Open Gromov–Witten invariants in dimension four

Given a closed orientable Lagrangian surface $L$ in a closed symplectic four-manifold $(X, \omega)$ together with a relative homology class $d \in H_2(X, L; \mathbb{Z})$ with vanishing boundary in $H_1(L; \mathbb{Z})$, we prove that the algebraic number of $J$-holomorphic discs with boundary on $L$, homologous to $d$ and passing through the adequate number of points neither depends on the choice of the points nor on the generic choice of the almost-complex structure $J$. We furthermore get analogous open Gromov–Witten invariants by counting, for every non-negative integer $k$, unions of $k$ discs instead of single discs.

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Published 17 March 2016