The Hurwitz Enumeration Problem of Branched Covers and Hodge Integrals

TL;DR

This paper develops algebraic methods to compute Hurwitz numbers and Hodge integrals for Riemann surfaces, providing new generating functions and reproducing known results for elliptic curves, advancing the understanding of enumerative geometry.

Contribution

It introduces a general algebraic framework for calculating Hurwitz numbers and Hodge integrals, including new generating functions related to symmetric group representations.

Findings

Derived a general generating function for simple Hurwitz numbers.

Reproduced known results for elliptic curve targets.

Established a new generating function for Hodge integrals on M_{g,2}.

Abstract

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S_n. We also find a generating function for Hodge integrals on the moduli space M_{g,2} of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.