H(\hom_{\widetilde{\mathcal{B}}}(U_{j},V_{l}))$

is always an isomorphism. For $j<l$, this follows from Proposition 8.18. For
$j=l$, this follows from strict unitality. For $j>l$, the domain is trivial by
definition, while $H(\hom_{\widetilde{\mathcal{B}}}(U_{j},V_{l}))$ is trivial
by Lemma 5.9. This implies that
$r_{l}(e_{x_{l}})\in\hom_{\mathcal{Q}_{r}}(\mathcal{U}_{l},\mathcal{V}_{l})$
is quasi-isomorphism. Thus we have completed the proof of Lemma 10.1 assuming
Lemma 10.3 and 10.4.

####  10.6. Proof of Lemma 10.2

Consider the commutative diagram:

(10.12)
${\hom_{\mathcal{A}}(S_{n},S_{l})}$${\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{l}\mathcal{S}_{n},\mathcal{G}^{l}\mathcal{S}_{l})}$${\hom_{\mathcal{A}}(S_{n},S_{l})\otimes\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{l}\mathcal{S}_{l},\mathcal{G}^{l}\mathcal{S}_{l})}$${\hom_{\mathcal{Q}_{r}}(D\hom_{\mathcal{A}}(S_{n},S_{l})\otimes\mathcal{G}^{l}\mathcal{S}_{l},\mathcal{G}^{l}\mathcal{S}_{l}).}$$\scriptstyle{(r_{\mathcal{G}^{l}\Delta})^{1}}$$\scriptstyle{a\mapsto
a\otimes
e_{\mathcal{G}^{l}\mathcal{S}_{l}}}$$\scriptstyle{\mu^{2}_{\mathcal{Q}_{r}}(\cdot,r_{n})}$

By Lemma 10.1, the $A_{\infty}$-homomorphism $r_{n}$ defined by (10.10) is a
quasi-isomorphism, and so is the right vertical arrow in (10.12) by Lemma 6.8.
The left vertical arrow in (10.12) is defined by the formula $a\mapsto
a\otimes e_{\mathcal{G}^{l}\mathcal{S}_{l}}$ and is also a quasi-isomorphism.
Indeed, by Lemma 10.1,

$H(\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{l}\mathcal{S}_{l},\mathcal{G}^{l}\mathcal{S}_{l}))\cong
H(\hom_{\mathcal{Q}_{r}}(\mathcal{U}_{l},\mathcal{U}_{l}))\cong
H(\hom_{\mathcal{B}}(U_{l},U_{l}))\cong\mathbb{K}$

is generated by the identity homomorphism
$e_{\mathcal{G}^{l}\mathcal{S}_{l}}$. This completes the proof of Lemma 10.2.

######  Remark 10.5.

The proof of Theorem 8.3 relies essentially only on a few formal properties of
the filtered bimodule $\Delta=\Delta_{R}$, namely, Lemma 9.3, Lemma 10.3 and
Lemma 10.4, whereas the explicit construction of $\Delta_{R}$ is somewhat
irrelevant. In particular, Lemma 10.3 and 10.4 are statements about the
cohomological categories. We have taken a detour in Section 9 to show that the
operations $\mu^{r|1|s}_{\Delta_{R}},r,s\geq 0$ preserve the geometric
filtration on $\Delta_{R}(U_{j},S_{k})$. Readers should feel free to propose a
more direct and less technical route as long as these properties can be
verified. ∎

###  11\. The Vertical Gluing Theorem

####  11.1. The idea of the vertical gluing theorem

To finish the proof of Theorem 8.3, we prove Lemma 10.3, 10.2 and Proposition
8.18 (on quasi-units) in this section using a vertical gluing theorem. To
illustrate, we focus on the case of Lemma 10.3 and first explain the basic
idea. Consider the $A_{\infty}$-subcategory $\mathcal{E}_{0}$ of
$\mathcal{E}_{R}=\mathcal{E}_{\Delta_{R}}$ with
$\operatorname{Ob}\mathcal{E}_{0}=(U_{j},S_{n},S_{l}),1\leq k<n\leq m$. One
may apply the rotational operation $\mathcal{R}$ in Remark 8.6 to
$\mathcal{E}_{0}$ and obtain a new $A_{\infty}$-category $\mathcal{E}_{1}$
with $\operatorname{Ob}\mathcal{E}_{1}=(S_{n},S_{l},U_{j}^{\prime})$,
$U_{j}^{\prime}\colonequals\Lambda_{x_{j},\eta_{j}-2\pi}$. $\mathcal{E}_{1}$
is modeled on the metric ribbon tree $\mathcal{T}^{2,1}_{R}$. Unwinding the
definition, Lemma 10.3 is equivalent to the statement that

(11.1)
$\mu^{2}_{\mathcal{E}_{1}}:\mathcal{G}^{l}\hom_{\mathcal{E}_{1}}(S_{l},U_{j}^{\prime})\otimes\hom_{\mathcal{E}_{1}}(S_{n},S_{l})\to\mathcal{G}^{l}\hom_{\mathcal{E}_{1}}(S_{n},U_{j}^{\prime}).$

is a quasi-isomorphism for all $R\gg\pi$. The roles between stable and
unstable thimbles are symmetric here. To ease our notation, we shall verify
this instead for the original $A_{\infty}$-category. By rotating Figure 30 by
$180^{\circ}$, it is clear that (11.1) is equivalent to the following lemma.

######  Lemma 11.1.

For any triple of objects $U_{j}\prec U_{l}\prec S_{k}$ in
$\mathcal{E}_{R}=\mathcal{E}_{\Delta_{R}}$ and $R\gg\pi$, the map

(11.2)
$\mu^{0|1|1}_{\Delta_{R}}:\mathcal{G}^{l}\Delta_{R}(U_{l},S_{k})\otimes\hom_{\mathcal{B}}(U_{j},U_{l})\to\mathcal{G}^{l}\Delta_{R}(U_{j},S_{k}).$

is a quasi-isomorphism. This lemma is non-trivial only when $k\leq l\leq j$;
see Figure 34.

\begin{overpic}[scale={.15}]{Pic54.png}
\put(-3.0,10.0){\small$U_{j}^{\prime}$} \put(39.0,30.0){\small$S_{n}$}
\put(39.0,22.0){\small$S_{l}$} \put(100.0,23.0){\small$S_{k}$}
\put(79.0,12.0){\small$x_{l}$} \put(58.0,0.0){\small$U_{j}$}
\put(58.0,5.0){\small$U_{l}$} \put(58.0,10.0){\small$U_{\star}$}
\put(100.0,10.0){\small$S_{\star}$} \put(40.0,15.0){rotate} \end{overpic}
Figure 34. The thimbles $U_{\star}$ and $S_{\star}$.

Lemma 11.1 is a property about the cohomological category, which depends only
on a single $(2+1)$-pointed disk. In this case, the choice of the perturbation
1-forms $\delta H^{R}$ can be more flexible: the norms in (9.11) may not tend
to zero as $R\to\infty$; it suffices to require that the left hand side of
(9.11) is bounded by a fixed constant $\epsilon^{\prime}$. If $R$ is chosen
sufficiently large depending on this $\epsilon^{\prime}$, the continuation
method can be used to show that $\eqref{VT.E.1}$ is independent of these
perturbations up to chain homotopy (however, the phase pairs are remained
fixed in this process). Lemma 11.1 is then proved by choosing some special
$\delta H^{R}$ so that the complex $\mathcal{G}^{l}\Delta(U_{j},S_{k})$ can be
described rather concretely. To set the stage, consider the thimbles

$U_{\star}=\Lambda_{x_{l},\pi}\text{ and
}S_{\star}\colonequals\Lambda_{x_{l},2\pi}.$

Choose admissible Floer data
$\mathfrak{a}_{j}^{un}=(\pi,\alpha_{j}^{un},\beta_{*},\epsilon_{*},\delta
H_{j}^{st})$ and
$\mathfrak{a}^{st}_{k}=(\pi,\alpha_{k}^{st},\beta_{*},\epsilon_{*},\delta
H_{k}^{st})$, one for each pair $(U_{j},U_{\star})$ and $(S_{\star},S_{k})$,
where $\alpha_{j}^{un}(s),\alpha_{k}^{st}(s)$ are given as in (9.8) and
$\beta_{*},\epsilon_{*}$ as in (9.10). The perturbation 1-forms $\delta
H_{j}^{st},\delta H_{k}^{un}$ are supported on $[0,\pi]_{s}$ as usual. Then
these Floer data can be concatenated to give a Floer datum
$\alpha_{jk}^{R}=(R+\pi,\alpha_{jk}^{R},\beta_{*},\epsilon_{*},\delta
H^{R}_{jk})$ for the pair $(U_{j},S_{k})$ with

(11.3) $\delta H^{R}_{jk}=\left\\{\begin{array}[]{ll}\delta H_{k}^{st}&\text{
if }s\in[\pi,R+\pi]_{s},\\\ \delta H_{j}^{un}&\text{ if }s\in[0,\pi]_{s},\\\
0&\text{ otherwise.}\end{array}\right.$

Under these assumptions, Lemma 9.1 can be refined as follows.

######  Lemma 11.2.

For all $k\leq l\leq j$ and $R\gg\pi$, there is a gluing bijection

$\displaystyle\mathfrak{C}(S_{\star},S_{k};\mathfrak{a}_{k}^{st})\times\mathfrak{C}(U_{j},U_{\star};\mathfrak{a}_{j}^{un})$
$\displaystyle\to\mathfrak{C}^{l}(U_{j},S_{k};\mathfrak{a}^{R}_{jk})$
$\displaystyle(p^{st}_{k},p^{un}_{j})$ $\displaystyle\mapsto
p^{st}_{k}\circ_{R}p^{un}_{j}$

such that $p^{st}_{k}\circ_{R}p^{un}_{j}\to(p^{st}_{k},p^{un}_{j})$ in the
sense of Lemma 9.1 as $R\to\infty$ and
$\operatorname{\mathfrak{gr}}(p^{st}_{k}\circ_{R}p^{un}_{j})=\operatorname{\mathfrak{gr}}(p^{st}_{k})+\operatorname{\mathfrak{gr}}(p^{un}_{j})$.
Thus there is an isomorphism between graded vector spaces

(11.4)
$\widehat{\Phi}_{jk}:\operatorname{CF}^{*}_{\natural}(S_{\star},S_{k};\mathfrak{a}_{k}^{st})\otimes\operatorname{CF}^{*}_{\natural}(U_{j},U_{\star};\mathfrak{a}_{j}^{un})\to\mathcal{G}^{l}\Delta(U_{j},S_{k}).$

\begin{overpic}[scale={.15}]{Pic55.png}
\put(12.0,16.0){\small$\mathbb{R}_{t}\times[\pi,R]_{s}$}
\put(5.0,7.0){\small$U_{j}$} \put(5.0,16.0){\small$x_{l}$}
\put(5.0,26.0){\small$S_{k}$} \put(65.0,5.0){\small$U_{j}$}
\put(65.0,13.0){\small$U_{\star}$} \put(65.0,20.0){\small$S_{\star}$}
\put(65.0,29.0){\small$U_{k}$} \put(75.0,5.0){\small$x_{j}$}
\put(75.0,13.0){\small$x_{l}$} \put(75.0,20.0){\small$x_{l}$}
\put(75.0,29.0){\small$x_{k}$} \put(54.0,25.0){\small$p^{st}_{k,-}$}
\put(86.0,25.0){\small$p^{st}_{k,+}$} \put(54.0,9.0){\small$p^{un}_{j,-}$}
\put(86.0,9.0){\small$p^{un}_{j,+}$} \put(-2.0,20.0){\small$p^{st}_{k,-}$}
\put(30.0,20.0){\small$p^{st}_{k,+}$} \put(-2.0,11.0){\small$p^{un}_{j,-}$}
\put(30.0,11.0){\small$p^{un}_{j,+}$} \put(-1.0,15.5){\small$\circ_{R}$}
\put(31.0,15.5){\small$\circ_{R}$} \put(39.0,19.0){\small converge}
\put(42.0,14.0){\small glue} \end{overpic} Figure 35. Gluing Floer
differentials.

######  Theorem 11.3 (The Vertical Gluing Theorem I).

The map $\widehat{\Phi}_{jk}$ is also an isomorphism between complexes. This
means that for any solitons
$p_{k,\pm}^{st}\in\mathfrak{C}(S_{\star},S_{k};\mathfrak{a}_{k}^{st})$ and
$p_{j,\pm}^{un}\in\mathfrak{C}(U_{j},U_{\star};\mathfrak{a}_{j}^{un})$, there
is a gluing bijection between finite sets

(11.5)
$\widecheck{\mathcal{M}}(p_{k,-}^{st}\circ_{R}p_{j,-}^{un},p_{k,+}^{st}\circ_{R}p_{j,+}^{un};\mathfrak{a}_{jk}^{R})\cong\left\\{\begin{array}[]{ll}\widecheck{\mathcal{M}}(p_{k,-}^{st},p_{k,+}^{st})&\text{
if }p_{j,-}^{un}=p_{j,+}^{un},\\\
\widecheck{\mathcal{M}}(p_{j,-}^{un},p_{j,+}^{un})&\text{ if
}p_{k,+}^{st}=p_{k,+}^{st},\\\ \emptyset&\text{ otherwise }\end{array}\right.$

where $\widecheck{\mathcal{M}}(\cdots;\mathfrak{a}_{jk}^{R})$ is the moduli
space contributing to the differential map (2.14) for the associated graded
complex $\mathcal{G}^{l}\Delta(U_{j},S_{k})$$($so the expected dimension of
$\widecheck{\mathcal{M}}=\mathcal{M}/\mathbb{R}$ $=0)$.

Theorem 11.3 says that not only $\alpha$-solitons can be glued vertically in
the direction of $s$ but so are $\alpha$-instantons. This is possible because
the drop of the action functional along such an $\alpha$-instanton does not
increase as $R\to\infty$, so we have uniform energy control. This implies that
any $\alpha^{R}_{jk}$-instanton must decay exponentially towards the critical
point $x_{l}\in\operatorname{Crit}(W)$ as
$z\in\mathbb{R}_{t}\times[\pi,R]_{s}$ approaches the middle line
$\mathbb{R}_{t}\times\\{\frac{R+\pi}{2}\\}$, and this convergence is uniform
in the time variable; see Lemma 11.6 below. Thus one may establish a
compactness theorem and compare the moduli space for a finite large $R$ with
that of $``R=\infty"$, i.e., when the neck is completely stretched.

Now we explain the case for cobordism maps. The $(2+1)$-pointed disk $S$
carries a unique $S$-compatible quadratical differential $\phi$ with
$\mathcal{T}_{\phi}=\mathcal{T}^{2,1}_{R}$. In terms of the decomposition
$S=S^{st}\cup Z_{R}\cup S^{un}$ in Figure 31, we label
$S^{st}\cong\mathbb{R}_{t}\times[0,2\pi]_{s}$ by $(S_{\star},S_{k})$ and the
$(2+1)$-pointed disk $S^{un}$ by $(U_{j},U_{l},U_{\star})$. When restricted on
$S^{un}$, the horizontal foliation of $\phi|_{S^{un}}$ is modeled on
$\mathcal{T}^{2,1}_{\pi}$. Fix an admissible continuation datum $\mathfrak{c}$
on $S^{st}$ and a cobordism datum $\mathfrak{b}$ on $S^{un}$ to define the
maps

$\displaystyle\operatorname{Cont}:$
$\displaystyle\operatorname{CF}^{*}_{\natural}(S_{\star},S_{k};\mathfrak{a}_{k}^{st})\to\operatorname{CF}^{*}_{\natural}(S_{\star},S_{k};\mathfrak{a}_{k}^{st}),$
$\displaystyle\mu^{2}:$
$\displaystyle\operatorname{CF}^{*}_{\natural}(U_{l},U_{\star};\mathfrak{a}_{l}^{un})\otimes\hom_{\mathcal{B}}(U_{j},U_{l})\to\operatorname{CF}^{*}_{\natural}(U_{j},U_{\star};\mathfrak{a}_{j}^{un}).$

One may simply take $\mathfrak{c}$ be the identity cobordism, so
$\operatorname{Cont}=\operatorname{Id}$; but this is unnecessary for the next
gluing theorem. Concatenate $\mathfrak{c}$ and $\mathfrak{b}$ to obtain a
rigid cobordism datum on $(S,\phi)$, denoted by
$\mathfrak{c}\circ_{R}\mathfrak{b}$. The next theorem says that any moduli
space contributing to (11.2) is also obtained by a gluing construction when
$R\gg\pi$.

######  Theorem 11.4 (The Vertical Gluing Theorem II).

There is a gluing bijection between finite sets:

$\mathcal{M}(p_{k,0}^{st}\circ_{R}p_{j,0}^{un},\ p_{1},\
p_{k,2}^{st}\circ_{R}p_{l,2}^{un}:\mathfrak{c}\circ_{R}\mathfrak{b})\xrightarrow{\cong}\mathcal{M}(p_{k,0}^{st},p_{k,2}^{st};\mathfrak{c})\times\mathcal{M}(p_{j,0}^{un},p_{1},p_{l,2}^{un};\mathfrak{b})$

where $\mathcal{M}(\cdots;\mathfrak{c}\circ_{R}\mathfrak{b})$ $($resp.
$\mathcal{M}(\cdots;\mathfrak{c})$ and $\mathcal{M}(\cdots;\mathfrak{b}))$ is
any moduli space contributing to the map $\mu_{\Delta}^{0|1|1}$ $($resp. to
$\operatorname{Cont}$ and $\mu^{2}$$)$; see Figure 36 below. Thus the chain
map (11.2) fits into a commutative diagram

(11.6)
${\operatorname{CF}^{*}_{\natural}(S_{\star},S_{k};\mathfrak{a}_{k}^{st})\otimes\operatorname{CF}^{*}_{\natural}(U_{l},U_{\star};\mathfrak{a}_{l}^{un})\otimes\hom_{\mathcal{B}}(U_{j},U_{l})}$${\operatorname{CF}^{*}_{\natural}(S_{\star},S_{k};\mathfrak{a}_{k}^{st})\otimes\operatorname{CF}^{*}_{\natural}(U_{j},U_{\star};\mathfrak{a}_{j}^{un})}$${\mathcal{G}^{l}\Delta(U_{l},S_{k})\otimes\hom_{\mathcal{B}}(U_{j},U_{l})}$${\mathcal{G}^{l}\Delta(U_{j},S_{k}).}$$\scriptstyle{\operatorname{Cont}\otimes\mu^{2}}$$\scriptstyle{\widehat{\Phi}_{lk}\otimes\operatorname{Id}}$$\scriptstyle{\widehat{\Phi}_{jk}}$$\scriptstyle{\mu^{0|1|1}_{\Delta}}$

Theorem 11.4 implies
$[\mu_{\Delta}^{0|1|1}]=[\operatorname{Cont}]\otimes[\mu^{2}]$. Since
$[\operatorname{Cont}]=\operatorname{Id}$, and
$\operatorname{HF}_{\natural}^{*}(U_{l},U_{\star})=H(\operatorname{CF}^{*}_{\natural}(U_{l},U_{\star};\mathfrak{a}_{l}^{un}))$
is generated by the quasi-unit $e_{x_{l}}$ at $x_{l}$, $[\mu^{2}]$ is an
isomorphism by Proposition 8.18, and so is $[\mu_{\Delta}^{0|1|1}]$. This
proves Lemma 11.1 and so Lemma 10.3.

Lemma 10.4 is in fact the special case of Lemma 11.1 with $j=l$. Finally,
Proposition 8.18 (the property about quasi-units) is proved by another
application of the vertical gluing theorem and is reduced the easier case,
Lemma 8.20, which has been verified directly.

The proof of Theorem 11.4 is almost identical to that of Theorem 11.3 and is
omitted in this paper. In fact, Theorem 11.4 is slightly simpler, since we are
gluing moduli spaces of index $0$, in which case the equation does not carry a
translation symmetry.

The rest of this section is therefore devoted to the technical proof of
Theorem 11.3. Readers who feel comfortable with this heuristic may proceed
directly to the next part.

\begin{overpic}[scale={.15}]{Pic56.png}
\put(12.0,23.0){\small$\mathbb{R}_{t}\times[\pi,R]_{s}$}
\put(5.0,13.0){\small$U_{j}$} \put(5.0,23.0){\small$x_{l}$}
\put(5.0,32.0){\small$S_{k}$} \put(65.0,9.0){\small$U_{j}$}
\put(65.0,20.0){\small$U_{\star}$} \put(65.0,27.0){\small$S_{\star}$}
\put(65.0,36.0){\small$S_{k}$} \put(75.0,2.0){\small$x_{j}$}
\put(75.0,20.0){\small$x_{l}$} \put(75.0,27.0){\small$x_{l}$}
\put(75.0,36.0){\small$x_{k}$} \put(54.0,32.0){\small$p^{st}_{k,0}$}
\put(86.0,32.0){\small$p^{st}_{k,2}$} \put(54.0,16.0){\small$p^{un}_{j,0}$}
\put(86.0,16.0){\small$p^{un}_{l,2}$} \put(-2.0,27.0){\small$p^{st}_{k,0}$}
\put(30.0,27.0){\small$p^{st}_{k,2}$} \put(-2.0,18.0){\small$p^{un}_{j,0}$}
\put(30.0,18.0){\small$p^{un}_{l,2}$} \put(30.0,8.0){\small$p_{1}$}
\put(86.0,6.0){\small$p_{1}$} \put(-1.0,22.5){\small$\circ_{R}$}
\put(31.0,22.5){\small$\circ_{R}$} \put(39.0,26.0){\small converge}
\put(42.0,20.0){\small glue} \end{overpic} Figure 36. Gluing the cobordism
map.

