Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS$_5$ blackholes

TL;DR

This paper rigorously analyzes the Cardy-like limit of the 4d $ abla=4$ superconformal index using elliptic hypergeometric integrals, revealing new blackhole saddle points and bifurcation phenomena related to AdS$_5$ blackholes.

Contribution

It provides a rigorous derivation of the index asymptotics, uncovers a new blackhole saddle point, and explores bifurcation phenomena in the index's behavior.

Findings

Identification of a new blackhole saddle point.

Discovery of bifurcation phenomena in index asymptotics.

Clarification of the role of supersymmetric Casimir energy.

Abstract

Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardy-like limit of the superconformal index of the 4d $\mathcal{N}=4$ theory with complexified fugacities which encodes the entropy of the dual supersymmetric AdS$_5$ blackholes. Here we study the Cardy-like asymptotics of the index within the rigorous framework of elliptic hypergeometric integrals, thereby filling a gap in their derivation of the blackhole entropy function, finding a new blackhole saddle-point, and demonstrating novel bifurcation phenomena in the asymptotics of the index as a function of fugacity phases. We also comment on the relevance of the supersymmetric Casimir energy to the blackhole entropy function in the present context.

Figures

Equations

$$\begin{array}{r}\mathcal{I}(p,q,u_k):=\mathrm{Tr}\left[(-1{)}^F{e}^{\hat{\beta }(\Delta -J_1-J_2-\frac{3}{2}r)}{p}^{J_1+\frac{r}{2}}{q}^{J_2+\frac{r}{2}}\prod _k{u}_k^{q_k}\right],\end{array}$$

$$\begin{array}{r}\mathcal{I}(p,q,y_{1,2,3})=\mathrm{Tr}\left[(-1{)}^F{p}^{J_1}{q}^{J_2}{y}_1^{Q_1}{y}_2^{Q_2}{y}_3^{Q_3}\right].\end{array}$$

$$\begin{split}\mathcal{I}(p,q,y_{1,2,3}):&=\mathrm{Tr}\left[(-1)^{F}p^{J_{1}}q^{J_{2}}y_{1}^{Q_{1}}y_{2}^{Q_{2}}y_{3}^{Q_{3}}\right]\\ &=\frac{\big{(}(p;p)(q;q)\big{)}^{N-1}}{N!}\prod_{k=1}^{3}\Gamma^{N-1}\big{(}y_{k}\big{)}\oint\prod_{j=1}^{N-1}\frac{\mathrm{d}z_{j}}{2\pi iz_{j}}\ \prod_{1\leq i<j\leq N}\frac{\prod_{k=1}^{3}\Gamma\big{(}y_{k}(z_{i}/z_{j})^{\pm 1}\big{)}}{\Gamma\big{(}(z_{i}/z_{j})^{\pm 1}\big{)}},\end{split}$$

$$(a;q):=\prod _{k=0}^{\infty }(1-a{q}^k),$$

$$\Gamma (z):=\prod _{j,k\ge 0}\frac{1-z^{-1}{p}^{j+1}{q}^{k+1}}{1-z{p}^j{q}^k},$$

$$\text{the CKKN limit:}|\sigma |,|\tau |,\mathrm{Im}{\Delta }_k\to 0,\text{with }\frac{\tau }{\sigma }\in {\mathbb{R}}_{>0},\mathrm{Re}{\Delta }_k\text{fixed},\text{and}\mathrm{Im}\tau ,\mathrm{Im}\sigma >0.$$

$$\Gamma (e^{2\pi ix})=e^{-2\pi i\frac{\kappa (x)}{12\tau \sigma }+\mathcal{O}(\frac{\tau +\sigma }{\tau \sigma })},$$

$$\begin{array}{rl}\kappa (x)& :=\{x\}(1-\{x\})(1-2\{x\})\\ & \left(=2x^3-3x|x|+x\text{for x∈[−1,1]}\right),\end{array}$$

$$\begin{array}{r}\mathcal{I}(p,q,y_{1,2,3})\stackrel{\text{in the CKKN limit}}{\to }{\int }_{-1/2}^{1/2}{\mathrm{d}}^{N-1}\mathbf{x}{e}^{-2\pi i\frac{Q_h(\mathbf{x};\mathrm{Re}{\Delta }_k)}{\tau \sigma }},\end{array}$$

$$\begin{array}{r}{Q}_h(\mathbf{x};\mathrm{Re}{\Delta }_k):=\frac{1}{12}\sum _{k=1}^3\left((N-1)\kappa (\mathrm{Re}{\Delta }_k)+\sum _{1\le i<j\le N}\kappa (\mathrm{Re}{\Delta }_k\pm (x_i-x_j))\right),\end{array}$$

$$\begin{array}{r}\log \mathcal{I}(p,q,y_{1,2,3})\approx -\frac{2\pi i}{\tau \sigma }{Q}_h(0;\mathrm{Re}{\Delta }_k)=-2\pi i\frac{N^2-1}{12\tau \sigma }\sum _{k=1}^3\kappa (\mathrm{Re}{\Delta }_k).\end{array}$$

$$\begin{array}{r}\log \mathcal{I}(p,q,y_{1,2,3})\approx -2\pi i\frac{N^2-1}{2\tau \sigma }\mathrm{Re}{\Delta }_1\mathrm{Re}{\Delta }_2\mathrm{Re}{\Delta }_3.\end{array}$$

$$\begin{array}{r}\log \mathcal{I}(p,q,y_{1,2,3})\approx -2\pi i\frac{N^2-1}{2\tau \sigma }{\Delta }_1{\Delta }_2{\Delta }_3,\end{array}$$

$$Q_{1}Q_{2}Q_{3}+\frac{N^{2}-1}{2}J_{1}J_{2}=\big{(}Q_{1}+Q_{2}+Q_{3}+\frac{N^{2}-1}{2}\big{)}\big{(}Q_{1}Q_{2}+Q_{2}Q_{3}+Q_{3}Q_{1}-\frac{N^{2}-1}{2}(J_{1}+J_{2})\big{)},$$

$$d(J_{1,2},Q_{1,2,3})=\oint \mathcal{I}(p,q,y_{1,2,3})p^{-J_1}{q}^{-J_2}(\prod _{k=1}^3{y}_k^{-Q_k})\frac{\mathrm{d}p}{2\pi ip}\frac{\mathrm{d}q}{2\pi iq}\left(\prod _{k=1}^2\frac{\mathrm{d}{y}_k}{2\pi i{y}_k}\right),$$

$$S(J_{1,2},Q_{1,2,3})\approx {\left(-2\pi i\frac{N^2-1}{2\tau \sigma }{\Delta }_1{\Delta }_2{\Delta }_3-2\pi i\sigma J_1-2\pi i\tau J_2-\sum _{k=1}^{3}2\pi i{\Delta }_k{Q}_k\right)}_{\mathrm{ext}},$$

$${\Delta }_k=-\frac{\frac{1}{S-2\pi i{Q}_k}}{{\sum }_{j=1}^3\frac{1}{S-2\pi i{Q}_j}-{\sum }_{l=1}^2\frac{1}{S+2\pi i{J}_l}},\{\sigma ,\tau \}=-\frac{\frac{1}{S+2\pi i{J}_{\{1,2\}}}}{{\sum }_{j=1}^3\frac{1}{S-2\pi i{Q}_j}-{\sum }_{l=1}^2\frac{1}{S+2\pi i{J}_l}},$$

$$S\approx 2\pi \sqrt{Q_1{Q}_2+Q_2{Q}_3+Q_3{Q}_1-\frac{N^2-1}{2}(J_1+J_2)},$$

$$S\approx 2\pi \sqrt{\frac{Q_1{Q}_2{Q}_3+\frac{N^2-1}{2}{J}_1{J}_2}{Q_1+Q_2+Q_3+\frac{N^2-1}{2}}}.$$

$${\Delta }_k=\frac{\frac{1}{S+2\pi i{Q}_k}}{{\sum }_{j=1}^3\frac{1}{S+2\pi i{Q}_j}-{\sum }_{l=1}^2\frac{1}{S-2\pi i{J}_l}},\{\sigma ,\tau \}=\frac{\frac{1}{S-2\pi i{J}_{\{1,2\}}}}{{\sum }_{j=1}^3\frac{1}{S+2\pi i{Q}_j}-{\sum }_{l=1}^2\frac{1}{S-2\pi i{J}_l}},$$

$$\text{the CKKN limit:}|\beta |,\mathrm{Im}{\Delta }_k\to 0,\text{with }b\in {\mathbb{R}}_{>0},\mathrm{Re}{\Delta }_k\text{fixed},\text{and}\mathrm{Re}\beta >0.$$

$$\begin{array}{r}\Gamma (e^{2\pi ix})=e^{2\pi i{Q}_{+}(x;\sigma ,\tau )}{\psi }_b(\text{−β2πix−2b+b−1})\prod _{n=1}^{\infty }\frac{{\psi }_b(-\frac{2\pi in}{\beta }-\frac{2\pi ix}{\beta }-\frac{b+b^{-1}}{2})}{{\psi }_b(-\frac{2\pi in}{\beta }+\frac{2\pi ix}{\beta }+\frac{b+b^{-1}}{2})},\end{array}$$

$$\begin{array}{rl}{Q}_{+}(x;\sigma ,\tau )=& -\frac{x^3}{6\tau \sigma }+\frac{\tau +\sigma +1}{4\tau \sigma }{x}^2-\frac{{\tau }^2+{\sigma }^2+3\tau \sigma +3\tau +3\sigma +1}{12\tau \sigma }x\\ & +\frac{1}{24}(\tau +\sigma +1)(1+{\tau }^{-1}+{\sigma }^{-1}),\end{array}$$

