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,

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):

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  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 $\beta$).

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  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. $b^{\prime}$).

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  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.