aainstitutetext: Università del Salento, Dipartimento di Matematica e Fisica
Ennio De Giorgi, and I.N.F.N. - sezione di Lecce, Via Arnesano, I-73100 Lecce,
Italybbinstitutetext: Instituut voor Theoretische Fysica, KU Leuven.
Celestijnenlaan 200D, B-3001 Leuven, Belgium ccinstitutetext: Asia Pacific
Center for Theoretical Physics, Postech, Pohang 37673, Korea

# Thermodynamics of black holes with probe D-branes

Alejandro Cabo-Bizet b Marina David c Alfredo González Lezcano
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Understanding how the thermodynamic properties of a black hole are modified
when probed by D-branes is an important problem in AdS/CFT. This work focuses
on a recently proposed black hole/D3-brane system in AdS${}_{5}\times$S5,
which is dual to four-dimensional $\mathcal{N}=4$ SYM in the presence of a
two-dimensional surface defect. The Laplace transform that extracts the
asymptotic growth of states in this defect CFT naturally defines a
thermodynamic approach in the gravitational side of the duality for which
charges and entropy are real. Studying the superconformal defect index in a
large-charge expansion for all values of $N$, we compute the leading
correction to the entropy of the combined system, which matches precisely with
its gravity counterpart.

##  1 Introduction and Summary

The AdS/CFT correspondence has allowed us to understand the Bekenstein-Hawking
entropy for a large class of supersymmetric black holes as a semiclassical
limit of the Boltzmann entropy of supersymmetric gauge theories Benini:2015eyy
; Benini:2016rke ; Choi:2018hmj ; Cabo-Bizet:2018ehj ; Benini:2018ywd ;
ArabiArdehali:2019tdm ; Honda:2019cio ; Cabo-Bizet:2019osg ; Kim:2019yrz ;
Cabo-Bizet:2019eaf ; Amariti:2019mgp ; GonzalezLezcano2019 ; Lanir:2019abx ;
Goldstein:2019gpz ; ArabiArdehali:2019orz ; Murthy:2020rbd ;
PhysRevD.105.L021903 ; Agarwal:2020zwm ; Benini:2020gjh ; Cabo-Bizet:2020nkr ;
Cabo-Bizet:2021plf ; Cassani:2021fyv ; Jejjala:2021hlt ; Jejjala:2022lrm ;
Aharony:2021zkr ; Cabo-Bizet:2021jar ; Goldstein:2020yvj ; Choi:2021rxi ;
Choi:2023tiq . Subleading corrections beyond the semiclassical result, which
could be perturbative or logarithmic in the semi-classical expansion, and
which could come from $\alpha^{\prime}$ corrections, as well as from other
quantum effects, have been also computed and exactly matched across both sides
of the duality, see for example Bhattacharyya:2012ye ; Liu:2017vbl ;
Liu:2017vll ; David:2021eoq ; GonzalezLezcano:2020yeb ; Amariti:2020jyx ;
David:2021qaa ; Cassani:2022lrk ; Bobev:2021qxx ; Bobev:2022bjm ;
Cassani:2023vsa ; Bobev:2023dwx . Despite these remarkable quantitatively
precise advances, not much is understood yet about more drastic quantum
gravity processes such as perturbing black holes with D-branes.

A pioneering attempt in this direction has been recently put forward in the
context of AdS5/CFT4 in Chen:2023lzq . In this reference the authors studied
the effect of perturbing a supersymmetric black hole in AdS5 Chong:2005da ;
Chong:2005hr ; Gutowski:2004ez ; Gutowski:2004yv ; Wu:2011gq ; Kunduri:2006ek
with a D$3$-brane111There are two non-trivial properties that need a closer
analysis regarding how to preserve supersymmetry when inserting the
D$3$-brane: The first is to check that it is possible for the D$3$-brane to be
supersymmetric and the second is whether or not this supersymmetry is
compatible with the one preserved by the black hole. Some indications that it
is possible for this D$3$-brane to be supersymmetric have been presented in
Chen:2023lzq . Here we assume this is true and leave a more rigorous check for
future work.  extending across the time, radial, one compact direction in AdS5
and one compact direction in the internal space $S^{5}$. In the dual gauge
theory, which is the four-dimensional $SU(N)$ $\mathcal{N}=4$ SYM, inserting
this probe D$3$-brane corresponds to inserting a surface operator Gukov:2006jk
, compatible with the supercharges used to construct the 4d superconformal
index.

In the probe approximation, the authors of Chen:2023lzq  found that the free
energy of the black hole/D$3$-brane system reduces to the sum of the free
energy of the unperturbed black hole solution and the Dirac-Born-Infeld on-
shell action of the D$3$-brane in the geometry of the unperturbed black hole
solution, respectively. It may seem natural to assume that in the very same
probe approximation the entropy of the total system reduces to the sum of the
entropies of the two unperturbed components. Indeed, if one commits to this
intuition then the charges of the D$3$-brane are fixed in terms of the charges
of the embedding black hole solution. Unfortunately, the charges and entropy
of the D$3$-brane fixed by this procedure turn out to be complex.222In
contradistinction to the unperturbed system, in the presence of the D$3$-brane
there is no non-linear constraint among real charges for which the extremal
value of the corresponding entropy function becomes real and thus identifiable
with the asymptotic value of a Boltzmann entropy. As also stated in
Chen:2023lzq , this result is intriguing, because the dual microstates that
one counts in the field theory do indeed have real charges, and certainly
their Boltzmann entropy is not complex.

This naive contradiction strongly suggests that another procedure must be used
to define the thermodynamic properties of the combined system. The goal of
this paper is to find such a procedure. Indeed, we propose that the entropy of
the system black hole/D$3$-brane is recovered by means of the natural
holographic translation of the method used to count states in the holographic
dual 4d-2d field theory system: the Laplace transform of the defect
superconformal index.333This Laplace transforms only depends on the charges of
the combined system and not on the charges of its individual components.

For the case of the superconformal index without the insertion of the surface
defect, the Laplace transform – in the leading order in the semiclassical
large charge approximation – picks up two leading complex conjugated saddle
points whose contributions add up to give a real entropy Agarwal:2020zwm ;
Cabo-Bizet:2020ewf ; Beccaria:2023hip . Similarly, as we show here, we find
this also to be the case when the defect is introduced in the system. In the
gravitational picture, these leading saddle points correspond to two complex
geometries that serve as saddle points of the Euclidean gravitational path
integral. Borrowing the field-theory procedure to the holographic dual setup
implicitly defines how to compute the corrections to the entropy that the
D$3$-brane induces when probing the black hole.

Using a Cardy-like expansion we confirm the results of Chen:2023lzq  for the
free energy of the 4d-2d field theory, and extend them, both to finite $N$ and
beyond the probe approximation. We find that at leading order in the Cardy-
like expansion, inserting the defect does not change the shape of the saddle
point governing the growth of the unperturbed 4d superconformal index.
Surprisingly, this tells us that the naive probe approximation is sufficiently
precise to exactly describe the 4d-2d system at large charges. In the string-
theory side of the duality, this result predicts a fully backreacted answer
for the entropy of the perturbed black hole at leading order in the Cardy-like
expansion. It would be very interesting to understand whether the
$\frac{1}{N}$ corrections induced by the presence of the D$3$-brane can be
understood, geometrically, as a change in the area of the horizon. In order to
answer this question we would need to understand how the D$3$-brane backreacts
the geometry in the bulk. We leave that problem for future work.

The paper is organized as follows. Section 2 summarizes the supergravity
theory, and other useful background information. We motivate and present our
prescription to study the thermodynamics of the combined black hole/D$3$-brane
system and perform the Laplace transform to extract the microcanonical
entropy. In Section 3 we study the field theory dual description and we
revisit the 4d computation in the absence of defect and recall how to evaluate
the asymptotic growth of the index in that case. In Section 4 we compute the
defect index. We analyze the 2d index of the surface defect at large charges
by implementing a systematic Cardy-like expansion. In Section 5 we conclude
with brief remarks and questions for the future. Appendix A reviews the
thermodynamic procedure used in Chen:2023lzq  while Appendix B summarizes
useful mathematical identities.

##  2 The gravity theory

In this section, we summarize the five-dimensional gauged black hole solution
of Chong:2005hr  and its thermodynamic properties. After reviewing its
supersymmetric limit we carry on to study the effects of adding the probe
D$3$-brane. Then we proceed to study the thermodynamics of the combined
system.

###  2.1 The black hole solution

We consider five-dimensional minimal gauged supergravity whose action takes
the form

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi G_{5}}\int
d^{5}x\left[(R+12g^{2})\star 1-\frac{2}{3g^{2}}F\wedge\star
F+\frac{8}{27g^{3}}F\wedge F\wedge A\right],$ (1)

where $F=dA$ and $g$ is the inverse length of AdS. The five-dimensional
coordinates describing the solution are $\\{t,r,\theta,\phi,\psi\\}$ where
$0\leq\phi,\psi\leq 2\pi$ and $0\leq\theta\leq\frac{\pi}{2}$. The equations of
motion can be derived from (1)

$\displaystyle 0$
$\displaystyle=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-6g^{2}g_{\mu\nu}-\frac{4}{3g^{2}}\left(\frac{1}{2}F^{2}_{\mu\nu}-\frac{1}{8}g_{\mu\nu}F^{2}\right),\qquad
0=d\star F+\frac{2}{3g}F\wedge F.$ (2)

We review the known nonextremal nonsupersymmetric black hole solution with one
electric charge and two rotations as was studied in Chong:2005hr . The
solution of the metric and gauge field are given by

$\displaystyle ds^{2}_{\text{AdS}_{5}}=$
$\displaystyle-\frac{\Delta_{\theta}\left[\left(1+g^{2}r^{2}\right)\rho^{2}dt+2q\nu\right]dt}{\Xi_{a}\Xi_{b}\rho^{2}}+\frac{2q\nu\omega}{\rho^{2}}+\frac{f}{\rho^{4}}\left(\frac{\Delta_{\theta}dt}{\Xi_{a}\Xi_{b}}-\omega\right)^{2}+\frac{\rho^{2}dr^{2}}{\Delta_{r}}+\frac{\rho^{2}d\theta^{2}}{\Delta_{\theta}}$
(3) $\displaystyle+\frac{r^{2}+a^{2}}{\Xi_{a}}\sin^{2}\theta
d\phi^{2}+\frac{r^{2}+b^{2}}{\Xi_{b}}\cos^{2}\theta d\psi^{2},$ $\displaystyle
A=$
$\displaystyle\frac{3q}{2\rho^{2}}\left(\frac{\Delta_{\theta}dt}{\Xi_{a}\Xi_{b}}-\omega\right)+\alpha_{5}dt,$

where we have added a pure gauge term $\alpha_{5}dt$ with $\alpha_{5}$ being a
constant. The remaining functions in the metric and 1-form are

$\displaystyle\nu$ $\displaystyle=b\sin^{2}\theta d\phi+a\cos^{2}\theta
d\psi,\quad$ $\displaystyle\Delta_{r}$
$\displaystyle=g^{2}\left(r^{2}+a^{2}\right)\left(r^{2}+b^{2}\right)\left(1+\frac{1}{g^{2}r^{2}}\right)+\frac{q^{2}+2abq}{r^{2}}-2m,$
(4) $\displaystyle\omega$
$\displaystyle=a\sin^{2}\theta\frac{d\phi}{\Xi_{a}}+b\cos^{2}\theta\frac{d\psi}{\Xi_{b}},\quad$
$\displaystyle\Delta_{\theta}$
$\displaystyle=1-a^{2}g^{2}\cos^{2}\theta-b^{2}g^{2}\sin^{2}\theta,$
$\displaystyle\rho^{2}$
$\displaystyle=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta,\quad$
$\displaystyle\Xi_{a}$ $\displaystyle=1-a^{2}g^{2},$ $\displaystyle f$
$\displaystyle=2m\rho^{2}-q^{2}+2abqg^{2}\rho^{2},\quad$
$\displaystyle\Xi_{b}$ $\displaystyle=1-b^{2}g^{2}.$

For the general non-extremal solution with no supersymmetry, there are four
independent parameters that characterize the black hole

$\displaystyle\\{a,b,m,q\\}.$ (5)