####  11.2. Proof of Lemma 11.2: some estimates about solitons

As usual, these gluing theorems are proved using the implicit function theorem
[AD14, Lemma 9.4.4] or [MS12, Propsoiton A.3.4], and the key ingredient is to
estimate the right inverse of the linearized operator at a pre-gluing
configuration. Although Lemma 11.2 is standard and follows immediately from
the argument of [AD14, Section 14], we summarize some key estimates as a
warmup for the more sophisticated case of Theorem 11.3. The upshot is that all
such estimates must be uniform in the stretching parameter $R$. From now on we
assume that $k=1,j=m$ to simplify our notations; the general case will follow
by the same argument. The subscripts $j,k$ are saved later for different
purposes. For simplicity, write

$\mathfrak{C}^{st}=\mathfrak{C}(S_{\star},S_{1};\mathfrak{a}^{st}_{1}),\
\mathfrak{C}^{un}=\mathfrak{C}(U_{m},U_{\star};\mathfrak{a}^{un}_{m})\text{
and }\mathfrak{C}^{R}=\mathfrak{C}^{l}(U_{m},S_{1};\mathfrak{a}^{R}_{m1}).$

To start, choose a cutoff function $\chi:\mathbb{R}_{s}\to[0,1]$ such that
$\chi(s)\equiv 0$ if $s\leq 0$ and $\equiv 1$ if $s\geq 1$, and define

$\displaystyle\chi^{un}_{R}(s)$
$\displaystyle\colonequals\chi\big{(}3-\frac{8s}{R+\pi}\big{)},$
$\displaystyle\chi^{st}_{R}(s)$
$\displaystyle\colonequals\chi\big{(}\frac{8s}{R+\pi}-5\big{)}.$

Then $(\chi^{un}_{R},\chi^{st}_{R})(s)\equiv(1,0)$ if $s\leq\frac{R+\pi}{4}$
and $\equiv(0,1)$ if $s\geq\frac{3(R+\pi)}{4}$ and $\equiv(0,0)$ for
$s\in[\frac{3(R+\pi)}{8},\frac{5(R+\pi)}{8}]_{s}$. For any pair of thimbles
$(\Lambda_{0},\Lambda_{1}),\ \Lambda_{n}=\Lambda_{q_{n},\theta_{n}}$, $n=1,2$,
let $\mathcal{P}_{r}(\Lambda_{0},\Lambda_{0}),r\geq 2$ denote the space of
smooths paths $\mathbb{R}_{s}\to M$ which connect $q_{0},q_{1}$ and have
finite $L^{r}_{1}$-distance to a model path; in contrast to (2.9), we use the
$L^{r}_{1}$-norm instead of $L^{2}_{k}$. Let $R_{0}>0$. If
$p^{un}\in\mathcal{P}_{r}(U_{j},U_{\star})$ and
$p^{st}\in\mathcal{P}_{r}(S_{\star},S_{k})$ satisfy that
$p^{un}(s),p^{st}(s)\in\mathcal{O}(x_{l})$ for all $s\geq R_{0}$, where
$\mathcal{O}(x_{l})$ is a normal neighborhood of $x_{l}$, then for $R\geq
4R_{0}$, there is a pre-gluing map

(11.7)
$\Phi_{R}:\mathcal{P}_{r}(S_{\star},S_{1})\times\mathcal{P}_{r}(U_{m},U_{\star})\dashrightarrow\mathcal{P}_{r}(U_{m},S_{1})$

such that $(p^{st},p^{un})$ is sent to the path

$\Phi_{R}(p^{st},p^{un})(s)\colonequals\left\\{\begin{array}[]{ll}p^{st}(s-R)&\text{
if }s\geq\frac{3(\pi+R)}{4},\\\
\exp_{x_{l}}\big{(}\chi^{un}_{R}(s)\cdot\exp_{x_{l}}^{-1}(p^{un}(s))+\chi^{st}_{R}(s)\cdot\exp_{x_{l}}^{-1}(p^{st}(s))\big{)}&\text{
otherwise, }\\\ p^{un}(s)&\text{ if }s\leq\frac{\pi+R}{4}.\end{array}\right.$

By construction, $\Phi_{R}(p^{st},p^{un})(s)\equiv x_{l}$ for
$s\in[\frac{3(R+\pi)}{8},\frac{5(R+\pi)}{8}]_{s}$. If $p^{st},p^{un}$ are
$\alpha^{st}_{1}$-and $\alpha^{un}_{m}$-solitons respectively, then
$p^{st}(-s),p^{un}(s)\to x_{l}$ exponentially as $|s|\to\infty$. We use
$p^{\operatorname{pre}}_{R}\colonequals\Phi_{R}(p^{st},p^{un})$ as the pre-
gluing configuration and construct an actual $\alpha_{m1}^{R}$-soliton which
is close to $p^{\operatorname{pre}}_{R}$ using Newton-Picard iteration when
$R\gg\pi$. The key ingredient in this argument is the following estimate.

######  Lemma 11.5.

For any $r\geq 2$, there exists a constant $C_{1}=C_{1}(r)>0$ with the
following property. For any $R\gg\pi$, $p^{st}\in\mathfrak{C}^{st}$ and
$p^{un}\in\mathfrak{C}^{un}$, the operator

$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p^{\operatorname{pre}}_{R}):L^{r}_{1}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{R})^{*}TM)\to
L^{r}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{R})^{*}TM)$

is invertible for $p^{\operatorname{pre}}_{R}=\Phi_{R}(p^{st},p^{un})$.
Moreover, its inverse is bounded by $C_{1}$ in the operator norm.

###### Sketch of Proof.

Consider the following truncations of $p^{st},p^{un}$:

$\displaystyle p^{un}_{R}(s)$
$\displaystyle=\left\\{\begin{array}[]{ll}\Phi_{R}(p^{st},p^{un})(s)&\text{ if
}s\leq\frac{3(R+\pi)}{8},\\\ x_{l}&\text{ otherwise, }\end{array}\right.$
$\displaystyle p^{st}_{R}(s)$
$\displaystyle=\left\\{\begin{array}[]{ll}\Phi_{R}(p^{st},p^{un})(s+R)&\text{
if }s+R\geq\frac{5(R+\pi)}{8},\\\ x_{l}&\text{ otherwise. }\end{array}\right.$

For any $R\gg\pi$, one can think of $p^{un}_{R}(s)$ as a section of
$L^{r}_{1}(\mathbb{R}_{s};(p^{un})^{*}TM)$ using the exponential map along
$p^{un}$, and $p^{un}_{R}\to p^{un}$ as $R\to\infty$ in
$L^{r}_{1}(\mathbb{R}_{s};(p^{un})^{*}TM)$ for all $r\geq 2$. The same holds
also for $p^{st}_{R}$. This shows that
$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}_{1}^{st}}(p^{st}_{R})$ and
$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}_{m}^{un}}(p^{un}_{R})$ are
invertible with uniformly bounded inverses.

\begin{overpic}[scale={.30}]{Pic58.png} \put(3.0,2.0){$R_{*}$}
\put(25.0,2.0){$\frac{R_{*}}{4}$} \put(36.0,2.0){$\frac{3R_{*}}{8}$}
\put(58.0,2.0){$\frac{5R_{*}}{8}$} \put(70.0,2.0){$\frac{3R_{*}}{4}$}
\put(90.0,2.0){$R_{*}\colonequals R+\pi$} \put(7.0,15.0){$\chi_{R}^{un}$}
\put(90.0,15.0){$\chi_{R}^{st}$} \put(35.0,15.0){$\kappa_{R}^{un}$}
\put(60.0,15.0){$\kappa_{R}^{st}$} \end{overpic} Figure 37. The cutoff
functions.

The pre-gluing configuration
$p^{\operatorname{pre}}_{R}=\Phi_{R}(p^{st},p^{un})$ is a “concatenation” of
$p^{un}_{R}$ and $p^{st}_{R}$. To deduce the invertibility of
$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}$ at
$p^{\operatorname{pre}}_{R}=\Phi_{R}(p^{st},p^{un})$, consider the cutoff
functions

$\kappa^{un}(s)\colonequals\cos\bigg{(}\frac{\pi}{2}\cdot\chi(s)\bigg{)}\text{
and }\kappa^{st}(s)\colonequals\sin\bigg{(}\frac{\pi}{2}\cdot\chi(s)\bigg{)}.$

Then $(\kappa^{un})^{2}+(\kappa^{st})^{2}=1$,
$(\kappa^{un},\kappa^{st})(s)=(1,0)$ if $s\leq 0$ and $=(0,1)$ if $s\geq 1$.

$\kappa^{un}_{R}(s)\colonequals\kappa^{un}(\frac{4s}{R+\pi}-\frac{3}{2})\text{
and }\kappa^{st}_{R}(s)\colonequals\kappa^{st}(\frac{4s}{R+\pi}-\frac{3}{2}).$

Then $(\kappa^{un}_{R},\kappa^{st}_{R})(s)\equiv(1,0)$ when
$s\leq\frac{3(R+\pi)}{8}$ and $\equiv(0,1)$ when $s\geq\frac{5(R+\pi)}{8}$.
Now consider the transformation

$\displaystyle U_{R}:L^{r}_{1}(\mathbb{R}_{s};(p^{st}_{R})^{*}TM)\oplus
L^{r}_{1}(\mathbb{R}_{s};(p^{un}_{R})^{*}TM)$ $\displaystyle\to
L^{r}_{1}(\mathbb{R}_{s};T_{x_{l}}M)\oplus
L^{r}_{1}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{R})^{*}TM)$ (11.8)
$\displaystyle\begin{pmatrix}v^{st}\\\ v^{un}\end{pmatrix}$
$\displaystyle\mapsto\begin{pmatrix}\kappa^{un}_{R}&-\kappa^{st}_{R}\\\
\kappa^{st}_{R}&\kappa^{un}_{R}\\\ \end{pmatrix}\begin{pmatrix}v^{st}(s-R)\\\
v^{un}(s)\end{pmatrix}$

with the inverse $U_{R}^{-1}$ defined by a similar matrix operator. Let

$D_{x_{l}}=J\partial_{s}-\operatorname{Hess}_{x_{l}}:L^{r}_{1}(\mathbb{R}_{s};T_{x_{l}}M)\to
L^{r}_{1}(\mathbb{R}_{s};T_{x_{l}}M),\ r\geq 2$

denote the self-adjoint operator associated to the constant path at $x_{l}$.
Then on the interval $[\frac{3(R+\pi)}{8},\frac{5(R+\pi)}{8}]_{s}$, the
operators

$D_{x_{l}}=\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}_{m}^{un}}(p^{un}_{R})=\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(\Phi_{R}(p^{st},p^{un}))$

are the same as
$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}_{1}^{st}}(p^{st}_{R})$ on
$[\frac{3(R+\pi)}{8}-R,\frac{5(R+\pi)}{8}-R]_{s}$. With this in mind, one
verifies that

$\displaystyle
U^{-1}_{R}\circ\big{(}D_{x_{l}}\oplus\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(\Phi_{R}(p^{st},p^{un}))\big{)}$
$\displaystyle\circ U_{R}$ (11.9)
$\displaystyle=\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}_{1}^{st}}(p^{st}_{R})\oplus\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}_{m}^{un}}(p^{un}_{R})+\delta_{R}$

where the error term $\delta_{R}:L^{r}_{1}\to L^{r}_{1}$ involves only the
derivatives of $\kappa_{R}^{un},\kappa_{R}^{st}$ and has norm bounded by $C/R$
for some constant $C>0$. Since the operator norms of $U_{R},U_{R}^{-1}$ are
uniformly bounded, this proves that
$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(\Phi_{R}(p^{st},p^{un}))$
is invertible, and its inverse is bounded in the operator norm by some uniform
$C_{1}>0$ in dependent of $R\gg\pi$. ∎

Using [AD14, Lemma 9.4.4] we can now construct a genuine soliton
$p^{st}\circ_{R}p^{un}\in\mathfrak{C}^{R}$ which is $L^{r}_{1}$-close to
$p^{\operatorname{pre}}_{R}=\Phi_{R}(p^{st},p^{un})$ for $R\gg\pi$. To
conclude the proof of Lemma 11.2, one has to verify an additional compactness
property: suppose that $p_{n}\in\mathfrak{C}^{R_{n}}$ is any sequence of
solitons with $R_{n}\to\infty$ and $p_{n}\to(p^{st},p^{un})$ in the sense of
Lemma 9.1, then for any $n\gg 1$, then $p_{n}$ is $C^{0}$-close to
$p^{\operatorname{pre}}_{n}\colonequals\Phi_{R_{n}}(p^{st},p^{un})$ and so can
be viewed as a section of $(p^{\operatorname{pre}}_{n})^{*}TM$ using the
exponential map along. With this understood, we have

(11.10)
$\|p_{n}-p_{n}^{\operatorname{pre}}\|_{L^{r}_{1}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{n})^{*}TM)}\to
0$

as $n\to\infty$ for all $r\geq 2$. This is a stronger statement than the
$C^{\infty}_{loc}$-convergence in Lemma 9.1 and implies that
$p_{n}=p^{st}\circ_{R_{n}}p^{un}$ for all $n\gg 1$ in this sequence, by the
uniqueness part of [AD14, Lemma 9.4.4]. In order to verify that
$\operatorname{\mathfrak{gr}}(p^{st}\circ_{R}p^{un})=\operatorname{\mathfrak{gr}}(p^{un})+\operatorname{\mathfrak{gr}}(p^{st})$,
note that replacing $p^{st}$ by $p^{st}_{R}$, $p^{un}$ by $p^{un}_{R}$ and
$p^{st}\circ_{R}p^{un}$ by $p^{\operatorname{pre}}_{R}$ does not introduce any
spectral flow to the Hessians for all $R\gg\pi$. One can perturb these paths
further to reduce this grading computation to the operators with Lagrangian
boundary conditions using the Axiom (A’-II). Finally, by Axiom (A-II), it
suffices to verify that if $\Pi^{un},\Pi^{st}$ are any graded linear
Lagrangian subspaces of $T_{x_{l}}M$ such that $\Pi^{un}\pitchfork\Pi^{st}$,
$\Pi^{un}\pitchfork T_{x_{l}}U_{\star}$ and
$T_{x_{l}}S_{\star}\pitchfork\Pi^{st}$, then

(11.11)
$i(T_{x_{l}}S_{\star},\Pi^{st})+i(\Pi^{un},T_{x_{l}}U_{\star})=i(\Pi^{un},\Pi^{st}).$

Note that
$i(T_{x_{l}}S_{\star},T_{x_{l}}U_{\star})=i(T_{x_{l}}\Lambda_{x_{l},2\pi},T_{x_{l}}\Lambda_{x_{l},\pi})=0$
by our grading convention. Then (11.11) follows from the general fact that

$i(\Pi_{0}^{\\#},\Pi_{1}^{\\#})+i(\Pi_{2}^{\\#},\Pi_{3}^{\\#})=i(\Pi_{0}^{\\#},\Pi_{3}^{\\#})+i(\Pi_{2}^{\\#},\Pi_{1}^{\\#})$

for any graded linear Lagrangian subspaces $\Pi_{j}^{\\#},0\leq j\leq 3$ of
$T_{x_{l}}M$ whenever these indices are defined; this follows from the
defining property of the Maslov index; see [Sei08, Section (11h)]. This
completes the proof of Lemma 11.2.

####  11.3. Proof of Theorem 11.3 Part I: Some estimates about instantons

From now on fix some solitons

$p^{st}_{\pm}\in\mathfrak{C}^{st},p^{un}_{\pm}\in\mathfrak{C}^{un},$

and let

$p^{\operatorname{pre}}_{R,\pm}=\Phi_{R}(p^{st}_{\pm},p^{un}_{\pm}),\
p_{R,\pm}=p^{st}_{\pm}\circ_{R}p^{un}_{\pm}\in\mathfrak{C}^{R}.$

$p_{R,\pm}$ are non-degenerate critical points of
$\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}$ (solitons), while
$p^{\operatorname{pre}}_{R,\pm}$ are the pre-gluing configuration
approximating $p_{R,\pm}$. By Lemma 11.5 and (11.10),

(11.12)
$\|\operatorname{Hess}^{-1}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p_{R,\pm})\|_{L^{r}\to
L^{r}_{1}}<C_{1}^{\prime}$

for some constant $C_{1}^{\prime}=C_{1}^{\prime}(r)$. Then by Lemma 9.5, the
drop of the action functional $\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}$

$0\leq\mathcal{A}_{W,\mathfrak{a}_{m1}^{R}}(p_{R,-})-\mathcal{A}_{W,\mathfrak{a}_{m1}^{R}}(p_{R,+})<C_{2}$

is bounded by a uniform constant $C_{2}>0$. This property is special to the
associated graded complex $\mathcal{G}^{l}\Delta(U_{m},S_{1})$. Combined with
Lemma 2.8, this implies that for some $C_{3}>0$, we have

(11.13) $\mathcal{E}_{an}(P_{R};I_{t}\times\mathbb{R}_{s})<C_{3},\
I_{t}\colonequals[t-1,t+1]_{t}$

for any $\alpha_{m1}^{R}$-instanton
$P_{R}\in\mathcal{M}(p_{R}^{-},p_{R}^{+};\mathfrak{a}^{R}_{m1})$ and
$t\in\mathbb{R}_{t}$. The next lemma allows us to control the solution $P_{R}$
over the strip $Z_{R}=\mathbb{R}_{t}\times[2\pi,R-\pi]_{s}$ on which the
$\alpha_{m1}^{R}$-instanton equation (2.12) takes the standard form

$\partial_{t}P_{R}-J\partial_{s}P_{R}-\nabla H=0.$

######  Lemma 11.6.

There exists constants $C_{4},R_{4},\zeta>0$ with the following property. For
any $R+\pi>2R_{4}$ and any $\alpha_{m1}^{R}$-instanton
$P_{R}\in\mathcal{M}(p_{R,-},p_{R,+};\mathfrak{a}_{m1}^{R})$, we have
$P_{R}(t,s)\in\mathcal{O}(x_{l})$ and

$0\leq u_{R}(t,s)\leq
C_{4}\cdot\frac{\cosh(\zeta\cdot(\frac{R+\pi}{2}-s))}{\cosh(\zeta\cdot\frac{R+\pi}{2})}$

if $s\in[R_{4},(R+\pi)-R_{4}]_{s}$, where $u_{R}$ is the energy density
function of $P_{R}$.

###### Proof of Lemma 11.6.

The proof follows the same line of arguments as in Proposition 3.6. To start,
we have to show a uniform energy decay as in Lemma 3.7, but this is easy: the
estimate (11.13) combined with our assumption that the angle $\theta=0$ is
admissible implies the analogue of Lemma 3.8, then we deduce the uniform
energy decay using the triviality of point-like solutions, i.e., Lemma 3.9. To
verify this exponential decay, we exploit the inequality

$0\geq(\Delta+\zeta^{2})|dP_{R}|^{2}$

as in the proof of Proposition 3.6 but now on the strip
$\mathbb{R}_{t}\times[R_{4},(R+\pi)-R_{4}]_{s}$ for some $R_{4}\gg\pi$. Then
we use the maximal principle from [Wan20a, Corollary A.4] to conclude. ∎

The next step is to control the behavior of $P$ when the drop of the action
functional $\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}$ is small on some
$I_{t}\times\mathbb{R}_{s}$, $I_{t}=[t-1,t+1]_{t}$; cf. [KM07, Proposition
13.4.7]. Chose $r_{0}>0$ such that $r_{0}$ is less then the injective radius
of $(M,g_{M})$ and

(11.14)
$r_{0}<\frac{1}{2}\cdot\min\\{\operatorname{dist}_{C^{0}(\mathbb{R}_{s})}(p^{st}_{0},p^{st}_{1}),\operatorname{dist}_{C^{0}(\mathbb{R}_{s})}(p^{un}_{0},p^{un}_{1}):p^{st}_{j}\in\mathfrak{C}^{st},\
p^{un}_{j}\in\mathfrak{C}^{un},\ j=0,1\\}.$

######  Lemma 11.7.

Let $I_{t}^{\prime}\colonequals[t-\frac{1}{2},t+\frac{1}{2}]_{t}$. For any
$r\geq 2$, there exist constants $\epsilon_{5},C_{5}$ and $C_{6}=C_{6}(r)>0$
with the following properties. For any solitons
$p_{R,\pm}\in\mathfrak{C}^{R}$, any instanton
$P_{R}\in\mathcal{M}(p_{R,-},p_{R,+};\mathfrak{a}^{R}_{m1})$ and
$t\in\mathbb{R}_{t}$, suppose that

(11.15)
$\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(P_{R}(t-1,\cdot))-\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(P_{R}(t+1,\cdot))<\epsilon_{5},$

then for some soliton $p_{R}\in\mathfrak{C}^{R}$, $P_{R}(t^{\prime},\cdot)$
has $C^{0}(\mathbb{R}_{s})$-distance $<r_{0}$ with $p_{R}$ for all
$t^{\prime}\in I_{t}^{\prime}$. This $p_{R}$ is unique due to our choice
(11.14) of $r_{0}$. Then $P_{R}|_{I_{t}^{\prime}\times\mathbb{R}_{s}}$ can be
viewed as a section of $(p_{R})^{*}TM$ using the exponential map along
$p_{R}$, and we have the estimate

(11.16)
$\|P_{R}|_{I_{t}^{\prime}\times\mathbb{R}_{s}}-p_{R}\|_{L^{r}_{1}(I_{t}^{\prime}\times\mathbb{R}_{s})}\leq
C_{6}(r)\bigg{(}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}\big{(}P_{R}(t-1,\cdot)\big{)}-\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}\big{(}P_{R}(t+1,\cdot)\big{)}\bigg{)},$

where $p_{R}$ is viewed as a constant trajectory on $I_{t}^{\prime}$.
Moreover, for all $t^{\prime}\in I^{\prime}_{t}$,

(11.17)
$\big{|}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}\big{(}P_{R}(t^{\prime},\cdot))-\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p_{R})\big{|}\leq
C_{5}\|\operatorname{grad}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}\big{(}P_{R}(t^{\prime},\cdot)\big{)}\|^{2}_{L^{2}(\mathbb{R}_{s})}.$

###### Proof.

This lemma follows from the standard analysis of Floer trajectories; see for
instance [KM07, Section 13.4]. The reason why $\epsilon_{5},C_{5},C_{6}$ can
be made uniform in $R$ is (11.12): the Hessians of
$\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}$ can be inverted uniformly at all
solitons in $\mathfrak{C}^{R}$ for $R\gg\pi$. ∎

####  11.4. Proof of Theorem 11.4 Part II: Compactness of instantons

With Lemma 11.6 and Lemma 11.7 at hands, we are ready to analyze the limit of
$P_{R}$ as $R\to\infty$. Recall that the moduli space
$\mathcal{M}(p_{R,-},p_{R,+};\mathfrak{a}_{m1}^{R})$ carries an
$\mathbb{R}_{t}$-action: let $\tau_{t}P=P(\cdot-t,\cdot),t\in\mathbb{R}_{t}$.

