For the programme of §3, it will be necessary to enlarge the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ of a Calabi–Yau $m$-fold $(M,J,g,\Omega)$ to include immersed Lagrangians. As a first step in doing this, Akaho and the author [2] explain how to generalize the Lagrangian Floer cohomology of Fukaya, Oh, Ohta and Ono [20] from embedded Lagrangians to immersed Lagrangians with transverse self-intersections. We now explain some of the main ideas in [2].

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, and $(L,E)$ a Lagrangian brane in $M$. As in §2.5, in the embedded case [20], a bounding cochain for $(L,E)$ is a singular $(m\!-\!1)$-chain $b\in C_{m-1}(L,\Lambda_{\rm nov}^{+})$ (or equivalence class of such chains), satisfying an equation (2.15) involving virtual chains for moduli spaces ${\mathbin{\smash{\,\,\overline{\!\!\mathcal{M}\!}\,}}}_{k+1}^{\rm main}(J,\beta)$ of $J$-holomorphic discs in $M$ with boundary in $L$.

In the immersed case [2], if $L$ has transverse self-intersections, a bounding cochain $b$ for $(L,E)$ consists of two pieces of data: a chain $b_{\rm ch}$ in $C_{m-1}(L,\Lambda_{\rm nov}^{+})$ as above, and also, for each point $p\in M$ at which two local sheets $L_{+},L_{-}$ of $L$ intersect transversely with $\mu_{L_{+},L_{-}}(p)=1$, an element

$$b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}|_{p},E_{-}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0},$$

(2.17)

where we write $E_{\pm}$ for the restriction of $E\rightarrow L$ to the local sheets $L_{\pm}$. These $b_{\rm ch},b_{p}$ must satisfy equations involving virtual chains for moduli spaces of $J$-holomorphic discs in $M$ with boundary in $L$, but now these $J$-holomorphic discs can be polygons with ‘corners’ at self-intersection points of $L$.

For example, suppose $(L_{1},E_{1}),(L_{2},E_{2})$ are embedded, transversely intersecting Lagrangian branes in $M$. Then $(L,E)=(L_{1},E_{1})\cup(L_{2},E_{2})$ is an immersed Lagrangian brane in $M$. A bounding cochain $b$ for $(L,E)$ could consist of $b_{\rm ch}=b_{1}\oplus b_{2}$, where $b_{i}\in C_{m-1}(L_{i},\Lambda_{\rm nov}^{+})$ for $i=1,2$ are embedded bounding cochains for $(L_{1},E_{1}),(L_{2},E_{2})$, together with elements $b_{p}$ in (2.17) for $p\in L_{1}\cap L_{2}$ with $\mu_{L_{1},L_{2}}(p)=1$ or $\mu_{L_{2},L_{1}}(p)=1$ which encode how the objects $(L_{1},E_{1},b_{1}),(L_{2},E_{2},b_{2})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ are glued together to make $(L,E,b)$. For instance, if we have a distinguished triangle in $D^{b}{\mathbin{\mathscr{F}}}(M)$

then the $b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ for $p\in L_{1}\cap L_{2}$ with $\mu_{L_{2},L_{1}}(p)=1$ form a chain in $CF^{1}\bigl((L_{2},E_{2}),(L_{1},E_{1})\bigr)$ representing $\beta$, and $b_{p}=0$ otherwise.

Note that $\beta$ is represented by $(b_{p})$ with $b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$, but to define a bounding cochain we require that $b_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$. This can be achieved by multiplying $\beta,b_{p}$ by $P^{\lambda}$ for $\lambda\gg 0$, which does not change $(L,E,b)$ up to isomorphism in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

The new cause of obstructions to $HF^{*}$ for immersed Lagrangians $L$ with transverse self-intersections is ‘teardrop-shaped’ $J$-holomorphic discs $\Sigma$ of the form shown in Figure 2.3, with one corner at $q\in M$, and with $\mu_{L_{+},L_{-}}(q)=2$, where $L_{\pm}$ are the local sheets of $L$ intersecting at $q$. As $\mu_{L_{+},L_{-}}(q)=2$ the moduli space of such discs has virtual dimension 0. Such $\Sigma$ only obstruct $HF^{*}$ if they have ‘small area’ (that is, $\mathop{\rm area}(\Sigma)$ is smaller than the areas of other relevant curves with boundary in $L$). Thus we deduce an analogue of Lemma 2.21:

In §2.5 we explained that if $(L^{t},E^{t}):t\in[0,1]$ is a smooth family of embedded Lagrangian branes with the $L^{t}$ Hamiltonian isotopic and the $E^{t}$ locally constant in $t$, and $b^{0}$ is a bounding cochain for $(L^{0},E^{0})$, then $b^{0}$ extends to bounding cochains $b^{t}:t\in[0,1]$ for $(L^{t},E^{t})$ by a kind of ‘parallel transport’, and the isomorphism class of $(L^{t},E^{t},b^{t})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ is independent of $t\in[0,1]$.