Moreover, we may sometimes find it convenient to swap one of the parameters,
namely, $q$ with the horizon radius, i.e.,

$\displaystyle q=-ab\pm
r_{+}\sqrt{-a^{2}\left(b^{2}g^{2}+g^{2}r_{+}^{2}+1\right)-b^{2}\left(g^{2}r_{+}^{2}+1\right)-g^{2}r_{+}^{4}+2m-r_{+}^{2}}.$
(6)

The electric charges and angular momentum can be computed via the Komar
integrals

$\displaystyle Q_{\text{BH}}$ $\displaystyle=\frac{1}{16\pi
G_{5}}\int_{S^{3}}\left(\frac{4}{3g^{2}}\right)\star F-\frac{8}{9g^{3}}F\wedge
A=\frac{1}{G_{5}}\frac{\pi q}{2g\Xi_{a}\Xi_{b}},$ (7) $\displaystyle
J_{1,\text{BH}}$ $\displaystyle=\frac{1}{16\pi G}\int_{S^{3}}\star
d\xi_{\phi}=\frac{1}{G_{5}}\frac{\pi\left[2am+qb\left(1+a^{2}g^{2}\right)\right]}{4\Xi_{a}^{2}\Xi_{b}},$
(8) $\displaystyle J_{2,\text{BH}}$ $\displaystyle=\frac{1}{16\pi
G}\int_{S^{3}}\star
d\xi_{\psi}=\frac{1}{G_{5}}\frac{\pi\left[2bm+qa\left(1+b^{2}g^{2}\right)\right]}{4\Xi_{b}^{2}\Xi_{a}},$
(9)

where $\xi_{\phi}$ and $\xi_{\psi}$ are dual to Killing vector
$-\partial_{\phi}$ and $-\partial_{\psi}$ respectively such that

$\displaystyle\xi_{\phi}=-g_{\mu\phi}dx^{\mu},\qquad\xi_{\psi}=-g_{\mu\psi}dx^{\mu}.$
(10)

The charges are evaluated at the asymptotic boundary and for this reason, the
Chern Simons term in the integral for the electric charge does not contribute
to the charge. In fact there are different notions of charge and we refer the
reader to Marolf:2000cb  for more details. The energy can be found from the
AMD formalism

$E_{\text{BH}}=\frac{1}{G_{5}}\frac{m\pi\left(2\Xi_{a}+2\Xi_{b}-\Xi_{a}\Xi_{b}\right)+2\pi
qabg^{2}\left(\Xi_{a}+\Xi_{b}\right)}{4\Xi_{a}^{2}\Xi_{b}^{2}}.$ (11)

The Hawking temperature is derived by requiring appropriate periodic
identifications in Euclidean time, which leads us to

$\displaystyle T_{\text{BH}}=\beta^{-1}_{\text{BH}}$
$\displaystyle=\frac{r_{+}^{4}\left[\left(1+g^{2}\left(2r_{+}^{2}+a^{2}+b^{2}\right)\right]-(ab+q)^{2}\right.}{2\pi
r_{+}\left[\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq\right]}.$
(12)

The angular velocities $\Omega_{1}$ and $\Omega_{2}$ are found to be

$\displaystyle\Omega_{1,\text{BH}}$
$\displaystyle=\frac{a\left(r_{+}^{2}+b^{2}\right)\left(1+g^{2}r_{+}^{2}\right)+bq}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq},\quad\Omega_{2,\text{BH}}=\frac{b\left(r_{+}^{2}+a^{2}\right)\left(1+g^{2}r_{+}^{2}\right)+aq}{\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq}.$
(13)

We can now define the null Killing vector field

$\displaystyle\chi^{\mu}=\partial_{t}+\Omega_{1}\partial_{\phi}+\Omega_{2}\partial_{\psi},$
(14)

and the electrostatic potential is

$\displaystyle\Phi_{\text{BH}}$
$\displaystyle=\left.\chi^{\mu}A_{\mu}\right|_{r\to
r_{+}}-\left.\chi^{\mu}A_{\mu}\right|_{r\to
r_{+}}=\frac{3gqr_{+}^{2}}{2\left(\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq\right)}.$
(15)

The entropy can be computed via the area of the horizon

$S_{\text{BH}}=\frac{1}{G_{5}}\frac{\pi^{2}\left[\left(r_{+}^{2}+a^{2}\right)\left(r_{+}^{2}+b^{2}\right)+abq\right]}{2\Xi_{a}\Xi_{b}r_{+}}.$
(16)

Once we have computed these thermodynamic quantities, we may deduce the on-
shell action from the quantum statistical relation

$\displaystyle\begin{split}I_{\text{BH}}&=\beta_{\text{BH}}E_{\text{BH}}-S_{\text{BH}}-\beta_{\text{BH}}\Omega_{1,\text{BH}}J_{1,\text{BH}}-\beta_{\text{BH}}\Omega_{2,\text{BH}}J_{2,\text{BH}}-\beta_{\text{BH}}\Phi_{\text{BH}}Q_{\text{BH}}\\\
&=\frac{\pi\beta}{4G_{5}\Xi_{a}\Xi_{b}}\left(m-g^{2}\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)-\frac{q^{2}r_{+}^{2}}{\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq}\right).\end{split}$
(17)

###  2.2 The supersymmetric limit

We are interested in solutions that admit a Killing spinor, i.e., preserve
$\mathcal{N}=2$ supersymmetry. The BPS bound

$\displaystyle
E_{\text{BH}}=gJ_{1,\text{BH}}+gJ_{2,\text{BH}}+\frac{3}{2}gQ_{\text{BH}}\,,$
(18)

is saturated for

$q=\frac{m}{1+(a+b)g}.$ (19)

This can be found by imposing (7), (8), (9) and (11) into (18). Keeping in
mind (6), we find that the parameter $q$ simplifies to the following

$\displaystyle\begin{split}q&=-ab+agr_{+}^{2}+bgr_{+}^{2}+r_{+}^{2}\pm
ir_{+}\left(abg+a+b-gr_{+}^{2}\right)\\\
&=(a-n_{0}ir_{+})(b-n_{0}ir_{+})(-1+n_{0}igr_{+}).\end{split}$ (20)

From now on, we denote $n_{0}=\pm 1$ for the upper/lower sign in (20)
respectively. This choice of sign can be interpreted as a choice in one of two
branches that dominate the path integral and denote a growth of states. We
shall come back to this point in great detail in Section 2.4. In the
supersymmetric limit, the temperature, angular velocities and electrostatic
potential in (12), (13) and (15) are complex

$\displaystyle T_{\text{BH}}$
$\displaystyle=\frac{g\left(r_{+}^{2}-r_{*}^{2}\right)\left(2r_{+}(ag+bg+1)+in_{0}g\left(r_{*}^{2}-3r_{+}^{2}\right)\right)}{2\pi\left(a-in_{0}r_{+}\right)\left(b-in_{0}r_{+}\right)\left(gr_{*}^{2}+in_{0}r_{+}\right)},$
(21) $\displaystyle\Omega_{1,\text{BH}}$
$\displaystyle=\frac{g\left(ar_{+}-in_{0}r_{*}^{2}\right)\left(1-in_{0}gr_{+}\right)}{\left(a-in_{0}r_{+}\right)\left(r_{+}-in_{0}gr_{*}^{2}\right)},$
(22) $\displaystyle\Omega_{2,\text{BH}}$
$\displaystyle=\frac{g\left(br_{+}-in_{0}r_{*}^{2}\right)\left(1-in_{0}gr_{+}\right)}{\left(b-in_{0}r_{+}\right)\left(r_{+}-in_{0}gr_{*}^{2}\right)},$
(23) $\displaystyle\Phi_{\text{BH}}$
$\displaystyle=\frac{3gr_{+}\left(1-in_{0}gr_{+}\right)}{2r_{+}-2in_{0}gr_{*}^{2}},$
(24)

and we can equivalently find a linear constraint among the angular velocities
and electrostatic potentials of the black hole

$\displaystyle\beta_{\text{BH}}\left(g+\Omega_{1,\text{BH}}+\Omega_{2,\text{BH}}-2\Phi_{\text{BH}}\right)=2\pi
in_{0}.$ (25)

Once (18) is imposed, we find that if we want to preserve the reality of the
parameters $a,b,q$ and $m$, we find that

$\displaystyle r^{\star}=\sqrt{\frac{a+b+abg}{g}},$ (26)

and this is the exact value where the discriminant of $r^{2}\Delta_{r}$ is
zero, i.e., the inner and outer horizons coincide and we land in the extremal
regime of the solution. This analysis leads us to conclude that supersymmetric
Lorentzian solutions must be extremal if we preserve the reality of roots of
$\Delta_{r}$ to avoid naked singularities.

Imposing both these conditions (19) and (26) leads to the following
thermodynamic relations

$\displaystyle Q^{\star}_{\text{BH}}=$
$\displaystyle-\frac{1}{G_{5}}\frac{\pi(a+b)}{2g(1-ag)(1-bg)},$ (27)
$\displaystyle J^{\star}_{1,\text{BH}}=$
$\displaystyle\frac{1}{G_{5}}\frac{\pi(a+b)(2a+b+abg)}{4g(1-ag)^{2}(1-bg)},$
$\displaystyle J^{\star}_{2,\text{BH}}=$
$\displaystyle\frac{1}{G_{5}}\frac{\pi(a+b)(a+2b+abg)}{4g(1-ag)(1-bg)^{2}},$
$\displaystyle E^{\star}_{\text{BH}}=$
$\displaystyle\frac{1}{G_{5}}\frac{\pi(a+b)}{4g(1-ag)^{2}(1-bg)^{2}}((1-ag)(1-bg)+(1+ag)(1+bg)(2-ag-
bg)),$ $\displaystyle S^{\star}_{\text{BH}}=$
$\displaystyle\frac{1}{G_{5}}\frac{\pi^{2}(a+b)\sqrt{a+b+abg}}{2g^{3/2}(1-ag)(1-bg)},$

which are now all real-valued expressions. We shall call the BPS limit the
limit of the solution where both extremal and supersymmetric conditions are
imposed and we denote this by $\star$. The family of solutions has now been
reduced to two free parameters, $a$ and $b$. Revisiting the quantum
statistical relation, we introduce the variables

$\displaystyle\omega_{1,\text{BH}}$
$\displaystyle=\frac{\beta_{\text{BH}}}{2\pi
i}(\Omega_{1,\text{BH}}-\Omega^{\star}_{1,\text{BH}}),\quad\omega_{2,\text{BH}}=\frac{\beta_{\text{BH}}}{2\pi
i}(\Omega_{2,\text{BH}}-\Omega^{\star}_{2,\text{BH}}),\quad\frac{3}{2}\varphi_{\text{BH}}=\frac{\beta_{\text{BH}}}{2\pi
i}(\Phi_{\text{BH}}-\Phi^{\star}_{\text{BH}}),$ (28)

with

$\displaystyle\Omega^{\star}_{1,\text{BH}}=g,\qquad\Omega^{\star}_{2,\text{BH}}=g,\qquad\Phi^{\star}_{\text{BH}}=\frac{3g}{2}.$
(29)

Note that the BPS values of the chemical potentials are independent of which
saddle we consider. Imposing these new variables into (25), we find the new
linear constraint among the chemical potentials takes the form

$\displaystyle\omega_{1,\text{BH}}+\omega_{2,\text{BH}}-3\varphi_{\text{BH}}=n_{0}.$
(30)

With some manipulation, as the supersymmetric limit must be taken carefully,
the quantum statistical relation can now be rewritten as a statement
independent of the temperature

$\displaystyle\begin{split}I_{\text{BH,
SUSY}}&=-S_{\text{BH}}-\omega_{1,\text{BH}}J_{1,\text{BH}}-\omega_{2,\text{BH}}J_{2,\text{BH}}-\varphi_{\text{BH}}Q_{\text{BH}}=\frac{\pi^{2}i}{2g^{3}G_{5}}\frac{\varphi^{3}_{\text{BH}}}{\omega_{1,\text{BH}}\omega_{2,\text{BH}}},\end{split}$
(31)

and via the holorgraphic dictionary, we arrive at the following on-shell
action

$\displaystyle I_{\text{BH, SUSY}}=\pi
iN^{2}\frac{\varphi^{3}_{\text{BH}}}{\omega_{1,\text{BH}}\omega_{2,\text{BH}}}.$
(32)

###  2.3 The combined system: black hole and probe D$3$-brane

Next we move on to introduce the methodology we follow to compute the
$\mathcal{O}(N)$ corrections to the entropy induced by the probe D$3$-brane.
It is important to emphasize that even when $\alpha^{\prime}$ corrections are
included, the entropy of the supersymmetric black hole receives corrections of
$\mathcal{O}(N^{0})$ Bobev:2021qxx ; Bobev:2022bjm ; Cassani:2022lrk ;
Cassani:2023vsa  and so for the purposes of this paper, they can be ignored.

