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).