Hodge integrals and $\lambda_{g}$ conjecture with target varieties

TL;DR

This paper introduces a new $ ext{lambda}_g$ conjecture for Hodge integrals with target varieties, establishes its relation to the Virasoro conjecture, and proves it in various cases, providing new universal constraints for Gromov-Witten invariants.

Contribution

It formulates the $ ext{lambda}_g$ conjecture for target varieties and proves it in all genus for certain classes of varieties, linking it to Virasoro conjecture and Gromov-Witten invariants.

Findings

Proved $ ext{lambda}_g$ conjecture in all genus for varieties with semisimple quantum cohomology.

Established genus zero case for all smooth projective varieties.

Derived new universal constraints for descendant Gromov-Witten invariants.

Abstract

In this paper, we propose $\lambda_{g}$ conjecture for Hodge integrals with target varieties. Then we establish relations between Virasoro conjecture and $\lambda_{g}$ conjecture, in particular, we prove $\lambda_{g}$ conjecture in all genus for smooth projective varieties with semisimple quantum cohomology or smooth algebraic curves. Meanwhile, we also prove $\lambda_{g}$ conjecture in genus zero for any smooth projective varieties. In the end, together with DR formula for $\lambda_g$ class, we obtain a new type of universal constraints for descendant Gromov-Witten invariants. As an application, we prove $\lambda_{g}$ conjecture in genus one for any smooth projective varieties.