Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy

TL;DR

This paper investigates the semiclassical evolution of spectral curves in the normal random matrix ensemble, demonstrating its equivalence to the Whitham hierarchy through compatibility equations and canonical forms.

Contribution

It establishes the connection between semiclassical limits of spectral curve evolution and the Whitham hierarchy, extending previous analyses of the normal random matrix ensemble.

Findings

Semiclassical limit of evolution equations matches Whitham hierarchy.

Spectral curve evolution described by compatibility equations.

Canonical differential forms characterize the semiclassical limit.

Abstract

We continue the analysis of the spectral curve of the normal random matrix ensemble, introduced in an earlier paper. Evolution of the full quantum curve is given in terms of compatibility equations of independent flows. The semiclassical limit of these flows is expressed through canonical differential forms of the spectral curve. We also prove that the semiclassical limit of the evolution equations is equivalent to Whitham hierarchy.