††institutetext: Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel

# The SUSY Index Beyond the Cardy Limit

Ohad Mamroud<EMAIL_ADDRESS>
###### Abstract

We analyze a set of contributions to the superconformal index of 4d
$\mathcal{N}=4$ $SU(N)$ super Yang-Mills using the Bethe Ansatz approach.
These contributions dominate at the large $N$ limit, where their leading order
in $N$ reproduces various supersymmetric Euclidean black hole saddles in the
dual theory, and they also dominate for finite $N$ in high temperature Cardy-
like limits. We compute the $O(N^{0})$ terms, including those exponentially
suppressed in the Cardy limit, and show that there are no $1/N$ corrections
beyond them. Under certain assumptions, it implies that the gravitational
perturbative series around these black hole saddles is 1-loop exact.

##  1 Introduction

The last few years have seen immense progress in the ability to compute the
entropy of black holes and quantum gravity partition functions through their
holographic dual conformal field theories, at least in supersymmetric cases.
The pioneering works of Benini:2015eyy ; Benini:2018ywd ; Cabo-Bizet:2018ehj ;
Choi:2018hmj  had succeeded in reproducing the leading contribution to the
black hole partition function and reproduced the black hole entropy, while
subsequent works had matched both perturbative and non-perturbative effects,
in various dimensions and settings Honda:2019cio ; ArabiArdehali:2019tdm ;
Kim:2019yrz ; Cabo-Bizet:2019osg ; Amariti:2019mgp ; Lezcano:2019pae ;
Lanir:2019abx ; Cabo-Bizet:2019eaf ; ArabiArdehali:2019orz ; Cabo-
Bizet:2020nkr ; Murthy:2020rbd ; Agarwal:2020zwm ; Benini:2020gjh ;
GonzalezLezcano:2020yeb ; Copetti:2020dil ; Cabo-Bizet:2020ewf ;
Amariti:2020jyx ; Hosseini:2016tor ; Benini:2016hjo ; Benini:2016rke ;
Hosseini:2016cyf ; Cabo-Bizet:2017jsl ; Azzurli:2017kxo ; Hosseini:2017fjo ;
Benini:2017oxt ; Hosseini:2018uzp ; Crichigno:2018adf ; Hosseini:2018usu ;
Suh:2018szn ; Fluder:2019szh ; Gang:2019uay ; Kantor:2019lfo ; Choi:2019zpz ;
Bobev:2019zmz ; Nian:2019pxj ; Benini:2019dyp ; David:2021qaa ; David:2020ems
; Amariti:2022nvn ; Hosseini:2021mnn ; Hong:2021bzg ; Goldstein:2020yvj ;
Jejjala:2021hlt ; Jejjala:2022lrm .

In this work, we will concentrate on the case of $\mathcal{N}=4$ $SU(N)$ super
Yang-Mills theory. With a specific choice of chemical potentials, the
partition function of this theory essentially becomes a protected quantity,
the superconformal index, which is sensitive to 1/16-BPS states in the theory.
On the other hand, the gravitational partition function of the dual theory can
be computed in a semi-classical manner as a sum over the Euclidean saddles of
type IIB supergravity on asymptotically $\text{AdS}_{5}\times S^{5}$ with
matching boundary conditions.

Previous works Benini:2018ywd ; Cabo-Bizet:2018ehj ; Choi:2019zpz ;
Choi:2021rxi  had shown various ways in which information about different
Euclidean gravitational saddles, describing black hole and orbifold
geometries, can be extracted from the superconformal index - either by showing
that at large $N$ specific contributions to the index scale as these
geometries, or by taking various “high temperature” Cardy-like limits where on
the gravity side one of these geometries becomes dominant, and calculating the
asymptotic behavior of the index in that limit.

In the present paper we concentrate on the field theory side, and compute a
set of discrete contributions111The very same ones that after taking the
aforementioned limits would have a gravitational interpretation. to the index
using the Bethe Ansatz approach, at finite $N$ and $\tau$, where $\tau$
denotes one of the chemical potentials, to be defined below. We then show that
in the large $N$ limit the perturbative contribution of our solution in $1/N$
truncates at $O(N^{0})$, and explicitly compute these terms. Moreover, we
compute the contribution at finite $N$ and in the Cardy limit $\tau\to 0$,
including $O(e^{-1/\tau})$ terms up to $O(e^{-N/\tau})$ and find that it takes
a particularly compact form. Since the leading order contributions we compute
match those of black hole saddles on the gravity side, we conjecture that the
gravitational perturbation theory around these saddles truncates as well.

Given some supercharge $\mathcal{Q}$, the superconformal index of the theory
is

$\mathcal{I}=\operatorname{Tr}\left[(-1)^{F}e^{-\beta\\{\mathcal{Q},\bar{\mathcal{Q}}\\}+2\pi
i\tau\left(J_{1}+\frac{1}{2}R_{3}\right)+2\pi
i\sigma\left(J_{2}+\frac{1}{2}R_{3}\right)+2\pi
i\Delta_{1}\mathfrak{q}_{1}+2\pi i\Delta_{2}\mathfrak{q}_{2}}\right]\;,$ (1.1)

where $J_{1,2}+\frac{1}{2}R_{3}$ and $\mathfrak{q}_{1,2}$ are conserved
charges that commute with $\mathcal{Q}$, for further details see Appendix A.
The index can be expressed as an integral over the holonomies $\\{u_{i}\\}$ of
the $SU(N)$ gauge field around the thermal cycle. As reviewed in Section 2 and
following Benini:2018mlo , when $\tau=\sigma$ and after some manipulation this
integral can be evaluated using the residue theorem. The residues are
solutions to some complicated equations, (2.9), termed the Bethe Ansatz
equations. Eventually, the index can be expressed as a sum over configurations
of the holonomies, schematically

$\mathcal{I}(\tau,\Delta_{1},\Delta_{2})=\kappa\sum_{\hat{u}\,\in\,\text{BAEs}}\mathcal{Z}(\hat{u};\Delta,\tau)\,H(\hat{u};\Delta,\tau)^{-1}+\text{vanishing
Jacobian}\;,$ (1.2)

where $\mathcal{Z}$ and $\kappa$ come from the integrand and $H$ is the
Jacobian relating the integration variable to the Bethe Ansatz equations. Some
residues might have a vanishing Jacobian, and their contribution to the index
will be described in an upcoming paper Aharony:2023 .

A general classification of the solutions is still beyond reach, but there is
a known family of solutions called the Hong-Liu solutions Hong:2018viz . They
are given by symmetric configurations of the $N$ holonomies $u_{i}$ on a
lattice of modular parameter $\tau$

$u_{j}=u_{\hat{\jmath}\hat{k}}=\bar{u}+\frac{\hat{\jmath}}{m}+\frac{\hat{k}}{n}\left(\tau+\frac{r}{m}\right)\
,\qquad\hat{\jmath}=0,\dots,m-1\;,\qquad\hat{k}=0,\dots,n-1\;,$ (1.3)

where $N=m\cdot n$, $r=0,\dots,n-1$ and $\bar{u}$ is chosen such that
$\sum_{j}u_{j}=0$, as the holonomies are of an $SU(N)$ matrix. These
configurations are sometimes denoted by the triplet $\\{m,n,r\\}$. Their
contribution to the index is denoted by $\mathcal{I}_{\\{m,n,r\\}}$, and there
could be additional contributions as well,

$\mathcal{I}=\sum_{\\{m,n,r\\}}\mathcal{I}_{\\{m,n,r\\}}+\text{others}\;.$
(1.4)

The partition function of the theory on $S^{1}\times S^{3}$ with a particular
choice of chemical potentials is proportional to its superconformal index
Closset:2013vra . Since the partition function should match that of the dual
gravitational theory, Aharony:2021zkr  argued222Following Benini:2018ywd  who
first used the method to match one of the gravitational saddles and reproduced
the entropy of the Lorentzian BPS black hole, and Cabo-Bizet:2018ehj ;
Choi:2018hmj  who achieved this by different methods. See also
GonzalezLezcano:2020yeb  for the expansion to this order of the BA solution.
that in the large $N$ limit each Hong-Liu configuration with $n\sim N$ and
$m,r\ll n$ reproduces the action of a different Euclidean geometry, in the
sense that the “action” of an HL configuration is

$\log\mathcal{I}_{\\{m,n,r\\}}=-c_{\\{m,n,r\\}}N^{2}+\log N+O(N^{0})\;,$ (1.5)

where $c_{\\{m,n,r\\}}N^{2}$ is the on-shell action of the corresponding
gravitational saddle. These geometries describe (complex) Euclidean black
holes and their orbifolds. The argument hinges on evaluating $\mathcal{Z}$ and
$\kappa$ in the large $N$ limit, and bounding the contribution of the Jacobian
$H$. Moreover, $\mathcal{Z}$ contains $O(e^{-c_{D_{3}}N})$ terms,
corresponding to the on-shell action of additional saddles containing
arbitrary number of wrapped supersymmetric D3 branes on top of these
geometries.

In this paper we exactly compute the Jacobian $H$ for the Hong-Liu solutions.
When $n=N\to\infty,r\ll n,m=1$, which is the large $N$ limit corresponding to
the various black hole saddles without orbifolds, we find that the series
truncates333This was conjectured in Aharony:2021zkr , but was not proven
there. at order $N^{0}$ up to exponential corrections

$\log\mathcal{I}_{\\{1,N,r\\}}\xrightarrow{N\to\infty}-c_{\\{1,N,r\\}}N^{2}+\log
N+c^{(0)}_{\\{1,N,r\\}}+O(e^{-Nc_{D3}})\;.$ (1.6)

We conjecture that this result carries over to the gravity side, meaning that
the gravitational perturbative series truncates at order $O(G_{N}^{0})$ and is
one-loop exact around these saddles. Moreover, the new exponentially small
corrections we have found are of the same order of magnitude as the
exponential correction found before, and therefore can again be considered as
arising from saddles with additional branes.

It should be stressed that what we actually compute in this paper is the
contribution of a set of Bethe Ansatz solutions to the index, whose leading
order behavior matches that of the gravitational saddles. We have not
performed any new gravitational computation. However, these are the leading
contributions to the index in various Cardy-like limits (see Cassani:2021fyv ;
ArabiArdehali:2021nsx ). As long as there is no other BA contribution whose
action differs from $\log\mathcal{I}_{\\{1,N,r\\}}$ by $O(N^{0})$, our
conjecture should hold. Amongst the Hong-Liu family this is indeed the case.
It seems likely that this is the case in general - any other BA configuration
would differ by at least one of the eigenvalues $u_{i}$, and thereby change
$4N$ out of the $4N^{2}$ terms in the relevant $\log\mathcal{Z}$, generically
changing it by $O(N)$.

We are also able to analyze the theory in the Cardy limit $\tau\to 0$, in
which the $\\{1,N,0\\}$ HL configuration dominates. Up to order
$O(e^{-1/\tau})$ our expression agrees with those of GonzalezLezcano:2020yeb ;
ArabiArdehali:2021nsx ; Cassani:2021fyv ; Cabo-Bizet:2019osg ; Kim:2019yrz .
The only corrections up to $O(e^{-N/\tau})$ come from the Jacobian. The final
expression takes the rather compact form444Again, we assume that the
$O(e^{-1/\tau})$ corrections come from the leading Bethe Ansatz configuration
and not from another, unknown, contribution.

