Recursion Formulas for Spin Hurwitz Numbers

TL;DR

This paper establishes a recursive formula for spin Hurwitz numbers, linking them to Gromov-Witten invariants of Kahler surfaces, and introduces novel techniques involving spectral flow and eigenfunction localization.

Contribution

It provides the first recursive formula for spin Hurwitz numbers, connecting them to Gromov-Witten invariants and employing new analytical methods.

Findings

Recursive formula for spin Hurwitz numbers

Connection to Gromov-Witten invariants of Kahler surfaces

Cancellation of maps with even ramification points

Abstract

The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These "spin Hurwitz numbers", recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors' previous work, they are also related to the Gromov-Witten invariants of Kahler surfaces. We prove a recursive formula for spin Hurwitz numbers, which then gives the dimension zero GW invariants of Kahler surfaces with positive geometric genus. The proof uses a degeneration of spin curves, an invariant defined by the spectral flow of certain anti-linear deformations of the d-bar operator, and an interesting localization phenomenon for eigenfunctions that shows that maps with even ramification points cancel in pairs.