[Paper Review] Abelian Hurwitz-Hodge integrals

Paul D. Johnson, Rahul Pandharipande+1 more|arXiv (Cornell University)|Mar 4, 2008

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper computes linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy groups by expressing them in terms of multiplication in wreath group algebras. For cyclic groups with faithful representations, the results reduce to double Hurwitz numbers; in the trivial group case, they recover the Ekedahl-Lando-Shapiro-Vainshtein formula for single Hurwitz numbers.

ABSTRACT

Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms of multiplication in an associated wreath group algebra. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers. In case G is trivial, the formula specializes to the well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge integrals over the moduli space of curves in terms of single Hurwitz numbers.

Motivation & Objective

• To evaluate linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy groups.
• To establish a connection between Hodge integrals and Hurwitz numbers through representation theory and stable map theory.
• To generalize the Ekedahl-Lando-Shapiro-Vainshtein formula for linear Hodge integrals on the moduli space of curves to the case of abelian covers.
• To provide a uniform framework for computing Hodge integrals using wreath group algebra structures.
• To extend the computation to unstable cases and verify consistency via degree and parity conditions.

Proposed method

• Utilizes the stable map perspective on moduli spaces of admissible covers, identifying them with moduli of stable maps to the classifying stack $\mathcal{B}G$.
• Associates Hodge classes $\lambda_i^R$ to irreducible representations $R$ of the abelian group $G$, forming vector bundles $\mathbb{E}^R$ on $\overline{\mathcal{M}}_{g,\gamma}(\mathcal{B}G)$.
• Applies localization techniques to compute top intersection numbers of $\lambda_i^R$ and $\bar{\psi}_j$ classes on the moduli stack.
• Reduces the computation to Hurwitz numbers via a forgetful map from $K$-covers to $\Sigma_d$-covers, using automorphism group corrections.
• Uses the wreath product structure of $G \wr \Sigma_d$ to express the integral as a product in the group algebra of $G \wr \Sigma_d$.
• Verifies consistency of degree and parity conditions between the maps $\rho$ and $\rho'$ in the proof of Theorem 3.

Experimental results

Research questions

• RQ1 How can linear Hodge integrals on moduli spaces of abelian admissible covers be expressed in terms of group algebra structures?
• RQ2 What is the precise relationship between Hodge integrals and double Hurwitz numbers when the monodromy group is cyclic and the representation is faithful?
• RQ3 How does the formula reduce to the Ekedahl-Lando-Shapiro-Vainshtein result in the trivial group case?
• RQ4 What is the role of automorphism group corrections in relating Hurwitz numbers with and without additional labeling data?
• RQ5 Under what conditions do unstable Hodge integrals appear, and how are they consistently defined in the formula?

Key findings

• Linear Hodge integrals over $\overline{\mathcal{M}}_{g,\gamma}(\mathcal{B}G)$ for abelian $G$ are evaluated via multiplication in the wreath group algebra $G \wr \Sigma_d$.
• For $G = \mathbb{Z}_a$ and a faithful representation, the Hodge integral reduces to a weighted sum of double Hurwitz numbers $H_{g,K}(\overline{\nu},\overline{\mu})$.
• When $G$ is trivial, the formula specializes to the Ekedahl-Lando-Shapiro-Vainshtein result, recovering single Hurwitz numbers.
• The degree of the forgetful map $\rho^\prime$ from $\overline{\mathcal{M}}_{g,\iota \cup \kappa}(\mathcal{B}K)$ to $\overline{\mathcal{M}}_{g,d/a + \ell(\mu)}$ matches the degree of $\rho$, ensuring consistency in the computation.
• The parity condition $\sum \kappa_j - dx = 0$ is equivalent for both maps $\rho$ and $\rho^\prime$, validating the reduction in Theorem 3.
• Unstable integrals, such as $\int_{\overline{\mathcal{M}}_{0,(0)}(\mathcal{B}G)} \frac{\sum (-a)^i \lambda_i^R}{1 - x\bar{\psi}_1} = \frac{1}{|G|} \cdot \frac{1}{x^2}$, are consistently defined and included in the formula.