Connecting Hodge integrals to Gromov-Witten invariants by Virasoro operators

TL;DR

This paper establishes a connection between Hodge integrals and Gromov-Witten invariants for algebraic varieties using Virasoro operators, extending previous work from the point case to general varieties.

Contribution

It introduces a series of Virasoro operators that relate Hodge integrals to Gromov-Witten invariants for any nonsingular projective variety, generalizing prior results.

Findings

Virasoro operators satisfy the Virasoro bracket relation

Generating functions are connected via differential operators

Extension of the point case to general varieties

Abstract

In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety $X$ can be connected to the generating function for Gromov-Witten invariants of $X$ by a series of differential operators $\{ L_m \mid m \geq 1 \}$ after a suitable change of variables. These operators satisfy the Virasoro bracket relation and can be seen as a generalization of the Virasoro operators appeared in the Virasoro constraints for Kontsevich-Witten tau-function in the point case. This result is an extension of the work in \cite{LW} for the point case which solved a conjecture of Alexandrov.