Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

TL;DR

This paper uncovers an integrable structure underlying the two-dimensional Dirichlet boundary problem, linking domain deformations to the dispersionless Toda hierarchy and matrix models.

Contribution

It reveals that the Dirichlet problem's variations are governed by an integrable hierarchy, connecting harmonic moments, tau-functions, and matrix model partition functions.

Findings

The Hadamard formula indicates an integrable structure.

Harmonic moments serve as independent variables for flows.

The tau-function relates to the dispersionless Toda hierarchy and matrix models.

Abstract

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model.