The Laplace transform of the cut-and-join equation of Mari\~no-Vafa formula and its applications

TL;DR

This paper computes the Laplace transform of the cut-and-join equation related to the Marino-Vafa formula, applies polynomial identities to prove conjectures, and derives explicit formulas for Hodge integrals.

Contribution

It introduces a method to compute the Laplace transform of the cut-and-join equation and applies it to prove the Bouchard-Marino conjecture for C^3, also deriving formulas for Hodge integrals.

Findings

Proved Bouchard-Marino conjecture for C^3.

Derived series of Hodge integral identities.

Provided explicit formulas for specific Hodge integrals.

Abstract

By the same method introduced in [9], we calculate the Laplace transform of the celebrated cut-and-join equation of Mari\~no-Vafa formula discovered by C. Liu, K. Liu and J. Zhou [17]. Then, we study the applications of the polynomial identity (1) obtained in theorem 1.1 of this paper. We show the proof Bouchard-Mari\~no conjecture for C^3 which was given by L. Chen [5] Firstly. Subsequently, we will present how to obtain series Hodge integral identities from this polynomial identity (1). In particular, the main result in [9] is one of special case in such series of Hodge integral identities. At last, we give a explicit formula for the computation of Hodge integral <\tau_{b_L}\lamda_{g}\lamda_{1}>_{g} where b_L = (b_1,..,b_l).