{00} - 2 B^i L_{0 i} \right]
\end{eqnarray}
The scalar potential features the following quantities,
\begin{eqnarray}
A = \epsilon^{r s t} K_{r s t} \hspace{1 in}\,\,B^r\, = \,\epsilon^{r s t} K_{s t 0}
\nonumber\\
C_I^t = \epsilon^{t r s} K_{r I s} \hspace{1 in} D_{I t}\,\, = \,\, K_{0 I t}
\end{eqnarray}

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with the so-called ``boosted structure constants" $K$ given by,
\begin{eqnarray}
K_{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}
\nonumber\\
K_{\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}
\end{eqnarray}
We 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
\begin{eqnarray}
{\cal N}_{\Lambda \Sigma} = L_\Lambda^{\,\,\,\,\alpha}\left( L^{-1}\right)_{\alpha \Sigma} -  L_\Lambda^{\,\,\,\,I}\left( L^{-1}\right)_{I \Sigma}
\end{eqnarray}
\subsection{Supersymmetry variations}
We 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.
In order to write the fermionic variations, it is first necessary to introduce a matrix $\gamma^7$ defined as
\begin{eqnarray}
\gamma^7=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4\gamma^5
\end{eqnarray}
and satisfying $(\gamma^7)^2=-\mathds{1}$. With this, the supersymmetry transformations of the fermions in the Lorentzian case can be given as
\begin{eqnarray}
\delta \chi_A = {i \over 2} \gamma^\mu \partial_\mu \sigma \varepsilon_A + N_{AB} \varepsilon^B
\nonumber\\\nonumber\\
\delta \psi_{A \mu} = {\cal D}_\mu \varepsilon_A + S_{AB} \gamma_\mu \varepsilon^B
\nonumber\\\nonumber\\
\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
\end{eqnarray}
where we have defined
\begin{eqnarray}
S_{AB}=\!\frac{i}{24}[Ae^{\sigma}\! +\!
6me^{-3\sigma}(L^{-1})_{00}]\varepsilon_{AB}\! -\!
\frac{i}{8}[B_te^{\sigma}-2me^{-3\sigma}(L^{-1})_{t0}]\gamma^7\sigma^t_{AB}\nonumber
\\\nonumber\\
N_{AB}=\!\frac{1}{24}[Ae^{\sigma}\! -\!
18me^{-3\sigma}(L^{-1})_{00}]\varepsilon_{AB}\! +\!
\frac{1}{8}[B_te^{\sigma}\! +\! 6me^{-3\sigma}(L^{-1})_{t0}]\gamma^7\sigma^t_{AB}\nonumber\\
\nonumber\\
M^I_{AB}=\!(-C^I_{~t}+2i\gamma^7D^I_{~t})e^{\sigma}\sigma^t_{AB}-
2me^{-3\sigma}(L^{-1})^I_{\ \ 0}\gamma^7\varepsilon_{AB},
\end{eqnarray}
In 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.
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\subsection{Mass deformations}
In 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
\begin{eqnarray}
L=\prod_{i=0}^3e^{\phi^i K^i}
\end{eqnarray}
where $K^i$ 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.
The scalar potential for this specific case can be obtained from and takes the following form
\begin{align}
V(\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
\nonumber\\
+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\\
+\cosh (2 (\phi^1+\phi^2))+2 \cosh (2 \phi^1)+2 \cosh (2 \phi^2)\bigg)\bigg]
\end{align}
The 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
\begin{eqnarray}
m^2l^2= \Delta(\Delta-5)
\end{eqnarray}
where $l$ is the curvature radius of the AdS$_6$ vacuum. For the scalars at hand, one finds
\begin{eqnarray}
m_\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
\end{eqnarray}
Hence the dimensions of the dual operators are
\begin{eqnarray}
\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
\end{eqnarray}
In  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,
\begin{eqnarray}
{\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
\end{eqnarray}
Note 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
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 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 ${\cal O}_{\phi^3}$.
\section{Euclidean theory and BPS solutions}
\setcounter{equation}{0}
In 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
\begin{eqnarray}
ds^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
\end{eqnarray}
with 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.
