$τ$-function for analytic curves

Kostov, I. K.; Krichever, I.; Mineev-Weinstein, M.; Wiegmann, P.; Zabrodin, A.

$τ$-function for analytic curves

Authors

Description

We review the concept of $τ$-function for simple analytic curves. The $τ$-function gives a formal solution to the 2D inverse potential problem and appears as the $τ$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $τ$-function also emerges in the context of topological gravity and enjoys an interpretation as a large $N$ limit of the normal matrix model.
13 pages, no figures, prepared for Proccedings of MSRI Workshop ``Matrix Models and Painlevé Equations'', Berkeley (USA) 1999