1. Introduction

Recently, it has been realized [1; 2] that conformal maps exhibit an integrable structure: conformal maps of compact simply connected domains bounded by analytic curves provide a solution to the dispersionless limit of the two-dimensional Toda hierarchy. As is well known from the theory of solitons, solutions of an integrable hierarchy are represented by τ-functions. The dispersionless limit of the τ-function emerges as a natural object associated with the curves. In this paper we discuss the τ-function for simple analytic curves and its connection to the inverse potential problem, area preserving diffeomorphisms, the Dirichlet boundary problem, and matrix models.

2. The Inverse Potential Problem

Define a closed analytic curve as a curve that can be parametrized by a function z = x + iy = z(w), analytic in a domain that includes the unit circle |w| = 1. Consider a closed analytic curve 7 in the complex plane and denote by D+ and D- the interior and exterior domains with respect to the curve. The point z = 0 is assumed to be in D+. Assume that the domain D+ is filled homogeneously with electric charge, with a density that we set to 1.