Enumerative mirror correspondence between MF invariants and open GW invariants

Show that for any Fano toric manifold X_Δ whose real locus X_Δ^R has the cohomology of an n-sphere, there exists a homomorphism φ_Δ : H_2(X_Δ, X_Δ^R; Z) → G_Δ such that for all k ≥ 1 and β ∈ G_Δ, the numerical invariants N^{M_Δ}_{β,k} of the Dirac matrix factorization M_Δ satisfy N^{M_Δ}_{β,k} = ∑_{\tilde β ∈ φ_Δ^{-1}(β)} OGW^{X_Δ^R}_{\tilde β,k}, where OGW^{X_Δ^R}_{\tilde β,k} are the open Gromov–Witten invariants of X_Δ with k boundary point constraints.

Background

The paper defines numerical invariants N^{M_Δ}_{β,k} from point-like bounding cochains of spherical, normed Calabi–Yau objects in MF(W_Δ), motivated as mirrors of open Gromov–Witten counts with boundary constraints. The group G_Δ underlies the completed group ring R_Δ.

They prove the conjecture for Δ = Δ^1 (CP^1, RP^1) and provide computer evidence for Δ = Δ^3 (CP^3, RP^3). A general proof would establish a precise enumerative mirror symmetry linking algebraic invariants from matrix factorizations with open Gromov–Witten invariants for real Lagrangian spheres in toric Fano manifolds.