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Equality of Hori–Vafa and Lagrangian Floer superpotentials for semi-Fano toric manifolds

Prove that for any semi-Fano toric manifold X (with anticanonical divisor −KX nef), the Hori–Vafa superpotential WHV and the Lagrangian Floer superpotential WLF coincide after applying the inverse toric mirror map t(q) = ψ−1(q), i.e., establish WHV_{t(q)} = WLF_q.

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Background

The text reviews mirror constructions for toric Fano and semi-Fano varieties, defining the Hori–Vafa (HV) superpotential on the B-side and the Lagrangian Floer (LF) superpotential on the A-side, and discusses the toric mirror map ψ relating complex and Kähler moduli.

The conjecture asserts the precise equality of these two superpotentials upon substituting the inverse mirror map, aligning enumerative data from closed Gromov–Witten theory (HV) with open/Lagrangian Floer counts (LF).

References

Conjecture

Let X be a semi-Fano toric manifold, namely, its anticanonical divisor, -K_{_X} is nef. Then,

{\cal W}{{t(q)}{{\text{HV}\ =\ {\cal W}_{_q}{{\text{LF}

via the inverse mirror map t(q)=\psi{{-1}(q).

— Homological Mirror Symmetry Course at SIMIS: Introduction and Applications (2506.14779 - Pasquarella, 23 May 2025) in Section “Open Mirror Theorems,” Conjecture