To go beyond mod 2 coefficients, we need to orient moduli spaces. Good references can be found in Seidel’s book [69] and Abouzaid [3, Appendix]. All Lagrangians are assumed to be graded, with second Stiefel-Whitney class equal to the restriction of a fixed class in $H^{2}(X,\mathbb{Z}/2)$, and we equip the Lagrangians with relative spin structures. At any transverse Lagrangian intersection point $p\in L_{+}\cap L_{-}$, there is a unique up to homotopy path $\Lambda_{p}$ of Lagrangian planes in $T_{p}X\simeq\mathbb{C}^{n}$ with graded lift interpolating $TL_{+}$ and $TL_{-}$. We fix a relative spin structure on $\Lambda_{p}$, compatible with the relative spin structure on $L_{\pm}$. We can associate a vector space $o_{p}$ as the determinant line of the Cauchy-Riemann operator $D_{p}$ on the upper half plane with boundary data $\Lambda_{p}$. The dual of $o_{p}$ is denoted $o_{p}^{\vee}$, namely $o_{p}\otimes o_{p}^{\vee}\simeq\mathbb{R}$ canonically. The orientation line $|o_{y}|$ is the free abelian group generated by the two possible orientations of $o_{y}$ with the relation that their sum vanishes. Furthermore, we equip the Lagrangians $L$ with (rank one) local systems $E$, and write the Floer cochain complex as the graded vector space

$$CF^{*}(L,L^{\prime})=\oplus_{p\in L\cap L^{\prime}}\text{Hom}(E|_{p},E^{\prime}|_{p})\otimes|o_{p}|.$$

Given a holomorphic polygon $u:\Sigma\to X$, with inputs $x_{1},\ldots x_{k}$ and output $x_{0}$ mapping to $p_{1},\ldots p_{k}$ and $q$, the det line of the linearized Cauchy-Riemann operator $D_{u}$ can be computed from gluing kernel and cokernels:

$$\det(D_{u}\#D_{p_{k}}\ldots\#D_{p_{1}})\simeq\det(D_{u})\otimes o_{p_{k}}\ldots\otimes o_{p_{1}}.$$

The role of the relative spin structure, is to specify a homotopically unique choice of isotopy between the glued operator $D_{u}\#D_{x_{k}}\ldots\#D_{x_{1}}$ and $D_{x_{0}}$ (i.e. an isotopy between Lagrangian boundary conditions), hence a preferred isomorphism

$$\det D_{u}\simeq o_{q}\otimes o_{p_{1}}^{\vee}\otimes\ldots o_{p_{k}}^{\vee}.$$

The tangent space of the moduli space of holomorphic polygons $\mathcal{M}(p_{1},\ldots p_{k},q)$ involves not only the linearized Cauchy-Riemann operator, but also the variation of the complex structure of the domain of the polygon, controlled by the Stasheff associahedron $\overline{\mathcal{R}}_{k+1}$. Denote $\lambda^{top}(V)$ as the top wedge product of a vector space $V$. Then there are preferred isomorphisms depending on the relative spin structure choice

$$\lambda^{top}(T\mathcal{M}(p_{1},\ldots p_{k},q))\simeq\lambda^{top}(\mathcal{R}_{k+1})\otimes o_{q}\otimes o_{p_{1}}^{\vee}\otimes\ldots o_{p_{k}}^{\vee}.$$

(69)

For $k\geq 2$, when the moduli spaces are zero dimensional, so carry canonical orientations, then a universal orientation choice for $\lambda^{top}(\mathcal{R}_{k+1})$ determines an operator

$$|c_{u}|:|o_{p_{k}}|\otimes\ldots\otimes|o_{p_{1}}|\to|o_{q}|.$$

In our degree conventions the corners $x_{0},x_{1},x_{2},\ldots x_{k}$ on the domain disc boundary are ordered clockwise, so a natural orientation on the Stasheff associahedron can be obtained by fixing $x_{0},x_{1},x_{2}$ and allowing the other corner points to move in the clockwise orientation. The parallel transports along the local systems contribute another factor

$$\Hom(E_{k-1},E_{k})|_{p_{k}}\otimes\ldots\Hom(E_{1},E_{2})|_{p_{2}}\otimes\Hom(E_{0},E_{1})|_{p_{1}}\to\Hom(E_{0},E_{k})|_{q}.$$

Each pseudoholomorphic polygon contributes to the operation

$$m_{k}:CF^{*}(L_{k-1},L_{k})\otimes\ldots CF^{*}(L_{0},L_{1})\to CF^{*}(L_{0},L_{k})[2-k]$$

via the tensor product of the orientation factor $|c_{u}|$ and the local system factor, multiplied by another sign factor depending only on the degrees (cf. [69, equation (12.24)])

$$(-1)^{\deg p_{1}+2\deg p_{2}+\ldots k\deg p_{k}}.$$

In the case of holomorphic strips, we have a natural isomorphism

$$T\mathcal{M}(p,q)=\mathbb{R}(-\partial_{s})\oplus T(\mathcal{M}(p,q)/\mathbb{R}).$$

(70)

where $-\partial_{s}$ is the translation vector field pointing towards the input point. When $\mathcal{M}(p,q)/\mathbb{R}$ consists of isolated points, it carries canonical orientations, whence by (69) we obtain

$$|c_{u}|:|o_{p}|\to|o_{q}|.$$

The local system parallel transport produces another factor

$$\Hom(E_{0},E_{1})|_{p}\to\Hom(E_{0},E_{1})|_{q}.$$

Each pseudoholomorphic strip contributes to the Floer differential

$$d:CF^{*}(L_{0},L_{1})\to CF^{*+1}(L_{0},L_{1})$$

by the product of these two factors. We write

$$m_{1}:CF^{*}(L_{0},L_{1})\to CF^{*+1}(L_{0},L_{1}),\quad m_{1}=(-1)^{\deg p_{1}}d.$$

When the signs and local system weighting factors are taken into account, the $A_{\infty}$-relation reads

$$\sum_{l=1}^{k}\sum_{j=0}^{k-l}(-1)^{\dagger}m_{k+1-l}(p_{k},\ldots,p_{j+l+1},m_{l}(p_{j+l},\ldots p_{j+1}),p_{j},\ldots p_{1})=0.$$

(71)

where $\dagger=j+\deg p_{1}+\ldots+\deg p_{j}$. The Fukaya category for the compact embedded Lagrangians comprises of the following data:

• •

  The objects are embedded Lagrangians (with additional brane data, such as grading, Lagrangian potential, orientation, relative spin structure, and local system).

• •

  The morphisms $\Hom^{*}(L,L^{\prime})$ are the vector spaces $CF^{*}(L,L^{\prime})$ (where $L$ can coincide with $L^{\prime}$).

• •

  The $A_{\infty}$-composition maps are the multilinear maps $m_{k}$ satisfying the $A_{\infty}$ relations.

The Fukaya category is an example of an $A_{\infty}$-category.

In particular, the Floer differential squares to zero, so we can define the Floer cohomology groups $HF^{*}(L,L^{\prime})$ for embedded exact Lagrangian branes. The Floer product on cohomology is given by

$$[p_{2}]\circ[p_{1}]=(-1)^{\deg p_{1}}m_{2}(p_{2},p_{1})$$

which is associative.