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.

It is desirable to take Floer cohomology of $L$ with itself. One major feature of $HF^{*}(L,L)$ is that it contains units, at least at cohomological level.

One challenge to implement self Floer cohomology is that $L$ is not transverse to itself, so the Cauchy-Riemann equation needs perturbation. There are many frameworks to address this problem, and one idea dating back to Floer is to use the Hamiltonian invariance of Floer cohomology, to think of self Floer cohomology via $HF^{*}(L,L)\simeq HF^{*}(L,\phi_{\epsilon H}(L))$ where $\phi_{\epsilon H}$ is the time one flow of the small generic Hamiltonian $\epsilon H$ [9, section 1.6]. For $\epsilon\ll 1$, the Lagrangian $\phi_{\epsilon H}(L)$ can be identified as a graph over $L$ inside $T^{*}L$, the transverse intersection $L\cap\phi_{\epsilon H}(L)$ are the critical points of $H|_{L}$, and a suitable setup of the Floer trajectories (65) can be identified as Morse flowlines of $H$. Thus $HF^{*}(L,L)$ is isomorphic to the Morse cohomology of $L$, so $HF^{*}(L,L)\simeq H^{*}(L)$. In the exact case, the ring structure on $HF^{*}(L,L)$ defined from perturbed holomorphic triangles agrees with the cup product ring structure on $H^{*}(L)$. The unit can be represented by the Morse generator of $H^{0}(L)$, or more non-perturbatively via the Piunikhin-Salamon-Schwarz map.

It takes some effort to promote the self Floer cohomology to the Fukaya category framework, and ensure the consistency in the perturbation schemes (cf. Auroux [9, section 2.1] for a sketch and Seidel [69] for details). In applications it is often more convenient to avoid Hamiltonian perturbations as much as possible.

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.

A fundamental problem of the embedded Fukaya category is that it lacks enough geometric objects. Morally, Fukaya category is a construction that inputs the symplectic geometry of Lagrangian branes, and outputs the representation theory of an $A_{\infty}$-category. Now the general feature of $A_{\infty}$-module categories is that one can take cones and idempotent summands, two properties which are useful for classifying such categories, and desirable for mirror symmetry. The problem is that cones and idempotent summands are not obviously represented by embedded Lagrangian objects under the Yoneda embedding. The common solution is to sideline this issue by the formal algebraic construction of twisted complexes and idempotent completions. This is not quite adequate for the Thomas-Yau conjecture. However, we will discuss how the introduction of immersed Lagrangian objects geometrizes the twisted complexes (cf. section 6.2). The geometric meaning of idempotents is an open problem.

The formal algebraic constructions are well explained in [9, section 3] and [73, section 4], to which we refer the reader for more details. Given objects $L_{1},\ldots L_{N}$ of the Fukaya category $\mathcal{A}$, a twisted complex $(L,b_{L})$ consists of

• •

  The formal shifted direct sum $L=\oplus_{1}^{N}L_{i}[k_{i}]$ with $k_{i}\in\mathbb{Z}$ formally keeping track of degrees (the geometric meaning of the shift $[1]$ is to add a constant $\pi$ to the Lagrangian phase, which reverses the orientation of the Lagrangian, with a corresponding twist to the spin structure),

• •

  and a strictly triangular differential $b_{L}\in\End(L)$, i.e. a collection of maps $b_{ij}\in\Hom^{k_{j}-k_{i}+1}(L_{i},L_{j})$ for $i>j$,⁶⁴⁶⁴ 64 In most symplectic references such as [9] the morphisms $b_{ij}$ go in the opposite direction $i<j$. This just amounts to reversing the ordering of $L_{1},\ldots,L_{N}$. We find our reversed convention a little more convenient for the Harder-Narasimhan decomposition.

satisfying the equation

$$\sum_{k\geq 1}m_{k}(b_{L},\ldots,b_{L})=0,\quad\emph{i.e.}$$

$$\sum_{k\geq 1}\sum_{i=i_{0}>\ldots>i_{k}=j}m_{k}(b_{i_{k-1}i_{k}},\ldots,b_{i_{0}i_{1}})=0.$$

Notice the strict triangularity implies the sum is finite. One can define morphisms between these twisted complexes, and assign $A_{\infty}$-structures to make twisted complexes into an $A_{\infty}$-category $Tw\mathcal{A}$, into which $\mathcal{A}$ naturally embeds fully faithfully. Using the $A_{\infty}$-structure, it makes sense to talk about closed morphisms and cohomologies, similar to the construction of Floer cohomology.

