Let $(X,\omega,J)$ be a Stein manifold, namely a Kähler manifold with $\omega=\sqrt{-1}\partial\bar{\partial}\phi$ for a plurisubharmonic exhaustion function $\phi$. In particular, $X$ is an exact symplectic manifold, meaning $\omega=d\lambda$, where $\lambda$ is the Liouville 1-form. All almost complex structure perturbations are assumed to agree with the fixed complex structure outside some compact set.

Given two transversely intersecting exact embedded⁶²⁶² 62 In our terminology, embedded Lagrangians are always connected, while immersed Lagrangians can have disconnected domains. Lagrangians $L,L^{\prime}$ with potential $f_{L},f_{L^{\prime}}$, namely $df_{L}=\lambda|_{L}$ and $df_{L^{\prime}}=\lambda|_{L^{\prime}}$, and some extra brane data, one can associate an algebraic invariant called the Floer cohomology. A general feature of Floer theory, is that the constructions depend on many auxiliary choices, but the invariants depend on only a small number of data, and should always be invariant under global Hamiltonian isotopies.

We assume $c_{1}(TX)=0$ and let $\Omega$ be a complex volume form on $TX$. We shall always assume the Lagrangians to be graded, namely the phase function $\theta=\arg\Omega|_{L}:L\to S^{1}$ lifts to a real valued function. The grading is part of the brane data. Working first with $\mathbb{Z}_{2}$ coefficients, the Floer cohomology can be defined as the cohomology of a complex $(CF^{*}(L,L^{\prime}),d)$. Here $CF^{*}(L,L^{\prime})$ is generated by the transverse intersection points $p\in L\cap L^{\prime}$, whose degrees are given by

$$\mu_{L,L^{\prime}}(p)=\frac{1}{\pi}(\sum\phi_{i}+\theta_{L}(p)-\theta_{L^{\prime}}(p))$$

(63)

where we put the tangent planes of $L,L^{\prime}$ inside $T_{p}X\simeq\mathbb{C}^{n}$ into the standard form

$$T_{p}L=\mathbb{R}^{n}\subset\mathbb{C}^{n},\quad T_{p}L^{\prime}=(e^{i\phi_{1}},\ldots e^{i\phi_{n}})\mathbb{R}^{n}\subset\mathbb{C}^{n},\quad 0<\phi_{i}<\pi.$$

Notice if we reverse the role of $L,L^{\prime}$, then we can regard $p\in CF^{*}(L^{\prime},L)$, but this affects the degree by $\mu_{L,L^{\prime}}(p)=n-\mu_{L^{\prime},L}(p)$. For alternative formulations of the degree in terms of Lagrangian Grassmannians, see [68].

We consider the moduli space $\mathcal{M}(p,q;J,[u])$ of finite energy holomorphic strips with ends at $p,q$ and boundary on $L,L^{\prime}$, in the homotopy class $[u]\in\pi_{2}(X,L\cup L^{\prime})$:

$$\begin{split}&u:\Sigma=\mathbb{R}\times[0,1]\to X,\quad u(s,0)\in L,\quad u(s,1)\in L^{\prime},\quad\lim_{s\to-\infty}u=p,\quad\lim_{s\to+\infty}u=q,\\ &\partial_{s}u+J(t,u)\partial_{t}u=0,\quad E(u)=\int u^{*}\omega=\iint|\partial u/\partial s|^{2}dsdt<\infty.\end{split}$$

Index theory of the Cauchy-Riemann operator with Lagrangian boundary conditions implies this moduli space has virtual dimension $\deg q-\deg p$. The holomorphic strip equation is invariant under domain translation in the $\mathbb{R}$ direction. Using generic domain dependent almost complex structures which are fixed outside a large compact set, one can achieve suitable transversality on the moduli spaces, and in particular $\mathcal{M}(p,q;J,[u])/\mathbb{R}$ are isolated points for $\deg q-\deg p=1$. A key advantage of the exact setting is that the energy can be computed a priori by the topological formula:

$$\int_{\Sigma}u^{*}\omega=\int_{\partial\Sigma}\lambda=\int_{-\infty}^{\infty}df_{L}-\int_{-\infty}^{\infty}df_{L^{\prime}}=(f_{L}-f_{L^{\prime}})(q)-(f_{L}-f_{L^{\prime}})(p).$$

(64)

By Gromov compactness, the number of isolated points is finite, and only finitely many homotopy classes $[u]$ admit holomorphic strips. To save some notations, we sometimes write $\mathcal{M}(p,q)=\bigcup_{[u]\in\pi_{2}(M,L\cup L^{\prime})}\mathcal{M}(p,q;J,[u])$.

