Relatively Open Gromov-Witten Invariants for Symplectic Manifolds of Lower Dimensions

Abstract

Let $(X,\omega )$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary $(\Sigma ,\partial \Sigma )$ to the pair $(X,L)$ satisfying Lagrangian boundary conditions and intersecting $V$. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If $L\cap V=\varnothing$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $\varphi$ on $X$ such that $L$ is the fixed point set of $\varphi$ and $V$ is $\varphi$-anti-invariant, then we define the so-called "relatively open" invariants for the tuple $(X,\omega ,V,\varphi )$ if $L$ is orientable and dim$X\le 6$. If $L$ is nonorientable, we define such invariants under the condition that dim$X\le 4$ and some additional restrictions on the number of marked points on each boundary component of the domain.

60 pages; Add more details on gluing construction; To appear in Journal of Symplectic Geometry

References in corpus (6)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• Higher genus symplectic invariants and sigma model coupled with gravity
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Gromov-Witten invariants of general symplectic manifolds
• Enumerative invariants of stongly semipositive real symplectic six-manifolds
• Degeneration and gluing of Kuranishi structures in Gromov-Witten theory and the degeneration/gluing axioms for open Gromov-Witten invariants under a symplectic cut

Cited by in corpus (1)

• Cyclic Homology of Fukaya Categories and the Linearized Contact Homology