To gauge or to double gauge? Matrix models, global symmetry, and black hole cohomologies

TL;DR

This paper investigates the structure of Hilbert spaces in gauged matrix models with global symmetries, simplifying complex models and analyzing non-graviton operators in supersymmetric theories, revealing insights into black hole microstates.

Contribution

It introduces a method to simplify double-gauged matrix models and analyzes non-graviton operators in the BMN sector of super Yang-Mills, connecting to black hole microstates.

Findings

Double gauging simplifies matrix models.

Most graviton operators are projected out at low N.

Closed form for superconformal index at N=3.

Abstract

We study the structure of the Hilbert space of gauged matrix models with a global symmetry. In the first part of the paper, we focus on bosonic matrix models with $U(2)$ gauge group and $SO(d)$ global symmetry, and consider singlets under both the gauge and global symmetry. We show how such "double-gauged'' matrix models can be described in terms of a simpler $SO(3)$ single-matrix model. In the second part of the paper, we consider the so-called BMN subsector of the $\mathcal{N}=4$ $SU(N)$ super Yang-Mills theory, which is closely related to the BMN matrix model. Among the 1/16 BPS operators in this sector, "non-graviton'' operators were recently discovered, which are expected to relate to the microstates of supersymmetric $AdS_5$ black holes. We show that a double gauging of this model, where one projects onto $SU(3)_R$ $R$-symmetry singlets, considerably simplifies the analysis of the non-graviton spectrum. In particular, for low values of $N$, we show that (almost) all graviton operators project out of the spectrum, while important classes of non-graviton operators remain. In the $N=3$ case, we obtain a closed form expression for the superconformal index of singlet non-gravitons, which reveals structural features of their spectrum.