Large $N$ matrix models for 3d ${\cal N}=2$ theories: twisted index, free energy and black holes

TL;DR

This paper derives formulas for the topologically twisted index of 3d ${ m f N}=2$ gauge theories with large $N$ duals, linking it to black hole entropy and large $N$ partition functions, revealing universal scaling behaviors.

Contribution

It introduces general formulae for the twisted index in large $N$ 3d ${ m f N}=2$ theories with M-theory or massive IIA duals, connecting black hole entropy and partition functions.

Findings

Index scales as $N^{3/2}$ for M-theory duals.

Index scales as $N^{5/3}$ for massive IIA duals.

Universal relation between the index and $S^3$ partition function.

Abstract

We provide general formulae for the topologically twisted index of a general three-dimensional ${\cal N}\geq 2$ gauge theory with an M-theory or massive type IIA dual in the large $N$ limit. The index is defined as the supersymmetric path integral of the theory on $S^2\times S^1$ in the presence of background magnetic fluxes for the R- and global symmetries and it is conjectured to reproduce the entropy of magnetically charged static BPS AdS$_4$ black holes. For a class of theories with an M-theory dual, we show that the logarithm of the index scales indeed as $N^{3/2}$ (and $N^{5/3}$ in the massive type IIA case). We find an intriguing relation with the (apparently unrelated) large $N$ limit of the partition function on $S^3$. We also provide a universal formula for extracting the index from the large $N$ partition function on $S^3$ and its derivatives and point out its analogy with the attractor mechanism for AdS black holes.