The assumptions on $(X,\omega)$ are as in the previous section. Immersed Lagrangians are immersions $\iota:L\to X$ with $\omega|_{L}=0$, and all self intersections are transverse. The domain of $L$ is allowed to be disconnected, so the union of finitely many transversely intersecting embedded Lagrangians are examples of immersed Lagrangians. Each self intersection point of two local sheets $L_{+},L_{-}$ corresponds to two different points $x_{\pm}$ on the domain of $L$. It is important to distinguish $x_{+}$ and $x_{-}$, because for the boundary of the holomorphic curve to pass through $x_{+}$ in the clockwise direction means crossing from $L_{+}$ to $L_{-}$, and $x_{-}$ signifies the opposite crossing.

We say $L$ is exact, if there is a function $f_{L}$ on the domain of $L$, such that $df_{L}$ agrees with the Liouville 1-form restricted to $L$. For energy reasons, this forbids nontrivial holomorphic disks with boundary on $L$ which never change local sheets at any boundary point. The caveat is that the relative homology class $[\omega]\in H_{2}(X,L)$ may still be nonzero. The brane structures on $L$ are as in the embedded case. The construction of $CF^{*}(L,L)$ depends on the approach, but a common feature is that it includes

$$CF^{*}_{self}(L,L)=\bigoplus_{\text{self intersection $p$}}CF^{*}(L_{+},L_{-})\oplus CF^{*}(L_{-},L_{+}).$$

generated by the local system factor $\text{Hom}(E_{+},E_{-})|_{p}$ (resp. $\text{Hom}(E_{-},E_{+})|_{p}$) tensored with the orientation line.

The Gromov compactness discussion is largely similar to the embedded case. A new phenomenon is the teardrop curves, namely the holomorphic curves with boundary on $L$ and a single output corner at a self intersection point $r\in CF^{*}(L_{+},L_{-})$. Of particular importance is the case with $\mu_{L_{+},L_{-}}(r)=2$. The number $2$ is intuitively explained by the 2 degrees of freedom of the domain Möbius transforms fixing the corner point $Aut(D^{2},1)$, modulo which such teardrop curves occur in dimension zero moduli spaces.

Now if we attempt to run the usual argument for $d^{2}=0$ in Floer cohomology, we would consider the moduli space of holomorphic strips between $p,q$ with $\deg q-\deg p=2$, modulo the translation $\mathbb{R}$. However, in addition to the usual strip breaking, the holomorphic strips can also break into a holomorphic triangle with inputs $p,r$ and output $q$, and a teardrop curve with corner at $r$. In summary, teardrop curves with corner at a degree 2 intersection point obstruct Floer cohomology.

The automorphism group $Aut(D^{2},1)$ forbids the naïve domain dependent perturbation schemes, which in turn causes transversality problems. In the literature there are two approaches to solve this problem: Joyce and Akaho [7] use virtual perturbation techniques for bordered Riemann surfaces, while Woodward et al. [81][82] circumvent the virtual perturbations by utilizing stabilising divisors. Both approaches assign curved $A_{\infty}$ algebra structures $(m_{0},m_{1},\ldots)$ to the Floer cochain spaces $CF^{*}(L,L)$ of immersed Lagrangians. In the exact setting, the $m_{0}\in CF_{self}^{2}(L,L)$ term amounts to a count of teardrop curves with corner at degree 2 self intersection points, with weighting factors coming from the holonomy of the local system. Since in the main text the emphasis is on the automatic transversality assumption, we shall not dwell on the details of perturbation schemes, but only identify a few simplifications in the exact setting.

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

• •

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

• •

  The Mauer-Cartan equation

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

  (72)

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:

• •

  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 $(m_{1}^{b,b^{\prime}})^{2}(p)$.

• •

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

• •

  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

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

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

(73)

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 $[\alpha]\circ[\beta]=1_{L^{\prime}}\in HF^{0}(L^{\prime},L^{\prime})$.

