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

TL;DR

This paper develops a framework for defining and studying relatively open Gromov-Witten invariants in symplectic manifolds of low dimensions, incorporating intersection data and orientability considerations.

Contribution

It introduces the concept of relatively open invariants for symplectic manifolds with Lagrangian and symplectic submanifolds, extending Gromov-Witten theory to new settings with boundary and intersection conditions.

Findings

Constructed moduli spaces of open pseudoholomorphic maps with intersection data.

Established orientability conditions for these moduli spaces.

Defined relatively open invariants under specific dimensional and orientability assumptions.

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 $L\cap V=\emptyset$, 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$X\le 6$. 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.