Homological mirror symmetry\\ for the quartic surface
Paul Seidel
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\begin{document}
\title[Homological mirror symmetry]{Homological mirror symmetry\\ for the quartic surface}
\author{Paul Seidel}
\date{This copy {\LaTeX}ed: \today}
\maketitle

\begin{quote}
{\bf Warning.} This preprint is in finished form, and should be
readable by itself. However, the argument uses certain general
properties of Fukaya categories, the proofs of which will appear in
\cite{seidel04} (under preparation). I apologize for this
unsatisfactory state of affairs.

\parskip0em
\tableofcontents
\end{quote}
%\input{todo}

\section{Introduction}

This paper deals with a special case of Kontsevich's ``homological
mirror symmetry'' conjecture \cite{kontsevich94}. Before formulating
the precise statement, it is necessary to introduce the relevant
coefficient rings.

\begin{notation}
Let $\Lambda_\N = \C[[q]]$ be the ring of formal power series in one
variable. We denote by $\Lambda_\Z$ its quotient field, obtained by
formally inverting $q$. By adjoining roots $q^{1/d}$ of all orders to
$\Lambda_\Z$, one obtains its algebraic closure $\Lambda_\Q$, known
in symplectic geometry as the rational Novikov field. Elements of
this field are formal series
\[
 f(q) = \sum_m a_m q^m
\]
where $m$ runs over all numbers in $(1/d)\Z \subset \Q$ for some $d
\geq 1$ which depends on $f$, and $a_m \in \C$ vanishes for all
sufficiently negative $m$. Geometrically, one should think of
$\Lambda_\N$ as a small disc, of $\Lambda_\Z$ as the punctured disc,
and of $\Lambda_\Q$ as its universal cover. Consider the semigroup
$End(\Lambda_\N)$ of $q$-adically continuous $\C$-algebra
endomorphisms of $\Lambda_\N$. More concretely, these are
substitutions
\[
 \psi^* : q \longmapsto \psi(q)
\]
with $\psi \in q \C[[q]]$. Each nonzero $\psi$ extends to an
endomorphism of the quotient field, and the $q$-adically continuous
Galois group of $\Lambda_\Z/\C$ is the group of invertibles
$End(\Lambda_\N)^\times$, consisting of those $\psi$ with $\psi'(0)
\neq 0$. Moreover, the Galois group of $\Lambda_\Q/\Lambda_\Z$ is the
profinite group $\hat{\Z}$, whose topological generator $\hat{1}$
takes $q^{1/d}$ to $e^{2\pi i/d} q^{1/d}$.
\end{notation}

On the symplectic side of mirror symmetry, take any smooth quartic
surface $X_0 \subset \CP{3}$, with its standard symplectic structure.
To this we associate a triangulated category linear over
$\Lambda_\Q$, the split-closed derived Fukaya category
$D^\pi\Fuk(X_0)$, defined using Lagrangian submanifolds of $X_0$ and
pseudo-holomorphic curves with boundary on them. The use of formal
power series to take into account the area of pseudo-holomorphic
curves is a familiar device. $\Lambda_\Q$ appears because we allow
only a certain kind of Lagrangian submanifolds, namely the rational
ones (there is also another restriction, vanishing of Maslov classes,
which is responsible for making the Fukaya category $\Z$-graded).

On the complex side, we start with the quartic surface in ${\mathbb
P}^3_{\Lambda_\Q}$ defined by
\begin{equation} \label{eq:fermat}
y_0 y_1 y_2 y_3 + q(y_0^4 + y_1^4 + y_2^4 + y_3^4) = 0.
\end{equation}
Note that here $q$ is a ``constant'' (an element of the ground
field). The group $\Gamma_{16} = \{ [diag(a_0,a_1,a_2,a_3)] \suchthat
a_k^4 = 1, \; a_0a_1a_2a_3 = 1 \} \subset PSL(V)$, $\Gamma_{16} \iso
\Z/4 \times \Z/4$, acts on this surface in the obvious way. We denote
by $Z_q^*$ the unique minimal resolution of the quotient orbifold.
Non-algebraic-geometers may wish to remind themselves that
$\Lambda_\Q$ is an algebraically closed field of characteristic zero,
so that all the geometric constructions like minimal resolutions go
through as in the familiar case of $\C$. In fact, $Z_q^*$ could
already be defined over the smaller field $\Lambda_\Z$, since the
fixed points of the action that need to be resolved are
$\Lambda_\Z$-points. The associated category is the bounded derived
category of coherent sheaves, $D^bCoh(Z_q^*)$, which is also a
triangulated category linear over $\Lambda_\Q$.

\begin{theorem} \label{th:main}
There is a $\psi \in End(\Lambda_\N)^\times$ and an equivalence of
triangulated categories,
\[
D^\pi\Fuk(X_0) \iso \hat{\psi}^*D^bCoh(Z_q^*).
\]
\end{theorem}

Here $\hat\psi$ is a lift of $\psi$ to an automorphism of
$\Lambda_\Q$, which we use to change the $\Lambda_\Q$-module
structure of $D^bCoh(Z_q^*)$, by letting $\hat\psi^*f$ act instead of
$f$. The theorem says that the outcome of this ``base change'' is
equivalent to $D^\pi\Fuk(X_0)$. The choice of lift $\hat\psi$ is
irrelevant because $D^bCoh(Z_q^*)$ carries a $\hat{\Z}$-action (and
so does $D^\pi\Fuk(X_0)$ by \cite{fukaya03}).

The function $\psi$ which occurs in the theorem is unknown at present
(or rather, we cannot verify that it agrees with the standard
``mirror map''). Nevertheless, certain consequences can already be
drawn from the result as given. For instance, let $\mathcal{M}^*$ be
the classifying space (or moduli stack, depending on the reader's
preference) for smooth $K3$ surfaces equipped with an ample
cohomology class $A$ whose square is $A \cdot A = 4$, and with a
choice of nonzero holomorphic two-form. There is a canonical
homomorphism $\pi_1(\mathcal{M}^*) \rightarrow \pi_0(Aut(X_0))$,
which is defined by symplectic monodromy of families of $K3$ surfaces
equipped with K{\"a}hler forms in the class $A$. On the other hand,
since $D^\pi\Fuk(X_0)$ is a symplectic invariant, we have a map
$\pi_0(Aut(X_0)) \rightarrow Aut(D^\pi\Fuk(X_0)/\Lambda_\Q)/[2]$,
where now $Aut$ on the right hand side is the group of isomorphism
classes of $\Lambda_\Q$-linear autoequivalences, and we divide by the
subgroup generated by even translations. Actually, the monodromy maps
obtained from loops in ${\mathcal M}^*$ are naturally graded, so the
shift ambiguity can be removed, and we get a homomorphism
\begin{equation} \label{eq:alg-monodromy}
\pi_1(\mathcal{M}^*) \longrightarrow Aut(D^bCoh(Z_q^*)/\Lambda_\Q).
\end{equation}
This had been postulated by several people, starting with Kontsevich
himself. Interestingly, even though both $\mathcal{M}^*$ and
$D^bCoh(Z_q^*)$ are objects from algebraic geometry, there seems to
be no definition of the map which avoids symplectic geometry and
mirror symmetry. Previous work on Dehn twists \cite{seidel-thomas99}
shows that \eqref{eq:alg-monodromy} is highly nontrivial, but the
exact kernel and cokernel remain unknown. Another direct implication
of Theorem \ref{th:main} is the equality of certain simpler
invariants, such as the Grothendieck groups
\[
K_0(D^\pi\Fuk(X_0)) \iso K_0(D^b Coh(Z_q^*)) = K_0(Z_q^*).
\]
Every object of the Fukaya category (in particular, every oriented
Lagrangian sphere in $X_0$) determines a class in
$K_0(D^\pi\Fuk(X_0))$, and $\pi_0(Aut(X_0))$ acts linearly on this
group. On the other hand $K_0(Z_q^*) \otimes \Q \iso CH_*(Z_q^*)
\otimes \Q$ by Grothendieck's theorem, so it contains $CH_0(Z_q^*)
\otimes \Q$ which is quite an interesting object.

Having stated our result, it is maybe time to review briefly some
other work on homological mirror symmetry. The first case to be
considered was that of elliptic curves, where Polishchuk and Zaslow
\cite{polishchuk-zaslow98} proved a result roughly similar to Theorem
\ref{th:main}, with an explicit mirror map $\psi$ but not taking into
account the triangulated structure (subsequent work on Massey
products \cite{polishchuk98} largely makes up for this deficiency).
There are many beautiful and deep results on abelian varieties
\cite{fukaya02b, kontsevich-soibelman00, fukaya02, nishinou03}, but
none of them comes close to describing the entire Fukaya category. A
common thread of all these papers is to establish an explicit
bijection between objects on both sides (Lagrangian submanifolds and
sheaves). The prevailing view is that this correspondence should be
set up as a Fourier-Mukai type transform associated to a
Strominger-Yau-Zaslow torus fibration. This is the approach of
\cite{kontsevich-soibelman00}, in the situation where the SYZ
fibration has no singularities, and where the Lagrangian submanifolds
are supposed to be transverse to the fibres. Both restrictions are
serious ones from the point of symplectic topology: work on removing
them has started, see in particular \cite{fukaya02b}, but it may be
some time before a complete thery emerges.

Actually, this entire circle of ideas is of little importance for our
proof (although the SYZ conjecture does serve as a source of
intuition in some places). We rely instead on techniques from
symplectic topology and homological algebra, which are maybe closer
to the spirit of Kontsevich's original paper. The computation on the
algebraic geometry side relies on a classical idea of Beilinson,
together with a Massey product computation which the author learned
from \cite{douglas-govindarajan-jayaraman-tomasiello01}. On the
symplectic side, where the overwhelming majority of the work is
invested, there are three steps, namely (slightly simplified, and in
reverse logical order):
\begin{itemize} \itemsep1em

\item
First, we prove that $D^\pi\Fuk(X_0)$ can be entirely reconstructed
from the full $A_\infty$-subcategory of $\Fuk(X_0)$ consisting of a
particular set of 64 Lagrangian two-spheres, which are vanishing
cycles for the standard Fermat pencil (the choice of these cycles is
best explained by physics considerations, compare for instance
\cite{hori-iqbal-vafa00}). The necessary ``split-generation''
argument resembles the ones in \cite{seidel04}, with some added
geometric input concerning the monodromy around the singular fibre at
infinity.

\item
By construction, all 64 vanishing cycles lie in an affine
Zariski-open subset $M_0 \subset X_0$. Following a proposal from
\cite{seidel02} the relation between the Fukaya categories
$\Fuk(M_0)$ and $\Fuk(X_0)$ can be formulated in terms of the
deformation theory of $A_\infty$-structures. This recourse to
abstract deformation theory is the reason why we cannot determine
$\psi$ explicitly.

\item
Having reduced our problem to a finite collection of Lagrangian
two-spheres in an affine $K3$ surface, we now apply a general
dimensional induction argument from \cite{seidel04} to compute the
relevant subcategory of $\Fuk(M_0)$. This starts with an application
of Picard-Lefschetz theory in one dimension lower than before,
fibering our $K3$ surface by a pencil of curves.
\end{itemize}

The last-mentioned argument involves a certain amount of computation,
of a combina\-torial (drawing loops on surfaces, and counting
polygons) and homological (determining morphisms between certain
twisted complexes in $A_\infty$-categories) nature. Although the
algorithm is simple, the computations themselves are not particularly
enlightening, and in fact a computer was used to do parts of them.
The reason why this step in the proof is less transparent is that,
unlike the previous two, it has no known meaning in terms of the
mirror. This is partially due to the fact that the fibres of our
pencil are curves of genus $3$, hence outside the scope of mirror
symmetry.

In principle, the approach taken here is fairly general. Suppose for
instance that one wanted to adapt it to the quintic threefold case.
The deformation theoretic part actually becomes easier since, unlike
the situation for $K3$s, there are no noncommutative deformations of
the mirror variety. On the other hand, the definition of the Fukaya
category is much more complicated \cite{fooo}, and one would have to
generalize the twist exact sequence \cite{seidel01} accordingly, but
these are on the whole technical issues. There are also some
elementary geometric complications, which arise where we have relied
on intuition provided by the SYZ conjecture. The final obstacle is in
the computational part, where the procedure remains precisely the
same but the complexity rises by a factor of at least 10. This raises
serious practical concerns, but it may still be feasible, or one can
try to find a more conceptual way of carrying it out (ideas of
Moishezon might prove helpful, see the comments and computations in
\cite{auroux-donaldson-katzarkov-yotov01}).

The paper is structured as follows. In the first two parts we
assemble the basic algebraic (Sections
\ref{sec:generators}--\ref{sec:sheaves}) and geometric (Sections
\ref{sec:geometry}--\ref{sec:induction}) techniques. Much of this is
not really new, but the application to mirror symmetry often results
a change of perspective, which means that it is hard to find the
theorems stated elsewhere in the form in which we need them. The
level of generality varies somewhat: we have tried to include
generalizations which were reasonably easy or natural, and to exclude
those which are either irrelevant to homological mirror symmetry, or
which would require a significant amount of additional techniques or
terminology. The last part of the paper (Sections
\ref{sec:4-64}--\ref{sec:computation}) contains the argument which is
really specific to the quartic surface, including figures and tables
which list the results of the computational step. The aim of that was
to give enough data so that the reader desirous of verifying the
computation independently could break up that process into manageable
steps.

\acknowledgments This paper could not have been written without the
assistance of Michael Douglas, who showed me the ``quiver
presentation'' of the derived category of coherent sheaves which
appears as the algebras $Q_4$ and $Q_{64}$ in this paper, and also
the crucial Massey product formula (Lemma \ref{th:massey}) from
\cite{douglas-govindarajan-jayaraman-tomasiello01}. Maxim Kontsevich
helped me in many ways, one of them being the Hochschild cohomology
computations for semidirect products (Lemma \ref{th:twisted-hkr}).
Simon Donaldson introduced me to the notion of matching cycle, on
which the main dimensional induction procedrure is based. To all of
them, thanks! ETH Z\"urich invited me to give a lecture course on
Fukaya categories, and provided a responsive audience on which parts
of this paper were tested. The following computer programs have been
used: {\sc singular} for elimination theory, {\sc mathematica} for
visualization, and {\sc maple} for the homological algebra part.


\section{$A_\infty$-categories\label{sec:generators}}

This section is designed to serve as quick reference for the basic
algebraic terminology, for the benefit of readers with a geometry
background. The main construction is that of the bounded derived
category of a $A_\infty$-category, following Kontsevich
\cite{kontsevich94}. We then consider the idempotent (Karoubi, or
split-closed) completion, and the corresponding notion of
split-generators. A brief look at twist functors and exceptional
collections completes the tour. Some references for more in-depth
treatments of various matters touched on here are \cite{kontsevich98,
keller99, lefevre, rudakov90, fooo, fukaya00, seidel04}.

\subsection{}
A non-unital $A_\infty$-category $\Cat$ consists of a set (all our
categories are small) of objects $Ob\,\Cat$, together with a graded
$\C$-vector space $hom_\Cat(X_0,X_1)$ for any two objects, and
multilinear maps
\begin{equation} \label{eq:mu}
 \mu^d_\Cat: hom_\Cat(X_{d-1},X_d) \otimes \dots \otimes
 hom_\Cat(X_0,X_1) \longrightarrow hom_\Cat(X_0,X_d)[2-d]
\end{equation}
for all $d \geq 1$ and all $(d+1)$-tuples of objects
$(X_0,\dots,X_d)$, satisfying the $A_\infty$-associativity equations
\begin{equation} \label{eq:ainfty}
 \sum_{e,i} (-1)^\ast \mu_\Cat^{d-e+1}(a_d,\dots,
 a_{i+e+1},\mu_\Cat^e(a_{i+e},\dots,a_{i+1}),a_i,\dots,a_1) = 0,
\end{equation}
where $\ast = |a_1|+\dots+|a_i|-i$ (the decreasing numbering of the
$a_k$ is the result of trying to bring together various standard
conventions). The underlying (non-unital) cohomological category
$H(\Cat)$ has the same objects as $\Cat$, with
\[
Hom^*_{H(\Cat)}(X_0,X_1) = H^*(hom_{\Cat}(X_0,X_1),\mu^1_\Cat)
\]
and composition induced by the cochain level map $a_2a_1 =
(-1)^{|a_1|}\mu^2_\Cat(a_2,a_1)$. There is also the full subcategory
$H^0(\Cat)$ which retains only the morphisms of degree zero. One says
that $\Cat$ is {\em c-unital} (cohomologically unital) if $H^0(\Cat)$
has identity morphisms. It is true, but by no means obvious, that
this is a good class of $A_\infty$-categories to work with: the
proofs of many properties rely on the fact that c-unitality turns out
to be equivalent to the less flexible but more visibly well-behaved
notion of strict unit \cite{lefevre,seidel04}. {\em From now on, all
$A_\infty$-categories are assumed to be c-unital}. Similarly, when
considering $A_\infty$-functors $\F: \Cat \rightarrow {\mathcal D}$,
we always require that $H(\F): H(\Cat) \rightarrow H({\mathcal D})$
be unital (carries identity morphisms to identity morphisms). $\F$ is
called a quasi-equivalence if $H(\F)$ is an equivalence. One can
prove that then, there is a $\G: {\mathcal D} \rightarrow \Cat$ such
that $H(\G)$ is an inverse equivalence to $H(\F)$.

