Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, which may be compact or noncompact, with Kähler form $\omega$. We now explain a little about (embedded) Lagrangian branes $(L,E)$ in $(M,\omega)$, bounding cochains $b$ for $(L,E)$ and obstructions to $HF^{*}$, Lagrangian Floer cohomology $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$, the Fukaya category ${\mathbin{\mathscr{F}}}(M)$, and the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$. Section 2.6 discusses the extension of all this to immersed Lagrangians.

The construction of $D^{b}{\mathbin{\mathscr{F}}}(M)$ in the generality we need may not yet be available in the literature. As this paper is wholly conjecture anyway, and clearly the theory will eventually work, this does not matter very much.

The version of bounding cochains, obstructions to $HF^{*}$, and Lagrangian Floer cohomology we need is in Fukaya, Oh, Ohta and Ono [20]. An early explanation of how to define the (derived) Fukaya category ${\mathbin{\mathscr{F}}}(M),D^{b}{\mathbin{\mathscr{F}}}(M)$ is Fukaya [18], and a more recent survey is Fukaya [19]. Floer [17] originally introduced Lagrangian Floer cohomology.

For exact Lagrangians in Liouville manifolds (a class of noncompact, exact symplectic manifolds), a simpler, more complete, and more satisfactory theory of Lagrangian Floer cohomology and Fukaya categories is given in Seidel [64], which we used in [31] to prove Theorems 2.6 and 2.14. In Seidel’s theory there are no bounding cochains or obstructions to $HF^{*}$.

However, for our purposes Seidel’s theory will not do: we need to extend the theory to immersed Lagrangians, and even for exact Lagrangians, bounding cochains and obstructions to $HF^{*}$ will then appear. Also, we wish to stress the idea that Lagrangian MCF is better behaved for Lagrangians with $HF^{*}$ unobstructed, and in Seidel’s framework this issue is hidden by restricting to exact, embedded Lagrangians, for which $HF^{*}$ is automatically unobstructed.

Here is the basic idea of Lagrangian Floer cohomology. Let $(L,E),(L^{\prime},E^{\prime})$ be Lagrangian branes in a Calabi–Yau $m$-fold $(M,J,g,\Omega)$, and suppose $L,L^{\prime}$ intersect transversely. The aim is to define a $\Lambda_{\rm nov}$-module $HF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ called the Lagrangian Floer cohomology, which is the cohomology of a complex of $\Lambda_{\rm nov}$-modules $\bigl(CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),{\rm d}\bigr)$ called the Floer complex.

Define a free, graded $\Lambda_{\rm nov}$-module $CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ by

Initially we define ${\rm d}:CF^{k}(L,L^{\prime})\rightarrow CF^{k+1}(L,L^{\prime})$ by

for $p\in L\cap L^{\prime}$ with $\mu_{L,L^{\prime}}(p)=k$ and $\alpha_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E|_{p},E^{\prime}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$, where ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ is the moduli space of stable $J$-holomorphic discs $\Sigma$ in $M$ with boundary in $L\cup L^{\prime}$, corners at $p,q$ and area $A$, of the form shown in Figure 2.2, where $\#_{\rm virt}{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}\in{\mathbin{\mathbb{Q}}}$ is the ‘virtual number of points’ in ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$, and the sum is weighted by composition with the parallel transport maps $PT_{\cdots}(E)\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E|_{q},E|_{p}\bigr)$ and $PT_{\cdots}(E^{\prime})\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E^{\prime}|_{p},E^{\prime}|_{q}\bigr)$ in the ${\mathbin{\mathbb{F}}}$-local systems $E,E^{\prime}$ along the two segments of $\partial\Sigma$. These $PT_{\cdots}(E),PT_{\cdots}(E^{\prime})$ are locally constant on ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$.

Constructing an appropriate geometric structure (‘Kuranishi space’ or ‘polyfold’) on ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$, and defining the virtual count $\#_{\rm virt}{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$, raise many complicated issues which we will not go into.

For exact Lagrangians in an exact symplectic manifold, as in Seidel [64], the differential ${\rm d}$ in (2.14) has ${\rm d}^{2}=0$, so $HF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr)$ is well-defined. However, in the non-exact case we may have ${\rm d}^{2}\neq 0$, because of contributions to the boundaries $\partial{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{p,q}^{A}$ from holomorphic discs with boundary in $L$ or in $L^{\prime}$.

