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

Andrew McIntyre, Leon A. Takhtajan

TL;DR

The paper extends Kronecker's first limit formula to higher genus by constructing a holomorphic, t-dependent basis of holomorphic n-differentials on genus g>1 surfaces and introducing a holomorphic F(n) on Schottky space. It proves a holomorphic factorization of the regularized Laplacian determinant into a Liouville-action term, a holomorphic factor |F(n)|^2, and a Gram-determinant factor, thereby generalizing Zograf's genus-one results and tying to Quillen metrics and the local index theorem for families. The approach combines Poincaré series for Green's functions, Eichler cohomology to select a natural n-differential basis, and meticulous holomorphic variation calculations to show that the determinant ratio equals a holomorphic-analytic object times a universal geometric term. This yields a higher-genus analogue of classical limit formulas and offers a pathway to generalized theta-function product formulas via twisting by characters. The results have implications for spectral geometry, moduli of Riemann surfaces, and connections to string-theoretic measures on moduli spaces.

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 \prod(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.

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

TL;DR

The paper extends Kronecker's first limit formula to higher genus by constructing a holomorphic, t-dependent basis of holomorphic n-differentials on genus g>1 surfaces and introducing a holomorphic F(n) on Schottky space. It proves a holomorphic factorization of the regularized Laplacian determinant into a Liouville-action term, a holomorphic factor |F(n)|^2, and a Gram-determinant factor, thereby generalizing Zograf's genus-one results and tying to Quillen metrics and the local index theorem for families. The approach combines Poincaré series for Green's functions, Eichler cohomology to select a natural n-differential basis, and meticulous holomorphic variation calculations to show that the determinant ratio equals a holomorphic-analytic object times a universal geometric term. This yields a higher-genus analogue of classical limit formulas and offers a pathway to generalized theta-function product formulas via twisting by characters. The results have implications for spectral geometry, moduli of Riemann surfaces, and connections to string-theoretic measures on moduli spaces.

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 \prod(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.

Paper Structure

This paper contains 11 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Necessary basic facts
  3. Kleinian groups
  4. The operators $\bar{\partial}_{n}$ and $\Delta_{n}$
  5. Teichmüller and Schottky spaces
  6. Classical Liouville action
  7. Poincaré series and the Green's function of $\bar{\partial}_{n}$
  8. Natural basis for $H^{0}(\mathfrak{S}_g,\Lambda_n)$
  9. Proof of Theorems 1 and 2
  10. Computation of $\partial\log\det\Delta_{n}$
  11. Computation of $\partial\log F(n)$

Key Result

Theorem 1

Let $g>1$, and let ${\det}'\Delta_{0}$, $\mathop{\mathrm{\mathrm{Im}}}\nolimits\boldsymbol\tau$ and $S$ be the functions on the Schottky space $\mathfrak{S}_{g}$ defined above. Then there exists a holomorphic function $F:\mathfrak{S}_{g}\to\mathbb{C}$ such that where $c_{g}$ is a constant depending only on $g$. For points in $\mathfrak{S}_{g}$ corresponding to Schottky groups $\Gamma$ with expone

Theorems & Definitions (26)

  • Theorem 1: P. Zograf
  • Theorem 2
  • Lemma 2.1
  • proof
  • Remark 1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 16 more