Gromov-Witten invariants of target curves via Symplectic Field Theory

TL;DR

This paper computes Gromov-Witten invariants of target curves at all genera using Symplectic Field Theory, deriving differential equations and explicit formulas, especially for low-degree invariants of P1 with descendants.

Contribution

It introduces a novel approach combining Symplectic Field Theory with quantum differential equations to compute invariants of target curves.

Findings

Explicit formulas for low-degree Gromov-Witten invariants of P1

Differential equations for the full descendant potential

Connection to quantization of the dispersionless KdV hierarchy

Abstract

We compute the Gromov-Witten potential at all genera of target smooth Riemann surfaces using Symplectic Field Theory techniques and establish differential equations for the full descendant potential. This amounts to impose (and possibly solve) different kinds of Schroedinger equations related to some quantization of the dispersionless KdV hierarchy. In particular we find very explicit formulas for the Gromov-Witten invariants of low degree of P1 with descendants of the Kaehler class.