Rotating Black Hole Entropy from M5-branes

TL;DR

This work leverages the 3d-3d correspondence to compute the superconformal index of N M5-branes on hyperbolic M3, relating it to perturbative SL(N, C) Chern-Simons invariants and reproducing the entropy function of dual rotating AdS$_4$ black holes at leading order and in the logarithmic $\log N$ correction. It demonstrates a precise match between field theory and 11d supergravity, including one-loop contributions, and reveals a vanishing refined twisted index that aligns with the non-existence of certain magnetically charged black holes in the universal sector. The analysis extends to general hyperbolic M3, reveals integrality and Bethe-vacua structures, and provides a robust framework for exploring subleading corrections and potential defects, linking gravity, topology, and number-theoretic structure in a deep, testable way.

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.