The claim that the $K$-theory of the derived Fukaya category of a symplectic Calabi-Yau manifold (exact with suitable convexity at infinity, or compact) factorizes through homology $H_{n}(X)$ modulo torsion seems well known to symplectic topology experts, although a precise reference seems rather difficult to find. We now sketch a high brow viewpoint explained to the author by P. Seidel and S. Rezchikov, and will later explain in more detail a more pedestrian approach in the exact setting. The claim is a formal consequence of the existence of maps

$$K_{0}(D^{b}Fuk(X))\xrightarrow{ch_{0}}HH_{0}(D^{b}Fuk(X))$$

$$HH_{0}(D^{b}Fuk(X))\xrightarrow{OC}QH^{n}(X;\Lambda_{nov})\simeq H_{n}(X)\otimes\Lambda_{nov}.$$

Here $HH_{0}(D^{b}Fuk(X))$ is the Hochschild homology in degree zero. The first map sends the K-theory class of an unobstructed Lagrangian brane $L$ (compact, graded, oriented, with spin and bounding cochain structure) to the unit $1_{L}\in HF^{0}(L,L)\to HH_{0}(D^{b}Fuk(X))$; the well definition of this map is an essentially algebraic fact. Suppose $L,L^{\prime}$ are isomorphic in $D^{b}Fuk(X)$, then there are closed morphisms $\alpha\in CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$ whose derived category compositions are equal to $1_{L}\in HF^{0}(L,L)$ and $1_{L^{\prime}}\in HF^{0}(L^{\prime},L^{\prime})$ in cohomology. The Hochschild differential of $\beta\otimes\alpha$ exhibits $1_{L}-1_{L^{\prime}}$ as a coboundary in the Hochschild chain complex, so $1_{L}=1_{L^{\prime}}$ in $HH_{0}(D^{b}Fuk(X))$. Some additional calculation shows the compatibility with distinguished triangles.

The second map is a special case of the open-closed string map, and in general requires working over the Novikov field. One then needs the claim that $1_{L}$ is sent to the homology class $[L]\in H_{n}(X)$, without quantum correction. The intuitive meaning of the open-closed string map is to consider holomorphic discs with boundary on $L$ with an unconstrained boundary marked point, and find the cycle in $X$ traced out by an interior marked point. The claim amounts to saying that the only contribution comes from constant maps. Unfortunately, the author is unable to locate a general reference. Granted this claim, we would get by composition a map from $K_{0}(D^{b}Fuk(X))$ to $H_{n}(X;\Lambda_{nov})$ which sends the K-theory class of $L$ to the homology class $[L]$.

We specialize to the setting of Stein manifolds, and all Lagrangians are assumed to be exact, graded and compact, and in particular carry an orientation (cf. the Appendix for some basic Floer theory). The local systems have holonomy in $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$. We consider two transverse embedded Lagrangians $L,L^{\prime}$ in the same derived Fukaya category class. By definition, we have closed morphisms $\alpha\in CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$; we sometimes view the Lagrangian intersections in $\beta$ as degree $n$ outputs. Morever, in terms of the product structure on cohomology

$$\begin{cases}HF^{0}(L,L^{\prime})\otimes HF^{0}(L^{\prime},L)\to HF^{0}(L^{\prime},L^{\prime}),\\ HF^{0}(L^{\prime},L)\otimes HF^{0}(L,L^{\prime})\to HF^{0}(L,L),\end{cases}$$

the composition $\alpha\circ\beta=1_{L^{\prime}}$ and $\beta\circ\alpha=1_{L}$. These conditions completely characterize isomorphism in $D^{b}Fuk(X)$. Our goal is to explain

The bordism current will be constructed from universal families of (perturbed) holomorphic strips with boundary on $L$ and $L^{\prime}$, and with ends at $\alpha$ and $\beta$ (meaning that the ends of the strip converge to intersection points of $L,L^{\prime}$ of degree $0$ and $n$ respectively, and $\alpha,\beta$ encode the weighting factors to the contribution of these intersection points). We will assume all the usual transversality assumptions in Floer theory are satisfied, so the compactified moduli space $\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}}$ of perturbed holomorphic strips up to domain translation is a smooth manifold with boundary and corners, of dimension $(n-1)$. The notation really stands for a formal sum of many moduli spaces, coming from the summands of $\alpha,\beta$. The universal family is fibred over this moduli space $\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}}$, whose fibres are the solutions to the Cauchy-Riemann equation (with domain dependent perturbations of the almost complex structure), which we call perturbed holomorphic curves. The fibres over the boundary of the moduli space are broken holomorphic curves. The orientation on the universal family is induced from the complex orientation on $\Sigma$ and the orientation on the moduli space, up to an extra minus sign (cf. Example 6.2 for conventions). Upon evaluation to $X$ we obtain an $(n+1)$-dimensional current $\mathcal{C}$.

