Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

Details Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

2001-09-06

arxiv.org/abs/hep-th/0109048v1

Commun.Math.Phys. 227 (2002) 131-153

Physics

High Energy Physics

High Energy Physics - Theory

25 pages, 2 figures, LaTeX

Scientific paper

10.1007/s002200200629

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.

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