Relative quantum cohomology of the Chiang Lagrangian

TL;DR

This work provides a complete computation of the open Gromov-Witten disk invariants for the Chiang Lagrangian $L_{\triangle}$ in $\mathbb{C}P^3$, revealing nontrivial corrections from bounding cochains since $L_{\triangle}$ is not fixed by an anti-symplectic involution. The authors identify three basic invariants that are equal to straightforward disk counts via axial-disk geometry, then use open WDVV equations and wall-crossing to recursively determine all higher invariants, both with boundary and interior constraints. They derive an explicit presentation of the small relative quantum cohomology, $QH^*(\mathbb{C}P^3,L_{\triangle}) \cong \mathbb{R}[[q^{1/4}]][x,y]/I$, with $I$ generated by concrete degree constraints, and show a surprising arithmetic structure where denominators are powers of $2$, despite potential contributions from bounding-cochain data. The analysis also demonstrates non-vanishing mixed disk–sphere invariants and a rich periodic behavior in degrees, and discusses how invariants depend on choices of left inverses to the boundary map, spin structures, and orientations. Overall, the paper advances the understanding of open-closed interactions in relative quantum cohomology for homogeneous Lagrangians and offers a framework potentially extensible to other Platonic and homogeneous Lagrangians.

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 $16.$

Figures (5)

• Figure 1: The choice of $\xi_v, \xi_f$ for the configuration $\triangle$.
• Figure 2: A Maslov 2 disk passing through $\triangle$.
• Figure 3: A Maslov 4 disk passing through $\triangle$.
• Figure 4: The triangle $u(w).$
• Figure 5: The rotations $\alpha$ and $\beta$ at $p$.

Theorems & Definitions (126)

• Theorem 1
• Theorem 2
• Corollary 1.1
• Proposition 1.2
• Corollary 1.3
• Theorem 3
• Theorem 4
• Proposition 2.1
• Proposition 2.2
• proof