Directed and wrapped Fukaya category interfaces

Fix a geometric realization $(M,\lambda ,w)$ with ordered exact vanishing cycles $L_1,\dots ,L_r$ as in the Weinstein Lefschetz fibration interface. Choose a coefficient field $k$, gradings and relative Spin structures as in the compact-brane interface, and finite-rank $k$-local systems on the cycles. The directed Fukaya interface specifies $\mathrm{hom}(L_i,L_j)=0$ for $i>j$, $\mathrm{hom}(L_i,L_i)=k{e}_i$ in degree zero, and a chosen transverse Floer cochain complex for $i<j$. For each increasing sequence $i_0<\cdots <i_d$ it specifies maps ${\mu }^d:\mathrm{hom}(L_{i_{d-1}},L_{i_d})\otimes \cdots \otimes \mathrm{hom}(L_{i_0},L_{i_1})\to \mathrm{hom}(L_{i_0},L_{i_d})$ of degree $2-d$, and the strict-unit extensions involving the $e_i$. Construction of these maps, compactifications, orientations and the signed A-infinity equations remains required; the ordered cycles alone supply none of these assertions.

For the wrapped interface, let $(M,\lambda )$ be a Weinstein domain of real dimension $2n$. Put $Y=\mathrm{\partial }M$, $\alpha =\lambda {\mathrm{\mid }}_Y$, and define the completion $\widehat{M}=M{\cup }_Y([1,\mathrm{\infty })\times Y)$ with primitive $r\alpha$ on the attached end. Here $\alpha \wedge (d\alpha {)}^{n-1}$ is nowhere zero. The Reeb vector field $R_{\alpha }$ is specified by $\alpha (R_{\alpha })=1$ and $d\alpha (R_{\alpha },-)=0$. A cylindrical brane has a proper smooth embedded oriented Lagrangian $L\subset \widehat{M}$ and a smooth function $f_L:L\to \mathbb{R}$ with $\lambda {\mathrm{\mid }}_L=d{f}_L$, such that outside a compact set $L=[r_L,\mathrm{\infty })\times {\mathrm{\Lambda }}_L$, where ${\mathrm{\Lambda }}_L\subset Y$ is a closed embedded $(n-1)$-manifold with $\alpha {\mathrm{\mid }}_{T{\mathrm{\Lambda }}_L}=0$. Thus $f_L$ is constant on each end component. It also has a finite-rank $k$-local system, a chosen relative Spin structure (a Spin structure on $TL\oplus B{\mathrm{\mid }}_L$ for a fixed oriented background bundle $B$ with $w_2(B){\mathrm{\mid }}_L=w_2(TL)$), and a grading lift of its Gauss map to a specified Maslov covering of the ambient Lagrangian Grassmannian bundle. Integer gradings require a covering with deck group $\mathbb{Z}$ and the chosen lift; their existence is part of the data. The compact-brane definition does not supply this noncompact object class.

Choose smooth Hamiltonians $H_j:[0,1]\times \widehat{M}\to \mathbb{R}$, $j\ge 1$, with ${\iota }_{X_{H_j}}d\lambda =-d{H}_j$, linear of slope $a_j$ in $r$ for large $r$, where $a_j\to +\mathrm{\infty }$. For each selected ordered brane pair require these slopes to avoid its Reeb-chord lengths and choose compact perturbations so the time-one chords are nondegenerate. Require a finite chord set ${\mathcal{X}}_j(L_0,L_1)$ at each level; simultaneous choices and the analytic no-escape estimates are construction data. With $x(0)\in L_0$, $x(1)\in L_1$ and $\dot{x}=X_{H_j}(t,x)$, the level-$j$ cochain module is $C{F}_j(L_0,L_1)={⨁}_{x\in {\mathcal{X}}_j(L_0,L_1)}{o}_x{\otimes }_k{\mathrm{Hom}}_k(E_{0,x(0)},E_{1,x(1)})\mathrm{.}$ The graded orientation line $o_x$ has the degree fixed by the two grading lifts; its gluing signs are fixed by the relative Spin structures. Choose compatible cylindrical almost-complex structures and strip data to specify a degree-one differential $d_j$ with $d_j^2=0$ and degree-zero continuation chain maps ${\kappa }_j:C{F}_j\to C{F}_{j+1}$. Composites of the consecutive ${\kappa }_j$ define the sequential diagram; any additional geometric continuation maps must be accompanied by the corresponding chain homotopies and higher coherences.

Let $C={⨁}_{j\ge 1}C{F}_j$ and let $T:C\to C$ be ${\kappa }_j$ on its $j$th summand, valued in the $(j+1)$st. A wrapped cochain model is the telescope $CW(L_0,L_1)=\mathrm{Cone}(1-T:C\to C)$, with $\mathrm{Cone}(f{)}^r=C^r\oplus C^{r+1}$ and differential $(b,a)\mapsto (db+fa,-da)$. Its target cohomology is for the induced continuation maps. The wrapped Fukaya interface further specifies maps ${\mu }^d:CW(L_{d-1},L_d)\otimes \cdots \otimes CW(L_0,L_1)\to CW(L_0,L_d)$ of degree $2-d$, with ${\mu }^1$ this differential, units and coherent continuation components. In suspended degree $\mathrm{\mid }x\mathrm{\mid }-1$, their bar coderivation must have square zero. These are typed data and required identities, not an existence theorem: compactness, regularization, finite counts, coherent gluing and independence comparisons must still be supplied. A cohomological direct limit alone does not define these chain operations.

Directed objects lie in the regular fibre and use its ordering; wrapped objects lie in the completed total space. Equivalences with coherent-sheaf categories are additional results.

Weinstein Lefschetz fibration data interface

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