$\tau$-function for analytic curves

TL;DR

This paper reviews the $ au$-function for simple analytic curves, highlighting its role in solving inverse potential problems, conformal mapping, and its connections to integrable hierarchies, topological gravity, and matrix models.

Contribution

It provides a comprehensive overview of the $ au$-function's applications in complex analysis, mathematical physics, and integrable systems, emphasizing its interdisciplinary significance.

Findings

$ au$-function solves the 2D inverse potential problem

It describes conformal maps of analytic domains to the unit disk

Connections to topological gravity and large N matrix models

Abstract

We review the concept of $\tau$-function for simple analytic curves. The $\tau$-function gives a formal solution to the 2D inverse potential problem and appears as the $\tau$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $\tau$-function also emerges in the context of topological gravity and enjoys an interpretation as a large $N$ limit of the normal matrix model.