\begin{remark}
A linear category with one object is the same as an associative
algebra, and similarly, an $A_\infty$-category with one object is an
$A_\infty$-algebra. Slightly more generally, given a category $C$
with only finitely many objects $X_1,\dots,X_m$, one can form the
{\em total morphism algebra}
\[
 A = \bigoplus_{j,k} Hom_C(X_j,X_k)
\]
which is linear over the semisimple ring $R_m = \C \oplus \dots
\oplus \C$ ($m$ summands). Again, the same holds for
$A_\infty$-categories versus $R_m$-linear $A_\infty$-algebras, and we
will use that frequently. Our notation will be that $u_1,\dots,u_m$
are the orthogonal idempotents in $R_m$, so $u_k A u_j =
Hom_C(X_j,X_k)$.
\end{remark}

\begin{remark}
For concreteness, we stick to the ground field $\C$ for our
exposition of the basic algebraic theory, but any other field would
do just as well (later on though, we will make use of deformation
theoretic ideas which work best in characteristic $0$). Defining
$A_\infty$-categories over a ring is a more difficult issue, and the
only simple solution is to assume that all the $hom$ spaces are
projective modules. That of course holds automatically for the
semisimple rings $R_m$. The only other ring of interest to us is
$\Lambda_\N$, which requires special treatment because of its
topology. Recall that for a $\Lambda_\N$-module $V_q$ which is
$q$-adically complete, the following conditions are equivalent: (1)
$V_q$ is torsion-free, which means that multiplication by $q$ is an
injective endomorphism; (2) $V_q$ is topologically free, which means
that projection $V_q \rightarrow V$ to the vector space $V = V_q
\otimes_{\Lambda_q} \C = V_q/qV_q$ can be lifted to a topological
isomorphism $V_q \rightarrow V \hat\otimes \Lambda_\N$. A {\em
topological $A_\infty$-algebra} over $\Lambda_\N$ consists of a
graded $\Lambda_\N$-module $\A_q$ such that each degree $k$ piece is
complete and torsion-free. If $H(\A_q)$ is torsion-free, it is
automatically also complete.
%
%, together with maps $\mu^d_{\A_q}$ which are $q$-adically
% continuous. This condition is automatic!!!
\end{remark}

\subsection{}
The twisted complex construction embeds $\Cat$ into a larger
$A_\infty$-category which is closed under (appropriately understood)
mapping cones. This can be broken into two steps:
\begin{itemize}
\item[(1)]
The additive enlargement $\Sigma\Cat$ has objects which are formal
sums
\[
X = \bigoplus_{f \in F} X_f[\sigma_f],
\]
where $F$ is some finite set, $X_f \in Ob\,\Cat$, and $\sigma_f \in
\Z$. The morphisms between any two objects are
\[
 hom_{\Sigma\Cat}\Big(\bigoplus_{f \in F} X_f[\sigma_f],
 \bigoplus_{g \in G} Y_g[\tau_g]\Big) =
 \bigoplus_{f,g} hom_{\Cat}(X_f,Y_g)[\tau_g-\sigma_f],
\]
and the compositions are defined using those of $\Cat$ and the
obvious ``matrix multiplication'' rule, with additional signs put in
as follows: for $a_1 \in hom(X_0[\sigma_0],X_1[\sigma_1]), \dots, a_d
\in hom(X_{d-1}[\sigma_{d-1}],X_d[\sigma_d])$,
\[
 \mu^d_{\Sigma\Cat}(a_d,\dots,a_1) = (-1)^{\sigma_0}
 \mu^d_\Cat(a_d,\dots,a_1).
\]

\item[(2)]
A twisted complex is a pair $(C,\delta)$ consisting of $C \in
Ob\,\Sigma\Cat$ and $\delta \in hom^1(C,C)$, with the ``upper
triangularity'' property that the indexing set $F$ for $C =
\bigoplus_{f \in F} C_f$ can be ordered in such a way that all
components $\delta_{fg}$ with $f \geq g$ are zero, and subject to the
generalized Maurer-Cartan equation
\[
 \mu^1_{\Sigma\Cat}(\delta) + \mu^2_{\Sigma\Cat}(\delta,\delta) + \dots = 0.
\]
Upper triangularity says that this is a finite sum, hence the
condition makes sense. We define an $A_\infty$-category $Tw\Cat$ of
which the $(C,\delta)$ are the objects. The spaces of morphisms are
the same as for $\Sigma\Cat$, but composition involves the $\delta$s:
for $a_k \in hom((C_{k-1},\delta_{k-1}),(C_k,\delta_k))$,
\[
 \mu^d_{Tw\Cat}(a_d,\dots,a_1) = \sum_{j_0,\dots,j_d \geq 0}
 \mu^d_{\Sigma\Cat}(\overbrace{\delta_d,\dots,\delta_d}^{j_d},a_d,
 \overbrace{\delta_{d-1},\dots,\delta_{d-1}}^{j_{d-1}},\dots).
\]
\end{itemize}
The c-unitality of $\Cat$ implies that of $Tw\Cat$. The underlying
cohomological category in degree zero, which means we retain only the
$H^0$ of the morphisms spaces, is (somewhat improperly) called the
{\em bounded derived category} $D^b(\Cat) = H^0(Tw\Cat)$. Given two
twisted complexes $(C_k,\delta_k)$, $k = 0,1$, and a morphism $a \in
hom^0(C_0,C_1)$ such that $\mu^1_{Tw\Cat}(a) = 0$, one can define the
mapping cone as usual:
\[
 Cone(a) = \Big\{C_0 \stackrel{a}{\rightarrow} C_1\Big\} \stackrel{\text{def}}{=}
 \Big(C_0[1] \oplus C_1, \begin{pmatrix} \delta_0 & 0 \\ a &
 \delta_1 \end{pmatrix}\Big)
 \in Ob\,Tw\,\Cat.
\]
The isomorphism class of this in the derived category depends only on
the cohomology class $\alpha = [a] \in Hom_{D^b(\Cat)}(C_0,C_1)$,
hence we will sometimes write sloppily $Cone(\alpha)$. Declare {\em
exact triangles} in the derived category to be those which are
isomorphic to the standard mapping cone triangles $C_0 \rightarrow
C_1 \rightarrow Cone(a) \rightarrow C_0[1]$. This makes $D^b(\Cat)$
into a triangulated category. It contains $H^0(\Cat)$ as a full
subcategory, and the objects of that subcategory {\em generate}
$D^b(\Cat)$ in the usual triangulated sense, that is, any object of
$D^b(\Cat)$ can be obtained from objects of $H^0(\Cat)$ by successive
mapping cones and isomorphism. However, this does not mean that
$H^0(\Cat)$ determines $D^b(\Cat)$. For instance, triangulated
categories have intrinsically defined Massey product operations
\cite{gelfand-manin}, and in our case these involve the higher order
$\mu^d_\Cat$.

The idempotent completion of an arbitrary linear category has objects
which are pairs $(X,p)$, where $X$ is an object of the original
category and $p \in Hom(X,X)$ an idempotent endomorphism. One thinks
of this as the ``direct summand $im(p)$'', and so
$Hom((X_0,p_0),(X_1,p_1)) = p_1 Hom(X_0,X_1) p_0$. It is proved in
\cite{balmer-schlichting01} that the idempotent completion of a
triangulated category is again triangulated, in a unique way which
makes the inclusion $X \mapsto (X,id_X)$ into an exact functor. The
result of applying this construction to the bounded derived category
of an $A_\infty$-category is called the {\em split-closed derived
category} $D^\pi(\Cat)$. Suppose that we have a set of objects in an
idempotent complete triangulated category; these are called {\em
split-generators} if one can obtain any object from them by
successive mapping cones, splitting off direct summands, and
isomorphism.

Here are some obvious but useful facts about derived categories:

\begin{lemma}
If two $A_\infty$-categories are quasi-equivalent, their derived
categories $D^b$ and $D^\pi$ are equivalent triangulated categories.
\end{lemma}

\proof Any $A_\infty$-functor $\F: \Cat \rightarrow {\mathcal D}$
admits a canonical extension $Tw\F: Tw\Cat \rightarrow Tw{\mathcal
D}$, and $D^b\F = H^0(Tw\F): D^b(\Cat) \rightarrow D^b({\mathcal D})$
is exact (it takes cones to cones). Suppose that $\F$ is a
quasi-equivalence, and consider the subcategory of $D^b(\Cat)$
consisting of those objects $X$ such that $D^b\F$ maps $Hom^*(X,Y)$
isomorphically to $Hom^*(D^b\F(X),\F(Y))$ for all $Y \in
Ob\,H^0(\Cat)$. By assumption this subcategory contains $H^0(\Cat)$,
and it is a standard exercise to show that it is triangulated, so it
is the whole of $D^b(\Cat)$. Repetition of the argument proves that
$D^b\F$ is full and faithful. Its image is a triangulated subcategory
of $D^b({\mathcal D})$ which, up to isomorphisms, contains all
objects of $H^0({\mathcal D})$. Since these are generators for
$D^b({\mathcal D})$, we conclude that $D^b(\F)$ is an equivalence.
The corresponding statement for $D^\pi$ follows immediately. \qed

\begin{lemma} \label{th:generating-subcategory}
Let $\Cat' \subset \Cat$ a full $A_\infty$-subcategory such that the
set of all objects of $\Cat'$ split-generates $D^\pi(\Cat)$. Then
$D^\pi(\Cat') \iso D^\pi(\Cat)$ as triangulated categories.
\end{lemma}

\proof Inclusion induces a full and faithful functor $D^\pi(\Cat')
\rightarrow D^\pi(\Cat)$, the image of which contains the objects of
$H^0(\Cat')$ and is closed under cones and idempotent splittings.
\qed

Combining the two previous Lemmas, we have

\begin{lemma} \label{th:derived-equivalence}
Let $\Cat,{\mathcal D}$ be two $A_\infty$-categories. Suppose that
they contain full $A_\infty$-subcategories $\Cat' \subset \Cat,
{\mathcal D}' \subset {\mathcal D}$ which are quasi-equivalent to
each other. If the objects of $\Cat',{\mathcal D}'$ split-generate
$D^\pi(\Cat)$ respectively $D^\pi({\mathcal D})$, then these two
derived categories are equivalent. \qed
\end{lemma}

\subsection{}
Let $C$ be a $\C$-linear triangulated category, and $X,Y$ two objects
such that $Hom_C^*(X,Y) = \bigoplus_{k \in \Z} Hom_C(X,Y[k])$ is
finite-dimensional. Then one can form two new objects $T_X(Y)$,
$T_Y^\vee(X)$ which fit into exact triangles
\[
\begin{aligned}
 & T_X(Y)[-1] \rightarrow Hom_C^*(X,Y) \otimes X \xrightarrow{ev} Y \rightarrow T_X(Y), \\
 & T_Y^\vee(X) \rightarrow X \xrightarrow{ev^\vee} Hom^*_C(X,Y)^\vee \otimes Y \rightarrow
 T_Y^\vee(X)[1].
\end{aligned}
\]
The notation means that we choose a homogeneous basis $\{f_k\}$ of
$Hom^*(X,Y)$, set $Hom^*(X,Y) \otimes X = \bigoplus_k X[-deg(f_k)]$,
and take the evaluation map $ev = \bigoplus_k f_k[-k]$. The dual map
$ev^\vee$ is defined in the same way. These algebraic {\em twist} and
{\em untwist} operations are well-defined up to non-unique
isomorphism, and the isomorphism type of the new objects depends only
on those of $X$ and $Y$.

The construction becomes somewhat more functorial in the case where
$C = D^b(\Cat)$ is the bounded derived category of an
$A_\infty$-category $\Cat$ such that (1) the cochain level morphism
space $hom_{Tw\Cat}(X,Y)$ is itself finite-dimensional, and (2) $X$
has a strict cochain level identity morphism $e_X$. One can then form
the tensor product $hom(X,Y) \otimes X \in Ob\,Tw(\Cat)$, which is a
finite direct sum of shifted copies of $X$ with a differential
$\mu^1_{hom(X,Y)} \otimes e_X$, and $T_X(Y)$ can be defined as a
twisted complex:
\begin{equation} \label{eq:explicit-twist}
 T_X(Y) = \{hom^*(X,Y) \otimes X \longrightarrow Y\}.
\end{equation}
In fact, if the finite-dimensionality condition holds for all $Y$,
one can realize $T_X$ by an $A_\infty$-endofunctor of $Tw\Cat$,
inducing an exact functor on $D^b(\Cat)$. Similarly, if (1) holds and
(2') $Y$ has a strict cochain level identity morphism $e_Y$, then
$T_Y^\vee(X) = \{X \rightarrow hom^*(X,Y)^\vee \otimes Y\}$ will do.
Fukaya categories are not in general strictly unital, so the previous
observation does not apply directly to them. This can be remedied,
but instead of entering into a full discussion, we just record the
technical statement that will be used later on:

\begin{lemma} \label{th:recognize-twist}
Let $\Cat$ be a c-unital $A_\infty$-category whose $hom$ spaces are
finite-dimensional, and $X,Y,Z$ objects of it. Any pair
\[
 k \in hom^0_\Cat(Y,Z), \quad
 h : hom_\Cat(X,Y) \longrightarrow hom_\Cat(X,Z)[-1]
\]
satisfying
\begin{equation} \label{eq:coupled-morphism}
 \mu^1_\Cat(k) = 0, \quad
 \mu^1_\Cat h(a) + h(\mu^1_\Cat a) + \mu^2_\Cat(k,a) = 0
\end{equation}
for all $a \in hom_\Cat(X,Y)$, determines a morphism $T_X(Y)
\rightarrow Z$ in $D^b(\Cat)$. This is an isomorphism if for all
objects $W$ of $\Cat$, the following complex of vector spaces is
acyclic:
\begin{equation} \label{eq:total-cone}
\begin{split}
 & (hom(X,Y) \otimes hom(W,X))[2] \oplus hom(W,Y)[1] \oplus hom(W,Z), \\
 & \partial(a_2 \otimes a_1,b,c) =
 \big(\mu^1_\Cat(a_2) \otimes a_1 +
 (-1)^{|a_2|-1} a_2 \otimes \mu^1_\Cat(a_1),
 \mu^1_\Cat(b) \\ & \qquad + \mu^2_\Cat(a_2,a_1),
 -\mu^1_\Cat(c) + \mu^2_\Cat(k,b) + \mu^3_\Cat(k,a_2,a_1) +
 \mu^2(h(a_2),a_1)\big).
\end{split}
\end{equation}
\end{lemma}

\proof Suppose first that $\Cat$ is strictly unital. Then $(k,h)$ do
indeed define a $\mu^1$-closed cochain level morphism from the object
$T_X(Y)$ as defined in \eqref{eq:explicit-twist} to $Z$. Composition
with this morphism is a map of complexes of vector spaces
\[
 hom_{Tw\Cat}(W,T_X(Y)) \longrightarrow hom_{\Cat}(W,Z)
\]
and \eqref{eq:total-cone} is the mapping cone of it, so the
assumption that this is acyclic means that our morphism induces
isomorphisms
\[
Hom^*_{D^b(\Cat)}(W,T_X(Y)) \iso Hom^*_{D^b(\Cat)}(W,Z)
\]
for all $W \in Ob\,\Cat$. A straightforward long exact sequence
argument extends this to objects $W$ in the derived category, and
then the Yoneda Lemma implies that $T_X(Y) \iso Z$.

In the c-unital case, one argues as follows. There is a differential
graded category $\Cat^\vee = {\mathcal Fun}(\Cat,Ch)$ of c-unital
$A_\infty$-functors from $\Cat$ to the dg category $Ch$ of
finite-dimensional chain complexes over $\C$, and a natural
$A_\infty$-functor $\Cat \rightarrow \Cat^\vee$, which on cohomology
is full and faithful. This extends to an $A_\infty$-functor $Tw\Cat
\rightarrow \Cat^\vee$, which induces a full embedding of
triangulated categories $D^b(\Cat) \hookrightarrow H^0(\Cat^\vee)$.
Since $\Cat^\vee$ is strictly unital, one can define the object
$T_X(Y)$ in $H^0(\Cat^\vee)$ as in \eqref{eq:explicit-twist}; of
course, this will be non-canonically isomorphic to the object of the
same name in the subcategory $D^b(\Cat)$. By unravelling the
definitions one sees that the pair $(h,k)$ gives a morphism $T_X(Y)
\rightarrow Z$ in $H^0(\Cat^\vee)$; one proves as before that this is
an isomorphism. \qed

\begin{lemma} \label{th:twist-generators}
Let $C$ be a $\C$-linear idempotent closed triangulated category,
such that $Hom^*_C(X,Y)$ is finite-dimensional for all $X,Y$. Take a
collection of objects $X_1,\dots,X_m$ and another object $Y$, and
consider the sequence of objects $Y_0 = Y$, $Y_{d+1} = T_{X_1} \dots
T_{X_m}(Y_d)$. Suppose there is a $d>0$ such that $Hom_C(Y,Y_d) = 0$
(that is, there are no nontrivial morphisms of degree zero). Then $Y$
lies in the idempotent closed triangulated subcategory of $C$
split-generated by $X_1,\dots,X_m$.
\end{lemma}

\proof By definition, we have exact triangles
\begin{equation} \label{eq:taut-triangle}
\xymatrix{
 {T_{X_{i+1}} \dots T_{X_m}(Y_j)} \ar[rr] &&
 {T_{X_i}\dots T_{X_m}(Y_j)} \ar_{[1]}[dl] \\
 & {\hspace{-5em}
 Hom^*_C(X_i,T_{X_{i+1}}\dots T_{X_m}(Y_j)) \otimes X_i
 \hspace{-5em} } \ar_-{ev}[ul] &
}
\end{equation}
The octahedral axiom allows us to conflate these, for all $i$ and all
$j < k$, into a single exact triangle
\begin{equation} \label{eq:new-triangle}
\xymatrix{
 Y \ar[rr] && Y_d \ar_-{[1]}[dl] \\
 & R_d \ar[ul] &}
\end{equation}
where $R_d$ is an object built by taking repeated mapping cones from
the objects in the same positions in \eqref{eq:taut-triangle}, hence
which lies in the triangulated subcategory generated by
$X_1,\dots,X_m$. The assumption says that the $\rightarrow$ in
\eqref{eq:new-triangle} must be zero, hence $R_d \iso Y \oplus
Y_d[-1]$. \qed

Our applications of this Lemma will be along the following line: if
$C = D^\pi(\Cat)$ and there are fixed $X_1,\dots,X_m \in
Ob\,H^0(\Cat)$ such that any $Y \in Ob\,H^0(\Cat)$ has the property
required above, then the $X_k$ are split-generators for
$D^\pi(\Cat)$.

\subsection{}
Let $C^\rightarrow$ be a linear graded category with only finitely
many objects $(X_1,\dots,X_m)$, ordered in a fixed way. We say that
$C^\rightarrow$ is {\em directed} if
\begin{equation} \label{eq:directedness}
 Hom^*_{C^\rightarrow}(X_i,X_j) = \begin{cases}
 0 & i>j, \\ \C \cdot id_{X_i} & i = j, \\
 \text{finite-dimensional} & i < j.
 \end{cases}
\end{equation}
If $C$ is a linear $\C$-graded category with finite-dimensional
$Hom^*$'s, and $\Gamma = (X_1,\dots,X_m)$ an ordered finite family of
nonzero objects in it, we can define the associated directed
subcategory to be the subcategory $C^\rightarrow \subset C$ with the
same objects, which is directed and satisfies
$Hom^*_{C^\rightarrow}(X_i,X_j) = Hom^*_C(X_i,X_j)$ for $i<j$. This
is a full subcategory iff the $X_i$ form an {\em exceptional
collection}, which means that their morphisms in $C$ already satisfy
the conditions of \eqref{eq:directedness}.