$\mathcal{I}\xrightarrow{\tau\to 0}\frac{e^{\frac{\pi
i}{12}}}{\tau\mathcal{I}_{U(1)}}Ne^{-\frac{\pi
iN^{2}[\Delta_{1}]_{\tau}[\Delta_{2}]_{\tau}[\Delta_{3}]_{\tau}}{\tau^{2}}}\prod_{k=1}^{\infty}\left[\frac{1-\tilde{q}^{k}}{\left(1-\tilde{y}_{1}^{k}\right)\left(1-\tilde{y}_{2}^{k}\right)\left(1-\tilde{y}_{3}^{k}\right)}\right]^{2}\\\
\times\left(1+O\left(\tilde{q}^{N},\tilde{y_{a}}^{N}\right)\right)\;.$ (1.7)

where $\tilde{q}=e^{-2\pi i/\tau}$ and $\tilde{y}_{a}=e^{2\pi
i[\Delta_{a}]_{\tau}/\tau}$. Exact details and definition of $[\Delta]_{\tau}$
are given in Section 3. The final 1-loop determinant looks vaguely
similar555We thank Francesco Benini for pointing our attention to the
similarity. to the multi-graviton partition function of Kinney:2005ej ,
$\mathcal{I}_{\infty}=\prod_{k=1}^{\infty}\frac{(1-q^{k})^{2}}{(1-y_{1}^{k})(1-y_{2}^{k})(1-y_{3}^{k})}$,
where $y_{a}=e^{2\pi i\Delta_{1}},q=e^{2\pi i\tau}$, but with the modular
transformed fugacities and with a different power of the denominator. It would
be interesting to understand if there’s a deeper reason for the similarity.

These results raise several questions. The first is, of course, can they be
verified by any of the other approaches used to analyze the superconformal
index? A direct Cardy limit analysis seems quite hard. One would need to re-
sum the exponentially suppressed terms to find the form (1.7). However,
computing the first few exponentially suppressed terms in the Cardy limit
would test our assumptions about the contribution of any other BA solutions.
Another method is the large $N$ saddle point analysis of the $\mathcal{N}=4$
index matrix model, initiated in Choi:2021rxi , in which one of the saddles
reproduces the exponential term in (1.7) for the equal angular velocities
case. Our results imply that the perturbative expansion around this saddle,
combined with terms of lower order in $N$ in the potential used in
Choi:2021rxi , should also truncate.

A second, more ambitious question would be whether there is an argument from
the gravity side for this apparent one loop exactness in $G_{N}$, maybe
through localization on the supergravity side Dabholkar:2014wpa ;
Hristov:2018lod ; Hristov:2019xku ; Hristov:2021zai ; Iliesiu:2022kny .
Lastly, we note that a truncation of similar yet different spirit was found
numerically in Bobev:2022eus ; Bobev:2022jte ; Bobev:2022wem  for the
topologically twisted index of the three dimensional ABJM theory and for its
superconformal index in the Cardy limit. It would be nice to understand
whether this is a generic feature of supersymmetric partition functions, or
merely a coincidence.

The paper is organized as follows. Section 2 contains a brief overview of the
Bethe Ansatz approach. Section 3 describes the explicit behavior of the Hong-
Liu solutions in the large $N$ and Cardy limits, up to terms exponentially
suppressed in $N$. Appendix A includes definitions of the index and our
conventions. Appendix B contains the explicit computation of the Hong-Liu
contributions. Appendix C contains definitions of the special functions we use
throughout the paper.

##  2 The Bethe ansatz approach

Here we review the computation of the superconformal index for $\mathcal{N}=4$
super Yang-Mills (SYM) using the so-called Bethe Ansatz approach. The theory
is an $SU(N)$ superconformal gauge theory, with an $SU(4)_{R}$ R-symmetry, and
when put on the cylinder $\mathbb{R}\times S^{3}$ it has an additional $SO(4)$
symmetry due to the isometries of the underlying manifold. It has six real
adjoint scalars in the $\mathbf{6}$ of the R-symmetry group and four adjoint
Weyl fermions in its $\mathbf{4}$. Turning on chemical potentials for two
elements of the Cartan which commute with a specific supercharge, and equal
chemical potentials for the two angular momenta (see Appendix A for exact
details) the superconformal index can be written in the integral form

$\mathcal{I}=\frac{\mathcal{I}_{U(1)}^{N-1}}{N!}\oint_{\mathbb{T}^{N-1}}\left(\prod_{i=1}^{N-1}\frac{dz_{i}}{2\pi
iz_{i}}\right)\mathcal{Z}\left(\\{u_{i}\\};\Delta,\tau\right)\;,$ (2.1)

where $\mathbb{T}^{N-1}$ is the torus $\\{\forall i:|z_{i}|=1\\}$,
$z_{i}=e^{2\pi iu_{i}}$, $u_{ij}=u_{i}-u_{j}$, and
$u_{N}=-\sum_{i=1}^{N-1}u_{i}$ so the $u$’s can be thought of the eigenvalues
of an $N\times N$ traceless matrix. We also use the fugacity $q=e^{2\pi
i\tau}$. The integrand and prefactor are

$\mathcal{Z}\left(\\{u_{i}\\};\Delta,\tau\right)=\frac{\prod_{a=1}^{3}\prod_{i\neq
j}^{N}\widetilde{\Gamma}(\Delta_{a}+u_{ij};\tau,\tau)}{\prod_{i\neq
j}^{N}\widetilde{\Gamma}(u_{ij};\tau,\tau)}\;,\qquad\mathcal{I}_{U(1)}=(q;q)_{\infty}^{2}\prod_{a=1}^{3}\widetilde{\Gamma}\left(\Delta_{a};\tau,\tau\right)\;.$
(2.2)

The chemical potentials satisfy $2\tau-\sum_{a=1}^{3}\Delta_{a}\in\mathbb{Z}$.
The function $\widetilde{\Gamma}$ is the elliptic Gamma function and
$(q;q)_{\infty}$ is the Pochhammer symbol, see Appendix C.

The Bethe Ansatz approach uses the quasi-periodicity of $\widetilde{\Gamma}$,
(C.16). There is an elliptic function666The explicit form of this function is
$\hat{Q}_{i}=\frac{Q_{i}}{Q_{N}}\;,\quad Q_{i}=e^{2\pi
i\sum_{j=1}^{N}u_{ji}}\prod_{j\neq
i}^{N}\prod_{a=1}^{3}\frac{\theta_{0}(\Delta_{a}+u_{ji};\tau)}{\theta_{0}(\Delta_{a}-u_{ji};\tau)}\;.$
(2.3) Not all the $Q_{i}$’s are independent, $\prod_{i=1}^{N}Q_{i}=1$.
$\hat{Q}_{i}\left(\\{u_{i}\\};\Delta,\tau\right)$ with periods $1,\tau$ for
which

$\hat{Q}_{i}\left(\\{u_{i}\\};\Delta,\tau\right)\mathcal{Z}\left(\\{u_{i}\\};\Delta,\tau\right)=\mathcal{Z}\left(\\{\tilde{u}_{i}\\};\Delta,\tau\right)\;,$
(2.4)

where
$\tilde{u}=(u_{1},\dots,u_{i-1},u_{i}-\tau,u_{i+1},\dots,u_{N-1},u_{N}+\tau)$.
Now comes the trick. Multiply both the numerator and the denominator of (2.1)
by the same factor $\prod_{i=1}^{N-1}(1-\hat{Q}_{i})$ to get

$\displaystyle\mathcal{I}$
$\displaystyle=\frac{\mathcal{I}_{U(1)}^{N-1}}{N!}\oint_{\mathbb{T}^{N-1}}\left(\prod_{i=1}^{N-1}\frac{dz_{i}}{2\pi
iz_{i}}\right)\frac{\prod_{i=1}^{N-1}(1-\hat{Q}_{i})}{\prod_{i=1}^{N-1}(1-\hat{Q}_{i})}\mathcal{Z}\left(\\{u_{i}\\};\Delta,\tau\right)\;,$
(2.5)
$\displaystyle=\frac{\mathcal{I}_{U(1)}^{N-1}}{N!}\oint_{\mathcal{C}}\left(\prod_{i=1}^{N-1}\frac{dz_{i}}{2\pi
iz_{i}}\right)\frac{\mathcal{Z}\left(\\{u_{i}\\};\Delta,\tau\right))}{\prod_{i=1}^{N-1}(1-\hat{Q}_{i})}\;,$

where in the last line we’ve used the property (2.4) to change the contour
integral into the new contour
$\mathcal{C}=\bigcup_{i=1}^{N-1}\left(\\{|z_{i}|=1\\}-\\{|z_{i}|=q^{-1}\\}\right)$,
where the sign denotes the direction of the contour. Evaluating the integral
by the residue theorem, one finds that poles of the numerator are canceled by
poles of the denominator, such that the only poles of the integrand come from
zeroes of the denominator. A more thorough analysis will appear in an upcoming
paper Aharony:2023 , but a simple sufficient condition is the vanishing of all
the terms of the denominator

$\hat{Q}_{i}=1\;,\qquad\forall i=1,\dots,N-1\;.$ (2.6)

These are called777Another interpretation for these equations was initially
given by Closset:2017bse . the _Bethe Ansatz equations_. If the solutions are
isolated first order zeros of $1-\hat{Q}_{i}$ and the Jacobian for the change
of variables from $\\{z_{i}\\}$ to $\\{\hat{Q}_{i}\\}$ is well defined, then
the contribution to the integral is the inverse of the Jacobian times the
residue at the pole (which is simply the numerator evaluated at the pole).
Finally, the Bethe Ansatz approach to evaluating the index takes the form

$\mathcal{I}(q,y_{1},y_{2})=\frac{\mathcal{I}_{U(1)}^{N-1}}{N!}\sum_{\hat{u}\,\in\,\text{BAEs}}\mathcal{Z}(\hat{u};\Delta,\tau)\,H(\hat{u};\Delta,\tau)^{-1}+\text{vanishing
Jacobian}\;,$ (2.7)

where the sum is over well-behaved solutions to the Bethe Ansatz equations,
and the last term denotes contributions where the Jacobian vanishes. Such
contributions will be discussed in Aharony:2023 . Here $\mathcal{Z}$ and
$\mathcal{I}_{U(1)}$ are defined as in (2.2), and $H$ is the Jacobian,

$H=\det\left[\frac{1}{2\pi i}\,\frac{\partial(Q_{i}/Q_{N})}{\partial
u_{j}}\right]_{i,j=1,\dots,N-1}\,\equiv\,\det(\mathcal{A}_{ij})\;,$ (2.8)

we note that $\mathcal{I}_{U(1)}$ is the index of the free $U(1)$
$\mathcal{N}=4$ theory.

In Benini:2021ano  it was argued that the contribution of many solutions to
the Bethe Ansatz equations cancel each other, such that effectively only the
solutions to the _reduced Bethe Ansatz equations_

$Q_{i}=(-1)^{N-1}\;,\qquad i=1,\dots,N-1$ (2.9)

contribute to the index. There is a known family of solutions to these
equations Hong:2018viz . In order to find them, note that $Q_{i}$ is invariant
to shifting any of the $u$’s by $1$ or by $\tau$, meaning that the $u$’s
naturally live on a torus. The equations are then automatically solved if one
distributes the $u$’s symmetrically on the torus, meaning that for every
$u_{i}$ and $u_{j}$, there exists a $u_{k}$ such that $u_{ij}=-u_{ik}$ on the
torus.888For odd $N$ the factors in $Q_{i}$ associated with $u_{ij}$ and
$u_{ik}$ then automatically cancel each other, and the proof is complete. For
even $N$, a symmetric configuration still leaves for each $u_{i}$ one $u_{j}$
for which $u_{ij}=u_{ji}$ on the torus, which are exactly the half periods of
the torus. Their contribution to $Q_{i}$ is exactly $-1$, so it also solves
the reduced Bethe Ansatz equations. These solutions are termed the _Hong-Liu
solutions_ , whose explicit form is (1.3).

##  3 The Cardy and large $N$ limits

The contribution of the $\\{m,n,r\\}$ Hong-Liu solution to the index is
computed in Appendix B, and turns out to be999We use a notation where notation
$\check{\tau}=m\tau+r$ for the $\\{m,n,r\\}$ solution, and we will use two
different conventions for the chemical potentials,
${\bar{\Delta}}_{1,2}=\Delta_{1,2}$ and
${\bar{\Delta}}_{3}={\bar{\Delta}}_{1}+{\bar{\Delta}}_{2}$ or
$\Delta_{3}=2\tau-\Delta_{1}-\Delta_{2}-1$, such that
$\Delta_{3}=2\tau-{\bar{\Delta}}_{3}-1$. Some computations are easier using
the barred notation and some using the unbarred one. Only the unbarred
notation is used in Aharony:2021zkr .

