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

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.