Assuming unequal black hole angular momentum and equal black hole electric
charges, Chen:2023lzq  found that the supersymmetric on-shell action of the
D$3$-brane is

$\displaystyle I_{\text{D3, SUSY}}$ $\displaystyle=-2\pi
iN\frac{\widetilde{\varphi}^{2}}{\widetilde{\omega}_{1}}.$ (33)

This result comes from regularizing the Dirac-Born-Infeld and the Wess-Zumino
contributions Chen:2023lzq .

The wide tilde denotes the variables in the perturbed system and not just of
the unperturbed black hole, e.g.,
$\widetilde{\varphi}\neq{\varphi}_{\text{BH}}\,$.

To understand physically the dependence on the chemical potentials in the on-
shell action, let us review how the brane is extended into the bulk. On the
AdS5 coordinates, the coordinates $\theta$ and $\psi$ are fixed

$\displaystyle\text{AdS}_{5}:$
$\displaystyle\quad(t,r,\theta=\theta_{0},\phi,\psi=\psi_{0}),$ (34)

while on the $S^{5}$, the D$3$-brane only wraps around one of the coordinates
while the others remain fixed

$\displaystyle\text{S}^{5}:$
$\displaystyle\quad(\phi_{1},\phi_{2}=\phi_{2,0},\phi_{3}=\phi_{3,0},\overline{\theta}=\overline{\theta}_{0},\overline{\psi}=\overline{\psi}_{0}).$
(35)

As the angular momentum of the system comes from the symmetries associated to
the Killing vectors $\partial_{\phi}$ and $\partial_{\psi}$, only one is set
to be free which means that the on-shell action may only depend on the
chemical potential conjugate to the angular momentum associated to
$\partial_{\phi}$. On the other hand, the electric charges come from the
$S^{5}$, in particular, from $\phi_{i}$ and we expect that the on-shell action
depends on the two potentials associated to the electric charges from the
fixed angles $\phi_{2}$ and $\phi_{3}$.

Although we have studied the black hole at equal electric charges and
potentials, once the D$3$-brane is introduced into the system, the potentials
acquire a subleading correction in the $1/N$-expansion. Denoting the perturbed
potentils as $\widetilde{\varphi}_{1},\widetilde{\varphi}_{2}$ and
$\widetilde{\varphi}_{3}$ we expect the on-shell actions for the fully refined
system to be

$\widetilde{I}\,=\,I_{\text{BH, SUSY}}\,+\,I_{\text{D3, SUSY}},$ (36)

where

$\displaystyle I_{\text{BH, SUSY}}$ $\displaystyle=\pi
iN^{2}\frac{\widetilde{\varphi}_{1}\widetilde{\varphi}_{2}\widetilde{\varphi}_{3}}{\widetilde{\omega}_{1}\widetilde{\omega}_{2}},\qquad
I_{\text{D3, SUSY}}=-2\pi
iN\frac{\widetilde{\varphi}_{2}\widetilde{\varphi}_{3}}{\widetilde{\omega}_{1}}.$
(37)

This expectation is reassured by microscopic computations of the fully refined
4d superconformal index Benini:2018ywd ; Choi:2018hmj ; Cabo-Bizet:2018ehj
and our calculation of the 2d index in Section 4.1. To prove (37), one would
need to study the BPS limit of the fully refined AdS5 black hole solution of
Wu:2011gq .

The quantum statistical relation of the perturbed system is

$\displaystyle\widetilde{I}$
$\displaystyle=-\widetilde{S}-\textstyle{\sum_{k=1}^{2}}\widetilde{\omega}_{k}\widetilde{J}_{k}-\textstyle{\sum_{I=1}^{3}}\widetilde{\varphi}_{I}\widetilde{Q}_{I}.$
(38)

###  2.4 The extremization

The entropy of the system can be found by extremizing the entropy function

$\displaystyle\widetilde{S}=-\widetilde{I}-2\pi
i\sum_{I=1}^{3}\widetilde{\varphi}_{I}\widetilde{Q}_{I}-2\pi
i\sum_{k=1}^{2}\widetilde{\omega}_{k}\widetilde{J}_{k}+2\pi
i\widetilde{\Lambda}\left(\sum_{I=1}^{3}\widetilde{\varphi}_{I}-\sum_{k=1}^{2}\widetilde{\omega}_{k}+n_{0}\right).$
(39)

We impose the linear constraint among the chemical potentials via the Lagrange
muliplier $\Lambda$.444We note that in the gauge theory side the two choices
$n_{0}=\pm 1$ correspond to two different saddle points of a multi-dimensional
Laplace transform used to exchange from canonical to microcanonical ensemble
as well as to impose the gauge-singlet constraint. We will further elaborate
on this in Section 4.

The extremization of (39) leads to the following constraints

$\displaystyle 0$ $\displaystyle=2\pi
i(\widetilde{\Lambda}-\widetilde{Q}_{I})-i\pi
N^{2}\frac{\widetilde{\varphi}_{1}\widetilde{\varphi}_{2}\widetilde{\varphi}_{3}}{\widetilde{\omega}_{1}\widetilde{\omega}_{2}\widetilde{\varphi}_{I}}+2i\pi
N\frac{\widetilde{\varphi}_{2}\widetilde{\varphi}_{3}}{\widetilde{\omega}_{1}(\varphi_{I}^{2}\widetilde{\varphi}_{2}+\varphi_{I}^{3}\widetilde{\varphi}_{3})},$
$\displaystyle\qquad I$ $\displaystyle=1,2,3,$ (40) $\displaystyle 0$
$\displaystyle=-2\pi i(\widetilde{\Lambda}+\widetilde{J}_{k})+i\pi
N^{2}\frac{\widetilde{\varphi}_{1}\widetilde{\varphi}_{2}\widetilde{\varphi}_{3}}{\widetilde{\omega}_{1}\widetilde{\omega}_{2}\widetilde{\omega}_{k}}-2i\pi
N\frac{\widetilde{\varphi}_{2}\widetilde{\varphi}_{3}}{\delta_{k}^{1}(\widetilde{\omega}_{1})^{2}},$
$\displaystyle\qquad k$ $\displaystyle=1,2\,.$ (41)

Solving for the charges and substituting back into (39) we find the extremal
value of the entropy function

$\displaystyle\widetilde{S}=2\pi in_{0}\widetilde{\Lambda}\,,$ (42)

which has the same structure as the result obtained in the absence of the
probe D$3$-brane, although the value of $\widetilde{\Lambda}$ as a function of
charges changes for the perturbed system.

In order to identify (42) with the entropy, which is a real-valued quantity,
one may constrain the charges of the system to the locus
$\text{Im}(\widetilde{S})=0$. In the absence of the D$3$-brane, this is the
well-known non-linear constraint among charges that happens to be equivalent
to the vanishing of the Bekenstein-Hawking temperature. In the presence of the
D$3$-brane and for real charges, there is no solution to the locus
$\text{Im}(\widetilde{S})=0\,$. As explained in the introduction, the field-
theoretic analysis will provide the solution to this puzzle: at large charges
of order $N^{2}$ for large $N$ the entropy of the system approaches,
asymptotically, to the real part of entropy functional $\widetilde{S}$ of the
dominating saddle points,

$\text{Entropy}\,\sim\,\text{Re}(\widetilde{S})\,,$ (43)

without the need of imposing further constraints on the charge locus. This
result comes from the addition of contributions coming from two complex
conjugated saddles, each of them with its own on-shell entropy functional.
This addition yields a real-valued asymptotic entropy. Let us elaborate. With
four of the extremization equations in (40) and (41), we solve for
$\widetilde{\varphi}_{1},\widetilde{\varphi}_{2},\widetilde{\varphi}_{3}$, and
$\widetilde{\omega}_{2}$

$\displaystyle\begin{split}\widetilde{\varphi}_{1}&=\frac{2\widetilde{\omega}_{1}(\widetilde{J}_{1}N+\widetilde{\Lambda}(\widetilde{\Lambda}+N-\widetilde{Q}_{3})+\widetilde{Q}_{2}(\widetilde{Q}_{3}-\widetilde{\Lambda}))^{2}}{N^{2}(\widetilde{J}_{2}+\widetilde{\Lambda})(\widetilde{Q}_{2}-\widetilde{\Lambda})(\widetilde{Q}_{3}-\widetilde{\Lambda})},\\\
\widetilde{\varphi}_{2}&=-\frac{\widetilde{\omega}_{1}(\widetilde{J}_{1}+\widetilde{\Lambda})}{\widetilde{Q}_{2}-\widetilde{\Lambda}},\\\
\widetilde{\varphi}_{3}&=-\frac{\widetilde{\omega}_{1}(\widetilde{J}_{1}+\widetilde{\Lambda})}{\widetilde{Q}_{3}-\widetilde{\Lambda}},\\\
\widetilde{\omega}_{2}&=\frac{\widetilde{\omega}_{1}(\widetilde{J}_{1}+\widetilde{\Lambda})(\widetilde{J}_{1}N+\widetilde{\Lambda}(\widetilde{\Lambda}+N-\widetilde{Q}_{3})+\widetilde{Q}_{2}(\widetilde{Q}_{3}-\widetilde{\Lambda}))}{(\widetilde{J}_{2}+\widetilde{\Lambda})(\widetilde{\Lambda}-\widetilde{Q}_{2})(\widetilde{\Lambda}-\widetilde{Q}_{3})}.\end{split}$
(44)

Note that the dependence on $\widetilde{\omega}_{1}$ is trivial. Imposing
these relations into the remaining extremization equation in (40) and (41)
leads to the following equation for $\widetilde{\Lambda}$

$\displaystyle 0$
$\displaystyle=2(\widetilde{\Lambda}-\widetilde{Q}_{1})(\widetilde{\Lambda}-\widetilde{Q}_{2})(\widetilde{\Lambda}-\widetilde{Q}_{3})-N^{2}(\widetilde{\Lambda}+\widetilde{J}_{1})(\widetilde{\Lambda}+\widetilde{J}_{2})+2N(\widetilde{\Lambda}+\widetilde{J}_{1})(\widetilde{\Lambda}-\widetilde{Q}_{1}).$
(45)

From (45), we can see that the first two terms have the same form as the cubic
equation for $\Lambda$ for the black hole in absence of the D$3$-brane, but
now the charges correspond to the combined system. The last term in (45) is of
order $\mathcal{O}(N)$ and can be treated perturbatively. To keep track of the
leading terms in $N$, we consider a rescaling of the form

$\displaystyle\widetilde{\Lambda}$
$\displaystyle=N^{2}\Lambda,\quad\widetilde{J}_{k}=N^{2}J_{k,\text{BH+D3}},\quad\widetilde{Q}_{I}=N^{2}Q_{I,\text{BH+D3}}.$
(46)

To ease presentation we remove the subscript “BH+D3” in the remaining of this
section. Then, we find

$\displaystyle\begin{split}0&=2(\Lambda-Q_{1})(\Lambda-Q_{2})(\Lambda-
Q_{3})-(\Lambda+J_{1})(\Lambda+J_{2})+\frac{2}{N}(\Lambda+J_{1})(\Lambda-
Q_{1}).\end{split}$ (47)

Moreover, the first two terms, with the probe D$3$-brane turned off, is the
usual cubic equation for $\Lambda$ that appears when only considering the
black hole Cabo-Bizet:2018ehj ; Choi:2018hmj ; Benini:2018ywd . We analyze the
cubic equation in a perturbative expansion in $N$ by first considering the
general form of the cubic polynomial

$\displaystyle P(a_{\ell},\Lambda)\equiv
a_{0}+a_{1}\Lambda+a_{2}\Lambda^{2}+a_{3}\Lambda^{3}=(\Lambda-\Lambda_{+})(\Lambda-\Lambda_{0})(\Lambda-\Lambda_{-})$
$\displaystyle=0\,,\quad a_{0,\,1,\,2,\,3}\,\in\mathbb{R}\,,$ (48)

where $\Lambda_{\pm}$ and $\Lambda_{0}$ are the three roots of the cubic
equation.