Our main task is to understand the boundary of $\mathcal{C}$. There are two sources of boundaries:

• •

  The holomorphic curves themselves have boundary along $L\cup L^{\prime}$. This boundary contribution is always supported on $L\cup L^{\prime}$.

• •

  The compactified moduli spaces have boundary due to holomorphic strip breaking.

In schematic notation, the boundary of the moduli space is described by

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

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Here $r$ can range from all intersection points of degree between $1$ and $n-1$. The notation $\mathcal{M}(\alpha,\beta)/\mathbb{R}$ stands for a weighted sum of moduli spaces, with weighting coming from the holonomy factors of the local systems.

Next comes a crucial observation. Although $2\leq\deg r\leq n-2$ give rise to boundaries of the moduli space $\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}}$, their contributions to $\partial\mathcal{C}$ are contained in the universal families associated to $\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}$ and $\overline{\mathcal{M}(r,\beta)/\mathbb{R}}$. These smaller moduli spaces have dimension at most $n-3$, and the corresponding universal families have dimension at most $n-1$. By rectifiability considerations, the $n$-dimensional current $\partial\mathcal{C}$ cannot receive contributions from at most $(n-1)$-dimensional supports, so such disc breakings do not contribute to $\partial\mathcal{C}$.

Now for $\deg r=1$, the moduli spaces $\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}$ are zero dimensional, so their presence merely amounts to some counting factors. The condition for $\alpha$ to be closed in $CF^{0}(L,L^{\prime})$ is equivalent to the weighted count of $\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}$ being zero. This weighted sum appears as the coefficient of the $n$-dimensional current defined by the universal family over $\overline{\mathcal{M}(r,\beta)/\mathbb{R}}$. Thus we see that $r\in CF^{1}(L,L^{\prime})$ does not contribute to $\partial\mathcal{C}$. Similarly, the condition for $\beta$ to be closed in $CF^{0}(L^{\prime},L)$ implies that $r\in CF^{1}(L^{\prime},L)$ (alternatively viewed as degree $n-1$ intersections from $L$ to $L^{\prime}$) does not contribute to $\partial\mathcal{C}$. In summary, $\partial\mathcal{C}$ must be an integration cycle supported on $L\cup L^{\prime}$.

Since $\partial\mathcal{C}$ is itself the boundary of a current, it must be closed. This explains why $\partial\mathcal{C}$ is a constant linear combination of the integration cycle of $L$ and $L^{\prime}$, instead of some nontrivial function times these cycles. The constant coefficients can be pinned down by counting the number of holomorphic strips passing through a given generic point on $L$ (resp. $L^{\prime}$), and the choice of the generic point does not matter. Such counts are precisely the geometric interpretation of the Floer product $HF^{0}(L,L^{\prime})\otimes HF^{0}(L^{\prime},L)\to HF^{0}(L^{\prime},L^{\prime}),$ and $HF^{0}(L^{\prime},L)\otimes HF^{0}(L,L^{\prime})\to HF^{0}(L,L)$ (cf. Example 6.1). When the moduli space orientations are taken into account, we obtain $\partial\mathcal{C}=L-L^{\prime}$ (cf. Example 6.2 for an exposition on signs).

Still working in the exact setting, we now allow $(L,b),(L^{\prime},b^{\prime})$ to be unobstructed immersed Lagrangians with transverse self intersections. Assume they intersect transversally, and define isomorphic objects in $D^{b}Fuk(X)$. We now wish to explain why Proposition 3.1 should continue to hold even in the immersed setting, without delving too deep into the specifics of the perturbation schemes and transversality issues. For some background on the immersed Floer theory, see the Appendix 6.2.

