Topological recursion relations and Gromov-Witten invariants in higher genus

TL;DR

This paper establishes topological recursion relations for higher genus Gromov-Witten invariants, enabling their computation in projective spaces by linking complex invariants to simpler ones.

Contribution

It proves a new recursion relation for higher genus invariants and demonstrates its sufficiency with Virasoro constraints to compute Gromov-Witten potentials.

Findings

Recursion relations express complex invariants via simpler ones.

Relations combined with Virasoro constraints allow full potential calculation.

First feasible computational method for higher genus invariants in projective spaces.

Abstract

We state and prove a topological recursion relation that expresses any genus-g Gromov-Witten invariant of a projective manifold with at least a (3g-1)-st power of a cotangent line class in terms of invariants with fewer cotangent line classes. For projective spaces, we prove that these relations together with the Virasoro conditions are sufficient to calculate the full Gromov-Witten potential. This gives the first computationally feasible way to determine the higher genus Gromov-Witten invariants of projective spaces.