With the combined system, each coefficient in the polynomial as well as the
Lagrange multiplier may also receive corrections

$\displaystyle
a_{\ell}=a_{\ell}^{(0)}+\frac{1}{N}a_{\ell}^{(1)},\qquad\Lambda=\Lambda^{(0)}+\frac{1}{N}\delta\Lambda^{(1)},\qquad\ell=0,1,2,3.$
(49)

The given values of $a_{\ell}^{(0)}$ and $a_{\ell}^{(1)}$ are

$\displaystyle a^{(0)}_{0}$ $\displaystyle=-2Q_{1}Q_{2}Q_{3}-J_{1}J_{2},$
$\displaystyle a^{(1)}_{0}$ $\displaystyle=2J_{1}Q_{1},\quad$ (50a)
$\displaystyle a^{(0)}_{1}$
$\displaystyle=2(Q_{1}Q_{2},+Q_{2}Q_{3}+Q_{1}Q_{3})-\textstyle\sum_{k=1}^{2}J_{k},$
$\displaystyle a^{(1)}_{1}$ $\displaystyle=2(Q_{1}-J_{1}),\quad$ (50b)
$\displaystyle a^{(0)}_{2}$ $\displaystyle=-2\textstyle\sum_{I=1}^{3}Q_{I}-1,$
$\displaystyle a^{(1)}_{2}$ $\displaystyle=-2,$ (50c) $\displaystyle
a^{(0)}_{3}$ $\displaystyle=2,$ $\displaystyle a^{(1)}_{3}$
$\displaystyle=0.\,$ (50d)

We can expect that the roots of (48) are shifted by a subleading correction in
$N$

$\displaystyle\sum_{l=0}^{3}\left(a^{(0)}_{l}+\frac{1}{N}a^{(1)}_{l}\right)\left(\Lambda^{(0)}+\frac{1}{N}\Lambda^{(1)}\right)^{l}$
$\displaystyle=0.$ (51)

Expanding for large $N$, we find

$\displaystyle
P(a_{\ell},\Lambda)=P(a^{(0)}_{\ell},\Lambda^{(0)})+\frac{1}{N}\sum_{\ell=0}^{3}\frac{\partial
P(a^{(0)}_{\ell},\Lambda^{(0)})}{\partial
a_{\ell}}a^{(1)}_{\ell}+\frac{1}{N}\frac{\partial
P(a^{(0)}_{\ell},\Lambda^{(0)})}{\partial\Lambda}\Lambda^{(1)}+\mathcal{O}(N^{-2}).$
(52)

Note that the expansion is only valid up to $\mathcal{O}(N^{-1})$ as higher
corrections must also be supplemented, for example, by higher derivative
corrections from the black hole. Evaluating at the roots
$\Lambda^{(0)}_{k=\pm,0}$, the zeroth order term vanishes, as expected, while
the subleading terms are in general nonzero

$\displaystyle\sum_{\ell=0}^{3}\frac{\partial
P(a^{(0)}_{\ell},\Lambda^{(0)}_{k})}{\partial a_{\ell}}a^{(1)}_{\ell}$
$\displaystyle=a^{(1)}_{0}+a^{(1)}_{1}\Lambda^{(0)}_{k}+a^{(1)}_{2}(\Lambda^{(0)}_{k})^{2}+a^{(1)}_{3}(\Lambda^{(0)}_{k})^{3},$
(53a) $\displaystyle\begin{split}\frac{\partial
P(a^{(0)}_{\ell},\Lambda^{(0)}_{k})}{\partial\Lambda}&=a^{(0)}_{1}+2a^{(0)}_{2}\Lambda^{(0)}_{k}+3a^{(0)}_{3}(\Lambda^{(0)}_{k})^{2}\\\
&=\frac{1}{2}\sum_{m\neq n\neq
k}(\Lambda^{(0)}_{k}-\Lambda^{(0)}_{m})(\Lambda^{(0)}_{k}-\Lambda^{(0)}_{n}).\end{split}$
(53b)

Using (53a) and (53b), we arrive at an expression for the subleading
correction to the roots of the cubic equation

$\displaystyle\Lambda^{(1)}_{k}=\frac{2(\Lambda^{(0)}_{k}+Q_{1})(\Lambda^{(0)}_{k}+J_{1})}{\frac{1}{2}\sum_{m\neq
n\neq
k}(\Lambda^{(0)}_{k}-\Lambda^{(0)}_{m})(\Lambda^{(0)}_{k}-\Lambda^{(0)}_{n})}.$
(54)

This solution is only valid when there are three distinct roots. In the case
of degenerate roots, the corrections are modified and the analysis must be
done carefully. Since the coefficients of the cubic equation for $\Lambda$ are
real, we expect two cases for the types of roots we may encounter: A) one real
root and two complex conjugated roots or B) all real roots.555The regime where
the roots are all real are defined by a constraint on the discriminant of
(48):
$\operatorname{disc}=-4a_{3}a_{1}^{3}+a_{2}^{2}a_{1}^{2}+18a_{0}a_{2}a_{3}a_{1}-a_{0}\left(4a_{2}^{3}+27a_{0}a_{3}^{2}\right)>0$.

Figure 1: Here we show the complex $\Lambda$ plane with several values of two
different sets of roots of (45) for $N=20$, $Q_{1}=Q_{2}=Q_{3}=Q$ and
$J_{1}=J_{2}=J$ satisfying the non-linear constraint that ensures a real
entropy in the pure black hole case. The first set of roots corresponds to the
black hole without D$3$-brane, whereas the second corresponds to the combined
system. The nonlinear constraint no longer remedies the two complex roots of
$\Lambda$ to be purely imaginary when a probe D$3$-brane is introduced in the
system.

We shall focus on case A first and for simplicity, we choose a regime of
charges, where all angular momenta are equal denoted by $J$ and likewise all
electric charges are equal denoted by $Q$. We now revisit the problem of the
nonlinear constraint

$\displaystyle\left(6Q+1\right)\left(3Q^{2}-J\right)=\left(2Q^{3}+J^{2}\right),$
(55)

in the case of no probe brane. Then we find that the roots $\Lambda_{k}$ are
given by

$\displaystyle\Lambda^{(0)}_{\pm}$ $\displaystyle=\pm
i\sqrt{3Q^{2}+\frac{1}{2}\left(-6Q+\sqrt{(1-4Q)^{3}}+1\right)}\,,\quad
Q<\frac{2}{9},$ (56a) $\displaystyle\Lambda^{(0)}_{0}$
$\displaystyle=\frac{1}{2}-3Q\,,$ (56b)

where $\Lambda^{(0)}_{\pm}$ are complex conjugated to each other and
$\Lambda^{(0)}_{0}$ is real. We plot the functions of these roots on the left
hand side of Figure 1. With the probe brane extended in the bulk, we impose
yet again the nonlinear constraint (55) with the charges promoted to the total
charge of the combined system. We then see that the two purely imaginary roots
pick up a real part and therefore get shifted while the real root also takes a
smaller value, as shown in the right plot of Figure 1.

The key observation here is that in contradistinction to the unperturbed black
holes, there is no generalization of the nonlinear constraint enforcing the
reality of the entropy when the D$3$-brane is introduced. Instead, the reality
of the entropy comes from the addition of the two leading gravitational
saddles corresponding to the constraints $n_{0}=1$ and $n_{0}=-1$.

Therefore, in the case of not imposing (55), we find two of the roots take the
general form

$\displaystyle\Lambda_{\pm}=\Lambda_{x}\pm i\Lambda_{y},$ (57)

where $\Lambda_{x}$ and $\Lambda_{y}$ are real and can be found by taking the
real and imaginary parts of (54). The entropy of the system can be found by
considering the sum of the two gravitational saddles, where the first saddle
corresponds to the constraint $n_{0}=1$ for the choice $\Lambda_{-}$ and the
second saddle corresponds to $n_{0}=-1$ for the choice $\Lambda_{+}$

$\displaystyle\begin{split}e^{S}&\sim e^{2\pi i\Lambda_{-}}+e^{-2\pi
i\Lambda_{+}}\sim e^{2\pi
i(\Lambda_{x}-i\Lambda_{y})}+e^{-2\pi(\Lambda_{x}+i\Lambda_{y})}=2e^{2\pi\Lambda_{y}}\cos(2\pi\Lambda_{x})\,\sim\,e^{2\pi\Lambda_{y}}\,,\end{split}$
(58)

where

$\displaystyle\Lambda_{y}=\frac{\gamma\left(\sqrt{3}\xi\left(3\delta^{2}(\gamma-3)+\gamma-1\right)+9\delta^{3}(3\gamma-1)\right)}{6\left(\xi\left(54\delta^{4}+18\delta^{2}+1\right)+6\sqrt{3}\left(27\delta^{2}+2\right)\delta^{3}\right)}$
(59)

and

$\displaystyle\gamma^{3}=6\delta^{2}\left(\sqrt{3}\xi\delta+9\delta^{2}+3\right)+1,\quad\delta^{2}=J+Q,\quad\xi^{2}=27J+27Q+2.$
(60)

We stress that this result is not assuming a non-linear constraint amongst
charges and we have assumed equal charges for simplicity.

Finally, we make a brief comment about scenario B. In this case, there is no
predicted growth of states as the extremized value of the entropy function is
purely imaginary. It would be interesting to study what happens to the growth
of BPS states in this regime of charges realized in the field theory dual.

##  3 The 4d superconformal index: A brief review

In this section we focus on the undeformed 4d superconformal index and its
integral representation. We also review the Laplace transform procedure that
extracts state-degeneracies at large charges and finite values $N$.

The superconformal index counts (with sign) BPS states that can not combine to
form long representations of the superconformal algebra. For $\mathcal{N}=1$
theories on $S^{1}\times S^{3}$, the superconformal index was defined in
Romelsberger:2007ec ; Kinney:2005ej  and takes the form

${\mathcal{I}_{4d}}(\omega;\xi)=\text{Tr}_{\mathcal{H}(S^{1}\times
S^{3})}\left[\left(-1\right)^{F}e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}}e^{2\pi
i\xi_{a}Q_{a}}e^{2\pi i{\sigma}(J_{1}+\frac{r}{2})}e^{2\pi
i{\tau}(J_{2}+\frac{r}{2})}\right],$ (61)

where $Q_{a}$ are the flavor charges with chemical potentials given by
$\xi_{a}$ that will be later traded by
$\Delta_{a}\equiv\xi_{a}+\frac{1}{2}r_{a}({\sigma}+{\tau})$, where $r_{a}$ is
the $R$-charge. The combination $J_{1,2}+\frac{r}{2}$, where $J_{1,2}$ are the
angular momenta on $S^{3}$ and $r$ is the R-charge, commute with the
supercharge $\mathcal{Q}$. The chemical potentials ${\sigma}$ and ${\tau}$ are
associated to $J_{1}+\frac{r}{2}$ and $J_{2}+\frac{r}{2}$, respectively.