\subsection{Euclidean action}
The 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,
\begin{eqnarray}
S_{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)
\end{eqnarray}
where 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
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which for the coset representative is given by
\begin{eqnarray}
G_{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)
\end{eqnarray}
In addition to performing the above Wick rotation, we also perform a Wick rotation on the sigma model
\begin{eqnarray}
{SO(4,1) \over SO(4)} \to{SO(4,1) \over SO(3,1)} \simeq dS_4
\end{eqnarray}
The 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,
\begin{eqnarray}
 \phi_r\rightarrow i \phi_r \hspace{0.5 in}r=1,2,3
 \end{eqnarray}
It 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.
\subsection{Euclidean supersymmetry}
The 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.
\begin{eqnarray}
\bar \psi_A^{(\mathbb{H}_6)} =  i \bar \psi_A^{(dS_6)}
\end{eqnarray}
Because 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
\begin{eqnarray}
\delta \chi_A = - {1\over 2} \gamma^\mu \partial_\mu \sigma \varepsilon_A + N_{AB} \varepsilon^B + \dots
\nonumber\\\nonumber\\
\delta \psi_{A \mu} = {\cal D}_\mu \varepsilon_A + i S_{AB} \gamma_\mu \varepsilon^B +\dots
\nonumber\\\nonumber\\
\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
\end{eqnarray}
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where $N_{AB}$, $S_{AB}$, 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.
A nice property of the variations above is the fact that they are consistent with the following $SO(6)$-invariant symplectic Majorana condition,
\begin{eqnarray}
\bar \psi_A = \epsilon^{AB} \psi_B^T {\cal C}
\end{eqnarray}
The 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.
As 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
consistent 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
\begin{eqnarray}
S_{AB} = i S_0 \epsilon_{AB} + i S_3 \gamma^7 \sigma^3_{AB}
\nonumber\\
N_{AB}= -N_0 \epsilon_{AB} -N_3 \gamma^7 \sigma^3_{AB}
\nonumber\\
M^I_{AB}= M_0 \gamma^7 \epsilon_{AB} + M_3  \sigma^3_{AB}
\end{eqnarray}
where we have defined
\begin{eqnarray}
S_0=\frac14 \left(g\cos \phi^3 e^\sigma+m e^{-3\sigma}\cosh \phi^0\right)
\nonumber\\
S_3=\frac14 i \,m ~e^{-3\sigma}\sinh \phi^0 \sin \phi^3
\nonumber\\
N_0=-\frac14 \left(g\cos \phi^3 e^\sigma-3m e^{-3\sigma}\cosh \phi^0\right)
\nonumber\\
N_3=-\frac34 i\, m e^{-3\sigma}\sinh \phi^0 \sin \phi^3
\nonumber\\
M_0=2m ~e^{-3\sigma}\cos \phi^3\sinh \phi^0
\nonumber\\
M_3=-2 i\, g ~e^{\sigma}\sin \phi^3
\end{eqnarray}
Importantly, 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.
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\subsection{BPS Equations}
We now use the vanishing of the fermionic variations to obtain BPS equations for the warp factor and the three non-zero scalars.
\subsubsection{Dilatino equation and projector}
We begin by imposing the vanishing of the dilatino variation, $\delta\chi_A = 0$, which implies
\begin{eqnarray}
{1\over 2} \gamma^5 \sigma' \varepsilon_A = N_0 \varepsilon_A + N_3 \gamma^7 {(\sigma^3)^B}_A \varepsilon_B
\end{eqnarray}
This equation can be interpreted as a projection condition on the spinors $\varepsilon_A$. Consistency of this projection condition then requires that
\begin{eqnarray}
 \sigma' = 2 \eta \sqrt{N_0^2 + N_3^2}
\end{eqnarray}
where $\eta = \pm1$. Plugging this BPS equation back into then yields a second form of the projection condition,
\begin{eqnarray}
\gamma^5 \varepsilon_A = G_0 \varepsilon_A - G_3 \gamma^7 {(\sigma^3)^B}_A \varepsilon_B
\end{eqnarray}
which is more useful in the derivation of the other BPS equations. In the above, we have defined
\begin{eqnarray}
G_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}}
\end{eqnarray}
\subsubsection{Gravitino equation}
The 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,
\begin{eqnarray}
f^{\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)
\end{eqnarray}
\subsubsection{Gaugino equations}
Finally, 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
\begin{eqnarray}
\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)
\end{eqnarray}
The right-hand sides of both equations are real, and thus give rise to real solutions when appropriate initial conditions are imposed.
