Hurwitz-Hodge integral identities from the cut-and-join equation

TL;DR

This paper derives new identities for Hurwitz-Hodge integrals using the cut-and-join equation's Laplace transform and confirms a conjecture by J. Zhou from 2008.

Contribution

It introduces novel Hurwitz-Hodge integral identities and proves a longstanding conjecture, advancing understanding in algebraic geometry and enumerative combinatorics.

Findings

Derived Hurwitz-Hodge integral identities from the cut-and-join equation.

Proved J. Zhou's conjecture on Hurwitz-Hodge integrals.

Connected Laplace transform techniques with orbifold Hurwitz numbers.

Abstract

In this paper, we present some Hurwitz-Hodge integral identities which are derived from the Laplace transform of the cut-and-join equation for the orbifold Hurwitz numbers. As an application, we prove a conjecture on Hurwitz-Hodge integral proposed by J. Zhou in 2008.