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  • 7 - Bergman Tau-Function: From Einstein Equations and Dubrovin-Frobenius Manifolds to Geometry of Moduli Spaces

  • Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
  • Book:
    Integrable Systems and Algebraic Geometry
    Published online:
    19 March 2020
    Print publication:
    02 March 2020, pp 215-287

  • Summary

    We review the role played by tau-functions of special type - called 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.