Conformal maps and dispersionless integrable hierarchies

TL;DR

This paper reveals a deep connection between conformal maps of simply connected domains and the dispersionless 2D Toda hierarchy, showing how these maps relate to integrable systems, inverse potential problems, and 2D gravity models.

Contribution

It establishes a novel link between conformal maps, integrable hierarchies, and inverse potential problems, introducing a new $ au$-function concept for analytic curves.

Findings

Conformal maps correspond to specific solutions of the dispersionless 2D Toda hierarchy.

The hierarchy solves the inverse potential problem locally, reconstructing domains from harmonic moments.

The same hierarchy describes 2D gravity coupled to c=1 matter.

Abstract

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as "string equations". The same hierarchy locally solves the 2D inverse potential problem, i.e. reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c=1 matter. We also introduce a concept of the $\tau$-function for analytic curves.