Hori–Vafa conjecture: hypersurface H mirror to a toric Calabi–Yau Landau–Ginzburg model

Establish that a smooth algebraic hypersurface H ⊂ (C*)^n defined by a Laurent polynomial is mirror to a toric Calabi–Yau manifold Y equipped with a toric Landau–Ginzburg superpotential W, in the sense that H corresponds to the LG model (Y, W) predicted by the Hori–Vafa construction.

Background

In constructing mirrors of hypersurfaces in algebraic tori, the text summarizes Hori–Vafa’s proposal that such hypersurfaces arise as mirrors of toric Calabi–Yau manifolds equipped with superpotentials. This connects A-side symplectic geometry to B-side Landau–Ginzburg models.

The conjecture sets the stage for subsequent categorical predictions connecting wrapped Fukaya categories to derived categories of singularities on the zero fiber Z = W−1(0).