$\displaystyle\log\mathcal{I}_{\\{m,n,r\\}}$
$\displaystyle=\sum_{a=1}^{3}\eta_{a}\left[m\log\widetilde{\Gamma}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n},\frac{\check{\tau}}{n}\right)+(m-N)\log\theta_{0}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n}\right)\right]$
(3.1)
$\displaystyle\qquad+N\log\left(\check{q}^{\frac{1}{n}},\check{q}^{\frac{1}{n}}\right)_{\infty}^{2}-\log\mathcal{I}_{U(1)}+\log
N-N\log n$ $\displaystyle\qquad-\left(N-1\right)\log
b_{N}-\log\det\left(1+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right)\;,$

where $\eta_{a}=(1,1,-1)$, $\check{q}=e^{2\pi i\check{\tau}}$, the special
functions are defined in Appendix C, and

$\displaystyle b_{N}$
$\displaystyle=-4+\frac{1}{\check{\tau}}\sum_{a=1}^{3}\left[2m\Delta_{a}+1+\mathcal{G}\left(0,\frac{N\Delta_{a}}{\check{\tau}};-\frac{n}{\check{\tau}}\right)\right]\;,$
(3.2) $\displaystyle\mathcal{B}_{ij}$
$\displaystyle=\delta_{ij}+1\;,\qquad\mathcal{B}^{-1}_{ij}=\delta_{ij}-\frac{1}{N}\;,$
$\displaystyle\mathcal{C}_{ij}$
$\displaystyle=\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{iN},\Delta_{a};\tau\right)+\mathcal{G}\left(u_{jN},\Delta_{a};\tau\right)-\mathcal{G}\left(u_{ij},\Delta_{a};\tau\right)-\mathcal{G}\left(0,\Delta_{a};\tau\right)\right]\;.$

We compute
$\log\det\left(1+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right)$ by
diagonalizing the matrix in Appendix B. The explicit result is given for the
cases $m=1$, where we find that the eigenvalues of the matrix
$\mathcal{B}^{-1}\mathcal{C}$ are practically the discrete Fourier transform
of $\mathcal{G}(\cdot,\frac{\Delta}{\check{\tau}};-\frac{1}{\check{\tau}})$,
which is closely related to the logarithmic derivative of $\theta_{0}$. We
comment on the $m\ll n$ case in the appendix.

#### Large $n$

Let us look at the $\\{m,n,r\\}$ configuration when $n\sim O(N)\to\infty$ and
$m,r\ll n$. Then using the modular identities for the various special
functions

$\displaystyle\log\theta_{0}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n}\right)$
$\displaystyle=\log\theta_{0}\left(\frac{N{\bar{\Delta}}_{a}}{\check{\tau}};-\frac{n}{\check{\tau}}\right)+\frac{\pi
i}{2}+\pi im{\bar{\Delta}}_{a}-\frac{\pi i\check{\tau}}{6n}$ (3.3)
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{n\pi
i}{\check{\tau}}\left(\left(m{\bar{\Delta}}_{a}\right)^{2}+\left(m{\bar{\Delta}}_{a}\right)+\frac{1}{6}\right)\;,$
$\displaystyle\log\widetilde{\Gamma}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n},\frac{\check{\tau}}{n}\right)$
$\displaystyle=-\pi
i\mathcal{Q}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n},\frac{\check{\tau}}{n}\right)-\log\theta_{0}\left(\frac{N{\bar{\Delta}}_{a}}{\check{\tau}};-\frac{n}{\check{\tau}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{k=0}^{\infty}\log\frac{\psi\left(\frac{k+1+m{\bar{\Delta}}_{a}}{\check{\tau}}n\right)}{\psi\left(\frac{k-m{\bar{\Delta}}_{a}}{\check{\tau}}n\right)}\;,$
$\displaystyle\log\left(\check{q}^{\frac{1}{n}},\check{q}^{\frac{1}{n}}\right)_{\infty}^{2}$
$\displaystyle=\frac{\pi i}{2}-\log\frac{\check{\tau}}{n}-\frac{\pi
i}{6}\left(\frac{\check{\tau}}{n}+\frac{n}{\check{\tau}}\right)+\log\left(\tilde{q}^{n};\tilde{q}^{n}\right)_{\infty}^{2}\;,$

where $\mathcal{Q}$ is defined in (C.21) and $\tilde{q}=e^{-2\pi
i/\check{\tau}}$.

If one shifts the ${\bar{\Delta}}$’s by integers to the regime
$-\operatorname{\mathbb{I}m}\frac{1}{\check{\tau}}>\operatorname{\mathbb{I}m}\frac{m{\bar{\Delta}}_{a}}{\check{\tau}}>0$
before performing the modular transformations then the special functions on
the right hand side above are exponentially suppressed in $n$. We denote

$[{\bar{\Delta}}]_{\tau}={\bar{\Delta}}\mod
1\;,\qquad-\operatorname{\mathbb{I}m}\frac{1}{\tau}>\operatorname{\mathbb{I}m}\frac{[{\bar{\Delta}}]_{\tau}}{\tau}>0\;.$
(3.4)

The special functions are now of order
$O(\tilde{q}^{n},\tilde{y}_{a}^{n},(\tilde{q}^{2}/\tilde{y}_{a})^{n})$
where101010We used $\Delta$ and not ${\bar{\Delta}}$ to define $y$, but it
doesn’t matter, as $\tilde{y}_{a}=\tilde{q}/e^{2\pi
i(\Delta_{1}+\Delta_{2}-1)/\check{\tau}}$, so a definition using
${\bar{\Delta}}$ would amount to exchanging $\tilde{y}_{3}$ and
$\tilde{q}/\tilde{y}_{3}$, both of which have magnitude smaller than 1 in our
limit. $\tilde{y}_{a}=e^{2\pi i[m\Delta_{a}]_{\check{\tau}}/\check{\tau}}$.

We now have to split into two cases, as in Aharony:2021zkr ; Benini:2018ywd :

$\left[m\Delta_{1}+m\Delta_{2}\right]_{\check{\tau}}=\begin{cases}\left[m\Delta_{1}\right]_{\check{\tau}}+\left[m\Delta_{2}\right]_{\check{\tau}}&\text{{1\textsuperscript{st}}\
case}\\\
\left[m\Delta_{1}\right]_{\check{\tau}}+\left[m\Delta_{2}\right]_{\check{\tau}}+1&\text{{2\textsuperscript{nd}}\
case}\end{cases}$ (3.5)

In order to compute the $b_{N}$ term in (3.1) up to exponential accuracy one
needs to shift its second argument using (C.27), which changes
$\Delta_{a}\to[\Delta_{a}]_{\check{\tau}}$ in $b_{N}$. Using
$[m\Delta_{3}]=2\check{\tau}-[m\Delta_{1}+m\Delta_{2}]-1$ we find that

$b_{N}=\pm\frac{1}{\check{\tau}}+\frac{1}{\check{\tau}}\sum_{a=1}^{3}\mathcal{G}\left(0;\frac{n[m\Delta_{a}]_{\check{\tau}}}{\check{\tau}},-\frac{n}{\check{\tau}}\right)\qquad\text{{1\textsuperscript{st}}/{2\textsuperscript{nd}}\
case}\;,$ (3.6)

where the upper sign is for the first case and the lower for the second. For
the first case

$\displaystyle\log\mathcal{I}_{\left\\{m,n,r\right\\}}\xrightarrow{n\sim
N\to\infty}$ $\displaystyle-\frac{\pi
iN^{2}}{m}\frac{\left[m\Delta_{1}\right]_{\check{\tau}}\left[m\Delta_{2}\right]_{\check{\tau}}\left[m\Delta_{3}\right]_{\check{\tau}}}{\check{\tau}^{2}}+\log
N-\log\mathcal{I}_{U(1)}$ (3.7) $\displaystyle\qquad+\frac{\pi
im}{12}-\log\check{\tau}-\log\det\left(1+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right)+O\left(\tilde{q}^{n},\tilde{y}_{a}^{n}\right)\;,$

where
$\left[m\Delta_{3}\right]_{\check{\tau}}=2\check{\tau}-\left[m\Delta_{1}+m\Delta_{2}\right]_{\check{\tau}}-1$.
The determinant was evaluated in (B.24), and using
$\tilde{q}=\tilde{y}_{1}\tilde{y}_{2}\tilde{y}_{3}$

$\displaystyle\det\left[\mathbf{1}+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right]\Bigg{|}_{{1\textsuperscript{st}},m=1}$
$\displaystyle=\prod_{k=1}^{\infty}\left[1-\sum_{a=1}^{3}\frac{\tilde{y}_{a}^{k}-\left(\tilde{q}/\tilde{y}_{a}\right)^{k}}{1-\tilde{q}^{k}}\right]^{2}+O\left(e^{-n}\right)$
(3.8)
$\displaystyle=\prod_{k=1}^{\infty}\left[\frac{(1-\tilde{y}_{1}^{k})(1-\tilde{y}_{2}^{k})(1-\tilde{y}_{3}^{k})}{\left(1-\tilde{q}^{k}\right)}\right]^{2}+O\left(e^{-n}\right)\;,$

where the corrections are of order
$O(\tilde{y}_{a}^{n},(\tilde{q}/\tilde{y}_{a})^{n})$. Overall, the large $N$
limit of the $\\{1,N,r\\}$ Hong-Liu solution is

$\mathcal{I}_{\\{1,N,r\\}}=\frac{e^{\frac{\pi
i}{12}}}{\check{\tau}\mathcal{I}_{U(1)}}Ne^{-\pi
iN^{2}\frac{\left[\Delta_{1}\right]_{\check{\tau}}\left[\Delta_{2}\right]_{\check{\tau}}\left[\Delta_{3}\right]_{\check{\tau}}}{\check{\tau}^{2}}}\prod_{k=1}^{\infty}\left[\frac{1-\tilde{q}^{k}}{(1-\tilde{y}_{1}^{k})(1-\tilde{y}_{2}^{k})(1-\tilde{y}_{3}^{k})}\right]^{2}\\\
\times\left[1+O\left(\tilde{q}^{N},\tilde{y}_{a}^{N}\right)\right]\,.$ (3.9)

For the second case, it is convenient to define
$[m\Delta_{a}]^{\prime}_{\check{\tau}}=[m\Delta_{a}]_{\check{\tau}}+1$.
Then111111In the case of even $N$ and odd $n$, the expression differs from
that in Aharony:2021zkr  by $\pi i$.