To get round this, Fukaya, Oh, Ohta and Ono [20, §3.6] introduce the notion of a bounding cochain $b$ for $(L,E)$, an element $b$ of the singular $(m-1)$-chains $C_{m-1}(L,\Lambda_{\rm nov}^{+})$ of $L$ with coefficients in $\Lambda_{\rm nov}^{+}\subset\Lambda_{\rm nov}$, satisfying an equation in $C_{m-2}(L,\Lambda_{\rm nov}^{+})$ which is (very roughly, and oversimplified) of the form

where ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}$ is the moduli space (as a Kuranishi space or polyfold, of virtual dimension $m+k-2$) of isomorphism classes $[\Sigma,\vec{z}]$ where $\Sigma$ is a stable $J$-holomorphic disc of area $A>0$ in $M$ with boundary in $L$, and $\vec{z}=(z_{0},z_{1},\ldots,z_{k})$ are cyclically ordered marked points in $\partial\Sigma$. Also ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}\times_{L^{k}}b^{k}$ is the moduli space of such $[\Sigma,\vec{z}]$ in which $z_{1},\ldots,z_{k}$ intersect the chain $b$ in $L$, and $\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}\times_{L^{k}}b^{k}\bigr]{}_{\rm virt}$ is a virtual chain for this. The sum is weighted by the holonomy $\mathop{\rm Hol}\nolimits_{\partial\Sigma}(E)\in{\mathbin{\mathbb{F}}}^{*}$ of the rank one ${\mathbin{\mathbb{F}}}$-local system $E\rightarrow L$ around $\partial\Sigma\subset L$, which depends only on $[\partial\Sigma]\in H_{1}(L,{\mathbin{\mathbb{Z}}})$, and is locally constant on ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{A}$.

If a bounding cochain $b$ exists for $(L,E)$, we say that $(L,E)$ has $HF^{*}$ unobstructed, otherwise we say that $(L,E)$ has $HF^{*}$ obstructed. Implicitly we will always consider bounding cochains $b$ up to the appropriate notion of equivalence.

To oversimplify even further, suppose that the terms for $k\geqslant 1$ in (2.15) are zero, and $\partial{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}=\emptyset$ for all $A>0$ when $k=0$, so that $\partial\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}\bigr]{}_{\rm virt}=0$, and write $b=\sum_{A>0}P^{A}\cdot b_{A}$ for $b_{A}\in C_{m-1}(L,{\mathbin{\mathbb{Q}}})$. Then (2.15) becomes $\partial b_{A}=\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}\bigr]{}_{\rm virt}\cdot\mathop{\rm Hol}\nolimits_{\partial\Sigma}(E)$ for all $A>0$. So a bounding cochain $b$ exists if $\bigl[\bigl[{\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{1}^{A}\bigr]{}_{\rm virt}\bigr]=0$ in $H_{m-2}(L,{\mathbin{\mathbb{Q}}})$ for all $A>0$. In particular, if $H_{m-2}(L,{\mathbin{\mathbb{Q}}})=0$ then a bounding cochain exists.

In the general case, if $H_{m-2}(L,{\mathbin{\mathbb{Q}}})=0$ then (2.15) may be solved for $b=\sum_{A>0}P^{A}\cdot b_{A}$ by an inductive procedure in increasing $A$, yielding:

Suppose $b,b^{\prime}$ are bounding cochains for $(L,E),(L^{\prime},E^{\prime})$. Then Fukaya et al. [20] define a modification ${\rm d}^{b,b^{\prime}}$ of ${\rm d}$ in (2.14) involving $b,b^{\prime}$ and satisfying $({\rm d}^{b,b^{\prime}})^{2}=0$. The Lagrangian Floer cohomology $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is the cohomology of $\bigl(CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),{\rm d}^{b,b^{\prime}}\bigr)$, which may depend on $b,b^{\prime}$. Here are some properties of Lagrangian Floer cohomology in the theory of Fukaya, Oh, Ohta and Ono [20]:

• (a)

  The Lagrangian Floer cohomology $HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is independent of the choice of almost complex structure $J$ up to canonical isomorphism, although $\bigl(CF^{*}\bigl((L,E),(L^{\prime},E^{\prime})\bigr),{\rm d}^{b,b^{\prime}}\bigr)$ does depend on $J$.

• (b)

  Let $(L^{t},E^{t}):t\in[0,1]$ be a smooth family of Lagrangian branes, with the $L^{t}$ Hamiltonian isotopic and the $E^{t}$ locally constant in $t$, and let $b^{0}$ be a bounding cochain for $L^{0}$. By a kind of ‘parallel transport’ we can extend $b^{0}$ to a family of bounding cochains $b^{t}$ for $(L^{t},E^{t})$ for $t\in[0,1]$. If $(L^{\prime},E^{\prime})$ is another Lagrangian brane with bounding cochain $b^{\prime}$ then $HF^{*}\bigl((L^{t},E^{t},b^{t}),(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is independent of $t\in[0,1]$ up to canonical isomorphism. Thus $HF^{*}\bigl((L,E,b),\allowbreak(L^{\prime},E^{\prime},b^{\prime})\bigr)$ is an invariant of Lagrangian branes up to Hamiltonian isotopy.

