Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair

TL;DR

This paper develops a framework for studying moduli spaces of $J$-holomorphic curves with Lagrangian boundary conditions, constructs a compatible Kuranishi structure, and defines an $S^1$-equivariant invariant in special cases.

Contribution

It introduces a Kuranishi structure with corners for these moduli spaces and defines an $S^1$-equivariant invariant, extending open Gromov-Witten theory.

Findings

Moduli space is compact and Hausdorff in Gromov topology.

Constructed an orientable Kuranishi structure when $L$ is spin.

Defined an $S^1$-equivariant Euler number conjecturally matching localization computations.

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 acts freely 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.