Bergman tau function: from Einstein equations and Dubrovin-Frobenius manifolds to geometry of moduli spaces

Abstract

We review the role played by tau functions of special type - called {\it Bergman} tau functions in various areas: theory of isomonodromic deformations, solutions of Einstein's equations, theory of Dubrovin-Frobenius manifolds, geometry of moduli spaces and spectral theory of Riemann surfaces. These tau functions are natural generalizations of Dedekind's eta-function to higher genus. Study of their properties allows to get an explicit form of Einstein's metrics, obtain new relations in Picard groups of various moduli spaces and derive holomorphic factorization formulas of determinants of Laplacians in flat singular metrics on Riemann surfaces, among other things.