Relative quantum cohomology of the Chiang Lagrangian

Abstract

We compute the open Gromov-Witten disk invariants and the relative quantum cohomology of the Chiang Lagrangian $L_\triangle \subset \mathbb{C}P^3$. Since $L_\triangle$ is not fixed by any anti-symplectic involution, the invariants may augment straightforward $J$-holomorphic disk counts with correction terms arising from the formalism of Fukaya $A_\infty$-algebras and bounding cochains. These correction terms are shown in fact to be non-trivial for many invariants. Moreover, examples of non-vanishing mixed disk and sphere invariants are obtained. We characterize a class of open Gromov-Witten invariants, called basic, which coincide with straightforward counts of $J$-holomorphic disks. Basic invariants for the Chiang Lagrangian are computed using the theory of axial disks developed by Evans-Lekili and Smith in the context of Floer cohomology. The open WDVV equations give recursive relations which determine all invariants from the basic ones. The denominators of all invariants are observed to be powers of $2$ indicating a non-trivial arithmetic structure of the open WDVV equations. The magnitude of invariants is not monotonically increasing with degree. Periodic behavior is observed with periods $8$ and $16.$