$$\log {\psi }_b(x)\sim 0,(\text{as }|x|\to \infty )$$

$$S^{+}:0<\mathrm{Re}(x{e}^{-i\arg \beta })<\mathrm{Re}(e^{-i\arg \beta }),$$

$$\Gamma (e^{2\pi ix})=e^{-2\pi i\frac{2x^3-3x^2+x}{12\tau \sigma }+\mathcal{O}(\frac{\tau +\sigma }{\tau \sigma })}(\text{for x∈S+}).$$

$$\eta (\tau )=e^{2\pi i\tau /24}(e^{2\pi i\tau };e^{2\pi i\tau }),$$

$$(p;p{)}^{N-1}(q;q{)}^{N-1}=e^{\mathcal{O}(\frac{\tau +\sigma }{\tau \sigma })}.$$

$$\begin{array}{r}-\frac{\sin (2\arg \beta )}{12}\sum _{1\le i<j\le N}\sum _{k=1}^3\kappa (\mathrm{Re}{\Delta }_k\pm x_{ij}),\end{array}$$

$$\begin{array}{r}{V}^Q(x_{ij};\arg \beta ,\mathrm{Re}{\Delta }_k)=-\sin (2\arg \beta )\cdot \sum _{k=1}^3\kappa (\mathrm{Re}{\Delta }_k\pm x_{ij}),\end{array}$$

Full text

1 Introduction

It has been a long-standing challenge in AdS5/CFT4 to reproduce the entropy of the charged, rotating, BPS, asymptotically AdS5 blackholes of GR:2004a ; GR:2004b ; KLR:2006 ; CCLP:2005 ; CGLP:2005b from a microscopic counting of BPS states in the 4d $\mathcal{N}=4$ CFT. Several attempts in this direction were made in the past fifteen years or so (e.g. Kinney:2005ej ; Berkooz:2006 ; Grant:2008 ; Berkooz:2008 ; Chang:2014 ), leading to various new lessons for holography and superconformal field theory (SCFT), but not to the desired microscopic count.

In particular, an index was devised in Kinney:2005ej ; Romelsberger:2005eg for counting the BPS states of general 4d SCFTs. The index counts all of the states—in the radial quantization of the SCFT—that are annihilated by a chosen supercharge. We adopt conventions in which such states satisfy the “BPS condition” $\Delta-J_{1}-J_{2}-\frac{3}{2}r=0$, where $\Delta,J_{1},J_{2},r$ are the quantum numbers of the 4d $\mathcal{N}=1$ superconformal group SU($2,2|1$). The index

[TABLE]

is thus independent of $\hat{\beta}$, but it does depend on the spacetime fugacities $p,q$, as well as the flavor fugacities $u_{k}$ associated with flavor quantum numbers $q_{k}$ commuting with the supercharge. In the case of the $\mathcal{N}=4$ theory, the SU(4) $R$-symmetry of the $\mathcal{N}=4$ superconformal algebra decomposes into SU($3)\times$U(1)r, so there is an SU(3) “flavor” symmetry group commuting with the chosen supercharge; hence there are three $q_{k}$ with $\sum_{k=1}^{3}q_{k}=0$, and three $u_{k}$ satisfying $\prod_{k=1}^{3}u_{k}=1$. It is customary to define $y_{k}:=(pq)^{1/3}u_{k}$ and $Q_{k}:=q_{k}+r/2$. Then, dismissing $\hat{\beta}$, we can rewrite the index of the $\mathcal{N}=4$ theory as

[TABLE]

This index was computed at finite rank for the U($N$) $\mathcal{N}=4$ theory in the original paper Kinney:2005ej . Then, in an initial attempt to make contact with holography, the large-$N$ limit of the index was evaluated for real-valued fugacities and was seen to be $O(N^{0})$; the result perfectly matched the index of the KK supergravity multi-particle states in the dual AdS5 theory, but clearly could not account for the $O(N^{2})$ entropy of the bulk supersymmetric AdS5 blackholes Kinney:2005ej . For some time this negative result was interpreted as an indication that the index does not encode the bulk blackhole microstates.

Very recently it has been discovered by Choi, Kim, Kim, and Nahmgoong (CKKN) Choi:2018b , and independently by Benini and Milan Benini:2018 , that allowing the five fugacities in the index to take complex values one can achieve the desired $O(e^{N^{2}})$ behavior in the large-$N$ limit of the index. Benini and Milan have succeeded in directly obtaining the AdS5 blackhole entropy function in the large-$N$ limit of the index Benini:2018 , while CKKN took a different route and derived the entropy function in a double-scaling—Cardy-like as well as large-$N$—limit Choi:2018b ; Choi:2018 . In the present paper we derive the entropy function in a Cardy-like limit of the index at finite rank; although our analysis is closely related to that of CKKN Choi:2018 , ours is more analogous to the Cardy-formula Cardy:1986ie derivations of blackhole entropy in AdS3/CFT2 (e.g. Strominger:1997nh ; Strominger:1996 ; Breckenridge:1997 ) where the central charge is kept fixed.

The study of the Cardy-like asymptotics of 4d superconformal indices had some history prior to Choi:2018 , but was again mostly limited to real-valued fugacities (e.g. Ardehali:2014 ; DiPietro:2014 ; Ardehali:2015b ; Ardehali:2015c ; Ardehali:thesis ; DiPietro:2017 ; Hwang:2018 ). The idea that blackhole microstate counting requires complex-valued fugacities in the $\mathcal{N}=4$ index was not properly appreciated until the recent work of Hosseini, Hristov, and Zaffaroni (HHZ) Hosseini:2017 . This work provided the impetus for the later investigations of CKKN Choi:2018b ; Choi:2018 and Benini-Milan Benini:2018 . HHZ started from the supergravity side and bridged half-way towards the CFT by presenting a “grand-canonical” function—henceforth the HHZ function—from which a Legendre transform gives the micro-canonical entropy of the AdS5 blackholes; the remaining challenge was to extract the HHZ function in an appropriate asymptotic regime from the index. In particular, it was understood by HHZ Hosseini:2017 (based on recent lessons from AdS4/CFT3 BHZ:2016 ; BHZ:2017 ) that complexified fugacities are needed in the index in order to make contact with the grand-canonical function of the AdS5 blackholes. As alluded to above, CKKN Choi:2018 and independently Benini and Milan Benini:2018 have recently completed the bridge between the CFT and the bulk by deriving the HHZ function through asymptotic analysis of the $\mathcal{N}=4$ theory index, the first group in a double-scaling limit and the second group in a large-$N$ limit.

In the present paper we analyze the Cardy-like asymptotics of the $\mathcal{N}=4$ theory index with complexified fugacities using the rigorous machinery of elliptic hypergeometric integrals vanD:2001 ; Spiridonov:2003 ; Dolan:2008 ; Rains:2009 in various Cardy-like regimes of parameters where the flavor fugacities approach the unit circle and the spacetime fugacities approach $1$. In particular, we fill a gap in the CKKN derivation of the HHZ function in this limit by showing that the eigenvalue configuration they chose in their asymptotic analysis of the matrix-integral expression for the index is indeed the dominant configuration in the regime of parameters pertaining to the blackhole saddle-point they considered (though we demonstrate that it in fact fails to be the dominant configuration in essentially half of the parameter-space, where a more intricate analysis is called for). Moreover, we discover a new blackhole saddle-point in a different regime of parameters, corresponding to fugacities that are complex conjugate to those at the CKKN saddle-point. We present intuitive arguments suggesting that no other blackhole saddle-points with such large entropies exist in the Cardy-like limit. We also illustrate interesting dependence of the qualitative behavior of the Cardy-like asymptotics of the index on the complex phases of the fugacities.

In the rest of this introduction we give a sketchy account of the asymptotic analysis extracting the blackhole entropy function from the appropriate Cardy-like limit of the superconformal index of the $\mathcal{N}=4$ theory. The main body of the paper starts in Section 2 where we elaborate on the sketchy derivation of the present section; we study the Cardy-like asymptotics of the $\mathcal{N}=4$ theory index with all its fugacities complexified, clarifying—and addressing a gap in—the CKKN derivation of the HHZ function. A thorough enough understanding of the asymptotics of the index in different Cardy-like regimes of parameters results in that section which reveals a second blackhole saddle-point in a regime complementary to that of CKKN, and moreover allows us to argue intuitively that no further saddle-points with such large entropies exist. In Section 3 we keep the spacetime fugacities real-valued, and demonstrate novel bifurcation phenomena in the Cardy-like asymptotics of the index as a function of the flavor-fugacity phases. Section 4 discusses the relation between the Hamiltonian superconformal index and the Lagrangian index computed through path-integration; the two differ by a Casimir-energy factor which is argued to be irrelevant to the blackhole entropy function in the present context. Finally, Section 5 discusses the important open ends of the present work.

1.1 Outline of the CKKN

derivation in the elliptic hypergeometric language

We now present an outline of the CKKN derivation Choi:2018 of the HHZ function Hosseini:2017 , translated to the language of elliptic hypergeometric integrals. More precisely, the problem we consider differs from that of Choi:2018 in two respects:

• •

while Choi:2018 considered the $\mathcal{N}=4$ theory with U($N$) gauge group, we consider the SU($N$) theory—the details are rather similar and the end results are related via $N^{2}\to N^{2}-1$ shifts;

• •

while in Choi:2018 a double-scaling—Cardy-like as well as large-$N$—limit is taken to simplify the analysis, here in analogy with the Cardy-formula derivations of blackhole entropy in AdS3/CFT2 we keep $N$ finite and only take a Cardy-like limit.

