Topological Recursion Relations on $\bar{\cal M}_{3,2}$

TL;DR

This paper introduces new genus-3 universal equations for Gromov-Witten invariants derived from relations in the tautological ring of moduli spaces, also discovering a genus-2 universal equation as a byproduct.

Contribution

It presents novel genus-3 universal equations for Gromov-Witten invariants and a new genus-2 equation, expanding the understanding of relations in the tautological ring.

Findings

New genus-3 universal equations for Gromov-Witten invariants

A new genus-2 universal equation was discovered

Relations in the tautological ring lead to these equations

Abstract

In this paper, we give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying new relations in the tautological ring of the moduli space of 2-pointed genus-3 stable curves. A byproduct of our search for genus-3 equations is a new genus-2 universal equation for Gromov-Witten invariants.