Similarly, let $\Cat^\rightarrow$ be an $A_\infty$-category with
objects $(X_1,\dots,X_m)$. We call it directed if the $hom$-spaces
satisfy the same conditions as in \eqref{eq:directedness}, with the
additional condition that there are generators $e_i$ of
$hom(X_i,X_i)$ which are units in the strict sense. Given a finite
collection of objects $(X_1,\dots,X_m)$ in an arbitrary
$A_\infty$-category $\Cat$ with finite-dimensional $hom$'s, one can
define an associated directed $A_\infty$-category $\Cat^\rightarrow$
by requiring that $hom_{\Cat^\rightarrow}(X_i,X_j) =
hom_\Cat(X_i,X_j)$ for $i<j$, and
$\mu^d_{\Cat^\rightarrow}(a_d,\dots,a_1) =
\mu^d_{\Cat}(a_d,\dots,a_1)$ for all $a_k \in
hom(X_{i_{k-1}},X_{i_k})$ with $i_0 < \dots < i_d$. This is not
really an $A_\infty$-subcategory of $\Cat$ (unless that happens to
have strict units), but after some more work one can still construct
an $A_\infty$-functor $\Cat^\rightarrow \rightarrow \Cat$ which maps
the $hom(X_i,X_j)$, $i<j$, trivially. We will therefore tolerate a
slight abuse of terminology, and call $\Cat^\rightarrow$ the {\em
directed $A_\infty$-subcategory} associated to $(X_1,\dots,X_m)$.

The main interest in exceptional collections and directed categories
comes from the notion of mutations. Occasional glimpses of this
theory will be visible later in the paper, in particular in Section
\ref{sec:induction}, but it is not strictly necessary for our
purpose, so we will just refer the interested reader to
\cite{rudakov90,kontsevich98,seidel00}.

\section{Deformation theory\label{sec:deformations}}

We need to put together some material (well-known in the right
circles) from the classification theory of $A_\infty$-structures.
Using the Homological Perturbation Lemma
\cite{gugenheim-lambe-stasheff90}, the ``moduli space'' of
quasi-isomorphism types of $A_\infty$-algebras with fixed cohomology
algebra can be written as the quotient of an infinite-dimensional
space of solutions to the $A_\infty$-equations, by the action of an
infinite-dimensional ``gauge'' group. There is a canonical base
point, represented by the trivial $A_\infty$-structure (the one where
the higher order composition maps are all zero). The formal ``Zariski
tangent space'' at this point is a product of suitable Hochschild
cohomology groups of the cohomology algebra. Moreover, the ``moduli
space'' carries a $\C$-action which contracts it to the base point.
As noted by Kadeishvili \cite{kadeishvili88}, this implies that the
``tangent space'' already carries a lot of information about the
``moduli space''. A more straightforward argument along the same
lines applies to the classification of infinitesimal deformations of
a fixed possibly nontrivial $A_\infty$-structure, using a suitable
generalization of Hochschild cohomology \cite{getzler-jones90}. We
will recall these and some similar ideas, aiming in each case for
uniqueness results with easily verifiable criteria, tailored to the
specific computations later on. The proofs are exercises in abstract
deformation theory \`a la Deligne \cite{goldman-millson88}, and we
postpone them to the end.

\subsection \label{subsec:classification}
Let $A$ be a graded algebra. Consider the set $\AA(A)$ of those
$A_\infty$-algebras $\A$ whose underlying graded vector space is $A$,
and where the first two composition maps are
\begin{equation} \label{eq:mu1zero}
 \mu^1_\A = 0, \quad
 \mu^2_\A(a_2,a_1) = (-1)^{|a_1|} a_2a_1.
\end{equation}
This means that the cohomology algebra is $H(\A) = A$ itself. $\A,\A'
\in \AA(A)$ are considered equivalent if there is an
$A_\infty$-homomorphism $\G: \A \rightarrow \A'$ whose underlying
linear map is $\G^1 = id_A$. After writing out the relevant
equations, of which the first one is
\begin{equation} \label{eq:pullback}
\begin{split}
 \mu^3_{\A'}(a_3,a_2,a_1)
 & = \mu^3_\A(a_3,a_2,a_1) \\ &
 + \G^2(a_3,\mu^2_\A(a_2,a_1)) + (-1)^{|a_1|-1} \G^2(\mu^2_\A(a_3,a_2),a_1) \\
 & - \mu^2_\A(a_3,\G^2(a_2,a_1))
 - \mu^2_\A(\G^2(a_3,a_2),a_1),
\end{split}
\end{equation}
one sees that $\A$ and $\G$ determine $\A'$ completely, and that
there are no constraints on $\G$. In other words, one can define a
group $\GG(A)$ of {\em gauge transformations} which are arbitrary
sequences of multilinear maps $\{\G^1 = id_A, \G^2: A \otimes A
\rightarrow A[-1], \dots\}$, and then \eqref{eq:pullback} and its
successors define an action of $\GG(A)$ on $\AA(A)$, whose quotient
is the set of equivalence classes (this becomes entirely transparent
in a geometric formulation, where it is the action of formal
noncommutative diffeomorphisms on integrable odd vector fields).
Additionally, there is a canonical action of the multiplicative
semigroup $\C$ on $\AA(A)$, by multiplying $\mu^d$ with
$\epsilon^{d-2}$, which we denote by $\A \mapsto \epsilon^*\A$. For
$\epsilon \neq 0$, it is still true that $\A$ is isomorphic to
$\epsilon^*\A$, even though not by a gauge transformation: the
isomorphism is multiplication by $\epsilon^k$ on the degree $k$
component. The $\C$-action obviously descends to $\AA(A)/\GG(A)$.

The main justification for imposing conditions \eqref{eq:mu1zero} is
the Homological Perturbation Lemma, which says the following. Let
$\B$ be an $A_\infty$-algebra with an isomorphism $F: A \rightarrow
H(\B)$. Then one can find an $A_\infty$-structure $\A$ on $A$
satisfying \eqref{eq:mu1zero} and an $A_\infty$-homomorphism $\F: \A
\rightarrow \B$ such that $H(\F) = F$. This establishes a bijection
\[
 \frac{\AA(A)}{\GG(A)} \iso \frac{
 \{\text{pairs $(\B,F: A \rightarrow H(\B))$}\}
 }
 {
 \{\text{$A_\infty$-quasi-isomorphisms compatible with $F$}\}
 }.
\]

\begin{remark} \label{th:explicit-hpt}
The construction of $\A$ is explicit: take maps
\begin{equation} \label{eq:hpt-data}
 \pi: \B \rightarrow A, \quad
 \lambda: A \rightarrow \B, \quad
 h: \B \rightarrow \B[-1]
\end{equation}
such that $\pi \mu^1_\B = 0$, $\mu^1_\B \lambda = 0$ (with the map
induced by $\lambda$ on cohomology being $F$), $\pi \lambda = id_A$,
$\lambda \pi = id_\B + \mu^1_\B h + h \mu^1_\B$. Then $\mu_\A^d$ is a
sum over planar trees with $d$ leaves and one root. The contribution
from each tree involves only the $\mu^k_\B$ for $k$ at most $d$, and
the auxiliary data \eqref{eq:hpt-data} (this particular formulation
is taken from \cite{kontsevich-soibelman00}). For instance, writing
$b_k = \lambda(a_k)$ one has
\begin{align} \label{eq:mu4-hpt}
 & \mu^4_\A(a_4,a_3,a_2,a_1) =
 \pi \mu^4_\B(b_4,b_3,b_2,b_1) + {} \\ \notag
 & \qquad \pi \mu^2_\B(h\mu^3_\B(b_4,b_3,b_2),b_1)
 + \pi \mu^2_\B(b_4,h\mu^3_\B(b_3,b_2,b_1)) + {} \\ \notag
 & \qquad \pi \mu^3_\B(h\mu^2_\B(b_4,b_3),b_2,b_1)
 + \pi \mu^3_\B(b_4,h\mu^2_\B(b_3,b_2),b_1) + {} \displaybreak[0] \\ \notag
 & \qquad \pi \mu^3_\B(b_4,b_3,h\mu^2_\B(b_2,b_1))
 + \pi \mu^2_\B(h\mu^2_\B(b_4,b_3),h\mu^2_\B(b_2,b_1)) + {} \\ \notag
 & \qquad \pi \mu^2_\B(h\mu^2_\B(h\mu^2_\B(b_4,b_3),b_2),b_1)
 + \pi \mu^2_\B(h\mu^2_\B(b_4,h\mu^2_\B(b_3,b_2)),b_1) + {} \\ \notag
 & \qquad \pi \mu^2_\B(b_4,h\mu^2_\B(h\mu^2_\B(b_3,b_2),b_1))
 + \pi \mu^2_\B(b_4,h\mu^2_\B(b_3,h\mu^2_\B(b_2,b_1))).
\end{align}
\end{remark}

The Hochschild cohomology of $A$ is a bigraded vector space. In our
slightly unusual notation, elements of $HH^{s+t}(A,A)^t$ are
represented by cocycles $\tau \in CC^{s+t}(A,A)^t$, which are
multilinear maps $\tau: A^{\otimes s} \rightarrow A$ of degree $t$.
Suppose that for $\A \in \AA(A)$, the compositions $\mu_\A^s$ vanish
for $2 < s < d$. Then $\mu^d_\A$ is a Hochschild cocycle; its class
\[
m_\A^d = [\mu^d_\A] \in HH^2(A,A)^{2-d},
\]
the order $d$ {\em deformation class} of $\A$, forms the obstruction
to making $\mu^d_\A$ trivial by a gauge transformation which is
itself equal to the identity to all orders $<d-1$. In particular, if
all the obstruction groups $HH^2(A,A)^{2-s}$, $s > 2$, are zero,
$\AA(A)/\GG(A)$ reduces to a point, which means that each
$A_\infty$-algebra structure on $A$ can be trivialized by a gauge
transformation (this is the {\em intrinsically formal} situation from
\cite{halperin-stasheff79, kadeishvili88}). A slightly more general
result allows the total obstruction group to be one-dimensional:

\begin{lemma} \label{th:versal-1}
Suppose that there is some $d>2$ such that
\[
\begin{cases}
HH^2(A,A)^{2-d} = \C, \\ HH^2(A,A)^{2-s} = 0 & \text{for all $s > 2$,
$s \neq d$}.
\end{cases}
\]
Suppose also that there exists an $\A \in \AA(A)$ such that $\mu_\A^s
= 0$ for $2 < s < d$, and whose deformation class $m_\A^d$ is
nonzero. Then any $\A' \in \AA(A)$ is equivalent to $\epsilon^*\A$
for some $\epsilon \in \C$, which means the $\C$-orbit of $[\A]$ is
the whole of $\AA(A)/\GG(A)$.
\end{lemma}

We say that $\A$ is {\em versal}. All other problems that we will
consider have the same structure: a set of equivalence classes
described as a quotient, often involving the appeal to some kind of
Perturbation Lemma; an additional group of reparametrizations acting
on the quotient; a suitable version of Hochschild cohomology, and
notion of deformation class; and ultimately, results expressing
intrinsic formality or versality.

\begin{remark} \label{th:versal-2}
Lemma \ref{th:versal-1} can be somewhat generalized, even though that
does not fit in as well with our general philosophy as the original
statement. The generalization goes as follows: suppose that
$HH^2(A,A)^{2-s} = 0$ for all $s>2$ except $s = d$, with
$HH^2(A,A)^{2-d}$ arbitrary. Then every point in $\AA(A)/\GG(A)$ has
a representative $\A$ with $\mu^s_\A = 0$ for $2<s<d$; and if two
such representatives $\A,\A'$ have the same order $d$ deformation
class, they are equivalent.
\end{remark}

\subsection{}
Let $\A$ be an $A_\infty$-algebra. A {\em one-parameter deformation}
of $\A$ is a topological $A_\infty$-algebra $\A_q$ over $\Lambda_\N$,
whose underlying graded vector space is $\A \hat\otimes \Lambda_\N$,
and whose composition maps reduce to those of $\A$ if one sets $q =
0$. More concretely,
\[
\mu_{\A_q}^d = \mu_{\A}^d + q \mu_{\A_q,1}^d + q^2 \mu_{\A_q,2}^d +
\dots =\mu_{\A}^d + O(q),
\]
where each coefficient is a $\C$-linear map $\A^{\otimes d}
\rightarrow \A[2-d]$. Two one-parameter deformations are equivalent
if there is a $\Lambda_\N$-linear $A_\infty$-homomorphism $\G_q: \A_q
\rightarrow \A'_q$ between them of the form
\begin{equation} \label{eq:zero-phi}
\G_q^d = \begin{cases} id + O(q) & d = 1, \\ O(q) & d>1. \end{cases}
\end{equation}
Let $\AA_q(\A)$ be the set of all one-parameter deformations, and
$\GG_q(\A)$ the group of sequences of multilinear maps $\{\G^1_q,
\G^2_q,\dots\}$ on $\A \hat{\otimes} \Lambda_\N$ which satisfy
\eqref{eq:zero-phi}. Then $\AA_q(\A)/\GG_q(\A)$ is the set of
equivalence classes. The semigroup $End(\Lambda_\N)$ acts on
$\AA_q(\A)$ by reparametrizations $q \mapsto \psi(q)$; we denote the
action by $\A_q \mapsto \psi^*\A_q$. In this context, versality
therefore means that the $End(\Lambda_\N)$-orbit of some equivalence
class $[\A_q]$ is the whole of $\AA_q(\A)/\GG_q(\A)$.

\begin{remark}
We will also occasionally use terminology from finite order
deformation theory. An order $d$ infinitesimal deformation of $\A$
would be an $A_\infty$-algebra $\A_q$ as before, except that the
ground ring is now $\C[[q]]/q^{d+1}$. The notion of equivalence is
obvious. Of course, a one-parameter deformation can be truncated to
any desired finite order in $q$.
\end{remark}

Hochschild cohomology generalizes to $A_\infty$-algebras
\cite{getzler-jones90} as a graded (no longer bigraded) space
$HH^*(\A,\A)$. Consider $CC^d(\A,\A) = \prod_{s+t = d}
Hom_\C(\A^{\otimes s},\A[t])$ with the Gerstenhaber product
\begin{equation} \label{eq:gerstenhaber}
 (\sigma \circ \tau)^d(a_d,\dots,a_1) =
 \sum_{i,j} (-1)^\sharp \sigma^{d-j+1}(a_d,\dots,
 \tau^j(a_{i+j},\dots,a_{i+1}),\dots,a_1),
\end{equation}
where $\sharp = (|\tau|-1)(|a_1|+\dots+|a_i|-i)$, and the
corresponding Lie bracket $[\sigma,\tau] = \sigma \circ \tau -
(-1)^{(|\sigma|-1)(|\tau|-1)} \tau \circ \sigma$. As $\mu_\A^*$
itself lies in $CC^2(\A,\A)$ and satisfies $\mu_A^* \circ \mu_A^* =
0$ by definition, one can define a differential $\delta\tau =
[\mu_\A^*,\tau]$, and this is the $A_\infty$ version of the
Hochschild differential. Filtration by length yields a spectral
sequence with
\[
E_2^{s,t} = HH^{s+t}(A,A)^t
\]
for $A = H(\A)$, and this converges to $HH^*(\A,\A)$ under suitable
technical assumptions (for instance, if the nonzero terms in the
$E_2$ plane are concentrated in a region cut out by $t \leq c_1$, $t
\geq c_2 - c_3 s$ for $c_1,c_2$ arbitrary and $c_3 < 1$). If
$\mu_{\A}^d$, $d \geq 3$, is the first nonzero higher order
composition, then the first potentially nontrivial differential in
the spectral sequence is
\begin{equation} \label{eq:bracket-differential}
 \delta_{d-1} = [m_{\A}^d,\cdot]:
 E_2^{s,t} \longrightarrow E_2^{s+d-1,t+2-d}
\end{equation}
where $[\cdot,\cdot]$ is the Gerstenhaber bracket on $HH^{*+1}(A,A)$,
and $m_{\A}^d$ is the order $d$ deformation class. Actually, the
spectral sequence is a quasi-isomorphism invariant, so that one can
study it under the assumption \eqref{eq:mu1zero}. Then, either $\A$
is equivalent to the trivial $A_\infty$-structure on $A$ and the
spectral sequence degenerates; or else, after using gauge
transformations to make as many higher order terms zero as possible,
one encounters a nontrivial deformation class $m_{\A}^d$ for some
$d>2$. In that case, taking the Euler element $\tau \in HH^1(A,A)^0$
given by the derivation which is $k$ times the identity on $A^k$, one
finds that
\begin{equation} \label{eq:euler-element}
 [m_\A^d,\tau] = (d-2) m_\A^d.
\end{equation}
By \eqref{eq:bracket-differential}, $\delta_{d-1}$ is nonzero and
kills $m_\A^d \in E_{d-1}^{d,2-d} = HH^2(A,A)^{2-d}$.

