
where
\begin{equation}
\gamma = \frac{m}{2}\left(\frac{m}{2}-2\pi i \right) - \left(\frac{\Delta}{2} + \pi i\right)\left(\frac{\Delta}{2} - \pi i\right)
= - \frac{(\Delta+ \hat{m})(\Delta - \hat{m})}{4}>0\,
\end{equation}
where we defined $\hat{m} = m-2\pi i$.

To summarize, the large $N$ and Cardy free energy of the index
\eqref{asymp-free-energy} is given by
\begin{equation}\label{free-energy}
\log Z \sim - i \frac{\sqrt{2}}{15} \frac{N^{\frac{5}{2}}}{\sqrt{8-N_f}} \frac{[(\Delta+\hat{m})(\Delta-\hat{m})]^{\frac{3}{2}}}{\omega_1 \omega_2}\,
\end{equation}
subject to the constraint
\begin{equation}
\Delta - \omega_1 -\omega_2 =  2\pi i\,
\end{equation}
in the Cardy-like limit $|\omega_i| \ll 1$.
$\Delta^2-\hat{m}^2$ appearing in the square-root is negative in our canonical chamber.
The expression (\ref{free-energy}) and similar expressions at the end of
section 2.2 are obtained with the convention $(-1)^{\frac{3}{2}}=-i$.
In section 2.3, we shall explain that this free energy counts the dual BPS black holes
in the background of warped $AdS_6\times S^4/\mathbb{Z}_2$ product. Here
we simply note that the leading large $N$ free energy $\propto N^{\frac{5}{2}}$ does not
see the flavor symmetries, e.g. $M_l$'s and $q$ for $SO(2N_f)\times U(1)_I
\subset E_{N_f+1}$.
This is natural in the bulk dual because the states charged under $E_{N_f+1}$ are
localized on a codimension $1$ wall, $S^3\sim\partial(S^4/\mathbb{Z}_2)$, so that the
leading large $N$ bulk physics does not see them. However, the value of $N_f$ itself is
visible in the leading free energy. This is because the number of D8-branes affects the
bulk dilaton field.



\subsection{$SU(2N)$ theories}\label{sec: Cardy-SU(N)}

Similar to the studies of section \ref{sec: Cardy-Sp(N)}, we analyze the large $N$ and
Cardy free energy of the index for 5d $\mathcal{N}=1$ gauge theory with
$SU(2N)$ gauge group, two rank 2 antisymmetric hypermultiplets, and $N_f$
fundamental hypermultiplets. The related geometric settings are explained at the
beginning of this section. The index is defined as \cite{Kim:2012gu}
\begin{equation}
\hspace*{-0.5cm}
Z(\omega_1, \omega_2, \Delta, m, M, q) = \textrm{Tr}\left[(-1)^F e^{-\beta\{Q,S\}} e^{-\omega_1 J_1} e^{-\omega_2 J_2} e^{-(\Delta-2\pi i) R} e^{-m h_M- b h_B}
e^{- \sum_{l} M_l H_l} q^k \right]\.
\end{equation}
We introduced the fugacities $e^{-m}, \, e^{-b}$ for the Cartans
$h_M, \, h_B$ of $SU(2)_M \times U(1)_B \cong U(2)$ acting on two
antisymmetric matters, and the fugacities $e^{-M_l}$ for
the Cartans $H_l$ of $U(N_f)$ acting on fundamental matters.
As before, the parameters should meet the constraint
$\Delta-\omega_1-\omega_2=2\pi i$ for $Z$ to be an index.
Again, the index is given by \cite{Kim:2012gu, Bergman:2013koa}
\begin{eqnarray}
\hspace*{-0.5cm}
Z(\omega_{1,2}, \Delta, m, b, M_l, q) &=& \oint [d\alpha] \textrm{PE} \big
[f_{vec} (\omega_{1,2}, \Delta, \alpha_a) + f_{mat}^{asym} (\omega_{1,2}, m, b, \alpha_a)
+ f_{mat}^{fund} (\omega_{1,2}, M_l, \alpha_a)\big]\nonumber
\\
&&\times\prod_{\pm}Z_{\rm inst} (\pm\omega_1, \pm\omega_2, \pm\Delta, \pm m,
\pm b, \pm M_l, \pm\alpha_a,q^{\pm 1})\.
\end{eqnarray}
The Haar measure is given by
$[d\alpha] = \frac{1}{(2N)!} \left[\prod_{a} \frac{d\alpha_a}{2\pi} \right] \prod_{a<b} \left[2 \sin \left(\frac{\alpha_a-\alpha_b}{2}\right) \right]^2$,
with $\sum_{a=1}^{2N} \alpha_a =0$.
Sum of all $\alpha_a$'s vanishes because the gauge group is
$SU(2N)$. The letter indices are given by
\begin{eqnarray}
\hspace*{-0.5cm}
\tilde{f}_{vec} &=& - \frac{2\cosh \frac{\tilde{\Delta}}{2}}{2\sinh \frac{\omega_1}{2}
\cdot 2\sinh \frac{\omega_2}{2}} \left[\sum_{a<b}^{2N} \left(e^{-i\alpha_a+i\alpha_b}+e^{i\alpha_a-i\alpha_b}\right) \right]
+ \left(\!1 - \frac{2\cosh \frac{\tilde{\Delta}}{2}}{2\sinh \frac{\omega_1}{2}
\cdot 2\sinh \frac{\omega_1}{2}} \!\right) (2N\!-\!1)\, \nonumber\\
\hspace*{-0.5cm}
f_{mat}^{asym}&=& \frac{2 \cosh \frac{m}{2}}{2\sinh \frac{\omega_1}{2} \cdot 2\sinh \frac{\omega_2}{2}} \sum_{a<b}^{2N} \left(e^{\frac{b}{2}+i\alpha_a+i\alpha_b}+e^{-\frac{b}{2}-i\alpha_a-i\alpha_b} \right) \, \nonumber\\
\hspace*{-0.5cm}
f_{mat}^{fund}&=& \frac{1}{2\sinh \frac{\omega_1}{2} \cdot 2\sinh \frac{\omega_2}{2}} \sum_{l=1}^{N_f} \sum_{a=1}^{2N} \left(e^{M_l+i\alpha_a}
+e^{-M_l-i\alpha_a} \right)\,
\end{eqnarray}
where $\tilde{\Delta}=\Delta-2\pi i$. As in section 2.1,
the Haar measure contribution is absorbed into
$\tilde{f}_{vec}$. The instanton part $Z_{\rm inst}$ can be
computed from the ADHM construction of multi-instantons \cite{Hwang:2014uwa}.
We assume $Z_{\rm inst}\approx 1$ at the large $N$ saddle point that we shall
present below. We believe this can be shown using the methods of \cite{Hwang:2014uwa}.
With this assumed, and following the steps similar to section \ref{sec: Cardy-Sp(N)},
PE of the letter indices in the Cardy limit $|\omega_i|\ll 1$ are given by
\begin{eqnarray}
  \textrm{PE}\left[\tilde{f}_{vec}\right] &\sim&
  \exp \Bigg[ - \frac{1}{\omega_1 \omega_2} \sum_{a<b}^{2N} \sum_{\pm,\pm} \textrm