To make sense of Floer cohomology one needs to cancel the obstructions by introducing bounding cochains $b\in CF_{self}^{1}(L,L)$, which represents a formal sum of $b_{p}\in\text{Hom}(E_{+},E_{-})|_{p}\otimes|o_{p}|$ associated to degree one intersection points $p\in CF^{1}(L_{+},L_{-})$. We require

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  The Novikov positivity condition $f_{L_{+}}(p)\geq f_{L_{-}}(p)$ for each of the intersection points appearing in $b$.

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  The Mauer-Cartan equation

  $$m_{0}^{b}=m_{0}+m_{1}(b)+m_{2}(b,b)+\ldots=0\in CF_{self}^{2}(L,L).$$

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Geometrically, the coefficients of $q\in CF_{self}^{2}(L,L)$ in the $m_{k}(b,\ldots b)$ term represent the zero dimensional counts of holomorphic polygons with the inputs at the summands $b_{p}$ of $b$, and the output at $q$, weighted by the holonomy and orientation factors. Using the Novikov positivity requirement of the bounding cochain, the topological energy formula (66) for the polygon then implies

$$\int_{\Sigma}\omega\leq f_{L_{-}}(q)-f_{L_{+}}(q),$$

where the boundary of $\Sigma$ passes from $L_{+}$ to $L_{-}$ at $q$ in the clockwise direction. By Gromov compactness, this uniform energy bound implies there are only finitely many terms involved in the Mauer-Cartan equation. When such a bounding cochain $b$ exists, we say $(L,b)$ defines an unobstructed Lagrangian brane. In this case, both the Akaho-Joyce and the Woodward-Palmer approaches assign self Floer cohomology groups $HF^{*}((L,b),(L,b))$, defined as the cohomology of a degree one operator

$$m_{1}^{b}:CF^{*}(L,L)\to CF^{*+1}(L,L),\quad m_{1}^{b}(x)=\sum_{k,k^{\prime}\geq 0}m_{k+k^{\prime}+1}(\underbrace{b,\ldots b}_{k^{\prime}},x,\underbrace{b,\ldots,b}_{k}).$$

This cohomology is invariant under global Hamiltonian deformations. Two bounding cochains $b,b^{\prime}$ on $L$ are said to be gauge equivalent, if there is $h\in CF^{0}(L,L)$ satisfying the Novikov positivity condition, such that

$$b-b^{\prime}=\sum_{k,k^{\prime}\geq 0}m_{k+k^{\prime}+1}(b^{\prime},\ldots b^{\prime},h,b,\ldots b).$$

Gauge equivalent bounding cochains give rise to isomorphic Floer cohomology.

Now suppose $(L,b)$ and $(L^{\prime},b^{\prime})$ are two unobstructed Lagrangian branes, intersecting transversally avoiding the self intersections of $L$ and $L^{\prime}$. Then we can define the Floer cohomology $HF^{*}((L,b),(L^{\prime},b^{\prime}))$. The Floer cochain space $CF^{*}(L,L^{\prime})$ is the same as in the embedded case, generated by the local system factor tensored with the orientation factor, associated to the transverse intersection points. The Floer differential is

$$m_{1}^{b,b^{\prime}}(p)=\sum_{k,k^{\prime}\geq 0}m_{k+k^{\prime}+1}(b^{\prime},\ldots b^{\prime},p,b,\ldots b),$$

where the sum has $k^{\prime}$ insertions of $b^{\prime}$, and $k$ insertions of $b$. The coefficient of $q\in CF^{*+1}(L,L^{\prime})$ are morally defined by the weighted count of holomorphic polygons with boundary marked points mapping to the summands of $b,\ldots p,b^{\prime},\ldots,q$, arranged in clockwise order. A similar a priori energy bound argument shows the sum is finite.

It is instructive to see why $(m_{1}^{b,b^{\prime}})^{2}=0$. We consider the breaking of one dimensional moduli spaces, associated with $p,r\in CF^{*}(L,L^{\prime})$ with $\deg r-\deg p=2$. There are several mechanisms for disc bubbling and disc splittings:

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  The polygon breaks into two parts, connected at a nodal point mapping to some $q\in CF^{*}(L,L^{\prime})$ with $\deg q-\deg p=1$. The sum of all such contributions give rise to $\langle m_{1}^{b,b^{\prime}}(p),q\rangle\langle m_{1}^{b,b^{\prime}}(q),r\rangle$, and summing over $q,r$ produces $(m_{1}^{b,b^{\prime}})^{2}(p)$.

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  The polygon bubbles off a teardrop curve at a self intersection point $q$ of degree 2 on either $L$ or $L^{\prime}$.

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  The polygon splits into two parts, connected at a node mapping to a degree 2 self intersection point $q$ on either $L$ or $L^{\prime}$.

The combined effect of the last two contributions, is a sum of the weighted counts of polygons with boundary mapping to $b,b,\ldots p,b^{\prime},\ldots q,b^{\prime}\ldots r$ multiplied by the coefficient of $q$ in $m_{0}^{b^{\prime}}=m_{0}+m_{1}(b^{\prime})+\ldots\in CF^{2}_{self}(L^{\prime},L^{\prime})$ in the case of $q\in CF^{2}_{self}(L^{\prime},L^{\prime})$ (the case with $q\in CF^{2}_{self}(L,L)$ gives an entirely similar contribution related to $m_{0}^{b}\in CF^{2}_{self}(L,L)$). By the unobstructed assumption $m_{0}^{b}=0$ and $m_{0}^{b^{\prime}}=0$, so these contributions vanish. But the grand sum of all contributions from all boundaries of the moduli spaces should be zero, which implies $(m_{1}^{b,b^{\prime}})^{2}=0$.

The generalization to many Lagrangians is a matter of bookkeeping. We have the $A_{\infty}$ compositions

$$m_{k}^{b_{0},\ldots b_{k}}:CF^{*}(L_{k-1},L_{k})\otimes\ldots CF^{*}(L_{0},L_{1})\to CF^{*}(L_{0},L_{k})[2-k],$$

$$m_{k}^{b_{0},\ldots b_{k}}(p_{k},\ldots p_{1})=\sum m_{l}(b_{k},\ldots b_{k},p_{k},b_{k-1},\ldots,p_{k-1},\ldots,p_{1},b_{0},\ldots b_{0}).$$

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In particular, this induces a product structure on Floer cohomology $HF^{*}(L_{1},L_{2})\otimes HF^{*}(L_{0},L_{1})\to HF^{*}(L_{0},L_{2})$ (with bounding cochains suppressed in the notation),

$$[\beta]\circ[\alpha]=(-1)^{\deg\alpha}m_{2}^{b_{0},b_{1},b_{2}}(\beta,\alpha).$$

We say two unobstructed Lagrangian branes $L,L^{\prime}$ are isomorphic in $D^{b}Fuk(X)$, if there exist $[\alpha]\in HF^{0}(L,L^{\prime})$ and $[\beta]\in HF^{0}(L^{\prime},L)$, such that their compositions are the cohomological units: $[\beta]\circ[\alpha]=1_{L}\in HF^{0}(L,L)$ and $[\alpha]\circ[\beta]=1_{L^{\prime}}\in HF^{0}(L^{\prime},L^{\prime})$.