Comparison of the superbosonization formula and the generalized Hubbard-Stratonovich transformation

TL;DR

This paper proves the equivalence of two advanced mathematical methods, the superbosonization formula and the generalized Hubbard-Stratonovich transformation, used to extend supersymmetry techniques in random matrix theory beyond Gaussian ensembles.

Contribution

It establishes the theoretical equivalence of the superbosonization and Hubbard-Stratonovich approaches, unifying two methods for analyzing rotation invariant ensembles.

Findings

Proves the mathematical equivalence of the two approaches.

Reduces complex integrals to simpler forms involving quadratic supermatrices.

Provides a foundation for applying either method interchangeably in future research.

Abstract

Recently, two different approaches were put forward to extend the supersymmetry method in random matrix theory from Gaussian ensembles to general rotation invariant ensembles. These approaches are the generalized Hubbard-Stratonovich transformation and the superbosonization formula. Here, we prove the equivalence of both approaches. To this end, we reduce integrals over functions of supersymmetric Wishart-matrices to integrals over quadratic supermatrices of certain symmetries.