$\tau$-function for analytic curves
  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. 
I.K.Kostov, I.Krichever, M.Mineev-Weinstein, P.Wiegmann, A.Zabrodin