The isomorphism condition gives us closed morphisms $\alpha\in CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$ whose cohomological compositions give the identities. At the chain level,

$$m_{2}^{b,b^{\prime}}(\beta,\alpha)=1_{L}-m_{1}^{b}(\gamma),\quad m_{2}^{b^{\prime},b}(\alpha,\beta)=1_{L^{\prime}}+m_{1}^{b^{\prime}}(\gamma^{\prime})$$

where $1_{L},1_{L^{\prime}}$ stand for the geometric units (represented by a sum of local maximum points of Hamiltonian functions on $L,L^{\prime}$ respectively), and $\gamma,\gamma^{\prime}$ are elements in $CF^{-1}(L,L)$, $CF^{-1}(L^{\prime},L^{\prime})$ respectively. Notice in the almost calibrated case, $\gamma,\gamma^{\prime}$ would be both zero, since there are no self intersections of degree $-1$.

As before, the bordism current shall be constructed from the universal family of (perturbed) holomorphic curves with boundary on $L$ and $L^{\prime}$. But instead of working only with holomorphic strips, we need holomorphic polygons with corners not only at intersection points in $\alpha,\beta$, but also at points in $b,b^{\prime}$. In addition to the holomorphic strip moduli space $\mathcal{M}(\alpha,\beta)/\mathbb{R}$, we also need the moduli space of polygons $\mathcal{M}(b,\ldots b,\alpha,b^{\prime},\ldots b^{\prime},\beta)$, and $\mathcal{M}(b,\ldots,b,\gamma)$, $\mathcal{M}(b^{\prime},\ldots b^{\prime},\gamma^{\prime})$. The notation here is a shorthand for a weighted sum of many moduli spaces of polygons. Since the bounding cochain elements have Floer degrees one, these moduli spaces all have dimension $n-1$. The energy of the polygons satisfies the topological formula (66), so by the Novikov positivity requirement of bounding cochains, there is a uniform a priori energy bound once $\alpha,\beta,\gamma,\gamma^{\prime}$ are given, whence there are in fact only finitely many moduli spaces involved. Each moduli space provides a universal family of holomorphic curves, and the sum of all the contributions defines an $(n+1)$-dimensional current $\mathcal{C}$. For sign conventions, see the Appendix 6.2, and Example 6.2.

The boundary of $\mathcal{C}$ comes from two sources: the boundary of the individual holomorphic curves which lie on $L\cup L^{\prime}$, and the boundary of the compactified moduli spaces. In the exact setting, there are no sphere bubbles. As in the embedded case, for support dimension reasons, the boundaries of the compactified moduli space that can contribute to $\partial\mathcal{C}$, is caused by curve breaking into two pieces arising in $0$ and $n-2$ dimensional moduli spaces. The cancellation of these contributions is very similar to the standard argument for the Floer differential to square to zero (cf. the Appendix 6.2):

• •

  For breakings at a nodal point mapping to $CF^{1}(L,L^{\prime})$ (resp. $CF^{1}(L^{\prime},L)$), the contributions vanish due to the closedness condition $m_{1}^{b,b^{\prime}}(\alpha)=0$ (resp. the closedness of $\beta$).

• •

  For breakings at degree 2 self intersection point on $L$ (resp. $L^{\prime}$), the contributions vanish due to the Mauer-Cartan equation on $b$ (resp. $b^{\prime}$).

• •

  A new way of disc breaking/splitting, is at a degree zero self intersection $r$ on $L$ (the $L^{\prime}$ case being entirely similar). The discs of $\mathcal{M}(b,\ldots b,\alpha,b^{\prime},\ldots b^{\prime},\beta)$ can break into a virtual dimension zero disc with input at $b,\ldots,\alpha,b^{\prime},\ldots\beta,b,\ldots b$ and output at $r$, and a virtual dimension $n-1$ disc with input corners at $b,\ldots b,r$. On the other hand, the discs of $\mathcal{M}(b,\ldots,b,\gamma)$ can break into a virtual dimension zero disc with input at $b,\ldots,\gamma,b,\ldots b$ and output at $r$, and a virtual dimension $n-1$ disc with input corners at $b,\ldots b,r$. These two effects cancel out.

After the cancellation of all moduli space boundaries, the only contributions $\partial\mathcal{C}$ are supported in $L\cup L^{\prime}$. As in the embedded case, $\partial\mathcal{C}$ is locally a constant multiple of the underlying cycles of $L,L^{\prime}$. The interpretation of the geometric unit pins down $\partial\mathcal{C}=L-L^{\prime}$ as in the embedded case.