Given twisted complexes $A=(L,b_{L}),B=(L^{\prime},b_{L^{\prime}})\in Tw\mathcal{A}$, and a closed morphism $f\in\Hom^{0}((L,b_{L}),(L^{\prime},b_{L^{\prime}}))$, the abstract mapping cone of $f$ is the twisted complex

$$\text{Cone}(f)=\left(L[1]\oplus L^{\prime},\left(\begin{matrix}b_{L}&0\\ f&b_{L^{\prime}}\end{matrix}\right)\right).$$

Generally, a mapping cone of $f$ is an object of $Tw\mathcal{A}$ quasi-isomorphic to $\text{Cone}(f)$. This gives rise to a distinguished triangle $A\to B\to\text{Cone}(f)\xrightarrow{[1]}A$. This illustrates the advantage of introducing twisted complexes: $Tw\mathcal{A}$ is a triangulated category.

The cohomological category of $Tw\mathcal{A}$ is commonly denoted $D^{b}Fuk(X)$. This has the same objects as $Tw\mathcal{A}$, but the Floer cochain spaces are replaced by their $H^{0}$, namely we remember the Floer cohomology.

Under the Yoneda embedding, $Tw\mathcal{A}$ embedds into its module category. The idempotent closure $Tw^{\pi}\mathcal{A}$ is obtained by formally adding the direct summands of the Yoneda image of twisted complexes in $Tw\mathcal{A}$. The cohomological category of $Tw^{\pi}\mathcal{A}$ is commonly denoted $D^{\pi}Fuk(X)$. In the variant setting of compact $X$, it is usually $D^{\pi}Fuk(X)$ instead of $D^{b}Fuk(X)$ that shows up in mirror symmetry, since the derived category of coherent sheaves is automatically idempotent closed.

The assumptions on $(X,\omega)$ are as in the previous section. Immersed Lagrangians are immersions $\iota:L\to X$ with $\omega|_{L}=0$, and all self intersections are transverse. The domain of $L$ is allowed to be disconnected, so the union of finitely many transversely intersecting embedded Lagrangians are examples of immersed Lagrangians. Each self intersection point of two local sheets $L_{+},L_{-}$ corresponds to two different points $x_{\pm}$ on the domain of $L$. It is important to distinguish $x_{+}$ and $x_{-}$, because for the boundary of the holomorphic curve to pass through $x_{+}$ in the clockwise direction means crossing from $L_{+}$ to $L_{-}$, and $x_{-}$ signifies the opposite crossing.

We say $L$ is exact, if there is a function $f_{L}$ on the domain of $L$, such that $df_{L}$ agrees with the Liouville 1-form restricted to $L$. For energy reasons, this forbids nontrivial holomorphic disks with boundary on $L$ which never change local sheets at any boundary point. The caveat is that the relative homology class $[\omega]\in H_{2}(X,L)$ may still be nonzero. The brane structures on $L$ are as in the embedded case. The construction of $CF^{*}(L,L)$ depends on the approach, but a common feature is that it includes

$$CF^{*}_{self}(L,L)=\bigoplus_{\text{self intersection $p$}}CF^{*}(L_{+},L_{-})\oplus CF^{*}(L_{-},L_{+}).$$

generated by the local system factor $\text{Hom}(E_{+},E_{-})|_{p}$ (resp. $\text{Hom}(E_{-},E_{+})|_{p}$) tensored with the orientation line.

The Gromov compactness discussion is largely similar to the embedded case. A new phenomenon is the teardrop curves, namely the holomorphic curves with boundary on $L$ and a single output corner at a self intersection point $r\in CF^{*}(L_{+},L_{-})$. Of particular importance is the case with $\mu_{L_{+},L_{-}}(r)=2$. The number $2$ is intuitively explained by the 2 degrees of freedom of the domain Möbius transforms fixing the corner point $Aut(D^{2},1)$, modulo which such teardrop curves occur in dimension zero moduli spaces.

Now if we attempt to run the usual argument for $d^{2}=0$ in Floer cohomology, we would consider the moduli space of holomorphic strips between $p,q$ with $\deg q-\deg p=2$, modulo the translation $\mathbb{R}$. However, in addition to the usual strip breaking, the holomorphic strips can also break into a holomorphic triangle with inputs $p,r$ and output $q$, and a teardrop curve with corner at $r$. In summary, teardrop curves with corner at a degree 2 intersection point obstruct Floer cohomology.

The automorphism group $Aut(D^{2},1)$ forbids the naïve domain dependent perturbation schemes, which in turn causes transversality problems. In the literature there are two approaches to solve this problem: Joyce and Akaho [7] use virtual perturbation techniques for bordered Riemann surfaces, while Woodward et al. [81][82] circumvent the virtual perturbations by utilizing stabilising divisors. Both approaches assign curved $A_{\infty}$ algebra structures $(m_{0},m_{1},\ldots)$ to the Floer cochain spaces $CF^{*}(L,L)$ of immersed Lagrangians. In the exact setting, the $m_{0}\in CF_{self}^{2}(L,L)$ term amounts to a count of teardrop curves with corner at degree 2 self intersection points, with weighting factors coming from the holonomy of the local system. Since in the main text the emphasis is on the automatic transversality assumption, we shall not dwell on the details of perturbation schemes, but only identify a few simplifications in the exact setting.

