Toric Fano Landau–Ginzburg mirror conjecture

Let ${\mathbb P}_\Sigma$ be an $n$-dimensional toric Fano manifold with fan $\Sigma$, and let $v_1,\dots,v_d$ be the generators of its one-dimensional cones. For an element $\varphi$ of the complexified Kähler cone, define the Laurent polynomial

$$f_\varphi(z)=\sum_{i=1}^d\exp(\varphi(v_i))^{-1}z^{v_i}$$

on $({\mathbb C}^*)^n$.

Toric Fano mirror conjecture. There should exist a primitive form for $f$ given by

$$\zeta^{(0)}=\left[\frac{dz_1}{z_1}\wedge\dots\wedge\frac{dz_n}{z_n}\right].$$

Then the pair $(({\mathbb C}^*)^n,f)$ is a mirror manifold of the toric Fano manifold ${\mathbb P}_\Sigma$.

This conjecture proposes a primitive-form construction whose Frobenius structure reproduces the mirror of toric Fano quantum cohomology. The source gives no resolution or further evidence for the conjecture.