######  Lemma 11.8.

For any sequence of $\alpha^{R_{n}}_{m1}$-instantons
$P_{n}\in\mathcal{M}(p_{R_{n},-},p_{R_{n},+};\mathfrak{a}_{m1}^{R_{n}})$ with
$R_{n}\to\infty$, there exists a subsequence converging to a broken instanton:

(11.18)
$(p^{st}_{0},p^{un}_{0})\xrightarrow{(P^{st}_{0},P^{un}_{0})}(p^{st}_{1},p^{st}_{1})\to\cdots\to(p^{st}_{k},p^{un}_{k})\xrightarrow{(P^{st}_{k_{1}},P^{un}_{k_{1}})}(p^{st}_{k_{1}+1},p^{un}_{k_{1}+1}),$

in the following sense:

  * •

for every $0\leq k\leq k_{1}$, $p^{st}_{k}\in\mathfrak{C}^{st}$ and
$p^{un}_{k}\in\mathfrak{C}^{un}$ are some solitons;

  * •

$(p^{st}_{0},p^{un}_{0})=(p^{st}_{-},p^{un}_{-})$ and
$(p^{st}_{k_{1}+1},p^{un}_{k_{1}+1})=(p^{st}_{+},p^{un}_{+})$;

  * •

for every $0\leq k\leq k_{1}$,
$P_{k}^{st}\in\mathcal{M}(p_{k}^{st},p_{k+1}^{st};\mathfrak{a}_{1}^{st})$ and
$P_{k}^{un}\in\mathcal{M}(p_{k}^{un},p_{k+1}^{un};\mathfrak{a}_{m}^{un})$ are
some instantons; one of them must be non-constant in time;

  * •

for any $n$, there exists a sequence of numbers:

$t^{0}_{n}<t^{1}_{n}<\cdots<t^{k_{1}}_{n}$

such that for all $0\leq k\leq k_{1}$, we have

$\tau_{t^{k}_{n}}P_{n}\to P_{k}^{un}\text{ and
}(\tau_{t^{k}_{n}}P_{n})(\cdot,\cdot-R_{n})\to P_{k}^{st}$

in $C^{\infty}_{loc}$-topology, and in addition
$|t^{k+1}_{n}-t^{k}_{n}|\to+\infty$ as $n\to\infty$.

###### Proof.

This lemma follows from Lemma 11.6 & 11.7 and the argument in [KM07, Section
16.1& 16.2]. Note that the divergence $|t^{k+1}_{n}-t^{k}_{n}|\to\infty$ may
be slower or faster than $R_{n}\to\infty$; they are not necessarily related. ∎

\begin{overpic}[scale={.25}]{Pic59.png} \put(12.0,73.0){$t_{n}^{0}$}
\put(52.0,73.0){$t_{n}^{1}$} \put(72.0,73.0){$t_{n}^{2}$}
\put(11.0,17.0){$P_{0}^{un}$} \put(11.0,53.0){$P_{0}^{st}$}
\put(51.0,17.0){$P_{1}^{un}$} \put(51.0,53.0){$P_{1}^{st}$}
\put(71.0,17.0){$P_{2}^{un}$} \put(71.0,53.0){$P_{2}^{st}$}
\put(30.0,65.0){$x_{1}$} \put(30.0,35.0){$x_{l}$} \put(30.0,5.0){$x_{m}$}
\put(0.0,17.0){$p_{0}^{un}$} \put(0.0,53.0){$p_{0}^{st}$}
\put(30.0,17.0){$p_{1}^{un}$} \put(30.0,53.0){$p_{1}^{st}$}
\put(62.0,17.0){$p_{2}^{un}$} \put(62.0,53.0){$p_{2}^{st}$}
\put(85.0,17.0){$p_{3}^{un}$} \put(85.0,53.0){$p_{3}^{st}$} \end{overpic}
Figure 38. Converging to a broken instanton ($k_{1}=2$).

By Lemma 11.2, for such a broken instanton (11.18), we must have

$\displaystyle\operatorname{\mathfrak{gr}}(p_{R_{n},-})-\operatorname{\mathfrak{gr}}(p_{R_{n},+})$
$\displaystyle=\big{(}\operatorname{\mathfrak{gr}}(p^{st}_{0})-\operatorname{\mathfrak{gr}}(p^{st}_{k_{1}+1})\big{)}+\big{(}\operatorname{\mathfrak{gr}}(p^{un}_{0})-\operatorname{\mathfrak{gr}}(p^{un}_{k_{1}+1})\big{)}$
(11.19)
$\displaystyle=\sum_{k=0}^{k_{1}}\big{(}\operatorname{\mathfrak{gr}}(p^{st}_{k})-\operatorname{\mathfrak{gr}}(p^{st}_{k+1})\big{)}+\big{(}\operatorname{\mathfrak{gr}}(p^{un}_{k})-\operatorname{\mathfrak{gr}}(p^{un}_{k+1})\big{)}.$

In the second line, each term in the summation is either the index of
$P_{k}^{st}$ or that of $P_{k}^{un}$. Since the Floer data
$\mathfrak{a}_{1}^{st}$ and $\mathfrak{a}_{m}^{un}$ are assumed to be
admissible, each of them must be $\geq 0$ and $=0$ only if the associated
instanton is a constant trajectory in time. Thus in the case that
$\operatorname{\mathfrak{gr}}(p_{R,-})-\operatorname{\mathfrak{gr}}(p_{R,+})=1$,
we must have $k_{1}=0$ and

(11.20)
$\big{(}\operatorname{\mathfrak{gr}}(p_{-}^{st})-\operatorname{\mathfrak{gr}}(p_{+}^{st}),\
\operatorname{\mathfrak{gr}}(p_{-}^{un})-\operatorname{\mathfrak{gr}}(p_{+}^{un})\big{)}=(0,1)\text{
or }(1,0)$

depending on whether $P_{0}^{st}$ or $P_{0}^{un}$ is a constant trajectory;
this explains the trichotomy in (11.5). In particular, we have proved that

######  Corollary 11.9.

Suppose that
$\big{(}\operatorname{\mathfrak{gr}}(p_{-}^{st})-\operatorname{\mathfrak{gr}}(p_{+}^{st})\big{)}+\big{(}\operatorname{\mathfrak{gr}}(p_{-}^{un})-\operatorname{\mathfrak{gr}}(p_{+}^{un})\big{)}=1$,
then the moduli space
$\mathcal{M}(p_{R,-},p_{R,+};\mathfrak{a}^{R}_{m1})=\emptyset$ for all
$R\gg\pi$ unless

  1. $($i$)$

$\big{(}\operatorname{\mathfrak{gr}}(p_{-}^{st})-\operatorname{\mathfrak{gr}}(p_{+}^{st}),\
\operatorname{\mathfrak{gr}}(p_{-}^{un})-\operatorname{\mathfrak{gr}}(p_{+}^{un})\big{)}=(0,1)$
and $p_{-}^{st}=p_{+}^{st}$; or

  2. $($ii$)$

$\big{(}\operatorname{\mathfrak{gr}}(p_{-}^{st})-\operatorname{\mathfrak{gr}}(p_{+}^{st}),\
\operatorname{\mathfrak{gr}}(p_{-}^{un})-\operatorname{\mathfrak{gr}}(p_{+}^{un})\big{)}=(1,0)$
and $p_{-}^{un}=p_{+}^{un}$.

Since the two cases in Corollary 11.9 are symmetric, we shall focus on the
first one from now on. In this special case, Lemma 11.8 is refined as follows.

######  Corollary 11.10.

Suppose that the first case of Corollary 11.9 happens. For any sequence of
instantons
$P_{n}\in\mathcal{M}(p_{R_{n},-},p_{R_{n},+};\mathfrak{a}_{m1}^{R_{n}})$ with
$R_{n}\to\infty$, there exists a $($non-constant$)$
$\alpha^{un}_{m}$-instanton
$P^{un}\in\mathcal{M}(p_{-}^{un},p_{+}^{un};\mathfrak{a}^{un}_{m})$ and a
sequence of real numbers $t^{n}$ such that

$\tau_{t_{n}}P_{n}\to P^{un}\text{ and }(\tau_{t_{n}}P_{n})(\cdot,\cdot-
R_{n})\to P^{st}$

in $C^{\infty}_{loc}(\mathbb{R}_{t}\times\mathbb{R}_{s})$-topology, where
$P^{st}$ is the constant trajectory at $p^{st}_{-}=p^{st}_{+}$.

The $C^{\infty}_{loc}$-convergence in Corollary 11.10 is not quite enough for
the proof of Theorem 11.3. A global estimate in the spirit of (11.10) is more
relevant. The proof of Lemma 11.8 implies the following stronger result.

######  Corollary 11.11.

Then there exists a constant $C_{7},K_{7}>0$ with the following properties.
For the converging subsequence in Corollary 11.10, we have

$\|\operatorname{grad}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(P_{R}(t,\cdot))\|_{L^{2}(\mathbb{R}_{s})}^{2}<\frac{\epsilon_{5}}{4},$

for all $|t|\geq K_{7}$ and $R\gg\pi$; so the condition (11.15) holds on
$I_{t}$ for all $t>K_{7}+1$. Then (11.17) implies that

$0\leq\mp\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p_{R,\pm})\pm\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(P_{R}(t,\cdot))\leq
C_{7}e^{\pm C_{5}t},\ \forall\pm t\geq K_{7}+1.$

Combined with (11.16), this means that the $L^{r}_{1}$-norm of
$P_{R}-p_{R,\pm}$ on $I_{t}^{\prime}\times\mathbb{R}_{s}$ decays exponentially
to zero as $\pm t\to\infty$. These estimates are uniform in the stretching
parameter $R$ and hold also for the limiting instanton
$P^{un}\in\mathcal{M}(p^{un}_{-},p^{un}_{+};\mathfrak{a}^{un}_{m})$ in
Corollary 11.10.

####  11.5. Proof of Theorem 11.3 Part III: A gluing scheme

First recall the Implicit Function Theorem we shall use later:

######  Theorem 11.12 ([MS12, Theorem A.3.3]).

Let $X$ and $Y$ be Banach spaces and $U\subset X$ be an open subset. If
$F:U\to Y$ is Fredholm $($i.e. the linear map $dF(x):X\to Y$ is bounded and
Fredholm for all $x\in X)$ and continuously differentiable, and $y$ is a
regular value of $F$, then $\mathcal{N}=F^{-1}(y)\subset U$ is a
$C^{1}$-manifold with $T_{x}\mathcal{N}=\ker dF(x)$ for all $x\in\mathcal{N}$.

In what follows, we shall apply this theorem to a family of Fredholm maps
$F:X\to Y$ which depend on the stretching parameter $R$. The next proposition,
which is used in the proof of Theorem 11.12, becomes more useful in order to
obtain estimates uniform in $R$.

######  Proposition 11.13 ([MS12, Proposition A.3.4]).

Under the assumptions of Theorem 11.12, suppose that the differential
$D=dF(x_{0}):X\to Y$ is surjective at some $x_{0}\in U$ and has a right
inverse $Q:Y\to X$. Choose $c,\delta>0$ such that $\|Q\|\leq c$ and
$\|dF(x)-D\|\leq\frac{1}{2c}$ for all $x\in B(x_{0},\delta)\subset U$. If
$x_{1}\in X$ satisfies

$\|F(x_{1})\|<\frac{\delta}{4c},\ \|x_{1}-x_{0}\|<\frac{\delta}{8},$

then there exists a unique $x\in B(x_{0},\delta)$ such that $F(x)=0$ and
$x-x_{1}\in\operatorname{Im}Q$. Moreover, $\|x-x_{1}\|\leq 2c\|f(x_{1})\|$.

In Proposition 11.13, one should think of $x_{0}$ as a reference point and
$x_{1}$ as the approximate solution.

We begin with a linear problem to explain our gluing scheme. Suppose that $V$
is a finite dimensional Euclidean space, $\sigma_{\pm}:V\to V$ are self-
adjoint and invertible, and $\sigma_{t}:V\to V,t\in\mathbb{R}_{t}$ is a smooth
family of self-adjoint operators such that
$\|\sigma_{t}-\sigma_{\pm}\|_{L^{r}(I_{t},\operatorname{End}(V))}\to 0$,
$I_{t}\colonequals[t-1,t+1]$ exponentially as $t\to\pm\infty$ for some $r>2$.
Then the linear map:

$\displaystyle D:C^{\infty}(\mathbb{R}_{s};V)$ $\displaystyle\to
C^{\infty}(\mathbb{R}_{s};V)$ $\displaystyle v(t)$
$\displaystyle\mapsto\partial_{t}v(t)+\sigma_{t}(v(t)).$

is Fredholm from $L^{r}_{1}(\mathbb{R}_{t};V)$ to $L^{r}(\mathbb{R}_{t};V)$.
If $D:L^{r}_{1}\to L^{r}$ is invertible, then any $\xi\in L^{r}$ has a unique
solution $v\in L^{r}_{1}$ such that $D(v)=\xi$.

However, we would like to solve this equation for $\xi$ in a larger subspace
of $C^{\infty}(\mathbb{R}_{s};\mathcal{V})$. Let
$\chi_{\pm}:\mathbb{R}_{t}\to[0,1]$ be a pair of cutoff functions such that
for some $K>1$,

(11.21) $\chi_{-}(t)=\chi_{+}(-t),\ \chi_{+}(t)\equiv 1\text{ if }t\geq
K+1\text{ and }\equiv 0\text{ if }t\leq K.$

Let $\mathcal{V}_{j},j=0,1$ denote the subspace of
$C^{\infty}(\mathbb{R}_{s};V)$ whose elements take the form

(11.22) $v=v^{-}\chi_{-}(s)+v^{+}\chi_{+}(s)+v_{0}\text{ with }v_{\pm}\in
V_{\pm},v_{0}\in L^{r}_{j}(\mathbb{R}_{s};V)$

where $V^{\pm}$ is another copy of $V$, and $\mathcal{V}_{j}$ is equipped with
the norm on $V_{-}\oplus V_{+}\oplus L^{r}_{j}(\mathbb{R}_{s};V)$. Then $D$
extends to a bounded operator $\mathcal{V}_{1}\to\mathcal{V}_{0}$ which takes
the form

(11.23) $\begin{pmatrix}v^{-}\\\ v^{+}\\\
v_{0}\end{pmatrix}\mapsto\begin{pmatrix}\sigma_{-}&0&0\\\ 0&\sigma_{+}&0\\\
\partial_{t}\chi_{-}+\chi_{-}(\sigma_{t}-\sigma_{-})&\partial_{t}\chi_{+}+\chi_{+}(\sigma_{t}-\sigma_{+})&\partial_{t}+\sigma_{t}\end{pmatrix}\begin{pmatrix}v^{-}\\\
v^{+}\\\ v_{0}\end{pmatrix}.$

Since all diagonal entries are invertible, so is the operator
$D:\mathcal{V}_{1}\to\mathcal{V}_{0}$. Thus any $\xi\in\mathcal{V}_{0}$ also
admit a unique solution $D(v)=\xi$ with $v\in\mathcal{V}_{1}$. One should
think of this $D$ as the differential map $dF(x_{1})$ in Proposition 11.13 and
$\xi=F(x_{1})$ is the error term to be corrected.

Returning to the proof of Theorem 11.3, we fix some $r>2$ from now on. suppose
that we are in the first case of Corollary 11.9,
$P^{un}\in\mathcal{M}(p_{-}^{un},p_{+}^{un};\mathfrak{a}^{un}_{m})$ is any
$\alpha^{un}_{m}$-instanton, and $P^{st}$ is the constant trajectory at
$p^{st}\colonequals p^{st}_{-}=p^{un}_{+}$. For any $R\gg\pi$, we have to
construct an instanton
$P_{R}\in\mathcal{M}(p^{-}_{R},p^{+}_{R};\mathfrak{a}^{R}_{m1})$ such that
$P_{R}\to(P^{un},P^{st})$ as $R\to\infty$ in the sense of Corollary 11.10.

The approximate solution $P^{\operatorname{pre}}_{R}$ is simply obtained by
applying the pre-gluing map (11.7) to $P^{st}$ and $P^{un}$ at each time
slice, i.e.,

(11.24)
$P^{\operatorname{pre}}_{R}(t,\cdot)=\Phi_{R}(P^{st}(t,\cdot),P^{un}(t,\cdot))=\Phi_{R}(p^{st},P^{un}(t,\cdot)),\forall
t\in\mathbb{R}_{t}.$

It is also convenient to choose a reference trajectory which is constant when
$|t|\gg 1$ so that our gluing scheme looks more similar to one above. Consider
an approximation $P^{un}_{K}$ of $P^{un}$ with $K\gg 1$ such that

  * •

$P^{un}_{K}(t,s)=P^{un}(t,s)$ when $|t|\leq K-1$;

  * •

$P^{un}_{K}(t,s)=p^{un}_{\pm}(s)$ when $\pm t\geq K$;

  * •

$\operatorname{dist}_{C^{0}}(P^{un},P^{un}_{K})<r_{0}$; identify $P^{un}$ as a
section of $L^{r}_{1}(\mathbb{R}_{t}\times\mathbb{R}_{s};(P^{un}_{K})^{*}TM)$,
then

(11.25)
$\delta_{K}\colonequals\|P^{un}-P^{un}_{K}\|_{L^{r}_{1}(\mathbb{R}_{t}\times\mathbb{R}_{s};(P^{un}_{K})^{*}TM)}<Ce^{-C_{5}K}$

for some $C>0$, and $C_{5}$ is the constant in Lemma 11.7 and Corollary 11.11.

This approximation $P^{un}_{K}$ is constructed using the exponential maps
along $p^{un}_{\pm}$ and some cutoff functions as in the case of the pre-
gluing map (11.7), and the required exponential decay follows from Corollary
11.11. Now we replace $P^{un}$ by $P^{un}_{K}$ in (11.24) to obtain a
reference trajectory denoted by $P^{\operatorname{ref}}_{R}$, which satisfies
the following properties:

  * •

$P^{\operatorname{ref}}_{R}(t,\cdot)\equiv
p_{R,\pm}^{\operatorname{pre}}=\Phi_{R}(p^{st},p^{un}_{\pm})$ when $\pm t\geq
K$;

  * •

$P^{\operatorname{ref}}_{R}=P^{\operatorname{pre}}_{R}\equiv x_{l}$ when
$s\in[\frac{3(R+\pi)}{8},\frac{5(R+\pi)}{8}]_{s}$;

  * •

$P^{\operatorname{ref}}_{R}=P^{\operatorname{pre}}_{R}$ when $|t|\leq K-1$.

Neither of $P^{\operatorname{pre}}_{R}$ and $P^{\operatorname{ref}}_{R}$
satisfy the boundary condition (2.13) yet, and they must be corrected as in
our model problem above. Let
$V^{\pm}_{j}=L^{r}_{1}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{R,\pm})^{*}TM)$
and
$V_{j}^{0}=L^{r}_{1}(\mathbb{R}_{t}\times\mathbb{R}_{s};(P^{\operatorname{ref}}_{R})^{*}TM),j=0,1.$
Then define

$\mathcal{V}_{j}\colonequals V^{-}_{j}\oplus V^{+}_{j}\oplus V^{0}_{j},\
j=0,1.$

Since $P^{\operatorname{ref}}_{R}$ is a constant trajectory when $|t|\geq K$,
we can identify $\mathcal{V}_{j}$ as a subspace of
$L^{r}_{loc}(\mathbb{R}_{t}\times\mathbb{R}_{s};(P^{\operatorname{ref}}_{R})^{*}TM)$
using the cutoff functions (11.21) and the formula (11.22).

For any $x\in M$ and $w\in T_{x}M$, let $\Psi_{x}(w):T_{x}M\to
T_{\exp_{x}(w)}M$ denote the parallel transportation along the geodesic
$t\mapsto\exp_{x}(tw),t\in[0,1]$. For any $v\in\mathcal{V}_{1}$, consider the
map $P_{v}(z)=\exp_{P^{\operatorname{ref}}_{R}(z)}(v(z)),z\in\mathbb{R}^{2}$.
Using the Floer datum $\mathfrak{a}_{m1}^{R}$, the formula (5.5) defines a
section $\mathfrak{F}(P_{v}(z))\in
L^{r}_{loc}(\mathbb{R}_{t}\times\mathbb{R}_{s};(P_{v})^{*}TM\otimes\Lambda^{0,1}\mathbb{C})$,
$\mathbb{C}_{z}=\mathbb{R}_{t}\times\mathbb{R}_{s}$ , which is turned into a
non-linear map using the parallel transportation $\Psi$:

(11.26) $\displaystyle\mathcal{F}:\mathcal{V}_{1}$ $\displaystyle\to
L^{r}_{loc}(\mathbb{R}_{t}\times\mathbb{R}_{s};(P^{\operatorname{ref}}_{R})^{*}TM)$
$\displaystyle v$
$\displaystyle\mapsto\Psi_{P^{\operatorname{ref}}_{R}(z)}(v(z))^{-1}\big{(}\mathfrak{F}(P_{v}(z))\big{)}.$

where the bundle $\Lambda^{0,1}\mathbb{C}$ has been trivialized using
$d\overline{z}=dt-is$. Although we have dropped the subscript, $\mathcal{F}$
depends on the stretching parameter $R\gg\pi$.

Suppose that under the exponential map, the approximate solution
$P^{\operatorname{pre}}_{R}$ is represented by the section
$v_{1}\in\mathcal{V}_{1}$, so
$P^{\operatorname{pre}}_{R}(z)=P_{v_{1}}(z)=\exp_{P^{\operatorname{ref}}_{R}(z)}(v_{1}(z))$.
Then in fact $v_{1}\in L^{r}_{1}$ and $v_{1}\equiv 0$ when $|t|\leq K-1$. To
apply Proposition 11.13 to the non-linear map (11.26), we take
$x_{0}=0\in\mathcal{V}_{1}$ as the reference point and $x_{1}=v_{1}$ as the
approximate solution. The conditions of Proposition 11.13 are verified by the
next lemma.

######  Lemma 11.14.

There exists constants $\delta_{1},c,c_{1}>0$ independent of $R,K$ so that the
following holds for $R\gg\pi$:

  1. $($i$)$

$\mathcal{F}(0)\in\mathcal{V}_{0}$ and
$(d\mathcal{F})(0):\mathcal{V}_{1}\to\mathcal{V}_{0}$ is bounded linear and
Fredholm of index $1$;

  2. $($ii$)$

for all $v_{2},v_{3}\in B(0,\delta_{1})\subset\mathcal{V}_{1}$,

$\|d\mathcal{F}(v_{2})-d\mathcal{F}(v_{3})\|_{\mathcal{V}_{1}\to\mathcal{V}_{0}}\leq
c_{1}\|v_{2}-v_{3}\|_{\mathcal{V}_{1}}.$

Combined with the first property, this implies that the image of
$\mathcal{F}(B(0,\delta_{1}))$ lies in the smaller space
$\mathcal{V}_{0}\subset L^{r}_{loc}$, and
$\mathcal{F}:\mathcal{V}_{1}\to\mathcal{V}_{0}$ is continuously
differentiable;

  3. $($iii$)$

$\|v_{1}\|_{L^{r}_{1}}=\|v_{1}\|_{\mathcal{V}_{1}}\leq 2\delta_{K}$ and
$\|\mathcal{F}(v_{1})\|_{\mathcal{V}_{0}}\to 0$ as $R\to\infty$, with
$\delta_{K}$ defined as in (11.25);

  4. $($iv$)$

$d\mathcal{F}(v_{1}):\mathcal{V}_{1}\to\mathcal{V}_{0}$ is surjective and has
a right inverse $Q_{R}$ with norm bounded by $c$.