Having come this far by straightforward adaptation of the previously
used notions, we encounter a slight snag, which is that the
deformation problem governed by $HH^*(\A,\A)$ reaches beyond the
class of $A_\infty$-algebras into that of ``curved'' or
``obstructed'' ones. To exclude these additional deformations, one
can proceed as follows. Let $\A$ be an $A_\infty$-algebra with
$\mu^1_{\A} = 0$. The {\em truncated Hochschild cohomology} is a
graded vector space $HH^*(\A,\A)^{\scriptscriptstyle \leq 0}$, built
from the subcomplex $CC^*(\A,\A)^{\scriptscriptstyle \leq 0}$ of
Hochschild cochains $\tau: \A^{\otimes s} \rightarrow \A[t]$ with $t
\leq 0$. There is a spectral sequence leading to it as before, where
now
\begin{equation} \label{eq:truncated-ss}
 E_2^{s,t} = \begin{cases}
 HH^{s+t}(A,A)^t & \text{$t \leq 0$}, \\
 0 & \text{otherwise.}
 \end{cases}
\end{equation}
All remarks made above carry over to this modification, including
\eqref{eq:euler-element} because the Euler element lies in
\eqref{eq:truncated-ss}. For the application to deformation theory,
consider the subset $\AA_q(\A)^{\scriptscriptstyle \leq 0} \subset
\AA_q(\A)$ of those one-parameter deformations $\A_q$ during which
the differential remains trivial, $\mu^1_{\A_q} = 0$. This still
carries actions of $\GG_q(\A)$ and of $End(\Lambda_\N)$. The order
$q$ term of a deformation of this kind defines a primary deformation
class
\[
m_{\A_q}^{\scriptscriptstyle \leq 0} = [\mu_{\A_q,1}^*] \in
HH^2(\A,\A)^{\scriptscriptstyle \leq 0},
\]
which is the obstruction to transforming $\A_q$ into a deformation of
order $q^2$ (or in other words, making the associated first order
deformation trivial). The desired versality result is:

\begin{lemma} \label{th:versal-3}
Suppose that $HH^2(\A,\A)^{\scriptscriptstyle \leq 0} \iso \C$. Let
$\A_q$ be a one-parameter deformation with $\mu^1_{\A_q} = 0$, whose
primary deformation class is nonzero. Then $\A_q$ is versal in
$\AA_q(\A)^{\scriptscriptstyle \leq 0}/\GG_q(\A)$.
\end{lemma}

Take an arbitrary $A_\infty$-algebra $\B$, and use the Perturbation
Lemma to find an $\A$ with $\mu^1_\A = 0$ and a quasi-isomorphism
$\F: \A \rightarrow \B$. Suppose that $\B_q$ is a one-parameter
deformation of $\B$ such that $H(\B_q)$ is torsion-free. Then
$H(\B_q) \iso H(\B) \hat\otimes \Lambda_\N$ non-canonically, and a
straightforward extension of the Perturbation Lemma allows one to
construct a one-parameter deformation $\A_q$ of $\A$ with vanishing
differential, and an $\F_q: \A_q \rightarrow \B_q$ which deforms
$\F$. This in fact yields a bijection
\begin{equation} \label{eq:deformation-comparison}
 \frac{\AA_q(\A)^{\scriptscriptstyle \leq 0}}{\GG_q(\A)} \iso
 \frac{\{\B_q \in \AA_q(\B), \text{$H(\B_q)$ torsion-free}\}}{\GG_q(\B)}.
\end{equation}

\subsection{}
Lemma \ref{th:versal-1} is essentially sufficient for our needs, but
Lemma \ref{th:versal-3} is not, since
$HH^2(\A,\A)^{\scriptscriptstyle \leq 0}$ turns out to be too large
in our example. This means that we need to restrict the class of
allowed deformations further, so as to whittle down the dimension of
the relevant ``moduli space'' to one. The solution adopted here
(largely motivated by the desire to minimize work on the symplectic
geometry side later on) is to consider $A_\infty$-algebras equipped
with an automorphism, and deformations of those. In fact, to turn
this into a standard deformation problem, we replace the automorphism
by its graph, which is an $A_\infty$-bimodule.

Some background (see \cite{tradler01} for a more detailed
exposition): to any $A_\infty$-algebra $\A$ one can associate the dg
category $\bimodA$ of $A_\infty$-bimodules. An object is a graded
vector space $\M$ with operations $\mu^{f,e}_\M: \A^{\otimes f}
\otimes \M \otimes \A^{\otimes e} \rightarrow \M[1-e-f]$, for all
$e,f \geq 0$, satisfying certain structural equation, and also
subject to a certain c-unitality condition (see below). The cochain
space of morphisms $\M \rightarrow \NN$ of degree $k$ is the space of
sequences of multilinear maps $\beta^{f,e}: \A^{\otimes f} \otimes \M
\otimes \A^{\otimes e} \rightarrow \NN[k-e-f]$. Note that
$\mu_\M^{*,*}$ itself is an element in $hom^1(\M,\M)$. To give
explicit formulae for the dg structure, it is best to start with
composition of morphisms, which is somewhat similar to the
Gerstenhaber product:
\begin{multline*}
 (\gamma \circ \beta)^{f,e}
 (b_f,\dots,b_1,m,a_e,\dots,b_1) = \\ =
 \sum_{i,j} (-1)^\dag \gamma^{f-j,i}(b_f,\dots,b_{j+1},
 \beta^{j,e-i}(b_j,\dots,m,\dots,a_{i+1}),a_i,\dots,a_1),
\end{multline*}
where $\dag = |\beta|(|a_i|+\dots+|a_1|-i)$. We also have a
preliminary version of the differential, involving only the
$A_\infty$-structure of $\A$:
\begin{multline*}
 (-1)^{|\beta|+1}(\delta_{\text{pre}}\beta)^{f,e}(b_f,\dots,b_1,m,a_e,\dots,a_1) = \\ =
 \sum_{i,j} (-1)^{\ast}
 \beta^{f,e-j+1}(b_f,\dots,m,\dots,a_{i+j+1},\mu^j_\A(a_{i+j},\dots,a_{i+1}),a_i,\dots,a_1) \\ +
 \sum_{i,j} (-1)^{\natural}
 \beta^{f-j+1,e}(b_f,\dots,b_{i+j+1},\mu^j_\A(b_{i+j},\dots,b_{i+1}),b_i,\dots,m,\dots,a_1),
\end{multline*}
with $\ast$ as in \eqref{eq:ainfty} and $\natural = |a_1| + \dots +
|a_e| + |m| + |b_1| + \dots + |b_i| - e - 1 - i$. This satisfies
$\delta_{\text{pre}}^2 = 0$ and $\delta_{\text{pre}}(\gamma \circ
\beta) = \delta_{\text{pre}}(\gamma) \circ \beta + (-1)^{|\gamma|}
\gamma \circ \delta_{\text{pre}}(\beta)$. Having that, one can write
the structural equations of an $A_\infty$-bimodule as
\begin{equation} \label{eq:module-equation}
\delta_{\text{pre}}\mu^{*,*} + \mu^{*,*} \circ \mu^{*,*} = 0,
\end{equation}
and the actual differential on $hom(\M,\NN)$ as $\delta \beta =
\delta_{\text{pre}}\beta + \mu_\NN^{*,*} \circ \beta - (-1)^{|\beta|}
\beta \circ \mu_\M^{*,*}$. The cohomology $M = H(\M,\mu_\M^{0,0})$ of
any $A_\infty$-bimodule is naturally a graded bimodule over $A =
H(\A)$, and we require that the unit of $A$ acts as the identity from
both sides. By using the length filtration of
$hom_{\bimodA}(\M,\NN)$, one gets a spectral sequence whose $E_1$
term can be identified with $E_1^{s,t} =
Hom_{A-A}(B^{-s}_{A-A}M,N[t])$, where $B^*_{A-A}M$ is the classical
bimodule bar resolution. This is compatible with the differential, so
that
\begin{equation} \label{eq:bimodule-ss}
 E_2^{s,t} = Ext_{A-A}^s(M,N[t]).
\end{equation}
In particular, if $\M$ or $\NN$ are acyclic, then so is
$hom_{\bimodA}(\M,\NN)$. If we allow ourselves to loosely borrow
language from from \cite{spaltenstein88}, the fact that the bar
resolution is built into the definition of our category ensures that
``all $A_\infty$-bimodules are $K$-projective (and $K$-injective)''.
It follows that any quasi-isomorphism $\M \rightarrow \NN$ has a
quasi-inverse, hence is an isomorphism in the category
$H^0(\bimodA)$.

There is a ``coupled Hochschild cohomology'' theory associated to a
pair $(\A,\M)$ consisting of an $A_\infty$-algebra and an
$A_\infty$-bimodule over it. It sits in a long exact sequence
\begin{multline} \label{eq:coupled-sequence}
 \quad \cdots \longrightarrow HH^{*-1}(\A,\A) \stackrel{d}{\longrightarrow}
 H^{*-1}(hom_{\bimodA}(\M,\M)) \stackrel{a}{\longrightarrow} \\ \longrightarrow
 HH^*(\A \ampersand \M, \A \ampersand \M) \stackrel{b}{\longrightarrow}
 HH^*(\A,\A) \longrightarrow \cdots \quad
\end{multline}
Later on, when we consider it from the point of view of deformation
theory, the meaning of the sequence will be clear: $b$ forgets $\M$
and concentrates on the deformation of $\A$, while the image of $a$
consists of deformations of $\M$ with respect to a fixed $\A$. To
define $HH^*(\A \ampersand \M,\A \ampersand \M)$ one can use the
triangular matrix trick, see for instance \cite{guccione-guccione01}.
Define an $A_\infty$-algebra
\begin{equation} \label{eq:t}
\T = \begin{pmatrix} \T_{11} & \T_{12} \\ \T_{21} & \T_{22}
\end{pmatrix} = \begin{pmatrix} \A & 0 \\ \M & \A \end{pmatrix}
\end{equation}
whose underlying graded vector space is the direct sum of two copies
of our $A_\infty$-algebra, $\T_{11} = \T_{22} = \A$, with one copy of
the bimodule, $\T_{12} = \M$. The only nonzero compositions are
\begin{align*}
 & \mu^d_\T = \mu^d_\A: \T_{kk}^{\otimes d} \longrightarrow \T_{kk},
 \;\;\quad k = 1,2, \\
 & \mu_\T^{e+f+1} = \mu^{f,e}_\M: \T_{22}^{\otimes f} \otimes \T_{21} \otimes \
 \T_{11}^{\otimes e} \longrightarrow \T_{21}.
\end{align*}
Consider the subcomplex $CC^*(\A \ampersand \M, \A \ampersand \M)
\subset CC^*(\T,\T)$ of cochains of the same form, meaning that they
are made up of maps $\T_{kk}^{\otimes d} \rightarrow \T_{kk}$, which
should be the same for $k = 1,2$, and maps $\T_{22}^{\otimes f}
\otimes \T_{21} \otimes \T_{11}^{\otimes e} \rightarrow \T_{21}$. The
cochains of the second kind form a subcomplex of $CC^*(\A \ampersand
\M, \A \ampersand \M)$ which is isomorphic to $hom(\M,\M)[-1]$, and
the quotient complex is just $CC^*(\A,\A)$, which makes
\eqref{eq:coupled-sequence} clear.

The bimodules that we will be interested in are graphs $\M =
Graph(\F)$ of $A_\infty$-homomorphisms $\F: \A \rightarrow \A$. The
underlying graded vector space of $\M$ is the same as for $\A$, and
the operations are
\begin{multline*}
 \mu^{f,e}_\M(b_f,\dots,b_1,m,a_e,\dots,a_1) = \\ = \!\!
 \sum_{i_1 + \dots + i_r = f} \!\!
 \mu_\A^{f+r+1}(b_f,\dots,b_1,m,\F^{i_r}(a_e,\dots,a_{e-i_r+1}),
 \dots,\F^{i_1}(a_{i_1},\dots,a_1)).
\end{multline*}

\begin{lemma} \label{th:graph}
For a quasi-isomorphism $\F$, one can map $H^*(hom_{\bimodA}(\M,\M))
\iso HH^*(\A,\A)$ in such a way that the differential $d$ in
\eqref{eq:coupled-sequence} becomes $id-\F^*$, where $\F^*$ is the
automorphism of Hochschild cohomology induced by $\F$.
\end{lemma}

Our proof of this will be rather pedestrian, but a more satisfactory
treatment requires all the machinery of $A_\infty$-categories of
$A_\infty$-functors, for which we have no real use elsewhere. To get
started, we need to recall how quasi-isomorphism invariance of
Hochschild cohomology for $A_\infty$-algebras is proved. The most
obvious scheme does not apply because $HH^*$ is neither co- nor
contravariantly functorial, and instead one passes through a
generalization, which is the {\em functor Hochschild cohomology}
$HH^*(\F,\F)$ of an $A_\infty$-homomorphism $\F: \A \rightarrow \B$.
We will not give the full definition of this, but the underlying
cochain complex is $CC^d(\F,\F) = \prod_{s+t = d} Hom(\A^{\otimes
s},\B[t])$, and the $E_2$ term of the filtration-by-length spectral
sequence is $Ext^s_{A-A}(A,B[t])$, where $B$ is considered as an
$A$-bimodule through pullback by $F = H(\F)$. There are natural maps
 $\F^{*,\text{left}}: HH^*(\A,\A) \rightarrow HH^*(\F,\F)$,
 $\F^{*,\text{right}}: HH^*(\B,\B) \rightarrow HH^*(\F,\F)$
defined on the cochain level by
\begin{align*}
 & (\F^{*,\text{left}}\tau)(a_d,\dots,a_1) =
 \sum_{i,j} (-1)^\sharp
 \F^{d-j+1}(a_d,\dots,\tau^j(a_{i+j},\dots,a_{i+1}),\dots,a_1), \\
 &
 (\F^{*,\text{right}}\tau)(a_d,\dots,a_1) = \\ & \qquad \qquad \qquad =
 \sum_{i_1 + \dots + i_r = d}
 \tau^r(\F^{i_r}(a_d,\dots,a_{d-i_r+1}),\dots,\F^{i_1}(a_{i_1},\dots,a_1)).
\end{align*}
with $\sharp$ as in \eqref{eq:gerstenhaber}. If $\F$ is a
quasi-isomorphism, a spectral sequence comparison argument shows that
both maps are isomorphisms. One then defines $\F^*$ to be the
composition $(\F^{*,\text{left}})^{-1} \circ \F^{*,\text{right}}:
HH^*(\B,\B) \rightarrow HH^*(\A,\A)$.

Suppose from now on that $\B = \A$, so we are in the situation of
Lemma \ref{th:graph}, and $\F^*$ is the automorphism of Hochschild
cohomology induced by $\F$. There is a natural isomorphism $c:
HH^*(\F,\F) \rightarrow H^*(hom_{\bimodA}(\M,\M))$ for $\M =
Graph(\F)$, given on cochains by
\begin{multline*}
 (c\tau)^{f,e}(b_f,\dots,b_1,m,a_e,\dots,a_1) =
 \\ = \sum_{\substack{k \leq r \\ i_1 + \dots + i_r = e}}
 (-1)^\flat \mu_\A^{f+1+r}(b_f,\dots,b_1,m,\F^{i_r}(a_e,\dots,a_{e-i_r+1}),
 \dots, \qquad \\[-1.5em] \tau^{i_k}(a_{i_1+\dots+i_k},\dots,a_{i_1+\dots+i_{k-1}+1}),
 ,\dots,\F^{i_1}(a_{i_1},\dots,a_1)).
\end{multline*}
$\flat = (|\tau|-1)(|a_1|+\dots+|a_{i_{k-1}}|-i_1-\dots-i_{k-1})$. To
prove that this is an isomorphism, one considers the $E_2$ term of
the usual spectral sequences, and checks that the action of $c$ on it
can be traced back to a comparison map between two different
resolutions of the bimodule $M = H(\M)$. Finally, the following
diagram, where $d$ is from \eqref{eq:coupled-sequence}, commutes:
\begin{equation} \label{eq:f-f}
\xymatrix{
 {HH^*(\A,\A)}
 \ar[rr]^-{\F^{*,\text{left}}-\F^{*,\text{right}}}
 \ar@/_1pc/[drr]_-{d}
 && {HH^*(\F,\F)} \ar[d]_{\iso}^{c} \\
 && {H^*(hom_{\bimodA}(\M,\M))}
}
\end{equation}
In view of the definition of $\F^*$, this proves Lemma
\ref{th:graph}. Actually \eqref{eq:f-f} does not strictly commute on
cochains, but the necessary homotopy can be easily provided,
\begin{multline*}
(h\tau)(b_f,\dots,m,\dots,a_1) = \\ = \!\! \sum_{i_1+\dots+i_r = e}
 \!\!\!
 \tau(b_f,\dots,m,\F^{i_r}(a_e,\dots,a_{e-i_r+1}),
 \dots,\F^{i_1}(a_{i_1},\dots,a_1)).
\end{multline*}

\subsection{}
Take a graded algebra $A$, and let $\A$ be an $A_\infty$-structure on
it as in \eqref{eq:mu1zero}. Given a graded $A$-bimodule $M$, we want
to look at $A_\infty$-bimodule structures $\M$ on it such that the
analogue of \eqref{eq:mu1zero} holds,
\begin{equation} \label{eq:m-mu2}
 \mu^{0,0}_\M = 0, \quad
 \mu^{1,0}_\M(a,m) = (-1)^{|a|} ma, \quad
 \mu^{0,1}_\M(m,a) = (-1)^{|m|} am.
\end{equation}

\begin{lemma} \label{th:formal-m}
Suppose that $Ext_{A \otimes A^{op}}^k(M,M[1-k]) = 0$ for all $k \geq
2$. Then, up to isomorphism, there is at most one $\M$ satisfying
\eqref{eq:m-mu2}.
\end{lemma}

It might seem that there is a straightforward ``linear'' approach to
this: take two $A_\infty$-bimodules $\M,\M'$ satisfying
\eqref{eq:m-mu2}, and look at the spectral sequence
\eqref{eq:bimodule-ss} associated to $hom(\M,\M')$. The assumption in
the Lemma ensures that $id_M \in E_2^{0,0}$ survives to $E_\infty$,
and if the spectral sequence converges, one could use a standard
comparison argument to show that the resulting element of
$H^0(hom(\M,\M'))$ is an isomorphism. However, there seems to be no
reason why convergence should hold in general, and our proof will
rely instead on the ``nonlinear'' methods of deformation theory. To
complete our discussion, it is maybe useful to say that there is a
version of the Perturbation Lemma for $A_\infty$-bimodules, which
shows that any such bimodule over an $A_\infty$-algebra $\A$ with
$\mu^1_\A = 0$ is quasi-isomorphic to one with $\mu^{0,0}_\M = 0$,
hence which is of the form \eqref{eq:m-mu2}. To put this on a more
formal footing, consider the dg subcategory
$\bimodA^{\scriptscriptstyle \leq 0} \subset \bimodA$ whose objects
are $A_\infty$-bimodules $\M$ with $\mu^{0,0}_\M = 0$, and where the
morphisms $\beta$ of degree $d$ must satisfy $\beta^{p,q} = 0$ for
$p+q < d$. By definition, the functor induced by inclusion is a full
embedding $H^0(\bimodA^{\scriptscriptstyle \leq 0}) \rightarrow
H^0(\bimodA)$. The bimodule Perturbation Lemma shows that it is an
equivalence.

