HCIZ integral and 2D Toda lattice hierarchy

TL;DR

This paper explores the large N HCIZ integral using integrable hierarchies, revealing its connection to the dispersionless Toda lattice hierarchy and deriving exact expansion results.

Contribution

It demonstrates that the large N HCIZ integral is governed by the dispersionless Toda hierarchy and derives its string equation, providing new insights into its structure.

Findings

HCIZ integral linked to dispersionless Toda hierarchy

Derived the string equation for the HCIZ integral

Obtained exact expansion results in moments

Abstract

The expression of the large $N$ Harish Chandra--Itzykson--Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large $N$ HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.