Take $F=\mathcal{F}$, $x_{0}=0$, $x_{1}=v_{1}$ and and
$\delta=\min\\{\delta_{1},\frac{1}{2cc_{1}}\\}$ in Proposition 11.13. We have
to verify in addition that

$\|\mathcal{F}(v_{1})\|_{\mathcal{V}_{0}}<\frac{\delta}{4c},\
\|v_{1}\|_{\mathcal{V}_{1}}<\frac{\delta}{8}.$

By Lemma 11.14, this can be arranged by taking a fixed $K\gg 1$ (use (11.25))
then by letting $R\to\infty$. Thus we obtain an $\alpha^{R}_{m1}$-instanton
$P_{R}^{\star}(z)=\exp_{P^{\operatorname{pre}}_{R}(z)}(v_{\star}(z))\in\mathcal{M}(p^{R}_{-},p^{R}_{+};\mathfrak{a}^{R}_{m1})$
with $v_{\star}-v_{1}\in\operatorname{Im}Q_{R}$ and
$\|v_{\star}-v_{1}\|_{\mathcal{V}_{1}}\leq
2c\|\mathcal{F}(v_{1})\|_{\mathcal{V}_{0}}\to 0$ as $R\to\infty$.

By Theorem 11.12, $v_{R}$ fits into a 1-parameter family of perturbed
$\alpha^{R}_{m1}$-instantons. The uniqueness part of Proposition 11.13 then
says that in a small neighborhood of $P_{R}^{\operatorname{pre}}$, this family
is obtained by translating $P_{R}^{*}$. In the meantime, the
$C^{\infty}_{loc}$-convergence in Corollary 11.10 can be improved as follows.

######  Lemma 11.15.

For the converging subsequence in Corollary 11.10, we have

$\|\tau_{t_{n}}P_{n}-P^{\operatorname{pre}}_{R_{n}}\|_{\mathcal{V}_{1}}\to 0$

as $n\to\infty$, where $\tau_{t_{n}}P_{n}$, $P^{\operatorname{pre}}_{R_{n}}$
are viewed as sections in $\mathcal{V}_{1}$ using the exponential maps. This
implies that $\tau_{t_{n}}P_{n}$ is a translated copy of $P_{R_{n}}^{\star}$.

Thus for $R\gg\pi$, any instanton in
$\mathcal{M}(p^{R}_{-},p^{R}_{+};\mathfrak{a}^{R}_{m1})$ is obtained up to a
translation by gluing some $P^{un}$ vertically with $P^{st}$. To complete the
proof of Theorem 11.3, it remains to prove Lemma 11.14 and 11.15.

###### Proof of Lemma 11.14 $($i$)$.

For the reference trajectory $P^{\operatorname{ref}}_{R}$, the section
$\mathcal{F}(0)$ is defined by the formula

$\partial_{t}P^{\operatorname{ref}}_{R}+J\partial_{s}P^{\operatorname{ref}}_{R}+\nabla\operatorname{Im}(e^{-i\alpha_{m1}^{R}(s)}W)+\nabla\delta
H^{m1}_{R}=0,$

so
$\mathcal{F}(0)(t,\cdot)\equiv\operatorname{grad}p^{\operatorname{pre}}_{R,\pm}\in
L^{r}_{1}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{R,\pm})^{*}TM)$ for $\pm
t\geq K$. The gradient vector
$\operatorname{grad}p^{\operatorname{pre}}_{R,\pm}$ is supported on
$[\frac{R+\pi}{4},\frac{3(R+\pi)}{4}]_{s}$. By construction the difference

$\mathcal{F}(0)-\chi_{-}(s)\operatorname{grad}p^{\operatorname{pre}}_{R,-}-\chi_{+}(s)\operatorname{grad}p^{\operatorname{pre}}_{R,+}.$

is supported on

$\\{K-1\leq|t|\leq K+1\\}\ \bigcup\
[-K-1,-K+1]_{t}\times[\frac{R+\pi}{4},\frac{3(R+\pi)}{4}]_{s}$

and lies in $L^{1}_{r}(\mathbb{R}_{t}\times\mathbb{R}_{s})$. The linearization
of $\mathcal{F}$ at $0$ takes a lower triangular form as in (11.23):

(11.27)
$\begin{pmatrix}\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p_{R,-}^{\operatorname{pre}})&0&0\\\
0&\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p_{R,+}^{\operatorname{pre}})&0\\\
\partial_{t}\chi_{-}&\partial_{t}\chi_{+}&D_{0}\end{pmatrix}$

where $D_{0}$ is $d\mathcal{F}(0)$ as a map between $L^{r}_{1}\to L^{r}$.
These entries are bounded linear, and so is
$d\mathcal{F}(0):\mathcal{V}_{1}\to\mathcal{V}_{0}$. The Fredholm index will
be computed later using $d\mathcal{F}(v_{1})$. ∎

###### Proof of Lemma $($ii$)$.

The second statement $($ii$)$ requires some work. Its predecessor can be found
in [Flo88b, Theorem 3a],[AD14, Lemma 9.4.8] and [MS12, Proposition 3.5.3], and
we follow the same argument.The key ingredient is the following pointwise
estimate.

######  Lemma 11.16.

Let $S$ be any Riemann surface, $M_{0}\subset M$ any compact subset, and
$P:S\to M_{0}\subset M$ any smooth map. Following the construction of (11.26),
for any smooth vector field $\mathcal{X}\in\Omega^{1}(S;C^{\infty}(M;TM))$,
one can define a non-linear map

$\displaystyle\mathcal{F}_{0}:C^{\infty}(S;P^{*}(TM))$ $\displaystyle\to
C^{\infty}(S;P^{*}(TM)\otimes\Lambda^{0,1}S)$ $\displaystyle\xi$
$\displaystyle\mapsto\Psi_{P}(\xi)^{-1}\big{(}\mathfrak{F}_{0}(\exp_{P}(\xi))\big{)}$

with $\mathfrak{F}_{0}(P_{\xi})\colonequals(dP_{\xi}-\mathcal{X})^{0,1}$,
$P_{\xi}=\exp_{P}(\xi)$. Then for any $c_{2}>0$, there exist some constant
$C>0$ such that for any smooth $\delta\xi_{1},\delta\xi_{2},\xi_{0},\xi\in
C^{\infty}(B(0,1);P^{*}TM)$ with $L^{\infty}$-norms $<c_{2}$, there are
pointwise bounds

(11.28)
$|(d\mathcal{F}_{0})(\xi_{0})\xi-(d\mathcal{F}_{0})(\xi_{0}+\delta\xi_{1})\xi|<C(|\delta\xi_{1}||\nabla\xi|+|\nabla\delta\xi_{1}||\xi|+|\delta\xi_{1}||\xi|),$

and

(11.29)
$\displaystyle\big{|}(d\mathcal{F}_{0})(\xi_{0})\xi-(d\mathcal{F}_{0})(\xi_{0}+\delta\xi_{1})\xi-(d\mathcal{F}_{0})(\xi_{0}+\delta\xi_{2})\xi+(d\mathcal{F}_{0})(\xi_{0}+\delta\xi_{2}+\delta\xi_{1})\xi\big{|}$
$\displaystyle\leq$ $\displaystyle
C\big{(}|\delta\xi_{1}||\delta\xi_{2}||\nabla\xi|+|\nabla\delta\xi_{1}||\delta\xi_{2}||\xi|+|\nabla\delta\xi_{1}||\delta\xi_{2}||\xi|+|\delta\xi_{1}||\delta\xi_{2}||\xi|\big{)}.$

on $B(0,\frac{1}{2})$. This constant $C$ depends on the geometry of $M$,
$c_{2}>0$, $\|dP\|_{\infty}$ and the $C^{2}$-norm of $\mathcal{X}$ on a
slightly larger compact subset of $M$ depending on $c_{2}$.

###### Proof of Lemma 11.16.

If one of $\xi_{0},\xi_{1}$ is zero, then (11.28) follows from the computation
in [MS12, Proposition 3.5.3]. In general, $\eqref{VT.E.26}$ and (11.28) follow
from a (well-known) estimate on the second and the third derivatives of
$\mathcal{F}_{0}$. For the sake of completeness, this estimate is summarized
and proved in Lemma B.1. ∎

To apply Lemma 11.16, let $P=P^{\operatorname{pre}}_{R}$ and $\mathcal{X}$ is
the Hamiltonian vector field associated to
$\operatorname{Im}(e^{-\alpha_{m1}^{R}(s)}W)ds+\delta
H_{m1}^{R}\in\Omega^{1}(S;C^{\infty}(M;\mathbb{R}))$. The compactness theorem
implies that $P^{\operatorname{pre}}_{R}$ lies a fixed compact subset of $M$.
The $L^{\infty}_{1}$-norm of $P^{\operatorname{pre}}_{R}$ and the $C^{2}$-norm
of $\mathcal{X}$ on this compact subset are uniformly bounded.

Let $\delta v=v_{3}-v_{2}$. We have to show that for all $v_{2},v_{3}\in
B(0,\delta_{1})\subset\mathcal{V}_{1}$ and $v\in\mathcal{V}_{1}$, the estimate

(11.30) $\|(d\mathcal{F})(v_{2}+\delta
v)(v)-(d\mathcal{F})(v_{2})(v)\|_{\mathcal{V}_{0}}\leq c_{1}\|\delta
v\|_{\mathcal{V}_{1}}\|v\|_{\mathcal{V}_{1}}$

holds for some $c_{1}>0$. By the nature of $\mathcal{V}_{1}$, decompose
$\delta v$ as $\chi_{-}\delta v^{-}+\chi_{+}\delta v^{+}+\delta v^{0}$. It
suffices to verify (11.30) when only one of $\delta v^{\pm},\delta v^{0}$ is
non-zero. Similarly write $v=\chi_{-}v^{-}+\chi_{+}v^{+}+v^{0}$ with
$v^{\pm}\in V_{1}^{\pm}$ an $v^{0}\in V_{1}^{0}.$

Case 1\. If $\delta v=\delta v^{0}\in
L^{r}_{1}(\mathbb{R}_{t}\times\mathbb{R}_{s})$, then we can prove a stronger
result:

$\|(d\mathcal{F})(v_{2}+\delta
v)(v)-(d\mathcal{F})(v_{2})(v)\|_{L^{r}(\mathbb{R}^{2})}\leq c_{1}\|\delta
v\|_{L^{r}_{1}}\|v\|_{\mathcal{V}_{1}}.$

This follows immediately from (11.28) by noting that $\mathcal{V}_{1}\subset
L^{\infty}$, and

$\nabla(\chi_{-}v^{-}+\chi_{+}v^{+})\in
L^{\infty}(\mathbb{R}_{t};L^{r}(\mathbb{R}_{s}))\text{ and
}L^{r}_{1}\hookrightarrow
L^{r}(\mathbb{R}_{t};L^{r}_{1}(\mathbb{R}_{s}))\hookrightarrow
L^{r}(\mathbb{R}_{t};L^{\infty}(\mathbb{R}_{s})).$

The multiplication $L^{\infty}(\mathbb{R}_{t};L^{r}(\mathbb{R}_{s}))\times
L^{r}(\mathbb{R}_{t};L^{\infty}(\mathbb{R}_{s}))\to L^{r}(\mathbb{R}^{2})$ is
continuous. Thus

$\displaystyle\|(d\mathcal{F})(v_{2}+\delta
v)(v)-(d\mathcal{F})(v_{2})(v)\|_{r}$ $\displaystyle\lesssim$
$\displaystyle\||\delta v||\nabla v^{0}|\|_{r}+\||\delta
v||\nabla(\chi_{-}v^{-}+\chi_{+}v^{+})|\|_{r}+\||\nabla\delta
v||v|\|_{r}+\||\delta v||v|\|_{r}$ $\displaystyle\lesssim$
$\displaystyle\|\delta v\|_{\infty}\|\nabla v^{0}\|_{r}+\|\delta
v\|_{L^{r}(L^{\infty})}\|\nabla(\chi_{-}v^{-}+\chi_{+}v^{+})\|_{L^{\infty}(L^{r})}+\|\nabla\delta
v\|_{r}\|v\|_{\infty}+\||\delta v\|_{r}\|v\|_{\infty}$ $\displaystyle\lesssim$
$\displaystyle\|\delta v\|_{L^{r}_{1}}\|v\|_{\mathcal{V}_{1}}.$

Case 2\. If $\delta v=\chi_{-}\delta v^{-}$, then we think of
$(d\mathcal{F})(v_{2}+\delta v)-(d\mathcal{F})(v_{2})$ as a lower triangular
matrix as in (11.23) and verify that each component is bounded. The same
argument as in Case 1 shows that this operator is $L^{r}_{1}\to L^{r}$ and it
acts trivially on $v=\chi_{+}v^{+}$, $v^{+}\in V_{1}^{+}$.

We focus on the case when $v=\chi_{-}v^{-}$ for some $v^{-}\in V^{-}_{1}$. The
$(1,1)$-entry of (11.23) is bounded using Lemma 11.16 (this is a property
about the real line $\mathbb{R}_{s}$). Since the section $v$ is supported on
$\\{t\leq-K\\}$, we may pretend that $P^{\operatorname{ref}}_{R}$ is the
constant trajectory at $p^{\operatorname{pre}}_{R,-}$ and
$v_{2}=\chi_{-}v_{2}^{-}+v_{2}^{0}$ for this computation. With this
understood, the $(3,1)$-entry of (11.23) for $(d\mathcal{F})(v_{2}+\delta
v)-(d\mathcal{F})(v_{2})$ applied to $v=\chi_{-}v^{-}$ is given by

$(d\mathcal{F})(v_{2}+\chi_{-}\delta
v^{-})(\chi_{-}v^{-})-(d\mathcal{F})(v_{2})(\chi_{-}v^{-})-\chi_{-}\big{(}(d\mathcal{F})(v_{2}^{-}+\delta
v^{-})(v^{-})-(d\mathcal{F})(v_{2}^{-})(v^{-})\big{)}$

which is bounded pointwise by

$\chi_{-}(|\delta v^{-}||v^{-}||v_{2}^{0}|+|\nabla\delta
v^{-}||v^{-}||v_{2}^{0}|+|\delta v^{-}||\nabla v^{-}||v_{2}^{0}|+|\delta
v^{-}||v^{-}||\nabla v_{2}^{0}|)$

up to a uniform constant. This is due to (11.29). Since $\nabla\delta
v^{-},\nabla v^{-}\in L^{\infty}(\mathbb{R}_{t};L^{r}(\mathbb{R}^{s}))$ and
$v_{2}^{0}\in L^{r}_{1}(\mathbb{R}_{t}\times\mathbb{R}_{s})\hookrightarrow
L^{r}(\mathbb{R}_{t};L^{\infty}(\mathbb{R}^{s}))$, we conclude that this
$(3,1)$-entry applied to $v=\chi_{-}v^{-}$ has $L^{r}$-norm bounded by

$\|v^{0}_{2}\|_{L^{r}_{1}(\mathbb{R}_{t}\times\mathbb{R}_{s})}\|\delta
v^{-}\|_{L^{r}_{1}(\mathbb{R}_{s})}\|v^{-}\|_{L^{r}_{1}(\mathbb{R}_{s})}.$

Case 3\. The case that $\delta v=\chi_{-}\delta^{+}$ is identical to Case 2.

This completes the proof of Lemma 11.14 $($ii$)$. ∎

###### Proof of Lemma 11.14 $($iii$)$.

Note that if $s\leq\frac{(R+\pi)}{4}$, then
$P^{\operatorname{pre}}_{R}=P^{un}$ and
$P^{\operatorname{ref}}_{R}=P^{un}_{K}$, so $v_{1}$ is equal to the section
$P^{un}-P^{un}_{K}$ in (11.25). Moreover, $v_{1}\equiv 0$ if
$s\geq\frac{3(R+\pi)}{8}$. For $s\in[\frac{R+\pi}{4},\frac{3(R+\pi)}{8}]_{s}$,
since the instanton $P^{un}$ decays exponentially to $x_{l}$ as $s\to\infty$,
one verifies that

$\|v_{1}\|_{L^{r}_{1}(\mathbb{R}_{t}\times[\frac{R+\pi}{4},\frac{3(R+\pi)}{8}]_{s})}\lesssim\|P^{un}-P^{un}_{K}\|_{L^{r}_{1}(\mathbb{R}_{t}\times[\frac{R+\pi}{4},\frac{3(R+\pi)}{8}]_{s})}\to
0$

as $R\to\infty$. This implies that $\|v_{1}\|_{L^{r}_{1}}\leq 2\delta_{K}$ for
$R\gg\pi$.

To estimate $\|\mathcal{F}(v_{1})\|_{\mathcal{V}_{0}}$, since
$\Psi_{P^{\operatorname{ref}}_{R}}(v_{1})$ is an isometry, it suffices to
estimate $\mathfrak{F}(P^{\operatorname{pre}}_{R})$. This section is supported
on $\mathbb{R}_{t}\times[\frac{R+\pi}{4},\frac{3(R+\pi)}{4}]_{s}$. Apply
Proposition 3.6 to $P^{un}$ and $P^{st}$; we conclude that the
$L^{r}_{1}$-norm of $\mathfrak{F}(P^{\operatorname{pre}}_{R})$ on
$[-R,R]_{t}\times[\frac{R+\pi}{4},\frac{3(R+\pi)}{4}]_{s}$ decays
exponentially as $R\to\infty$. The limits of
$\mathfrak{F}(P^{\operatorname{pre}}_{R})$ at $\pm\infty$ are given precisely
by
$\operatorname{grad}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p^{\operatorname{pre}}_{R,\pm})$
which also decays exponentially as $R\to\infty$ in
$L^{r}_{1}(\mathbb{R}_{s})$. Finally, the convergence
$P^{\operatorname{pre}}_{R}(t,\cdot)\to p^{\operatorname{pre}}_{R,\pm}$ is
also exponentially in $L^{r}_{1}(I_{t}^{\prime}\times\mathbb{R}_{s})$ as
$t\to\pm\infty$, so
$\mathfrak{F}(P^{\operatorname{pre}}_{R})-\operatorname{grad}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p^{\operatorname{pre}}_{R,\pm})\to
0$ exponentially in $L^{r}(I_{t}^{\prime}\times\mathbb{R}_{s})$. We conclude
that $\|\mathcal{F}(v_{1})\|_{\mathcal{V}_{0}}\to 0$ as $R\to\infty$. ∎

###### Proof of Lemma 11.14$($iv$)$.

Let $D^{un},D^{st}$ denote the linearization of the $\alpha$-instanton
equation at $P^{st}$ and $P^{un}$. Then $D^{st}$ is invertible, and $\ker
D^{un}$ is 1-dimensional and is generated by $\partial_{t}P^{st}$. Choose a
smooth section $v^{\prime}$ of $(P^{un})^{*}TM\to\mathbb{R}^{2}$ supported on
$[-1,1]_{t}\times[-1,1]_{s}$ such that the $L^{2}$-pairing
$f=\langle\cdot,v^{\prime}\rangle$ is non-zero on $\ker D^{un}$, so

$D^{un}\oplus f:L^{r}_{1}(\mathbb{R}^{2};(P^{un})^{*}TM)\to
L^{r}(\mathbb{R}^{2};(P^{un})^{*}TM\oplus\mathbb{R}.$

is invertible. Such a section $v^{\prime}$ exists due to a unique continuation
property of $\partial_{t}P^{st}$. Since
$P^{\operatorname{ref}}_{R}=P^{\operatorname{pre}}_{R}=P^{un}$ on
$[-1,1]_{t}\times[-1,1]_{s}$, $f$ can be viewed also a map
$\mathcal{V}_{1}\to\mathbb{R}$. We first consider the linearization of
$\mathfrak{F}$ at $P^{\operatorname{pre}}_{R}$:

$D^{\operatorname{pre}}_{R}:L^{r}_{1}(\mathbb{R}^{2};(P^{\operatorname{pre}}_{R})^{*}TM)\to
L^{r}(\mathbb{R}^{2};(P^{\operatorname{pre}}_{R})^{*}TM).$

Since $P^{\operatorname{pre}}_{R}\equiv x_{l}$ when
$s\in[\frac{3(R+\pi)}{8},\frac{5(R+\pi)}{8}]_{s}$, the same excision argument
using the transformation (11.8) on $\mathbb{R}_{t}\times\mathbb{R}_{s}$ allows
us to compare $D^{st}\oplus D^{un}\oplus f$ with the operator

$(\partial_{t}+D_{x_{l}})\oplus(D^{\operatorname{pre}}_{R}\oplus f).$

and obtain an equation similar to (11.2). Again the error term $\delta_{R}$
has operator norm $\to 0$ as $R\to\infty$, so
$D^{\operatorname{pre}}_{R}\oplus f:L^{r}_{1}\to L^{r}$ is also invertible,
and its inverse is bounded uniformly. Returning to the reference trajectory
$P^{\operatorname{ref}}_{R}$, this implies that $d\mathcal{F}(v_{1})\oplus
f:L^{r}_{1}\to L^{r}$ is invertible. As a map from
$\mathcal{V}_{1}\to\mathcal{V}_{0}$, $d\mathcal{F}(v_{1})\oplus f$ is cast
into a lower triangular matrix as in (11.23), and we just verified that the
last diagonal entry is invertible. The first two are given by
$\operatorname{Hess}\mathcal{A}_{W,\mathfrak{a}^{R}_{m1}}(p^{\operatorname{pre}}_{R,\pm})$,
which are invertible by Lemma 11.5. Off-diagonal entries are bounded by
$($ii$)$. This proves that $d\mathcal{F}(v_{1})\oplus
f:\mathcal{V}_{1}\to\mathcal{V}_{0}\oplus\mathbb{R}$ is invertible. We take
$Q_{R}=(d\mathcal{F}(v_{1})\oplus f)^{-1}|_{\mathcal{V}_{0}}$. Then the
operator norm of $Q_{R}$ is bounded uniformly and $\operatorname{Im}Q_{R}=\ker
f$. ∎

###### Proof of Lemma 11.15.

For any $\epsilon>0$, we prove that
$\|\tau_{t_{n}}P_{n}-P^{\operatorname{pre}}_{R_{n}}\|_{\mathcal{V}_{1}}<100\epsilon$
for $R_{n}\gg\pi$. The limits of $\tau_{t_{n}}P_{n}$ and
$P^{\operatorname{pre}}_{R_{n}}$ at $\pm\infty$ are given respectively by
$p_{R_{n},\pm}\colonequals p^{st}_{\pm}\circ_{R_{n}}p^{un}$ and
$p^{\operatorname{pre}}_{R_{n},\pm}$. Write

