Linear Hyperelliptic Hodge Integrals

TL;DR

This paper derives a closed-form combinatorial formula for linear Hodge integrals on the hyperelliptic locus, utilizing Atiyah-Bott localization on a stack of stable maps into an orbifold.

Contribution

It introduces a succinct combinatorial formula for intersection numbers involving one lambda-class and powers of psi-classes on the hyperelliptic locus.

Findings

Closed-form expression for linear Hodge integrals on hyperelliptic locus

Utilizes Atiyah-Bott localization on orbifold stack

Provides explicit combinatorial formulas

Abstract

We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one $\lambda$-class, and powers of a $\psi$-class pulled back along the branch map. This is achieved by using Atiyah-Bott localization on a stack of stable maps into the orbifold $[\mathbb{P}^1/\mathbb{Z}_2]$.