Polynomiality of $\mathbb{Z}_2$ Hurwitz-Hodge Integrals

TL;DR

This paper proves that hyperelliptic $ ext{Hurwitz-Hodge}$ integrals are polynomial in genus and develops recursions and PDEs to compute them, revealing new structural properties and conjectures about coefficient log-concavity.

Contribution

It introduces recursions and PDEs for hyperelliptic Hodge integrals, establishing their polynomiality in genus and proposing a conjecture on coefficient log-concavity.

Findings

Hyperelliptic Hodge integrals are polynomial in genus g.

Recursions determine all such integrals from initial conditions.

Generated PDEs encode the integrals' structure.

Abstract

Using Atiyah-Bott localization on the space of stable maps to the stack quotient $[\mathbb{P}^1/\mathbb{Z}_2]$, we find recursions that determine all Hodge integrals with descendent insertions at one marked point on the hyperelliptic locus $\overline{\mathcal{H}}_{g, 2g + 2} \subseteq \overline{\mathcal{M}}_{g, 2g + 2}$. The initial conditions required for our recursions are gravitational descendents at one marked point, which are known to be $\frac{1}{2}$. We discover a new structure concerning these intersection numbers: for a fixed monomial of $\lambda$-classes, the resulting family of hyperelliptic Hodge integrals is polynomial in $g$. We formulate a conjecture concerning the log-concavity of the coefficients of these polynomials. Lastly, we turn our recursions into a non-linear system of partial differential equations for the generating functions of hyperelliptic Hodge integrals.