We now turn to the ``coupled'' deformation problem which is our main
interest in this matter. Let $A$, $\A$ and $M$ be as before, and
suppose that we have some fixed $A_\infty$-bimodule structure $\M$
satisfying \eqref{eq:m-mu2}. Let $\AA_q(\A \ampersand
\M)^{\scriptscriptstyle \leq 0}$ be the set of pairs $(\A_q,\M_q)$
where $\A_q$ is a one-parameter deformation of $\A$, and $\M_q$ a
corresponding one-parameter deformation of $\M$ (that is, an
$\A_q$-bimodule structure on $\M \hat{\otimes}_\C \Lambda_\N$ having
the obvious kind of $q$-expansion), such that
\begin{equation} \label{eq:truncated-pair}
 \mu^1_{\A_q} = 0, \quad \mu^{0,0}_{\M_q} = 0.
\end{equation}
The formal gauge group $\GG_q(\A \ampersand \M)$ for this problem can
be defined as a suitable subgroup of the corresponding group for
deformations of the triangular algebra $\T$. For us, what matters is
that gauge equivalence of pairs implies equivalence of the
$A_\infty$-deformations $\A_q$, and we will later state the versality
result in a correspondingly weak form. Let $HH^*(\A \ampersand \M,\A
\ampersand \M)^{\scriptscriptstyle \leq 0}$ be the truncated version
of coupled Hochschild cohomology, defined by looking only at cochains
$\tau_{11} = \tau_{22}: \A^{\otimes s} \rightarrow \A[t]$,
$\tau_{12}: \A^{\otimes f} \otimes \M \otimes \A^{\otimes e}
\rightarrow \M[t]$ with $t \leq 0$. This sits in a suitable version
of \eqref{eq:coupled-sequence},
\begin{multline} \label{eq:coupled-sequence-2}
 \dots \longrightarrow HH^{*-1}(\A,\A)^{\leq 0} \xrightarrow{d}
 H^{*-1}(hom_{\bimodA^{\scriptscriptstyle\leq 0}}(\M,\M)) \longrightarrow
 \\ \rightarrow
 HH^*(\A \ampersand \M, \A \ampersand \M)^{\scriptscriptstyle \leq 0} \longrightarrow
 HH^*(\A,\A)^{\scriptscriptstyle \leq 0} \xrightarrow{d} \dots
\end{multline}
In the case where $\M = Graph(\F)$, the obvious variation of Lemma
\ref{th:graph} applies. Any pair $(\A_q,\M_q) \in \AA_q(\A \ampersand
\M)^{\scriptscriptstyle \leq 0}$ has a primary deformation class in
truncated coupled Hochschild cohomology. Again, the important thing
for us is that this maps to the deformation class of $\A_q$ under
\eqref{eq:coupled-sequence-2}.

\begin{lemma} \label{th:versal-4}
Suppose that $HH^2(\A \ampersand \M,\A \ampersand
\M)^{\scriptscriptstyle \leq 0} \iso \C$, and that there is a
deformation $(\A_q,\M_q)$ satisfying \eqref{eq:truncated-pair} with
nonzero $m_{\A_q}^{\scriptscriptstyle \leq 0} \in
HH^2(\A,\A)^{\scriptscriptstyle \leq 0}$. Then, for any other pair
$(\A_q',\M_q') \in \AA_q(\A \ampersand \M)^{\scriptscriptstyle \leq
0}$, there is a $\psi \in End(\Lambda_\N)$ such that $\A_q' \iso
\psi^*\A_q$ as a one-parameter deformation of $A_\infty$-algebras.
\end{lemma}

\subsection{}
The framework of abstract one-parameter deformation theory is as
follows. Let $\g$ be a dg Lie algebra. We consider the space
$\AA_q(\g)$ of elements $\alpha_q \in \g^1 \hat{\otimes}\,
q\Lambda_\N$ which satisfy the Maurer-Cartan equation
\begin{equation} \label{eq:classical-mc}
 \delta\alpha_q + \half [\alpha_q,\alpha_q] = 0.
\end{equation}
In particular, if we expand $\alpha_q = \alpha_{q,1}q + \alpha_{q,2}
q^2 + \cdots$, the leading order term defines a primary deformation
class $[\alpha_{q,1}] \in H^1(\g)$. An infinitesimal abstract gauge
transformation $\gamma_q \in \g^0 \hat{\otimes} q\Lambda_\N$ defines
a formal vector field $X_q$ on $\AA_q(\g)$, $X_q(\alpha_q) =
\delta\gamma_q + [\alpha_q,\gamma_q]$. These vector fields
exponentiate to an action of a group $\GG_q(\g)$ on $\AA_q(\g)$,
which we denote by $\exp(\gamma_q)(\alpha_q)$. The primary
deformation class is an invariant of this action. As usual, there is
also an action of $End(\Lambda_\N)$ on $\AA_q(\g)$ by
reparametrizations, hence a notion of versality. The abstract
formality and versality theorems are

\begin{lemma} \label{th:formal-abstract}
If $H^1(\g) = 0$, $\AA_q(\g)/\GG_q(\g)$ is a point.
\end{lemma}

\begin{lemma} \label{th:versal-abstract}
Suppose that $H^1(\g) \iso \C$. Then, every $\alpha_q \in \AA_q(\g)$
whose primary deformation class is nonzero is versal.
\end{lemma}

These are both standard results, but for the convenience of the
reader we reproduce the conventional induction-by-order proof of the
second one. Take an arbitrary $\beta_q \in \AA_q(\g)$. Suppose that
for some $d \geq 1$, we have $\psi^d \in End(\Lambda_\N)$ and
$\beta^d_q \in \AA_q(\g)$ gauge equivalent to $\beta_q$, such that
\begin{equation} \label{eq:discrepancy}
 \beta^d_q - (\psi^d)^*\alpha_q \stackrel{\text{def}}{=} \epsilon_q \in O(q^d).
\end{equation}
Because $\delta\epsilon_q = \half
[(\psi^d)^*\alpha_q+\beta^d_q,\epsilon_q] \in O(q^{d+1})$, the
leading order term $\epsilon_{q,d} \in \g^1$ is closed. By
assumption, one can therefore write $\beta^d_q = (\psi^d)^*\alpha_q +
\epsilon_q = (\psi^d)^* \alpha_q + c q^d \alpha^{d,1} + q^d \delta b
+ O(q^{d+1})$ for some $c \in \C$, $b \in \g^0$. With
\begin{equation} \label{eq:new-approximation}
 \psi^{d+1}(q) = \psi^d(q) + c q^d, \quad
 \beta^{d+1}_q = \exp(-q^d b)(\beta^d_q)
\end{equation}
one finds that \eqref{eq:discrepancy} holds for $d+1$ as well. Since
the parameter change and gauge transformation in
\eqref{eq:new-approximation} are equal to the identity up to
$O(q^d)$, the process converges to $(\psi^\infty)^*\alpha_q =
\beta^\infty_q$, with $\beta^\infty_q$ gauge equivalent to $\beta_q$.

When applied to $\g_\A = CC^{*+1}(\A,\A)^{\scriptscriptstyle \leq
0}$, this yields Lemma \ref{th:versal-2}. Similarly, for the coupled
case one sets $\g_{\A \ampersand \M} = CC^{*+1}(\A \ampersand \M,\A
\ampersand \M)^{\scriptscriptstyle \leq 0}$ and obtains Lemma
\ref{th:versal-4}. The other results stated above, which lack the
parameter $q$, require modifications of the basic framework. Let $A$
be a graded algebra, and $\g_A = CC^{*+1}(A,A)$ its shifted
Hochschild complex with the Gerstenhaber bracket. One can reformulate
the bigrading of Hochschild cohomology as a $\C^*$-action on $\g_A$,
which has weight $t$ on $CC^{s+t}(A,A)^t$. We combine this with the
$\C^*$-action on $\Lambda_\N$ which has weight $d$ on $\C q^d$, and
obtain an action on $\g_A \otimes q\Lambda_\N$. Given an
$A_\infty$-structure $\A$ on $A$ satisfying \eqref{eq:mu1zero}, one
can define a one-parameter deformation $\A_q$ of the trivial
$A_\infty$-structure by $\mu^d_{\A_q} = q^{d-2}\mu^d_\A$, and this is
given by a $\C^*$-invariant element of $\g_A \otimes q\Lambda_\N$.
The only possible changes of parameter in this context are the
homogeneous ones $q \mapsto \epsilon q$, $\epsilon \in \C^*$, and
with that in mind the proof of Lemma \ref{th:versal-1} is by a
straightforward adaptation of Lemma \ref{th:versal-abstract}. In
contrast, Lemma \ref{th:versal-3} requires a more substantial
tweaking of the theory, where the basic object one considers is a dg
Lie algebra $\g$ with a complete decreasing filtration $F_0\g = \g
\supset F_1\g \supset \dots$ such that $\delta(F_i\g) \subset
F_{i+1}\g$ and $[F_j\g,F_k\g] \subset F_{j+k}\g$. Take a solution
$\alpha \in (F_1\g)^1$ of \eqref{eq:classical-mc}, and consider the
corresponding deformed differential
\[
\delta_\alpha = \delta + [\alpha,\cdot].
\]
There is a series of obstruction groups, which are the terms
$E_2^{d,1-d}$, $d \geq 2$, in the spectral sequence leading to the
cohomology of $(\g,\delta_\alpha)$. A slightly refined version of the
computation in the proof of Lemma \ref{th:versal-abstract} shows that
if all of them vanish, then any other solution of
\eqref{eq:classical-mc} with $\beta-\alpha \in F_2\g$ is equivalent
to $\alpha$ by a gauge transformation, obtained by exponentiating the
action of $(F_1\g)^0$. For our purpose, one applies this to $\g =
\bigoplus_{e,f} Hom^*(A^{\otimes e} \otimes M \otimes A^{\otimes
f},A)$, with $\delta$ and the bracket as suggested by
\eqref{eq:module-equation}, and $F_i\g$ the filtration by length $i =
e+f$. The obstruction groups are precisely those which occur in Lemma
\ref{th:versal-3}.

\subsection{}
The previous discussion generalizes effortlessly to
$A_\infty$-categories. For didactic reasons, start with an ordinary
category $C$ having finitely many objects $X_1,\dots,X_m$, and let
$A$ be its total morphism algebra, which is linear over $R_m$.
Consider Hochschild cochains that preserve this structure, which
means that they are $R_m$-bimodule maps $A \otimes_{R_m} \dots
\otimes_{R_m} A \rightarrow A$. By writing this out, one sees that
the relevant complex can be generalized to arbitrary categories,
leading to a Hochschild cohomology theory $HH^*(C,C)$. The same
applies to Hochschild cohomology of $A_\infty$-categories, and all
the other notions of deformation theory.

\begin{remark}
For any category $C$ with finitely many objects, the Hochschild
cohomology $HH^*(C,C)$ is the same as $HH^*(A,A)$ of the total
morphism algebra. In other words, the inclusion of subcomplex of
$R_m$-bilinear Hochschild cochains into the whole cochain complex is
a quasi-isomorphism. The reason is that the relative bar resolution
\[
\cdots \longrightarrow A \otimes_{R_m} A \otimes_{R_m} \otimes A
\longrightarrow A \otimes_{R_m} A \longrightarrow A
\]
is also a resolution by projective $A$-bimodules in the non-relative
sense. By quasi-isomorphism invariance of abstract deformation
theories, this shows that any $A_\infty$-structure $\A$ on $A$
satisfying \eqref{eq:mu1zero} comes, up to gauge equivalence, from an
$A_\infty$-category structure $\Cat$ on $C$. The previous observation
combined with the standard spectral sequence argument also proves
that for any finite $A_\infty$-category $\Cat$ with total morphism
$A_\infty$-algebra $\A$, $HH^*(\Cat,\Cat) \iso HH^*(\A,\A)$, and
similarly for the truncated version. This implies that the
one-parameter deformation theory of $\A$ as an $A_\infty$-algebra is
the same as that of $\Cat$ as an $A_\infty$-category. On occasion,
these facts will allow us to be somewhat sloppy about the distinction
between finite categories and their total morphism algebras.
\end{remark}

To make the connection with Section \ref{sec:generators}, we should
relate the deformation theory of $\Cat$ with that of its derived
categories. However, this cannot be done in a truly satisfactory way
within the restrictions of the present paper, because we have not
introduced an $A_\infty$-category underlying $D^\pi(\Cat)$, and so we
will limit ourselves to a single rudimentary statement, Let $\Cat_q$
be a one-parameter deformation of $\Cat$. Suppose that we have a
finite set $(X_1,\dots,X_m)$ of objects of $\Cat$ which are
split-generators for $D^\pi(\Cat)$. Let $\A$ be their total morphism
$A_\infty$-algebra, and $\A_q$ its one-parameter deformation induced
from $\Cat_q$.

\begin{lemma} \label{th:deformation-generators}
Assume that there are $Y_0,Y_1 \in Ob\, \Cat$ such that
$H(hom(Y_0,Y_1))$ changes at first order in the deformation. More
precisely, this means that the natural map
\begin{equation} \label{eq:q2}
 H(hom_{\Cat_q}(Y_0,Y_1) \otimes_{\Lambda_\N} \C[q]/q^2, \mu^1_{\Cat_q}) \otimes_{\C[q]/q^2} \C
 \longrightarrow H(hom_\Cat(Y_0,Y_1),\mu^1_\Cat)
\end{equation}
is not an isomorphism. Then $\A_q$ is nontrivial at first order in
$q$.
\end{lemma}

\proof Suppose that the statement is false. After an $R_m$-linear
gauge transformation, we can assume that $\A_q$ is a deformation of
order $q^2$. Then the same holds for the induced one-parameter
deformation $Tw\A_q$ of $Tw\A$. But the natural $A_\infty$-functor
$\A_q \rightarrow \Cat_q$ extends to a $\Lambda_\N$-linear
$A_\infty$-functor $Tw\F_q: Tw\A_q \rightarrow Tw\Cat_q$, which is
full and faithful on cohomology. This means that \eqref{eq:q2} must
be an isomorphism for all objects $Y_0,Y_1$ which are twisted
complexes constructed from the $X_k$. Since that property carries
over to direct summands, we have a contradiction. \qed

\section{Group actions\label{sec:group-actions}}

Semidirect products $\Lambda V \semidirect \Gamma$ of an exterior
algebra with a finite abelian group appear naturally in the context
of homological mirror symmetry for hypersurfaces in projective
spaces. This is a consequence of Serre duality and Beilinson's
description of $D^bCoh(\CP{n})$ \cite{beilinson78}. The Hochschild
cohomology of such algebras can be computed by a version of the
classical Hochschild-Kostant-Rosenberg (HKR) theorem
\cite{hochschild-kostant-rosenberg62}, and this will later form the
basis for applying the deformation theory from the previous section.
Besides proving that theorem, we also use the opportunity to assemble
some elementary facts about semidirect products, both of algebras and
$A_\infty$-algebras. The exposition is by no means as general as
possible, in particular, we shamelessly restrict ourselves to abelian
groups whenever this makes things simpler. Dissatisfied readers may
want to turn to \cite{montgomery92} and references therein.

\subsection{\label{subsec:actions-generalities}}
Let $A$ be a graded algebra carrying an action $\rho$ of a finite
group $\Gamma$. The semidirect product $A \semidirect \Gamma$ is the
tensor product $A \otimes \C \Gamma$ with the twisted multiplication
\[
 (a_2 \otimes \gamma_2) (a_1 \otimes \gamma_1) =
 a_2 \rho(\gamma_2)(a_1) \otimes \gamma_2\gamma_1.
\]

The purpose of this construction is to turn the automorphisms
$\rho(\Gamma)$ into inner ones: in fact, $(1 \otimes \gamma) (a
\otimes e) (1 \otimes \gamma)^{-1} = \rho(\gamma)(a) \otimes e$.
Therefore, it is not surprising that $A \semidirect \Gamma$ depends
only on the ``outer part'' of the group action, in the following
sense: suppose that $\tilde{\rho}$ is another action related to the
previous one by
\begin{equation} \label{eq:conjugate-action}
 \tilde{\rho}(\gamma) = \iota(\gamma) \rho(\gamma) \iota(\gamma)^{-1},
\end{equation}
where $\iota: \Gamma \rightarrow (A^0)^\times$ is a map satisfying
$\iota(\gamma_2\gamma_1) = \iota(\gamma_2)
\rho(\gamma_2)(\iota(\gamma_1))$. Then $A \semidirect_\rho \Gamma$
and $A \semidirect_{\tilde\rho} \Gamma$ are isomorphic by $a \otimes
\gamma \mapsto a \iota(\gamma)^{-1} \otimes \gamma$. This kind of
consideration appears naturally in connection with the following
construction: suppose that there is a normal subgroup $\Gamma_0
\subset \Gamma$ such that the projection $\Gamma \rightarrow
\Gamma/\Gamma_0$ admits a right inverse $\Sigma$. Then $\Gamma \iso
\Gamma_0 \semidirect (\Gamma/\Gamma_0)$, and there is a corresponding
isomorphism
\begin{equation} \label{eq:in-stages}
 A \semidirect \Gamma
 \iso (A \semidirect \Gamma_0) \semidirect (\Gamma/\Gamma_0).
\end{equation}
More precisely, let $\rho_\Gamma$ be the given $\Gamma$-action, and
$\rho_{\Gamma_0}$ its restriction. Through the splitting $\Sigma$ we
have an action $\rho_{\Gamma/\Gamma_0}$ of the quotient on $A
\semidirect \Gamma_0$:
\begin{equation} \label{eq:quotient-action}
 \rho_{\Gamma/\Gamma_0}(\bar\gamma)(a \otimes \gamma_0) =
 \rho_{\Gamma}(\Sigma(\bar\gamma))(a) \otimes
 \Sigma(\bar\gamma) \gamma_0 \Sigma(\bar\gamma)^{-1},
\end{equation}
and these are the three actions involved in \eqref{eq:in-stages}.
Different choices of $\Sigma$ lead to actions
\eqref{eq:quotient-action} which differ by inner automorphisms as in
\eqref{eq:conjugate-action}, and the discussion there shows that the
right hand side of \eqref{eq:in-stages} is independent of the choice,
as indeed it ought to be.

