A formula for the Gromov-Witten potential of an elliptic curve

TL;DR

This paper derives an explicit formula for the Gromov-Witten potential of an elliptic curve by solving Virasoro constraints, building on prior algorithms for computing invariants of smooth projective curves.

Contribution

It provides a closed-form expression for the Gromov-Witten potential of elliptic curves, explicitly solving Virasoro constraints for this case.

Findings

Explicit formula for Gromov-Witten potential of elliptic curves

Solution of Virasoro constraints for elliptic case

Connection between stationary invariants and full potential

Abstract

An algorithm to determine all the Gromov-Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro type constraints that allow to determine all the other Gromov-Witten invariants in terms of the stationary ones. In the case of an elliptic curve, we show that these Virasoro type constraints can be explicitly solved leading to a very explicit formula for the full Gromov-Witten potential in terms of the stationary invariants.