In particular, the superconformal index
${\mathcal{I}_{4d}}({\sigma},{\tau};\Delta)$ counts $\frac{1}{16}-$BPS states
for $\mathcal{N}=4$ SYM theory and we shall focus on this theory from now on.
The matter content is given by the three chiral fields $\Phi_{1,2,3}$
appearing in the superpotential

$\displaystyle
W=\text{Tr}\left(\Phi_{1}\left[\Phi_{2},\Phi_{3}\right]\right),$ (62)

with the associated chemical potentials being $\Delta_{1,2,3}$. For an $SU(N)$
gauge group, ${\mathcal{I}_{4d}}({\sigma},{\tau};\Delta)$ can be written as a
multidimensional contour integral over the gauge holonomies $u_{ij}\equiv
u_{i}-u_{j}$ that imposes the gauge singlet constraint

$\displaystyle{\mathcal{I}_{4d}}\left({\sigma},{\tau};\Delta\right)$
$\displaystyle=$
$\displaystyle\int_{\text{SU(N)}}[\mathcal{D}U]\mathcal{Z}_{4d}(u,{\sigma},{\tau};\Delta)$
$\displaystyle=$
$\displaystyle\kappa_{N}\int_{0}^{1}\prod_{k=1}^{N-1}du_{k}\frac{\prod_{a=1}^{3}\prod_{i\neq
j}\widetilde{\Gamma}\left(u_{ij}+\Delta_{a};{\sigma},{\tau}\right)}{\prod_{i\neq
j}\widetilde{\Gamma}\left(u_{ij};{\sigma},{\tau}\right)},$

where

$\displaystyle\kappa_{N}$ $\displaystyle=\frac{\left(e^{2\pi
i{\sigma}};e^{2\pi i{\sigma}}\right)_{\infty}^{N-1}\left(e^{2\pi
i{\tau}};e^{2\pi
i{\tau}}\right)_{\infty}^{N-1}}{(N-1)!}\prod_{a=1}^{3}\left(\widetilde{\Gamma}(\Delta_{a};{\sigma},{\tau})\right)^{N-1}.$
(64)

We have used a modified version of the elliptic gamma function
$\widetilde{\Gamma}(u;{\tau},{\sigma})$, as described in Appendix B. The
evaluation of (3) has been the subject of several works Choi:2018hmj ; Cabo-
Bizet:2018ehj ; Benini:2018ywd ; ArabiArdehali:2019tdm ; Honda:2019cio ; Cabo-
Bizet:2019osg ; Kim:2019yrz ; Cabo-Bizet:2019eaf ; Amariti:2019mgp ;
GonzalezLezcano2019 ; Lanir:2019abx ; ArabiArdehali:2019orz ; Murthy:2020rbd ;
PhysRevD.105.L021903 ; Agarwal:2020zwm ; Benini:2020gjh ; Cabo-Bizet:2020nkr ;
Cabo-Bizet:2021plf ; Cassani:2021fyv ; Aharony:2021zkr ; Cabo-Bizet:2021jar ;
Choi:2021rxi ; Choi:2023tiq ; GonzalezLezcano:2020yeb ; Lezcano:2021qbj ;
Benini:2021ano . The final outcome is of the following form

$\displaystyle{\mathcal{I}_{4d}}({\sigma},{\tau};\Delta)$
$\displaystyle=N\exp\left[-(N^{2}-1)\frac{i\pi}{{\sigma}{\tau}}\prod_{I=1}^{3}\left(\left\\{\Delta_{I}\right\\}_{\omega}-\frac{1-n_{0}}{2}\right)\right],\,$
(65)

where we set $\omega=s_{1}^{-1}{\sigma}=s_{2}^{-1}{\tau}$ for some coprime
integers $s_{1},s_{2}$ (a more generic relation ). We have ignored
exponentially suppressed corrections in $1/|\omega|$. The function
$\\{\cdot\\}_{\omega}$ is defined in (109) and the value of $n_{0}=\pm 1$
indicates the domain of chemical potentials

$\displaystyle\text{Im}\left(-\frac{1}{\omega}\right)$
$\displaystyle>\text{Im}\left(\frac{\Delta}{\omega}\right)>0,\quad\quad\quad$
$\displaystyle n_{0}$ $\displaystyle=1,$ (66a)
$\displaystyle\text{Im}\left(-\frac{1}{\omega}\right)$
$\displaystyle<\text{Im}\left(\frac{\Delta}{\omega}\right)<0,\quad\quad\quad$
$\displaystyle n_{0}$ $\displaystyle=-1.$ (66b)

Moreover, the chemical potentials satisfy the constraint

$\displaystyle\sum_{I=1}^{3}\\{\Delta_{I}\\}_{\omega}={\sigma}+{\tau}+\frac{3-n_{0}}{2}.$
(67)

In case of real chemical potentials with $|\Delta_{I}|<1$, for $I=1,2,3$, the
leading contribution obtained in the large $N$ limit of (65) gives

$\displaystyle\begin{split}{\mathcal{I}_{4d}}({\sigma},{\tau};\Delta)&=N\exp\left[-\frac{\pi
i(N^{2}-1)}{{\sigma}{\tau}}\Delta_{1}\Delta_{2}\Delta_{3}\right]\,,\end{split}$
(68)

where the linear constraint (67) simplifies to

$\displaystyle\sum_{I=1}^{3}\Delta_{I}-{\tau}-{\sigma}$
$\displaystyle=-n_{0}.$ (69)

As we see, the 4d index crucially depends on the domain of chemical
potentials, i.e., the value of $n_{0}$.

###  3.1 Extracting degeneracies: changing ensemble

Now we would like to extract the degeneracies of the $\frac{1}{16}$-BPS states
counted by the superconformal index, see for example Agarwal:2020zwm ; Cabo-
Bizet:2020ewf ; Beccaria:2023hip . This means that we have to change (65) from
the grand canonical ensemble — with fixed chemical potentials – to the
microcanonical ensemble. To do so, we are instructed to perform the following
Laplace transformation

$\displaystyle d(J;Q)$ $\displaystyle=\int d\Delta
d{\tau}d{\sigma}e^{-\log{\mathcal{I}_{4d}}-2\pi
i\sum_{I=1}^{3}\Delta_{I}Q_{I}-2\pi i({\sigma}J_{1}+{\tau}J_{2})+2\pi
i\Lambda\left(\sum_{I=1}^{3}\Delta_{I}-{\tau}-{\sigma}+n_{0}\right)}\,.$ (70)

This integration can be approximately solved in the large $N$ limit or in the
Cardy-like limit ($|{\tau}|\,,|{\sigma}|\ll 1$)666See ArabiArdehali:2019tdm ;
Honda:2019cio ; Cabo-Bizet:2019osg ; Kim:2019yrz ; Cabo-Bizet:2019eaf ;
Amariti:2019mgp ; Amariti:2021ubd  for extensive work on the Cardy-like limit
of the superconformal index. using the saddle point approach. We then must
find the extrema of the exponent and sum over the saddle points that dominate.

$\hskip 9.95845pt1$$\hskip 9.95845pt0$$-1\hskip 24.18483pt$$\hskip
4.2679pt\omega$$\omega-1\hskip 4.2679pt$$\hskip
4.2679pt\omega+1$$\vspace{1.5mm}\Delta_{+}$$\vspace{1.5mm}\Delta_{-}$

Figure 2: The Figure shows the complex plane of chemical potentials for a
generic $\Delta$ where the region corresponding to $n_{0}=1$ (66a) is shown in
gray and the region specified by $n_{0}=-1$ (66b) is shown in light blue. The
thick blue arrows running over the interval $\text{Re}(\Delta)\in(-1;\,1)$
represent the integration contour for $\Delta_{I}$. With dashed curves we
schematically show a deformation of the initial contour such that it passes
through the critical values of chemical potentials $\Delta_{\pm}$.

Let us consider a family of critical points
$\\{\Delta_{n_{0}},\,\omega_{n_{0}}\\}$, as shown schematically in Figure 2,
such that the effective action has the same real part when evaluated at these
points. In other words, both saddles are equally leading in the saddle point
approximation which then gives us

$\displaystyle d(J;\,Q)$ $\displaystyle\approx\sum_{n_{0}=\pm
1}\left(e^{-\log{\mathcal{I}_{4d}}-2\pi i\sum_{I=1}^{3}\Delta_{I}Q_{I}-2\pi
i({\sigma}J_{1}+{\tau}J_{2})+2\pi
i\Lambda\left(\sum_{I=1}^{3}\Delta_{I}-{\tau}-{\sigma}+n_{0}\right)}\right)\Big{|}_{\Delta_{n_{0}}}\,.$
(71)

Note that here we identify the critical points using only $\Delta_{n_{0}}$
because the constraint (69) already determines the critical values of
$\omega_{n_{0}}$. The extremization leads us to the following relations

$\displaystyle\frac{\partial\log\mathcal{I}_{4d}}{\partial\Delta_{I}}$
$\displaystyle=2\pi i(\Lambda-Q_{I}),\,\quad I=1,\cdots 3\,,$ (72a)
$\displaystyle\frac{\partial\log\mathcal{I}_{4d}}{\partial{\sigma}}$
$\displaystyle=2\pi i(J_{1}+\Lambda),\,$ (72b)
$\displaystyle\frac{\partial\log\mathcal{I}_{4d}}{\partial{\tau}}$
$\displaystyle=2\pi i(J_{2}+\Lambda),$ (72c)

under the constraint (69). This implies

$\displaystyle 2\prod_{I=1}^{3}(Q_{I}-\Lambda)$
$\displaystyle=N^{2}(J_{1}+\Lambda)(J_{2}+\Lambda).$ (73)

The relation (73) have precisely the same structure as (45), without the last
term that accounts for the D$3$-brane contributions and assuming the charges
are the ones of the 4d $\frac{1}{16}$-BPS states counted by the superconformal
index. The degeneracy of states is then given by

$\displaystyle d(J;\,Q)$ $\displaystyle\,\sim\,\sum_{n_{0}=\pm
1}\sum_{k=\pm,0}e^{-2\pi in_{0}\Lambda_{k}}\,.$ (74)

Generically, if we require a growth of states, then (73) must have two complex
conjugated solutions for $\Lambda$, which we have called $\Lambda_{\pm}$ in
(57), that upon application of (74) generate a dominant saddle for each value
of $n_{0}$. These dominant saddles correspond to the roots satisfying
$\text{Re}(2\pi in_{0}\Lambda_{\pm})>0$. If we appropriately label
$\Lambda_{\pm}$ such that the subindex corresponds to the sign of its
imaginary part, we have

$\displaystyle d(J;\,Q)$ $\displaystyle\sim e^{-2\pi i\Lambda_{+}}+e^{2\pi
i\Lambda_{-}}=2e^{2\pi\Lambda_{y}}\cos\left(2\pi\Lambda_{x}\right)\,\sim\,e^{2\pi\Lambda_{y}}.$
(75)

If the imaginary part of $\Lambda_{\pm}$ vanishes, then we see that the
microcanonical expression for the index is a pure oscillatory term that does
not probe the growth of states compatible with black hole entropy.

##  4 The defect superconformal index

The computation of the defect index requires the consistent embedding of the
2d $\mathcal{N}=(2,2)$ superconformal algebra into the 4d $\mathcal{N}=4$
superconformal algebra. This has been done in detail for $\mathcal{N}=2$
Gadde:2013dda  as well as for $\mathcal{N}=4$ Chen:2023lzq . In these works,
the fugacities used in the 2d description are related to those in 4d. For this
reason, we write the full defect index purely in terms of the 4d chemical
potentials.

The defect worldvolume $\mathcal{M}_{2d}$ extends along the $S^{1}$ and wraps
a circle inside the $S^{3}$. The surface operator with support on this
$\mathcal{M}_{2d}$ will be such that it commutes with the supercharge selected
to construct the 4d superconformal index, and this is ensured in practice by
appropriately choosing the orientation of the surface operator during the
embedding procedure. The defect index is then given by

$\displaystyle\mathcal{I}_{\text{D}}$
$\displaystyle=\int_{\text{SU(N)}}[\mathcal{D}U]\mathcal{Z}_{4d}(u,{\sigma},{\tau};\Delta)\mathcal{Z}_{2d}(u,{\sigma},{\tau};\Delta)~{}.$
(76)

It has been proposed that (76) provides the microscopic definition of a dual
gravity system which includes black holes interacting with a probe D$3$-brane
Chen:2023lzq . We revisit this matter carefully in this section.

