Random super matrices with an external source

TL;DR

This paper extends the study of Gaussian random matrix ensembles with external sources to supermatrices, deriving explicit formulas and dualities that lead to new geometric insights in 2D gravity models.

Contribution

It introduces a supermatrix extension of external source models, providing explicit expressions and dualities that generalize previous results to include surfaces with boundaries.

Findings

Derived explicit k-point functions for supermatrix models.

Established a duality in supermatrix models with external sources.

Obtained new geometric results in 2D gravity from these models.

Abstract

In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich's Airy matrix models and generalizations. The techniques relied on explicit computations of the k-point functions for arbitrary N (the size of the matrices) and on an N-k duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to include surfaces with boundaries, we have extended these techniques to supermatrices. Again we have obtained quite remarkable explicit expressions for the k-point functions, as well as a duality. Although supermatrix models a priori lead to the same matrix models of 2d-gravity, the external source extensions considered in this article lead to new geometric results.