Any semidirect product $A \semidirect \Gamma$ carries a natural
action $\rho_{\Gamma^*}$ of the group of scalar characters $\Gamma^*
= Hom(\Gamma,\C^*)$; $\rho_{\Gamma^*}(\chi)(a \otimes g) = a\chi(g)
\otimes g$. Additional automorphisms can be produced from those of
$A$. We have already seen an instance of this in
\eqref{eq:quotient-action}, and the general construction goes as
follows: let $U$ be an automorphism of $A$, and suppose that there is
an automorphism $\xi$ of $\Gamma$ as well as a map $\iota: \Gamma
\rightarrow (A^0)^\times$ satisfying $\iota(\gamma_2\gamma_1) =
\rho(\xi(\gamma_2))(\iota(\gamma_2)) \cdot \iota(\gamma_1)$, such
that
\begin{equation} \label{eq:inner-twist}
 \rho(\xi(\gamma))(U(a))
 = \iota(\gamma) U(\rho(\gamma)(a)) \iota(\gamma)^{-1}
\end{equation}
holds. Informally, this means that $U$ commutes with the group
actions in a twisted sense given by $\xi$, and up to inner
automorphism which are conjugations with the $\iota$'s. Then one can
define an induced automorphism of $A \semidirect \Gamma$ by
\begin{equation} \label{eq:twisted-u}
 (U \semidirect \xi)(a \otimes \gamma) = U(a)\iota(\gamma)^{-1}
 \otimes \xi(\gamma).
\end{equation}
Of course, $\iota $ is not unique: for instance, passing from
$\iota(\gamma)$ to $\iota(\gamma)\chi(\gamma)$ for some $\chi \in
\Gamma^*$ amounts to composing $U \semidirect \xi$ with
$\rho_{\Gamma^*}(\chi)$.

In a slightly different direction, note that $A \semidirect \Gamma$
is linear over the semisimple ring $\C \Gamma$. For simplicity,
assume that $\Gamma$ is abelian, so that $\C \Gamma \iso
R_{|\Gamma|}$ has a basis of orthogonal idempotents, which are the
normalized characters $u_\chi = |G|^{-1}\chi$. This means that we can
consider $A \semidirect \Gamma$ as a category with objects $\chi$ and
morphism spaces
\[
 Hom(\chi_1,\chi_2) = u_{\chi_2} (A \semidirect \Gamma)
 u_{\chi_1} = A^{\chi_1/\chi_2} \otimes \C u_{\chi_1}.
\]
where $\chi_2/\chi_1$ is the (pointwise) division in $\Gamma^*$, and
$A^{\chi}$ is the subspace on which $G$ acts as on the representation
corresponding to $\chi$. This suggests an obvious generalization,
which is to consider an action of an abelian group $\Gamma$ on a
linear graded category $C$; here and in related discussions below,
the standing assumption is that the action on the set of objects is
trivial. Then, the semidirect product category $C \semidirect \Gamma$
has objects $(X,\chi) \in Ob\,C \times \Gamma^*$, and morphisms
$Hom((X_1,\chi_1),(X_2,\chi_2)) = Hom(X_1,X_2)^{\chi_1/\chi_2}$. By
construction, if $C$ has only finitely many objects and $A$ it its
total morphism algebra, then $A \semidirect \Gamma$ is the total
morphism algebra of $C \semidirect \Gamma$.

\subsection{\label{subsec:ainfty-actions}}
A $\Gamma$-action on an $A_\infty$-algebra $\A$ is a linear action on
$A$, such that all the $\mu_\A^d$ are equivariant. The semidirect
product $A_\infty$-algebra $\A \semidirect \Gamma$ is then defined as
$\A \otimes \C\Gamma$ with the compositions
\begin{multline*}
 \qquad \mu_{\A \semidirect \Gamma}^d(a_d \otimes \gamma_d, \dots, a_1 \otimes \gamma_1) = \\ =
 \mu_\A^d(a_d, \rho(\gamma_d)(a_{d-1}), \rho(\gamma_d\gamma_{d_1})(a_{d-2}), \dots)
 \otimes \gamma_d \cdots \gamma_1. \qquad
\end{multline*}
This generalizes to $A_\infty$-categories with $\Gamma$-actions; if
there are only finitely many objects, these are the same as
$R_m$-linear $\Gamma$-actions on $R_m$-linear $A_\infty$-algebras.
Most of the discussion above applies to the $A_\infty$-case as well,
the only problematic parts being those involving invertible elements
and the associated inner automorphisms, since that notion does not
carry over in a simple way. In the $R_m$-linear situation there is at
least a rudimentary substitute. Namely, for $r = (r_1,\dots,r_m) \in
I = (\C^*)^m$ and $a \in u_j A u_k$, define $r \cdot a = r_j a$ and
$a \cdot r = a r_k$. In a slight abuse of terminology, we call $a
\mapsto r \cdot a \cdot r^{-1}$ an {\em elementary inner
automorphism}. One can use these instead of actual $xax^{-1}$'s in
formulae like \eqref{eq:conjugate-action}, \eqref{eq:inner-twist},
and this gives a valid $A_\infty$-version of the relevant
observations for algebras, even though it is so reduced in scope as
to be basically trivial.

Given an $A_\infty$-category $\Cat$ with a $\Gamma$-action, one can
define the $A_\infty$-category $Tw_{\Gamma} \Cat$ of equivariant
twisted complexes. This also carries a $\Gamma$-action, and if we
forget that action, it becomes equivalent to a full
$A_\infty$-subcategory of the ordinary $Tw\,\Cat$. The construction
requires the following changes in the two basic steps: (1) the
equivariant version $\Sigma_\Gamma\Cat$ of the additive completion
has objects which are formal direct sums
\begin{equation} \label{eq:bigoplus-f}
 \bigoplus_{f \in F} (X_f \otimes V_f)[\sigma_f]
\end{equation}
where $V_f$ is some finite-dimensional representation of $\Gamma$.
The morphism spaces are correspondingly
\begin{multline*}
 \qquad hom_{\Sigma_\Gamma\Cat}\Big(\bigoplus_f (X_f \otimes V_f)[\sigma_f],
 \bigoplus_g (Y_g \otimes W_g)[\tau_g]\Big) = \\ =
 \bigoplus_{f,g} hom_{\Cat}(X_f,Y_g)[\tau_g-\sigma_f] \otimes Hom_\C(V_f,W_g) \qquad
\end{multline*}
and they carry the obvious tensor product $\Gamma$-actions. (2) In
the definition of an equivariant twisted complex, the differential
$\delta$ has to belong to the $\Gamma$-invariant part of $hom^1$. The
``equivariant derived category'' $D^b_\Gamma(\Cat) =
H^0(Tw_\Gamma\,\Cat)$ carries, in addition to the usual shift functor
$[1]$, automorphisms $- \otimes \chi$ for any character $\chi \in
\Gamma^*$, which change all $V_f$ in \eqref{eq:bigoplus-f} by
tensoring with the corresponding one-dimensional representation.

\begin{remark} \label{th:twist-is-equivariant}
$D^b_\Gamma(\Cat)$ is {\em not} triangulated, since mapping cones
exist only for the $\Gamma$-invariant morphisms. Fortunately, the
evaluation maps $ev$, $ev^\vee$ used in the definition of the twist
and untwist operations are invariant, so $T_X(Y)$, $T_Y^\vee(X)$ are
well-defined in $D^b_\Gamma(\Cat)$ (at least assuming that the
necessary finite-dimensionality conditions hold). In particular, the
theory of exceptional collections and mutation works well in the
equivariant context.
\end{remark}

\subsection{}
Because we are working over a characteristic zero field, taking
$\Gamma$-invariants is an exact functor. This means that the
classification theory from the previous section extends in a
straightforward way to the equivariant situation: one simply replaces
all ``obstruction groups'' by their $\Gamma$-fixed parts. More
concretely, suppose that we have a finite $\Gamma$ acting on a graded
algebra $A$. Then $HH^*(A,A)^\Gamma$ is the cohomology of the
subcomplex of $CC^*(A,A)^\Gamma$ of equivariant Hochschild cochains.
There is a canonical map $J^\Gamma: HH^*(A,A)^\Gamma \rightarrow
HH^*(A \semidirect \Gamma, A \semidirect \Gamma)$, induced by the
cochain homomorphism
\begin{equation} \label{eq:invariant-include}
\begin{aligned}
 & (J^\Gamma\tau)(a_s \otimes \gamma_s,\dots,a_1 \otimes \gamma_1) = \\
 & \qquad \qquad = \tau(a_s,\rho(\gamma_s)(a_{s-1}),\rho(\gamma_s
 \gamma_{s-1})(a_{s-2}),\dots) \otimes \gamma_s \cdots \gamma_1.
\end{aligned}
\end{equation}
This is compatible with the Lie bracket on $HH^{*+1}$, and has an
obvious deformation theoretic meaning: $\Gamma$-equivariant
$A_\infty$-deformations of $A$ induce deformations of the semidirect
product. In fact, \eqref{eq:invariant-include} is always injective.
One can see this in several different ways. For a classical
homological algebra argument, note that if $P$ is an $A$-bimodule
carrying a $\Gamma$-action which is compatible with that on $A$, then
$P \semidirect \Gamma$ is an $A \semidirect \Gamma$-bimodule, which
for any $A \semidirect \Gamma$-bimodule $M$ satisfies
\[
 Hom_{A \semidirect \Gamma-A \semidirect \Gamma}(P \semidirect \Gamma, M)
 \iso Hom_{A-A}(P,M)^\Gamma;
\]
here the $\Gamma$-action on the right hand side is obtained from that
on $P$ and the diagonal action on $M$, which sends $x$ to $(1 \otimes
\gamma)x(1 \otimes \gamma^{-1})$. In particular, if $P$ is projective
then so is $P \semidirect \Gamma$. Applying this observation to a
projective $A$-bimodule resolution $P^* \rightarrow A$ which carries
a compatible $\Gamma$-action, one obtains
\[
\begin{aligned}
 HH^*(A \semidirect \Gamma, A \semidirect \Gamma)
 & = Ext_{A \semidirect \Gamma - A \semidirect \Gamma}^*(A \semidirect \Gamma,
 A \semidirect \Gamma) \\
 & = Hom_{A \semidirect \Gamma - A \semidirect \Gamma}^*(P^* \semidirect \Gamma,A
 \semidirect \Gamma) \\
 & = Hom^*_{A-A}(P^*, A \semidirect \Gamma)^\Gamma.
\end{aligned}
\]
Under the diagonal action, $A \semidirect \Gamma$ splits into
summands corresponding to conjugacy classes $K \subset \Gamma$, and
this induces a splitting of $HH^*(A \semidirect \Gamma,A \semidirect
\Gamma)$. In particular, the trivial class contributes
$Hom^*_{A-A}(P^*,A)^\Gamma = HH^*(A,A)^\Gamma$, which is precisely
the image of \eqref{eq:invariant-include}. More generally, the
summand corresponding to $K$ is the direct sum of the graph bimodules
$Graph(\rho(\gamma))$ for $\gamma \in K$, so that
\begin{equation} \label{eq:summands}
 HH^*(A \semidirect \Gamma,A \semidirect \Gamma) =
 \bigoplus_K Ext^*_{A-A}\Big(A,\bigoplus_{\gamma \in K} Graph(\rho(\gamma))\Big)^\Gamma.
\end{equation}
Altenatively, one can approach the injectivity of $J^\Gamma$ directly
in terms of the Hochschild complex. This involves looking at $A
\semidirect \Gamma$ as an algebra linear over the semisimple ring $\C
\Gamma$. In the abelian case, this can be put more forcefully by
saying that we consider the $\Gamma^*$-action on $HH^*(A \semidirect
\Gamma, A \semidirect \Gamma)$ induced from that on $A \semidirect
\Gamma$ itself. The decomposition into different types of
$\Gamma^*$-modules is precisely \eqref{eq:summands}, with the
invariant part being the image of $J^\Gamma$. This ultimately finds a
natural explanation in terms of the duality theory for group actions,
which says in particular that $(A \semidirect \Gamma) \semidirect
\Gamma^*$ is Morita equivalent to $A$ \cite{blattner-montgomery85}.

The HKR theorem describes the Hochschild cohomology of polynomial
algebras, and more general smooth commutative algebras. There is a
generalization to graded algebras \cite[Proposition 5.4.6]{loday},
including exterior algebras as a special case, for which the
statement is as follows. Take $A = \Lambda V$ for some
finite-dimensional complex vector space $V$, with the usual grading.
Let $SV^\vee$ be the symmetric algebra of the dual space. Then there
is an explicit isomorphism
\begin{equation} \label{eq:classical-hkr}
 \phi_{HKR}: S^sV^\vee \otimes \Lambda^{s+t}V
 \longrightarrow HH^{s+t}(A,A)^t.
\end{equation}
For instance, the cocycle $[\tau] = \phi_{HKR}(p \otimes 1)$
corresponding to a homogeneous polynomial $p$ of degree $s$ is (in
terms of a basis $v_j$ and dual basis $v_j^\vee$)
\[
 \tau(a_s, \dots, a_1) = \frac{1}{s!} \sum_{j_s,\dots,j_1}
 \frac{\partial p}{\partial v_{j_1} \dots \partial v_{j_s}}
 (v_{j_s}^\vee \inner a_s) \wedge \dots \wedge (v_{j_1}^\vee \inner a_1);
\]
and conversely, if $\tau \in CC^0(A,A)^{-s}$ is any cocycle, we
recover a homogeneous polynomial $p \otimes 1 =
\phi_{HKR}^{-1}([\tau])$ by setting
\begin{equation} \label{eq:reverse-hkr}
 p(v) = \tau(v,\dots,v).
\end{equation}
$\phi_{HKR}$ is compatible with the actions of $GL(V)$ on both sides
(where the action on $V^\vee$ is contragredient to that on $V$).
Moreover, it carries the canonical bracket on $HH^{*+1}(A,A)$ to the
Schouten bracket on $SV^\vee \otimes \Lambda V$, which in particular
satisfies
\begin{equation} \label{eq:schouten}
 [v_{i_1}^\vee \dots v_{i_r}^\vee \otimes 1,
 v_{i_{r+1}}^\vee \dots v_{i_s}^\vee \otimes a] =
 \sum_{k \leq r}
 v_{i_1}^\vee \dots \widehat{v_{i_k}^\vee} \dots v_{i_s}
 \otimes v_{i_k}^\vee \inner a
\end{equation}
for $a \in \Lambda V$. The standard proof of the HKR formula is via
the Koszul bimodule resolution $P^* \rightarrow A$, which in our case
consists of $P^s = A \otimes S^s V \otimes A[-s]$ with the
differential
\begin{multline*}
 \qquad \partial(1 \otimes v_{i_1} \dots v_{i_s} \otimes 1) =
 \sum_k v_{i_k} \otimes v_{i_1} \dots \widehat{v_{i_k}} \dots v_{i_s}
 \otimes 1 + \\
 +(-1)^s 1 \otimes
 v_{i_1} \dots \widehat{v_{i_k}} \dots v_{i_s} \otimes v_{i_k}. \qquad
\end{multline*}
Because $A$ is supercommutative, the induced differential on
$Hom(P^*,A)$ vanishes, which shows that an abstract isomorphism
\eqref{eq:classical-hkr} exists; of course, the explicit formula and
other properties have to be verified separately, by exhibiting a map
between the Koszul and bar resolutions.

We now look at \eqref{eq:summands} in the special case where $A$ is
an exterior algebra, and $\rho$ the action of a finite abelian group
$\Gamma \subset GL(V)$. Taking the Koszul resolution as $P^*$, one
sees that the $\gamma$-summand of $HH^*(A \semidirect \Gamma,A
\semidirect \Gamma)$ can be obtained by computing the cohomology of
the complex $Hom_{\C}(SV,\Lambda V) \iso SV^\vee \otimes \Lambda V$
with the differential
\[
 (\delta_\gamma\eta)(v_{i_1} \dots v_{i_s}) =
 (-1)^s \sum_k \eta(v_{i_1} \dots \widehat{v_{i_k}} \dots
 v_{i_s}) \wedge (\gamma(v_{i_k}) - v_{i_k}),
\]
and then restricting to the $\Gamma$-invariant part of that. To carry
out the computation, we decompose $V$ into eigenspaces of $\gamma$,
and consider the corresponding tensor product decomposition of $S
V^\vee \otimes \Lambda V$. On the fixed part $V^\gamma$, the
differential vanishes as before; the remaining components can be
identified with classical Koszul complexes, which have
one-dimensional cohomology. After chasing the bidegrees and
$\Gamma$-action through the argument, this amounts to a proof of the
following:

\begin{prop} \label{th:twisted-hkr}
The Hochschild cohomology of $A = \Lambda V \semidirect \Gamma$, for
$\Gamma$ a finite abelian subgroup of $GL(V)$, is
\begin{multline*}
 HH^{s+t}(A, A)^t \iso \\ \iso
 \bigoplus_{\gamma \in \Gamma} \Big( S^s(V^\gamma)^\vee \otimes
 \Lambda^{s+t-codim(V^\gamma)}(V^\gamma) \otimes
 \Lambda^{codim(V^\gamma)}(V/V^\gamma)
 \Big)^\Gamma. \qed
\end{multline*}
\end{prop}

\section{Coherent sheaves\label{sec:sheaves}}

To apply the theory from Section \ref{sec:deformations} to the
derived category of coherent sheaves, one needs to introduce an
underlying differential graded category. Concretely, this means
choosing a class of cochain complexes which compute $Ext$ groups,
with cochain maps realizing the Yoneda product. Out of the many
possibilities offered by the literature, we take the lowest-tech path
via {\v C}ech cohomology, which is easy to work with but slightly
more difficult to relate to the usual derived category (a few readers
may prefer to skip that argument, and to take the {\v C}ech version
as definition of the derived category; in which case some of the
properties discussed later on have to be taken for granted).