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\subsubsection{Summary of first-order equations}
To summarize, the first-order equations for the warp factor $f$ and the scalars $\sigma,\phi^0,\phi^3$ are found to be
\begin{eqnarray}
f'=2\left(G_0S_0+ G_3S_3\right)
\nonumber \\%
 \sigma'=2 \eta \sqrt{N_0^2 + N_3^2}
\nonumber \\%
\cos \phi^3 \left(\phi^0\right)'=-\left(G_0M_0+G_3M_3\right)
\nonumber\\%
 \left(\phi^3\right)'=i \left(G_3 M_0-G_0M_3\right)
\end{eqnarray}
Furthermore, for consistency these were required to satisfy the algebraic constraint
\begin{eqnarray}
e^{-2f}=4\left(G_0S_0+G_3S_3\right)^2-4 \left(S_0^2+S_3^2\right)
\end{eqnarray}
The various functions featured in these equations were defined in and .
\subsection{Numeric solutions}
In 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,
\begin{eqnarray}
 \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]
\nonumber\\
\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]
\end{eqnarray}
We 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
\begin{eqnarray}
|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)
\end{eqnarray}
Noting that
\begin{eqnarray}
\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}
\end{eqnarray}
and also that $f(\phi^0_0)$ is monotonically increasing, i.e.
\begin{eqnarray}
{d f \over d \phi^0_0} >0\hspace{0.3in} \forall \phi^0_0 \in \mathbb R
\end{eqnarray}
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allows 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 $\phi^0_0$.
With 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 $e^{2 f_k}$.
\subsection{UV asymptotic expansions}
As 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
\begin{eqnarray}
f(z) = -\log z +f_k -  \left({1\over 4}e^{-2f_k} + {1\over 16}\alpha^2\right)  \,z^2 + O(z^4) \nonumber\\
\sigma(z) = {3\over 8} \alpha^2 \,z^2 +  {1\over 4} e^{f_k} \alpha \beta \,z^3 + O(z^4)
\nonumber\\
\phi^0(z) = \alpha \, z - \left({5\over 4}\alpha \,e^{-2f_k} + {23\over 48} \alpha^3\right) \,z^3 + O(z^4)
\nonumber\\
\phi^3(z) = e^{-f_k} \alpha  z^2 + \beta \, z^3 + O(z^4)
\end{eqnarray}
We have obtained the expansions up to $O(z^8)$, but we display only the first few terms here.
\section{Holographic sphere free energy}
\setcounter{equation}{0}
The 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,
\begin{eqnarray}
\vphantom{.} \hspace{1.7 in} S = S_{\mathrm{6D}} + S_{\mathrm{GH}}
\nonumber\\\nonumber\\
\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}
\end{eqnarray}
where $S_{\mathrm{6D}}$ 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
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(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
\begin{eqnarray}
{\cal K}_{i}=\frac{d}{du} \gamma_{i}
\end{eqnarray}
In 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.
\subsection{Finite counterterms}
In 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.
\subsubsection{Approximate superpotential}
In 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
\begin{eqnarray}
V(\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
\nonumber
\end{eqnarray}
 This scalar potential can in fact be rewritten as
\begin{eqnarray}
V = 4 (N_0^2 + N_3^2) + {1 \over 4} (M_0^2 +M_3^2) - 20 (S_0^2 + S_3^2)
\end{eqnarray}
Then for BPS solutions, implies that
\begin{eqnarray}
V = (\sigma')^2 + {1 \over 4}\left(-({\phi^3}')^2 + \cos^2 \phi^3 ({\phi^0}')^2\right)- 20 (S_0^2 + S_3^2)
\end{eqnarray}
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This motivates us to define a superpotential $W$ as
\begin{eqnarray}
W = \sqrt{S_0^2 + S_3^2}
\end{eqnarray}
\iffalse
This 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.
\begin{eqnarray}
[D_m , D_n] \epsilon_A = 2 W^2 \,\gamma_{mn} \epsilon_A
\end{eqnarray}
and indeed one finds.
