Symmetrized cut-join equation of Marino-Vafa formula

TL;DR

This paper presents a symmetrized version of the cut-join equation related to the Marino-Vafa formula, enabling new recursion formulas for Hodge integrals through polynomial equations and discussing some applications.

Contribution

It introduces a symmetrized cut-join equation via a transcendental change of variable, expanding the tools for deriving recursion formulas for Hodge integrals.

Findings

Derived new recursion formulas for Hodge integrals.

Provided applications of the symmetrized equation.

Enhanced understanding of the Marino-Vafa formula's computational aspects.

Abstract

In this note, we symmetrized the cut-join equation from the proof of Marino-Vafa formula by applying a transcendental changing of variable. One can derive more recursion formulas of Hodge integrals out of this polynomial equations. We also give some applications.