Comments on Supersymmetric Vector and Matrix Models

TL;DR

This paper extends results from random matrix theory to supermatrices, showing that for polynomial potentials, supermatrix models behave like ordinary matrix models after addressing charge annihilation, but differences may arise with more complex potentials.

Contribution

It demonstrates the equivalence of supermatrix and matrix models for polynomial potentials and discusses potential differences with more general potentials.

Findings

Supermatrix integration reduces to eigenvalue integration.

Volume element corresponds to a Coulomb gas with positive and negative charges.

Supermatrix models match ordinary models for polynomial potentials after removing instabilities.

Abstract

Some results in random matrices are generalized to supermatrices, in particular supermatrix integration is reduced to an integration over the eigenvalues and the resulting volume element is shown to be equivalent to a one dimensional Coulomb gas of both positive and negative charges.It is shown that,for polynomial potentials, after removing the instability due to the annihilation of opposite charges, supermatrix models are indistinguishable from ordinary matrix models, in agreement with a recent result by Alvarez-Gaume and Manes. It is pointed out however that this may not be true for more general potentials such as for instance the supersymmetric generalization of the Penner model.