Requiring that the warp factor BPS equation may be re-expressed as an equation linear in the superpotential, i.e.
\begin{eqnarray}
f' = 2 (G_0 S_0 + G_3 S_3)
\nonumber\\
= 2 \, \gamma \,W
\end{eqnarray}
gives us a factor $\gamma$ defined as
\begin{eqnarray}
\gamma = {G_0 S_0 + G_3 S_3 \over \sqrt{S_0^2 + S_3^2}}
\end{eqnarray}
We 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,
\begin{eqnarray}
\sigma ' =2 \eta \, \sqrt{N_0^2 + N_3^2}
\nonumber\\
= -2 \left({- S_0 N_0 + S_3 N_3 \over G_0 S_0 + G_3 S_3}\right)
\nonumber\\
= -2\, \gamma^{-1} \,\partial_\sigma W
\end{eqnarray}
exactly as expected.
\fi
Unfortunately, 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)
\begin{eqnarray}
{\phi^0}' = -8 \, \partial_{\phi^0} W|_{\phi^3=0} \hspace{1 in} {\phi^3}' =8 \, \partial_{\phi^3} W|_{\phi^0=0}
\end{eqnarray}
respectively. In either truncation, the BPS equations for the warp factor and dilaton $\sigma$ can be put in the following form,
\begin{eqnarray}
f'=2\,W\hspace{1 in} \sigma'=2 \,\partial_\sigma W
\end{eqnarray}
An 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
\begin{eqnarray}
W =  {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
\end{eqnarray}
where the dots represent terms of order $O(z^6)$. This is the approximate superpotential we will use in what follows.
\subsubsection{Bogomolnyi trick}
We 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.
To do so, we begin by inserting. We find that
\begin{eqnarray}
{\cal L} =- {1 \over 4}R -20 W^2 + 2 {\cal L}_{\mathrm{kin}}
\end{eqnarray}
where we've defined
\begin{eqnarray}
{\cal L}_{\mathrm{kin}} = (\sigma')^2 + {1 \over 4}\left[-({\phi^3}')^2 + \cos^2 \phi^3 ({\phi^0}')^2\right]
\end{eqnarray}
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The non-zero components of the Ricci tensor are
\begin{eqnarray}
R_{uu} = - 5 \left(f'' + (f')^2 \right) \hspace{1 in} R_{mn} =  -g_{mn} \left(f'' + 5(f')^2 \right)
\end{eqnarray}
while the Ricci scalar is given by
\begin{eqnarray}
R = - 10 f'' - 30(f')^2
\end{eqnarray}
Furthermore, 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
\begin{eqnarray}
S = \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]
\end{eqnarray}
The restriction to the flat case was not strictly necessary so far, but it will be crucial in the next step.
The gradient flow equations, together with the chain-rule, allows us to rewrite
\begin{eqnarray}
{\cal L}_{\mathrm{kin}} =-2\,  W'
\end{eqnarray}
Plugging this into and using the BPS equation of the warp factor, we find
\begin{eqnarray}
S = -4  \int d^5 x \sqrt{g}\,e^{5f} W \Big|_0^\Lambda
\end{eqnarray}
where $\Lambda$ 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.
Removing 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
\begin{eqnarray}
e^{5f} \sim {1 \over \varepsilon^5}
\end{eqnarray}
This 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.
We may thus legitimately insert the approximate superpotential into to get the counterterms,
\begin{eqnarray}
S^{(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]
\end{eqnarray}
where $\gamma$ 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.
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\subsection{Infinite counterterms}
We 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
\begin{enumerate}
\item Check that in the flat limit, they reduce to the divergent pieces of the flat counterterms found above.
\item Add to them the finite pieces found in but missing in the holographic renormalization procedure.
\end{enumerate}
For simplicity, we will perform holographic renormalization on supersymmetric solutions only, and thus the infinite counterterms we obtain are universal for supersymmetric solutions only.
We begin by using the expression for the on-shell Ricci scalar,
\begin{eqnarray}
R = 4 (\sigma')^2 + \left[-({\phi^3}')^2 + \cos^2 \phi^3 ({\phi^0}')^2 \right] + 6 V
\end{eqnarray}
to rewrite the action as
\begin{eqnarray}
S_{\mathrm{6D}} = -{1 \over 2} \int du\, d^5 x \sqrt{g} \,e^{5f}  V
\end{eqnarray}
We 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
\begin{eqnarray}
S_{\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.
