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

TL;DR

The authors examine how projecting matrix models onto singlet sectors of global symmetries (double-gauging) simplifies their Hilbert spaces and dynamics. Using Molien–Weyl methods, they compute refined partition functions for bosonic multi-matrix models with $U(2)$ or $SU(2)$ gauge groups and $SO(d)$ global symmetry, revealing that the $SO(d)$ singlet sector often reduces to a simple single-matrix (or small multi-component) model and identifying primary and secondary invariants via a Hironaka decomposition. In the BMN sector of $\mathcal{N}=4$ SYM, projecting onto $SU(3)_R$ singlets drastically simplifies the analysis of non-graviton cohomologies, yielding closed-form expressions for singlet non-graviton indices in low-$N$ theories and showing absence (or near absence) of singlet gravitons in several cases. For $SU(2)$ and $SU(3)$ gauge groups, the authors obtain explicit graviton indices and, where possible, closed-form non-graviton indices, including threshold cohomologies and explicit towers generated by operators like $\operatorname{tr}(f^2)$ and $\operatorname{tr}(f^3)$. These results illuminate the spectrum of BPS states and their potential relation to black hole microstates, while providing practical tools (Molien–Weyl projections and invariant theory) to study similar double-gauged systems and higher-$N$ extensions.

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.