Unitary matrix models, free fermions, and the giant graviton expansion

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) =∑ n≥1an qn, with an ∈ Z. Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on S³ 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 qαN+β where α, β do not depend on N. This pro-vides a matrix-model explanation of the giant graviton expansion that has been observed recently in the literature.