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

Let $(X,ω)$ 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 $(Σ,\partialΣ)$ 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 $L\cap V=\emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $ϕ$ on $X$ such that $L$ is the fixed point set of $ϕ$ and $V$ is $ϕ$-anti-invariant, then we define the so-called "relatively open" invariants for the tuple $(X,ω,V,ϕ)$ if $L$ is orientable and dim$X\le 6$. If $L$ is nonorientable, we define such invariants under the condition that dim$X\le4$ and some additional restrictions on the number of marked points on each boundary component of the domain.