The open Gromov–Witten comparison conjecture for Dirac matrix factorizations

Let $X_\triangle$ be a Fano toric manifold with real locus $X_\triangle^\mathbb{R}$, and suppose

$$H^*(X_\triangle^\mathbb{R};\mathbb{R})=H^*(S^n;\mathbb{R}).$$

Let $M_\triangle$ be the associated Dirac matrix factorization, let $N^{M_\triangle}_{\beta,k}$ denote its numerical invariants, let $G_\triangle$ be the group underlying the completed group ring $R_\triangle$, and let $\operatorname{OGW}^{X_\triangle^\mathbb{R}}_{\widetilde\beta,k}$ denote the open Gromov–Witten invariant with class $\widetilde\beta$ and $k$ point constraints.

Open Gromov–Witten comparison conjecture. There is a homomorphism

$$\varphi_\triangle:H_2(X_\triangle,X_\triangle^\mathbb{R};\mathbb{Z})\to G_\triangle$$

such that, for all $k\geq 1$ and $\beta\in G_\triangle$,

$$N^{M_\triangle}_{\beta,k}=\sum_{\widetilde\beta\in\varphi_\triangle^{-1}(\beta)}\operatorname{OGW}^{X_\triangle^\mathbb{R}}_{\widetilde\beta,k}.$$

This conjecture proposes that the numerical invariants of the Dirac matrix factorization recover open Gromov–Witten invariants of the real toric locus after grouping relative homology classes through $\varphi_\triangle$. The supplied text presents it as the main motivation for the preceding constructions and gives no resolution.