$\displaystyle\log\mathcal{I}_{\left\\{m,n,r\right\\}}\xrightarrow{n\sim
N\to\infty}$ $\displaystyle-\frac{\pi
iN^{2}}{m}\frac{\left[m\Delta_{1}\right]^{\prime}_{\check{\tau}}\left[m\Delta_{2}\right]^{\prime}_{\check{\tau}}\left[m\Delta_{3}\right]^{\prime}_{\check{\tau}}}{\check{\tau}^{2}}+\log
N$ (3.10) $\displaystyle\qquad-\log\mathcal{I}_{U(1)}-\frac{\pi im}{12}+\pi
i+\pi i(n+N)$
$\displaystyle\qquad-\log\check{\tau}-\log\det\left(1+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right)+O\left(\tilde{q}^{n},\tilde{y}_{a}^{n}\right)\;,$

where
$[m\Delta_{3}]_{\check{\tau}}^{\prime}=2\check{\tau}-[m\Delta_{1}]_{\check{\tau}}^{\prime}-[m\Delta_{2}]_{\check{\tau}}^{\prime}+1$.
We find it convenient to define the fugacities $\tilde{x}_{a}=e^{-2\pi
i[\Delta_{a}]^{\prime}_{\tau}}=\tilde{q}/\tilde{y}_{a}$ which satisfy
$\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}=\tilde{q}$ and
$|\tilde{q}|<|\tilde{x}_{a}|<1$. Picking the bottom sign in (B.24) leaves us
with the final expression for the deteminant in the second case

$\displaystyle\det\left[\mathbf{1}+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right]\Bigg{|}_{{2\textsuperscript{nd}},m=1}$
$\displaystyle=\prod_{k=1}^{\infty}\left[1+\sum_{a=1}^{3}\frac{\tilde{y}_{a}^{k}-\left(\tilde{q}/\tilde{y}_{a}\right)^{k}}{1-\tilde{q}^{k}}\right]^{2}+O\left(e^{-n}\right)$
(3.11)
$\displaystyle=\prod_{n=1}^{\infty}\left[\frac{(1-\tilde{x}_{1}^{k})(1-\tilde{x}_{2}^{k})(1-\tilde{x}_{3}^{k})}{1-\tilde{q}^{k}}\right]^{2}+O\left(e^{-n}\right)\;,$

where the corrections are of order
$O(\tilde{x}_{a}^{n},(\tilde{q}/\tilde{x}_{a})^{n})$. Finally, for the second
case the contribution of the $\\{1,N,r\\}$ Hong-Liu solution to the index is

$\mathcal{I}_{\\{1,N,r\\}}=-\frac{e^{-\frac{\pi
i}{12}}}{\check{\tau}\mathcal{I}_{U(1)}}Ne^{-\pi
iN^{2}\frac{\left[\Delta_{1}\right]_{\check{\tau}}^{\prime}\left[\Delta_{2}\right]_{\check{\tau}}^{\prime}\left[\Delta_{3}\right]_{\check{\tau}}^{\prime}}{\check{\tau}^{2}}}\prod_{k=1}^{\infty}\left[\frac{1-\tilde{q}^{k}}{(1-\tilde{x}_{1}^{k})(1-\tilde{x}_{2}^{k})(1-\tilde{x}_{3}^{k})}\right]^{2}\\\
\times\left[1+O\left(\tilde{q}^{N},\tilde{x}_{a}^{N}\right)\right]\,.$ (3.12)

#### The Cardy limit

In the Cardy limit $\tau\to 0$ the dominant contribution to the index comes
from the $\\{m,n,r\\}=\\{1,N,0\\}$ solution Benini:2018ywd ;
GonzalezLezcano:2020yeb ; Cabo-Bizet:2019eaf , where the holonomies approach
$u_{i}=0$. In order to get a good expansion121212Note that we don’t need to
expand in large $N$ before taking the Cardy limit - terms that are suppressed
in $N$ are also suppressed at high temperatures. in $1/\tau$ one should use
the modular transformations (3.3) on $\mathcal{I}_{U(1)}$ as well as on the
rest of (3.1),

$\displaystyle\log\mathcal{I}_{U\left(1\right)}$ $\displaystyle=\frac{\pi
i}{2}-\log\tau-\frac{\pi
i}{6}\left(\tau+\frac{1}{\tau}\right)+\log\left(\tilde{q};\tilde{q}\right)_{\infty}^{2}$
(3.13) $\displaystyle\quad-\sum_{a=1}^{3}\left[\pi
i\mathcal{Q}\left([\Delta_{a}]_{\tau};\tau;\tau\right)-\log\theta_{0}\left(\frac{[\Delta_{a}]_{\tau}}{\tau};-\frac{1}{\tau}\right)+\sum_{k=0}^{\infty}\log\frac{\psi\left(\frac{k+1+[\Delta_{a}]_{\tau}}{\tau}\right)}{\psi\left(\frac{k-[\Delta_{a}]_{\tau}}{\tau}\right)}\right]\;,$

so the index in the Cardy limit is

$\log\mathcal{I}\xrightarrow{\tau\to 0}\begin{cases}-\pi
i(N^{2}-1)\frac{[\Delta_{1}]_{\tau}[\Delta_{2}]_{\tau}[\Delta]_{\tau}}{\tau^{2}}+\log
N+O\left(\tilde{q},\tilde{y}_{a}\right)&\text{{1\textsuperscript{st}}\
case}\\\ -\pi
i(N^{2}-1)\frac{[\Delta_{1}]_{\tau}^{\prime}[\Delta_{2}]_{\tau}^{\prime}[\Delta]_{\tau}^{\prime}}{\tau^{2}}+\log
N+O\left(\tilde{q},\tilde{x}_{a}\right)&\text{{2\textsuperscript{nd}}\
case}\end{cases}\;.$ (3.14)

When $\Delta_{1}=\Delta_{2}=\Delta_{3}=\frac{2\tau-1}{3}$ the expression
agrees with ArabiArdehali:2021nsx ; Cassani:2021fyv ; Cabo-Bizet:2019osg ;
Kim:2019yrz ; GonzalezLezcano:2020yeb . Specifically, note that the prefactor
$N^{2}$ changes in this limit to $N^{2}-1$ and that the $\log\tau$ term
cancels out, both are necessary implications of the EFT argument of
Cassani:2021fyv . But we can do better. Given the exact expression for the
determinant term (B.24) and for the special functions involved, we can
systematically write down the leading corrections to
$\mathcal{I}_{\\{1,N,0\\}}$. The rather simple expression for the first case
is

$\boxed{\begin{aligned}
\mathcal{I}_{\\{1,N,0\\}}\big{|}_{{1\textsuperscript{st}}}\xrightarrow{\tau\to
0}\frac{e^{\frac{\pi i}{12}}}{\tau\mathcal{I}_{U(1)}}Ne^{-\pi
iN^{2}\frac{[\Delta_{1}]_{\tau}[\Delta_{2}]_{\tau}[\Delta_{3}]_{\tau}}{\tau^{2}}}\prod_{k=1}^{\infty}&\left[\frac{1-\tilde{q}^{k}}{\left(1-\tilde{y}_{1}^{k}\right)\left(1-\tilde{y}_{2}^{k}\right)\left(1-\tilde{y}_{3}^{k}\right)}\right]^{2}\\\
&\qquad\times\left(1+O\left(\tilde{y}_{a}^{N},\tilde{q}^{N},(\tilde{q}/\tilde{y}_{a})^{N}\right)\right)\end{aligned}}$
(3.15)

while for the second case it is

$\boxed{\begin{aligned}
\mathcal{I}_{\\{1,N,0\\}}\big{|}_{{2\textsuperscript{nd}}}\xrightarrow{\tau\to
0}-\frac{e^{-\frac{\pi i}{12}}}{\tau\mathcal{I}_{U(1)}}Ne^{-\pi
iN^{2}\frac{[\Delta_{1}]_{\tau}^{\prime}[\Delta_{2}]_{\tau}^{\prime}[\Delta_{3}]_{\tau}^{\prime}}{\tau^{2}}}\prod_{k=1}^{\infty}&\left[\frac{1-\tilde{q}^{k}}{\left(1-\tilde{x}_{1}^{k}\right)\left(1-\tilde{x}_{2}^{k}\right)\left(1-\tilde{x}_{3}^{k}\right)}\right]^{2}\\\
&\qquad\times\left(1+O\left(\tilde{x}_{a}^{N},\tilde{q}^{N},(\tilde{q}/\tilde{x}_{a})^{N}\right)\right)\end{aligned}}$
(3.16)

where $\tilde{y}_{a}=e^{2\pi i[\Delta]_{a}/\tau}$, $\tilde{q}=e^{-2\pi
i/\tau}$, $x_{a}=\tilde{q}/\tilde{y}_{a}$. These formulas receive corrections
at order $O(\tilde{x}^{N},\tilde{y}_{a}^{N},\tilde{q}^{N})$. Note that
$\frac{e^{\frac{\pi i}{12}}}{\tau\mathcal{I}_{U(1)}}$ changes the exponent to
$N^{2}-1$ and adds $O(\tilde{q},\tilde{y}_{a})$ terms to the action, so it
should also be considered if one wants to expand the order $O(e^{-1/\tau})$
terms in the Cardy limit order by order. $\mathcal{I}_{U(1)}$ is defined in
(2.2).

Assuming no other Bethe Ansatz configuration contributes in this limit, these
are also the leading asymptotics of the index itself. This assumption applies
for all other HL configurations, as they are smaller by
$O(e^{-N^{2}/\tau^{2}})$. We assume that in the Cardy limit other
contributions are also suppressed by at least $O(e^{-N/\tau})$.

#### Acknowledgment

The author thanks Erez Urbach and Tal Sheaffer for useful discussions, and is
grateful to Francesco Benini and Ofer Aharony for many useful conversations
and for comments on the manuscript. The work of the author was supported in
part by an Israel Science Foundation (ISF) center for excellence grant (grant
number 2289/18), by ISF grant no. 2159/22, by Simons Foundation grant 994296
(Simons Collaboration on Confinement and QCD Strings), by grant no. 2018068
from the United States-Israel Bi- national Science Foundation (BSF), by the
Minerva foundation with funding from the Federal German Ministry for Education
and Research, by the German Research Foundation through a German-Israeli
Project Cooperation (DIP) grant “Holography and the Swampland”, and by a
research grant from Martin Eisenstein. The author would like to thank SISSA
for its hospitality, funded by ERC-COG grant NP-QFT No. 864583. The author
also thanks the Simons Center for Geometry and Physics, Stony Brook University
for organizing the “Supersymmetric Black Holes, Holography and Microstate
Counting” workshop which helped facilitate this paper.

##  Appendix A The superconformal index of $\mathcal{N}=4$ SYM

Using $\mathcal{N}=1$ language, $\mathcal{N}=4$ SYM theory consists of a
vector multiplet and three chiral multiplets in the adjoint representation.
Its R-symmetry is $SU(4)_{R}$, whose Cartan is $U(1)^{3}$. We pick generators
$R_{1,2,3}$, each giving R-charge 2 to a single chiral multiplet and zero to
the other two, in a symmetric way. Local operators in the theory (and likewise
states on $S^{3}$) are labeled by two half-integer angular momenta $J_{1,2}$,
each rotating an $\mathbb{R}^{2}\subset\mathbb{R}^{4}$ in which $S^{3}$ is
embedded. The fermion number is defined as $F=2J_{1}$. Note that all fields in
the theory (and thus all states) have integer charges under $R_{1,2,3}$ and
obey

$F=2J_{1,2}=R_{1,2,3}\pmod{2}\;.$ (A.1)

The superconformal index of ${\cal N}=4$ SYM theory counts, with sign,
1/16-BPS states on $S^{3}$ preserving one complex supercharge $\mathcal{Q}$,
which we choose to be associated with a specific $U(1)_{R}$ symmetry the
generator $r=\frac{1}{3}(R_{1}+R_{2}+R_{3})$. The index can also keep track of
some combinations of the R-charges and the two angular momenta $J_{1,2}$. It
is useful to introduce two flavor generators
$\mathfrak{q}_{1,2}=\frac{1}{2}(R_{1,2}-R_{3})$ that commute with the
supercharge and with $r$. The superconformal index Romelsberger:2005eg ;
Kinney:2005ej  is then defined by the trace131313Often the powers in the index
are written as $p^{J_{1}+\frac{1}{2}r}\,q^{J_{2}+\frac{1}{2}r}$. Compared to
this convention, we have swallowed a power of $(pq)^{1/3}$ into $y_{1}$ and
$y_{2}$ in order to obtain a single-valued function. The relation of our
variables to those of Kinney:2005ej  is $p=t^{3}y\big{|}_{\text{there}}$,
$q=t^{3}/y\big{|}_{\text{there}}$, $y_{1}=t^{2}v\big{|}_{\text{there}}$,
$y_{2}=t^{2}w/v\big{|}_{\text{there}}$.

