Moduli of J-holomorphic curves with Lagrangian boundary conditions ‎and open Gromov-Witten invariants for an S1-equivariant pair

Let (X,ω) be a symplectic manifold, J be an ω-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∞-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 S1-action on the pair (X,L) which preserves J and has no fixed points on L, we define the Euler number for this S1-equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given S1-action.