\nonumber\\
\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\\
\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]
\end{eqnarray}
where 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,
\begin{eqnarray}
25 f_5 + 2 \,\phi^0_1 \phi^0_4 - 3 \,\phi^3_2 \phi^3_3 + 12 \sigma_2 \sigma_3 = 0
\end{eqnarray}
The absence of the logarithmic term is to be expected, since any dual five-dimensional field theory is anomaly-free. The Gibbons-Hawking term is
\begin{eqnarray}
S_{\mathrm{GH}} = -{5 \over 2} \int d^5 x \sqrt{g}\, e^{5f} f'
\end{eqnarray}
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We again use the asymptotic expansions to write
\begin{eqnarray}
S_{\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]
\end{eqnarray}
Adding the two together, we find in total that
\begin{eqnarray}
S_{\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\\
\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]
\end{eqnarray}
We 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
\begin{eqnarray}
\gamma_{i j}=  e^{2 f}\big|_{z = \epsilon} \,g^{(S^5)}_{i j}
\end{eqnarray}
The Ricci tensor and Ricci scalar are given by
\begin{eqnarray}
R_{ij}[\gamma] = 4e^{-2f}\gamma_{ij}\big|_{z = \epsilon}~~~~~~~~~~~~~~R[\gamma] = 20 \,e^{- 2 f}\big|_{z = \epsilon}
\end{eqnarray}
In terms of these quantities, we find that the inverted form of the divergent part of the on-shell action is
\begin{eqnarray}
S = - \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.
\nonumber\\
\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]
\end{eqnarray}
We 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
\begin{eqnarray}
S_{\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.
\nonumber\\
\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]
\end{eqnarray}
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The 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 $R_{ij}[\gamma]R[\gamma]^{ij}=\frac15 R[\gamma]^2$.
Note 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
\begin{eqnarray}
\varphi^0 \rightarrow -\varphi^0 \hspace{0.7in}\mathrm{and} \hspace{0.7in} \varphi^3 \rightarrow -\varphi^3
\end{eqnarray}
(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
\begin{eqnarray}
(\phi^3)^2 - {1 \over 20} R[\gamma] (\phi^0)^2 = 2\, e^{-f_k} \beta\ alpha \,z^5 + O(z^6)
\end{eqnarray}
This freedom is fixed by demanding that the vevs of the dual operators stay finite. We will simply quote the result here, \begin{eqnarray}
S_{\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.
\nonumber\\
\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]
\end{eqnarray}
and postpone showing that this gives finite vacuum expectation values to the next subsection.
At 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
\begin{eqnarray}
I_0 \equiv 5 \sigma^2 + {45 \over 64} (\varphi^0)^4 - {15 \over 4} (\varphi^0)^2 \sigma =  O(z^6)
\end{eqnarray}
we 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.
\subsection{Vevs and free energy}
The renormalized on-shell action is given by
\begin{eqnarray}
S_{\mathrm{ren}} = S_{\mathrm{6D}} + S_{\mathrm{GH}} + S_{\mathrm{ct}} + \Omega \int d^5 x~  \sqrt{\gamma}\, I_0
\end{eqnarray}
where 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 $\Omega$
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since $I_0$ 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.
While 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.