The special function as the starting point

The superconformal index of the SU($N$) $\mathcal{N}=4$ theory is given by the following elliptic hypergeometric integral (see e.g. Spiridonov:2010sv ):

[TABLE]

with the unit-circle contour for the $z_{j}=e^{2\pi ix_{j}}$ while $\prod_{j=1}^{N}z_{j}=1$, and with $p,q,y_{k}$ strictly inside the unit circle such that $\prod_{k=1}^{3}y_{k}=pq$. The two special functions $(\cdot;\cdot)$ and $\Gamma(\cdot)$ are respectively the Pochhammer symbol and the elliptic gamma function Ruijsenaars:1997 :

[TABLE]

[TABLE]

and $\Gamma(z^{\pm 1})$ stands for $\Gamma(z)\Gamma(z^{-1})$.

The integral expression gives the index as a meromorphic function of $p,q,y_{k}$ in the domain $0<|p|,|q|,|y_{k}|<1$. A contour deformation can presumably allow meromorphic continuation of the index to $0<|p|,|q|<1$, $y_{k}\in\mathbb{C}^{\ast}$ (c.f. Rains:2005 ).

Asymptotic analysis in the limit encoding blackholes

The Cardy-type limit analyzed prior to the work of CKKN Choi:2018 was of the form $p,q,y_{k}\to 1$; more precisely, it was what in the mathematics literature is referred to as the hyperbolic limit of the elliptic hypergeometric integral Rains:2009 ; Ardehali:2018a . CKKN considered instead limits of the type $p,q\to 1$, $y_{i}\to e^{i\theta_{i}}$, with $\theta_{i}\notin 2\pi\mathbb{Z}$: they correctly recognized that giving finite (non-vanishing) phases to the flavor fugacities can obstruct the bose-fermi cancelations111A similar obstruction mechanism is at work in the AdS3/CFT2 context, where the entropy of the AdS3 blackholes is derived from a Cardy-like limit of the CFT2 elliptic genus $\chi(q,y)$: the limit $q,y\to 1$ does not encode the bulk blackholes, but the limit $q\to 1$, $y\to e^{i\theta}$ with $\theta\notin 2\pi\mathbb{Z}$ does. However, note that while in the AdS3/CFT2 context $q$ can be kept real, in AdS5/CFT4 the spacetime fugacities $p,q$ should take off the real line to meet the blackhole saddle-points. See Benini:2018 ; BHZ:2016 ; BHZ:2017 for related discussions of “$\mathcal{I}$-extremization” in the large-$N$ analysis. occurring in the hyperbolic limit. For future reference we define $\sigma,\tau,\Delta_{k}$ through $p=e^{2\pi i\sigma},q=e^{2\pi i\tau},y_{k}=e^{2\pi i\Delta_{k}}$, and write the appropriate limit explicitly as

[TABLE]

Note that the “balancing condition” $\prod_{k=1}^{3}y_{k}=pq$ implies $\sum_{k=1}^{3}\Delta_{k}-\sigma-\tau\in\mathbb{Z}$, and that the restriction $\mathrm{Im}\tau,\mathrm{Im}\sigma>0$ keeps us in the domain of meromorphy of the index.

The asymptotic analysis of the integral (3) now proceeds as follows. As will be explained in Section 2, the leading asymptotics comes from the elliptic gamma functions $\Gamma(\cdot)$, so the Pochhammer symbols $(\cdot;\cdot)$ and the $N!$ in the pre-factor can be neglected. The required estimate, reviewed in Section 2, follows from Proposition 2.11 of Rains Rains:2009 :

[TABLE]

for $|\tau|,|\sigma|\to 0$, with $\mathrm{Im}\tau,\mathrm{Im}\sigma>0$, and $\frac{\tau}{\sigma}\in\mathbb{R}_{>0}$, $x\in\mathbb{R}/\mathbb{Z}$. Here $\kappa(\cdot)$ is the continuous, odd, piecewise cubic222Hence the “k”appa symbol introduced for it in Ardehali:2015c ., periodic function

[TABLE]

with $\{x\}=x-\lfloor x\rfloor$.

In order to apply the estimate (6) to the gamma functions in (3) we have to identify the phase of the arguments with $2\pi x$; then, for instance, we can apply (6) to the gamma function in the numerator of the integrand of (3) by identifying $x$ with $\mathrm{Re}\Delta_{k}\pm(x_{i}-x_{j})$. This way we can simplify (3) to

[TABLE]

with $Q_{h}$ given by333Compare with the $Q_{h}$ function defined in Ardehali:2015c ; gauge anomaly cancelation implies that both are piecewise “Q”uadratic as a function of $\boldsymbol{x}$. The subscript $h$ is used because the CKKN limit is a variant of the “h”yperbolic limit of the elliptic hypergeometric integral.

[TABLE]

where $\kappa(A\pm B)$ stands for $\kappa(A+B)+\kappa(A-B)$. It only remains to evaluate the asymptotics of the integral (8).

Note that we are assuming $\mathrm{Im}(\tau\sigma)\neq 0$; this corresponds to complexifying the “temperature” as explained below. When $\mathrm{Im}(\tau\sigma)=0$ the integrand of (8)—or already the RHS of (6)—would be a pure phase, and not sufficient to describe the exponential growth of the blackhole microstates. The $\mathrm{Im}(\tau\sigma)=0$ case is therefore not directly relevant to the AdS5 blackhole physics, but it exhibits some interesting asymptotic bifurcation phenomena that are discussed in Section 3.

The last step of the asymptotic analysis of the index involves arguing that in the appropriate range of parameters the dominant small-$|\tau|,|\sigma|$ configurations in (8) have $x_{i}-x_{j}=0$ (implying $x_{j}=\mathrm{const}$, which in our SU($N$) case would mean that all the holonomies are equal to $\frac{n}{N}$ for some $n\in\{0,1,\dots,N-1\}$). CKKN simply assumed Choi:2018 this to be the case. In Section 2 we will prove that for the range of parameters relevant to the AdS5 blackholes (e.g. for $\mathrm{Im}(\tau\sigma)>0$ & $\mathrm{Re}\Delta_{1,2},-1-\mathrm{Re}\Delta_{1}-\mathrm{Re}\Delta_{2}\in]-1,0[$) their assumption is correct. Hence the asymptotics of the index becomes444Compare with Eq. (2.34) of CKKN Choi:2018 ; note that $2\pi i\Delta^{\mathrm{here}}_{k}=-\Delta_{k}^{\mathrm{CKKN}}$, while $2\pi i\{\sigma,\tau\}=-\omega_{\{1,2\}}^{\mathrm{CKKN}}$.

[TABLE]

The right-hand-side is a nonanalytic function of the $\mathrm{Re}\Delta_{k}$, manifestly invariant under $\mathrm{Re}\Delta_{k}\to\mathrm{Re}\Delta_{k}+1$ as it should be.

To match the grand-canonical function of HHZ Hosseini:2017 we now pick a particular chamber in the parameter-space so that an analytic expression can be written down. Specifically, assuming $\mathrm{Im}(\tau\sigma)>0$, going into the chamber $-1<\mathrm{Re}\Delta_{1,2,3}<0$ with $\mathrm{Re}\Delta_{3}=-1-\mathrm{Re}\Delta_{1}-\mathrm{Re}\Delta_{2}$, we can simplify $\sum_{k=1}^{3}\kappa(\mathrm{Re}\Delta_{k})$ to $6\mathrm{Re}\Delta_{1}\mathrm{Re}\Delta_{2}\mathrm{Re}\Delta_{3}$, and arrive at

[TABLE]

Analytic continuation of (11) to complex $\Delta_{k}$ (i.e. replacing every $\mathrm{Re}\Delta_{k}$ with $\Delta_{k}$) allows recovering the subleading terms in the CKKN limit and connecting with the complex HHZ function Hosseini:2017 (see the end of Section 2 for more details):

[TABLE]

with $\Delta_{1}+\Delta_{2}+\Delta_{3}-\tau-\sigma=-1$.

So far in this subsection we have been essentially rephrasing the developments due to CKKN Choi:2018 . One of the novel contributions of the present paper is to demonstrate in Section 2 that when $\mathrm{Im}(\tau\sigma)<0$ another chamber with $0<\mathrm{Re}\Delta_{1,2,3}<1$ and $\mathrm{Re}\Delta_{3}=1-\mathrm{Re}\Delta_{1}-\mathrm{Re}\Delta_{2}$ yields the asymptotics (12) in the CKKN limit, this time with $\Delta_{1}+\Delta_{2}+\Delta_{3}-\tau-\sigma=+1$.

Legendre transform and blackhole entropy

Thinking of the index (2) as the generating function of the degeneracies $d(J_{1,2},Q_{1,2,3})$ of the BPS states555Although at first glance it appears that because of the $(-1)^{F}$ factor in it the index (2) counts the number of bosonic states minus the number of fermionic states, as argued in Benini:2018 , on the blackhole saddle-points essentially all the states are expected to be bosonic, so the index counts a degeneracy. in the $\mathcal{N}=4$ theory, methods of elementary analytic combinatorics can be used to extract the large-$J_{1,2},Q_{1,2,3}$ asymptotics of $d(J_{1,2},Q_{1,2,3})$ from the Cardy-like asymptotics of the index. The CKKN limit of the index encodes the degeneracy of the BPS states as $Q_{1,2,3}\sim\Lambda,\ J_{1,2}\sim\Lambda^{3/2},\ \Lambda\to\infty$, such that the charge relation Choi:2018 ; Cabo-Bizet:2018

[TABLE]

of the bulk AdS5 blackholes is satisfied Choi:2018 .

