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)

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

(17)

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

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

(18)

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.