$p^{st}_{\pm}\circ_{R_{n}}p^{un}=\exp_{p^{\operatorname{pre}}_{R_{n},\pm}}(v_{R_{n}}^{\pm})$

with $v_{R_{n}}^{\pm}\in
L^{r}_{1}(\mathbb{R}_{s};(p^{\operatorname{pre}}_{R_{n},\pm})^{*}TM)=V_{1}^{\pm}$,
then by (11.10)

$\|v_{R_{n}}^{\pm}\|_{L^{r}_{1}(\mathbb{R}_{s})}<\epsilon$

for $R_{n}\gg\pi$. Write

$\tau_{t_{n}}P_{n}=\exp_{P^{\operatorname{ref}}_{R}}(v_{n}),\
v_{n}=\chi_{-}v_{R_{n}}^{-}+\chi_{+}v_{R_{n}}^{+}+v_{n}^{0}\in\mathcal{V}_{1}.$

We have to show that $\|v_{n}^{0}-v_{1}\|_{L^{r}_{1}}\to 0$ for $R_{n}\gg\pi$.
Recall that

$P^{\operatorname{pre}}_{R_{n}}=\exp_{P^{\operatorname{ref}}_{R}}(v_{1}),\
v_{1}\in L^{r}_{1}=V_{1}^{0}.$

Since the convergences $P^{\operatorname{pre}}_{R_{n}}\to
p^{\operatorname{pre}}_{R_{n},\pm}$ and $\tau_{t_{n}}P_{n}\to p_{R_{n},\pm}$
are exponential as $t\to\pm\infty$ (and uniform in $R_{n}$), for some
$K_{\epsilon}\gg 0$, we have

$\|v_{n}^{0}\|_{L^{r}_{1}(\\{|t|\geq K_{\epsilon}\\})}<\epsilon,\
\|v_{1}\|_{L^{r}_{1}(\\{|t|\geq K_{\epsilon}\\})}<\epsilon.$

By Lemma 11.6 and Proposition 3.6, for some $R_{\epsilon}>0$,

$\|v_{n}^{0}\|_{L^{r}_{1}(\\{|s|>R_{\epsilon}\\}\cap\\{|s-R|\geq
R_{\epsilon}\\})}<\epsilon,\
\|v_{1}\|_{L^{r}_{1}(\\{|s|>R_{\epsilon}\\}\cap\\{|s-R|\geq
R_{\epsilon}\\})}<\epsilon.$

Thus it suffices to estimate the $L^{r}_{1}$-norm of $|v_{n}^{0}-v_{1}|$ on
the rectangles

$\Omega_{\epsilon}=[-K_{\epsilon},K_{\epsilon}]_{t}\times[-R_{\epsilon},R_{\epsilon}]_{s}\
\bigcup\
[-K_{\epsilon},K_{\epsilon}]_{t}\times[R-R_{\epsilon},R+R_{\epsilon}]_{s}.$

However, $\|v_{n}^{0}-v_{1}\|_{L^{r}_{1}(\Omega_{\epsilon})}\to 0$ as
$R_{n}\to\infty$ by the $C^{\infty}_{loc}$-convergence in Corollary 11.11.
This completes the proof of Lemma 11.15. ∎

####  11.6. Quasi-units revisited

Finally we prove Proposition 8.18 using the vertical gluing theorem. We focus
on the second case when $q_{0}=q_{1}$ and set up the neck-stretching picture.
As we will see shortly, this case is identical to the one addressed in Theorem
11.4. Since $l_{q_{2},\theta_{2}}$ intersects with $l_{q_{0},\theta_{0}}$ at
$y_{0}^{\prime}$, we must have $\theta_{0}-\pi<\theta_{2}$. Choose another
angle $\theta_{-1}=\theta_{0}+\epsilon$ for some $0<\epsilon\ll 1$ such that

(11.31)
$\theta_{0}-\pi<\theta_{-1}-\pi<\theta_{2}<\theta_{1}<\theta_{0}<\theta_{-1}.$

Moreover, if $l_{q_{2},\theta_{2}}$ intersects with the ray
$l_{q,\theta_{-1}}$ at $y_{-1}^{\prime}$, then we require that the triangle
$(y^{\prime}_{-1},y_{1}^{\prime},W(q))$ contains no critical values of $W$ in
the interior (think of $l_{q,\theta_{-1}}$ as a slight perturbation of
$l_{q,\theta_{0}})$. Now consider the thimbles:

$U_{\star}=\Lambda_{q,\theta_{-1}-\pi}\text{ and
}S_{\star}=\Lambda_{q,\theta_{-1}}.$

and write for convenience $\Lambda_{j}=\Lambda_{q_{j},\theta_{j}},j=0,1,2$.
For some $0<\delta\ll 1$, fix smooth monotone functions
$\alpha_{0},\alpha_{1},\alpha_{2}:\mathbb{R}_{s}\to\mathbb{R}$ such that

(11.36) $\displaystyle\alpha_{j}(s)$
$\displaystyle=\left\\{\begin{array}[]{ll}\theta_{-1}-\pi&\text{if
}s\geq\pi-\delta,\\\ \theta_{j}-\pi&\text{if }s\leq\delta,\end{array}\right.$
$\displaystyle\alpha_{2}(s)$
$\displaystyle=\left\\{\begin{array}[]{ll}\theta_{2}&\text{if
}s\geq\pi-\delta,\\\ \theta_{-1}-\pi&\text{if }s\leq\delta.\end{array}\right.$

Choose admissible Floer data
$\mathfrak{a}_{j}=(\pi,\alpha_{j}(s),\beta,\epsilon,\delta H_{j}\equiv 0)$ for
the pair $(\Lambda_{j},U_{\star})$, $j=0,1$ and
$\mathfrak{a}_{2}=(\pi,\alpha_{2}(s),\beta,\epsilon,\delta H_{2})$ for
$(S_{\star},\Lambda_{2})$. Then they can be concatenated as in Section 11.1 to
give a Floer datum
$\mathfrak{a}_{j2}^{R}=(R+\pi,\alpha_{j2}^{R}(s),\beta,\epsilon,\delta
H_{j2}^{R})$ for the pair $(\Lambda_{j},\Lambda_{2})$ with
$\alpha_{j2}^{R}:\mathbb{R}_{s}\to\mathbb{R}$ defined similarly as in (9.8)
and $\delta H_{j2}^{R}$ as in (11.3) for all $R\geq\pi$ and $j=0,1$. Finally,
choose any admissible Floer data
$\mathfrak{a}_{01}=(\pi,\alpha_{01},\beta_{01},\epsilon,\delta H_{01}\equiv
0)$ for the pair $(\Lambda_{0},\Lambda_{1})$.

\begin{overpic}[scale={.15}]{Pic60.png} \put(-20.0,20.0){$q=q_{0}=q_{1}$}
\put(5.0,65.0){$q_{2}$} \put(96.0,45.0){$\Lambda_{1}$}
\put(60.0,65.0){$\Lambda_{0}$} \put(100.0,25.0){$\Lambda_{2}$}
\put(45.0,70.0){$S_{\star}$} \put(-5.0,0.0){$U_{\star}$}
\put(23.0,54.0){$y_{-1}^{\prime}$} \put(49.0,52.0){$y_{0}^{\prime}$}
\put(70.0,42.0){$y_{1}^{\prime}$} \end{overpic} Figure 39. The dashed curve is
the projection of an $\alpha_{12}^{R}$-soliton. As $R\to\infty$, it converges
to the constant soliton at $q$ followed by an $\alpha_{2}$-soliton.

By (11.31), the function $\alpha_{j2}^{R}(s):\mathbb{R}_{s}\to\mathbb{R}$ is
also monotone for all $R\geq\pi$ and $j=0,1$. Combined with our condition on
the critical values of $W$, this implies that the complex

$\operatorname{CF}_{\natural}^{*}(\Lambda_{j},\Lambda_{2};\mathfrak{a}_{j2}^{R}),j=0,1$

only has one filtration level when $R\gg\pi$, and in fact

$\operatorname{CF}_{\natural}^{*}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{j2}^{R})\cong\operatorname{CF}_{\natural}^{*}(S_{\star},\Lambda_{2};\mathfrak{a}_{2})\otimes\operatorname{CF}_{\natural}^{*}(\Lambda_{j},U_{\star};\mathfrak{a}_{j}).$

This means any soliton
$p_{R}\in\mathfrak{C}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{j2}^{R})$ is
obtained by gluing some
$p_{2}\in\mathfrak{C}(S_{\star}.\Lambda_{2};\mathfrak{a}_{2})$ with the
constant soliton
$e_{q}\in\mathfrak{C}(\Lambda_{j},U_{\star};\mathfrak{a}_{j})$. As in the case
of Theorem 11.4, the vertical gluing theorem in this case identifies the
cobordism map

(11.37)
$\mu^{2}:\operatorname{CF}_{\natural}^{*}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})\otimes\operatorname{CF}_{\natural}^{*}(\Lambda_{0},\Lambda_{1},\mathfrak{a}_{01})\to\operatorname{CF}_{\natural}^{*}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{02}^{R})$

with the tensor product of the identity continuation map

$\operatorname{Id}:\operatorname{CF}_{\natural}^{*}(S_{\star},\Lambda_{2};\mathfrak{a}_{2})\to\operatorname{CF}_{\natural}^{*}(S_{\star},\Lambda_{2};\mathfrak{a}_{2})$

and the cobordism map

$\mu^{2}_{q}:\operatorname{CF}_{\natural}^{*}(\Lambda_{1},U_{\star};\mathfrak{a}_{1})\otimes\operatorname{CF}_{\natural}^{*}(\Lambda_{0},\Lambda_{1},\mathfrak{a}_{01})\to\operatorname{CF}_{\natural}^{*}(\Lambda_{0},U_{\star};\mathfrak{a}_{0}).$

For the very last map $\mu^{2}_{q}$, all thimbles are associated to the same
$q\in\operatorname{Crit}(W)$; so each Floer complex has rank 1 and is
generated by the quasi-unit $e_{q}$. Then $\mu^{2}_{q}$ is an isomorphism by
Lemma 8.20. This implies that the cobordism map $\mu^{2}$ in(11.37) is a chain
isomorphism for $R\gg\pi$. This completes the proof of Proposition 8.18. One
may compare Figure 40 below with Figure 36.

\begin{overpic}[scale={.15}]{Pic56.png}
\put(12.0,23.0){\small$\mathbb{R}_{t}\times[\pi,R]_{s}$}
\put(5.0,13.0){\small$\Lambda_{0}$} \put(5.0,23.0){\small$q$}
\put(5.0,32.0){\small$\Lambda_{2}$} \put(65.0,9.0){\small$\Lambda_{0}$}
\put(65.0,20.0){\small$U_{\star}$} \put(65.0,27.0){\small$S_{\star}$}
\put(65.0,36.0){\small$\Lambda_{2}$} \put(75.0,2.0){\small$q$}
\put(75.0,20.0){\small$q$} \put(75.0,27.0){\small$q$}
\put(75.0,36.0){\small$q_{2}$} \put(54.0,32.0){\small$p_{2}$}
\put(86.0,32.0){\small$p_{2}$} \put(54.0,16.0){\small$e_{q}$}
\put(86.0,16.0){\small$e_{q}$} \put(-2.0,27.0){\small$p_{2}$}
\put(30.0,27.0){\small$p_{2}$} \put(-2.0,18.0){\small$e_{q}$}
\put(30.0,18.0){\small$e_{q}$} \put(30.0,8.0){\small$e_{q}$}
\put(86.0,6.0){\small$e_{q}$} \put(-1.5,22.5){\small$\circ_{R}$}
\put(30.5,22.5){\small$\circ_{R}$} \put(39.0,26.0){\small converge}
\put(42.0,20.0){\small glue} \end{overpic} Figure 40. Quasi-Units.

####  11.7. Seidel’s long exact sequence

Proposition 8.18 is concerned with the case when there is no critical values
in interior of $(y_{0}^{\prime},y_{1}^{\prime},W(q))$, so the thimble
$\Lambda_{q,\theta_{0}}$ can be deformed continuous into
$\Lambda_{q,\theta_{1}}$ without violating the condition (2.8) or the weaker
version in Remark 2.7. But what if there is such a critical value?

\begin{overpic}[scale={.15}]{Pic61.png} \put(-20.0,20.0){$q=q_{0}=q_{1}$}
\put(5.0,65.0){$q_{2}$} \put(96.0,45.0){$\Lambda_{1}$}
\put(60.0,65.0){$\Lambda_{0}$} \put(100.0,25.0){$\Lambda_{2}$}
\put(103.0,35.0){$\Lambda_{3}$} \put(45.0,70.0){$S_{\star}$}
\put(-5.0,0.0){$U_{\star}$} \put(42.0,33.0){$q_{3}$} \end{overpic} Figure 41.
The dashed curves are the projection of some $\alpha_{12}^{R}$-solitons. There
are two filtration levels now.

Suppose that this critical value is associated to
$q_{3}\in\operatorname{Crit}(W)$. Choose an angle $\theta_{3}$ such that

$\theta_{-1}-\pi<\theta_{2}<\theta_{3}<\theta_{1}<\theta_{0}<\theta_{-1}$

and let $\Lambda_{3}\colonequals\Lambda_{q_{3},\theta_{3}}$. Figure 39 is then
replaced by Figure 41 above. One can also choose a function
$\alpha_{3}:\mathbb{R}_{s}\to\mathbb{R}$ such that for some $0<\delta\ll 1$,

$\displaystyle\alpha_{3}(s)$
$\displaystyle=\left\\{\begin{array}[]{ll}\theta_{-1}-\pi&\text{if
}s\geq\pi-\delta,\\\ \theta_{3}-\pi&\text{if }s\leq\delta,\end{array}\right.$

to define a filtered complex
$\operatorname{CF}^{*}_{\natural}(\Lambda_{3},\Lambda_{2};\mathfrak{a}_{32}^{R})$
for $R\gg\pi$ (replace $\Lambda_{0}$ by $\Lambda_{3}$ throughout the
construction of Section 11.6 ). In this case, there are two filtration levels
for
$\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})$:

$\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})=\mathcal{G}^{0}\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})\oplus\mathcal{G}^{1}\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})$

where $\mathcal{G}^{0}$ (resp. $\mathcal{G}^{1}$) is associated to $q_{3}$
(resp. $q$). The summand $\mathcal{G}^{0}$ has higher energy level. The
geometric filtration on
$\operatorname{CF}^{*}_{\natural}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{02}^{R})$
then gives a short sequence sequence of chain complexes:

(11.38)
$0\to\mathcal{G}^{0}\to\operatorname{CF}^{*}_{\natural}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{02}^{R})\to\mathcal{G}^{1}\to
0.$

Meanwhile,

$\operatorname{CF}^{*}_{\natural}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{32}^{R})=\mathcal{G}^{1}\operatorname{CF}^{*}_{\natural}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{02}^{R})\text{
and
}\operatorname{CF}^{*}_{\natural}(\Lambda_{3},\Lambda_{2};\mathfrak{a}_{32}^{R})=\mathcal{G}^{0}\operatorname{CF}^{*}_{\natural}(\Lambda_{3},\Lambda_{2};\mathfrak{a}_{32}^{R}).$

The chain map induced by the $(2+1)$-disk preserves the filtration, so we
arrive at a diagram

(11.39)
${\mathcal{G}^{0}\operatorname{CF}^{*}_{\natural}(\Lambda_{3},\Lambda_{2};\mathfrak{a}_{32}^{R})\otimes\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{3};\mathfrak{a}_{13})}$${\mathcal{G}^{0}\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})}$${\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{32}^{R})}$${\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{32}^{R})}$${\mathcal{G}^{1}\operatorname{CF}^{*}_{\natural}(\Lambda_{0},\Lambda_{2};\mathfrak{a}_{32}^{R})}$${\mathcal{G}^{1}\operatorname{CF}^{*}_{\natural}(\Lambda_{1},\Lambda_{2};\mathfrak{a}_{12}^{R})}$$\scriptstyle{\mu^{2}}$$\scriptstyle{\mu^{2}}$$\scriptstyle{\mu^{2}(\cdot,e_{q})}$$\scriptstyle{\mu^{2}(\cdot,e_{q})}$$\scriptstyle{\cong}$

where the right column is given by (11.38). As in the case of (11.37), the
bottom horizontal arrow of (11.39) is a chain isomorphism, whereas the top
arrow is only a quasi-isomorphism by the vertical gluing theorem and
Proposition 8.18. Nevertheless, this proves that the left column forms an
exact triangle and hence recovers a special case of Seidel’s long exact
sequence [Sei03].

######  Proposition 11.17.

Suppose we are in the scenario of Figure 41, then there is a long exact
sequence:

${\operatorname{HF}_{\natural}^{*}(\Lambda_{3},\Lambda_{2})\otimes\operatorname{HF}_{\natural}^{*}(\Lambda_{1},\Lambda_{3})}$${\operatorname{HF}_{\natural}^{*}(\Lambda_{1},\Lambda_{2})}$${\operatorname{HF}_{\natural}(\Lambda_{0},\Lambda_{2}).}$$\scriptstyle{m_{*}}$$\scriptstyle{m_{*}(e_{q},\cdot)}$$\scriptstyle{[1]}$

Thus the group
$\operatorname{HF}_{\natural}^{*}(\Lambda_{3},\Lambda_{2})\otimes\operatorname{HF}_{\natural}^{*}(\Lambda_{1},\Lambda_{3})$
measures the failure of $m_{*}(e_{q},\cdot)$ being an isomorphism.

######  Remark 11.18.

The fact that the composition
$m_{*}(e_{q},m_{*}(\cdot,\cdot)):\operatorname{HF}_{\natural}^{*}(\Lambda_{3},\Lambda_{2})\otimes\operatorname{HF}_{\natural}^{*}(\Lambda_{1},\Lambda_{3})\to\operatorname{HF}_{\natural}(\Lambda_{0},\Lambda_{2})$
is trivial be also seen as follows: since $m_{*}$ is associative, this map
factorizes through the group
$\operatorname{HF}_{\natural}^{*}(\Lambda_{0},\Lambda_{3})$, which is trivial,
since $\Lambda_{0}\cap\Lambda_{3}=\emptyset$; see Figure 41. ∎

##  Part IV Lagrangian Submanifolds

Having verified the Koszul duality pattern between $\mathcal{A}$ and
$\mathcal{B}$, our next step is to explore its implication for a pair of
$(X,Y)$ of Lagrangian submanifolds. Section 12 is devoted to a generalization
of $\operatorname{HF}_{\natural}^{*}$ where one entry is replaced by an exact
graded Lagrangian submanifold, which allows us to define the
$A_{\infty}$-modules ${}_{\mathcal{A}}X$ and $Y_{\mathcal{B}}$. In Section 13,
we prove our main result Theorem1.9 by verifying the geometric filtration on
the Floer complex of $(X,Y)$ is isomorphic to the algebraic filtration of
Seidel [Sei08, Corollary 18.27]. This proof will follow the same argument as
in Theorem 8.3.

###  12\. Floer Cohomology for Lagrangian Submanifolds

####  12.1. The setup

Our proof of the compactness theorem so far is based on the energy method; in
this regard, it is not realistic to work with arbitrary non-compact Lagrangian
submanifolds. We have to make some mild conditions on the projection
$W(X),W(Y)\subset\mathbb{C}$ in order to make this energy method work, which
are spelled out as follows.

######  Assumption 12.1.

In this paper, we shall only consider Lagrangian submanifolds $X$ of $M$
satisfying the following properties:

  1. $($L1$)$

$X$ is exact, i.e., there exists a smooth function $h_{X}:X\to\mathbb{R}$ such
that $dh_{X}=\lambda_{M}|_{X}$;

  2. $($L2$)$

$X$ is graded, i.e., the Maslov class $\mu_{X}\in H^{1}(X,\mathbb{Z})$
vanishes, and a grading $(X,\xi_{X}^{\\#})$ is already fixed.

  3. $($L3$)$

there exist some $\beta_{X}\in\mathbb{R}$ and $C_{X}>0$ such that

(12.1) $\operatorname{Re}(e^{-i\beta_{X}}W)|_{X}<C_{X},$

i.e., the projection $W(X)\subset\mathbb{C}$ is bounded above in the direction
of $e^{i\beta_{X}}$. If $W(X)$ is compact, then the angle $e^{i\beta_{X}}$ in
(12.1) can be arbitrary. But we may also see the case when $e^{i\beta_{X}}$ is
unique. We assume that a choice of such $\beta_{X}\in\mathbb{R}$ has been
chosen, once and for all, for this $X$. Unlike the case of thimbles, the
grading of $X$ is not related to $\beta_{X}$.

  4. $($L4$)$

$X$ has bounded geometry as a totally real submanifold of
$(M,J_{M},g_{M},\omega_{M})$. ∎

Let $\Lambda_{q,\theta},q\in\operatorname{Crit}(W)$ be any thimble satisfying
(2.8) and with

(12.2) $e^{i(\theta-\beta_{X})}>0.$

Under this assumption, one can construct the Floer cohomology
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta})$ following the scheme
in Section 2. In this case, a Floer datum
$\mathfrak{a}_{X}=(R,\alpha_{X}(s),\beta_{X},\epsilon_{X},\delta H)$ consists
of a constant $R\geq\pi$, a real-valued function
$\alpha_{X}:\mathbb{R}^{+}_{s}\to\mathbb{R}$, an angle
$\beta_{X}\in\mathbb{R}$ as in $($L3$)$, $\epsilon_{X}>0$, and a perturbation
1-form $\delta H=\delta H_{s}ds\in\Omega^{1}(\mathbb{R}_{s}^{+};\mathcal{H})$,
which satisfy the following properties:

  * •

for some $0<\delta\ll 1$, $\alpha_{X}(s)=\theta$ for all $s\geq R-\delta$;

  * •

$\operatorname{Re}(e^{i(\alpha_{X}(s)-\beta_{X})})\geq\epsilon_{X}$ for all
$s\in\mathbb{R}^{+}_{s}$ (this condition requires (12.2));

  * •

$\delta H$ is supported on $[0,R]_{s}$ and satisfies (2.7).