$\mathcal{I}(p,q,y_{1},y_{2})=\operatorname{Tr}\left[(-1)^{F}\,e^{-\beta\\{\mathcal{Q},\bar{\mathcal{Q}}\\}}\,p^{J_{1}+\frac{1}{2}R_{3}}\,q^{J_{2}+\frac{1}{2}R_{3}}\,y_{1}^{\mathfrak{q}_{1}}\,y_{2}^{\mathfrak{q}_{2}}\right]$
(A.2)

over the Hilbert space on $S^{3}$, where we normalized the vacuum contribution
to 1. Here $p$, $q$, $y_{1,2}$ are fugacities, and it is convenient to
introduce chemical potentials $\sigma$, $\tau$, $\Delta_{1,2}$ such that

$p=e^{2\pi i\sigma}\;,\qquad\qquad q=e^{2\pi i\tau}\;,\qquad\qquad
y_{1,2}=e^{2\pi i\Delta_{1,2}}\;.$ (A.3)

The index is well-defined for $|p|,|q|<1$, namely for
$\operatorname{\mathbb{I}m}(\tau),\operatorname{\mathbb{I}m}(\sigma)>0$. Given
(A.1), the index is a single-valued function of the fugacities (A.3) - it is
periodic under integer shifts of the chemical potentials $\sigma$, $\tau$,
$\Delta_{1}$ and $\Delta_{2}$. By standard arguments Witten:1982df , the index
only counts states annihilated by $\mathcal{Q}$ and $\bar{\mathcal{Q}}$ and is
thus independent of $\beta$.

In addition to the index, one could also compute the thermal partition
function of the theory on $S^{1}\times S^{3}$ in the presence of general
chemical potentials

$Z_{S^{1}\times S^{3}}=\operatorname{Tr}\left[e^{-\beta
H+\sum_{i=1}^{2}\beta\Omega_{i}J_{i}+\frac{1}{2}\sum_{a=1}^{3}\beta\Phi_{a}R_{a}}\right]\;.$
(A.4)

When the chemical potentials satisfy the constraint Cabo-Bizet:2018ehj

$\beta\left(1+\Omega_{1}+\Omega_{2}-\Phi_{1}-\Phi_{2}-\Phi_{3}\right)=2\pi
in\;,\qquad n\in\mathbb{Z}\;,$ (A.5)

there is a well defined complex Killing spinor on this space. The Killing
spinor is periodic around the thermal cycle when $n$ is even, and it is anti-
periodic when $n$ is odd, and so are the other fermions in the theory. The
existence of the Killing spinor implies the existence of a complex supercharge
$\mathcal{Q}$ on the manifold, whose algebra is Closset:2013vra

$\\{\mathcal{Q},\bar{\mathcal{Q}}\\}=H-J_{1}-J_{2}-\frac{3}{2}r\;.$ (A.6)

Combining this relation with the constraint (A.5), the thermal partition
function becomes

$Z_{S^{1}\times S^{3}}=\operatorname{Tr}\Big{[}(-1)^{(n+1)F}e^{\pi
inR_{3}}\exp\Big{\\{}-\beta\\{\mathcal{Q},\bar{\mathcal{Q}}\\}+2\pi
i\tau\left(J_{1}+\frac{1}{2}R_{3}\right)\\\ +2\pi
i\sigma\left(J_{2}+\frac{1}{2}R_{3}\right)+\pi i\Delta_{1}(R_{1}-R_{3})+\pi
i\Delta_{2}(R_{2}-R_{3})\Big{\\}}\Big{]}\;,$ (A.7)

the additional $(-1)^{(n+1)F}$ coming from the periodicity of the fermions,
and we defined

$\tau=\frac{\beta(\Omega_{1}-1)}{2\pi
i}\;,\qquad\sigma=\frac{\beta(\Omega_{2}-1)}{2\pi
i}\;,\Delta_{a}=\frac{\beta(\Phi_{a}-1)}{2\pi i}\;,$ (A.8)

for convenience. The constraint (A.5) is now

$\tau+\sigma-\Delta_{1}-\Delta_{2}-\Delta_{3}=n\;,\qquad n\in\mathbb{Z}\;.$
(A.9)

Further simplification arises after applying (A.1), and the partition function
turns out to be

$Z_{S^{1}\times
S^{3}}=\operatorname{Tr}\left[(-1)^{F}e^{-\beta\\{\mathcal{Q},\bar{\mathcal{Q}}\\}+2\pi
i\tau\left(J_{1}+\frac{1}{2}R_{3}\right)+2\pi
i\sigma\left(J_{2}+\frac{1}{2}R_{3}\right)+2\pi
i\Delta_{1}\mathfrak{q}_{1}+2\pi i\Delta_{2}\mathfrak{q}_{2}}\right]\;,$
(A.10)

which is precisely the index (A.2) up to the normalization of the vacuum.

The index is often written when $\tau,\sigma$ couple to
$J_{1,2}+\frac{1}{2}r$, instead as in (A.2). A redefinition of the flavor
chemical potentials, $\xi_{1,2}=\Delta_{1,2}+\frac{\tau+\sigma}{3}$,
transforms the index to that form.

We note that for the particular choice $\Delta_{1,2}=\frac{2\tau-1}{3}$, which
fits $n=1$, the partition function equals (up to the vacuum contribution) to
the index used in Cassani:2021fyv , also termed the R-charge index or the
superconformal index on the second sheet,

$Z_{S^{1}\times
S^{3}}\sim\mathcal{I}_{R}=\operatorname{Tr}\left[(-1)^{R}p^{J_{1}+\frac{1}{2}r}q^{J_{2}+\frac{1}{2}r}\right]\;.$
(A.11)

##  Appendix B Simplifying the Hong-Liu contribution

Let us compute the contribution of the $\\{m,n,r\\}$ Hong-Liu solution (1.3)
to the index. Each such solution contributes as

$\mathcal{I}_{\\{m,n,r\\}}=N\cdot
N!\cdot\frac{\mathcal{I}_{U(1)}^{N-1}}{N!}\cdot\mathcal{Z}(u_{\\{m,n,r\\}};{\bar{\Delta}},\tau)\cdot
H^{-1}\left(u_{\\{m,n,r\\}};{\bar{\Delta}},\tau\right)\;,$ (B.1)

Here $\mathcal{Z}$, $H$, and $\mathcal{I}_{U(1)}$ are defined in (2.1), (2.8).
The factor $N\cdot N!$ takes into account the multiplicity of each solution -
a factor of $N!$ comes from the Weyl group action that permutes the
eigenvalues $\\{u_{i}\\}$, and the factor of $N$ comes from the inequivalent
choice of points on the torus that give rise to the same $u_{ij}$’s, and
correspond to the action of the center symmetry as it shifts the center of
mass of the first $N-1$ holonomies by $\frac{\ell_{1}\tau+\ell_{2}}{N}$, where
$\ell_{1}=0,\dots,m-1$ and $\ell_{2}=0,\dots,n-1$.

We also remind that in our notation $\check{\tau}=m\tau+r$ for the
$\\{m,n,r\\}$ solution, and we will use two different conventions for the
chemical potentials, ${\bar{\Delta}}_{1,2}=\Delta_{1,2}$ and
${\bar{\Delta}}_{3}={\bar{\Delta}}_{1}+{\bar{\Delta}}_{2}$ or
$\Delta_{3}=2\tau-\Delta_{1}-\Delta_{2}-1$, such that
$\Delta_{3}=2\tau-{\bar{\Delta}}_{3}-1$. Some computations are easier using
the barred notation and some using the unbarred one. The paper Aharony:2021zkr
uses exclusively unbarred notation.

###  B.1 Computing $\mathcal{Z}$

We need to compute

$\log\mathcal{Z}(u;{\bar{\Delta}},\tau)=\gamma_{{\bar{\Delta}}_{1}}+\gamma_{{\bar{\Delta}}_{2}}-\gamma_{{\bar{\Delta}}_{1}+{\bar{\Delta}}_{2}}-\gamma_{0}\;,\qquad\gamma_{\bar{\Delta}}\equiv\sum_{i\neq
j}^{N-1}\log\widetilde{\Gamma}(u_{ij}+{\bar{\Delta}};\tau,\tau)\;.$ (B.2)

For any function $f$, $\sum_{i\neq
j=0}^{M-1}f(u_{i}-u_{j})=\sum_{i,j=0}^{M-1}f(u_{i}-u_{j})-Mf(0)$, and thus
summation over regular shifts along the two cycles of the $(1,\tau)$ torus is
given by141414One can check that the series expansions are valid for some
domain for the arguments, and then analytically continued to generic argument.
For the first identity the domain is
$0<\operatorname{\mathbb{I}m}({\bar{\Delta}})<2\operatorname{\mathbb{I}m}(\tau)$,
while for the second the domain is
$0<\operatorname{\mathbb{I}m}\left(\frac{n-1}{n}\tau\right)<\operatorname{\mathbb{I}m}({\bar{\Delta}})<\operatorname{\mathbb{I}m}\left(\frac{n+1}{n}\tau\right)$.

$\sum_{\hat{\imath}\neq\hat{\jmath}=0}^{m-1}\log\widetilde{\Gamma}\left(u+\frac{\hat{\imath}-\hat{\jmath}}{m};\tau,\tau\right)+m\log\widetilde{\Gamma}(u;\tau,\tau)=\sum_{\hat{\imath},\hat{\jmath}=0}^{m-1}\log\widetilde{\Gamma}\left(u+\frac{\hat{\imath}-\hat{\jmath}}{m};\tau,\tau\right)\\\
=\sum_{\hat{\imath},\hat{\jmath}=0}^{m-1}\sum_{\ell=1}^{\infty}\frac{1}{\ell}\frac{z^{\ell}e^{2\pi
i\ell(\hat{\imath}-\hat{\jmath})/m}-q^{2\ell}z^{-\ell}e^{-2\pi
i\ell(\hat{\imath}-\hat{\jmath})/m}}{(1-q^{\ell})^{2}}=m^{2}\sum_{\ell=1}^{\infty}\frac{1}{m\ell}\frac{z^{m\ell}-q^{2m\ell}z^{-m\ell}}{(1-q^{m\ell})^{2}}\\\
=m\log\widetilde{\Gamma}(mu;m\tau,m\tau)\;,$ (B.3)

and

$\sum_{\hat{\imath}\neq\hat{\jmath}}^{n-1}\log\widetilde{\Gamma}\left(u+\frac{\hat{\imath}-\hat{\jmath}}{n}\tau;\tau,\tau\right)+n\log\widetilde{\Gamma}(u;\tau,\tau)=\sum_{\hat{\imath},\hat{\jmath}=0}^{n-1}\log\widetilde{\Gamma}\left(u+\frac{\hat{\imath}-\hat{\jmath}}{n}\tau;\tau,\tau\right)\\\
=\sum_{\hat{\imath},\hat{\jmath}=0}^{n-1}\sum_{\ell=1}^{\infty}\frac{1}{\ell}\frac{z^{\ell}q^{(\hat{\imath}-\hat{\jmath})\ell/n}-q^{2\ell}z^{-\ell}q^{-(\hat{\imath}-\hat{\jmath})\ell/n}}{(1-q^{\ell})^{2}}=\sum_{\ell=1}^{\infty}\frac{1}{\ell}\frac{z^{\ell}q^{-\ell+\frac{\ell}{n}}\frac{(1-q^{\ell})^{2}}{(1-q^{\frac{\ell}{n}})^{2}}-q^{2\ell}z^{-\ell}q^{-\ell+\frac{\ell}{n}}\frac{(1-q^{\ell})^{2}}{(1-q^{\frac{\ell}{n}})^{2}}}{(1-q^{\ell})^{2}}\\\
=\sum_{\ell=1}^{\infty}\frac{1}{\ell}\frac{z^{\ell}q^{-\ell+\frac{\ell}{n}}-q^{\ell+\frac{\ell}{n}}z^{-\ell}}{(1-q^{\frac{\ell}{n}})^{2}}=\log\widetilde{\Gamma}\left(u-(n-1)\frac{\tau}{n};\frac{\tau}{n},\frac{\tau}{n}\right)\;,$
(B.4)