\subsection{}
Let $Y$ be a noetherian scheme over an algebraically closed field
($\C$ or $\Lambda_\Q$, in the cases of interest here). Fix a finite
affine open cover $\UU$. We introduce a differential graded category
$\cech{\SS}(Y)$ whose objects are locally free coherent sheaves
(algebraic vector bundles), and where the morphism spaces are {\v
C}ech cochain complexes with values in $\SHom$ sheaves,
\[
hom_{\cech{\SS}(Y)}(E,F) = \cech{C}^*(\UU,\SHom(E,F)).
\]
Composition is the shuffle product together with $\SHom(F,G)
\otimes_{\OO_Y} \SHom(E,F) \rightarrow \SHom(E,G)$. One can think of
any dg category as an $A_\infty$-category with vanishing compositions
of order $\geq 3$, and therefore define a triangulated category
$D^b\cech{\SS}(Y)$ in the way explained in Section
\ref{sec:generators}, as the cohomological category of twisted
complexes (we emphasize that there is no ``inverting of
quasi-isomorphisms'' involved in this process). By refining covers
and using standard results about {\v C}ech cohomology, one sees that
$\cech{\SS}(Y)$ is independent of $\UU$ up to quasi-isomorphism,
hence that $D^b\cech{\SS}(Y)$ is well-defined up to equivalence of
triangulated categories.

\begin{lemma} \label{th:derived-is-derived}
$D^b\cech{\SS}(Y)$ is equivalent to the full triangulated subcategory
of the usual bounded derived category of coherent sheaves,
$D^bCoh(Y)$, which is generated by locally free sheaves. If $Y$ is a
regular projective variety, this is the whole of $D^bCoh(Y)$.
\end{lemma}

\proof The usual approach to dg structures on derived categories of
coherent sheaves is through injective resolutions, see for instance
\cite[Section 2]{seidel-thomas99}. In our current language, this can
be formulated as follows. For each locally free coherent sheaf $E$,
fix a resolution $I^*_E$ by quasi-coherent injective sheaves.
Consider a differential graded category $\I(Y)$ with the same objects
as $\cech{\SS}(Y)$, but where $hom_{\I(Y)}(E,F) =
hom_{\O_Y}(I^*_E,I^*_F)$ is the space of sheaf homomorphisms $I^*_E
\rightarrow I^*_F$ of all degrees, with the obvious differential. If
we then define $D^b\I(Y) = H^0(Tw\,\I(Y))$ through twisted complexes,
it is equivalent to the subcategory of $D^bCoh(Y)$ appearing in the
Lemma. To see that, one associates to each object of $Tw\,\I(Y)$ its
total complex; this defines an exact, fully faithful functor from
$D^b\I(Y)$ to the chain homotopy category ${\mathcal K}^+(Y)$ of
bounded below complexes of injective quasi-coherent sheaves. The
image of this functor is the triangulated subcategory generated by
resolutions of locally free sheaves, and by standard properties of
derived categories, this is equivalent to the triangulated
subcategory of $D^bCoh(Y)$ generated by such sheaves. If $Y$ is
smooth and projective, all coherent sheaves have finite locally free
resolutions, so we get the whole of $D^bCoh(Y)$.

To mediate between {\v C}ech and injective resolutions, we use a
triangular matrix construction. Given two locally free sheaves $E$
and $F$, one can consider the {\v C}ech complex
$\cech{C}^*(\SHom(E,I_F^*))$ with coefficients in the complex of
sheaves $\SHom(E,I_F^*) \iso E^\vee \otimes_{\O_Y} I_F^*$. There are
canonical cochain maps
\begin{equation} \label{eq:t-maps}
 hom_{\O_Y}(I_E^*,I_F^*) \longrightarrow
 \cech{C}^*(\SHom(E,I_F^*)) \longleftarrow \cech{C}^*(\SHom(E,F)).
\end{equation}
The first one is defined by composing a sheaf homomorphism $I_E^*
\rightarrow I_F^*$ with $E \rightarrow I_E^0$, and then restricting
the associated section of $\SHom(E,I_F^*)$ to open subsets in $\UU$.
The second one is just the map induced from $F \rightarrow I_F^0$.
Both maps are quasi-isomorphisms: for the first map, note that the
{\v C}ech complex computes the hypercohomology of $E^\vee \otimes
I_F^*$, which on the other hand that is the ordinary cohomology of
$hom_{\O_Y}(E^*,I_F^*)$ because each $E^\vee \otimes I_F^k$ is
injective. For the second map one uses a standard spectral sequence
comparison argument, together with the fact that the cohomology of
$hom_{\O_U}(E|U, I_F^*|U)$ for each open subset $U$ is
$Hom_{\O_U}(E|U,F|U)$. Along the same lines, there are canonical
homomorphisms
\begin{equation} \label{eq:t-bistructure}
\begin{split}
 & hom_{\O_Y}(I_F^*,I_G^*) \otimes \cech{C}^*(\SHom(E,I_F^*))
 \longrightarrow \cech{C}^*(\SHom(E,I_G^*)), \\
 & \cech{C}^*(\SHom(F,I_G^*)) \otimes \cech{C}^*(\SHom(E,F))
 \longrightarrow \cech{C}^*(\SHom(E,I_G^*)).
\end{split}
\end{equation}
We introduce another dg category $\T$, still keeping the same objects
as before, and where the $hom$s are the direct sums of three
components
\[
 hom_\T(E,F) =
 \begin{pmatrix}
 hom_{\O_Y}(I_E^*,I_F^*) & \cech{C}^*(\SHom(E,I_F^*))[1] \\
 0 & \cech{C}^*(\SHom(E,F)).
 \end{pmatrix}
\]
The differential consists of the given differentials on each summand
together with the chain homomorphisms \eqref{eq:t-maps}. Similarly,
composition in $\T$ combines the usual compositions of morphisms in
$\I(Y)$ and $\cech{\SS}(Y)$ with the maps \eqref{eq:t-bistructure}.
By definition, this comes with dg functors $\T \rightarrow \I(Y)$,
$\T \rightarrow \cech{\SS}(Y)$, both of which are quasi-isomorphisms,
and therefore one has induced exact equivalences $D^b\cech{\SS}(Y)
\iso D^b\T \iso D^b\I(Y)$. \qed

\begin{remark} \label{th:equivariant-sheaves}
At least if one supposes that the ground field has characteristic
zero, the entire argument carries over to the derived category of
equivariant coherent sheaves with respect to the action of a finite
group $\Gamma$ on $Y$. First, by taking intersections over
$\Gamma$-orbits one can find an invariant affine open cover. One then
introduces the dg category $\cech{S}_\Gamma(Y)$ whose objects are
locally free sheaves with $\Gamma$-actions, and whose morphism spaces
are the $\Gamma$-invariant summands of the {\v C}ech complexes. In
the regular projective case,
\[
 D^b\cech{\SS}_\Gamma(Y) \iso D^bCoh_\Gamma(Y),
\]
where $Coh_\Gamma(Y)$ is the abelian category of $\Gamma$-equivariant
coherent sheaves. We will now make the connection between this
observation and the material from Section \ref{sec:group-actions}.
Suppose for simplicity that $\Gamma$ is abelian. If $E$ is a coherent
sheaf with a $\Gamma$-action, then so is $E \otimes W$ for any
finite-dimensional $\Gamma$-module $W$. Suppose for simplicity that
$\Gamma$ is abelian, and take equivariant locally free sheaves
$E_1,\dots,E_m$. Then the full dg subcategory $\Cat \subset
\cech{\SS}(Y)$ with objects $E_k$ carries a $\Gamma$-action, and the
semidirect product $\Cat \semidirect \Gamma$ can be identified with
the full dg subcategory of $\cech{\SS}_\Gamma(Y)$ whose objects are
$E_k \otimes W_\chi$ for all $k$ and all one-dimensional
representations $W_\chi$.
\end{remark}

\subsection{\label{subsec:formal-schemes}}
Let $Y_q$ be a noetherian formal scheme, flat over $\Lambda_\N$. By
definition \cite[ch.\ II \S 9]{hartshorne} there is a finite cover
$\UU$ by open subsets, on each of which the scheme is ``given by''
(actually, the formal spectrum of) a commutative algebra $A_{q,U}$
over $\Lambda_\N$ which is complete and torsion-free as a
$\Lambda_\N$-module. We consider locally free coherent sheaves over
$Y_q$, which are $\O_{Y_q}$-module sheaves given over each $U$ by
sections of a finitely generated free $A_{q,i}$-module. By the same
procedure as before, one can make these into objects of a dg category
$\cech{\SS}(Y_q)$ whose morphism spaces are again complete
torsion-free $\Lambda_\N$-modules.

Geometrically, $Y_q$ is an infinitesimal one-parameter family of
schemes. One can specialize the parameter $q$ to two different
``values''. Setting $q = 0$ yields the {\em special fibre} $Y = Y_q
\times_{\Lambda_\N} \C$, and on the other hand, inverting $q$ and
introducing formal roots of it gives the {\em general fibre} $Y_q^* =
Y_q \times_{\Lambda_\N} \Lambda_\Q$. Both of them are noetherian
schemes over algebraically closed fields. There are parallel
constructions in the world of our differential graded categories,
namely full and faithful dg functors
\begin{equation} \label{eq:specialization}
 \cech{\SS}(Y_q) \otimes_{\Lambda_\N} \C \longrightarrow \cech{\SS}(Y_0), \qquad
 \cech{\SS}(Y_q) \otimes_{\Lambda_\N} \Lambda_\Q \longrightarrow \cech{\SS}(Y_q^*).
\end{equation}
To see what this means concretely, pick a finite collection of
locally free sheaves $E_{0,1},\dots,E_{0,m}$ on $Y_0$, and suppose
that each $E_{0,k}$ admits an extension $E_{q,k}$ to a locally free
sheaf on $Y_q$, whose restriction to $Y_q^*$ we denote by
$E_{q,k}^*$. The fact that the first map in \eqref{eq:specialization}
is full and faithful on the cochain level means that the full dg
subcategory of $\cech{\SS}(Y_q)$ with objects $E_{q,k}$ is a
one-parameter deformation of the full dg subcategory of
$\cech{\SS}(Y_0)$ with objects $E_{0,k}$. Moreover, if we take the
first of these subcategories and tensor with $\Lambda_\Q$, we obtain
the full dg subcategory of $\cech{\SS}(Y_q^*)$ with objects
$E_{q,k}^*$. In this way, deformations of varieties and their effects
on coherent sheaves have been brought into the fold of the abstract
theory from Section \ref{sec:deformations}. This is still quite
crude, for instance it excludes totally those sheaves on $Y_0$ that
might not admit extensions to $Y_q$. One can give a much more natural
and satisfactory treatment using obstructed dg or $A_\infty$
structures, see for instance \cite{fukaya00}, but for our application
the rudimentary version is sufficient.

\subsection{}
We end our discussion by listing two properties of $D^bCoh(Y)$ for a
regular projective variety over a field, which are well-known to
specialists (I am indebted to Drinfeld for suggesting simplified
proofs and giving references). The first result is classical, and
explains why passage to $D^\pi$ is unnecessary.

\begin{lemma}
$D^bCoh(Y)$ is idempotent closed.
\end{lemma}

One way to prove this is to consider the chain homotopy category
$\K^+(Y)$, which is idempotent closed by an infinite direct sum trick
\cite[Proposition 1.6.8]{neeman} or \cite[Lemma 2.4.8.1]{levine}.
Having done that, one recognizes that the direct summand of any
object of $D^b(Y) \subset \K^+(Y)$ again lies in $D^b(Y)$, because
its cohomology sheaves are coherent and almost all zero
\cite[Proposition 2.4]{seidel-thomas99}. Alternatively, one can use
the stronger result from \cite{bondal-vandenbergh02} which says that
the bounded derived category is {\em saturated}.

\begin{lemma} \label{th:kontsevich}
If $\iota: Y \hookrightarrow {\mathbb P}^N$ is a projective
embedding, and $F_1,\dots,F_m$ are split-generators for
$D^bCoh({\mathbb P}^N)$, their derived restrictions $E_k =
L\iota^*(F_k)$ are split-generators for $D^bCoh(Y)$.
\end{lemma}

This is due to Kontsevich, and the proof follows the same pattern as
Lemma \ref{th:twist-generators}. By assumption, on ${\mathbb P}^N$
one can express $\O(m)$ for any $m$ as direct summand of a complex
built from the $F_k$. Hence, on $Y$ one can express $\O(m)$ in the
same way using $E_k$. It therefore suffices to show that all the
$\O(m)$ together are split-generators for $D^bCoh(Y)$. Any locally
free coherent sheaf $E$ on $Y$ has a finite left resolution of the
form
\[
 0 \rightarrow E' \rightarrow \O(m_l)^{\oplus r_l} \rightarrow
 \cdots \rightarrow \O(m_1)^{\oplus r_1} \rightarrow E \rightarrow 0
\]
for some $E'$, and one can make $l$ arbitrarily large. For $l >
dim\,Y$ we have $Ext^l(E,E') = H^l(E^\vee \otimes E') = 0$, hence the
resulting exact triangle
\[
 \xymatrix{
 \!\!\!\! {\big\{ \O(m_l)^{\oplus r_l} \rightarrow \dots \rightarrow
 \O(m_1)^{\oplus r_1} \big\}} \ar[r] & {E} \ar[d] \\
 & {E'[l]} \ar@/^1pc/[ul]^-{[1]}
}
\]
splits in such a way that $E \oplus E'[l-1]$ is isomorphic to the top
left complex. Once one has split-generated all locally free sheaves
from the $\O(m)$, the rest follows as in Lemma
\ref{th:derived-is-derived}.


\section{Symplectic terminology\label{sec:geometry}}

The purpose of this section is to assemble the necessary nuts and
bolts from elementary symplectic geometry. The origin of the main
notions, to the author's best knowledge, is as follows. Rational
Lagrangian submanifolds have a long history in quantization (going
back to Einstein); the first place where they occur in connection
with Floer homology is probably Fukaya's paper \cite{fukaya93}.
Lagrangian submanifolds with zero Maslov class also originated in
(Maslov's) quantization; the essentially equivalent notion of grading
was introduced by Kontsevich \cite{kontsevich94}. The symplectic
viewpoint on Lefschetz pencils and higher-dimensional linear systems
emerged gradually from the work of Arnol'd \cite{arnold95}, Donaldson
\cite{donaldson98,donaldson02}, Gompf \cite{gompf-stipsicz}, and
Auroux \cite{auroux99,auroux00}. In particular, matching cycles were
invented by Donaldson (unpublished).

\subsection{}
By a {\em projective K{\"a}hler manifold} we mean a compact complex
manifold $X$ carrying a unitary holomorphic line bundle $o_X$, which
is positive in the sense that the curvature of the corresponding
connection $A_X$ defines a K{\"a}hler form
\[
\o_X = \frac{i}{2\pi} F_{A_X}.
\]
From a symplectic viewpoint, this integrality condition on the
symplectic class leads one to make corresponding restrictions on the
other objects which live on $X$. A Lagrangian submanifold $L \subset
X$ is called {\em rational} if the monodromy of the flat connection
$A_X|L$ consists of roots of unity. This means that one can choose a
covariantly constant multisection $\lambda_L$ of $o_X|L$ which is of
unit length everywhere (as a multisection, this is the multivalued
$d_L$-th root of a section $\lambda_L^{d_L}$ of $o_X^{\otimes
d_L}|L$, for some $d_L \geq 1$). Rationality is invariant under exact
Lagrangian isotopy. More precisely, take such an isotopy $L_t$, and
embed it into a Hamiltonian isotopy $\phi^t$, with time-dependent
Hamiltonian $H$ and vector field $Y$. We obtain covariantly constant
multisections $\lambda_{L_t}$ by starting with $\lambda_{L_0}$ and
solving the ODE
\begin{equation} \label{eq:deform-lambda}
\nabla_{Y_t} \lambda_{L_t} = - 2\pi i (H_t|L_t) \, \lambda_{L_t}.
\end{equation}

Rationality takes its name from the effect on the periods of action
functionals. Take two rational Lagrangian submanifolds $L_0,L_1$. For
each of them, choose a covariantly constant unit length
$d_{L_k}$-fold multisection $\lambda_{L_k}$. The {\em mod $\Q$
action} $\bar{A}(x)$ of a point $x \in L_0 \cap L_1$ is defined by
\begin{equation} \label{eq:modq-action}
e^{2\pi i \bar{A}(x)} = \lambda_{L_1}(x)/\lambda_{L_0}(x).
\end{equation}
It is unique up to adding a number in $\lcm(d_{L_0},d_{L_1})^{-1} \Z
\subset \Q$, whence the name. Now suppose that we have two
intersection points $x_0,x_1$ and a connecting disc between them:
this is a smooth map $u: \R \times [0;1] \rightarrow X$ with $u(\R
\times \{k\}) \subset L_k$, such that $u(s,\cdot)$ converges to the
constant paths at $x_0,x_1$ as $s \rightarrow -\infty,+\infty$. By
definition of curvature,
\begin{equation} \label{eq:modq-energy}
\int u^*\o_X \in \bar{A}(x_0) - \bar{A}(x_1) +
\lcm(d_{L_0},d_{L_1})^{-1}\Z.
\end{equation}

Similarly, a symplectic automorphism $\phi$ of $X$ is rational if the
monodromy of the induced flat connection on $Hom(o_X,\phi^*o_X)$
consists of roots of unity. This is invariant under Hamiltonian
isotopy. We write $Aut(X)$ for the group of such automorphisms, which
acts in the obvious way on the set of rational Lagrangian
submanifolds.

\subsection{}
Equally important for us will be the relative situation, where in
addition to the previous data we have a holomorphic section
$\sigma_{X,\infty}$ of $o_X$ such that $X_\infty =
\sigma_{X,\infty}^{-1}(0)$ is a divisor with normal crossings, along
which $\sigma_{X,\infty}$ vanishes with multiplicity one, and such
that each irreducible component $C \subset X_\infty$ is smooth (no
self-intersections). We call $M = X \setminus X_\infty$ an {\em
affine K{\"a}hler manifold}. One can use
$\sigma_{X,\infty}/||\sigma_{X,\infty}||$ to trivialize $o_X|M$, and
by writing the connection with respect to this trivialization as
$A_X|M = d - 2\pi i \theta_M$, one gets a one-form $\theta_M$ which
satifies $d\theta_M = \o_M = \o_X|M$, so that $M$ is an exact
symplectic manifold in the usual sense of the word.