In practice, one may simply take $\alpha_{X}(s)\equiv\theta$ to be constant,
but it is harmless to make the setup a bit more flexible. Let
$p_{\operatorname{model}}:\mathbb{R}_{s}^{+}\to M$ be a reference map such
that $p_{\operatorname{model}}(s)\equiv q$ when $s\geq R$. Consider the path
space

$\mathcal{P}_{k}(X,\Lambda_{q,\theta})\colonequals\\{p\in
C^{\infty}(\mathbb{R}_{s}^{+};M):p(0)\in X,\ p\text{ has finite
$L^{2}_{k}$-distance with }p_{\operatorname{model}}\\},k\geq 1.$

Then any Floer datum $\mathfrak{a}_{X}$ defines an action functional
$\mathcal{A}_{X,\mathfrak{a}_{X}}$ on $\mathcal{P}_{k}(X,\Lambda_{q,\theta})$
by the formula:

(12.3) $\displaystyle\mathcal{A}_{X,\mathfrak{a}_{X}}(p)$
$\displaystyle\colonequals-
h_{X}(p(0))+\int_{\mathbb{R}_{s}^{+}}-p^{*}\lambda_{M}+\delta H_{s}(p(s))ds$
$\displaystyle+\int_{\mathbb{R}_{s}^{+}}\big{(}\operatorname{Im}(e^{-i\alpha_{X}}(W\circ
p(s)))-\chi_{[R,+\infty)}\operatorname{Im}(e^{-i\theta}W(q))\big{)}ds$

whose gradient vector at $p\in\mathcal{P}_{k}(X,\Lambda_{q,\theta})$ is an
$L^{2}$-section of $p^{*}TM$:

$\operatorname{grad}\mathcal{A}_{X,\mathfrak{a}_{X}}(p)=J\partial_{s}p(s)+\nabla\operatorname{Im}(e^{-i\alpha_{X}(s)}W)\circ
p(s)+\nabla(\delta H_{s})\in L^{2}\cap
C^{\infty}(\mathbb{R}_{s}^{+},p^{*}TM).$

\begin{overpic}[scale={.15}]{Pic51.png} \put(45.0,31.0){\small$S_{4}$}
\put(45.0,34.0){\small$S_{3}$} \put(45.0,37.0){\small$S_{2}$}
\put(45.0,40.0){\small$S_{1}$} \put(-10.0,22.0){\small$H(x_{4})$}
\put(-10.0,16.0){\small$H(x_{3})$} \put(-10.0,9.0){\small$H(x_{2})$}
\put(-10.0,3.0){\small$H(x_{1})$} \put(35.0,5.0){\small$X$}
\put(48.0,10.0){\small$e^{i\beta_{X}}$}
\put(60.0,35.0){\small$\operatorname{Re}\langle
w,e^{i\beta_{X}}\rangle=C_{X}$} \put(95.0,40.0){\small$l_{q,\theta_{0}}$}
\put(102.0,24.0){\small$l_{q,\theta_{1}}$} \put(66.0,6.0){\small$W(q)$}
\end{overpic} Figure 42. The left $\mathcal{A}$-module ${}_{\mathcal{A}}X$
(left) and quasi-units (right).

Let $\mathfrak{C}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$ denote the space of
critical points of $\mathcal{A}_{X,\mathfrak{a}_{X}}$, also called
$\alpha_{X}$-solitons (this is an abuse of notations, because they also rely
on the perturbation 1-form $\delta H$). The Floer cohomology
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$ is
defined by counting the $\alpha_{X}$-instantons on the upper half plane
$\mathbb{R}_{t}\times\mathbb{R}_{s}^{+}$ with a boundary condition along
$\mathbb{R}_{t}\times\\{0\\}$.

(12.4)
$\left\\{\begin{array}[]{rl}P:\mathbb{R}_{t}\times\mathbb{R}_{s}^{+}&\to M,\\\
\partial_{t}P+J(\partial_{s}P+\nabla\operatorname{Re}(e^{-i\alpha_{X}(s)}W))+\nabla(\delta
H)_{s}&=0,\\\ \lim_{s\to\infty}P(t,s)&=q,\\\
\lim_{t\to\pm\infty}(P,s)&=p_{\pm}(s),\\\ P(t,0)&\in X.\end{array}\right.$

with $p_{\pm}\in\mathfrak{C}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$. The
boundary condition should be interpreted as in (2.13). A solution of (12.4) is
formally a downward gradient flowline of $\mathcal{A}_{X,\mathfrak{a}_{X}}$.

####  12.2. Compactness

We explain quickly the necessary adaption in order to prove the compactness
theorem in this context. Let $\gamma_{X}:\mathbb{R}^{+}_{s}\to\mathbb{C}$,
$\partial_{s}\gamma_{X}(s)=-e^{i\alpha_{X}(s)}$ be a characteristic curve of
$\alpha_{X}(s)$. To drive the energy estimate, consider the phase function
$\Xi$

$\Xi(s,t)=-i\epsilon_{X}e^{i\beta_{X}}\cdot t+\gamma_{X}(s),\
(t,s)\in\mathbb{H}^{+}=\mathbb{R}_{t}\times\mathbb{R}_{s}^{+}$

and set $\delta\kappa=\operatorname{Im}(i\epsilon_{X}e^{-i\beta_{X}}dt\cdot
W)$. For any $\alpha_{X}$-instanton $P$, apply the energy estimate of Lemma
5.2 first to the rectangle $[t_{0},t_{1}]_{t}\times[0,R_{1}]_{s}$, $R_{1}\gg
R$ and then let $R_{1}\to\infty$ to obtain that

(12.5)
$\displaystyle\mathcal{A}_{X,\mathfrak{a}_{X}}\big{(}P(t_{0},\cdot)\big{)}-\mathcal{A}_{X,\mathfrak{a}_{X}}\big{(}P(t_{1},\cdot)\big{)}+\text{
some terms involving }\delta H$
$\displaystyle+\epsilon_{X}\int_{t_{0}}^{t_{1}}\operatorname{Re}\big{(}e^{-i\beta_{X}}\big{(}W(p(t,0))-W(q)\big{)}\big{)}\geq\epsilon_{X}^{\prime}\mathcal{E}_{an}(P;[t_{0},t_{1}]\times\mathbb{R}_{s}^{+}).$

for a constant $\epsilon_{X}^{\prime}>0$. This is the analogue of Lemma 2.8 in
the presence of a Lagrangian boundary condition. The condition (12.1) is used
here to upper bound $\operatorname{Re}(e^{-i\beta_{X}}W(p(t,0)))$ on the left
hand side of (12.5). At this point, we need a boundary version of Lemma 3.1.
For future reference, we record this as follows.

######  Lemma 12.2 (Local Compactness II: the Boundary Case).

Let $\Omega\subset\mathbb{H}^{+}$ denote any bounded open subset and $C>0$.
Suppose that $\xi:\Omega\to\mathbb{C}$ is any smooth function such that
$|\xi(z)|<C$, and $H^{t}_{z},H^{s}_{z}:M\to\mathbb{R},\ z\in\Omega$ are
families of Hamiltonian functions on $M$ with

$\|\nabla H^{s}_{z}\|_{L^{\infty}(M)},\|\nabla H^{t}_{z}\|_{L^{\infty}(M)}<C$

for all $z\in\Omega$. Then any sequence of solutions $P_{n}:\Omega\to M$ to
the Floer equation

(12.6) $(\partial_{t}P_{n}-J\nabla
H_{z}^{t})+J(\partial_{s}P_{n}-\nabla\operatorname{Re}(\overline{\xi(z)}W)-J\nabla
H_{z}^{s})=0$

subject to the boundary condition
$P(\Omega\cap\mathbb{R}_{t}\times\\{0\\})\subset X$ and the energy bound

(12.7) $\mathcal{E}_{an}(P_{n};\Omega)=\int_{\Omega}|\nabla P_{n}|^{2}+|\nabla
H\circ P_{n}|^{2}<C,$

has a subsequence that converges in $C^{\infty}_{loc}$-topology on $\Omega$.

This lemma is deduced again using a diameter estimate for “almost”
$J$-holomorphic curves in an almost Hermitian manifold with boundary lying in
a totally real submanifold.

######  Proposition 12.3.

Suppose that $(M,J_{M},g_{M},\omega_{M})$ is an almost Hermitian manifold with
bounded geometry $($$\omega_{M}$ is not necessarily closed$)$, and $X\subset
M$ is a totally real submanifold with
$2\dim_{\mathbb{R}}X=\dim_{\mathbb{R}}M$. Denote by $\bar{\partial}_{J}$ the
operator $\partial_{t}+J\partial_{s}$ and by
$B^{+}(0,R)=B(0,R)\cap\mathbb{H}^{+},R>0$ the upper half disk. If $X$ also has
bounded geometry, then for any $r>2$, there exists a function
$\varphi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that for any $C>0$ and for any
map $P:B^{+}(0,1)\to M$ if

$\int_{B^{+}(0,1)}|\nabla P|^{2}+|\bar{\partial}_{J}P|^{r}\leq C\text{ and
}P(t,0)\in X\text{ for all }t\in(-1,1)_{t},$

then $\operatorname{diam}(P(B^{+}(0,\frac{1}{2})))\leq\varphi(C)$.

The proof of this proposition is by repeating the argument in Appendix A and
using the reflection principle; the complete argument is omitted in this
paper. With this diameter estimate in hand, one can deduce the compactness
theorem of (12.4), both the local and global versions, following the argument
in Section 3. Hence the Floer cohomology
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$ is
well-defined if the perturbation 1-form $\delta H$ is chosen generically, in
which case we say that $\mathfrak{a}_{X}$ is admissible. Finally, the grading
function $\operatorname{\mathfrak{gr}}$ introduced in Section 4.4 is
generalized rather trivially to this case, so
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$ is
canonically $\mathbb{Z}$-graded.

####  12.3. Continuation Maps

Given any pair of admissible Floer data

$\mathfrak{a}_{X}^{\pm}=(R^{\pm},\alpha_{X}^{\pm},\beta_{X},\epsilon_{X}^{\pm},\delta
H^{\pm}),$

the continuation method in Section 5.2 shows that
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X}^{\pm})$
are canonically isomorphic, but there is a caveat about the proof: the energy
inequality (5.18) is replaced in this case by

(12.8)
$\displaystyle\mathcal{A}_{X,\mathfrak{a}^{-}_{X}}(P(0,\cdot))-\mathcal{A}_{X,\mathfrak{a}^{+}_{X}}(P(K,\cdot))+\text{
some terms involving }\delta H$ $\displaystyle+$
$\displaystyle\int_{0}^{K}\operatorname{Im}(W(P(t,0))\cdot\overline{\partial_{t}g_{0}(t)})dt-\int_{0}^{K}\operatorname{Im}(W(q)\cdot\overline{\partial_{t}g_{1}(t)})dt>\epsilon_{\mathfrak{c}}^{\prime}\int_{[0,K]\times\mathbb{R}_{s}}u.$

Thus for the embedding
$\Xi^{\dagger}:\mathbb{R}_{t}\times[0,\pi]_{s}\to\mathbb{C}$ considered in
Section 5.2 (again we have assumed that $R^{\pm}=\pi$), we have to insist that
$\partial_{t}g_{0}(t)$ is always in the direction of $-ie^{i\beta_{X}}$, i.e.,

(12.9) $g_{0}(t)\colonequals\Xi^{\dagger}(t,0)=-ie^{i\beta_{X}}\cdot
h(t)+\gamma^{-}(0),$

for a real-valued function $h:\mathbb{R}_{t}\to\mathbb{R}$ such that
$\partial_{t}h(t)\equiv\epsilon_{X}^{-}$ for $t\leq 0$ and
$\equiv\epsilon_{X}^{+}$ for $t\gg 0$. Otherwise, we can not estimate the term

$\int_{0}^{K}\operatorname{Im}(W(P(t,0))\cdot\overline{\partial_{t}g_{0}(t)})dt$

on the left hand side of (12.8) using (12.1). It is for the same reason that
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$ is
defined only when the condition (12.2) holds. This limitation may be get rid
of using more sophisticated analytic tools in the future, but in that case the
invariance of this Floer cohomology can not be verified using the continuation
method. From now on we shall use
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta})$ to denote the
isomorphism class of
$\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})$ for
any admissible $\mathfrak{a}_{X}$.

####  12.4. Stable- and Unstable-type Lagrangian submanifolds

We say that a Lagrangian submanifold $X$ is of unstable-type (resp. stable-
type) if $\operatorname{Re}(e^{i\beta_{X}})>0$ (resp. $<0$), in which case we
shall always assume that $\beta_{X}\in(-\frac{\pi}{2},\frac{\pi}{2})$ (resp.
$\in(\frac{\pi}{2},\frac{3\pi}{2})$). Let $\mathcal{A}$ and $\mathcal{B}$
denote the Fukaya-Seidel category of $(M,W)$ and its dual category
respectively. If $X$ is of unstable-type, then we require further that
$\mathcal{A}=\mathcal{E}(\mathbf{Th}_{0})$ is defined using an admissible set
of thimbles $\mathbf{Th}_{0}=(S_{1},\cdots,S_{m})\in\Gamma_{0},\
S_{k}=\Lambda_{x_{k},\theta_{k}},1\leq k\leq m$ such that

(12.10)
$0<\theta_{m}<\cdots<\theta_{2}<\theta_{1}<\min\\{\theta_{\star},\beta_{X}+\frac{\pi}{2}\\}.$

Under this condition the Floer cohomology
$\operatorname{HF}_{\natural}^{*}(X,S_{k})$ is well-defined for all
$S_{k}\in\operatorname{Ob}\mathcal{A}$. Choose admissible Floer data
$\mathfrak{a}_{X,k}$, one for each pair $(X,S_{k})$. Then there is a finite
directed $A_{\infty}$-category $\mathcal{E}_{X}$ with

$\operatorname{Ob}\mathcal{E}_{X}:X\prec S_{1}\prec S_{2}\prec\cdots\prec
S_{m},$

containing $\mathcal{A}$ as a full $A_{\infty}$-subcategory and with

$\hom_{\mathcal{E}_{X}}(X,S_{k})=\operatorname{CF}^{*}_{\natural}(X,S_{k};\mathfrak{a}_{X,k}),$

which is the Floer complex associated to $(X,S_{k};\mathfrak{a}_{X,k})$ for
all $k$. The construction of $\mathcal{E}_{X}$ is identical to the one in
Section 8: we shall use $\mathcal{T}^{m+1}$ as the metric ribbon tree; if a
$(d+1)$-pointed disk $S$ is equipped with a compatible quadratic differential
$\phi$ and labeled by $(X,S_{k_{1}},\cdots,S_{k_{d}})$, then we remove the
lower half plane $\mathbb{H}^{-}_{0}$ from $\hat{S}$ and consider the partial
completion:

$\hat{S}_{X}\colonequals\hat{S}\setminus\sigma_{0}(\mathbb{H}_{0}^{-})=S\
\bigcup\ \mathbb{H}^{-}_{1}\cup\cdots\cup\mathbb{H}^{-}_{d}.$

To specify the phase pair on $\hat{S}_{X}$, pretend that $X$ is a thimble
$\Lambda_{q,\theta_{0}}$ with $\theta_{0}=\pi$ and use
$\Lambda_{q,\theta_{0}}\sqcup\mathbf{Th}_{0}$ as the set of labels to
determine a phase pair $(\Xi,\delta\kappa)$ on $\hat{S}$ (as in Section 8.2);
then restricts this pair to the partial completion
$\hat{S}_{X}\subset\hat{S}$. As we already noted, in order to make the energy
method work when replacing the planar end $\mathbb{H}^{-}_{0}$ in $\hat{S}$ by
a Lagrangian boundary condition along $\partial\hat{S}_{X}$, we have to
require that

(12.11)
$g_{0}\colonequals\Xi^{\dagger}\circ\sigma_{0}(\cdot,0):\mathbb{R}_{t}\to\mathbb{C}$

be a straight line in the direction of $-ie^{i\beta_{X}}$; compare (12.9).
Figure 43 below illustrates such a phase function for a cobordism map

$\operatorname{HF}_{\natural}^{*}(\Lambda_{q_{0},\theta_{0}},\Lambda_{q_{1},\theta_{1}})\otimes\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q_{0},\theta_{0}})\to\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q_{1},\theta_{1}})$

with $\theta_{1}<\theta_{0}<\theta_{1}+2\pi$. With this convention understood,
one can prove the energy estimate as usual. Finally, suppose that this
construction is carried out for another
$\mathbf{Th}_{0}^{\prime}\in\Gamma_{0}$ which satisfies (12.10) and
$\mathbf{Th}_{0}^{\prime}\xrightarrow{s}\mathbf{Th}_{0}^{\prime}$, then the
$A_{\infty}$-category defined using $X\sqcup\mathbf{Th}_{0}^{\prime}$, which
we denote by $\mathcal{E}_{X}^{\prime}$, is quasi-isomorphic to
$\mathcal{E}_{X}$. This is seen by building up a larger category
$\tilde{\mathcal{E}}_{X}$ with

$\operatorname{Ob}\tilde{\mathcal{E}}_{X}:X\prec S_{1}\prec
S_{2}\prec\cdots\prec S_{m}\prec S_{1}^{\prime}\prec
S_{2}^{\prime}\prec\cdots\prec S_{m}^{\prime}.$

and by repeating the proof of Proposition 8.24. Thus the left
$\mathcal{A}$-module ${}_{\mathcal{A}}X$ associated to $\mathcal{E}_{X}$,
which assigns to each $S_{k}\in\operatorname{Ob}\mathcal{A}$ the Floer complex
$\hom_{\mathcal{E}_{X}}(X,S_{k})$, is well-defined up to canonical quasi-
isomorphism. The following property about quasi-units is used in this proof.

\begin{overpic}[scale={.15}]{Pic52.png}
\put(-5.0,22.0){\small$-e^{i\theta_{1}}$}
\put(2.0,-2.0){\small$-e^{i\theta_{0}}$}
\put(101.0,10.0){\small$-ie^{i\beta_{X}}$} \end{overpic} Figure 43. The phase
function on a pair-of-pants cobordism map. The green curve is $g_{0}(t)$,
while red curves are the images of $g_{1}(t)$ and $g_{2}(t)$. The blue region
corresponds to the incoming end, while the red ones correspond to outgoing
ends.

######  Proposition 12.4.

Let $(\Lambda_{q,\theta_{0}},\Lambda_{q,\theta_{1}})$ be any thimbles with
$\theta_{1}<\theta_{0}<\theta_{1}+2\pi$ and such that (12.2) holds with
$\theta=\theta_{0},\theta_{1}$. Suppose in addition that the triangle bounded
by $l_{q,\theta_{0}},\ l_{q,\theta_{1}}$ and the equation
$\operatorname{Re}\langle w,e^{i\beta_{X}}\rangle\leq C_{X}$ contains no
critical values of $W$ other than the vertex $W(q)$, then the multiplication
map

$m_{*}(e_{q},\cdot):\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta_{0}})\to\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta_{1}})$

is an isomorphism.

The proof of this result follows the same line of arguments in Section 11.6
(using the vertical gluing theorem) and is omitted here.

In the same vein, for any stable-type Lagrangian submanifold
$Y=(Y,h_{Y},\xi_{Y}^{\\#},\beta_{Y})$, one may construct a finite directed
$A_{\infty}$-category $\mathcal{E}_{Y}$ with

$\operatorname{Ob}\mathcal{E}_{Y}:Y\prec U_{m}\prec U_{m-1}\cdots\prec U_{1}.$

if the admissible set of thimbles $\mathbf{Th}_{\pi}\in\Gamma_{\pi}$ satisfies
the relation

(12.12)
$\pi<\eta_{1}<\eta_{2}<\cdots<\eta_{m}<\min\\{\pi+\theta_{\star},\beta_{Y}+\frac{\pi}{2}\\},\beta_{Y}\in(\frac{\pi}{2},\frac{3\pi}{2}),$

and the left $\mathcal{B}$-module ${}_{\mathcal{B}}Y$ associated to
$\mathcal{E}_{Y}$ is well-defined up to canonical quasi-isomorphism.

####  12.5. Lower half planes

As an alternative route, one may switch the role of $X$ and
$\Lambda_{q,\theta}$ and define the Floer cohomology
$\operatorname{HF}_{\natural}^{*}(\Lambda_{q,\theta},X)$ by counting
instantons on the lower upper plane $\mathbb{H}^{-}$. As the theory is
identical to the previous one, we only explain the basic setup. In this case,
a Floer datum $\mathfrak{a}^{\prime}_{X}$ is a quintuple
$(R,\alpha_{X}^{\prime}(s),\beta_{X}-\pi,\epsilon_{X},\delta H^{\prime})$ as
usual, but the function $\alpha_{X}^{\prime}$ is now defined on
$(-\infty,R]_{s}$ such that for some $0<\delta\ll 1$,

$\alpha_{X}(s)\equiv\theta-\pi\text{ for all }s\leq\delta,$

and $\beta_{X}$ is replaced by $\beta_{X}-\pi$. The reference path
$p_{\operatorname{model}}:(-\infty,R]_{s}\to M$ satisfies that
$p_{\operatorname{model}}(s)\equiv q$ when $s\leq 0$. Consider the path space

$\mathcal{P}_{k}(\Lambda_{q,\theta},X)\colonequals\\{p\in
C^{\infty}((-\infty,R]_{s};M):p(R)\in X,\ p\text{ has finite
$L^{2}_{k}$-distance with }p_{\operatorname{model}}\\},k\geq 1.$

Then any a Floer datum $\mathfrak{a}^{\prime}_{X}$ defines an action
functional on $\mathcal{P}_{k}(\Lambda_{q,\theta},X)$ by the formula:

(12.13) $\displaystyle\mathcal{A}_{X,\mathfrak{a}^{\prime}_{X}}(p)$
$\displaystyle\colonequals
h_{X}(p(R))+\int_{-\infty}^{R}-p^{*}\lambda_{M}+\delta H_{s}(p(s))ds$
$\displaystyle+\int_{-\infty}^{R}\big{(}\operatorname{Im}(e^{-i\alpha_{X}(s)}(W\circ
p(s)))+\chi_{(-\infty,0]}\operatorname{Im}(e^{-i\theta}W(q))\big{)}ds.$

If $\mathfrak{a}^{\prime}_{X}$ is admissible, then the Floer cohomology
$\operatorname{HF}_{\natural}^{*}(\Lambda_{q,\theta},X;\mathfrak{a}^{\prime}_{X})$
is defined as the Morse cohomology of
$\mathcal{A}_{X,\mathfrak{a}^{\prime}_{X}}$. This construction is not new.
Given any such $\mathfrak{a}^{\prime}_{X}$, one can construct a Floer datum
$\mathfrak{a}_{X}=(R,\alpha_{X},\beta_{X},\epsilon_{X},\delta H)$ for the pair
$(X,\Lambda_{q,\theta})$ by taking

$\alpha_{X}=\tau^{*}\alpha_{X}^{\prime}+\pi,\delta H=\tau^{*}\delta
H^{\prime}$

with $\tau:\mathbb{R}^{+}_{s}\to(-\infty,R]_{s}$ defined by $\tau(s)=R-s$.
Thus
$\operatorname{HF}_{\natural}^{*}(\Lambda_{q,\theta},X;\mathfrak{a}^{\prime}_{X})$
is isomorphic to the Morse homology of $\mathcal{A}_{X,\mathfrak{a}_{X}}$ up
to a degree shifting

$\operatorname{HF}_{\natural}^{*}(\Lambda_{q,\theta},X;\mathfrak{a}_{X}^{\prime})\cong\operatorname{HF}^{\natural}_{\mathfrak{n}-*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X}),$

or equivalently,
$\operatorname{HF}_{\natural}^{*}(\Lambda_{q,\theta},X;\mathfrak{a}_{X}^{\prime})\cong
D\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta};\mathfrak{a}_{X})[-\mathfrak{n}]$,
where $DV$ stands for the dual of a finite graded vector space $V$ and
$\mathfrak{n}=\dim_{\mathbb{C}}(TM,J_{M})$. If $X$ is of unstable-type, then
one can construct a right $\mathcal{A}$-module $X_{\mathcal{A}}$ using
$\operatorname{CF}^{*}_{\natural}(S_{k},X;\mathfrak{a}_{k,X}^{\prime})$,
$1\leq k\leq m$, which is obtained from ${}_{\mathcal{A}}X$ by applying the
duality functor
$\mathcal{D}:\mathcal{P}_{l}^{\operatorname{opp}}\to\mathcal{P}_{r}$ in
(6.13).