In our convention, an immersed Lagrangian can be made up of several connected components. A prototypical situation is when $L^{\prime}$ is the union of two immersed Lagrangians $L_{1}$ and $L_{2}$, with some degree one intersections in $CF^{1}(L_{2},L_{1})$ arising as part of the bounding cochain data of $L^{\prime}$. When the brane structure is taken into account, we can view $L^{\prime}$ as a twisted complex built from $L_{1},L_{2}$ (with bounding cochains $b_{1},b_{2}$ suppressed in the notation) and a closed morphism $\gamma\in CF^{1}(L_{2},L_{1})=CF^{0}(L_{2}[-1],L_{1})$. Inside $D^{b}Fuk(X)$,

$$L\simeq L^{\prime}\simeq\left(\begin{matrix}(L_{2},b_{2})&\\ \gamma&(L_{1},b_{1})\end{matrix}\right).$$

We have a distinguished triangle

$$L_{2}[-1]\xrightarrow{\gamma}L_{1}\to\text{Cone}(\gamma)\xrightarrow{}L_{2},$$

and $L\simeq L^{\prime}\simeq\text{Cone}(\gamma)$. Rotating the triangles, we get another distinguished triangle

$$L_{1}\to L\to L_{2}\xrightarrow{\gamma}L_{1}[1].$$

The bordism current between $L$ and $L^{\prime}$ is an $(n+1)$-dimensional integration current, with $\partial\mathcal{C}=L-L_{1}-L_{2}.$ In particular, this explains that the Grothendieck group of $D^{b}Fuk(X)$ should factorize through $H_{n}(X)$.

Here is a more geometric perspective on the bordism currents arising from distinguished triangles, which is very close to Thomas and Yau’s original viewpoint, where the fundamental phenomenon is Lagrangian breaking. In the simplest case, we can imagine $L$ is isomorphic in $D^{b}Fuk(X)$ to the Lagrangian connected sum $L_{1}\#L_{2}$ (beware our convention for $L_{1}\#L_{2}$ is the same as Thomas-Yau [65] but different from many symplectic texts), so that we can construct a bordism current between $L$ and $L_{1}\#L_{2}$. Now when $L_{1}\#L_{2}$ deforms, the Lagrangian handle part can shrink, and in the limit $L_{1}\#L_{2}$ can break into two components $L_{1}\cup L_{2}$ (cf. Example 2.12). The bordism current between $L$ and $L_{1}\cup L_{2}$ should simply be the limit of the sequence of bordism currents. This picture illustrates that even when the topology of the Lagrangians can change under non-smooth convergence, the bordism currents should persist in a continuous way.

One can proceed with the case of many Lagrangians, namely we take the immersed Lagrangian $L^{\prime}$ to be the twisted complex (cf. the Appendix section 6.2)

$$\left(\begin{matrix}(L_{N},b_{N})&&\\ b_{N,N-1}&(L_{N-1},b_{N-1})&\\ \ldots\\ b_{N,1}&b_{N-1,1}&\ldots&b_{2,1}&(L_{1},b_{1})\end{matrix}\right).$$

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In this case, assuming $L$ is isomorphic to $L^{\prime}$ in $D^{b}Fuk(X)$, the bordism current between $L$ and $L^{\prime}$ amounts to a bordism current between $L$ and $L_{1}\cup L_{2}\ldots\cup L_{N}$.

This multi-Lagrangian situation is built out of many distinguished triangles: for $1\leq k\leq N$, let $\mathcal{E}_{k}$ be the immersed Lagrangian $L_{1}\cup\ldots\cup L_{k}$ corresponding to the twisted complex

$$\left(\begin{matrix}(L_{k},b_{k})&&\\ b_{k,k-1}&(L_{k-1},b_{k-1})&\\ \ldots\\ b_{k,1}&b_{k-1,1}&\ldots&b_{2,1}&(L_{1},b_{1})\end{matrix}\right).$$

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Then (suppressing bounding cochains in the notation) we have a sequence in $D^{b}Fuk(X)$,

$$0=\mathcal{E}_{0}\to\mathcal{E}_{1}\to\ldots\to\mathcal{E}_{N}\simeq L,$$

with distinguished triangles

$$\mathcal{E}_{i-1}\to\mathcal{E}_{i}\to L_{i}\to\mathcal{E}_{i-1}[1].$$

The morphism from $L_{i}$ to $\mathcal{E}_{i-1}$ comes from $b_{i,j}$ for $j<i$. This setup should be reminiscent of Harder-Narasimhan decompositions (4), although at this stage we have not yet brought in stability conditions, which shall be discussed further in section 3.6.