Covers counting via Feynman Calculus

TL;DR

This paper introduces a novel method using Feynman calculus to count principal G-bundles over surfaces and connects it to generating functions for double Hurwitz numbers via Frobenius algebras.

Contribution

It develops a new counting tool for principal G-bundles and links it to algebraic structures related to symmetric groups, providing a fresh perspective on Hurwitz numbers.

Findings

Provides a formula for counting principal G-bundles over surfaces.

Expresses generating functions for double Hurwitz numbers as integrals over Frobenius algebras.

Establishes a connection between topological counting problems and algebraic structures.

Abstract

Let $G$ be a finite group. In this paper we present a tool for counting the number of principle $G$-bundles over a surface. As an application, we express (non-standard) generating functions for double Hurwitz numbers as integrals over commutative Frobenius algebras, associated with symmetric groups.