New topological recursion relations

TL;DR

This paper introduces new topological recursion relations for cotangent line classes on moduli spaces of stable curves, using virtual localization, leading to universal equations in Gromov-Witten theory and confirming recent conjectures.

Contribution

It derives simple boundary expressions for cotangent line classes for k >= 2g using virtual localization, enabling new tautological relations and Gromov-Witten identities.

Findings

Derived boundary expressions for cotangent line classes.

Constructed nontrivial tautological classes in the kernel of push-forward maps.

Proved Gromov-Witten identities conjectured by Liu and Xu.

Abstract

Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective line. As a consequence, nontrivial tautological classes in the kernel of the push-forward map associated to the irreducible boundary divisor of the moduli space of stable g+1 curves are constructed. The geometry of genus g+1 curves then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu.