Abstract

I will discuss a class of integrals over the unitary group U(N) with an infinite set of couplings characterized by a series f(q)= \sum_n a(n) q^{n, that arise in the context of counting U(N) invariants. I will show that any such model can be expressed in terms of a system of free fermions in an ensemble parameterized by the above 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}(aN+b) where a, b do not depend on N. I will explain the motivation and interpretation of the formula, which comes from the physics of AdS/CFT. In this context, the matrix integral is the superconformal index of Super Yang-Mills theory, and the terms in the expansion correspond to giant gravitons in anti de Sitter space.