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

Chiu-Chu Melissa Liu|ArXiv.org|Oct 17, 2002

Geometry and complex manifoldsMathematics28 references78 citations

TL;DR

This paper constructs a Kuranishi structure with corners on the moduli space of J-holomorphic curves with Lagrangian boundary conditions, proving compactness and Hausdorffness in the C^∞-topology. For S¹-equivariant pairs with vanishing virtual dimension, it defines an open Gromov-Witten invariant as a rational Euler number, conjectured to match localization computations.

ABSTRACT

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 $C^\infty$-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.

Motivation & Objective

• To rigorously define open Gromov-Witten invariants for J-holomorphic curves with boundary in a Lagrangian submanifold.
• To establish a compact, Hausdorff moduli space of stable maps with Lagrangian boundary conditions in the C^∞-topology.
• To construct a Kuranishi structure with corners on this moduli space, proving orientability when the Lagrangian is spin.
• To define an Euler number invariant for S¹-equivariant pairs when the virtual dimension is zero, conjectured to match localization results.
• To provide a foundational framework for open Gromov-Witten invariants that can be used to test physical predictions in string theory.

Proposed method

• Uses Gromov's C^∞-topology to compactify the moduli space of J-holomorphic curves with boundary in a Lagrangian submanifold.
• Constructs a Kuranishi structure with corners via deformation theory of bordered Riemann surfaces and gluing techniques for nodal curves.
• Applies the Kuranishi method to stabilize maps, using W^{k,p} maps and virtual dimension formulas to control the moduli space.
• Imposes S¹-equivariance on the pair (X,L) to define an Euler number invariant via localization, assuming the virtual dimension is zero.
• Relies on the Schwartz reflection principle to relate holomorphic maps with boundary in L to holomorphic maps on compact Riemann surfaces.
• Conjectures that the Euler number matches localization computations involving Hodge bundles and ψ-classes on moduli spaces of curves.

Experimental results

Research questions

• RQ1 Can a well-defined, compact, and oriented moduli space be constructed for J-holomorphic curves with Lagrangian boundary conditions?
• RQ2 Does a Kuranishi structure with corners exist on this moduli space, and is it orientable when the Lagrangian is spin?
• RQ3 Can an open Gromov-Witten invariant be defined for S¹-equivariant pairs when the virtual dimension is zero?
• RQ4 Is the Euler number invariant computed via this construction equivalent to the one obtained via localization techniques?
• RQ5 How can boundary conditions be generalized to remove the restrictive assumption of S¹-equivariance in the general case?

Key findings

• The moduli space of J-holomorphic curves with Lagrangian boundary conditions is compact and Hausdorff in the C^∞-topology.
• A Kuranishi structure with corners is constructed on this moduli space, and it is orientable if the Lagrangian submanifold is spin.
• For S¹-equivariant pairs with virtual dimension zero, an Euler number invariant is defined as a rational number.
• The invariant $ C(g;h|d;n_1,\ldots,n_h|a) $ is conjectured to satisfy a duality relation: $ (-1)^{d-h}C(g;h|d;n_1,\ldots,n_h|a) = C(g;h|d;n_1,\ldots,n_h|1-a) $.
• For genus 0, the invariant is given by $ (a(1-a))^{h-1} \prod_{i=1}^{h} \binom{n_i a - 1}{n_i - 1} d^{h-3} $, matching localization results.
• For higher genus, the invariant is expressed as an integral over $ \overline{M}_{g,h} $ involving Chern classes of the Hodge bundle and ψ-classes.