The argument of the resulting elliptic Gamma function can be shifted,

$\displaystyle\log\widetilde{\Gamma}$
$\displaystyle\left(u-(n-1)\frac{\tau}{n};\frac{\tau}{n},\frac{\tau}{n}\right)-\log\widetilde{\Gamma}\left(u;\frac{\tau}{n},\frac{\tau}{n}\right)$
(B.5)
$\displaystyle=-\sum_{k=1}^{n-1}\log\theta_{0}\left(u-k\frac{\tau}{n};\frac{\tau}{n}\right)$
$\displaystyle=-(n-1)\log\theta_{0}\left(u;\frac{\tau}{n}\right)-\sum_{k=1}^{n-1}\left[\pi
ik+\pi ik\left(2u-(k+1)\frac{\tau}{n}\right)\right]$
$\displaystyle=-(n-1)\log\theta_{0}\left(u;\frac{\tau}{n}\right)-\frac{\pi
i(n-1)}{3}(3nu-(n+1)\tau)-\pi i(n-1)\;.$

Overall, for the $\\{m,n,r\\}$ HL solution and when ${\bar{\Delta}}\neq 0$ we
find

$\displaystyle\gamma_{\bar{\Delta}}$
$\displaystyle=\sum_{\hat{k}\neq\hat{\ell}}^{n-1}\sum_{\hat{\imath},\hat{\jmath}=0}^{m-1}\log\widetilde{\Gamma}\left(\frac{\hat{\imath}-\hat{\jmath}}{m}+\frac{\hat{k}-\hat{\ell}}{n}\left(\tau+\frac{r}{m}\right)+{\bar{\Delta}};\tau,\tau\right)$
(B.6)
$\displaystyle\quad+n\sum_{\hat{\imath}\neq\hat{\jmath}}^{m-1}\log\widetilde{\Gamma}\left(\frac{\hat{\imath}-\hat{\jmath}}{m}+{\bar{\Delta}};\tau,\tau\right)$
$\displaystyle=m\sum_{\hat{k}\neq\hat{\ell}}^{n-1}\log\widetilde{\Gamma}\left(\frac{\hat{k}-\hat{\ell}}{n}\check{\tau}+m{\bar{\Delta}};\check{\tau},\check{\tau}\right)+N\log\frac{\widetilde{\Gamma}\left(m{\bar{\Delta}};\check{\tau},\check{\tau}\right)}{\widetilde{\Gamma}\left({\bar{\Delta}};\tau,\tau\right)}$
$\displaystyle=m\log\widetilde{\Gamma}\left(m{\bar{\Delta}}-(n-1)\frac{\check{\tau}}{n};\frac{\check{\tau}}{n},\frac{\check{\tau}}{n}\right)-N\log\widetilde{\Gamma}\left({\bar{\Delta}};\tau,\tau\right)$
$\displaystyle=m\log\widetilde{\Gamma}\left(m{\bar{\Delta}};\frac{\check{\tau}}{n},\frac{\check{\tau}}{n}\right)-(N-m)\log\theta_{0}\left(m{\bar{\Delta}};\frac{\check{\tau}}{n}\right)-N\log\widetilde{\Gamma}\left({\bar{\Delta}};\tau,\tau\right)$
$\displaystyle\qquad-\frac{\pi
i(N-m)}{3}(3N{\bar{\Delta}}-(n+1)\frac{\check{\tau}}{n})-\pi i(n-1)\;,$

where $\check{\tau}=m\tau+r$. The limit ${\bar{\Delta}}\to 0$ can now be taken
as well. This is most conveniently done by expressing the elliptic Gamma and
theta functions as an infinite product

$\displaystyle\lim_{u\to 0}\log\Gamma\left(u;\tau,\tau\right)$
$\displaystyle=\lim_{z\to
1}\log\left[\frac{\prod_{k,\ell=0}^{\infty}\left(1-q^{k+1}q^{\ell+1}z^{-1}\right)}{\prod_{k,\ell=0}^{\infty}\left(1-q^{k}q^{\ell}z\right)}\right]$
(B.7) $\displaystyle=\lim_{z\to
1}\log\left(\frac{1}{1-z}\right)+2\log\left[\prod_{\ell=1}^{\infty}\frac{1}{1-q^{\ell}}\right]$
$\displaystyle=\lim_{z\to
1}\log\left(\frac{1}{1-z}\right)-2\log(q;q)_{\infty}\;,$

and

$\displaystyle\lim_{u\to 0}\log\theta_{0}\left(u;\tau\right)$
$\displaystyle=\lim_{z\to
1}\log\left[\prod_{k=0}(1-zq^{k})(1-z^{-1}q^{k+1})\right]$ (B.8)
$\displaystyle=\lim_{z\to
1}\log(1-z)+\log\left[\prod_{k=1}(1-q^{k})(1-q^{k})\right]$
$\displaystyle=\lim_{z\to 1}\log(1-z)+2\log(q;q)_{\infty}$

so

$\gamma_{0}=\lim_{{\bar{\Delta}}\to 0}\gamma_{{\bar{\Delta}}}=\frac{\pi
i(N-m)(n+1)}{3}\frac{\check{\tau}}{n}-\pi i(n-1)-N\log
m+2N\log\frac{\left(q;q\right)_{\infty}}{\left(\check{q}^{\frac{1}{n}};\check{q}^{\frac{1}{n}}\right)_{\infty}}\;.$
(B.9)

Finally, we can put it all together to compute the total contribution
$\mathcal{Z}$ with $\check{q}=e^{2\pi i\check{\tau}}$

$\boxed{\begin{aligned}
\mathcal{Z}\big{(}u_{\\{m,n,r\\}},&{\bar{\Delta}}_{1},{\bar{\Delta}}_{2},\tau\big{)}\\\
&=\sum_{a=1}^{3}\eta_{a}\Big{[}m\log\widetilde{\Gamma}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n},\frac{\check{\tau}}{n}\right)+(m-N)\log\theta_{0}\left(m{\bar{\Delta}}_{a};\frac{\check{\tau}}{n}\right)\\\
&\qquad\qquad-N\log\widetilde{\Gamma}({\bar{\Delta}}_{a};\tau,\tau)\Big{]}+N\log
m-2N\log\frac{\left(q,q\right)_{\infty}}{\left(\check{q}^{\frac{1}{n}},\check{q}^{\frac{1}{n}}\right)_{\infty}}\end{aligned}}$
(B.10)

where we have denoted
${\bar{\Delta}}_{3}={\bar{\Delta}}_{1}+{\bar{\Delta}}_{2}$ and
$\eta_{a}=(1,1,-1)$.

###  B.2 The Jacobian $H$

The Jacobian can be written as (see appendix B.3 in Aharony:2021zkr )

$\log H=N\log N+\left(N-1\right)\log
b_{N}+\log\det\left(1+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right)\;,$
(B.11)

where

$\displaystyle b_{N}$
$\displaystyle=-4+\frac{1}{\check{\tau}}\sum_{a=1}^{3}\left[2m\Delta_{a}+1+\mathcal{G}\left(0,\frac{N\Delta_{a}}{\check{\tau}};-\frac{n}{\check{\tau}}\right)\right]\;,$
(B.12) $\displaystyle\mathcal{B}_{ij}$
$\displaystyle={\bar{\Delta}}_{ij}+1\;,\qquad\mathcal{B}^{-1}_{ij}={\bar{\Delta}}_{ij}-\frac{1}{N}\;,$
$\displaystyle\mathcal{C}_{ij}$
$\displaystyle=\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{iN},\Delta_{a};\tau\right)+\mathcal{G}\left(u_{jN},\Delta_{a};\tau\right)-\mathcal{G}\left(u_{ij},\Delta_{a};\tau\right)-\mathcal{G}\left(0,\Delta_{a};\tau\right)\right]\;.$

We will now compute
$\log\det\left(1+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right)$ by
diagonalizing the $(N-1)\times(N-1)$ matrix
$\mathcal{X}=\mathcal{B}^{-1}\mathcal{C}$. Only the expression for $m=1$ will
be reproduced here151515The more general case $n\gg m>1$ is interesting and
relevant for the orbifold saddles. While the eigenvalues for that case can be
found as a formal power series, it is not one with a good large $N$ limit, in
the sense that it contains powers of $q^{1/n}$. Perhaps some modular
properties could be used to get a good large $N$ limit, but we were not able
to do that.. However, we will restrict to $m=1$ only when needed, at (B.21).
Let’s begin. Using (B.26) and (B.29) in Aharony:2021zkr

$\Upsilon\equiv\sum_{j=1}^{N}\mathcal{G}\left(u_{j_{1}j},{\bar{\Delta}};\tau\right)=\sum_{\hat{\jmath}=0}^{m-1}\sum_{k=0}^{n-1}\mathcal{G}\left(\frac{\hat{\jmath}_{1}-\hat{\jmath}}{m}+\frac{k_{1}-k}{n}\frac{\check{\tau}}{m},{\bar{\Delta}};\tau\right)\\\
=\frac{N}{\check{\tau}}\left[2m{\bar{\Delta}}_{a}+1-\check{\tau}+\mathcal{G}\left(0,\frac{N{\bar{\Delta}}}{\check{\tau}};-\frac{n}{\check{\tau}}\right)\right]=m-N+m\mathcal{G}\left(0,m{\bar{\Delta}};\frac{\check{\tau}}{n}\right)$
(B.13)

is independent of $j_{1}$. Suppressing the $\tau$ argument of $\mathcal{G}$
for brevity,

$\sum_{\ell=1}^{N-1}\mathcal{C}_{\ell
j}=\sum_{\ell=1}^{N-1}\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{\ell
N},\Delta_{a}\right)+\mathcal{G}\left(u_{jN},\Delta_{a}\right)-\mathcal{G}\left(u_{\ell
j},\Delta_{a}\right)-\mathcal{G}\left(0,\Delta_{a}\right)\right]\\\
=\sum_{a=1}^{3}\Big{[}\Upsilon-\mathcal{G}\left(u_{NN},\Delta_{a}\right)+\left(N-1\right)\mathcal{G}\left(u_{jN},\Delta_{a}\right)-\left(\Upsilon-\mathcal{G}\left(u_{Nj},\Delta_{a}\right)\right)-\left(N-1\right)\mathcal{G}\left(0,\Delta_{a}\right)\Big{]}\\\
=N\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{jN},\Delta_{a}\right)-\mathcal{G}\left(0,\Delta_{a}\right)\right]\;.$
(B.14)

where we used the fact that $\mathcal{G}(u,\Delta)$ is an even function of
$u$, and so

$\mathcal{X}_{ij}=\mathcal{C}_{ij}-\frac{1}{N}\sum_{\ell=1}^{N-1}\mathcal{C}_{\ell
j}=\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{iN},\Delta_{a}\right)-\mathcal{G}\left(u_{ij},\Delta_{a}\right)\right]$
(B.15)

Now let’s find the eigenvalues $\lambda$ and eigenvectors $v$ of
$\mathcal{X}$. Denote $\bar{v}=\sum_{j=1}^{N-1}v_{j}$. The eigenvalue
equations are

$\lambda
v_{i}=\sum_{j=1}^{N-1}\mathcal{X}_{ij}v_{j}=\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{iN},\Delta_{a}\right)\bar{v}-\sum_{j=1}^{N-1}\mathcal{G}\left(u_{ij},\Delta_{a}\right)v_{j}\right]\;,\qquad
i=1,\dots,N-1\;,$

Now define a vector of length $N$,
$w=\left(v_{1},v_{2},\dots,v_{N-1},-\bar{v}\right)$, for which

$\lambda
w_{i}=-\sum_{j=1}^{N}\sum_{a=1}^{3}\mathcal{G}\left(u_{ij},\Delta_{a}\right)w_{j}\;,\qquad
i=1,\dots,N-1\;.$ (B.16)

By summing all the equations and multiplying by $-1$ we also find the
equations for $w_{N}$

