Unitary matrix models, free fermion ensembles, and the giant graviton expansion

Abstract

We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q)={\sum }_{n\ge 1}{a}_n{q}^n$, with $a_n\in \mathbb{Z}$. Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on $S^3$ and, in particular, as the superconformal index of super Yang-Mills theory. We show that any such model can be expressed in terms of a system of free fermions in an ensemble parameterized by the infinite set of couplings. Integrating out the fermions in a given quantum state leads to a convergent expansion as a series of determinants, as shown by Borodin-Okounkov many years ago. By further averaging over the ensemble, we obtain a formula for the matrix integral as a $q$-series with successive terms suppressed by $q^{\hat{I}\pm N+{\hat{I}}^2}$ where $\hat{I}\pm$, ${\hat{I}}^2$ do not depend on $N$. This provides a matrix-model explanation of the giant graviton expansion that has been observed recently in the literature.

v2: typos corrected, references added. Corresponds to published version in Pure and Applied Mathematics Quarterly, special issue in honor of Don Zagier