To make sense of Floer cohomology one needs to cancel the obstructions by introducing bounding cochains $b\in CF_{self}^{1}(L,L)$, which represents a formal sum of $b_{p}\in\text{Hom}(E_{+},E_{-})|_{p}\otimes|o_{p}|$ associated to degree one intersection points $p\in CF^{1}(L_{+},L_{-})$. We require

• •

  The Novikov positivity condition $f_{L_{+}}(p)\geq f_{L_{-}}(p)$ for each of the intersection points appearing in $b$.

• •

  The Mauer-Cartan equation

  $$m_{0}^{b}=m_{0}+m_{1}(b)+m_{2}(b,b)+\ldots=0\in CF_{self}^{2}(L,L).$$

  (72)

Geometrically, the coefficients of $q\in CF_{self}^{2}(L,L)$ in the $m_{k}(b,\ldots b)$ term represent the zero dimensional counts of holomorphic polygons with the inputs at the summands $b_{p}$ of $b$, and the output at $q$, weighted by the holonomy and orientation factors. Using the Novikov positivity requirement of the bounding cochain, the topological energy formula (66) for the polygon then implies

$$\int_{\Sigma}\omega\leq f_{L_{-}}(q)-f_{L_{+}}(q),$$

where the boundary of $\Sigma$ passes from $L_{+}$ to $L_{-}$ at $q$ in the clockwise direction. By Gromov compactness, this uniform energy bound implies there are only finitely many terms involved in the Mauer-Cartan equation. When such a bounding cochain $b$ exists, we say $(L,b)$ defines an unobstructed Lagrangian brane. In this case, both the Akaho-Joyce and the Woodward-Palmer approaches assign self Floer cohomology groups $HF^{*}((L,b),(L,b))$, defined as the cohomology of a degree one operator

$$m_{1}^{b}:CF^{*}(L,L)\to CF^{*+1}(L,L),\quad m_{1}^{b}(x)=\sum_{k,k^{\prime}\geq 0}m_{k+k^{\prime}+1}(\underbrace{b,\ldots b}_{k^{\prime}},x,\underbrace{b,\ldots,b}_{k}).$$

This cohomology is invariant under global Hamiltonian deformations. Two bounding cochains $b,b^{\prime}$ on $L$ are said to be gauge equivalent, if there is $h\in CF^{0}(L,L)$ satisfying the Novikov positivity condition, such that

$$b-b^{\prime}=\sum_{k,k^{\prime}\geq 0}m_{k+k^{\prime}+1}(b^{\prime},\ldots b^{\prime},h,b,\ldots b).$$

Gauge equivalent bounding cochains give rise to isomorphic Floer cohomology.

Now suppose $(L,b)$ and $(L^{\prime},b^{\prime})$ are two unobstructed Lagrangian branes, intersecting transversally avoiding the self intersections of $L$ and $L^{\prime}$. Then we can define the Floer cohomology $HF^{*}((L,b),(L^{\prime},b^{\prime}))$. The Floer cochain space $CF^{*}(L,L^{\prime})$ is the same as in the embedded case, generated by the local system factor tensored with the orientation factor, associated to the transverse intersection points. The Floer differential is

$$m_{1}^{b,b^{\prime}}(p)=\sum_{k,k^{\prime}\geq 0}m_{k+k^{\prime}+1}(b^{\prime},\ldots b^{\prime},p,b,\ldots b),$$

where the sum has $k^{\prime}$ insertions of $b^{\prime}$, and $k$ insertions of $b$. The coefficient of $q\in CF^{*+1}(L,L^{\prime})$ are morally defined by the weighted count of holomorphic polygons with boundary marked points mapping to the summands of $b,\ldots p,b^{\prime},\ldots,q$, arranged in clockwise order. A similar a priori energy bound argument shows the sum is finite.

It is instructive to see why $(m_{1}^{b,b^{\prime}})^{2}=0$. We consider the breaking of one dimensional moduli spaces, associated with $p,r\in CF^{*}(L,L^{\prime})$ with $\deg r-\deg p=2$. There are several mechanisms for disc bubbling and disc splittings:

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  The polygon breaks into two parts, connected at a nodal point mapping to some $q\in CF^{*}(L,L^{\prime})$ with $\deg q-\deg p=1$. The sum of all such contributions give rise to $\langle m_{1}^{b,b^{\prime}}(p),q\rangle\langle m_{1}^{b,b^{\prime}}(q),r\rangle$, and summing over $q,r$ produces $(m_{1}^{b,b^{\prime}})^{2}(p)$.

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  The polygon bubbles off a teardrop curve at a self intersection point $q$ of degree 2 on either $L$ or $L^{\prime}$.

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  The polygon splits into two parts, connected at a node mapping to a degree 2 self intersection point $q$ on either $L$ or $L^{\prime}$.