Lagrangian Floer cohomology is only the beginning of a more general theory of Fukaya categories, which may be still incomplete in the general case. Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold. The idea is to define the Fukaya category ${\mathbin{\mathscr{F}}}(M)$ of $M$, an $A_{\infty}$-category whose objects are triples $(L,E,b)$ of a Lagrangian brane $(L,E)$ in $M$ with $HF^{*}$ unobstructed, and a bounding cochain $b$ for $(L,E)$, such that the morphisms $\mathop{\rm Hom}\nolimits\bigl((L_{0},E_{0},b_{0}),(L_{1},E_{1},b_{1})\bigr)$ in ${\mathbin{\mathscr{F}}}(M)$ are the graded $\Lambda_{\rm nov}$-modules $CF^{*}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)$ from above, with $A_{\infty}$-operations

for $k\geqslant 1$, with $\mu^{1}:CF^{a_{1}}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)\allowbreak\rightarrow CF^{a_{1}+1}\bigl((L_{0},E_{0}),(L_{1},E_{1})\bigr)$ the differential ${\rm d}^{b_{0},b_{1}}$ in the Floer complex. The coefficients in the $\Lambda_{\rm nov}$-multilinear map $\mu^{k}$ in (2.16) are obtained by ‘counting’ $J$-holomorphic $(k\!+\!1)$-gons in $M$ with boundary in $L_{0}\cup\cdots\cup L_{k}$, weighted by parallel transport maps in $E_{0},\ldots,E_{k}$.

By a category theory construction, one then defines the derived Fukaya category, a triangulated category. There are two versions, which we will write $D^{b}{\mathbin{\mathscr{F}}}(M)$ and $D^{\pi}{\mathbin{\mathscr{F}}}(M)$. For $D^{b}{\mathbin{\mathscr{F}}}(M)$, the objects are twisted complexes, as in Seidel [64, §3l]. Roughly speaking, a twisted complex consists of objects $(L_{1},E_{1},b_{1}),\ldots,(L_{n},E_{n},b_{n})$ in ${\mathbin{\mathscr{F}}}(M)$ together with Floer cochains $b_{ij}\in CF^{*}\bigl((L_{i},E_{i}),(L_{j},E_{j})\bigr)$ for $1\leqslant\penalty i<j\leqslant\penalty n$ satisfying an equation related to the bounding cochain equation. In particular, objects $(L,E,b)$ in ${\mathbin{\mathscr{F}}}(M)$ are also objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

The translation functor $[1]$ in the triangulated category $D^{b}{\mathbin{\mathscr{F}}}(M)$ acts on objects $(L,E,b)$ by reversing the orientation of $L$ and changing the grading $\theta_{L}$ to $\theta_{L}+\pi$. The (graded) morphisms of objects $(L,E,b),(L^{\prime},E^{\prime},b^{\prime})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ are $\mathop{\rm Hom}\nolimits^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)=HF^{*}\bigl((L,E,b),(L^{\prime},E^{\prime},b^{\prime})\bigr)$.

The second version $D^{\pi}{\mathbin{\mathscr{F}}}(M)$, called the idempotent completion, Karoubi completion, or split closure of $D^{b}{\mathbin{\mathscr{F}}}(M)$, is obtained by applying a further category theory construction to $D^{b}{\mathbin{\mathscr{F}}}(M)$, which adds direct summands (idempotents) of objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$ as extra objects, as in Seidel [64, §4].

Kontsevich’s Homological Mirror Symmetry Conjecture [44], motivated by String Theory, says (very roughly) that if $M,\check{M}$ are ‘mirror’ Calabi–Yau $m$-folds then there should be an equivalence of triangulated categories

This has driven much research in the area.

For Mirror Symmetry, one must use $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ rather than $D^{b}{\mathbin{\mathscr{F}}}(M)$, as the mirror category $D^{b}\mathop{\rm coh}(\check{M})$ is automatically idempotent complete. In §3.1 we will conjecture that in the situation we are interested in, our enlarged version of $D^{b}{\mathbin{\mathscr{F}}}(M)$ should be idempotent complete, so that $D^{\pi}{\mathbin{\mathscr{F}}}(M)\simeq D^{b}{\mathbin{\mathscr{F}}}(M)$.