###  13\. Seidel’s Spectral Sequence

####  13.1. Filtered bimodules

In the most ideal case, one would like to establish Seidel’s algebraic
spectral sequence for any pair of Lagrangian submanifolds $(X,Y)$, where
$X=(X,h_{X},\beta_{X},\xi_{X}^{\\#})$ is of unstable-type, and
$Y=(Y,h_{Y},\beta_{Y},\xi_{Y}^{\\#})$ of stable-type. In this section, we
shall assume something stronger to make our method work:

  1. (S1)

$X$ is of unstable-type, so $\beta_{X}\in(-\frac{\pi}{2},\frac{\pi}{2})$, and
$H|_{X}\geq C_{X}$;

  2. (S2)

$Y$ is of stable-type, so $\beta_{Y}\in(\frac{\pi}{2},\frac{3\pi}{2})$, and
$H|_{Y}\leq C_{Y}$.

See Figure 44 below. Choose admissible sets of thimbles
$\mathbf{Th}_{0}\in\Gamma_{0}$ and $\mathbf{Th}_{\pi}\in\Gamma_{\pi}$, which
satisfy the conditions (12.10) (12.12), and denote by
$\mathcal{A}=\mathcal{E}(\mathbf{Th}_{0})$ and
$\mathcal{B}=\mathcal{E}(\mathbf{Th}_{\pi})$ the Fukaya-Seidel category and
its dual category respectively. Under (S1) and (S2), for some $C^{\prime}>0$,
we have

(13.1) $\displaystyle\operatorname{Re}(e^{-\beta_{X}^{\prime}}W)|_{X}$
$\displaystyle\leq C^{\prime}\text{ for all
}\beta_{X}^{\prime}\in[-\frac{\pi}{2},\beta_{X}]$
$\displaystyle\operatorname{Re}(e^{-\beta_{Y}^{\prime}}W)|_{Y}$
$\displaystyle\leq C^{\prime}\text{ for all
}\beta_{Y}^{\prime}\in[\frac{\pi}{2},\beta_{Y}].$

\begin{overpic}[scale={.12}]{Pic64.png} \put(95.0,60.0){\small$S_{k}$}
\put(0.0,0.0){\small$U_{j}$} \put(20.0,45.0){\small$Y$}
\put(75.0,15.0){\small$X$} \end{overpic} Figure 44. The projection of
thimbles, $X$ and $Y$ onto $\mathbb{C}$ (with $\beta_{X}=0$ and
$\beta_{Y}=\pi$).

Thus one can also define a left $\mathcal{A}$-module ${}_{\mathcal{A}}Y$ for
$Y$ and a $\mathcal{B}$-module $X_{\mathcal{B}}$ for $X$. In fact, there is a
finite directed $A_{\infty}$-category $\mathcal{E}_{\Delta_{*}}$ with

$\operatorname{Ob}\mathcal{E}_{\Delta_{*}}:\underbrace{Y\prec U_{m}\prec\cdots
U_{2}\prec U_{1}}_{\operatorname{Ob}\mathcal{E}_{Y}}\prec\underbrace{X\prec
S_{1}\prec S_{2}\prec\cdots\prec S_{m}}_{\operatorname{Ob}\mathcal{E}_{X}},\
S_{k}\in\operatorname{Ob}\mathcal{A},U_{j}\in\operatorname{Ob}\mathcal{B},$

and which contains $\mathcal{E}_{X},\mathcal{E}_{Y}$ and
$\mathcal{E}_{\Delta}$ as full $A_{\infty}$-subcategories. This
$A_{\infty}$-category $\mathcal{E}_{\Delta_{*}}$ depends on an extra
stretching parameter $R\gg\pi$ (which is omitted from our notations) and is
defined using the metric ribbon tree $\mathcal{T}^{m+1,m+1}_{R}$. For
$R\gg\pi$, the induced $(\mathcal{E}_{X},\mathcal{E}_{Y})$-bimodule
$\Delta_{*}=\Delta_{*,R}$ carries an increasing filtration:

$0=\Delta^{(0)}_{*}\subset\Delta^{(1)}_{*}\subset\cdots\subset\Delta^{(m)}_{*}=\Delta_{*},$

which is induced by the action functionals and agrees with the increasing
filtration (9.2) on the bimodule $\Delta$ when restricted to
$\mathcal{A},\mathcal{B}$.

We elaborate on the construction of $\Delta_{*}$. Suppose that the functions
$\alpha_{j}^{un},\alpha_{k}^{st}$ have been chosen as in (9.7); take
$\frac{\pi}{2}<\beta_{*}<\frac{\pi}{2}+\min\\{\theta_{m},\eta_{1}-\pi\\}$ and
$\epsilon_{*}>0$ as in (9.10). Combined with (12.10)(12.12), this implies that

$0<\beta_{*}-\frac{\pi}{2}<\theta_{m}<\cdots<\theta_{1}<\beta_{X}+\frac{\pi}{2}<\pi<\beta_{*}+\frac{\pi}{2}<\eta_{1}<\cdots<\eta_{m}<\beta_{Y}+\frac{\pi}{2}<2\pi.$

with $S_{k}=\Lambda_{x_{k},\theta_{k}}$ and $U_{j}=\Lambda_{x_{j},\eta_{j}}$.
We require further that

(13.2)
$0<\epsilon_{*}<\min\\{\operatorname{Re}(e^{i\beta_{X}}),-\operatorname{Re}(e^{i\beta_{Y}})\\}.$

\begin{overpic}[scale={.15}]{Pic63.png} \put(10.0,5.0){$Y$}
\put(45.0,5.0){$U_{j_{2}}$} \put(85.0,5.0){$U_{j_{1}}$} \put(85.0,40.0){$X$}
\put(5.0,40.0){$S_{k_{3}}$} \put(35.0,40.0){$S_{k_{2}}$}
\put(60.0,40.0){$S_{k_{1}}$} \end{overpic} Figure 45. The diagonal bimodule
$\Delta_{*}.$ The green region is stretched vertically.

As a label, we think of $X$ as $\Lambda_{q,\pi}$ and $Y$ as $\Lambda_{q,2\pi}$
for any choice of $q\in\operatorname{Crit}(W)$. This helps us determine Floer
data as well as cobordism data on pointed disks. Now for any $R\geq\pi$,
consider the functions

$\alpha^{R}_{Y,k}(s)=\alpha_{k}^{st}(s-R),\ s\in[0,+\infty)_{s}\text{ and
}\alpha^{R}_{j,X}(s)=\alpha_{j}^{un}(s),\ s\in(-\infty,R+\pi]_{s}.$

Then for any $s\in[\pi,R]_{s}$, $\alpha^{R}_{Y,k}(s)=\alpha^{R}_{j,X}(s)=\pi$.
The Floer data are then defined as

$\displaystyle\mathfrak{a}_{Y,k}^{R}$
$\displaystyle=(R+\pi,\alpha^{R}_{Y,k}(s),\beta_{*},\epsilon_{*},\delta
H^{R}_{Y,k})$ $\displaystyle\text{ for }(Y,S_{k}),1\leq k\leq m,$
$\displaystyle\mathfrak{a}_{j,X}^{R}$
$\displaystyle=(R+\pi,\alpha^{R}_{j,X}(s),\beta_{*},\epsilon_{*},\delta
H^{R}_{j,X})$ $\displaystyle\text{ for }(U_{j},X),1\leq j\leq m,$
$\displaystyle\mathfrak{a}^{R}_{Y,X}$
$\displaystyle=(R+\pi,\alpha^{R}_{Y,X}(s),\beta_{*},\epsilon_{*},\delta
H_{Y,X})$ $\displaystyle\text{ for }(Y,X)$

with the usual assumptions on perturbation 1-forms; cf. Section 9.2. For
simplicity we shall take the function $\alpha^{R}_{Y,X}(s)\equiv\pi$ to be
constant.

Let $S$ be any $(d+1)$-pointed disk labeled by a subset
$A\subset\operatorname{Ob}\mathcal{E}_{\Delta_{*}}$ with $|A|=d+1\geq 3$, and
let $\phi$ be any compatible quadratic differential which is $\epsilon$-close
to $\mathcal{T}^{d_{1},d_{2}}_{R}$ with
$d_{1}=|A\cap\operatorname{Ob}\mathcal{E}_{Y}|$ and
$d_{2}=|A\cap\operatorname{Ob}\mathcal{E}_{X}|$. If $Y\in A$, then $Y$ is
attached the first component $C_{0}\subset\partial S$. Similar to (12.11), the
curve
$g_{0}(t)\colonequals\Xi^{\dagger}\circ\sigma_{0}(t,0):\mathbb{R}_{t}\to\mathbb{C}$
must satisfy some additional properties:

  * •

If $d_{1}=1$, then $A\setminus\\{Y\\}\subset\operatorname{Ob}\mathcal{E}_{X}$,
and $\partial_{t}g_{0}(t)\equiv-i\epsilon_{*}e^{i\beta_{*}}$;

  * •

If $d_{2}=0$, then $A\subset\operatorname{Ob}\mathcal{E}_{Y}$, and
$\partial_{t}g_{0}(t)\equiv-i\epsilon_{*}e^{i\beta_{X}}$;

  * •

If $d_{1}\geq 2$ and $d_{2}\geq 1$, then $\partial_{t}g_{0}(t)$ is in the
direction of $-ie^{i\beta_{*}}$ for $t\leq 0$ and $-ie^{i\beta_{Y}}$ for $t\gg
0$. In general, this direction lies in the interval
$[\beta_{*}-\frac{\pi}{2},\beta_{Y}-\frac{\pi}{2}]$ for all
$t\in\mathbb{R}_{t}$. The condition (S2) ensures that
$\operatorname{Re}(e^{-i\beta}W)|_{Y}$ is bounded above for any
$\beta\in[\beta_{*},\beta_{Y}]\subset[\frac{\pi}{2},\beta_{Y}]$.

If $X\in A$, then $X$ is attached to the $d_{1}$-th component
$C_{d_{1}}\subset\partial S$, and for the curve
$g_{d_{1}}(t)\equiv\Xi^{\dagger}\circ\sigma_{d_{1}}(t,0):\mathbb{R}_{t}\to\mathbb{C}$,
we require that

  * •

If $d_{2}=1$, then $A\setminus\\{X\\}\subset\operatorname{Ob}\mathcal{E}_{Y}$,
and $\partial_{t}g_{d_{1}}(t)\equiv i\epsilon_{*}e^{i\beta_{*}}$;

  * •

If $d_{1}=0$, then $A\subset\operatorname{Ob}\mathcal{E}_{X}$, and
$\partial_{t}g_{d_{1}}(t)\equiv-i\epsilon_{*}e^{i\beta_{X}}$;

  * •

If $d_{2}\geq 2$ and $d_{1}\geq 1$, then $\partial_{t}g_{d_{1}}(t)$ is in the
direction of $i\epsilon_{*}e^{i\beta_{*}}$ for $t\ll 0$ and
$-i\epsilon_{*}e^{i\beta_{X}}$ for $t\gg 0$. In general, this direction lies
in the interval $[\beta_{*}-\frac{3\pi}{2},\beta_{X}-\frac{\pi}{2}]$. The
condition (S1) ensures that $\operatorname{Re}(e^{-i\beta}W)|_{X}$ is bounded
above for any
$\beta\in[\beta_{*}-\pi,\beta_{X}]\subset[-\frac{\pi}{2},\beta_{X}]$.

With these conventions understood, one can repeat the argument in Section 9 to
construct the filtered bimodule $\Delta_{*}=\Delta_{*,R}$. See Figure 46 below
for a typical phase function on the partial completion of a $(6+1)$ pointed
disk (compare Figure 45). The green region is the image of the strip
$Z_{R}=\mathbb{R}_{t}\times[2\pi,R-\pi]_{s}$, which is stretched as
$R\to\infty$. The blue (resp. pink) regions are associated to the incoming end
and the $d_{1}$-th outgoing end (resp. other outing ends). Each gray curve is
the translated copy of some characteristic curves. The two red curves are the
images of $g_{0}(t)$ and $g_{d_{1}}(t)$ respectively.

\begin{overpic}[scale={.15}]{Pic65.png} \put(10.0,50.0){$Y$}
\put(38.0,60.0){$U_{j_{2}}$} \put(60.0,60.0){$U_{j_{1}}$} \put(90.0,10.0){$X$}
\put(20.0,-2.0){$S_{k_{3}}$} \put(43.0,-2.0){$S_{k_{2}}$}
\put(60.0,-2.0){$S_{k_{1}}$} \put(18.0,40.0){$g_{0}(t)$}
\put(75.0,25.0){$g_{d_{1}}(t)$} \put(17.0,58.0){$-ie^{i\beta_{Y}}$}
\put(72.0,5.0){$-ie^{i\beta_{X}}$} \put(0.0,30.0){-$ie^{i\beta_{*}}$}
\put(95.0,30.0){$ie^{i\beta_{*}}$} \put(101.0,52.0){$-ie^{i\beta_{*}}$}
\put(101.0,40.0){$-ie^{i\beta_{*}}$} \end{overpic} Figure 46. A typical phase
function on a $(6+1)$-pointed disk with $d_{1}=3$ and $d_{2}=4$ ($\beta_{X}=0$
and $\beta_{Y}=\pi$).

######  Remark 13.1.

Readers may wonder how these properties about $g_{0}$ and $g_{d_{1}}$ are
certified when we construct the family of phase functions on
$\mathcal{S}^{|A|}\to\mathcal{R}^{|A|}$ using Lemma 9.14, or Lemma 8.13. The
problem may occur in the proof of Lemma 8.7 as we “push” these curves
outwards. Since one may always increase the constant $K$ to construct the
null-homotopy, these properties can be preserved if this push-out is designed
carefully enough (the proof outlined in Step 2 of Lemma 5.6 is just one
possible way for this push-out). We leave the details to interested readers. ∎

####  13.2. Statements

Now we are already to state the algebraic formula to rebuild the Floer
cohomology
$\operatorname{HF}^{*}_{\natural}(X,Y)=H(D\Delta_{*}(Y,X)[-\mathfrak{n}])$ in
terms of the $A_{\infty}$-modules ${}_{\mathcal{A}}X$, $Y_{\mathcal{B}}$ and
the bimodule ${}_{\mathcal{A}}\Delta_{\mathcal{B}}$. Applying the cyclic
operation $\mathcal{C}$ defined in (6.15), one can transform
$\mathcal{E}_{\Delta_{*}}$ into another category with

$\operatorname{Ob}\mathcal{C}\mathcal{E}_{\Delta_{*}}:U_{m}\prec\cdots\prec
U_{2}\prec U_{1}\prec X\prec S_{1}\prec S_{2}\prec\cdots\prec S_{m}\prec Y.$

This defines an $A_{\infty}$-functor into
$\mathcal{Q}_{r}\colonequals\operatorname{rfmod}(\mathcal{B})$:

(13.3)
$r_{X,Y}:\mathcal{C}\mathcal{E}_{\Delta_{*}}\xrightarrow{r_{\mathcal{C}\mathcal{E}_{\Delta_{*}}}}\operatorname{rfmod}(\mathcal{C}\mathcal{E}_{\Delta_{*}})\to\mathcal{Q}_{r}.$

such that $r_{X,Y}|_{\mathcal{A}}$ is the functor $r_{\Delta}$ in Theorem 8.3.
While $r_{X,Y}$ is not cohomologically full and faithful in general, one can
still prove the following result.

######  Theorem 13.2.

For any $V\prec
V^{\prime}\in\operatorname{Ob}\mathcal{C}\mathcal{E}_{\Delta_{*}}$, the first
order map of $r_{X,Y}$

$r^{1}_{X,Y}:\mathcal{C}\mathcal{E}_{\Delta_{*}}(V,V^{\prime})\to\hom_{\mathcal{Q}_{r}}(r_{X,Y}(V),r_{X,Y}(V^{\prime}))$

is a quasi-isomorphism.

If $V\in\operatorname{Ob}\mathcal{B}$, then this statement follows from the
Yoneda embedding theorem. If $V,V^{\prime}\in\operatorname{Ob}\mathcal{A}$,
then this follows from Theorem 8.3. Let
$X_{\mathcal{B}},Y_{\mathcal{B}},\mathcal{S}_{k}\in\mathcal{Q}_{r}$ denote the
images of $X,Y,S_{k}$ under $r_{X,Y}$ respectively. Thus the non-trivial part
of Theorem 13.2 is when the map $r^{1}_{X,Y}$ is between

(13.4)
$\displaystyle\mathcal{C}\mathcal{E}_{\Delta_{*}}(X,Y)=\mathcal{D}\Delta_{*}(X,Y)\colonequals
D(\Delta_{*}(X,Y))[-\mathfrak{n}]$
$\displaystyle\to\mathcal{C}\mathcal{E}_{\Delta_{*}}(S_{k},Y)=\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}},Y_{\mathcal{B}}),$
(13.5) $\displaystyle\mathcal{D}\Delta_{*}(S_{k},Y)\colonequals
D(\Delta_{*}(S_{k},Y))[-\mathfrak{n}]$
$\displaystyle\to\hom_{\mathcal{Q}_{r}}(\mathcal{S}_{k},Y_{\mathcal{B}}),$
(13.6) $\displaystyle\Delta_{*}(X,S_{k})$
$\displaystyle\to\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}},\mathcal{S}_{k}),$

Note that for $R\gg\pi$, the right $\mathcal{B}$-module $X_{\mathcal{B}}$
inherits an increasing filtration from $\Delta_{*}$:

$0=X_{\mathcal{B}}^{(0)}\subset\cdots X_{\mathcal{B}}^{(m)}=X_{\mathcal{B}},$

while the dual complex $\mathcal{D}\Delta_{*}(X,Y)$ inherits a decreasing
filtration

(13.7)
$0=(\mathcal{D}\Delta_{*})^{(m+1)}(X,Y)\subset\cdots\subset(\mathcal{D}\Delta_{*})^{(1)}(X,Y)=(\mathcal{D}\Delta_{*})(X,Y)$

with

$(\mathcal{D}\Delta_{*})^{(n)}(X,Y)\colonequals
D(\Delta_{*}(Y,X)/\Delta_{*}^{(n-1)}(Y,X))[-\mathfrak{n}],\ 1\leq n\leq m+1.$

While $X_{\mathcal{B}}^{(n)}$ involves the first $n$-critical points
$x_{1},\cdots,x_{n}$ of $W$, $(\mathcal{D}\Delta_{*})^{(n)}(X,Y)$ involves
$x_{n},\cdots,x_{m}$. Theorem 13.2 can be refined as follows.

######  Theorem 13.3.

Since the geometric filtration is preserved under the bimodule action of
$\Delta_{*}$, $r^{1}_{X,Y}$ restricts to a chain map for all $0\leq n\leq m$:

$(\mathcal{D}\Delta_{*})^{(n)}(X,Y)\to\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}}/X_{\mathcal{B}}^{(n-1)},Y_{\mathcal{B}}).$

This map is a quasi-isomorphism. The same holds if $X$ is replaced by any
stable thimble $S_{k}\in\operatorname{Ob}\mathcal{A}$.

The setup we have developed so far is symmetric between $\mathcal{A}$ and
$\mathcal{B}$, so one may work instead with the
$(\mathcal{E}_{Y},\mathcal{E}_{X})$-bimodule $\mathcal{D}\Delta_{*}$, which is
the dual of $\Delta_{*}$; see for instance Lemma 6.11. $\mathcal{D}\Delta_{*}$
is defined by the relation
$\mathcal{E}_{\mathcal{D}\Delta^{*}}\colonequals\mathcal{C}^{m+1}\mathcal{E}_{\Delta^{*}}$
and is filtered as in (13.7) where $X$ is replaced by any $S_{k}$ and $Y$ by
$U_{j}$ in general. Thus
$\mathcal{C}\mathcal{E}_{\mathcal{D}\Delta_{*}}=\mathcal{C}^{m+2}\mathcal{E}_{\Delta_{*}}$
has objects ordered by

$\operatorname{Ob}\mathcal{C}\mathcal{E}_{\mathcal{D}\Delta_{*}}:S_{1}\prec
S_{2}\prec\cdots\prec S_{m}\prec Y\prec U_{m}\prec\cdots\prec U_{2}\prec
U_{1}\prec X.$

Let
$r_{Y,X}:\mathcal{C}\mathcal{E}_{\mathcal{D}\Delta_{*}}\to\mathcal{P}_{r}\colonequals\operatorname{rfmod}(\mathcal{A})$
denote the induced functor. At this point, there are four $A_{\infty}$-modules
associated to $X$ (and resp. to $Y$)

$\displaystyle{}_{\mathcal{A}}X=\mathcal{D}X_{\mathcal{A}}$
$\displaystyle\text{ and }{}_{\mathcal{B}}X=\mathcal{D}(X_{\mathcal{B}})\text{
(with filtrations)},$
$\displaystyle{}_{\mathcal{B}}Y=\mathcal{D}Y_{\mathcal{B}}$
$\displaystyle\text{ and }{}_{\mathcal{A}}Y=\mathcal{D}(Y_{\mathcal{A}})\text{
(with filtrations)},$

where $\mathcal{D}$ denotes the duality functor defined by (6.13). Then
Theorem 13.2 and 13.3 implies the following.