The degeneracies can be obtained from the generating function through

[TABLE]

with all of the contours slightly inside the unit circle; note that $y_{3}$ is not independent, so is not integrated over on the RHS (c.f. Section 5 of Benini:2018 ). The asymptotic degeneracy can be obtained using a saddle-point evaluation of the integral on the right-hand side. Using the Cardy-like asymptotics in (12), the result for the asymptotic entropy $S(J_{1,2},Q_{1,2,3})=\log d(J_{1,2},Q_{1,2,3})$ becomes

[TABLE]

with $\sum\Delta_{k}-\tau-\sigma=-1$, as well as $-1<\mathrm{Re}\Delta_{1,2,3}<0$, $\mathrm{Im}\tau\sigma>0$. The subscript “ext” on the RHS means picking its extremized value on the saddle-point.

The extremization problem was addressed for the $\mathrm{Im}\tau\sigma>0$ case by HHZ Hosseini:2017 , but was made completely explicit and analytic by CKKN Choi:2018 (and independently in Appendix B of Cabo-Bizet:2018 by Cabo-Bizet, Cassani, Martelli, and Murthy), who found the blackhole saddle-point at

[TABLE]

satisfying $\Delta_{1}+\Delta_{2}+\Delta_{3}-\tau-\sigma=-1$, and giving the entropy

[TABLE]

which thanks to the charge relation (13) can be written in the alternative form

[TABLE]

Both of the relations (17), (18) correctly reproduce the Bekenstein-Hawking entropy of the BPS AdS5 blackholes of GR:2004a ; GR:2004b ; KLR:2006 ; CCLP:2005 ; CGLP:2005b , upon using the AdS/CFT dictionary $N^{2}-1=\frac{\pi\ell_{\mathrm{AdS}_{5}}}{2G_{\mathrm{AdS}_{5}}}$, with $\ell_{\mathrm{AdS}_{5}},G_{\mathrm{AdS}_{5}}$ respectively the radius and the Newton constant of the bulk $\mathrm{AdS}_{5}$.

Note that the term $\frac{N^{2}-1}{2}(J_{1}+J_{2})$ in (17) and the term $\frac{N^{2}-1}{2}$ in (18) are subleading in the CKKN scaling limit $J^{2}\sim Q^{3}\to\infty$; capturing them was the sole purpose of keeping the subleading terms in (12) (compared to (11)), as well as in (13), (15), and (16).

In Section 2 we show that (12) is valid in the CKKN limit also when $0<\mathrm{Re}\Delta_{1,2,3}<1$, $\mathrm{Im}\tau\sigma<0$, though this time with $\Delta_{1}+\Delta_{2}+\Delta_{3}-\tau-\sigma=+1$, and find a new blackhole saddle-point at

[TABLE]

with the same entropy $S$ as that of the CKKN saddle-point666The observation that the HHZ function with $\Delta_{1}+\Delta_{2}+\Delta_{3}-\tau-\sigma=+1$ also happens to give the right entropy has been made previously in Appendix B of Cabo-Bizet:2018 . Our contribution here is to clarify the physics of this observation by showing that the HHZ function with $\Delta_{1}+\Delta_{2}+\Delta_{3}-\tau-\sigma=+1$ actually arises as the asymptotics of the index in a regime of parameters separate from that considered by CKKN.. We moreover argue that besides the two just described—having complex conjugate fugacities $p,q,y_{1,2,3}$—no other blackhole saddle-points with such large entropies exist in the Cardy-like asymptotics of the $\mathcal{N}=4$ theory index.

Final remarks

A remaining gap for unequal $\mathbf{Q_{k}}$. A rather serious gap in the above derivation is revealed upon closer inspection of the critical $\Delta_{k}$ in (16) and (19): while our asymptotic analysis is valid only in the limit $\mathrm{Im}\Delta_{k}\to 0,$ the blackhole saddle-points have nonzero $\mathrm{Im}\Delta_{k}$ unless $Q_{1}=Q_{2}=Q_{3}$. It is therefore only in the special case with equal—or approximately equal—charges that the above derivation (augmented with the refinements of Section 2) is satisfactory. CKKN assumed in a leap of faith Choi:2018 that the asymptotics (12) remains valid away from the limit $\mathrm{Im}\Delta_{k}\to 0,$ and thus the blackhole entropy derivation can be extended to the general case with unequal charges. In Section 2 we present a partial justification for this extrapolation; the complete justification is beyond the scope of the present paper, and its absence constitutes the most important open end of this work.

Cardy-like versus large-$\mathbf{N}$. The above derivation extracts the AdS5 blackhole entropy from a “high-temperature” (Cardy-like) limit of the 4d superconformal index at finite $N$. This is analogous to how the classic papers of Strominger-Vafa Strominger:1996 , BMPV Breckenridge:1997 , and Strominger Strominger:1997nh derived the Bekenstein-Hawking entropy of certain blackholes in what nowadays might be called an AdS3/CFT2 context.

From the holographic perspective, a more conceptually satisfying derivation would involve the large-$N$ limit of the index. In AdS3/CFT2 such conceptually satisfactory derivations can be found in Sen:2011 ; Hartman:2014 . In the AdS5/CFT4 context this was achieved very recently by Benini and Milan Benini:2018 , leveraging the Bethe Ansatz formula of Closset, Kim, and Willett Closset:2017Bethe . Curiously, although the derivation in Benini:2018 is not limited to the equal-charge blackholes, because of certain technical obstacles it so far applies only to the case with equal angular momenta $J_{1}=J_{2}$ and the general case with $J_{1}\neq J_{2}$ is still open. The more general Bethe Ansatz formula of Benini:2018Bethe seems promising in that direction.

2 Complexified temperature and AdS5 blackholes

($|\beta|\to 0$, $0<|\mathrm{arg}\beta|<\frac{\pi}{2}$)

In this section we fill in the gaps of Subsection 1.1. In particular, we give a rigorous derivation of the estimate (6) for the elliptic gamma function, fill the gap in the CKKN asymptotic analysis in the region $\mathrm{Im}\tau\sigma>0$, extend the analysis to the region $\mathrm{Im}\tau\sigma<0$ where we find a new blackhole saddle-point, and argue that no further blackhole saddle-points with such large entropies exist.

The elliptic gamma function estimate (6)

Let us define the parameters $b,\beta$ through $\tau=\frac{i\beta b^{-1}}{2\pi}$, $\sigma=\frac{i\beta b}{2\pi}$. For $p,q\in\mathbb{R}$, the parameter $\beta$ defined as such was referred to as the inverse-temperature in Ardehali:2015b ; Ardehali:2015c ; here we similarly refer to $\beta$ as the complexified inverse-temperature. Throughout the present work we assume $b\in\mathbb{R}_{>0}$ (i.e. $\tau/\sigma\in\mathbb{R}_{>0}$); this simplifies the analysis and suffices for making contact with blackhole physics in the Cardy-like limit. We also take $\mathrm{Re}\beta>0$ (i.e. $|\mathrm{arg}\beta|<\frac{\pi}{2}$) to stay within the domain of meromorphy of the index (3). In terms of $b,\beta$ we have

[TABLE]

The starting point for deriving the estimate (6) is the following identity, essentially due to Narukawa Narukawa:2004 :

[TABLE]

where

[TABLE]

and $\psi_{b}(x)$ a function [see Appendix A of Ardehali:2015c for its definition in terms of the hyperbolic gamma function] with the important property that for $\mathrm{arg}x$ inside compact subsets of $(-\pi,0)$, and fixed $b>0$

[TABLE]

with an exponentially small error, of the type $e^{-|x|}$—see Corollary 2.3 of Rains Rains:2009 for the precise statement and see Appendix B of Ruijsenaars:1999 for an earlier analysis in a different notation. This property of $\psi_{b}$ guarantees that the infinite product in (20) is convergent when $\mathrm{Re}\beta>0$.

For $x$ strictly inside the strip

[TABLE]

as $|\beta|\to 0$ with $|\mathrm{arg}\beta|<\pi/2$ and $b>0$ fixed, all the $\psi_{b}$ functions on the RHS of (20) approach unity exponentially fast. Moreover, the dominant piece of $Q_{+}$ in the limit is of order $\frac{1}{\tau\sigma}$ and gives

[TABLE]

Since the LHS of the above relation is periodic in $x\to x+1$, we can extend it beyond $x\in S^{+}$ by replacing every $x$ on the RHS with its horizontal shift $\{x\}:=x-\lfloor\mathrm{Re}x+\mathrm{Im}x\cdot\tan(\mathrm{arg}\beta)\rfloor$ to inside $S^{+}$. For $x\in\mathbb{R}$ we have $\{x\}=x-\lfloor x\rfloor$; this yields our desired estimate (6).

Equivalently, we could use Proposition 2.11 of Rains Rains:2009 , after identifying $v_{\mathrm{there}}$ with $|\beta|/2\pi$, and $\omega_{1,2\ \mathrm{there}}$ with $ib^{\pm 1}e^{i\mathrm{arg}\beta}$.

A somewhat subtle point is that the estimate (6) is not uniform with respect to $x$ when applied to the (“vector multiplet”) gamma functions in the denominator of the RHS of (3)—or more generally (24) is not uniform when $x$ approaches the boundaries of the strip $S^{+}$. We need a uniform estimate because we want to apply the estimate in the integrand of the index; c.f. the paragraph of Eq. (3.15) in Ardehali:2015c . We expect though that an argument similar to that in the paragraph below Eq. (3.30) of Ardehali:2015c can be given implying that the non-uniform estimate introduces a negligible error on the leading asymptotics of the index; we postpone the rigorous analysis of this point to the future.

