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

Arash Arabi Ardehali

TL;DR

This work provides a rigorous Cardy-like analysis of the 4d N=4 superconformal index in the CKKN regime using elliptic hypergeometric integrals, clarifying the derivation of the HHZ blackhole entropy function. It identifies a second blackhole saddle with complex-conjugate fugacities and reveals bifurcations in the index as fugacity phases vary, while establishing that the dominant holonomy configuration in the Carty-like regime yields the known entropy expressions. The study shows the supersymmetric Casimir energy does not affect the leading entropy in the equal-charge case and discusses open problems for unequal charges away from the unit circle, outlining the path toward a complete holographic microstate count. By connecting finite-N Cardy-like asymptotics with AdS5 blackhole entropy and relating to HHZ and CKKN frameworks, the paper lays groundwork for future large-N and Bethe-Ansatz analyses of the microstate counting problem.

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.

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

TL;DR

This work provides a rigorous Cardy-like analysis of the 4d N=4 superconformal index in the CKKN regime using elliptic hypergeometric integrals, clarifying the derivation of the HHZ blackhole entropy function. It identifies a second blackhole saddle with complex-conjugate fugacities and reveals bifurcations in the index as fugacity phases vary, while establishing that the dominant holonomy configuration in the Carty-like regime yields the known entropy expressions. The study shows the supersymmetric Casimir energy does not affect the leading entropy in the equal-charge case and discusses open problems for unequal charges away from the unit circle, outlining the path toward a complete holographic microstate count. By connecting finite-N Cardy-like asymptotics with AdS5 blackhole entropy and relating to HHZ and CKKN frameworks, the paper lays groundwork for future large-N and Bethe-Ansatz analyses of the microstate counting problem.

Abstract

Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardy-like limit of the superconformal index of the 4d theory with complexified fugacities which encodes the entropy of the dual supersymmetric AdS 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.

Paper Structure

This paper contains 6 sections, 56 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of the CKKN derivation in the elliptic hypergeometric language
  3. Complexified temperature and AdS$_5$ blackholes ($|\beta|\to 0$, $0<|\mathrm{arg}\beta|<\frac{\pi}{2}$)
  4. Real-valued temperature ($\beta\to0$, $\beta\in\mathbb{R}_{>0}$)
  5. Supersymmetric Casimir energy with complex chemical potentials
  6. Summary and open problems

Figures (4)

  • Figure 1: The qualitative behavior of $V^Q(x;\mathrm{arg}\beta,\mathrm{Re}\Delta_k)$ as a function of $x$ for fixed $\mathrm{Re}\Delta_{1,2}$ and fixed $\mathrm{arg}\beta\in(-\pi/2,0)$, in various regions of the space of the control-parameters $\mathrm{Re}\Delta_{1,2}$.
  • Figure 2: The qualitative behavior of $V^Q(x;\mathrm{arg}\beta,\mathrm{Re}\Delta_k)$, as a function of $x$ for fixed $\mathrm{Re}\Delta_{1,2}$ and fixed $\mathrm{arg}\beta\in(-\pi/2,0)$, in the two complementary regions $-1<\mathrm{Re}\Delta_{1},\mathrm{Re}\Delta_{2},-1-\mathrm{Re}\Delta_1-\mathrm{Re}\Delta_2<0$ (lower-left) and $0<\mathrm{Re}\Delta_{1},\mathrm{Re}\Delta_{2},1-\mathrm{Re}\Delta_1-\mathrm{Re}\Delta_2<1$ (upper-right) of the space of the control-parameters $\mathrm{Re}\Delta_{1,2}$. The $M$ and $W$ wings switch places if $\mathrm{arg}\beta$ is taken to be inside $(0,\pi/2)$ instead.
  • Figure 3: The $L_h$ function (\ref{['eq:genRainsN']}) in the $N=2$ case, as a function of $x_1$, for $T_{1,2}=-1/9$ (left), $T_{1,2}=-1/3$ (middle), and $T_{1,2}=-4/9$ (right).
  • Figure 4: Contours of $L_h(x_1=\pm1/4,r_k=2/3;T_k)$. The blue curve and dots correspond to zero value, inside the blue curve except at the origin corresponds to positive values, while outside the blue curve and away from the blue dots corresponds to negative values.