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

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