It is possible to work in the approximation where the saddles of (3) are not
affected by the insertion of $\mathcal{Z}_{2d}$ in (76). This regime
corresponds holographically to the probe limit of the black hole/D$3$-brane
system. In this probe limit we can write

$\displaystyle\mathcal{I}_{\text{D}}$ $\displaystyle=\sum_{\hat{u}\in
4d-\text{saddles}}\mathcal{Z}_{4d}(\hat{u},{\sigma},{\tau};\Delta)\mathcal{Z}_{2d}(\hat{u},{\sigma},{\tau};\Delta)+\cdots\,,$
(77)

where the $\cdots$ correspond to the subleading saddles. Instead of directly
implementing the probe limit (77), we first study the 2d index in the context
of a systematic Cardy-like expansion along the lines of
GonzalezLezcano:2020yeb ; GonzalezLezcano:2022hcf . This enables us to have
better control over the effect of backreaction coming from the 2d defect on
the 4d index. If we denote the fundamental domains of chemical potentials
$\Delta^{(n_{0})}$, $n_{0}=\pm 1$ then we will see in Subsection 4.1 that for
$n_{0}=1$, the integrand $\mathcal{Z}_{2d}$ in (76) becomes independent of the
holonomies up to corrections exponentially suppressed in $1/|\omega|$

$\displaystyle\mathcal{I}^{(1)}_{\text{D}}$
$\displaystyle=\mathcal{Z}_{2d}({\sigma},{\tau};\Delta^{(1)})\sum_{\hat{u}\in
4d-\text{saddles}}\mathcal{Z}_{4d}(\hat{u},{\sigma},{\tau};\Delta^{(1)})\,.$
(78)

Moreover, for the other domain of chemical potentials labelled by $n_{0}=-1$,
the leading order in the Cardy-like limit is given by

$\displaystyle\mathcal{I}^{(-1)}_{\text{D}}\Big{|}_{\omega\rightarrow 0}$
$\displaystyle=\mathcal{Z}_{2d}({\sigma},{\tau};\Delta^{(-1)})\sum_{\hat{u}\in
4d-\text{saddles}}\mathcal{Z}_{4d}(\hat{u},{\sigma},{\tau};\Delta^{(-1)})\,.$
(79)

We now turn to study the 2d index in the systematic Cardy-like expansion.

###  4.1 The Cardy-like expansion of the 2d index

Following Chen:2023lzq ; Gadde:2013dda , we start with the 2d index given as

$\displaystyle\mathcal{Z}_{2d}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\exp\Biggl{[}\sum_{j\neq
i}\log\frac{\theta_{0}(-u_{ij}-\Delta_{2}+{\sigma};\,{\sigma})}{\theta_{0}(-u_{ij}+\Delta_{1}-{\tau};\,{\sigma})}+\log\frac{\theta_{0}(u_{ij}-\Delta_{1}-\Delta_{2}+{\tau}+{\sigma};\,{\sigma})}{\theta_{0}(u_{ij};\,{\sigma})}\Biggr{]}.$
(80)

Now using the elliptic theta functions in (106a) and (106b), the 2d index can
be recast in the form

$\displaystyle\begin{split}\mathcal{Z}_{2d}&=\sum_{i=1}^{N}\exp\left[\sum_{j\neq
i}\log\frac{\left(\,{\rm
e}^{2\pi\mathrm{i}(-u_{ij}-\Delta_{2}+{\sigma})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}\left(\,{\rm
e}^{2\pi\mathrm{i}(u_{ij}+\Delta_{2})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}}{\left(\,{\rm
e}^{2\pi\mathrm{i}(-u_{ij}+\Delta_{1}-{\tau})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}\left(\,{\rm
e}^{2\pi\mathrm{i}(u_{ij}-\Delta_{1}+{\tau}+{\sigma})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}}\right.\\\
&\left.\quad\quad+\sum_{j\neq i}\log\frac{\left(\,{\rm
e}^{2\pi\mathrm{i}(u_{ij}-\Delta_{1}-\Delta_{2}+{\tau}+{\sigma})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}\left(\,{\rm
e}^{2\pi\mathrm{i}(-u_{ij}+\Delta_{1}+\Delta_{2}+{\tau})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}}{\left(\,{\rm
e}^{2\pi\mathrm{i}(u_{ij})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}\left(\,{\rm
e}^{2\pi\mathrm{i}({\sigma}-u_{ij})};\,{\rm
e}^{2\pi\mathrm{i}{\sigma}}\right)_{\infty}}\right].\end{split}$ (81)

We consider $u_{ij}=(x_{ij}{\sigma}+y_{ij}{\tau}$) and
${\tau}=\frac{s_{1}}{s_{2}}{\sigma}$ such that
$u_{ij}={\sigma}(x_{ij}+\frac{s_{1}}{s_{2}}y_{ij})\equiv{\sigma}z_{ij}$. These
change of variables allows us to implement the systematic Cardy-like
expansion.777See GonzalezLezcano:2022hcf  for a similar implementation to the
3d superconformal index. Upon using the asymptotic expansion (107), we find

$\displaystyle\begin{split}\mathcal{Z}_{2d}&=\sum_{i=1}^{N}\exp\left[\frac{1}{2\pi\mathrm{i}{\sigma}}\sum_{j\neq
i}\sum_{r=0}^{\infty}(-1)^{r}\frac{(2\pi\mathrm{i}{\sigma})^{r}}{r!}\left(B_{r}(z_{ij})\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\Delta_{2})}\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.+B_{r}\left(1-z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\Delta_{2})}\right)+B_{r}\left(-\frac{s_{1}}{s_{2}}-z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\Delta_{1}-\Delta_{2})}\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.+B_{r}\left(1+\frac{s_{1}}{s_{2}}+z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\Delta_{1}+\Delta_{2})}\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.-B_{r}\left(1+\frac{s_{1}}{s_{2}}+z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\Delta_{1})}\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.-B_{r}\left(-\frac{s_{1}}{s_{2}}-z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\Delta_{1})}\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.-B_{r}\left(1-z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\epsilon)}\right)-B_{r}\left(z_{ij}\right)\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\epsilon)}\right)\right)\right].\end{split}$ (82)

In the last line we have regulated the Polylogarithms via a small $\epsilon>0$
regulator. We will see that we can safely take $\epsilon\rightarrow 0$ at the
end of the manipulations. Utilizing the property of Bernoulli polynomials,

$\displaystyle B_{r}(1-x)$ $\displaystyle=(-1)^{r}B_{r}(x),\,\quad r\geq 0\,,$
(83a) $\displaystyle B_{r}(0)$ $\displaystyle=(-1)^{r}B_{r}(1)\,,$ (83b)

the 2d index is simplified to

$\displaystyle\begin{split}\mathcal{Z}_{2d}&=\sum_{i=1}^{N}\exp\left[\frac{1}{2\pi\mathrm{i}{\sigma}}\sum_{j\neq
i}\sum_{r=0}^{\infty}\frac{(2\pi\mathrm{i}{\sigma})^{r}}{r!}\left[B_{r}(z_{ij})\left((-1)^{r}\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\Delta_{2})}\right)+\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\Delta_{2})}\right)\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.+B_{r}(-\frac{s_{1}}{s_{2}}-z_{ij})\left((-1)^{r}\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\Delta_{1}-\Delta_{2})}\right)+\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\Delta_{1}+\Delta_{2})}\right)\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.-B_{r}(-\frac{s_{1}}{s_{2}}-z_{ij})\left(\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\Delta_{1})}\right)+(-1)^{r}\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\Delta_{1})}\right)\right)\right.\right.\\\
&\quad\quad\quad\quad\left.\left.-B_{r}(z_{ij})\left(\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(\epsilon)}\right)+(-1)^{r}\text{Li}_{2-r}\left({\rm
e}^{2\pi\mathrm{i}(-\epsilon)}\right)\right)\right]\right]\,.\end{split}$ (84)

The final simplification requires us to use the property of Polylogarithm
functions

$\displaystyle\text{Li}_{n}({\rm
e}^{2\pi\mathrm{i}z})+(-1)^{n}\text{Li}_{n}({\rm e}^{-2\pi\mathrm{i}z})$
$\displaystyle=-\frac{(2\pi\mathrm{i})^{n}}{n!}B_{n}(\\{z\\}),\quad
n=1,2,3,\cdots\,,$ (85a) $\displaystyle\text{Li}_{-n}({\rm
e}^{2\pi\mathrm{i}z})+(-1)^{n}\text{Li}_{-n}({\rm e}^{-2\pi\mathrm{i}z})$
$\displaystyle=0,\quad n=0,1,2,3,\cdots\,,$ (85b)

for $0\leq\text{ Re}(z)<1$ and $\text{Im}(z)\geq 0$ or $0<\text{ Re}(z)\leq 1$
and $\text{Im}(z)<0$. Now (84) becomes

$\displaystyle\begin{split}\mathcal{Z}_{2d}&=\sum_{i=1}^{N}\exp\left[\frac{1}{2\pi\mathrm{i}{\sigma}}\sum_{j\neq
i}\sum_{r=0}^{1}\frac{(2\pi\mathrm{i})^{2-r}}{(2-r)!}\frac{(2\pi\mathrm{i}{\sigma})^{r}}{r!}\left(-B_{r}(z_{ij})\left(B_{2-r}\left(\\{\Delta_{2}\\}\right)-B_{2-r}\right)\right.\right.\\\
&\left.\left.-(-1)^{2-r}B_{r}(-\frac{s_{1}}{s_{2}}-z_{ij})\left(B_{2-r}\left(\\{-\Delta_{1}-\Delta_{2}\\}\right)-B_{2-r}\left(\\{-\Delta_{1}\\}\right)\right)+\right)\right]\\\
&=\sum_{i=1}^{N}\exp\left\\{\frac{2\pi\mathrm{i}(N-1)}{{\sigma}}\left[\prod_{a=2}^{3}\left(\\{\Delta_{a}\\}-n\right)+\delta_{n_{0},-1}\left(\sum_{j\neq
i}\frac{u_{ij}}{N-1}-\frac{{\sigma}}{2}\right)\right]\right\\}\,,\end{split}$
(86)

where we have recovered the original holonomy variables, namely
$u_{ij}=\frac{z_{ij}}{{\sigma}}$ and for compactness we have defined
$n\equiv\frac{1-n_{0}}{2}$. The function $\\{z\\}$ defined in (110) is such
that it forces the Bernoulli polynomials to have the same periodicity
properties as the Polylogarithm functions. Note that (85a) ensures that the
terms with the $\epsilon$ regulator produce a finite result as the right hand
side is a finite quantity at $z=0$.

For the leading correction in the Cardy-like limit, the last terms in (86)
vanishes and we have

$\displaystyle\mathcal{Z}_{2d}=N\exp\left[\frac{2\pi\mathrm{i}(N-1)}{{\sigma}}\left(\prod_{a=2}^{3}\left(\\{\Delta_{a}\\}-n\right)\right)\right]\,.$
(87)

There are two main observations to make regarding the transition from (86) to
(87):

  * •

For the fundamental domain of chemical potentials labelled by $n_{0}=1$, the
holonomies drop from the expression (86), rendering our result (87) valid at
finite $N$ up to exponentially supressed corrections in $1/|\omega|$. This
domain of chemical potentials corresponds to (78) and clearly in this case
there is no need to work in the probe limit.

  * •

For $n_{0}=-1$ there is a linear dependence of the holonomies in (86) that
accounts for backreaction of the D$3$-brane when considering the combined
4d-2d system. However, this term vanishes in the leading term of the Cardy-
like limit. This allow us to factor out the 2d index contribution to the 4d
defect superconformal index, as anticipated in (79).

The next step is to explicitly evaluate (76) and extract the microcanonical
degeneracies implementing the Laplace transformation.

$\hskip 11.95013pt1$$\hskip 9.95845pt0$$-1\hskip 24.18483pt$$\hskip
4.2679pt\omega$$\omega-1\hskip 4.2679pt$$\hskip
4.2679pt\omega+1$$\vspace{1.5mm}\Delta_{+}$$\vspace{1.5mm}\Delta_{-}$$\vspace{1.5mm}\widetilde{\Delta}_{+}$$\vspace{2.5mm}\widetilde{\Delta}_{-}$

Figure 3: The Figure shows the complex plane of chemical potentials for a
generic $\Delta$ where the region corresponding to $n_{0}=1$ (66a) is shown in
gray and the region specified by $n_{0}=-1$ (66b) is shown in light blue. The
thick blue arrows running over the interval $\text{Re}(\Delta)\in(-1;\,1)$
represent the integration contour for $\Delta_{I}$. Now we represent the
deformed contour in the presence of the D$3$-brane in black dashed lines
passing throught the new saddles labeled as $\widetilde{\Delta}_{\pm}$. We
have kept the contour (dashed orange curve) for the case of the black hole in
the absence of D$3$-brane just for reference.

