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

A bstract We consider the superconformal index of three-dimensional N = 2 supersymmetric field theories computed via localization on S 1 × 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 /β and powers of β i =0 , 1 , 2 where β is the ratio of radii. As we depart from the β → 0 limit for finite N , we find indications of non-perturbative contributions of the form e − 1 /β . We present an explicit evaluation of the superconformal index for the ABJM theory in the large- N limit that is exact in β 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 AdS 4 dual black hole. Interestingly, we find a closed form for the entropy that includes all perturbative corrections in β and takes the form of the leading order expression but with shifted charges.