Genus Expanded Cut-and-Join operators and generalized Hurwtiz numbers

TL;DR

This paper introduces genus expanded cut-and-join operators to analyze generalized Hurwitz numbers, providing differential equations for their generating functions based on the source Riemann surface's genus.

Contribution

It defines new genus expanded operators and derives differential equations for generating functions, advancing the understanding of Hurwitz numbers with varying genus.

Findings

Derived differential equations for generating functions.

Expressed generating functions using the new operators.

Enhanced analysis of Hurwitz numbers across different genera.

Abstract

To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, we define the genus expanded cut-and-join operators by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants. As an application, we get some differential equations for the generating functions of the generalized Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.