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

Abstract

Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PS{L}_2(\mathbb{C})$ 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.

28 pages. Improved version containing results of arXiv:1409.1302v1

References in corpus (3)

• Classical Chern-Simons theory, Part 1
• The Taylor expansion of Ruelle L-function at the origin and the Borel regulator
• A special value of Ruelle L-function and the theorem of Cheeger and Muller