Virasoro constraints and polynomial recursion for the linear Hodge integrals

TL;DR

This paper derives explicit Virasoro constraints for the linear Hodge tau-function, connects them to polynomial recursion formulas, and explores their implications for generating linear Hodge integrals within the KP hierarchy.

Contribution

It provides a new explicit form of Virasoro constraints for the Hodge tau-function and establishes their equivalence with polynomial recursion formulas.

Findings

Explicit Virasoro constraints expressed via Lambert W function coefficients

Simplified Virasoro constraints for the Hodge partition function using Gamma function

Equivalence between Virasoro constraints and polynomial recursion for linear Hodge integrals

Abstract

The Hodge tau-function is a generating function for the linear Hodge integrals. It is also a tau-function of the KP hierarchy. In this paper, we first present the Virasoro constraints for the Hodge tau-function in the explicit form of the Virasoro equations. The expression of our Virasoro constraints is simply a linear combination of the Virasoro operators, where the coefficients are restored from a power series for the Lambert W function. Then, using this result, we deduce a simple version of the Virasoro constraints for the linear Hodge partition function, where the coefficients are restored from the Gamma function. Finally, we establish the equivalence relation between the Virasoro constraints and polynomial recursion formula for the linear Hodge integrals.