Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

TL;DR

This paper provides an explicit formula for counting surface-kernel epimorphisms from co-compact Fuchsian groups to cyclic groups, with applications to string theory, QFT, and automorphism groups of Riemann surfaces.

Contribution

It introduces a practical formula for counting these epimorphisms, connecting to Harvey's theorem and utilizing recent advances in solving restricted linear congruences.

Findings

Derived an explicit formula for surface-kernel epimorphisms count

Connected the formula to Harvey's theorem on Riemann surface automorphisms

Utilized properties of Ramanujan sums and Fourier transforms in the derivation

Abstract

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch, Ramgoolam, and Wen [Nuclear Phys.\,B {\bf 870} (2013), 530--581] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an `equivalent' form of the famous Harvey's theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions.