Cardy expansion of 3d superconformal indices and corrections to the dual black hole entropy

We consider the superconformal index of three-dimensional N = 2 sup ersymmetric field theories computed via localization on S-1 x S-2. We systematically develop an expansion where the ratio of the radius of S-1 to the radius of S-2 is taken very small -a Cardy-like expansion. We emphasize the sub-leading structures in this Cardy-like expansion as well as their interplay with the large -N limit for theories with gauge group of the form product of U(N) factors. We demonstrate that taking the large -N limit first leads to an expression for the effective action yielding the index that only includes terms proportional to 1/beta and powers of beta(i=0,1,2) where beta is the ratio of radii. As we depart from the beta -> 0 limit for finite N, we find indications of non-perturbative contributions of the form e(-1/beta). We present an explicit evaluation of the superconformal index for the ABJM theory in the large -N limit that is exact in beta up to non-perturbative corrections. For the ABJM theory we explore the implications of the Cardy-like expansion for corrections to the entropy of the rotating, electricallly charged, asymptotically AdS4 dual black hole. Interestingly, we find a closed form for the entropy that includes all perturbative corrections in beta and takes the form of the leading order expression but with shifted charges.

Automated summary

In this study, we examine the superconformal index of three-dimensional N = 2 supersymmetric field theories computed via localization on S-1 x S-2. We develop a Cardy-like expansion where the ratio of S-1 radius to S-2 radius is very small. We highlight the sub-leading structures in this expansion and their interaction with the large-N limit for U(N) gauge group theories.