Tau-functions on spaces of abelian differentials and higher genus generalizations of ray-singer formula

Abstract

Let w be an Abelian differential on a compact Riemann surface of genus g > 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1. © 2009 Applied Probability Trust.