Relative Fukaya pre-category interface for the toric mirror

Let $N$ be the fan lattice and take the mirror torus $T^{\vee }=\mathrm{Hom}(N,{\mathbb{C}}^{\ast })\cong ({\mathbb{C}}^{\ast }{)}^n$, the potential $W_{\mathrm{\Sigma }}$, and a selected smooth zero fibre $M_0=W_{\mathrm{\Sigma }}^{-1}(0)$ from the toric fan and Laurent-potential datum. Fix an ordered integral basis $(e_1,\dots ,e_n)$ of $N$ and its dual basis of $M$ as part of the symplectic datum. Set $z_j=t(e_j)$ for $t\in T^{\vee }$ and write $z_j=\exp (q_j+i{\theta }_j)$, with $q\in M_{\mathbb{R}}$ and $\theta \in M_{\mathbb{R}}\mathrm{/}2\pi M$, and equip $T^{\vee }$ with the exact symplectic data $\lambda ={\sum }_j{q}_{j}d{\theta }_j$, $\omega =d\lambda ={\sum }_{j}d{q}_j\wedge d{\theta }_j$. The chosen dual basis declares a Euclidean inner product on $M_{\mathbb{R}}$ and the displayed sums use that inner product. A different basis with a newly declared identity Gram matrix generally defines different symplectic forms. When merely changing coordinates on this fixed datum, transform the inner-product matrix as a tensor and retain the same $\lambda ,\omega$; no identification of $N$ with $M$ is implicit. A declared positive rescaling must rescale both forms. For a source construction using a deformed potential, additionally specify its smooth map $W:T^{\vee }\to \mathbb{C}$, its fibre $M_0=W^{-1}(0)$ and an almost-complex structure $J$ tamed by $\omega$ for which $W$ is holomorphic near $M_0$; for the Laurent polynomial take its ordinary complex structure. Require $W$ to be a submersion near $M_0$ and its regular fibres to be symplectic. On that locus use the symplectic connection $H=(\ker dW{)}^{{\perp }_{\omega }}$ and require its lifts to exist throughout each collar segment that is selected. For a smooth embedded segment $\gamma :[0,ϵ)\to \mathbb{C}$, with $\gamma (0)=0$ and nonzero initial derivative, its parallel transport means the flow solving $\dot{x}(t)=(dW{\mathrm{\mid }}_{H_{x(t)}}{)}^{-1}\dot{\gamma }(t)$. Thus the primitive and transport used below are specified data; smoothness of $M_0$ alone supplies neither. The source relative Fukaya pre-category has as objects compact Lagrangian branes $L\subset T^{\vee }$ with boundary in $M_0$, a chosen grading and spin structure, a specified smooth primitive $f_L$ with $\lambda {\mathrm{\mid }}_L=d{f}_L$, and a collar near $\mathrm{\partial }L$ obtained by parallel transport along a segment in the base of the potential. For a specified object class and coefficient ring, morphism complexes use transverse interior intersections. An allowed sequence must have its collar segments in the open left half-plane with distinct counterclockwise ordered initial directions, and must have regularized, compactified polygon moduli for every subsequence. Higher operations are supplied only on those allowed sequences. This pre-category convention allows some sequences to have no operation defined; it is not silently replaced by an ordinary category.

The toric sub-pre-category used in the theorem further chooses a smoothing $Q$ of the moment polytope component in the complement of the logarithmic amoeba of $M_0$ and restricts to admissible Lagrangian sections over $Q$ of $\mathrm{Log}:T^{\vee }\to M_{\mathbb{R}}=\mathrm{Hom}(N,\mathbb{R})$, where $\mathrm{Log}(t)(a)=\log \mathrm{\mid }t(a)\mathrm{\mid }$. Their boundaries lie in $M_0$. The choice of $Q$, the existence of its sections, and the full analytic A-infinity construction are additional inputs, not consequences of the fan definition. The compact multi-object brane interface does not itself prove this relative-boundary construction.

Toric fan and Laurent potential data for the line-bundle case

Landau–Ginzburg pair and distinct categories

Blocked construction interface for multi-object Fukaya categories and polygon morphisms