How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?

Question

For Calabi-Yau variety $X$ which is a complete intersection $$ f_1=f_2=\ldots=f_r=0 $$ in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the following way. One can consider $X$ as zeroes of a section of vector bundle $$ {\mathcal O}_{{\mathbb P }^n}(d_1)\oplus\ldots\oplus{\mathcal O}_{{\mathbb P }^n}(d_r) $$ with $d_i=\mathrm{deg}\,(f_i)$. This section defines a function $W$ on the (total space of) dual bundle $$ {\mathcal X}={\mathrm{Total\ space\ of\ }}{\mathcal O}_{{\mathbb P }^n}(-d_1)\oplus\ldots\oplus{\mathcal O}_{{\mathbb P }^n}(-d_r). $$ One can check that $$ W=y_1f_1+\ldots+y_rf_r $$ for $y_i$ defined in some way. One has Landau-Ginzburg model: $$ W:{\mathcal X} \to {\mathbb C}. $$

Question: Is it possible to construct another Landau-Ginsburg model $$ W':{\mathcal X}'\to {\mathbb C} $$ such that (1) ${\mathcal X}'$ is a toric variety (as well as ${\mathcal X}$) and (2) one can not get ${\mathcal X}'$ from $\mathcal X$ by a sequence of toric blow-ups and contractions.

Comment 1. (a) It is not necessary that $\mathcal X'$ is also a total space of some vetor bundle, (b) it is not necessary that ${\mathrm{dim}}\,{\mathcal X}={\mathrm{dim}}\,{\mathcal X}'$.

Comment 2. In other words, I am looking for a toric variety $\mathcal X'$ such that the support of its fan (it should be a cone and I would like to have this cone simplicial) would be different from support of the fan for $\mathcal X$.

Answer 1

Yes. One way outlined in the work of Abouzaid-Auroux-Katzarkov's work (http://arxiv.org/pdf/1205.0053.pdf) is to look at the space $\mathbb{P}^n\times\mathbb{C}^r$ blown up along the codimension 2 subvarieties $H_i\times\mathbb{C}^{r-1}$ with $i=1,\cdot\cdot\cdot,r$. Here $r$ is the number of polynomials defining you complete intersection Calabi-Yau subvariety $X\subset\mathbb{P}^n$ and $X=\cap_{i=1}^rH_i$. We denote the resulting space after the blow ups by $Y$.

The beautiful thing is that after taking away an anticanonical divisor $D\cong-K_Y$, $Y\setminus D$ admits a (possibly piecewise smooth) Lagrangian torus fibration $\pi:Y\rightarrow B$. Then one can apply the SYZ procedure to construct the mirror of $(Y,D)$. The mirror manifold $Y^\vee\setminus D^\vee$ is constructed by studying the wall-crossing structure of the genus 0 Maslov index 0 open Gromov-Witten invariants associated to Lagrangian torus fibers of $\pi$ in $Y\setminus D$. One can also construct a superpotential $W^\vee:Y^\vee\setminus D^\vee\rightarrow\mathbb{K}$ by counting Maslov index 2 discs in $Y$, i.e. holomorphic discs with intersection number 1 with the divisor $D\subset Y$. In the above, $D^\vee\subset Y^\vee$ is an anticanonical divisor mirror to $D$.

The above process can be reversed to passing from $Y^\vee$ to $Y$, now the anticanonical divisor $D^\vee$ gives rise to the superpotential $W:Y\setminus D\rightarrow\mathbb{K}$. In many cases, $W$ admits an analytic continuation to the whole space $Y$, from this we get a Landau-Ginzburg model $(Y,W)$.

If you want $Y\setminus D$ to be a toric variety, then up to my knowledge you should impose some assumption on $X$ and $Y\setminus D$ and your local Calabi-Yau $\mathcal{X}$ should only differ by some anticanonical divisor. I see this from explicit computations for local complete intersections.

Another way, which is less geometric and more combinatorial is outlined in the work of Gross-Katzarkov-Ruddat (http://arxiv.org/pdf/1202.4042v3.pdf). The two methods described here are essentially the same. However, the geometric approach, namely the SYZ construction, requires a lot of technical assumptions, otherwise the open Gromov-Witten invariants are not well-defined or not computable by usual methods. The combinatorial approach is very easy to apply and less involved, however, you loose geometric understanding of mirror symmetry when using this.

The geometry of $X$ is equivalent to $(Y,W)$ basically because Knorrer periiodicity is expected here: the complete intersection $X$ is actually the critical locus of $W$. To give you a very simple example, consider $\mathbb{C}^2$ with superpotential $xy$, in this case your complete intersection $X$ is a single point in $\mathbb{P}^1$. Note that in the SYZ approach we actually treat $X$ and $(Y,W)$ as A-models. In this case, there is also an expected version of symplectic Knorrer periodicity, which is a work in progress of Abouzaid-Auroux, which assertes that the Fukaya category $\mathcal{F}(X)$ should be equivalent to the twisted partially wrapped Fukaya category $\mathcal{F}_s(Y,W)$. This latter category actually arises from Auroux's work on the symplectic interpretation of Heegard-Floer cohomology:https://math.berkeley.edu/~auroux/papers/fuksymg.pdf.