Cardy-like asymptotics of the index (10)

It follows from the relation between the Pochhammer symbol and the Dedekind eta function

[TABLE]

and the modular property $\eta(-1/\tau)=\sqrt{-i\tau}\eta(\tau)$ of the eta function that in the Cardy-like limit the Pochhammer symbols on the RHS of (3) contribute an exponential growth of

[TABLE]

They can hence be neglected, along with the $N!$ in the denominator of (3), in the Cardy-like limit when $0<|\mathrm{arg}\beta|<\pi/2$. We thus end up with (8) as promised.777More precisely, we also have to show that the complex phase of the integrand, arising from the subleading terms that we have ignored, does not cause a completely destructive interference modifying the leading asymptotics; c.f. the paragraph below that of Eq. (3.15) in Ardehali:2015c . For non-chiral SCFTs without flavor fugacities, the absence of such cancelations was established in Ardehali:thesis . For the case at hand, we expect such “unnatural” completely destructive interferences to be absent at least for generic $\mathrm{Re}\Delta_{1,2}$; we postpone the rigorous analysis of this point to the future.

We remind the reader that if $\beta\in\mathbb{R}_{>0}$ the integrand of (8) becomes a pure phase, and the more precise asymptotic analysis of Section 3 has to be performed.

For $0<|\mathrm{arg}\beta|<\frac{\pi}{2}$, in the small-$|\beta|$ limit the integral $(\ref{eq:EHIsimplified})$ is localized around the minima of $-\sin(2\mathrm{arg}\beta)\cdot Q_{h}(\boldsymbol{x};\mathrm{Re}\Delta_{k})$, whose $\boldsymbol{x}$-dependent part can be read from (9) to be

[TABLE]

with $x_{ij}=x_{i}-x_{j}$. The expression

[TABLE]

is thus roughly a pair-wise potential for the “holonomies” $x_{i}$.

We take $\mathrm{arg}\beta$ and $\mathrm{Re}\Delta_{1,2}$ to be our control-parameters; $\mathrm{Re}\Delta_{3}$ is determined (mod $\mathbb{Z}$ to be precise, which is enough) by the balancing condition. We take the fundamental region of $\mathrm{Re}\Delta_{1,2}$ to be $[-1/2,1/2]$. The two qualitatively different behaviors that the function $V^{Q}$ can exhibit in various regions of the space of the control-parameters $\mathrm{Re}\Delta_{1,2}$ are shown in Figure 1 for $-\pi/2<\mathrm{arg}\beta<0$. This figure can be deduced either by numerically scanning (using Mathematica for instance) the fundamental region $\mathrm{Re}\Delta_{1,2}\in[-1/2,1/2]$, for some fixed $\mathrm{arg}\beta\in(-\pi/2,0)$, or by analytically investigating the function $\sum_{k=1}^{3}\kappa(\mathrm{Re}\Delta_{k}\pm x_{ij})$ in its various regions of analyticity. Note that an $M$-type potential means $x_{ij}=0$ is preferred in the small-$|\beta|$ limit, while a $W$-type potential means some $x_{ij}\neq 0$ (always a neighborhood of $x_{ij}=\pm 1/2$ it turns out) is preferred. Since Figure 1 is a bit too featureful, we use the equivalence $\mathrm{Re}\Delta_{1,2}\to\mathrm{Re}\Delta_{1,2}\pm 1$ to shift its triangular regions so that the equivalent Figure 2 is obtained, which is one of the main results of the present paper. It should be clear from the $\sin(2\mathrm{arg}\beta)$ factor in (27) that the $M$ and $W$ wings in Figure 2 switch places if $\mathrm{arg}\beta$ is taken to be inside $(0,\pi/2)$ instead.

To be specific, let us continue with the $\mathrm{arg}\beta\in(-\pi/2,0)$ case for the moment. Then on the $M$ wing of Figure 2 the minimum value of $V^{Q}$ occurs at $x=0$. Moreover, since $V^{Q}$ is stationary at $x=0$, the phase of the integral (8) is stationary there. We conclude that the $x_{ij}=0$ configurations—which in our SU($N$) case correspond to $x_{j}=\frac{n}{N}$ for some $n\in\{0,1,\dots,N-1\}$ independent of $j$—indeed dominate the leading small-$\beta$ asymptotics of the index for $-\frac{\pi}{2}<\mathrm{arg}\beta<0$ and $-1<\mathrm{Re}\Delta_{1},\mathrm{Re}\Delta_{2},-1-\mathrm{Re}\Delta_{1}-\mathrm{Re}\Delta_{2}<0$; this proves that CKKN’s conjecture in Choi:2018 is valid in the range of parameters just mentioned, and fills the gap in their derivation of the HHZ function in this region of the parameter-space. (Note that in our SU($N$) case each choice of $n$ completely breaks the $\mathbb{Z}_{N}$ center symmetry generated by $x_{j}\to x_{j}+\frac{1}{N}$, thus meeting deconfinement expectations.)

On the bifurcation set, indicated by the dashed lines in Figure 2, the functions $V^{Q}$ and $Q_{h}$ vanish; a more precise analysis using the techniques of Section 3 is then required, but in any case it is clear that the asymptotic growth of the index is much slower (with $\mathrm{Re}\log\mathcal{I}=\mathcal{O}(\frac{1}{|\beta|})$) there, so we do not discuss this set any further.

The question we would like to address now is: do we have a faster or a slower asymptotic growth on the $W$ wing? Here by scanning (using Mathematica for instance) the whole range $-1/2<\mathrm{Re}\Delta_{1,2}<1/2$ we realize that the minima $\sum\kappa(+1/3\pm 1/2)=-5/9$ at $\mathrm{Re}\Delta_{1,2}=+1/3$ and $x_{ij}=\pm 1/2$ are lower than the minimum $\sum\kappa(-1/3\pm 0)=-4/9$ at $\mathrm{Re}\Delta_{1,2}=-1/3$ and $x_{ij}=0$. Does this mean that the index exhibits a faster asymptotic growth at $\mathrm{Re}\Delta_{1,2}=+1/3$? The answer turns out to be no for $N=2$, and seems to be no for all $N>2$ as well.

For $N=2$ the reason is that the $\boldsymbol{x}$-independent piece of $Q_{h}$ in (9) moves $\sum\kappa(-1/3\pm 0)=-4/9$ further down by $-2/9$, while it moves $\sum\kappa(+1/3\pm 1/2)=-5/9$ further up by $+2/9$. Thus in the CKKN limit with $-\pi/2<\mathrm{arg}\beta<0$ we have

[TABLE]

while

[TABLE]

In short, for $N=2$ the fastest asymptotic growth in the CKKN limit with $-\pi/2<\mathrm{arg}\beta<0$ occurs on the $M$ wing of Figure 2.

For higher ranks there is a more important reason why points on the $W$ wing can not compete with the fastest asymptotic growth on the $M$ wing. That is because for $N>2$ it is impossible to distribute $N$ holonomies $x_{i}$ on the fundamental region $[-1/2,1/2]$ (with $-1/2$ and $1/2$ identified of course, and with $x_{N}$ determined from the rest via $\sum_{i=1}^{N}x_{i}\in\mathbb{Z}$) and have all of them at equal distance $|x_{ij}|=1/2$ from each other. Colloquially speaking, it is not possible to capitalize on the minima of $V^{Q}$ on the $W$ wing at $|x_{ij}|=1/2$ with all the holonomies, whereas it is possible to do so on the minima at $|x_{ij}|=0$ on the $M$ wing; hence as we increase the rank it becomes more and more intuitively likely that the fastest asymptotic growth of the index should occur on the $M$ wing, and so we expect that only this region potentially bears entropy functions of the AdS5 blackholes.

Let us recapitulate our findings so far. We have demonstrated that the $|x_{ij}|=0$ points are preferred in the CKKN limit on the $M$ wing of the space of the control-parameters $\mathrm{Re}\Delta_{1,2}$, and thus the asymptotic result (10) is valid there. We have also argued intuitively that the $W$ wing yields slower asymptotic growth and is not expected to bear entropy functions as large as those of the AdS5 blackholes.

It is straightforward to deduce the analogous statements for $0<\mathrm{arg}\beta<\frac{\pi}{2}$. In that case the $M$ and $W$ wings of Figure 2 are swapped. Hence this time it is on the upper-right wing that the $|x_{ij}|=0$ configurations are preferred in the CKKN limit, and the asymptotic result (10) is valid, though this time with $\mathrm{Re}\Delta_{3}=1-\mathrm{Re}\Delta_{1}-\mathrm{Re}\Delta_{2}$. We also know that for $N=2$ the fastest asymptotic growth of the index occurs on the upper-right wing, and we expect the same to be true as $N$ increases.

