Rotating Black Hole Entropy from M5-branes

TL;DR

This paper calculates the superconformal index of 3d theories from M5-branes on hyperbolic 3-manifolds, linking microscopic field theory results to black hole entropy and exploring mathematical structures in quantum gravity.

Contribution

It introduces a novel computation of the superconformal index using 3d-3d correspondence and perturbative invariants, connecting black hole entropy to field theory and mathematical results.

Findings

Matching of supergravity one-loop and field theory $ ext{log} N$ terms.

Microscopic derivation of rotating black hole entropy.

Evidence for vanishing of the refined topologically twisted index.

Abstract

We compute the superconformal index of 3d $\mathcal{N}=2$ superconformal theories obtained from $N$ M5-branes wrapped on a hyperbolic 3-manifold. Exploiting the 3d-3d correspondence, we use perturbative invariants of $SL(N,\mathbb{C})$ Chern-Simons theory to determine the superconformal index in the large $N$ limit, including corrections logarithmic in $N$. The leading order partition function provides a microscopic foundation for the entropy function of the dual rotating asymptotically AdS$_4$ black holes. We also verify that the supergravity one-loop contribution to the $\log N$ term coincides with the field theoretic result. We propose a 3d-3d formulation for the refined topologically twisted index, and provide strong evidence in support of its vanishing --- which agrees with the fact that the expected dual rotating magnetically-charged black hole does not exist. This provides an interesting link between gravity and a tantalizing mathematical result.