In our convention an immersed Lagrangian can have several components. Of particular interest is the case where $L$ is the union of transverse immersed Lagrangians $L_{1},\ldots L_{N}$ with bounding cochains $b_{1},\ldots,b_{N}$ respectively, and we have morphisms $b_{ij}\in CF^{1}((L_{i},b_{i}),(L_{j},b_{j}))$ for $i>j$. The key assumption here is that the morphisms only go in one direction from $L_{i}$ to $L_{j}$, not vice versa. We assume that $b=\sum b_{i}+\sum_{i>j}b_{ij}$ is a bounding cochain for the immersed Lagrangian $L$, and in particular all intersection points in $b_{ij}$ satisfy the Novikov positivity condition $f_{L_{i}}\geq f_{L_{j}}$. We can write out the Mauer-Cartan equation

$$m_{0}+m_{1}(b)+m_{2}(b,b)+\ldots=0$$

in component form: for any $i>j$,

$$\sum_{l}\sum_{k\leq l}\sum_{i=i_{0}>\ldots>i_{k}=j}m_{l}(b_{i_{k}},\ldots b_{i_{k}},b_{i_{k-1}i_{k}},\ldots,b_{i_{1}},\ldots b_{i_{1}},b_{i_{0}i_{1}},b_{i_{0}},\ldots,b_{i_{0}})=0.$$

The key observation is that this is precisely how one would define twisted complexes built on $L_{1},\ldots L_{N}$, in the presence of the bounding cochains $b_{1},\ldots b_{N}$ and the data $b_{ij}$, when no further degree shifts are involved (cf. the exact setting in section 6.1). In this sense, we say that ‘immersed Lagrangians geometrises twisted complexes’. In other words, if the unobstructed immersed Lagrangians are admitted into the Fukaya category, then there is no need to formally add twisted complexes.

In section 3.1, 3.1.2 we encountered the $(n-1)$-dimensional moduli spaces such as $\mathcal{M}(b,\ldots,b,\alpha,b^{\prime},\ldots,\beta)$ and $\mathcal{M}(b,\ldots,b,\gamma)$. The special case of holmorphic strips was already mentioned in Example 6.2.

We now consider the moduli $\mathcal{M}(p_{1},\ldots p_{k})$ of polygons with at least 3 corners $p_{1},\ldots p_{k}$, all regarded as inputs, arranged in clockwise order on $\partial\Sigma$, each carrying the local system factors $\Hom(E_{+},E_{-})|_{p_{i}}$ and the orientation factors $|o_{p_{i}}|$. The clockwise composition of the local system hom factors and the parallel transport along $\partial\Sigma$, produces a holonomy factor around $\partial\Sigma$, which is a number in $\mathbb{Q},\mathbb{R},\mathbb{Z}$ depending on the coefficient ring choice. Using (69) and Remark 6.9, as well as the clockwise orientation convention on the Stasheff associahedron, we acquire a (naïve) orientation on $T\mathcal{M}(p_{1},\ldots p_{k})$. To assign orientation and weighting factors to $\mathcal{M}(p_{1},\ldots p_{k})$, we take the product of the holonomy factor, the naïve orientation on $T\mathcal{M}(p_{1},\ldots p_{k})$, and another universal sign factor

$$(-1)^{\deg p_{1}+2\deg p_{2}+\ldots+k\deg p_{k}}(-1)^{\deg p_{k}}.$$

The appearance of this universal sign adjustment is a familiar convention in the open-closed map, cf. [2, eqn 5.24]. The notation $\mathcal{M}$ is a shorthand for the weighted sum of all the $(n-1)$-dimensional moduli spaces involved in the construction of the bordism current.

We equip the domain $\Sigma$ with the complex orientation, and together with an extra minus sign, the orientation on $\mathcal{M}$ induces the orientation on $\mathcal{C}$. This minus sign arises for the same reason as in Example 6.2, namely the discrepancy between our clockwise convention on $\partial\Sigma$, with the standard complex orientation on $\Sigma$.