Twisted determinants on higher genus Riemann surfaces

TL;DR

This paper computes determinants of Dirac and Laplacian operators with twisted boundary conditions on higher genus Riemann surfaces, expressing results explicitly via Schottky parameters and extending classical theta-function formulas.

Contribution

It provides explicit formulas for determinants with twists on higher genus surfaces using Schottky parametrization, and generalizes classical theta-function product formulas to multi-loop cases.

Findings

Explicit determinant formulas in terms of Schottky parameters

Multi-loop generalization of g=1 theta-function product formulas

Applications to perturbative open charged string theory

Abstract

We study the Dirac and the Laplacian operators on orientable Riemann surfaces of arbitrary genus g. In particular we compute their determinants with twisted boundary conditions along the b-cycles. All the ingredients of the final results (including the normalizations) are explicitly written in terms of the Schottky parametrization of the Riemann surface. By using the bosonization equivalence, we derive a multi-loop generalization of the well-known g=1 product formulae for the Theta-functions. We finally comment on the applications of these results to the perturbative theory of open charged strings.