$\lambda
w_{N}=-\sum_{a=1}^{3}\left[\Upsilon\bar{v}-\mathcal{G}\left(0,\Delta_{a}\right)\bar{v}-\sum_{j=1}^{N-1}\left(\Upsilon-\mathcal{G}\left(u_{jN},\Delta_{a}\right)\right)v_{j}\right]\\\
=-\sum_{j=1}^{N-1}\sum_{a=1}^{3}\left[\mathcal{G}\left(u_{jN},\Delta_{a}\right)v_{j}-\mathcal{G}\left(0,\Delta_{a}\right)\bar{v}\right]=-\sum_{j=1}^{N}\sum_{a=1}^{3}\mathcal{G}\left(u_{jN},\Delta_{a}\right)w_{j}$

so the eigenvalue equations are (note the range of $i$)

$\lambda
w_{i}=-\sum_{j=1}^{N}\sum_{a=1}^{3}\mathcal{G}\left(u_{ij},\Delta_{a};\tau\right)w_{j}\;,\qquad
i=1,\dots,N\;.$ (B.17)

Writing the indices using the $u_{i}\equiv u_{\hat{\imath}\hat{\ell}}$
notation reduces us to the form

$\lambda
w_{\hat{\imath}\hat{\ell}}=-\sum_{\hat{\jmath}=0}^{m-1}\sum_{\hat{k}=0}^{n-1}f(\hat{\imath}-\hat{\jmath},\hat{\ell}-\hat{k})w_{\hat{\jmath}\hat{\ell}}\;,\qquad
f(\hat{\jmath},\hat{k})=\sum_{a=1}^{3}\mathcal{G}\left(\frac{\hat{\jmath}}{m}+\frac{\hat{k}}{n}(\tau+\frac{r}{m}),\Delta_{a};\tau\right)\;.$
(B.18)

These equations can be solved by a discrete Fourier transform, defined by

$\widetilde{w}_{b,c}=\sum_{\hat{\jmath}=0}^{m-1}\sum_{\hat{k}=0}^{n-1}w_{\hat{\jmath}\hat{k}}e^{-2\pi
i\hat{\jmath}b/m}e^{-2\pi i\hat{k}c/n}\;,$ (B.19)

such that the eigenvalue equations become161616The Fourier transform of the
discrete convolution is
$\sum_{\hat{k}=0}^{N-1}\sum_{\hat{\jmath}=0}^{N-1}f(\hat{k}-\hat{\jmath})g(\hat{\jmath})e^{-2\pi
i\hat{k}a/N}=\sum_{\hat{k}=0}^{N-1}\sum_{\hat{\jmath}=0}^{N-1}\sum_{b=0}^{N-1}\frac{\tilde{f}(b)}{N}e^{2\pi
ib(\hat{k}-\hat{\jmath})/N}g(\hat{\jmath})e^{-2\pi i\hat{k}a/N}\\\
=\sum_{\hat{\jmath}=0}^{N-1}\tilde{f}(a)e^{-2\pi
ia\hat{\jmath}/N}g(\hat{\jmath})=\tilde{f}(a)\tilde{g}(-a)\;.$

$\lambda\widetilde{w}_{b,c}=-\widetilde{f}_{b,c}\widetilde{w}_{b,c}\;.$ (B.20)

The eigenvalues are thus $\lambda_{b,c}=-\widetilde{f}_{b,c}$ for
$b=0,\dots,m-1$, $c=0,\dots,n-1$, besides $\widetilde{f}_{0,0}$, as
$\widetilde{w}_{0,0}=\sum_{j=1}^{N}w_{j}=\sum_{j=1}^{N-1}v_{j}-\bar{v}=0$, the
equation for $a=b=0$ is always satisfied, and there are indeed only $N-1$
eigenvalues as expected for our $(N-1)\times(N-1)$ matrix $\mathcal{X}$.

#### The case $m=1$

Let us now restrict ourselves to the case $m=1,\check{\tau}=\tau+r$. The
eigenvalues are

$\lambda_{n}=-\sum_{j=0}^{N-1}\sum_{a=1}^{3}\mathcal{G}\left(\frac{j\check{\tau}}{N},\Delta_{a};\check{\tau}\right)e^{2\pi
ijn/N}\;,\qquad n=1,\dots,N-1$ (B.21)

By using the modular properties of $\mathcal{G}$,
$\mathcal{G}\left(u,\Delta;\check{\tau}\right)=\frac{1}{\check{\tau}}\mathcal{G}\left(\frac{u}{\check{\tau}},\frac{\Delta}{\check{\tau}};-\frac{1}{\check{\tau}}\right)-1-\frac{2\Delta}{\check{\tau}}-\frac{1}{\check{\tau}}$
and the series expansion
$\mathcal{G}\left(u,\Delta;\tau\right)=\sum_{\ell=1}^{\infty}\frac{\left(z^{\ell}+z^{-\ell}\right)\left(y^{\ell}-\left(q/y\right)^{\ell}\right)}{1-q^{\ell}}$
we note that for $\tilde{q}=e^{-2\pi i/\check{\tau}}$, $\tilde{y}_{a}=e^{-2\pi
i\Delta_{a}/\check{\tau}}$

$\displaystyle-\lambda_{n}\check{\tau}$
$\displaystyle=\sum_{a=1}^{3}\sum_{j=0}^{N-1}\sum_{\ell=1}^{\infty}\frac{\left(e^{2\pi
ij\ell/N}+e^{-2\pi
ij\ell/N}\right)\left(\tilde{y}_{a}^{\ell}-\left(\tilde{q}/\tilde{y}_{a}\right)^{\ell}\right)}{1-\tilde{q}^{\ell}}e^{2\pi
ijn/N}$ (B.22)
$\displaystyle=\sum_{a=1}^{3}\sum_{\ell=1}^{\infty}\sum_{j=0}^{N-1}\frac{\left(e^{2\pi
ij\left(\ell+n\right)/N}+e^{-2\pi
ij\left(\ell-n\right)/N}\right)\left(\tilde{y}_{a}^{\ell}-\left(\tilde{q}/\tilde{y}_{a}\right)^{\ell}\right)}{1-\tilde{q}^{\ell}}$
$\displaystyle=N\sum_{a=1}^{3}\left[\sum_{k=1}^{\infty}\frac{\tilde{y}_{a}^{kN-n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{kN-n}}{1-\tilde{q}^{kN-n}}+\sum_{k=0}^{\infty}\frac{\tilde{y}_{a}^{kN+n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{kN+n}}{1-\tilde{q}^{kN+n}}\right]$

so at the large $N$ limit
$\lambda_{n}=-\frac{N}{\check{\tau}}\sum_{a=1}^{3}\left[\frac{\tilde{y}_{a}^{n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{n}}{1-\tilde{q}^{n}}+\frac{\tilde{y}_{a}^{N-n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{N-n}}{1-\tilde{q}^{N-n}}+O\left(\tilde{y}_{a}^{N},\left(\tilde{q}/\tilde{y}_{a}\right)^{N}\right)\right]$
where the corrections are exponentially small in $N$. Moreover171717For the
first branch we used here the fact that
$\Delta_{3}=2\tau-\Delta_{1}-\Delta_{2}-1$, the shift property of
$\mathcal{G}$ (C.27) such that we first shift $\Delta_{1,2}$ inside
$\mathcal{G}$ to their bracketed value and shift $\Delta_{3}\to
2\tau-[{\bar{\Delta}}_{1}]_{\check{\tau}}-[{\bar{\Delta}}_{2}]_{\check{\tau}}-1$
accordingly to cancel that, and then shift it again by $\Delta_{3}$ by
additional $2r$ to get to its bracketed value,
$[{\bar{\Delta}}_{3}]_{\check{\tau}}=2\check{\tau}-[{\bar{\Delta}}_{1}]_{\check{\tau}}-[{\bar{\Delta}}_{1}]_{\check{\tau}}-1$.
For the second branch, there’s one additional shift at the end, resulting in
the relative minus sign.,
$b_{N}=\pm\frac{1}{\check{\tau}}+\frac{1}{\check{\tau}}\sum_{a=1}^{3}\mathcal{G}\left(0,\frac{N[\Delta_{a}]}{\check{\tau}};-\frac{N}{\check{\tau}}\right)$,
where the upper sign is for the first case (3.5) and the lower for the second.
Note that all the exponents here are of $\frac{\Delta_{a}}{\check{\tau}}$,
like the combination that match the D-branes that wrap an $S^{3}\subset
S^{5}$, but without the corresponding factor of $N$. We can now evaluate the
the determinant and find that there are no perturbative corrections in $1/N$.

$\displaystyle\log\det\left[\mathbf{1}+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right]\Bigg{|}_{m=1}$
$\displaystyle=\sum_{n=1}^{N-1}\log\left(1+\frac{b_{N}^{-1}}{N}\lambda_{n}\right)$
(B.23)
$\displaystyle=\sum_{n=1}^{N-1}\log\left[1\mp\sum_{a=1}^{3}\left(\frac{\tilde{y}_{a}^{n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{n}}{1-\tilde{q}^{n}}+\frac{\tilde{y}_{a}^{N-n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{N-n}}{1-\tilde{q}^{N-n}}\right)\right]$
$\displaystyle\qquad\qquad+O\left(\tilde{y}_{a}^{N},\left(\tilde{q}/\tilde{y}_{a}\right)^{N}\right)$
$\displaystyle=2\sum_{n=1}^{N-1}\log\left[1\mp\sum_{a=1}^{3}\frac{\tilde{y}_{a}^{n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{n}}{1-\tilde{q}^{n}}\right]+O\left(\tilde{y}_{a}^{N},\left(\tilde{q}/\tilde{y}_{a}\right)^{N}\right)$
$\displaystyle=2\sum_{n=1}^{\infty}\log\left[1\mp\sum_{a=1}^{3}\frac{\tilde{y}_{a}^{n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{n}}{1-\tilde{q}^{n}}\right]+O\left(\tilde{y}_{a}^{N},\left(\tilde{q}/\tilde{y}_{a}\right)^{N}\right)\;.$

Eventually, one finds

$\boxed{\begin{aligned}
\det\left[\mathbf{1}+\frac{b_{N}^{-1}}{N}\mathcal{B}^{-1}\mathcal{C}\right]\Bigg{|}_{m=1}&=\prod_{n=1}^{\infty}\left[1\mp\sum_{a=1}^{3}\frac{\tilde{y}_{a}^{n}-\left(\tilde{q}/\tilde{y}_{a}\right)^{n}}{1-\tilde{q}^{n}}\right]^{2}\\\
&\times\left(1+O\left(\tilde{y}_{a}^{N},\left(\tilde{q}/\tilde{y}_{a}\right)^{N}\right)\right)\;,\quad\text{{1\textsuperscript{st}}/{2\textsuperscript{nd}}\
case.}\end{aligned}}$ (B.24)

##  Appendix C Special functions

Throughout we will sometimes use181818Note that this convention is more common
in the literature concerning the modular functions and transformations. In
some of the literature concerning elliptic functions one uses
$q^{\prime}=e^{\pi i\tau}=\sqrt{q}$, even though it is denoted by $q$ there.

$q=e^{2i\pi\tau}\;,\qquad p=e^{2\pi i\sigma}\;,\qquad z=e^{2\pi iu}$ (C.1)

#### q-Pochhammer symbol

The q-Pochhamemer symbol is

$(z;q)_{n}=\prod_{k=0}^{n-1}(1-zq^{k})\;,\qquad(z;q)_{\infty}=\prod_{k=0}^{\infty}(1-zq^{k})\qquad\text{for
}|q|<1\;.$ (C.2)

There are also series expansion and Plethystic representation for
$(z;q)_{\infty}$,

$(z;q)_{\infty}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\frac{1}{2}n(n-1)}}{(q;q)_{n}}z^{n}=\exp\left[-\sum_{k=1}^{\infty}\frac{1}{k}\frac{z^{k}}{1-q^{k}}\right]\;,$
(C.3)

where the first converges for $|q|<1$ while the second converges for
$|z|,|q|<1$.

