Gromov–Witten invariants of target curves via Symplectic Field Theory

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. We need to impose (and possibly solve) different kinds of Schrödinger equations related to some quantization of the dispersionless KdV hierarchy. In particular, we find explicit formulas for the Gromov–Witten invariants of low degree of P1P1 with descendants of the Kähler class.