The Floer differential $d:CF^{k}(L,L^{\prime})\to CF^{k+1}(L,L^{\prime})$ is $dp=\sum n_{p,q}q,$ where $n_{p,q}$ is the mod 2 count of $\mathcal{M}(p,q)/\mathbb{R}$. The key fact of Floer theory is that $d^{2}=0$. For this, one considers the holomorphic strips between $p,r$ with $\deg r-\deg p=2$, modulo the translation invariance $\mathbb{R}$ direction. This moduli space $\mathcal{M}(p,r)/\mathbb{R}$ is one-dimensional. Generally in Floer theory, the boundary of the compactified moduli spaces comes from disc breaking and disc and sphere bubbling. The latter is ruled out for energy reasons by the exactness assumption, and the former gives

$$\partial(\overline{\mathcal{M}(p,r)/\mathbb{R}})=\bigcup_{q}\mathcal{M}(p,q)/\mathbb{R}\times\mathcal{M}(q,r)/\mathbb{R}.$$

In terms of mod 2 counts, $\sum_{q}n_{p,q}n_{q,r}=0$, namely $d^{2}=0$. This fact allows one to take the cohomology, which is $HF^{*}(L,L^{\prime})$. Although suppressed in this notation, the homotopy classes of discs are additive under disc breaking. This fact allows one to introduce some extra weighting factors involving energy and holonomy of local systems.

The general strategy to show the Floer cohomology is independent of the choices of almost complex structures and Hamiltonian perturbations, is to consider continuity equations, whose counts define chain maps at the level of $CF^{*}$, so descend to comparison maps between Floer cohomologies defined by different auxiliary data (cf. Auroux [8, section 1.5]).

Floer cohomology admits rich algebraic structures, but the deeper structure is better set up at the chain level $CF^{*}(L,L^{\prime})$. We temporarily avoid the issue of signs and self Floer cohomology. The Fukaya category can be seen as the generalization of Floer cohomology in two directions:

• •

  We allow the interplay of many (transverse) Lagrangians. Each Lagrangian is labelled by an object in the Fukaya category. This labelling is the main difference between an algebra and a category.

• •

  The holomorphic strips are replaced by holomorphic polygons, with boundary segments mapped to a clockwise ordered sequence of at least three Lagrangians $L_{0},L_{1},\ldots L_{k}$, and clockwise ordered boundary marked points $x_{0},x_{1},\ldots x_{k}$ mapped to the Lagrangian intersection points $q\in L_{0}\cap L_{k},p_{1}\in L_{0}\cap L_{1},\ldots p_{k}\in L_{k-1}\cap L_{k}$. We distinguish $x_{0}$ as the output, and regard $x_{1},\ldots x_{k}$ as inputs. The marked points $x_{0},x_{1},x_{2}$ on the boundary of the domain disc are fixed, while the other $k-2$ marked points are allowed to move freely preserving their cyclic ordering.

  The moduli spaces of such polygons (with suitably domain dependent perturbations) are denoted as $\mathcal{M}(p_{1},\ldots p_{k},q)$. Moduli spaces with at least three marked points do not have the domain translation invariance, so there is no need to divide by $\mathbb{R}$.

The energy formula (64) generalizes to the holomorphic polygon case:

$$\int_{\Sigma}u^{*}\omega=(f_{L_{0}}-f_{L_{k}})(q)-\sum_{1}^{k}(f_{L_{i-1}}-f_{L_{i}})(p_{i}).$$

(66)

The virtual dimension formula is

$$\text{vdim}\mathcal{M}(p_{1},\ldots p_{k},q)=\deg q-\sum_{1}^{k}\deg p_{i}+k-2.$$

(67)

Here $\deg q-\sum_{1}^{k}\deg p_{i}$ comes from the index theory of the Cauchy-Riemann operator, and $k-2$ comes from the freedom to move the marked points on the boundary. Under suitable domain dependent perturbation schemes, in this exact setting one can ensure transversality, so that the moduli space is smooth. For setting up the Fukaya category, the zero dimensional moduli spaces are particularly important, since counting points give rise to operations, and 1-dimensional moduli spaces are important for producing $A_{\infty}$-relations. In the main text, we have also given considerable attention to $(n-1)$-dimensional moduli spaces, since these are relevant for producing bordism currents.