By relating the symbol to the Dedekind eta function,

$\eta(\tau)=e^{\frac{\pi i\tau}{12}}(q;q)_{\infty}\;,$ (C.4)

one obtains the properties of the q-Pochhammer symbol under modular
transformations

$(\tilde{q};\tilde{q})_{\infty}=\sqrt{-i\tau}e^{\frac{\pi
i}{12}(\tau+1/\tau}(q;q)_{\infty}\;,$ (C.5)

where $\tilde{q}=e^{-2\pi i/\tau}$. Finally, we have the asymptotic behaviour

$(z;q)_{\infty}\sim 1-z\qquad\text{for }q\to
0\;,\qquad\qquad\log(z;q)_{\infty}\sim-\frac{z}{1-q}\quad\text{for }z\to 0\;.$
(C.6)

#### Elliptic theta function

The elliptic theta function is

$\theta_{0}(u;\tau)=(z;q)_{\infty}(q/z;q)_{\infty}=\prod_{k=0}^{\infty}(1-zq^{k})(1-z^{-1}q^{k+1})\;,$
(C.7)

which gives an analytic function on $|q|<1$ with simple zeroes at $z=q^{k}$
for $k\in\mathbb{Z}$ and no singularities. The infinite product is convergent
in the whole domain. We can also give a plethystic expansion

$\theta_{0}(u;\tau)=\exp\left[-\sum_{k=1}^{\infty}\frac{1}{k}\frac{z^{k}+(qz^{-1})^{k}}{1-q^{k}}\right]\;,$
(C.8)

which converges for $|q|<|z|<1$. The periodicity relations are

$\displaystyle\theta_{0}(u+n+m\tau;\tau)$ $\displaystyle=(-1)^{m}e^{-\pi
im(2u+(m-1)\tau)}\theta_{0}(u;\tau)\;,\qquad\qquad m,n\in bZ$ (C.9)
$\displaystyle\theta_{0}(u;\tau)$
$\displaystyle=\theta_{0}(\tau-u;\tau)=-e^{2\pi iu}\theta_{0}(-u;\tau)\;,$

and under modular transformations

$\theta_{0}(u;\tau+1)=\theta_{0}(u;\tau)\;,\qquad\theta_{0}\left(\frac{u}{\tau};-\frac{1}{\tau}\right)=-ie^{\frac{\pi
i}{\tau}(u^{2}+u+\frac{1}{6})-\pi iu+\frac{\pi
i\tau}{6}}\theta_{0}(u;\tau)\;.$ (C.10)

One could define the logarithmic derivative of the elliptic theta function

$\Theta_{0}(u;\tau)=\partial_{u}\log\theta_{0}(u;\tau)=\frac{\partial_{u}\theta_{0}(u;\tau)}{\theta_{0}(u;\tau)}\;,$
(C.11)

with the series expansion

$\Theta_{0}(u;\tau)=2\pi
i\sum_{k=1}^{\infty}\left[\frac{q^{k}z^{-k-1}}{1-q^{k}}-\frac{z^{k-1}}{1-q^{k}}\right]=\frac{2\pi
i}{z}\sum_{m=0}^{\infty}\left[\frac{z^{-1}q^{m+1}}{1-z^{-1}q^{m+1}}-\frac{zq^{m}}{1-zq^{m}}\right]\;.$
(C.12)

Finally, the elliptic theta function can be related to the Jacobi theta
functions via

$\theta_{1}\left(u;\tau\right)=ie^{\pi
i(\frac{\tau}{4}-u)}\left(q;q\right)_{\infty}\theta_{0}(u;\tau)\;.$ (C.13)

#### The elliptic Gamma function

The elliptic Gamma function is defined by

$\widetilde{\Gamma}(u;\sigma,\tau)=\prod_{m,n=0}^{\infty}\frac{1-p^{m+1}q^{n+1}z^{-1}}{1-p^{m}q^{n}z}\;.$
(C.14)

This definition gives a meromorphic single value function on $|p|,|q|<1$ with
simple zeroes at $z=p^{m+1}q^{n+1}$ and simple poles at $z=p^{-m}q^{-n}$ for
$m,n\geq 0$. The infinite product is convergent on the whole domain. We can
also give a plethystic definition

$\widetilde{\Gamma}(u;\sigma,\tau)=\exp\left[\sum_{k=1}^{\infty}\frac{1}{k}\frac{z^{k}-(pqz^{-1})^{k}}{(1-p^{k})(1-q^{k})}\right]\;,$
(C.15)

which converges for $|pq|<|z|<1$. The function has the following periodicy
relations

$\displaystyle\widetilde{\Gamma}(u;\sigma,\tau)$
$\displaystyle=\widetilde{\Gamma}(u;\tau,\sigma)\;,$ (C.16)
$\displaystyle\widetilde{\Gamma}(u;\sigma,\tau)=\widetilde{\Gamma}(u+1;\sigma,\tau)$
$\displaystyle=\widetilde{\Gamma}(u;\sigma+1,\tau)=\widetilde{\Gamma}(u;\sigma,\tau+1)\;,$
$\displaystyle\widetilde{\Gamma}(u+\sigma;\sigma,\tau)$
$\displaystyle=\theta_{0}(u;\tau)\widetilde{\Gamma}(u;\sigma,\tau)\;,$
$\displaystyle\qquad\widetilde{\Gamma}(u+\tau;\sigma,\tau)$
$\displaystyle=\theta_{0}(u;\sigma)\widetilde{\Gamma}(u;\sigma,\tau)\;.$

Moreover,

$\widetilde{\Gamma}(u;\sigma,\tau)\widetilde{\Gamma}(\sigma+\tau-u;\sigma,\tau)=1\;.$
(C.17)

The elliptic Gamma function has $SL(3,\mathbb{Z})$ modular properties. For
$\sigma,\tau,\sigma/\tau,\sigma+\tau\in\mathbb{C}\setminus\mathcal{R}$ there
is a ”modular formula”: Felder:2000

$\widetilde{\Gamma}(u;\sigma,\tau)=e^{-\pi
i\mathcal{Q}(u;\sigma,\tau)}\frac{\widetilde{\Gamma}\left(\frac{u}{\tau};\frac{\sigma}{\tau},-\frac{1}{\tau}\right)}{\widetilde{\Gamma}\left(\frac{u-\tau}{\sigma};-\frac{1}{\sigma},-\frac{\tau}{\sigma}\right)}=e^{-\pi
i\mathcal{Q}(u;\sigma,\tau}\frac{\widetilde{\Gamma}\left(\frac{u}{\sigma};-\frac{1}{\sigma},\frac{\tau}{\sigma}\right)}{\widetilde{\Gamma}\left(\frac{u-\sigma}{\tau};-\frac{\sigma}{\tau},-\frac{1}{\tau}\right)}\;,$
(C.18)

where $\mathcal{Q}(u;\sigma,\tau)$ is the cubic polynomial

$\mathcal{Q}(u;\sigma,\tau)=\frac{u^{3}}{3\sigma\tau}-\frac{\sigma+\tau-1}{2\sigma\tau}u^{2}+\frac{\sigma^{2}+\tau^{2}+3\sigma\tau-3\sigma-3\tau+1}{6\sigma\tau}u+\frac{(\sigma+\tau-1)(\sigma+\tau-\sigma\tau)}{12\sigma\tau}\;.$
(C.19)

In the degenerate case $\sigma=\tau$ the formula above is not valid. For
$u\in\mathbb{C}\setminus(\mathbb{Z}+\tau\mathbb{Z})$, however, there is a
degenerate relation

$\widetilde{\Gamma}(u;\tau,\tau)=\frac{e^{-\pi
i\mathcal{Q}(u;\tau,\tau)}}{\theta_{0}(\frac{u}{\tau};-\frac{1}{\tau})}\prod_{k=0}^{\infty}\frac{\psi\left(\frac{k+1+u}{\tau}\right)}{\psi\left(\frac{k-u}{\tau}\right)}\;,$
(C.20)

the function $\psi$ is the elliptic digamma function defined below and the
polynomial $\mathcal{Q}$ reduces to

$\mathcal{Q}(u;\tau,\tau)=\frac{(2u-2\tau+1)(2u(u+1)-2\tau(2u+1)+\tau^{2})}{12\tau^{2}}\;.$
(C.21)

Using

$\mathcal{Q}(u+1;\tau,\tau)-\mathcal{Q}(u;\tau,\tau)=\frac{(u+1)(u+1-2\tau)}{\tau^{2}}+\frac{5}{6}\;,$
(C.22)

one can check that $\widetilde{\Gamma}(u;\tau,\tau)$ is invariant under $u\to
u+1$.

#### Function $\boldsymbol{\mathcal{G}}$.

We find it convenient to define a function, $\mathcal{G}$, which is closely
related to the logarithmic derivative of $\theta_{0}$,

$\mathcal{G}(u,\Delta;\tau)=\frac{1}{2\pi i}\,\frac{\partial}{\partial
u}\log\left(\frac{\theta_{0}(\Delta-u;\tau)}{\theta_{0}(\Delta+u;\tau)}\right)\;.$
(C.23)

It has the series expansion

$\mathcal{G}(u,\Delta;\tau)=\sum_{\ell=1}^{\infty}\frac{(z^{\ell}+z^{-\ell})\bigl{(}y^{\ell}-(q/y)^{\ell}\bigr{)}}{1-q^{\ell}}$
(C.24)

which converges for $|q|<|yz|<1$ and $|q|<|y/z|<1$, with $z=e^{2\pi iu}$ and
$y=e^{2\pi i\Delta}$.

The function has the properties

$\mathcal{G}(u,\Delta;\tau)=\mathcal{G}(-u,\Delta;\tau)=-\mathcal{G}(u,\tau-\Delta;\tau)\;.$
(C.25)

and also the modular properties, which follow from the ones of $\theta_{0}$:

$\mathcal{G}(u,\Delta;\tau+1)=\mathcal{G}(u,\Delta;\tau)\;,\qquad\mathcal{G}\left(\frac{u}{\tau},\frac{\Delta}{\tau};-\frac{1}{\tau}\right)=\tau-2\Delta-1+\tau\,\mathcal{G}(u,\Delta;\tau)\;.$
(C.26)

Its periodicities are

$\displaystyle\mathcal{G}(u,\Delta;\tau)$
$\displaystyle=\mathcal{G}(u+1,\Delta;\tau)=\mathcal{G}(u+\tau,\Delta;\tau)=\mathcal{G}(u,\Delta+1;\tau)$
(C.27) $\displaystyle\mathcal{G}(u,\Delta+\tau;\tau)$
$\displaystyle=2+\mathcal{G}(u,\Delta;\tau)\;.$

In particular, $\mathcal{G}$ is an elliptic function of $u$, and quasi-
elliptic of $\Delta$.

#### Function $\boldsymbol{\psi}$.

Define, for $\operatorname{\mathbb{I}m}(t)<0$, the function

$\psi(t)=\exp\left[t\log\bigl{(}1-e^{-2\pi it}\bigr{)}-\frac{1}{2\pi
i}\text{Li}_{2}(e^{-2\pi
it})\right]=\exp\left[-\sum_{\ell=1}^{\infty}\left(\frac{t}{\ell}+\frac{1}{2\pi
i\,\ell^{2}}\right)e^{-2\pi it\,\ell}\right]\;.$ (C.28)

The branch of the logarithm is determined by its series expansion
$\log(1-z)=-\sum_{\ell=1}^{\infty}z^{\ell}/\ell$, whereas
$\text{Li}_{2}(z)=\sum_{\ell=1}^{\infty}z^{\ell}/\ell^{2}$ is the dilogarithm.
One can show that the branch cut discontinuities of the logarithm and the
dilogarithm cancel in the definition of $\psi(t)$, therefore the latter
extends to a meromorphic function on the whole complex plane. Some useful
properties of $\psi(t)$ are:

$\psi(t)\,\psi(-t)=e^{-\pi i(t^{2}-1/6)}\;,\qquad\qquad\psi(t+n)=(1-e^{-2\pi
it})^{n}\,\psi(t)\qquad\text{for}\quad n\in\mathbb{Z}\;.$ (C.29)

In particular, from (C.28), $\psi(0)=e^{\pi i/12}$.

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