The blackhole saddle-points (16),(19)

Now we ask: in the case $-\pi/2<\mathrm{arg}\beta<0$ does the lower-left wing, and in the case $0<\mathrm{arg}\beta<\frac{\pi}{2}$ does the upper-right wing contain blackhole saddle-points? To make contact with the AdS5 blackholes we have to find the critical points of the Legendre transform of $\log\mathcal{I}$ in the CKKN limit. In both cases it turns out that one blackhole saddle-point exists. The latter saddle-point seems to have been overlooked in Choi:2018 , but can be obtained with minor modification of the computations in their Section 2.3 as we now outline. Recall that when $0<\mathrm{arg}\beta<\frac{\pi}{2}$ we impose $\sum_{k}\Delta_{k}=\tau+\sigma+1$ rather than $\sum_{k}\Delta_{k}=\tau+\sigma-1$; while CKKN Choi:2018 (following HHZ Hosseini:2017 ) impose the latter relation via

[TABLE]

the former relation can be simply imposed by putting

[TABLE]

We now would like to argue that the $z^{\ast}_{1,2,3,4}$ which solve the extremization problem for $0<\mathrm{arg}\beta<\frac{\pi}{2}$ are indeed the complex conjugates of the $z_{1,2,3,4}$ that CKKN found solving the extremization problem for $-\pi/2<\mathrm{arg}\beta<0$. To demonstrate this, we present some of the details of the extremization problem, in parallel with Section 2.3 of CKKN Choi:2018 . Setting the derivatives of (15) with respect to $z^{\ast}_{1,2,3,4}$ to zero, we get

[TABLE]

similar to the CKKN case—c.f. their Eq. (2.75). However, extremization with respect to $z^{\ast}_{4}$ yields

[TABLE]

with a different sign on the RHS compared to the CKKN case—c.f. their Eq. (2.79). As a result, the equations for $z^{\ast}_{1,2,3,4}$ following from the above relations read

[TABLE]

with only an $S\to-S$ difference compared to the CKKN case—c.f. their Eq. (2.88). In particular, since $S$ is real, our $z^{\ast}$s are complex conjugates of their $z$s, as claimed. The relations (16) and (19) follow rather effortlessly from (30) and (31) respectively.

To obtain $S$, we can follow CKKN and write things in terms of $f^{\ast}:=\frac{N^{2}-1}{2}\frac{z^{\ast}_{1}z^{\ast}_{2}z^{\ast}_{3}}{z^{\ast}_{4}}$, so that Eq. (32) becomes

[TABLE]

and then use the definition of $f^{\ast}$ to obtain

[TABLE]

which is a cubic relation for $f^{\ast}$. The cubic equation that follows for $S=2\pi i(f^{\ast}+J_{2})$ will then have the entropy functions (17) and (18) as its solutions.

To demonstrate the self-consistency of our computations we need to show that the CKKN/HHZ saddle-point (16) is indeed on the lower-left wing of Figure 2 and has $-\pi/2<\mathrm{arg}\beta<0$, while the new saddle-point lies on the upper-right wing and has $0<\mathrm{arg}\beta<\frac{\pi}{2}$. We show only the second statement, as the first follows using the fact that the two saddle-points have their $\Delta_{k},\tau,\sigma$ negative complex conjugate of each other.

A quick way to the desired result is to note that $S+2\pi iQ_{k}$ are on a straight line in the complex plane, so that their reciprocals are on a circle. This observation motivates the change of variables $\frac{1}{S+2\pi iQ_{k}}=\frac{1}{2S}(1+e^{-i\phi_{k}})$, with $\phi_{k}\in(0,\pi)$. Then the desired ranges of $\mathrm{Re}\Delta_{k}$ and $\mathrm{arg}\beta$ follow easily from (19) and the vector representation of the complex numbers $1+e^{-i\phi_{k}}$, after neglecting the subleading terms in the CKKN scaling limit $J^{2}\sim Q^{3}\to\infty$.

In summary, we have shown that when $0<\mathrm{arg}\beta<\frac{\pi}{2}$ a blackhole saddle-point exists on the upper-right wing of Figure 2; as comparison of Eqs. (30) and (31) shows, the new saddle-point has fugacities $p^{\ast},q^{\ast},y_{k}^{\ast}$ that are complex conjugates of the fugacities at the CKKN/HHZ saddle-point. Moreover, we have argued that besides this and the CKKN/HHZ saddle-point no other saddle-points with such large entropies exist in the Cardy-like limit.

Moving the flavor fugacities away from the unit circle

As we noted at the end of Subsection 1.1, unless $Q_{1}=Q_{2}=Q_{3}$, the critical $\Delta_{k}$ have nonzero imaginary parts, and thus the critical fugacities $u_{k}$ (and also $y_{k}$) lie away from the unit circle. Hence to complete the blackhole entropy derivation for the general case with unequal $Q_{k}$, we need to be able to justify the Cardy-like asymptotics (12) when $\mathrm{Im}\Delta_{k}$ are not sent to zero.

A partial justification is as follows. Let us assume that $\mathrm{Im}\Delta_{k}$ are small enough so that the integral (3) still represents the index, albeit with a slightly deformed contour of integration888It appears like we might only need the contour-deformation to be small near $x_{j}=\frac{n}{N}$, which are the dominant eigenvalue configurations in the regime of parameters pertaining to the blackhole saddle-points.. We can then use (24) to arrive at the following variant of (8):

[TABLE]

with

[TABLE]

where $\kappa(x)$ is still defined as in (7), but with $\{x\}:=x-\lfloor\mathrm{Re}x+\mathrm{Im}x\cdot\tan(\mathrm{arg}\beta)\rfloor$ as discussed around (24).

We expect that for fixed $\mathrm{arg}\beta$ (either in $(-\pi/2,0)$ or in $(0,\pi/2)$), and for small enough $\mathrm{Im}\Delta_{k}$, the catastrophic behavior of the pair-wise potential for the holonomies to remain similar to that discussed above, with the two complementary “wings” $\Delta_{1,2},1-\Delta_{1}-\Delta_{2}\in S^{+}$ and $\Delta_{1,2},-1-\Delta_{1}-\Delta_{2}\in S^{+}-1$ being associated to $M$- or $W$-type behaviors, with one or the other having $x=0$ as its preferred configuration depending on the sign of $\mathrm{arg}\beta$. Then for $\mathrm{arg}\beta\in(0,\pi/2)$ one can use (24) on the wing $\Delta_{1,2},1-\Delta_{1}-\Delta_{2}\in S^{+}$ to arrive at (12) with $\sum_{k}\Delta_{k}=\tau+\sigma+1$, while for $\mathrm{arg}\beta\in(-\pi/2,0)$ one can use (24) with $x\to x+1$ on the wing $\Delta_{1,2},-1-\Delta_{1}-\Delta_{2}\in S^{+}-1$ to arrive at (12) this time with $\sum_{k}\Delta_{k}=\tau+\sigma-1$.

Beyond a small neighborhood of $\mathrm{Im}\Delta_{k}=0$ the methods of the present paper do not seem powerful enough to demonstrate (12). Whether the fascinating formalism of Closset:2017Bethe ; Benini:2018Bethe can help addressing the general case with nonzero $\mathrm{Im}\Delta_{k}$ is currently being investigated.

3 Real-valued temperature ($\beta\to 0$, $\beta\in\mathbb{R}_{>0}$)

In this section we keep the spacetime fugacities $p,q$ real-valued and define $b,\beta\in\mathbb{R}_{>0}$ through $p=e^{-\beta b},q=e^{-\beta b^{-1}}$. We also keep the flavor fugacities $u_{k}=e^{2\pi iT_{k}}$ on the unit circle (hence $T_{k}\in\mathbb{R}$), and study the effect of finite nonzero $T_{k}$ on the small-$\beta$ asymptotics of the index.

In order to provide some conceptual context for the somewhat technical analysis in the rest of this section we now briefly discuss the path-integral interpretation of the index with real-valued $p,q$. We will still be analyzing the Hamiltonian index $\mathcal{I}$, and only importing intuition from the path-integral picture—until the next section where the path-integral partition function is analyzed.

The superconformal index with real $p,q$ can be obtained via the path-integral SUSY partition function of the theory on $S_{b}^{3}\times S^{1}_{\beta}$, where $S_{b}^{3}$ is the squashed three-sphere with unit radius and squashing parameter $b$, while $S^{1}_{\beta}$ is the circle with circumference $\beta$ Assel:2014 . The integration variables $z_{i}$ in the index (3) correspond to the eigenvalues of the holonomy matrix $P\exp(i\oint_{S^{1}_{\beta}}A_{0})$, with $A_{0}$ the component along $S^{1}_{\beta}$ of the SU($N$) gauge field. The $u_{k}$ correspond to the eigenvalues of the background holonomy matrix $P\exp(i\oint_{S^{1}_{\beta}}A^{u}_{0})$, with $A^{u}_{0}$ the component along $S^{1}_{\beta}$ of the background gauge field $A^{u}$ associated to the “flavor” SU($3$) of the $\mathcal{N}=4$ theory. The path-integral partition function is actually a Casimir-energy factor different from the index; this factor is irrelevant for the present analysis and we postpone its discussion to the next section. Interpreting the $S_{b}^{3}$ as the spatial manifold and the $S^{1}_{\beta}$ as the Euclidean time circle, we refer to $\beta$ as the inverse-temperature in analogy with thermal quantum physics—even though our fermions have supersymmetric (i.e. periodic) boundary conditions around $S^{1}_{\beta}$.

Next, we note that while large-$N$ QFTs ($N\to\infty$) on compact spatial manifolds can have finite-temperature phases associated to large-$N$ saddle-points, in the present work we are considering a finite-$N$ QFT on a compact spatial manifold (namely $S_{b}^{3}$), which can not be assigned a phase at any finite temperature. In the high-temperature limit ($\beta\to 0$), however, infinite-temperature phases can be associated to the small-$\beta$ saddle-points. In particular, we will say that the infinite-temperature phase of the index is Higgsed if the dominant small-$\beta$ saddle-point(s) of its matrix-integral lie away from the “origin” $\boldsymbol{x}=0$. For example, the infinite-temperature phase of the index of the SU($2$) ISS model is Higgsed, but that of the $\mathcal{N}=1$ SU($N$) SQCD (say in the conformal window) is not Ardehali:2015c .

