Matrix ensembles with global symmetries and 't Hooft anomalies from 2d gauge theory

TL;DR

The paper constructs a bulk 2D gravity framework augmented by a dynamical G gauge field to realize sectorized random matrix ensembles on the boundary. In the anomaly-free case, each representation sector r exhibits a spectrum with ground energy ∝ c2(r) and a density enhanced by (dim r)2, revealing a second G action beyond degeneracy. Introducing 't Hooft anomalies via discrete theta terms restricts boundary sectors to projective representations of fixed N-ality, yielding new matrix-model sectors. Time-reversal symmetry and unorientable surfaces further modify the ensemble, producing GOE/GSE blocks or degenerate GUE pairs depending on representation properties and anomalies. Collectively, the work clarifies how global symmetries and anomalies imprint precise, calculable structures in boundary random-matrix statistics, with potential relevance to SYK-like systems with symmetry and to broader holographic dualities.

Abstract

The Hilbert space of a quantum system with internal global symmetry $G$ decomposes into sectors labelled by irreducible representations of $G$. If the system is chaotic, the energies in each sector should separately resemble ordinary random matrix theory. We show that such "sector-wise" random matrix ensembles arise as the boundary dual of two-dimensional gravity with a $G$ gauge field in the bulk. Within each sector, the eigenvalue density is enhanced by a nontrivial factor of the dimension of the representation, and the ground state energy is determined by the quadratic Casimir. We study the consequences of 't Hooft anomalies in the matrix ensembles, which are incorporated by adding specific topological terms to the gauge theory action. The effect is to introduce projective representations into the decomposition of the Hilbert space. Finally, we consider ensembles with $G$ symmetry and time reversal symmetry, and analyze a simple case of a mixed anomaly between time reversal and an internal $\mathbb{Z}_2$ symmetry.

Figures (1)

• Figure 1: A cartoon of the geometry that computes $Z_{g}(\beta_1,r_1; \ldots; \beta_n,r_n)$. There is a genus $g$ surface glued to $n$ "trumpets" associated to the insertions of $\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_r}(e^{-\beta H})$ on the RMT side. The boundary conditions at the ends of the trumpets are determined by $\beta$ and $r$.