Integration of Grassmann variables over invariant functions on flat superspaces

TL;DR

This paper develops methods for integrating invariant functions over superspaces, deriving new integral theorems and applying them to random matrix theory and supermatrix functions.

Contribution

It provides a compact expression for Berezin integration over Grassmann variables and extends integral theorems for invariant functions on supervectors and supermatrices.

Findings

Derived a compact differential operator for Berezin integration.

Extended Cauchy-like integral theorems for invariant functions.

Applied results to compute generating functions in random matrix theory.

Abstract

We study integration over functions on superspaces. These functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables. We get a compact expression for the differential operator with respect to the commuting variables which results from Berezin integration over all Grassmann variables. Also, we derive Cauchy--like integral theorems for invariant functions on supervectors and symmetric supermatrices. This extends theorems partly derived by other authors. As an physical application, we calculate the generating function of the one--point correlation function in random matrix theory. Furthermore, we give another derivation of supermatrix Bessel--functions for U(k_1/k_2).