Moreover, we will say that the infinite-temperature phase of the index is exponentially growing if for the leading small-$\beta$ asymptotics we have $\mathrm{Re}\log\mathcal{I}\approx A/\beta$ with $A>0$; in other words if the index exhibits exponential growth in the high-temperature limit.

Below we will see that for generic non-zero $T_{k}\in\mathbb{R}$ the infinite-temperature phase of the index of the SU($N$) $\mathcal{N}=4$ theory is Higgsed, and in the $N=2$ case for some specific range of $T_{k}$ also exponentially growing.

Taking $p,q$ to be real means taking $\tau,\sigma$ to be pure imaginary. Then we have $\mathrm{Im}(\tau\sigma)=0$, so that the estimate (6) gives only a pure phase; we thus have to consider the subleading terms in the exponent of its RHS to get information about the modulus of the index. The improved estimate is Ardehali:2015c

[TABLE]

where the continuous, positive, even, periodic function

[TABLE]

is defined after Rains Rains:2009 .

In order to apply the estimate (39) to the gamma functions in (3) we have to interpret the modulus of the arguments of the gamma functions as $(pq)^{r/2}$, and interpret the phase of the arguments as $2\pi x$; then, for instance, we can apply (39) to the gamma function in the numerator of the integrand of (3) by identifying $r$, $x$ as $r=2/3$, $x=T_{k}\pm(x_{i}-x_{j})$; note that the balancing condition $\prod_{k=1}^{3}y_{k}=pq$ implies $\sum_{k=1}^{3}T_{k}\in\mathbb{Z}$. Since the Pochhammer symbols in (3) yield asymptotics that cancel the contribution of the gamma functions from the third term on the RHS of (39) DiPietro:2014 ; Ardehali:2015c , applying (39) to (3) we get

[TABLE]

where we have used $\tau=i\beta b^{-1}/2\pi$ and $\sigma=i\beta b/2\pi$. The functions $L_{h}$ and $Q_{h}$ are the natural generalizations of those defined in Ardehali:2015c for $T_{k}=0$, and are explicitly given by999Because of the ABJ U(1)R anomaly cancelation $L_{h}$ is a piecewise “L”inear function of $\boldsymbol{x}$ (c.f. Ardehali:2015c ). See Figure 3.

[TABLE]

[TABLE]

Note that for $T_{k}=0$ both functions identically vanish, as in Ardehali:2015c .

Since the $1/\beta^{2}$ term in the exponent of the RHS of (41) gives a pure phase, the dominant contribution to the integral presumably comes from the locus of minima of $L_{h}(\boldsymbol{x},r_{k}=2/3;T_{k})$. One has to make sure that $Q_{h}(\boldsymbol{x};T_{k})$ is stationary at that locus though, otherwise a more careful analysis is required.

The SU$(2)$ case

Take for example the $N=2$ case. Figure 3 shows the $L_{h}$ function of the SU($2$) $\mathcal{N}=4$ theory for sample values of $T_{k}$. As the picture clearly shows, at the point $x_{1}=0$ the integrand is maximally suppressed.

It is easy to check that the dominant configuration for $N=2$ is $|x_{1}|=1/4$ (Figure 3 is suggestive of this also); not only $L_{h}$ is minimized there, but also $Q_{h}$ is stationary as desired. Moreover, we see from Figure 3 that depending on $T_{k}$ the minimum of $L_{h}$ can be positive, negative, or zero. Only when the minimum is negative the infinite-temperature phase is exponentially growing. The contours of $L_{h}(x_{1}=\pm 1/4,r_{k}=2/3;T_{k})$ are shown in Figure 4: outside the blue contour we have $L_{h}(x_{1}=\pm 1/4,r_{k}=2/3;T_{k})<0$, so the index is exponentially growing, except on the blue dots at $T_{1}=T_{2}=\pm 1/3$ where $L_{h}(x_{1}=\pm 1/4,r_{k}=2/3;T_{k})$ vanishes.

Let us review what we have observed. While for $T_{k}=0$ both functions $L_{h}$ and $Q_{h}$ are zero and the index has a power-law asymptotics (more precisely an $\mathcal{I}\approx 1/\beta$ behavior as $\beta\to 0$ Ardehali:2015c ), finite nonzero $T_{k}$ can induce Mexican-hat potentials for the holonomies in the high-temperature limit, triggering an infinite-temperature exponential growth in the index.

Higher ranks

We now show that the integrand of the index is maximally suppressed at $\boldsymbol{x}=0$ in fact for arbitrary $N\geq 2$ and $T_{k}\in\mathbb{R}/\mathbb{Z}$.

Let us study the behavior of the $L_{h}$ function in (42) with respect to $x_{i}$. For this purpose, we use the following equality derived in Ardehali:2015c (c.f. Eq. (3.51) there), valid for $-1/2\leq u_{i}\leq 1/2$:

[TABLE]

We will use the above identity with $M=4$ and $u_{1,2,3}=T_{1,2,3}$; we would moreover like to take $u_{4}=x_{i}-x_{j}$, but this is not allowed since the range $-1<x_{i}-x_{j}<1$ is incompatible with $-1/2\leq u_{4}\leq 1/2$; to fix that we put instead $u_{4}=\{x_{i}-x_{j}+1/2\}-1/2$. Using (44) we can now rewrite the $L_{h}$ function in (42) such that its only $\boldsymbol{x}$-dependent piece is

[TABLE]

The above expression is obviously negative-semi-definite as a function of $x_{i}$, and it is maximized when $x_{i}-x_{j}=0$. So the index is Higgsed for any $T_{k}\in\mathbb{R}/\mathbb{Z}$ at infinite temperature.

Whether (or for which range of $T_{k}\in\mathbb{R}$) the infinite-temperature phase of the index can be exponentially growing when $N>2$, is an interesting problem which seems to require an intricate analysis.

4 Supersymmetric Casimir energy with complex chemical

potentials

When all the fugacities $p,q,u_{k}$ are real-valued, the index $\mathcal{I}(p,q,u_{k})$ is related to the path-integral SUSY partition function $Z(\beta,b,m_{k})$ of the theory on $S_{b}^{3}\times S^{1}_{\beta}$ via Bobev:2015

[TABLE]

where $E_{\mathrm{SUSY}}(b,m_{k})$ is known as the supersymmetric Casimir energy, $\beta,b,m_{k}$ are defined through

[TABLE]

and $S_{b}^{3}$ is the squashed three-sphere with unit radius and squashing parameter $b$, while $S^{1}_{\beta}$ is the circle with circumference $\beta$. (The special case of (46) with $m_{k}=0$ was understood already in Ardehali:2015b ; Assel:2015s , based on earlier slightly contrasting computations of Assel:2014 .)

As made clear by HHZ Hosseini:2017 (and further elucidated in Choi:2018 ; Choi:2018b ; Benini:2018 ; Cabo-Bizet:2018 ) making contact with the AdS5 BPS blackholes requires considering complex fugacities $p,q,u_{k}$ in the index. With the goal of understanding the role of the supersymmetric Casimir energy in the blackhole entropy discussion, in this section we study the relation between $Z$ and $\mathcal{I}$ for complex fugacities such that $b\in\mathbb{R}_{>0}$ and $\beta\in\mathbb{C}$ with $\mathrm{Re}\beta>0$ as in Section 2, while $u_{k}$ are on the unit circle as in Section 3. Rather than modifying the background geometry to achieve such complexified $\beta$ (c.f. Assel:2014 ), we simply analytically continue the results obtained for real $p,q$.

Let us consider a free chiral multiplet to begin with; as in Assel:2014 ; Ardehali:2015b , we expect that solving this case leads to the solution of the interacting non-abelian case as well.

Following Appendix A of Ardehali:2015b , we start with the one-loop determinant of the $n$th KK mode on $S^{3}\times S^{1}$. Eq. (A.15) in Ardehali:2015b now generalizes to

[TABLE]

where $\ell_{b}$ is the special function discussed in Ardehali:2015b , the $R$-charge of the multiplet is denoted by $R$, and $T_{k}:=\frac{i\beta m_{k}}{2\pi}\in\mathbb{R}$, with $m_{k}$ the only chemical potential the chiral multiplet couples to.

Define $X:=(R-1)\frac{b+b^{-1}}{2}$ for notational convenience. Following Ardehali:2015b step by step, we now rewrite $\log Z^{(n)}$ in terms of $\psi_{b}$ which has a simple asymptotic behavior. Eq. (A.2) of Ardehali:2015b implies that in terms of $\psi_{b}$:

[TABLE]

Now, using the fact that $\ell_{b}(-x)=-\ell_{b}(x)$, we can rewrite

[TABLE]

One way to check the above equation is to check it separately for $\mathrm{sgn}(n+T_{k})=+1$ and $\mathrm{sgn}(n+T_{k})=-1$, using $\ell_{b}(-x)=-\ell_{b}(x)$ and Eq. (A.2) in Ardehali:2015b . The reason for this rewriting is to divide $\psi_{b}$s into the numerator and denominator of $Z$, so we can eventually relate $Z$ to $\mathcal{I}$ using expressions such as (20).

Finally, we sum (50) over $n\in\mathbb{Z}$. In doing so, we use the relations Di Pietro and Honda used DiPietro:2017 for analyzing the high-temperature asymptotics of the index:

[TABLE]

The definitions are $\{x\}:=x-\lfloor x\rfloor$, $\vartheta(x):=\{x\}(1-\{x\})$, $\kappa(x):=\{x\}(1-\{x\})(1-2\{x\})$.

