Relatively open Gromov-Witten invariants for symplectic manifolds of lower dimensions

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 pseudoholomorphic maps from a Riemann surface with a fixed conformal structure and 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 semipositivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If $L \cap V = \emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $\phi$ on $X$ such that $L$ is the fixed point set of $\phi$ and $V$ is $\phi$-anti-invariant, then we define the so-called “relatively open” invariants for the tuple $(X, \omega, V, \phi)$ if $L$ is orientable and $\dim X \leq 6$. If $L$ is nonorientable, we define such invariants under the condition that $\dim X \leq 4$ and some additional restrictions on the number of marked points on each boundary component of the domain.

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Published 8 April 2015