Connecting Hodge Integrals to Gromov–Witten Invariants by Virasoro Operators

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 $$\{ L_m \mid m \ge 1 \}$$ 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 a generalization of the work in Liu and Wang [Commun. Math. Phys. 346(1):143–190, 2016] for the point case which solved a conjecture of Alexandrov.