Cardy Expansion of 3d Superconformal Indices and Corrections to the Dual Black Hole Entropy

We consider the superconformal index of three-dimensional ${\cal N}=2$ supersymmetric field theories computed via localization on $S^1\times S^2$. 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 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 $\beta\to 0$ limit for finite $N$, we find indications of non-perturbative contributions of the form $e^{-1/\beta}$. 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 that the corrections in $\beta$ to the entropy can be accounted for by appropriately shifting the electric charges and the angular momentum