In the immersed case, things are more complicated. Firstly, there are two notions of Hamiltonian isotopy. Let $\iota^{t}:L\rightarrow M$ for $t\in[0,1]$ be a smooth family of compact, immersed Lagrangians in $M$, where we also write $\iota^{t}:L\rightarrow M$ as $L^{t}$. We call the family globally Hamiltonian isotopic if $\frac{{\rm d}}{{\rm d}t}\iota^{t}$ for $t\in[0,1]$ is Hamiltonian flow by $H^{t}\circ\iota^{t}$ for some smooth $H^{t}:M\rightarrow{\mathbin{\mathbb{R}}}$. We call the family locally Hamiltonian isotopic if $\frac{{\rm d}}{{\rm d}t}\iota^{t}$ for $t\in[0,1]$ is Hamiltonian flow by some smooth $H^{t}:L\rightarrow{\mathbin{\mathbb{R}}}$, where there may exist $p_{+},p_{-}\in L$ with $\iota^{t}(p_{+})=\iota^{t}(p_{-})$ but $H^{t}(p_{+})\neq H^{t}(p_{-})$, so that $H^{t}$ does not descend from $L$ to $M$.

There is a notion of ‘parallel transport’ for bounding cochains $b^{t}$ along such local Hamiltonian isotopies, but it does not work all the time. Suppose for simplicity that $L^{t}$ has only transverse self-intersections for all $t\in[0,1]$. Then the self-intersection points of $L^{t}$ in $M$ depend smoothly on $t\in[0,1]$, so we can write $p^{t}=\iota^{t}(p^{t}_{+})=\iota^{t}(p^{t}_{-})$ for the intersection of local sheets $L^{t}_{+}\ni p^{t}_{+}$, $L^{t}_{-}\ni p^{t}_{-}$ of $L^{t}$ for $t\in[0,1]$, where $p^{t}_{\pm},L^{t}_{\pm}$ depend smoothly on $t$. Then $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})$ is independent of $t$.

Let $b^{t}$ be a bounding cochain for $(L^{t},E^{t})$ depending smoothly on $t$, with $(L^{t},E^{t},b^{t})\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$. Then $b^{t}$ evolves in time by a kind of ‘parallel transport’. Let $p^{t},p^{t}_{\pm},L^{t}_{\pm}$ be as above with $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})=1$. As above, $b^{t}$ includes an element $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$. Since the local systems $E^{t}$ are locally constant in $t$, we can identify the fibres $E_{+}^{t}|_{p^{t}}$ for $t\in[0,1]$, and the fibres $E_{-}^{t}|_{p^{t}}$ for $t\in[0,1]$, and so regard $\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ as being independent of $t$. Then $b^{t}_{p^{t}}$ is not constant, but evolves by

$$\frac{{\rm d}}{{\rm d}t}b^{t}_{p^{t}}=\bigl(H^{t}(p^{t}_{+})-H^{t}(p^{t}_{-})\bigr)\cdot\log P\cdot b^{t}_{p^{t}}.$$

(2.18)

Integrating this over $[0,t]$ gives

$$b^{t}_{p^{t}}=P^{\textstyle\int_{0}^{t}(H^{t}(p^{s}_{+})-H^{t}(p^{s}_{-}){\rm d}s}\cdot b^{0}_{p^{0}}.$$

(2.19)

Suppose $b^{0}_{p^{0}}\neq 0$, and write $b^{0}_{p^{0}}=\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ with $a_{i}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}(E_{+}^{0}|_{p^{0}},E_{-}^{0}|_{p^{0}})$, $a_{0}\neq 0$, and $0\leqslant\penalty\lambda_{0}<\lambda_{1}<\lambda_{2}<\cdots$. Then

$$b^{t}_{p^{t}}=\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}+\textstyle\int_{0}^{t}\bigl(H^{t}(p^{s}_{+})-H^{t}(p^{s}_{-})\bigr){\rm d}s}.$$

(2.20)

Thus $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}\subset\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$, required for $b^{t}$ to be a bounding cochain by (2.17), if and only if

$$\lambda_{0}+\int_{0}^{t}\bigl(H^{t}(p^{s}_{+})-H^{t}(p^{s}_{-})\bigr){\rm d}s\geqslant 0.$$

(2.21)

Hence we have the following situation, which will be important in §3.4. Let $(L^{t},E^{t}),$ $t\in[0,1]$ be a local Hamiltonian isotopy of Lagrangian branes in $M$, and $b^{0}$ a bounding cochain for $(L^{0},E^{0})$. We may extend $b^{0}$ to a family of bounding cochains $b^{t}:t\in[0,T]$ for $(L^{t},E^{t}),t\in[0,T]$ for some $T\in[0,1]$, so that $(L^{t},E^{t},b^{t})\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$. But at time $t=T$ we may cross a ‘wall’ when the l.h.s. of (2.21) becomes zero, and we cannot define $b^{t}$ for $t>T$. Either $(L^{t},E^{t})$ for $t>T$ may have $HF^{*}$ obstructed, or a bounding cochain $\tilde{b}^{t}$ may exist but $(L^{t},E^{t},\tilde{b}^{t})\not\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

The Lagrangian $h$-principle, due to Gromov [23, p. 60-61] and Lees [46], says that two Lagrangians $L,L^{\prime}$ are locally Hamiltonian isotopic in $(M,\omega)$ if and only if they are homotopic in a weak sense, which can be well understood using homotopy theory, and is weaker than isomorphism in $D^{b}{\mathbin{\mathscr{F}}}(M)$. So we should expect local Hamiltonian isotopies to connect Lagrangians with $HF^{*}$ unobstructed and with $HF^{*}$ obstructed, or to connect non-isomorphic Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M)$.