Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula

2004/10/31 by Andrew McIntyre, Leon A. Takhtajan · 2 citations
Mathematics · Physics and Astronomy · #hep-th #math.CV #math.SP #msc:11M36 #msc:14H15 #msc:30F10 #msc:32G15 #msc:58J52

paper · pdf

published as Geom. and Funct. Analysis, v. 16 (2006), 1291--1323 · LaTex, 31 pages, no figures; minor typos corrected, final version to appear in Geometric and Functional Analysis

arxiv created 2006/09/29 · arxiv updated 2015/01/12

Abstract

For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the Schottky space S_g, we define a natural basis for the holomorphic n-differentials on X_t which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n=1. We introduce a holomorphic function F(n) on S_g which generalizes the classical product ∏(1-q^m)² appearing in the Dedekind eta function for n=1 and g=1. We prove a holomorphic factorization formula expressing the regularized determinant of the Laplacian as a product of |F(n)|², a holomorphic anomaly depending on the classical Liouville action (a Kahler potential of S_g), and the determinant of the Gram matrix of the natural basis. The factorization formula reduces to Kronecker's first limit formula when n=1 and g=1, and to Zograf's factorization formula for n=1 and g>1.

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