###  4.2 The combined 4d-2d system: extracting degeneracies

For simplicity, we continue to restrict ourselves to the leading term in the
Cardy-like limit when we consider the combined 4d-2d system. From (65), (78),
(79) and (87), the total defect index is given by

$\displaystyle\mathcal{I}_{\text{D}}=N\text{exp}\left[\frac{i\pi(N-1)}{{\sigma}}\left(-\frac{(N+1)}{{\tau}}\prod_{I=1}^{3}\left(\left\\{\Delta_{I}\right\\}_{\omega}-n\right)+2\prod_{a=2}^{3}\left(\\{\Delta_{a}\\}-n\right)\right)\right].$
(88)

Note that for purely imaginary ${\sigma},{\tau}$ as well as for purely real
arguments, the functions $\\{\cdot\\}_{\omega}$ and $\\{\cdot\\}$ coincide,
which allows the expressions for $\mathcal{Z}_{2d}$ and $\mathcal{Z}_{4d}$ to
be written in terms of the same combinations of chemical potentials. The
expression (88) considerably simplifies in the regimes of real chemical
potentials $\Delta_{I}$ such that $|\Delta_{I}|<1$

$\displaystyle\mathcal{I}_{\text{D}}$
$\displaystyle=N\text{exp}\left[\frac{i\pi(N-1)}{{\sigma}}\left(-\frac{(N+1)}{{\tau}}\Delta_{1}\Delta_{2}\Delta_{3}+2\Delta_{2}\Delta_{3}\right)\right]\,,$
(89) $\displaystyle\Delta_{3}$
$\displaystyle={\sigma}+{\tau}-\Delta_{1}-\Delta_{2}-n_{0}\,.$ (90)

From now on we proceed to extract the microcanonical degeneracies from the
grandcanonical expression for the defect index given in (89). Implementing the
Laplace transform using the saddle point method requires the following
extremization process

$\displaystyle\frac{\partial\log\mathcal{I}_{\text{D}}}{\partial\Delta_{I}}$
$\displaystyle=2\pi i(\Lambda-Q_{I}),\,\quad I=1,\cdots 3\,,$ (91)
$\displaystyle\frac{\partial\log\mathcal{I}_{\text{D}}}{\partial{\sigma}}$
$\displaystyle=-2\pi i(J_{1}+\Lambda)\,,$ (92)
$\displaystyle\frac{\partial\log\mathcal{I}_{\text{D}}}{\partial{\tau}}$
$\displaystyle=-2\pi i(J_{2}+\Lambda)\,,$ (93)

under the constraint (69). It is worth pointing out that in this case the
charges $Q_{1,2,3}$ and $J_{1,2}$ are the total charge of 4d and 2d states.

From this point on, the mathematical problem is essentially equivalent to
(40). In the field theoretical language we have to repeat the calculation of
Section 3.1 just replacing $\mathcal{I}_{4d}$ by $\mathcal{I}_{\text{D}}$ in
(70). If we focus on the large $N$ regime, following the logic of Section 2.4,
we find a new set of saddle points through solving a modified cubic equation
completely equivalent to (45). In Figure 3, we show a schematic picture of how
the new saddles in the complex domain of chemical potentials can be changed by
the $1/N$ corrections introduced by considering the contribution of the 2d
defect states to the degeneracy.

##  5 Final comments and open questions

We have presented the thermodynamic analysis for the combined black
hole/D$3$-brane system in a way that ensures the reality of the charges and
the entropy. To do so, we took into account both leading saddles of the
gravitational path integral and likewise, on the gauge dual, the two leading
saddles of the integral over the holonomies that represent the defect
superconformal index. Following this procedure we obtain real charges and
entropy without the need of imposing a nonlinear constraint among the charges.

There are various interesting extensions of this work. The first is to compute
the backreaction of the D$3$-brane in the geometry. One first step in this
direction may be solving the Killing spinor equations for the combined system
of a black hole/D$3$-brane. Presumably, while doing so, we would be forced
into computing relevant backreaction effects. This may allow us to answer
questions like whether the exact change in entropy predicted by the field
theory side of the duality can be interpreted as a change in the area of the
black hole horizon.

## Acknowledgments

We thank Davide Cassani, Robie Hennigar and Enrico Turetta for fruitful for
discussions. MD is supported in part by the Odysseus grant (G0F9516N Odysseus)
from the as well as the the Postdoctoral Fellows of the Research Foundation -
Flanders grant (1235324N). AGL is supported by an appointment to the JRG
Program at the APCTP through the Science and Technology Promotion Fund and
Lottery Fund of the Korean Government, by the Korean Local Governments -
Gyeongsangbuk-do Province and Pohang City, and by the National Research
Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.
2021R1F1A1048531). The authors would would like to thank the Isaac Newton
Institute for Mathematical Sciences, Cambridge, for support and hospitality
during the programme Black holes: bridges between number theory and
holographic quantum information where work on this paper was undertaken. This
work was supported by EPSRC grant no EP/R014604/1.

##  Appendix A Revisiting a previous approach

In this appendix we briefly review the procedure used in Chen:2023lzq  to
compute the entropy of the combined system. Recalling that at the level of the
on-shell action

$I=I_{\text{BH}}+I_{\text{D3}}$ (94)

and requiring that

$S=S_{\text{BH}}+S_{\text{D3}},$ (95)

then the first law of thermodynamic of the total system would unequivocally
constrain the thermodynamic charges of the D3-brane to be

$\displaystyle J_{1,\text{D3}}$ $\displaystyle=-\frac{1}{\beta}\frac{\partial
I_{\text{D3}}}{\partial\Omega_{1}},\quad
J_{2,\text{D3}}=-\frac{1}{\beta}\frac{\partial
I_{\text{D3}}}{\partial\Omega_{2}},\quad
Q_{\text{D3}}=-\frac{1}{\beta}\frac{\partial I_{\text{D3}}}{\partial\Phi}\,,$
(96)

provided the chemical potentials do not receive subleading corrections in
$1/N$, or equivalently, that they are the very same chemical potentials of the
unperturbed black hole. To evaluate the right hand sides of the equations in
(96), we must invert the Jacobian matrix

$\displaystyle\frac{\partial(\Omega_{1},\Omega_{2},\Phi,\beta)}{\partial(a,b,q,r_{+})}.$
(97)

In this way we find the expressions for the electric charges and angular
momentum of the D3-brane reported in Chen:2023lzq .

###  A.1 Legendre transform with just the D3-brane

The entropy of the D3-brane is defined from the Legendre transform

$\displaystyle\begin{split}S_{\text{D3}}&=-I_{\text{D3}}-2\pi
i\sum_{I=1}^{3}\varphi_{I,\text{BH}}Q_{I,\text{D3}}-2\pi
i\sum_{k=1}^{2}\omega_{k,\text{BH}}J_{k,\text{D3}}+2\pi
i\Lambda\left(\sum_{I=1}^{3}\varphi_{I,\text{BH}}-\sum_{k=1}^{2}\omega_{k,\text{BH}}+n_{0}\right),\end{split}$
(98)

where $\Lambda$ is a Lagrange multiplier implementing the corresponding linear
constraint, and we have reinstated the subindex BH to recall that these are
the very same potential as the black hole solution. The extremization leads to
the following equations

$\displaystyle 0$ $\displaystyle=-\frac{\partial
I_{\text{D3}}}{\partial\varphi_{I,\text{BH}}}-2\pi
i(Q_{I,\text{D3}}-\Lambda)=2\pi
i\left(N\frac{\varphi_{2,\text{BH}}\varphi_{3,\text{BH}}}{\varphi_{I,\text{BH}}\omega_{1,\text{BH}}}(\delta^{I}_{2}+\delta^{I}_{3})-Q_{I,\text{D3}}+\Lambda\right),$
$\displaystyle I$ $\displaystyle=1,2,3,$ $\displaystyle 0$
$\displaystyle=-\frac{\partial
I_{\text{D3}}}{\partial\omega_{k,\text{BH}}}-2\pi
i(J_{k,\text{D3}}+\Lambda)=-2\pi
i\left(N\frac{\varphi_{2,\text{BH}}\varphi_{3,\text{BH}}}{\omega_{k,\text{BH}}\omega_{1,\text{BH}}}\delta^{k}_{1}+J_{k,\text{D3}}+\Lambda\right),$
$\displaystyle k$ $\displaystyle=1,2.$ (99)

Imposing (A.1), we find that the entropy is given by

$\displaystyle\begin{split}S_{\text{D3}}&=2\pi in_{0}\Lambda.\end{split}$
(100)

Solving for $\varphi_{2,\text{BH}}$ and $\varphi_{3,\text{BH}}$ in the
equation for $I=2$ and $I=3$, we find

$\displaystyle\begin{split}\varphi_{2,\text{BH}}&=-\omega_{1,\text{BH}}\frac{\Lambda-
Q_{3,\text{D3}}}{N},\qquad\varphi_{3,\text{BH}}=-\omega_{1,\text{BH}}\frac{\Lambda-
Q_{2,\text{D3}}}{N}.\end{split}$ (101)

and imposing this into the equation for $k=1$, we have

$\displaystyle\begin{split}0&=\left(\Lambda-
Q_{2,\text{D3}}\right)\left(\Lambda-
Q_{3,\text{D3}}\right)+N(J_{1,\text{D3}}+\Lambda)\\\
&=\Lambda^{2}+\Lambda(-Q_{2,\text{D3}}-Q_{3,\text{D3}}+N)+(Q_{2,\text{D3}}Q_{3,\text{D3}}+NJ_{1,\text{D3}}).\end{split}$
(102)

As this is a quadratic polynomial with real coefficients, the solutions can
either be two real roots or two complex roots, conjugate to each other.
Therefore, we have

$\displaystyle\Lambda_{\pm}$
$\displaystyle=\frac{1}{2}\left(Q_{2,\text{D3}}+Q_{3,\text{D3}}-N\pm\sqrt{\left(-Q_{2,\text{D3}}-Q_{3,\text{D3}}+N\right){}^{2}-4\left(NJ_{1,\text{D3}}+Q_{2,\text{D3}}Q_{3,\text{D3}}\right)}\right).$
(103)

In general, using the expression for the charges found in Chen:2023lzq ,
$\Lambda_{\pm}$ are complex valued roots, but not necessarily complex
conjugate to each other. Moreover, we have the extremization equations

$\displaystyle\Lambda=Q_{1,\text{D3}}\,,\qquad\Lambda=-J_{2,\text{D3}}\,.$
(104)

In the regime that (103) and (104) are satisfied, we find a complex-valued
entropy.

##  Appendix B Elliptic functions and their asymptotic behavior

Here we gather definitions and useful identities of elliptic functions.

The Pochhammer symbol is defined as

$(z;q)_{\infty}=\prod_{k=0}^{\infty}(1-zq^{k}).$ (105)

The elliptic theta functions have the following product forms

$\displaystyle\theta_{0}(u;\tau)$
$\displaystyle=\prod_{k=0}^{\infty}(1-e^{2\pi i(u+k\tau)})(1-e^{2\pi
i(-u+(k+1)\tau)})$ (106a) $\displaystyle=\left(e^{2\pi iu};\,e^{2\pi
i\tau}\right)_{\infty}\left(e^{2\pi i(\tau-u)};\,e^{2\pi
i\tau}\right)_{\infty}\,.$ (106b)

Consider an asymptotic expansion in $\tau$ with fixed $0<\arg\tau<\pi$ as
given in Garoufalidis:2018qds :

$\displaystyle(z{\rm e}^{a\pi\mathrm{i}\tau};{\rm
e}^{2\pi\mathrm{i}\tau})_{\infty}$
$\displaystyle=\exp\left(\frac{1}{2\pi\mathrm{i}\tau}\sum_{r=0}^{\infty}(-1)^{r}B_{r}\left(1-\frac{a}{2}\right)\frac{(2\pi\mathrm{i}\tau)^{r}}{r!}\text{Li}_{2-r}(z)\right)\,.$
(107)

The elliptic gamma function and the ‘tilde’ elliptic gamma function are
defined as

$\displaystyle\Gamma(z;p,q)$
$\displaystyle=\prod_{j,k=0}^{\infty}\frac{1-p^{j+1}q^{k+1}z^{-1}}{1-p^{j}q^{k}z},$
(108a) $\displaystyle\widetilde{\Gamma}(u;{\sigma},\tau)$
$\displaystyle=\prod_{j,k=0}^{\infty}\frac{1-e^{2\pi
i[(j+1){\sigma}+(k+1)\tau-u]}}{1-e^{2\pi i[j{\sigma}+k\tau+u]}}.$ (108b)

To study asymptotic behaviors of elliptic functions, we introduce a
$\tau$-modded value of a complex number $u$, namely $\\{u\\}_{\tau}$, as

$\\{u\\}_{\tau}\equiv
u-\lfloor\mbox{Re}u-\cot(\arg\tau)\mbox{Im}u\rfloor\quad(u\in\mathbb{C}),$
(109)

and define $\\{x\\}$ such that

$\\{x\\}\equiv x-\lfloor\text{Re}x\rfloor\,,$ (110)

where $\\{x\\}_{\tau}=\\{x\\}$ for purely imaginary $\tau$.