Within the exact setting, disc and sphere bubbling is impossible. After compactification, the moduli space of holomorphic polygons can have two kinds of boundaries, due to two kinds of disc breaking:

• •

  (Disc breaking at the corners) The disc may break at $L_{i}\cap L_{i+1}$. The polygons near the breaking limit are obtained from gluing polygons with corners mapped to $p_{1},\ldots,p_{l-1},r,p_{l+1},\ldots,q$, and strips with boundary on $L_{l-1},L_{l}$ and two ends mapped to $p_{l},r$. (Of course, disc breaking can also happen at the outgoing corner $q$.)

• •

  (Disc splitting at the edges) When there are at least 4 Lagrangians, the domain disc can split into two discs with $l+1\geq 3$ and $k+1-l\geq 3$ marked points. The edges of one disc map to $L_{0},\ldots L_{j},L_{j+l},\ldots L_{k}$, with cyclically marked points mapping to $p_{1},\ldots p_{j},r\in L_{j}\cap L_{l+j},p_{l+j+1},\ldots p_{k}$ and $q$. The edges of the other disc map to $L_{j},\ldots L_{l+j}$, with marked points mapping to $p_{j+1},\ldots p_{l+j}$ and $r\in L_{j}\cap L_{l+j}$.

When disc breaking and disc splitting are taken into account, the moduli spaces can be compactified into $\overline{\mathcal{M}}(p_{1},\ldots p_{k},q)$. We then have

$$\begin{split}\partial\overline{\mathcal{M}}(p_{1},\ldots p_{k},q)=&\bigcup\overline{\mathcal{M}(p_{l},r)/\mathbb{R}}\times\overline{\mathcal{M}}(p_{1},\ldots p_{l-1},r,p_{l+1},\ldots p_{k},q)\\ &\cup\bigcup\overline{\mathcal{M}}(p_{1},\ldots,p_{j},r,p_{l+j+1},\ldots p_{k},q)\times\overline{\mathcal{M}}(p_{j+1},\ldots p_{l+j},r).\end{split}$$

(68)

In particular, the (virtual) dimensions of both sides are equal, which constrains $\deg r$.

The $A_{\infty}$-structure is the algebraization of the disc breaking/splitting phenomenon. It consists of multilinear maps

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

satisfying the $A_{\infty}$-relation

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

Here $m_{1}$ is the degree one Floer differential $d$, and for $k\geq 2$ the operation $m_{k}$ is defined by counting holomorphic polygons in moduli spaces of virtual dimension zero,

$$m_{k}(p_{k},\ldots,p_{1})=\sum_{q}\#\mathcal{M}(p_{1},\ldots p_{k},q)q.$$

Virtual dimension zero requires $\deg q=\sum_{1}^{k}\deg p_{i}+2-k$, which explains the degree of $m_{k}$. The $A_{\infty}$-relation is the direct translation of (68), with disc breaking at corners contributing the $m_{1}$ terms, and disc splitting contributing the other terms.

The first few $A_{\infty}$-relations have clear geometric meanings:

• •

  The Floer differential squares to zero.

• •

  The Floer product $m_{2}$ satisfies the Leibniz rule. As such $m_{2}$ descends to a product structure on the mod 2 coefficient Floer cohomology $HF^{*}(L_{1},L_{2})\otimes HF^{*}(L_{0},L_{1})\to HF^{*}(L_{0},L_{2})$.

• •

  The $m_{2}$ is associative up to a homotopy given by the $m_{3}$ terms. In particular the Floer product is associative on cohomology.

The higher structures naturally lead to the Fukaya category of embedded exact Lagrangians. This requires some discussion on signs, brane structures, and self Floer cohomologies.