Start with dual nef partitions on $\Delta$ and $\nabla$.

Now, pick generic section $f_i \in O(2 \Delta_i)$, and $g_i \in O(2 \nabla_i)$.

Put bundle $\oplus_i O(-\Delta_i)$ on $X_\Delta$, do LG model $$ W_\Delta = \sum_i ev_{O(-\Delta_i)} (f_i) $$

Put bundle $\oplus_i O(-\nabla_i)$ on $X_\nabla$, do LG model $$ W_\Delta = \sum_i ev_{O(-\nabla_i)} (g_i) $$

Now, there is no hope this gonna work, simply because one sides has too much selection of monomials, and the other side has the same selection of divisors.

How to get more divisors? We can do 'stacky Fano'.

For example, consider A-side, we use superpotential $$ W = z^2 + 1/z^2 $$ Then B-side, we consider compactification of $\C^*$, where at $0$ and $\infty$, we put in $\mu_2$ orbifold points.

Before we even talk about BB, we can talk about Batyrev construction, just upgrade CY hypersurface to something else.

Suppose $\Delta$ is a smooth Fano polytope, and $\nabla = \Delta^\vee$.

One side, we have $Y_\Delta = Tot(O(-\Delta)) \to X_{\Delta}$, with $W_\Delta = ev(s)$, where $s \in \Gamma(O(\Delta))$ is a section in anti-canonical bundle.

• divisors: $1$ plus vertices of $\nabla$.
• monomials: lattice points in $\Delta$.

On the other side

• divisors: $1$ plus vertices of $\Delta$. Same as lattice points in $\Delta$
• monomials: lattice points in $\nabla$. Same as $1$ plus vertices in $\nabla$.

Suppose we do geometric double cover, what we do?

• take a section $f \in \Gamma(O(2\Delta))$.
• write down an equation in the total space $Y$ of $O(\Delta)$, fiber coord $y$, where we write $ y^2 = f $
• If we want to do an LG realization, we look at the defining 'section', $y^2 - f$, of some line bundle $L$ over $Y$, and we view it as a function $W$ on $Tot_Y(L^{-1})$.

Say, in the case we have $X = \P^1$, and $f \in O(4H)$, $Y$ is $Tot(O(2H))$, $y^2 - f$ is a section of $\pi^*(O(4H)).