Moduli of $J$-holomorphic curves with Lagrangian boundary conditions ‎and open Gromov-Witten invariants for an $S^1$-equivariant pair

Abstract

Let $(X,omega)$ be a symplectic manifold‎, ‎$J$ be an $omega$-tame‎ ‎almost complex structure‎, ‎and $L$ be a Lagrangian submanifold‎. ‎The stable compactification of the moduli space of parametrized $J$-holomorphic‎ ‎curves in $X$ with boundary in $L$ (with prescribed topological data)‎ is compact and Hausdorff in Gromov's $C^infty$-topology‎. ‎We construct a Kuranishi structure with corners in the sense of Fukaya and‎ ‎Ono‎. ‎This Kuranishi structure is orientable if $L$ is spin‎. ‎In the special case where the expected dimension of the moduli space‎ ‎is zero‎, ‎and there is an $S^1$-action on the pair $(X,L)$ which‎ ‎preserves $J$ and has no fixed points on $L$‎, ‎we define the ‎Euler number for this $S^1$-equivariant pair and the prescribed‎ ‎topological data‎. ‎We conjecture that this rational number is‎ ‎the one computed by localization techniques using the given $S^1$-action‎.