## References

  * (1) F. Benini, K. Hristov and A. Zaffaroni, _Black hole microstates in AdS 4 from supersymmetric localization_, _JHEP_ 05 (2016) 054 [1511.04085].
  * (2) F. Benini, K. Hristov and A. Zaffaroni, _Exact microstate counting for dyonic black holes in AdS 4_, _Phys. Lett._ B771 (2017) 462 [1608.07294].
  * (3) S. Choi, J. Kim, S. Kim and J. Nahmgoong, _Large AdS black holes from QFT_ , 1810.12067.
  * (4) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes_, 1810.11442.
  * (5) F. Benini and E. Milan, _Black Holes in 4D $\mathcal{N}$=4 Super-Yang-Mills Field Theory_, _Phys. Rev. X_ 10 (2020) 021037 [1812.09613].
  * (6) A. Arabi Ardehali, _Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS5 blackholes_, _JHEP_ 06 (2019) 134 [1902.06619].
  * (7) M. Honda, _Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula_ , 1901.08091.
  * (8) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _The asymptotic growth of states of the 4d N=1 superconformal index_ , _Submitted to: J. High Energy Phys._ (2019) [1904.05865].
  * (9) J. Kim, S. Kim and J. Song, _A 4d $N=1$ Cardy Formula_, 1904.03455.
  * (10) A. Cabo-Bizet and S. Murthy, _Supersymmetric phases of 4d $\mathcal{N}$ = 4 SYM at large $N$_, _JHEP_ 09 (2020) 184 [1909.09597].
  * (11) A. Amariti, I. Garozzo and G. Lo Monaco, _Entropy function from toric geometry_ , 1904.10009.
  * (12) A. González Lezcano and L. A. Pando Zayas, _Microstate counting via bethe ansätze in the 4d $\mathcal{N}=1$ superconformal index_, 1907.12841.
  * (13) A. Lanir, A. Nedelin and O. Sela, _Black hole entropy function for toric theories via Bethe Ansatz_ , 1908.01737.
  * (14) K. Goldstein, V. Jejjala, Y. Lei, S. van Leuven and W. Li, _Probing the EVH limit of supersymmetric AdS black holes_ , _JHEP_ 02 (2020) 154 [1910.14293].
  * (15) A. Arabi Ardehali, J. Hong and J. T. Liu, _Asymptotic growth of the 4d $\mathcal{N}$ = 4 index and partially deconfined phases_, _JHEP_ 07 (2020) 073 [1912.04169].
  * (16) S. Murthy, _The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM_, 2005.10843.
  * (17) S. Murthy, _Growth of the $\frac{1}{16}$-bps index in 4d $\mathcal{N}=4$ supersymmetric yang-mills theory_, _Phys. Rev. D_ 105 (2022) L021903.
  * (18) P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong, _AdS black holes and finite N indices_ , _Phys. Rev. D_ 103 (2021) 126006 [2005.11240].
  * (19) F. Benini, E. Colombo, S. Soltani, A. Zaffaroni and Z. Zhang, _Superconformal indices at large $N$ and the entropy of AdS5 $\times$ SE5 black holes_, _Class. Quant. Grav._ 37 (2020) 215021 [2005.12308].
  * (20) A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, _The large- $N$ limit of the 4d $\mathcal{N}$ = 1 superconformal index_, _JHEP_ 11 (2020) 150 [2005.10654].
  * (21) A. Cabo-Bizet, _On the 4d superconformal index near roots of unity: bulk and localized contributions_ , _JHEP_ 02 (2023) 134 [2111.14941].
  * (22) D. Cassani and Z. Komargodski, _EFT and the SUSY Index on the 2nd Sheet_ , _SciPost Phys._ 11 (2021) 004 [2104.01464].
  * (23) V. Jejjala, Y. Lei, S. van Leuven and W. Li, _SL(3, $\mathbb{Z}$) Modularity and New Cardy limits of the $\mathcal{N}$ = 4 superconformal index_, _JHEP_ 11 (2021) 047 [2104.07030].
  * (24) V. Jejjala, Y. Lei, S. van Leuven and W. Li, _Modular factorization of superconformal indices_ , _JHEP_ 10 (2023) 105 [2210.17551].
  * (25) O. Aharony, F. Benini, O. Mamroud and E. Milan, _A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index_, _Phys. Rev. D_ 104 (2021) 086026 [2104.13932].
  * (26) A. Cabo-Bizet, _Quantum phases of 4d SU(N) $\mathcal{N}$ = 4 SYM_, _JHEP_ 10 (2022) 052 [2111.14942].
  * (27) K. Goldstein, V. Jejjala, Y. Lei, S. van Leuven and W. Li, _Residues, modularity, and the Cardy limit of the 4d $\mathcal{N}$ = 4 superconformal index_, _JHEP_ 04 (2021) 216 [2011.06605].
  * (28) S. Choi, S. Jeong, S. Kim and E. Lee, _Exact QFT duals of AdS black holes_ , _JHEP_ 09 (2023) 138 [2111.10720].
  * (29) S. Choi, S. Kim and J. Song, _Large $N$ Universality of 4d $\mathcal{N}=1$ Superconformal Index and AdS Black Holes_, 2309.07614.
  * (30) S. Bhattacharyya, A. Grassi, M. Marino and A. Sen, _A One-Loop Test of Quantum Supergravity_ , _Class. Quant. Grav._ 31 (2014) 015012 [1210.6057].
  * (31) J. T. Liu, L. A. Pando Zayas, V. Rathee and W. Zhao, _One-Loop Test of Quantum Black Holes in anti–de Sitter Space_ , _Phys. Rev. Lett._ 120 (2018) 221602 [1711.01076].
  * (32) J. T. Liu, L. A. Pando Zayas, V. Rathee and W. Zhao, _Toward Microstate Counting Beyond Large N in Localization and the Dual One-loop Quantum Supergravity_ , _JHEP_ 01 (2018) 026 [1707.04197].
  * (33) M. David, V. Godet, Z. Liu and L. A. Pando Zayas, _Non-topological logarithmic corrections in minimal gauged supergravity_ , _JHEP_ 08 (2022) 043 [2112.09444].
  * (34) A. González Lezcano, J. Hong, J. T. Liu and L. A. Pando Zayas, _Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions_ , _JHEP_ 01 (2021) 001 [2007.12604].
  * (35) A. Amariti, M. Fazzi and A. Segati, _The SCI of $\mathcal{N}$ = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral_, _JHEP_ 06 (2021) 132 [2012.15208].
  * (36) M. David, A. Lezcano González, J. Nian and L. A. Pando Zayas, _Logarithmic corrections to the entropy of rotating black holes and black strings in AdS 5_, _JHEP_ 04 (2022) 160 [2106.09730].
  * (37) D. Cassani, A. Ruipérez and E. Turetta, _Corrections to AdS 5 black hole thermodynamics from higher-derivative supergravity_, _JHEP_ 11 (2022) 059 [2208.01007].
  * (38) N. Bobev, K. Hristov and V. Reys, _AdS 5 holography and higher-derivative supergravity_, _JHEP_ 04 (2022) 088 [2112.06961].
  * (39) N. Bobev, V. Dimitrov, V. Reys and A. Vekemans, _Higher derivative corrections and AdS5 black holes_ , _Phys. Rev. D_ 106 (2022) L121903 [2207.10671].
  * (40) D. Cassani, A. Ruipérez and E. Turetta, _Boundary terms and conserved charges in higher-derivative gauged supergravity_ , _JHEP_ 06 (2023) 203 [2304.06101].
  * (41) N. Bobev, M. David, J. Hong, V. Reys and X. Zhang, _A compendium of logarithmic corrections in AdS/CFT_ , 2312.08909.
  * (42) Y. Chen, M. Heydeman, Y. Wang and M. Zhang, _Probing Supersymmetric Black Holes with Surface Defects_ , 2306.05463.
  * (43) Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, _Five-dimensional gauged supergravity black holes with independent rotation parameters_ , _Phys. Rev. D_ 72 (2005) 041901 [hep-th/0505112].
  * (44) Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, _General non-extremal rotating black holes in minimal five-dimensional gauged supergravity_ , _Phys. Rev. Lett._ 95 (2005) 161301 [hep-th/0506029].
  * (45) J. B. Gutowski and H. S. Reall, _Supersymmetric AdS(5) black holes_ , _JHEP_ 0402 (2004) 006 [hep-th/0401042].
  * (46) J. B. Gutowski and H. S. Reall, _General supersymmetric AdS(5) black holes_ , _JHEP_ 04 (2004) 048 [hep-th/0401129].
  * (47) S.-Q. Wu, _General Nonextremal Rotating Charged AdS Black Holes in Five-dimensional $U(1)^{3}$ Gauged Supergravity: A Simple Construction Method_, _Phys. Lett. B_ 707 (2012) 286 [1108.4159].
  * (48) H. K. Kunduri, J. Lucietti and H. S. Reall, _Supersymmetric multi-charge AdS(5) black holes_ , _JHEP_ 04 (2006) 036 [hep-th/0601156].
  * (49) S. Gukov and E. Witten, _Gauge Theory, Ramification, And The Geometric Langlands Program_ , hep-th/0612073.
  * (50) A. Cabo-Bizet, _From multi-gravitons to Black holes: The role of complex saddles_ , 2012.04815.
  * (51) M. Beccaria and A. Cabo-Bizet, _Large black hole entropy from the giant brane expansion_ , 2308.05191.
  * (52) D. Marolf, _Chern-Simons terms and the three notions of charge_ , in _International Conference on Quantization, Gauge Theory, and Strings: Conference Dedicated to the Memory of Professor Efim Fradkin_ , pp. 312–320, 6, 2000, hep-th/0006117.
  * (53) C. Romelsberger, _Calculating the Superconformal Index and Seiberg Duality_ , 0707.3702.
  * (54) J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, _An Index for 4 dimensional super conformal theories_ , _Commun. Math. Phys._ 275 (2007) 209 [hep-th/0510251].
  * (55) A. G. Lezcano, J. Hong, J. T. Liu and L. A. Pando Zayas, _The Bethe-Ansatz approach to the $\mathcal{N}$ = 4 superconformal index at finite rank_, _JHEP_ 06 (2021) 126 [2101.12233].
  * (56) F. Benini and G. Rizi, _Superconformal index of low-rank gauge theories via the Bethe Ansatz_ , _JHEP_ 05 (2021) 061 [2102.03638].
  * (57) A. Amariti, M. Fazzi and A. Segati, _Expanding on the Cardy-like limit of the SCI of 4d $\mathcal{N}$ = 1 ABCD SCFTs_, _JHEP_ 07 (2021) 141 [2103.15853].
  * (58) A. Gadde and S. Gukov, _2d Index and Surface operators_ , _JHEP_ 03 (2014) 080 [1305.0266].
  * (59) A. González Lezcano, M. Jerdee and L. A. Pando Zayas, _Cardy expansion of 3d superconformal indices and corrections to the dual black hole entropy_ , _JHEP_ 01 (2023) 044 [2210.12065].
  * (60) S. Garoufalidis and D. Zagier, _Asymptotics of Nahm sums at roots of unity_ , 1812.07690.