Precision Microstate Counting for the Entropy of Wrapped M5-branes

TL;DR

The paper develops a precise large-N framework for counting microstates of magnetically charged AdS4 black holes arising from N M5-branes wrapped on a hyperbolic 3-manifold M3 by exploiting the 3d-3d correspondence. By mapping twisted partition functions of 3d N=2 theories to PSL(N,C) Chern-Simons invariants and analytic torsions on M3, the authors derive an all-orders 1/N expansion that, remarkably, terminates at finite order. The leading $N^3$ term reproduces the Bekenstein-Hawking entropy, while the subleading $\log N$ term matches the one-loop supergravity correction, providing a stringent check of AdS4/CFT3 and precise predictions for higher M-theory corrections. The work further establishes integrality of twisted indices, clarifies the role of Bethe vacua tied to hyperbolic geometry, and highlights universal features governed by vol(M3) and topological data, with robust gravity-side interpretations via logarithmic corrections. Together, these results constitute a high-precision holographic test for wrapped M5-branes and offer a powerful route to subleading quantum gravity data in AdS4/CFT3.

Abstract

We study the large $N$ expansion of twisted partition functions of 3d $\mathcal{N}=2$ superconformal field theories arising from $N$ M5-branes wrapped on a hyperbolic 3-manifold, $M_3$. Via the 3d-3d correspondence, the partition functions of these 3d ${\cal N}=2$ superconformal field theories are related to simple topological invariants on the 3-manifold. The partition functions can be expressed using only classical and one-loop perturbative invariants of $PSL(N,\mathbb{C})$ Chern-Simons theory around irreducible flat connections on $M_3$. Using mathematical results on the asymptotics of the invariants, we compute the twisted partition functions in the large $N$ limit including perturbative corrections to all orders in $1/N$. Surprisingly, the perturbative expansion terminates at finite order. The leading part of the partition function is of order $N^3$ and agrees with the Bekenstein-Hawking entropy of the dual black holes. The subleading part, in particular the $\log N$-terms in the field theory partition function is found to precisely match the one-loop quantum corrections in the dual eleven dimensional supergravity. The field theory results of other terms in $1/N$ provide a stringent prediction for higher order corrections in the holographic dual, which is M-theory.

Figures (1)

• Figure 1: Figure-eight knot, $K = \mathbf{4}_1$