######  Corollary 13.4.

By (13.6), the $A_{\infty}$-homomorphism

$t^{l}_{X}:{}_{\mathcal{A}}X\to\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}},{}_{\mathcal{A}}\Delta_{\mathcal{B}})$

induced by $\Delta_{*}$ is a quasi-isomorphism in
$\mathcal{P}_{l}\colonequals\operatorname{lfmod}(\mathcal{A})$. Indeed, its
first order map is given precisely (13.6). Similarly, Theorem 13.3 (with $X$
replaced by $S_{k}$) implies that for any $0\leq n\leq m$,

$t^{r}_{Y}:Y_{\mathcal{A}}^{(n)}\to\hom_{\mathcal{Q}_{r}}({}_{\mathcal{A}}\Delta_{\mathcal{B}}/{}_{\mathcal{A}}\Delta^{(n-1)}_{\mathcal{B}},Y_{\mathcal{B}})\xrightarrow[\cong]{\text{
duality
}}\hom_{\mathcal{Q}_{l}}({}_{\mathcal{B}}Y,{}_{\mathcal{B}}(\mathcal{D}\Delta)_{\mathcal{A}}^{(n)})$

is a quasi-isomorphism in
$\mathcal{P}_{r}\colonequals\operatorname{rfmod}(\mathcal{A})$.

######  Corollary 13.5.

Switching the roles of $\Delta$ and $\mathcal{D}\Delta$ in Corollary 13.4, we
obtain that

$t_{X}^{r}:X_{\mathcal{B}}^{(n)}\to\hom_{\mathcal{P}_{l}}({}_{\mathcal{A}}X,{}_{\mathcal{A}}\Delta_{\mathcal{B}}^{(n)}),$

is a quasi-isomorphism between finite right $\mathcal{B}$-modules for all
$0\leq n\leq m$.

######  Corollary 13.6.

By Lemma 6.9, $t_{X}^{r}$ admits a quasi-inverse $(t_{X}^{r})^{-1}$ in
$\mathcal{Q}_{r}$. By composing $r^{1}_{X,Y}$ with
$\mu^{2}_{\mathcal{Q}_{r}}(\cdot,(t_{X}^{r})^{-1})$, we obtain a quasi-
isomorphism:

$\mathcal{D}\Delta_{*}(X,Y)\xrightarrow{r^{1}_{X,Y}}\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}},Y_{\mathcal{B}})\xrightarrow{\mu^{2}_{\mathcal{Q}_{r}}(\cdot,(t_{X}^{r})^{-1})}\hom_{\mathcal{Q}_{r}}\big{(}\hom_{\mathcal{P}_{l}}({}_{\mathcal{A}}X,{}_{\mathcal{A}}\Delta_{\mathcal{B}}),Y_{\mathcal{B}}\big{)}.$

Switching the roles of $X$ and $Y$, we also have

$\Delta_{*}(Y,X)\xrightarrow{r^{1}_{Y,X}}\hom_{\mathcal{P}_{r}}(Y_{\mathcal{A}},X_{\mathcal{A}})\xrightarrow{\mu^{2}_{\mathcal{P}_{r}}(\cdot,(t_{Y}^{r})^{-1})}\hom_{\mathcal{P}_{r}}\big{(}\hom_{\mathcal{Q}_{l}}({}_{\mathcal{B}}Y,{}_{\mathcal{B}}(\mathcal{D}\Delta)_{\mathcal{A}}),X_{\mathcal{A}}\big{)}.$

####  13.3. Proof of Theorem 13.2 and 13.3: Step 1

Theorem 13.3 is proved by a few quick reductions. First, since the chain map
$r^{1}_{X,Y}$ preserves the filtrations, it induces a map between two spectral
sequences. To show that $[r^{1}_{X,Y}]$ is an isomorphism between
$E_{\infty}$-pages, one can work instead with the $E_{1}$-page. To put it
another way, for any $0\leq n\leq m-1$, we have a map between exact triangles,
which goes vertically in the diagram below,

(13.8)
${H\big{(}(\mathcal{D}\Delta_{*})^{(n+1)}(X,Y)\big{)}}$${H\big{(}(\mathcal{D}\Delta_{*})^{(n)}(X,Y)\big{)}}$${H\big{(}\mathcal{G}^{n}(\mathcal{D}\Delta_{*})(X,Y)\big{)}}$${H\big{(}\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}}/X_{\mathcal{B}}^{(n)},Y_{\mathcal{B}})\big{)}}$${H\big{(}\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}}/X_{\mathcal{B}}^{(n-1)},Y_{\mathcal{B}})\big{)}}$${H\big{(}\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}X_{\mathcal{B}},Y_{\mathcal{B}})\big{)}}$

where $\mathcal{G}^{n}$ denotes the associated graded complexes/modules. Thus
it suffices to verify the quasi-isomorphism for each associated graded
component:

(13.9)
$r^{1}_{n}:\mathcal{G}^{n}(\mathcal{D}\Delta_{*})(X,Y)\to\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}X_{\mathcal{B}},Y_{\mathcal{B}}),\
1\leq n\leq m.$

####  13.4. Proof of Theorem 13.2 and 13.3: Step 2

For the next reduction, we apply Lemma 6.1 to the $A_{\infty}$-functor
$r_{X,Y}:\mathcal{C}\mathcal{E}_{\Delta_{*}}\to\mathcal{Q}_{r}$ and to the
triple
$(X,S_{n},Y)\subset\operatorname{Ob}\mathcal{C}\mathcal{E}_{\Delta_{*}}$. Then
(6.4) says that the following diagram is commutative up to chain homotopy
(take the $n$-th graded component $\mathcal{G}^{n}$):

(13.10)
${\mathcal{G}^{n}(\mathcal{D}\Delta_{*})(S_{n},Y)\otimes\hom_{\mathcal{E}_{X}}(X,S_{n})}$${\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}\mathcal{S}_{n},Y_{\mathcal{B}})\otimes\hom_{\mathcal{E}_{X}}(X,S_{n})}$${\mathcal{G}^{n}(\mathcal{D}\Delta_{*})(X,Y)}$${\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}X_{\mathcal{B}},Y_{\mathcal{B}}).}$

The next lemma implies that the vertical arrows in (13.10) are quasi-
isomorphisms.

######  Lemma 13.7.

The $A_{\infty}$-homomorphism

$\mathcal{G}^{n}X_{\mathcal{B}}\to\mathcal{G}^{n}\mathcal{S}_{n}\otimes
D\hom_{\mathcal{E}_{X}}(X,S_{n})$

induced by
$r_{X,Y}^{1}:\hom_{\mathcal{E}_{X}}(X,S_{n})\to\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}X_{\mathcal{B}},\mathcal{G}^{n}\mathcal{S}_{n})$
is a quasi-isomorphism meaning that for any
$U_{j}\in\operatorname{Ob}\mathcal{B}$, the cobordism map induced by a pair-
of-pants

$\mathcal{G}^{n}\Delta_{*}(U_{j},X)\to\mathcal{G}^{n}\Delta_{*}(U_{j},S_{n})\otimes
D\hom_{\mathcal{E}_{X}}(X,S_{n})$

is a quasi-isomorphism. The same holds if $U_{j}$ is replaced by $Y$.

######  Remark 13.8.

Lemma 13.7 is the analogue of Lemma 10.3 where $S_{n}$ is replaced by $X$ and
$S_{l}$ by $S_{n}$. ∎

Thus it suffices to show that the top row of (13.10) is a quasi-isomorphism,
so we may replace $X$ by $S_{n}$ in (13.9) and verify instead that

(13.11)
$r^{1}_{n}:\mathcal{G}^{n}(\mathcal{D}\Delta_{*})(S_{n},Y)\to\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}\mathcal{S}_{n},Y_{\mathcal{B}}).$

is a quasi-isomorphism. From now on the problem is not related to $X$ any
longer.

####  13.5. Proof of Theorem 13.2 and 13.3: Step 3

For the third reduction, let $V_{n}=\Lambda_{x_{n},\eta_{n}^{\prime}}$ denote
the thimble in the direction of $\eta_{n}^{\prime}\in(0,\eta_{1})$. As in Step
2 of the proof of Lemma 10.1, we consider an enlargement of
$\mathcal{E}^{\prime}$ of $\mathcal{E}_{\Delta_{*}}$ whose objects are ordered
as

$Y\prec U_{1}\prec U_{2}\prec\cdots\prec U_{m}\prec V_{n}\prec S_{n}.$

where we have omitted $X$ and other stable thimbles for simplicity. This
defines a filtered
$(\mathcal{E}_{X}^{\prime},\mathcal{E}_{Y}^{\prime})$-bimodule
$\Delta_{*}^{\prime}$, where
$\operatorname{Ob}\mathcal{E}_{X}^{\prime}=(S_{n})$ and
$\mathcal{E}_{Y}^{\prime}$ consists of all other objects of
$\mathcal{E}_{\Delta_{*}^{\prime}}$. Then the cyclic rotation
$\mathcal{C}\mathcal{E}^{\prime}$ of $\mathcal{E}^{\prime}$ has objects
ordered by

$U_{1}\prec U_{2}\prec\cdots\prec U_{m}\prec V_{n}\prec S_{n}\prec Y$

and defines an $A_{\infty}$-functor

$r^{\prime}:\mathcal{C}\mathcal{E}^{\prime}\xrightarrow{r_{\mathcal{C}\mathcal{E}^{\prime}}}\operatorname{rfmod}(\mathcal{C}\mathcal{E}^{\prime})\xrightarrow{\text{restriction}}\mathcal{Q}_{r}=\operatorname{rfmod}(\mathcal{B}).$

Apply Lemma 6.1 to $r^{\prime}$ and the triple $(V_{n},S_{n},Y)$; then (6.5)
says that the following diagram is commutative up to a chain-homotopy (take
the $n$-th graded component $\mathcal{G}^{n}$):

(13.12)
${\mathcal{G}^{n}\hom_{\mathcal{C}\mathcal{E}^{\prime}}(S_{n},Y)}$${\hom_{\mathcal{Q}_{r}}(\mathcal{G}^{n}\mathcal{S}_{n},Y_{\mathcal{B}})}$${\hom_{\mathcal{C}\mathcal{E}^{\prime}}(V_{n},Y)\otimes
D\mathcal{G}^{n}\hom_{\mathcal{C}\mathcal{E}^{\prime}}(V_{n},S_{n})}$${\hom_{\mathcal{Q}_{r}}(\mathcal{V}_{n},Y_{\mathcal{B}})\otimes
D\mathcal{G}^{n}\hom_{\mathcal{C}\mathcal{E}^{\prime}}(V_{n},S_{n}).}$

The next lemma says that the vertical arrows in (13.12) are quasi-
isomorphisms.

######  Lemma 13.9.

The $A_{\infty}$-homomorphism

$\mathcal{G}^{n}\hom_{\mathcal{E}^{\prime}}(V_{n},S_{n})\otimes\mathcal{V}_{n}\to\mathcal{G}^{n}\mathcal{S}_{n}$

induced by
$(r^{\prime})^{1}:\mathcal{G}^{n}\hom_{\mathcal{E}^{\prime}}(V_{n},S_{n})\to\hom_{\mathcal{Q}_{r}}(\mathcal{V}_{n},\mathcal{G}^{n}\mathcal{S}_{n})$
is a quasi-isomorphism meaning that for any
$U_{j}\in\operatorname{Ob}\mathcal{B}$, the cobordism map induced by a pair-
of-pants

$\mathcal{G}^{n}\hom_{\mathcal{E}^{\prime}}(V_{n},S_{n})\otimes\hom_{\mathcal{E}^{\prime}}(U_{j},V_{n})\to\mathcal{G}^{n}\hom_{\mathcal{E}^{\prime}}(U_{j},S_{n})$

is a quasi-isomorphism. The same holds if $U_{j}$ is replaced by $Y$.

######  Remark 13.10.

The first part of Lemma 13.9 is exactly Lemma 10.4. Only the case when $U_{j}$
is replaced by $Y$ is new. ∎

####  13.6. Proof of Theorem 13.2 and 13.3: Step 4

Finally, it remains to verify the second row of (13.12)

$r^{\prime}:\hom_{\mathcal{C}\mathcal{E}^{\prime}}(V_{n},Y)\to\hom_{\mathcal{Q}_{r}}(\mathcal{V}_{n},Y_{\mathcal{B}})$

is a quasi-isomorphism. This follows from the fact that $\mathcal{V}_{n}$ is
quasi-isomorphic to the Yoneda image $\mathcal{U}_{n}$ of $U_{n}$ in
$\mathcal{Q}_{r}$. At this point, there is no energy filtration involved in
the chain map. Apply Lemma 6.1 again to the functor
$r^{\prime}:\mathcal{C}\mathcal{E}^{\prime}\to\mathcal{Q}_{r}$ but now to the
triple $(U_{n},V_{n},Y)$. Then (6.4) implies that the following diagram is
commutative up to a chain-homotopy:

(13.13)
${\hom_{\mathcal{E}^{\prime}}(V_{n},Y)\otimes\hom_{\mathcal{E}^{\prime}}(U_{n},V_{n})}$${\hom_{\mathcal{Q}_{r}}(\mathcal{V}_{n},Y_{\mathcal{B}})\otimes\hom_{\mathcal{E}^{\prime}}(U_{n},V_{n})}$${\hom_{\mathcal{E}^{\prime}}(U_{n},Y)}$${\hom_{\mathcal{Q}_{r}}(\mathcal{U}_{n},Y_{\mathcal{B}}).}$

Note that $\hom_{\mathcal{E}^{\prime}}(U_{n},V_{n})$ is 1-dimensional and is
generated by the quasi-unit $e_{x_{n}}$ (the constant soliton at $x_{n}$). By
Proposition 12.4 and Step 3 in the proof of Lemma 10.1, the vertical arrows of
(13.13) are quasi-isomorphisms. By the Yoneda embedding theorem, the second
row of (13.13) is a quasi-isomorphism, and so is the first row.

Lemma 13.7 and 13.9 are proved in the same way as Lemma 10.3 and Lemma 10.4
using the vertical gluing theorem (the proof is identical and hence omitted).
This completes the proof of Theorem 13.3. Note that (13.4) and (13.5) follow
from Theorem 13.3 by taking $n=0$.

Finally, (13.6) is proved by repeating the argument in Theorem 8.3 with
$S_{1}$ replaced by $X$ and Lemma 10.1 by Lemma 13.7. This completes the proof
of Theorem 13.2.

####  13.7. Identifying the geometric/algebraic filtrations

We would like to compare the geometric filtration on
$(\mathcal{D}\Delta_{*})(X,Y)$ with the algebraic filtration on
$\hom_{\mathcal{Q}_{r}}(X_{\mathcal{B}},Y_{\mathcal{B}})$ as defined in
(6.26). By Theorem 13.3 and the argument in Section 6.6, it suffices to show
that the ladder

(13.14)
${X_{\mathcal{B}}/X_{\mathcal{B}}^{(m)}}$${0}$${X_{\mathcal{B}}/X_{\mathcal{B}}^{(m-1)}}$${\mathcal{G}^{m}X_{\mathcal{B}}}$${\cdots}$${\cdots}$${X_{\mathcal{B}}/X_{\mathcal{B}}^{(2)}}$${\mathcal{G}^{3}X_{\mathcal{B}}}$${X_{\mathcal{B}}/X_{\mathcal{B}}^{(1)}}$${\mathcal{G}^{2}X_{\mathcal{B}}}$${X_{\mathcal{B}}=X_{\mathcal{B}}/X_{\mathcal{B}}^{(0)}}$${\mathcal{G}^{1}X_{\mathcal{B}}}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$

defines a Postnikov decomposition of $X_{\mathcal{B}}$; cf. (6.25)$(L)$.
Combining Lemma 13.7 and Lemma 13.9, we have
$\mathcal{G}^{n}X_{\mathcal{B}}\cong\mathcal{U}_{n}\otimes
D\hom_{\mathcal{S}_{X}}(X,S_{n})$. An inductive argument using the exact
triangles in (13.14) shows that
$X_{\mathcal{B}}/X_{\mathcal{B}}^{(n)}\in\mathcal{Q}_{n,r}$ and the property
(6.23) holds for every $\mathcal{N}\in\mathcal{Q}_{n,r},0\leq k\leq m$.
Although we have worked constantly with chain complexes in Section 6.6, this
spectral sequence relies essentially on the cohomological category
$H(\mathcal{Q}_{r})$, especially if one thinks in terms of exact couples
[GM03, P.263]. Thus we have proved that

######  Theorem 13.11.

The spectral sequence induced by the geometric filtration on
$\mathcal{D}\Delta_{*}(X,Y)$ is isomorphic to Seidel’s algebraic spectral
sequence defined in Theorem 6.12. This means that there is an isomorphism
between each page of the spectral sequences, starting from $E_{1}$, which
commutes with the differential maps.

####  13.8. Seidel’s long exact sequence: revisited

We end this section with some applications of the vertical gluing theorem.
Proposition 12.4 is concerned with the case when the thimble
$\Lambda_{q,\theta_{1}}$ can be continuously deformed into
$\Lambda_{q,\theta_{0}}$ without violating the condition (2.8). As in the case
of thimbles, if the triangle in Proposition 12.4 (compare Figure 42 with
Figure 47 below) contains a single critical point, then the multiplication map
$m_{*}(e_{q},\ \cdot\
):\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta_{0}})\to\operatorname{HF}_{\natural}^{*}(X,\Lambda_{q,\theta_{1}})$
might not be an isomorphism and will fit into a long exact sequence, as first
proved by Seidel [Sei03]. The next proposition generalizes Proposition 11.17
to the case of Lagrangian submanifolds. To see their resemblance, we now put
$X$ in the second entry.

######  Proposition 13.12.

Choose angles $\theta_{3}<\theta_{1}<\theta_{0}<\theta_{3}+\pi$ such that
$e^{i(\theta_{j}-\beta_{X})}>0$ for all $j=0,1,3$. Let
$\Lambda_{j}=\Lambda_{q,\theta_{j}},j=0,1$, $q\in\operatorname{Crit}(W)$, and
suppose that the closed triangle bounded by
$l_{q,\theta_{0}},l_{q,\theta_{1}}$ and the equation $\operatorname{Re}\langle
w,e^{i\beta_{X}}\rangle\leq C_{X}$ only contains the critical value $W(q)$ on
the boundary and a critical value $W(q_{3})$ in the interior. Let
$\Lambda_{3}=\Lambda_{q_{3},\theta_{3}}$. Then we have a long exact sequence :

${\operatorname{HF}_{\natural}^{*}(\Lambda_{3},X)\otimes\operatorname{HF}_{\natural}^{*}(\Lambda_{1},\Lambda_{3})}$${\operatorname{HF}_{\natural}^{*}(\Lambda_{1},X)}$${\operatorname{HF}_{\natural}(\Lambda_{0},X).}$$\scriptstyle{m_{*}}$$\scriptstyle{m_{*}(\cdot,e_{q})}$$\scriptstyle{[1]}$

The proof of Proposition 13.12 follows the same line of arguments as in
Proposition 11.17 with $\Lambda_{2}$ replaced by $X$ and hence is omitted
here.

\begin{overpic}[scale={.15}]{Pic62.png} \put(35.0,5.0){\small$X$}
\put(100.0,5.0){\small$X$} \put(60.0,35.0){\small$\operatorname{Re}\langle
w,e^{i\beta_{X}}\rangle=C_{X}$}
\put(-5.0,35.0){\small$\operatorname{Re}\langle
w,e^{i\beta_{X}}\rangle=C_{X}$} \put(95.0,38.0){\small$\Lambda_{0}$}
\put(92.0,43.0){\small$S_{\star}$} \put(58.0,0.0){\small$U_{\star}$}
\put(66.0,6.0){\small$q$} \put(1.0,6.0){\small$q$}
\put(14.0,20.0){\small$q_{3}$} \put(30.0,38.0){\small$\Lambda_{0}$}
\put(35.0,21.0){\small$\Lambda_{1}$} \put(45.0,9.0){\small$\Lambda_{3}$}
\end{overpic} Figure 47. Seidel’s long exact sequence (left) and the
isomorphism between of $\operatorname{HF}_{\natural}^{*}$ and
$\operatorname{HF}^{*}$ (right).

####  13.9. The isomorphism between $\operatorname{HF}_{\natural}^{*}$ and
$\operatorname{HF}^{*}$

Let $\Lambda_{0}=\Lambda_{q,\theta_{0}}$. Suppose that $W(X)\subset\mathbb{C}$
is bounded above in the direction of $e^{i\beta_{X}}$ and
$e^{i(\theta_{0}-\beta_{X})}>0$. Another application of the vertical gluing
theorem is to show that $\operatorname{HF}_{\natural}^{*}(\Lambda_{0},X)$ is
isomorphic to the $\operatorname{HF}^{*}(\underline{\Lambda}_{0},X)$, the
Floer cohomology defined using Lagrangian boundary condition (counting
solutions on strips). We used an underline here to emphasis that the
Lagrangian boundary condition along $\underline{\Lambda}_{0}$ is used here to
define this group. At this point, we have to assume that the thimble
$\Lambda_{0}$ has bounded geometry as a totally real submanifold of
$(M,J_{M},\omega_{M},g_{M})$. This is a technical point which we will not
address in this paper.

Suppose that we are in the scenario of Figure 47, and
$\theta_{-1}=\theta_{0}+\epsilon,0<\epsilon\ll 1$ is a slight perturbation of
$\theta_{0}$. Let $U_{\star}=\Lambda_{q,\theta_{-1}-\pi}$ and
$S_{\star}=\Lambda_{q,\theta_{-1}}$. Choose a function
$\alpha_{0}(s):\mathbb{R}_{s}\to\mathbb{R}$ as in (11.36) to define an
admissible Floer datum $\mathfrak{a}_{0}$ for the pair
$(\Lambda_{0},U_{\star})$. If $\mathfrak{a}_{X}$ is any admissible Floer datum
for $(S_{\star},X)$, then they can be concatenated to give a Floer datum
$\mathfrak{a}_{0,X}^{R}$ for $(\Lambda_{0},X)$ (cf. Section 11.1). There are
two ways to define the Floer cohomology of $(\Lambda_{0},X)$, either by
counting instantons on the lower half planes,
$\mathbb{R}_{t}\times(-\infty,R+\pi]$, which gives
$\operatorname{CF}^{*}_{\natural}(\Lambda_{0},X;\mathfrak{a}_{0,X}^{R})$, or
by restricting the Floer datum $\mathfrak{a}_{0,X}^{R}$ on
$[0,R+\pi]_{s}\times\mathbb{R}_{t}$ and counting instantons on strips, which
gives
$\operatorname{CF}^{*}(\underline{\Lambda}_{0},X;\mathfrak{a}_{0,X}^{R})$. For
any $R\gg\pi$, the Floer complex
$\operatorname{CF}^{*}(\Lambda_{0},X;\mathfrak{a}_{0,X}^{R})$ has only one
filtration level. The proof of Theorem 11.3 is adapted to this case to show
that