\subsubsection{One-point functions}
By the usual AdS/CFT dictionary, the one-point functions of the operators dual to the three scalar fields and the metric are given by
\begin{align}
\langle {\cal O}_{\sigma} \rangle = \lim_{\epsilon \rightarrow 0}  \frac{1}{\epsilon^3} {1 \over \sqrt{\gamma}} {\delta S_{ren} \over \delta \sigma}~~~~~~~~~~
\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}
\nonumber\\
 \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}~~~~~~~~~~
 \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}}
\end{align}
We 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
\begin{align}
\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 \\
\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\\
~~~~~~~~~~~~~~~~~~~~+\Omega\left(\frac{45}{16}\left(\phi^0\right)^3-\frac{15}{2}\phi^0\sigma\right)\bigg]\nonumber \\
\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  \\
\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]
\end{align}
Evaluating the limits, we get the following one-point functions
\begin{align}
\langle {\cal O}_{\sigma} \rangle = \frac52 e^{f_k}\alpha\beta\,\Omega ~~~~~~~~~~~
\langle {\cal O}_{\phi^0} \rangle = \frac32 e^{-f_k} \beta -\frac{15}{8} e^{f_k}\alpha^2\beta\, \Omega
\nonumber\\
 \langle {\cal O}_{\phi^3} \rangle = \frac12 \beta~~~~~~~~~~~~~~~~~~~ \langle {T^i}_i \rangle  =  -\frac52 e^{-f_k} \alpha\beta
\end{align}
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The 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,
\begin{eqnarray}
\langle{T^i}_i\rangle = -\sum_{\cal O}(d-\Delta_{\cal O})\,\phi_{\cal O}\,\langle{\cal O}\rangle
\end{eqnarray}
\subsubsection{Derivative of the free energy}
Following , we may now compute the derivative of $F$ with respect to $\alpha$ as follows. First we note that
\begin{eqnarray}
{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}
\end{eqnarray}
In our case, the terms appearing in the sum over fields are
\begin{align}
{\delta \left(\sqrt{\gamma} {\cal L}_{\mathrm{ren}}\right) \over \delta \sigma } =  \sqrt{\gamma}\,  \langle O_\sigma \rangle \epsilon^3 + \dots~~~~~~~~~
{\delta \left(\sqrt{\gamma} {\cal L}_{\mathrm{ren}}\right) \over \delta \phi^0 }  = \sqrt{\gamma} \, \langle O_\phi^0 \rangle \epsilon^4 + \dots
\nonumber\\
{\delta \left(\sqrt{\gamma} {\cal L}_{\mathrm{ren}}\right) \over \delta \phi^3 } = \sqrt{\gamma}\,  \langle O_\phi^3 \rangle \epsilon^3+ \dots~~~~~~~~~
{\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
\end{align}
The dots represent terms of strictly lower order in $\epsilon$. Furthermore, from the form of the UV asymptotic expansions , we have
\begin{align}
{d\sigma \over d \alpha} =  {3 \over 4}\alpha \epsilon^2 + O(\epsilon^3)~~~~~~~~~~~~~~~~~~~~~~~~~~~
{d\phi^0 \over d \alpha} = \epsilon + O(\epsilon^3)~~~~~~~~\nonumber\\
{d\phi^3 \over d \alpha} = \left(1- \alpha {d f_k \over d \alpha}\right)e^{-f_k}\epsilon^2 + O(\epsilon^3)~~~~~~~~
\frac{d\gamma^{ij}}{d\alpha} = -2\frac{df_k}{d\alpha}e^{-2f_k}\epsilon^2 + O(\epsilon^2)
\end{align}
Combining 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
\begin{eqnarray}
{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]
\nonumber\\
= \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)
\end{eqnarray}
where $\mathrm{vol}_0(S^5)=\pi^3$ 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 $\Omega$.
We thus obtain the final result
\begin{eqnarray}
{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)
\end{eqnarray}
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Note 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.
Treating $\b(\alpha)$ 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 .
\section{Field theory calculation}
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Localization  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.
A 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.
One 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.
\subsection{The D4-D8 system}
The 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
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the 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).
However, 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.
Having 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 .
To 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 .
\subsection{Undeformed $USp(2N)$ gauge theory}
In , 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
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result given by
\begin{eqnarray}
Z = {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)
\nonumber\\
\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)
\end{eqnarray}
where $r$ is the radius of $S^5$, $\sigma$ is a dimensionless matrix, and $f$ is defined as
\begin{eqnarray}
f(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}
\end{eqnarray}
The 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.
 One may rewrite the partition function in terms of the free energy as
\begin{eqnarray}
Z = {1 \over |{\cal W}|} \int_{\mathrm{Cartan}}[d\sigma] \,e^{-F({\sigma})}+ O\left(e^{- 16 \pi^3 r \over g_{YM}^2}\right)
\nonumber\\
F(\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)
\end{eqnarray}
The definitions of $F_V(\sigma)$ and $F_H(\sigma)$ follow simply from , and using  one may obtain the following large argument expansions
\begin{eqnarray}
F_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|
\end{eqnarray}
It 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 $F_H(\sigma)$.
The 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
\begin{eqnarray}
\sigma = \mathrm{diag}\{\lambda_1,\dots, \lambda_N, - \lambda_1, \dots, -\lambda_N \}
\end{eqnarray}
since $USp(2N)$ 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
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 $e_i \pm e_j$ 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
\begin{eqnarray}
F(\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]
\nonumber\\
\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]
\end{eqnarray}
The 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.
