Riemann-Roch isomorphism, Chern-Simons invariant and Liouville action

TL;DR

This paper links the Riemann-Roch isomorphism, Chern-Simons invariant, and Liouville action using arithmetic Schottky uniformization, providing explicit formulas and applications to zeta value rationality.

Contribution

It introduces an explicit formula for the Riemann-Roch isomorphism involving the Chern-Simons invariant and determines a key constant in the Liouville action's factorization.

Findings

Explicit formula for Riemann-Roch isomorphism as an infinite product.

Determination of the constant in the holomorphic factorization formula.

Proof of the rationality of Ruelle zeta values for certain 3-manifolds.

Abstract

Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PSL_{2}({\mathbb C})$ Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Riemann-Roch isomorphism as Zograf-Mcintyre-Takhtajan's infinite product for families of algebraic curves. By this formula, we determine the unknown constant which appears in the holomorphic factorization formula of determinant of Laplacians on Riemann surfaces via the classical Liouville action. As an application, we show the rationality of Ruelle zeta values for Schottky uniformized $3$-manifolds.