With this regularization—combining techniques from Ardehali:2015b and DiPietro:2017 —we obtain

[TABLE]

Putting $T_{k}=0$ we can compare with Eq. (A.16) in Ardehali:2015b , noting that $\kappa(0)=\vartheta(0)=0$, so that the only surviving term on the second line of the RHS of the above relation gives the Di Pietro-Komargodski asymptotics DiPietro:2014 as $\beta\to 0$; the first and the third lines combine to give the first and the third terms on the RHS of Eq. (A.16) in Ardehali:2015b .

We are done with our regularization. We believe our method of regularization is correct because we have been careful with the convergence of the infinite product appearing in $Z$—or equivalently the convergence of the infinite sum appearing in $\log Z$—after regularization, and because we have used well-established tools of analytic continuation101010See Chapter VII of Weil:1975 for some context. for evaluating the sums (51)–(53). As a byproduct, from the second line on the RHS of (54) we can read off the high-temperature asymptotics of the partition function of a chiral multiplet with a flavor fugacity on the unit circle.

We now would like to relate $Z$ as obtained in (54) to the index $\mathcal{I}$. We use (20) and the fact that the index of the chiral multiplet is $\Gamma((pq)^{R/2}u_{k})$. For simplicity we assume $0<T_{k}<1$, and replace all $\{T_{k}\}$ in (54) with $T_{k}$. Then we set (54) equal to

[TABLE]

The end result is that $E_{\mathrm{SUSY}}$ comes out just as in Ardehali:2015b ; Assel:2015s : there is no dependence on $T_{k}$!

In other words, for $u_{k}=e^{2\pi iT_{k}}$ on the unit circle (which is relevant to the equal-charge AdS5 blackholes) we have $E_{\mathrm{SUSY}}(b,m_{k})=E_{\mathrm{SUSY}}(b,0)$. Since in the small-$|\beta|$ limit with $b>0$ fixed we have $\beta E_{\mathrm{SUSY}}(b,0)\to 0$, we conclude that on the saddle-point associated to the equal-charge blackholes the supersymmetric Casimir energy has no significance in the leading Cardy-like asymptotics of the partition function $Z$. In particular, the Casimir-energy factor relating $Z$ and $\mathcal{I}$ is irrelevant to the blackhole entropy function arising in the Cardy-like limit of either.

The relation between the above discussion and the interesting proposal of Cabo-Bizet:2018 which seems to involve analytic continuation of $Z$ with respect to $\tau$ and $\sigma$ is currently under study.

5 Summary and open problems

We have presented a careful analysis of the asymptotics of the SU($N$) $\mathcal{N}=4$ theory index (3) in the CKKN limit where the flavor fugacities approach the unit circle and the spacetime fugacities approach $1$. We emphasize that compared to the previous work Ardehali:2015c the Cardy-like limit studied here is more general in two respects: $i)$ instead of sending the flavor fugacities to $1$ as in Ardehali:2015c , following CKKN here we allowed the flavor fugacities to approach the unit circle; $ii)$ although in Section 3 we kept the “inverse-temperature” $\beta$ on the positive real axis as in Ardehali:2015c , in Section 2 we complexified $\beta$ and let $|\mathrm{arg}\beta|\in(0,\pi/2)$ which was necessary for making contact with the HHZ function and the AdS5 blackholes.

For complexified temperature (with $0<|\mathrm{arg}\beta|<\pi/2$), we have demonstrated that in the CKKN limit, the dominant holonomy configurations in the index are dictated by the pairwise potential (27). We explained that depending on the sign of $\mathrm{arg}\beta$, the pair-wise potential has $M$- or $W$-type behavior on complementary wings of the space of the control-parameters $\mathrm{Re}\Delta_{1,2}$—see Figure 2. On the $M$ wings the potential is minimized at the origin, so the holonomies condense (at either of $N$ possibilities $x_{j}=0,\frac{1}{N},\dots,\frac{N-1}{N}$ breaking the $\mathbb{Z}_{N}$ center), thereby giving rise to the HHZ function (12) (with $\Delta_{3}=\tau+\sigma-\Delta_{1}-\Delta_{2}+\mathrm{sgn}(\mathrm{arg}\beta)$) in the leading asymptotics of the index, from which we extracted two blackhole saddle-points (one for each sign of $\mathrm{arg}\beta$). On the other hand, on the $W$ wings, except for the $N=2$ case, the quantitative analysis seems difficult; we only presented intuitive arguments suggesting that for $N>2$ the index has a slower asymptotic growth there and therefore no blackhole saddle-points with entropies as large as the two just mentioned are expected in those regions.

Problem 1)

In the CKKN limit

[TABLE]

find the asymptotics of the SU($N$) $\mathcal{N}=4$ theory index for $N>2$ on the $W$ wings; that is, for $\tau,\sigma$ inside the 2nd quadrant and $\mathrm{Re}\Delta_{1,2}$ on the lower-left wing of Figure 2, or for $\tau,\sigma$ inside the 1st quadrant and $\mathrm{Re}\Delta_{1,2}$ on the upper-right wing of Figure 2. In particular, show that the fastest asymptotic growth in those regions is slower than the fastest growth on the complementary $M$ wings.

Even without addressing the above problem, we have successfully derived two blackhole saddle-points in the CKKN limit. However, the saddle-points have flavor fugacities that are away from the unit circle unless the three charges $Q_{k}$ are (approximately) equal [in which case the critical $\Delta_{1,2,3}$ are in fact (approximately) simply $\mathrm{sgn}(\mathrm{arg}\beta)\times\frac{1}{3}$]. Therefore our derivation of the blackhole entropy function is incomplete for the general blackholes with unequal $Q_{k}$. To complete the analysis for the general case we have to derive the asymptotic relation (12) when $\mathrm{Im}\Delta_{k}$ are not sent to zero.

Problem 2)

For complexified temperature ($0<|\mathrm{arg}\beta|<\pi/2$) perform the asymptotic analysis of the index in the limit

[TABLE]

In particular, derive the HHZ function, once with $\sum_{k}\Delta_{k}=\tau+\sigma-1$ and once with $\sum_{k}\Delta_{k}=\tau+\sigma+1$, in two separate regimes of parameters.

We would like to emphasize that although we have not given a complete derivation of the entropy function for the general case with unequal charges, our analysis in the equal-charge case already allows addressing various conceptual issues in the derivation. One such conceptual issue has been the significance of the rather special relation $\sum_{k}\Delta_{k}=\tau+\sigma-1$ in the HHZ function Hosseini:2017 . In the present paper we have shown that a similar asymptotics arises with $\sum_{k}\Delta_{k}=\tau+\sigma+1$ in a separate region of parameters, leading to a second blackhole saddle-point with fugacities that are complex conjugate to those of the HHZ saddle-point. (See Benini:2018 for related statements in the large-$N$ analysis.)

Another conceptual point that we were able to clarify in the special case with equal charges was the insignificance of the supersymmetric Casimir energy to the blackhole entropy function in the Cardy-like limit. Generalizing that discussion to the case with the flavor fugacities away from the unit circle constitutes another important open problem related to the present work.

Problem 3)

Study the supersymmetric Casimir energy of the $\mathcal{N}=4$ theory with flavor fugacities away from the unit circle. In particular, investigate its relevance to the blackhole entropy function in the Cardy-like limit.

Note added:

While this work was nearing completion the preprint Honda:2019 appeared on arXiv which has some overlap with our Section 2. Reference Honda:2019 seems to suggest that extra hairy-blackhole Markeviciute:2018a ; Markeviciute:2018b saddle-points might reside on the $W$ wings of the parameter-space. As discussed in Section 2, we find it more likely that no such extra blackhole saddle-points with entropies as large as the ones discussed here (as expected to be the case for the hairy blackholes of Markeviciute:2018a ; Markeviciute:2018b ) exist in the Cardy-like limit of the index. The existence/interpretation of extra saddle-points in the large-$N$ analysis Benini:2018 is of course a separate issue.

Also, note that Honda:2019 gives around its Eq. (2.11) a neat analytic proof for the $M$-type behavior of the pairwise potential in a specific subset of the parameter-space for $\mathrm{arg}\beta>0$ (i.e. $\mathrm{Re}(i/\tau\sigma)<0$). However, the argument below (2.9) there that the periodicity of the pairwise potential can be used to extend the analytic proof of the $M$-type behavior to the whole parameter-space does not seem applicable; as illustrated in Figure 2, in essentially half of the parameter-space the pairwise potential is in fact maximized at the origin (this can actually be seen from Honda’s analytic proof in the appropriate region of the parameter-space, together with the oddity of the potential under $\Delta_{k}\to-\Delta_{k}$). Consequently, the $W$-type behavior of the pairwise potential in half of the parameter-space when $\mathrm{arg}\beta>0$, as well as the $M$-type behavior in half of the parameter-space when $\mathrm{arg}\beta<0$ (and thus also the second blackhole saddle-point) seem to have been overlooked in Honda:2019 .

Acknowledgements.

It is a pleasure to thank F. Benini, F. Bouya, D. Cassani, S. Kim, P. Milan, V. Reys, and especially M. Hosseini for valuable discussions and correspondences related to this work, as well as J. Liu and P. Szepietowski for enjoyable collaboration on related projects. I am grateful to F. Larsen, L. Pando Zayas, and M. M. Sheikh-Jabbari for encouraging me to study related developments, and indebted to F. Benini and P. Milan for their comments on an earlier draft of this work. All the graphs in this paper were produced by Mathematica. This work was supported by the Knut and Alice Wallenberg Foundation under grant Dnr KAW 2015.0083, and by the National Elites Foundation of Iran.