To 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
\begin{eqnarray}
\rho(x) = {1 \over N} \sum_{i=1}^N \delta(x-x_i)
\end{eqnarray}
which in the continuum limit should approach an $L^1$ function normalized as
\begin{eqnarray}
\int dx\, \rho(x) = 1
\end{eqnarray}
In terms of this density function, one finds that
\begin{eqnarray}
F \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
\end{eqnarray}
where 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
\begin{eqnarray}
F \approx - {9 \sqrt{2} \pi N^{5/2} \over 5 \sqrt{8-N_f}}
\end{eqnarray}
This 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,
\begin{eqnarray}
G_6= \frac{5\pi\sqrt{8-N_f}}{27\sqrt{2}} ~N^{-5/2}
\end{eqnarray}
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\subsection{Mass-deformed $USp(2N)$ gauge theory}
As 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
\begin{eqnarray}
F(\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]
\nonumber\\
\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.
\nonumber\\
\vphantom{.}\hspace{1in}\left. + (N_f-1) F_H(\lambda_i) + (N_f-1) F_H(-\lambda_i)  \right]\,\,\,\,
\end{eqnarray}
As 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
\begin{eqnarray}
F(\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
\nonumber\\
\vphantom{.} \hspace{0.3 in} - {\pi \over 6} N^{1 + 3 \alpha} \int dx \,\rho(x) \left[ |x + \mu|^3 + |x-\mu|^3 \right]
\end{eqnarray}
where 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
\begin{eqnarray}
\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)}\,\,\,
\end{eqnarray}
with $\rho(x)$ having support only on the interval $x \in [0,x_*]$. Inserting this result back into  then gives our final result,
\begin{eqnarray}
F(\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}
\end{eqnarray}
We may check that when $\mu=0$, we reobtain the result of the undeformed case .
With 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
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\section{Gamma matrix and spinor conventions }
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For concreteness, we take the following basis of gamma matrices
\begin{eqnarray}
\gamma_1 =\sigma_2 \otimes \mathds{1}_2 \otimes \sigma_3
\nonumber\\
\gamma_2 =\sigma_2 \otimes \mathds{1}_2 \otimes \sigma_1
\nonumber\\
\gamma_3 = \mathds{1}_2  \otimes \sigma_1  \otimes \sigma_2
\nonumber\\
\gamma_4 = \mathds{1}_2 \otimes \sigma_3  \otimes \sigma_2
\nonumber\\
\gamma_5 =\sigma_1 \otimes \sigma_2 \otimes \mathds{1}_2
\nonumber\\
\gamma_6 = \sigma_3 \otimes \sigma_2 \otimes \mathds{1}_2
\end{eqnarray}
These gamma matrices satisfy the Clifford algebra
\begin{eqnarray}
\{ \gamma_\mu, \gamma_\nu\} = 2 \delta_{\mu \nu}
\end{eqnarray}
as appropriate for a positive definite Euclidean spacetime. All matrices are purely imaginary and satisfy
\begin{eqnarray}
(\gamma_\mu)^\dagger = \gamma_\mu \hspace{1 in} \left(\gamma_\mu \right)^2 =  \mathds{1}
\end{eqnarray}
We 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
\begin{eqnarray}
x_1^2 + \dots +x_6^2 - x_7^2 = - L^2
\end{eqnarray}
The seven-dimensional Clifford algebra is made up of the set of matrices $\{ \gamma_1, \dots, \gamma_6, \gamma_7\}$, with $\gamma_7$ satisfying
\begin{eqnarray}
(\gamma_7)^2 = - \mathds{1} \hspace{1 in} \{ \gamma_\mu, \gamma_7\} = 0 \,\,\,\forall \mu \neq 7
\end{eqnarray}
As usual, we use the notation $\gamma^7 = (\gamma_7)^{-1}$, so that by the above we have $\gamma^7 = - \gamma_7$.
We now discuss Dirac spinors in $d=7$. We define the Dirac conjugate of $\psi_A$ to be
\begin{eqnarray}
\bar \psi_A = \psi_A^\dagger G^{-1}
\end{eqnarray}
for some matrix $G$. There are two possible choices for $G$ , which in the particular case of the ambient space above are
\begin{eqnarray}
G_1 = \gamma^7 \hspace{1 in} G_2 = \gamma^1 \dots \gamma^6
\end{eqnarray}
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