Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions

TL;DR

This paper introduces a new family of open Gromov-Witten invariants for pseudoholomorphic curves with boundary in Lagrangian submanifolds, linking intersection theory with real curve counts and providing explicit calculations.

Contribution

It defines novel invariants for curves with boundary in Lagrangians arising from anti-symplectic involutions, extending intersection theory and connecting to real enumerative geometry.

Findings

New invariants coincide with Welschinger's counts in specific cases

Explicitly computed the invariant for the real quintic threefold

Established intersection-theoretic framework for open Gromov-Witten invariants

Abstract

We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30.