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

## TL;DR

This paper explores Bergman tau functions and their applications across geometry, physics, and spectral theory, revealing new relations and explicit formulas in the context of moduli spaces and Einstein metrics.

## Contribution

It connects Bergman tau functions to Einstein equations, Frobenius manifolds, and moduli space geometry, providing explicit formulas and new relations.

## Key findings

- Explicit form of Einstein's metrics derived
- New relations in Picard groups established
- Holomorphic factorization formulas for Laplacian determinants obtained

## 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.