Abelian Hurwitz-Hodge integrals

TL;DR

This paper provides explicit formulas for Hodge integrals over moduli spaces of admissible covers with abelian monodromy, connecting them to wreath group algebra and Hurwitz numbers, generalizing known results.

Contribution

It introduces a unified approach to evaluate linear Hodge integrals for abelian monodromy groups using wreath group algebra, extending classical Hurwitz number formulas.

Findings

Explicit formulas for abelian Hurwitz-Hodge integrals

Connection to wreath group algebra and Hurwitz numbers

Generalization of known results for cyclic and trivial groups

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.