Cohomology of the Dirac matrix factorization matches the real locus

Prove that for each combinatorially relatively spin Delzant polytope Δ such that the associated toric manifold X_Δ is Fano, the cohomology of the degree-zero part of the endomorphism algebra of the Dirac matrix factorization M_Δ satisfies H^*(End(M_Δ)_0) ≅ H^*(X_Δ^R; (R_Δ)_0).

Background

The authors construct an object M_Δ in MF(W_Δ), called the Dirac matrix factorization, under a combinatorial relative spin condition on Δ. This object is expected to mirror the real locus X_ΔR of the toric manifold X_Δ.

They verify the conjectural cohomology identification for Δn with n odd (in particular RPn ⊂ CPn) and conjecture it holds more generally for combinatorially relatively spin Δ when X_Δ is Fano. Establishing this identification would strengthen the categorical–geometric mirror correspondence at the cohomological level.

References

Conjecture. For each combinatorially spin Delzant polytope \triangle such that X_\triangle is Fano, we have

H*(\End(M_{\triangle})_0)\cong H*(X_\triangleR; (R_{\triangle})_0).

— Numerical invariants of normed matrix factorizations (2412.04437 - Sela et al., 5 Dec 2024) in Construction from toric geometry (Conjecture \ref{conj:cohom})