Superconformal Indices of 3d $\mathcal{N}=2$ SCFTs and Holography

TL;DR

This paper analyzes the superconformal index of 3d $ ext{N}=2$ SCFTs in a specific limit, revealing a relation to the Bethe Ansatz and deriving formulas relevant to holography and black hole entropy.

Contribution

It establishes a connection between the superconformal index and the Bethe Ansatz formulation, providing explicit formulas for holographic theories in the Cardy-like limit.

Findings

Leading terms in the index are determined by the Bethe Ansatz.

Derived all-order expressions for the index in large N theories.

Discussed implications for black hole entropy and supergravity corrections.

Abstract

We study the superconformal index of 3d $\mathcal{N}=2$ superconformal field theories on $S^1\times_{\omega} S^2$ in the Cardy-like limit where the radius of the $S^1$ is much smaller than that of the $S^2$. We show that the first two leading terms in this Cardy-like expansion are dictated by the Bethe Ansatz formulation of the topologically twisted index of the same theory. We apply this relation to 3d $\mathcal{N}=2$ holographic superconformal field theories describing the low-energy dynamics of $N$ M2-branes and derive closed form expressions, valid to all orders in the $1/N$ expansion, for the two leading terms in the Cardy-like expansion of the superconformal index. We also discuss the implications of our results for the entropy of supersymmetric Kerr-Newman black holes in AdS$_4$ and the four-derivative corrections to 4d gauged supergravity.