We look at how one can construct from the data of a dimer model a Lagrangian submanifold in \((\mathbb {C}^*)^n\) whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori \(L_{T^2}\) in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair \((\mathbb {CP}^2{\setminus } E, W), {\check{X}}_{9111}\). We find a symplectomorphism of \(\mathbb {CP}^2{\setminus } E\) interchanging \(L_{T^2}\) and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on \({\check{X}}_{9111}\).

我们看一看如何从二聚体模型的数据构造\（（\ mathbb {C} ^ *）^ n \）中的拉格朗日子流形，其估值预测近似于热带超曲面。二聚体的每个面都对应于一个拉格朗日圆盘，在我们的热带拉格朗日子流形上具有边界，形成一个拉格朗日突变种子。使用此函数，我们发现热带拉格朗日花托\（L_ {T ^ 2} \）与复曲面del-Pezzo的光滑反典除数的补码互补，后者的壁交转换与单调SYZ纤维的壁交转换相同。为镜像对\（（\ mathbb {CP} ^ 2 {\ setminus} E，W），{\ check {X}} _ {9111} \）给出一个示例。我们发现\（\ mathbb {CP} ^ 2 {\ setminus} E \）的辛同构交换\（L_ {T ^ 2} \）和SYZ纤维。提供的证据表明，这种辛同构性是\（{\ check {X}} _ {9111} \）上的光纤傅里叶-穆凯变换的镜像。