New Solvable Matrix Integrals

TL;DR

This paper introduces a new framework for solving complex matrix integrals by expressing them as tau functions, enabling reduction to eigenvalue integrals and generalizing several classical integrals.

Contribution

It generalizes classical matrix integrals using tau functions, providing a unified approach and extending the Kontsevich integral.

Findings

Reduction of matrix integrals to eigenvalue tau functions

Generalization of Harish-Chandra-Itzykson-Zuber integral

Extension of Kontsevich integral

Abstract

We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the integral over eigenvalues, which in turn is certain tau function. We also consider a generalization of the Kontsevich integral.