Topological recursion relations in genus 2

TL;DR

This paper investigates genus 2 topological recursion relations in Gromov-Witten theory, combining virtual fundamental class techniques with intersection theory on moduli spaces to derive new recursive formulas.

Contribution

It introduces explicit topological recursion relations in genus 2 for Gromov-Witten invariants, extending previous genus 0 and 1 results.

Findings

Derived two new genus 2 recursion relations

Applied Behrend-Fantechi virtual class construction

Performed intersection theory on moduli space ar{M}_{2,2}

Abstract

In Part 1 of this paper, we study gravitational descendents of Gromov-Witten invariants for general projective manifolds, applying the Behrend-Fantechi construction of the virtual fundamental classes. In Part 2, we calculate the topological recursion relations in genus 2. There are two of these, one for the second descendent of a field, and one for the first descendents of two fields. The proof uses the results of Part 1 together with a thorough study of intersection theory on the moduli space $\bar{M}_{2,2}$.