Intersection theory on $\Mbar_{1,4}$ and elliptic Gromov-Witten invariants

TL;DR

This paper introduces a new universal equation for genus 1 elliptic Gromov-Witten invariants, enabling their determination for projective spaces, and explores its geometric significance and uniqueness among such equations.

Contribution

It presents a novel universal equation for elliptic Gromov-Witten invariants and demonstrates its sufficiency in computing invariants for projective spaces.

Findings

The new equation determines elliptic Gromov-Witten invariants of projective spaces.

The equation is more complex than WDVV and its geometric meaning is unclear.

It is unique among similar equations relating elliptic and rational invariants in genus 1.

Abstract

The WDVV equation is satisfied by the genus 0 correlation functions of any topological field theory in two dimensions coupled to topological gravity, and may be used to determine the genus 0 (rational) Gromov-Witten invariants of many projective varieties (as was done for projective spaces by Kontsevich). In this paper, we present an equation of a similar universal nature for genus 1 (elliptic) Gromov-Witten invariants -- however, it is much more complicated than the WDVV equation, and its geometric significance is unclear to us. (Our prove is rather indirect.) Nevertheless, we show that this equation suffices to determine the elliptic Gromov-Witten invariants of projective spaces. In a sequel to this paper, we will prove that this equation is the only one other than the WDVV equation which relates elliptic and rational correlation functions for two-dimensional topological field theories coupled to topological gravity. It is unclear if there are any further equations of this type on the small phase space in higher genus, but we think it unlikely. (The genus 0 and 1 cases are special, since the correlation functions on the small phase space determine those on the large phase space.)