New Insights into Superintegrability from Unitary Matrix Models

Abstract

Some eigenvalue matrix models possess an interesting property: one can manifestly define the basis where all averages can be explicitly calculated. For example, in the Gaussian Hermitian and rectangular complex models, averages of the Schur functions are again expressed through the Schur functions. However, so far this property remains restricted to very particular (e.g. Gaussian) measures. In this paper, we extend this observation to unitary matrix integrals, where one could expect that this restriction is easier to lift. We demonstrate that this is indeed the case, only this time the Schur averages are linear combinations of the Schur functions. Full factorization to a single item in the sum appears only on the Miwa locus, where at least one half of the time-variables is expressed through matrices of the same size. For unitary integrals, this is a manifestation of the de Wit-t'Hooft anomaly, which prevents the answer to be fully analytic in the matrix size $N$. Once achieved, this understanding can be extended back to the Hermitian model, where the phenomenon looks very similar: beyond Gaussian measures superintegrability requires an additional summation.