Exact generating function of a zero-dimensional supersymmetric non-linear sigma model

Abstract

We compute exactly the generating function of a supersymmetric non-linear sigma model describing random matrices belonging to the unitary class. Although an arbitrary source explicitly breaks the supersymmetry, a careful analysis of the invariance of the generating function allows us to show that it depends on only three invariant functions of the source. This generating function allows us to recover various results found in the literature. It also questions the possibility of a functional renormalization group study of the three-dimensional Anderson transition.