Cardy-like asymptotics of the 4d $$ \mathcal{N}=4 $$ index and AdS5 blackholes

Abstract

Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardylike limit of the superconformal index of the 4d 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.

Black hole entropy function for toric theories via Bethe Ansatz

“…This important result was subsequently extended to various 4d N = 1 SCFTs [37][38][39][40][41]. For the purposes of the present paper, the most relevant is the observation made in [41], where the authors considered a Cardy-like limit of the superconformal index for the class of toric quiver gauge theories.…”

Supersymmetric phases of 4d $$ \mathcal{N} $$ = 4 SYM at large N

“…Following the insightful observation of [3], this puzzle has been revisited recently [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] The emergent picture shows that there is actually an exponential growth of states contained in the index, which is captured by the saddle-point estimate if one allows the potentials σ, τ , ϕ to take values in the complex plane away from the pure imaginary values assumed in [2]. 1 There are essentially three strands of analysis contained in the recent progress:…”

“…2. The second strand, referred to as the Cardy-like limit [5,10,11,13,14] is the analysis of the index I in the limit when the charges are much larger than all other parameters of the theory including N . In the bulk gravitational dual, this corresponds to a black hole that "fills up all of AdS space".…”

“…by demanding that it vanishes in the first fundamental region and extending it by demanding periodicity of Q. 11 We have explicitly parameterized the ambiguity by a doubly-periodic real function Ψ Q (z) which vanishes at z 2 = 0, which we fix in section 3.3.…”

General supersymmetric $$\hbox {AdS}_5$$ black holes with squashed boundary

“…A solution to the five-dimensional enigma has finally been proposed in [24][25][26][27][28] (see also [29] for previous progress) by reconsidering the field theory partition function dual to the black hole allowing for complex values of the fugacities. The approaches used in the above-mentioned papers are quite different and the relation between them remains an interesting puzzle.…”