A compact Lagrangian submanifold $L \subset M$ is called {\em exact}
if $\theta_M|L = dK_L$ for some function $K_L$. This condition is
invariant under exact Lagrangian isotopy. Note that if $L$ is exact,
\begin{equation} \label{eq:k-lambda}
\lambda_L = \exp(2 \pi i K_L )
\frac{\sigma_{X,\infty}}{||\sigma_{X,\infty}||}
\end{equation}
is a covariantly constant section of $o_X|L$, so that must be the
trivial flat line bundle. Conversely, suppose that $L$ is some
Lagrangian submanifold of $M$, such that $o_X|L$ is trivial and
admits a covariantly constant section which can be written in the
form \eqref{eq:k-lambda} for some $K_L$. Then $dK_L = \theta_M|L$,
hence $L$ is exact. We have proved:

\begin{lemma} \label{th:check-exactness}
$L$ is exact iff $o_X|L$ is trivial, and its nonzero covariantly
constant sections lie in the same homotopy class of nowhere zero
sections as $\sigma_{X,\infty}|L$. \qed
\end{lemma}

Let $L_0$, $L_1$ be exact Lagrangian submanifolds of $M$, and choose
functions $K_{L_0}$, $K_{L_1}$. One can then define the action of an
intersection point $x \in L_0 \cap L_1$ to be
\begin{equation} \label{eq:action}
A(x) = K_{L_1}(x) - K_{L_0}(x) \in \R.
\end{equation}
For any two such points $x_0,x_1$ and any connecting disc $u$ in $X$,
the intersection number $u \cdot X_\infty$ is well-defined, and
\begin{equation} \label{eq:energy}
\int u^*\o_X = A(x_0) - A(x_1) + (u \cdot X_\infty).
\end{equation}
In particular, the action functional is indeed exact if we allow only
connecting discs which lie in $M$.

Let $\phi$ be a symplectic automorphism of $M$, which extends
smoothly to $X$ in such a way that $X_\infty$ remains pointwise fixed
(we denote the extension equally by $\phi$). We say that $\phi$ exact
if $\phi^*\theta_M - \theta_M = dK_\phi$ for some function $K_\phi$
on $M$. This condition is invariant under Hamiltonian isotopies
keeping $X_\infty$ fixed. Exact symplectic automorphisms form a group
$Aut(M)$, which acts naturally on the set of exact Lagrangian
submanifolds. The analogue of Lemma \ref{th:check-exactness} is

\begin{lemma} \label{th:check-exactness-2}
$\phi$ is exact iff $Hom(o_X,\phi^*o_X)$ is the trivial flat bundle,
and every covariantly constant nonzero section has the property that
its restriction to $M$ lies in the same homotopy class of nowhere
zero sections as $\phi^*\sigma_{X,\infty}/\sigma_{X,\infty}$. \qed
\end{lemma}

\begin{remark} \label{th:h1}
The conditions for exactness which appear in the Lemma are
automatically satisfied if $H_1(X) = 0$. This is obvious for the
first one, and for the second one argues as follows: take the
meridian $\gamma$ in $M$ around a component of $X_\infty$, and bound
it by a small disc going through $X_\infty$ once. The homotopy class
of covariantly constant sections of $Hom(o_X,\phi^*o_X)|\gamma$ is
characterized by the fact that they extend to nowhere zero sections
over the disc. On the other hand, $\sigma_{X,\infty}\gamma$ extends
to a section with precisely one zero, and so does
$\phi^*\sigma_{X,\infty}|\gamma$. Hence,
$(\phi^*\sigma_{X,\infty}/\sigma_{X,\infty})|\gamma$ lies in the same
homotopy class of nowhere zero sections as the covariantly constant
ones.
\end{remark}

%The way one goes from $K_\phi$ to $\gamma_\phi$ and back is by
%writing
%\[
%\gamma_\phi|M = \exp(2 \pi i K_\phi) \cdot
%\frac{\phi^*\sigma_{X,\infty}}{||\phi^*\sigma_{X,\infty}||} \left(
%\frac{\sigma_{X,\infty}}{||\sigma_{X,\infty}||} \right)^{-1}.
%\]
%
The relation between the affine and projective notions is clear:
Lemma \ref{th:check-exactness} shows that an exact $L \subset M$ is a
rational Lagrangian submanifold of $X$, and Lemma
\ref{th:check-exactness-2} does the same for automorphisms. Given a
pair of exact Lagrangian submanifolds $L_0,L_1$, the multivalued
action \eqref{eq:modq-action} is a reduction of the single-valued one
\eqref{eq:action}, and \eqref{eq:energy} is a refinement of
\eqref{eq:modq-energy}.

\subsection{}
Let $M$ be an affine K{\"a}hler manifold, coming from $X$ and
$\sigma_{X,\infty}$ as always. Suppose that we have a second
holomorphic section $\sigma_{X,0}$ of $o_X$, linearly independent of
$\sigma_{X,\infty}$. Let $\{X_z\}_{z \in \CP{1}}$ be the pencil of
hypersurfaces generated by these two sections. This means that we set
\[
\sigma_{X,z} = \sigma_{X,0} - z\sigma_{X,\infty} \quad \text{ and }
\quad X_z = \sigma_{X,z}^{-1}(0)
\]
for $z \in \C$. Assume that $X_0 = \sigma_{X,0}^{-1}(0)$ is smooth in
a neighbourhood of the base locus $X_{0,\infty} = X_0 \cap X_\infty$,
and that it intersects each stratum of $X_\infty$ transversally; then
the same will hold for each $X_z$, $z \in \C$. Every nonsingular
$X_z$, $z \in \C$, is a projective K{\"a}hler manifold with ample
line bundle $o_{X_z} = o_X|X_z$, and it carries a preferred
holomorphic section $\sigma_{X_z} = \sigma_{X,\infty}|X_z$ whose zero
set is $X_{0,\infty}$. By assumption, this is a divisor with normal
crossings, so that $M_z = X_z \setminus X_{0,\infty}$ is an affine
K{\"a}hler manifold. One can also see the $M_z$ as the regular fibres
of the holomorphic function
\[
\pi_M = \sigma_{X,0}/\sigma_{X,\infty} :  M \longrightarrow \C.
\]
We denote by $Critv(\pi_M)$ its critical value set, which is finite
by Bertini. Given $z,z' \in \C \setminus Critv(\pi_M)$, we denote by
$Iso(M_z,M_{z'})$ the space of symplectic isomorphisms $\phi: M_z
\rightarrow M_{z'}$ which extend to $X_z \rightarrow X_{z'}$ in such
a way that $X_0 \cap X_\infty$ remains pointwise fixed, and such that
$\phi^*\theta_{M_{z'}} - \theta_{M_z}$ is exact. These spaces form a
topological groupoid under composition, and for $z = z'$ they reduce
to the groups $Aut(M_z)$ defined before. Take a smooth path $c: [0;l]
\rightarrow \C \setminus Critv(\pi_M)$. The $c'(t)$ can be lifted in
a unique way to vector fields $Z_t \in \smooth(TM|M_{c(t)})$ which
are horizontal (orthogonal to $ker(D\pi_M)$ with respect to the
symplectic form). By integrating these one gets an exact {\em
symplectic parallel transport} map
\[
h_c \in Iso(M_{c(0)},M_{c(l)}).
\]
To show that $h_c$ is well-defined, one embeds $M$ into the graph
$\Gamma = \{(z,x) \in \C \times X_z \suchthat x \in X_z\}$. The
projection $\pi_\Gamma: \Gamma \rightarrow \C$ is a properification
of $\pi_M$. It is a standard fact that parallel transport with
respect to the product symplectic form $\o_\Gamma = (\o_X +
\o_{\C})|\Gamma$ gives a symplectic isomorphism $X_{c(0)} \rightarrow
X_{c(l)}$. This extends the original $h_c$, because adding the
pullback of $\o_{\C}$ to $\o_M$ does not change the vector fields
$Z_t$. Moreover, at a point $(z,x) \in \C \times X_{0,\infty}$ we
have
\[
 T\Gamma_{(z,x)} = \C \times \{ \xi \in TX_x \suchthat
 D\sigma_0(x)\xi - z\, D\sigma_\infty(x)\xi = 0 \},
\]
and since this splitting is $\o_\Gamma$-orthogonal, the parallel
transport vector field will lie in $\C \times \{0\}$. This implies
that the extension of $h_c$ to $X_{c(0)}$ leaves $X_{0,\infty}$
pointwise fixed. Finally, the exactness of $h_c^*\theta_{M_{c(l)}} -
\theta_{M_{c(0)}}$ follows from the Cartan formula
$
 (L_{Z_t} \theta_M) \,|\, M_{c(t)} =
 d(i_{Z_t}\theta_M) \,|\, M_{c(t)} +
 i_{Z_t}\o_M \,|\, M_{c(t)}
$, where the last term vanishes because of the horizontality of
$Z_t$.

\subsection{}
We say that $\{X_z\}$ is an {\em quasi-Lefschetz pencil} if $\pi_M$
is nondegenerate in the sense of Picard-Lefschetz theory, meaning
that it has only nondegenerate critical points, no two of which lie
in the same fibre. This is the same as saying that the set of
critical points $Crit(\pi_M)$ is regular ($D\pi_M$ is transverse to
the zero-section) and $\pi_M: Crit(\pi_M) \rightarrow Critv(\pi_M)$
is bijective. Lefschetz pencils are the special case where $X_\infty$
is smooth.

An {\em embedded vanishing path} for a quasi-Lefschetz pencil is an
embedded path $c: [0;1] \rightarrow \C$ with $c^{-1}(Critv(\pi_M)) =
\{1\}$. By considering the limiting behaviour of parallel transport
along $c|[0;t]$ as $t \rightarrow 1$, one defines an embedded
Lagrangian disc $\Delta_c \subset M$, the {\em Lefschetz thimble},
whose boundary is a Lagrangian sphere $V_c \subset M_{c(0)}$, the
associated {\em vanishing cycle}. The Picard-Lefschetz formula
\cite[Proposition 1.15]{seidel01} says that the monodromy around a
loop $\gamma$ which doubles around $c$ in positive sense is the Dehn
twist associated to the vanishing cycle, up to isotopy inside the
group of exact symplectic automorphisms:
\[
h_\gamma \htp \tau_{V_c} \in Aut(M_{c(0)}).
\]
To be precise, $V_c$ carries a small additional piece of structure,
called a framing in \cite{seidel00,seidel01}, and the definition of
Dehn twist uses that too. However, framings contain no information if
the vanishing cycles are of dimension $\leq 3$, so omitting them will
not matter for our applications.

Now choose a base point $z_* \in \C \setminus Critv(\pi_M)$ and base
path $c_*: [0;\infty) \rightarrow \C \setminus Critv(\pi_M)$, meaning
an embedded path with $c_*(0) = z_*$ which goes off to infinity, in
the sense that there is a $\zeta \in \C^*$ such that $c_*(t) = \zeta
t$ for all $t \gg 0$. By a {\em distinguished basis} of embedded
vanishing paths, we mean an ordered collection of such paths
$(c_1,\dots,c_r)$, $r = |Critv(\pi_M)|$, starting at $z_*$ and with
the following properties: the tangent directions $\Rgeq c_k'(0)$ are
pairwise distinct and clockwise ordered, and different $c_k$ do not
intersect except at $z_*$. Moreover, $\Rgeq c_*'(0)$ should lie
strictly between $\Rgeq c_r'(0)$ and $\Rgeq c_1'(0)$, and no $c_k$
may intersect $c_*$ except at $z_*$. The composition of the
corresponding loops $\gamma_k$ is freely homotopic in $\C \setminus
Critv(\pi_M)$ to a circle $\gamma_\infty(t) = R\zeta e^{it}$ of some
large radius $R \gg 0$, and hence the product of all the associated
Dehn twists is the monodromy around that circle. More precisely,
taking into account the change of base point, one has
\begin{equation} \label{eq:large-small-loop}
 \tau_{V_{c_1}} \dots \tau_{V_{c_r}} \htp
 (h_{c_*|[0;R]})^{-1} h_{\gamma_\infty} h_{c_*|[0;R]} \in Aut(M_{z_*}).
\end{equation}
This becomes more familiar for actual Lefschetz pencils, where one
can extend the graph to $\CP{1}$,
\begin{equation} \label{eq:bar-gamma}
\bar{\Gamma} = \{(z,x) \in \CP{1} \times X \suchthat x \in X_z\}
\xrightarrow{\pi_{\bar{\Gamma}}} \CP{1},
\end{equation}
so that $\infty$ is a regular value of $\pi_{\bar{\Gamma}}$.
Proceeding as before, one can define symplectic parallel transport
for paths in $\CP{1} \setminus Critv(\pi_M)$. In particular, since
$\gamma_\infty$ can be contracted by passing over $\infty$, its
monodromy is isotopic to the identity, so that
\eqref{eq:large-small-loop} simplifies to the following relation
between Dehn twists in the symplectic mapping class group
$\pi_0(Aut(M_{z_*}))$:
\begin{equation} \label{eq:relation-in-mcg}
\tau_{V_{c_1}} \circ \dots \circ \tau_{V_{c_r}} \htp id.
\end{equation}

\begin{remark} \label{th:multiple-fibres}
It is no problem (except terminologically) to allow several
nondegenerate critical points to lie in the same fibre. One should
then choose a vanishing path for every critical point. Two paths
$c_k,c_l$ going to the same critical value should either coincide or
else satisfy $c_k(t) \neq c_l(t)$ for $t \in (0;1)$, and $\Rgeq
c_k'(0) \neq \Rgeq c_l'(0)$. In the first case, the associated
vanishing cycles in $M_{z_*}$ will be disjoint.
\end{remark}

\subsection{}
Let $\{X_z\}$ be a quasi-Lefschetz pencil. Consider a smooth embedded
path $d: [-1;1] \rightarrow \C$ such that $d^{-1}(Critv(\pi_M)) =
\{-1;1\}$. Split it into a pair of vanishing paths $c_-(t) = d(-t)$,
$c_+(t) = d(t)$, $t \in [0;1]$, and consider the associated vanishing
cycles
\[
V_{c_-},V_{c_+} \subset M_{d(0)}.
\]
We say that $d$ is a {\em matching path} if $V_{c_-}$, $V_{c_+}$ are
isotopic as exact Lagrangian spheres. Actually, there is an
additional condition concerning the framings, which we do not want to
spell out here, since it is vacuous in the dimensions that we will be
interested in. If $d$ is a matching path, one can glue together the
Lefschetz thimbles $\Delta_{c_-}$, $\Delta_{c_+}$ to obtain a
Lagrangian sphere in the total space, the {\em matching cycle}
$\Sigma_d \subset M$. In the naive situation where the two vanishing
cycles coincide, this would simply be the set-theoretic union of the
Lefschetz thimbles. In general, the construction involves a choice of
isotopy between the cycles, and $\Sigma_d$ might theoretically depend
on that choice. Note however that this can never happen if the fibres
are Riemann surfaces with boundary, since there, the space of exact
Lagrangian submanifolds (circles) having a fixed isotopy class is
contractible.

\subsection{\label{subsec:braid-monodromy}}
Let $\sigma_{X,0},\sigma_{X,\infty}$ be two sections of $o_X$ which
generate a quasi-Lefschetz pencil $\{X_z\}$. Suppose that we have yet
another section $\sigma_{X,0}'$, linearly independent from the
previous ones. We require that its zero set should be smooth in a
neighbourhood of $X_0 \cap X_\infty$, and should intersect each
stratum of that base locus transversally. Consider the associated map
\begin{equation} \label{eq:bm-map}
M \xrightarrow{b_M = \left(\frac{\sigma_{X,0}}{\sigma_{X,\infty}},
\frac{\sigma_{X,0}'}{\sigma_{X,\infty}}\right)} \C^2,
\end{equation}
its critical point set $Crit(b_M)$ and critical value set $C =
Critv(b_M)$. Concerning this, we make the following additional
assumptions:
\begin{itemize}
\item
$Crit(b_M)$ is regular, and $b_M: Crit(b_M) \rightarrow C$ is an
embedding away from finitely many points.
\end{itemize}
Regularity means that $Db_M \in \Gamma(Hom(TM,\C^2))$ is never zero,
and is transverse to the subset of rank one linear maps. The second
statement implies that for generic $z \in \C \setminus Critv(\pi_M)$,
the curve $Crit(b_M)$ is transverse to $b_M^{-1}(\{z\} \times \C)$,
and the projection $b_M|Crit(b_M) \times b_M^{-1}(\{z\} \times \C)
\rightarrow C \times (\{z\} \times \C)$ is a bijection. In this case,
the sections $(\sigma_{X,0}'|X_z,\sigma_{X,\infty}|X_z)$ generate a
quasi-Lefschetz pencil on $X_z$, and the critical value set of the
associated holomorphic function $q_{M_z}: M_z \rightarrow \C$ is
precisely
\begin{equation} \label{eq:z-critical}
Critv(q_{M_z}) = C \cap (\{z\} \times \C) \subset \C.
\end{equation}
There is a finite subset of $\C \setminus Critv(\pi_M)$ where this
fails, and we denote it by $Fakev(\pi_M)$. These ``fake critical
values'' will play a role similar to the real ones, see
\eqref{eq:braid-monodromy} below.
\begin{itemize}
\item
Let $x$ be a critical point of $\pi_M$, $\pi_M(x) = z$. Then the
restriction of $D^2\pi_M$ to the complex codimension one subspace
$ker(D_xb_M)$ is nondegenerate. Moreover, all critical points of
$q_{M_z}|(M_z \setminus \{x\})$ are nondegenerate, and their
$q_{M_z}$-values are pairwise distinct and different from
$q_{M_z}(x)$.
\end{itemize}
As a consequence, $b_M: Crit(b_M) \rightarrow C$ is an embedding on
$Crit(b_M) \cap M_z$, and moreover the projection $C \subset \C^2
\rightarrow \C$ to the first variable has an ordinary (double) branch
point at $b_M(x)$, and is a local isomorphism at all other points of
$C \cap (\{z\} \times \C)$.

If the two conditions stated above are satisfied, we say that
$\sigma_{X,0}'$ is a {\em generic auxiliary section} of $o_X$.
Practically, the situation is that we have a quasi-Lefschetz pencil
on $X$, almost every regular fibre $X_z$ of which again admits a
quasi-Lefschetz pencil. Moreover, we have a good understanding of how
these pencils degenerate as $X_z$ becomes singular. We will now
explain the consequences of this for the vanishing cycles of $\pi_M$.
For that, suppose that our base point $z_* \notin Fakev(\pi_M)$.
Alongside the ordinary monodromy of the pencil,
\begin{align}
 \notag \pi_1(\C \setminus Critv(\pi_M),z_*) & \longrightarrow
 \pi_0(Aut(M_{z_*})), \\
\intertext{there is a ``relative'' or ``braid'' monodromy
homomorphism}
 \label{eq:braid-monodromy}
 \pi_1(\C \setminus (Critv(\pi_M) \cup Fakev(\pi_M)),z_*)
 & \longrightarrow
 \pi_0(\Diff^c(\C,(C \cap \{z_*\} \times \C))),
\end{align}
which describes how the finite set \eqref{eq:z-critical} moves in
$\C$ as $z$ changes. Take an embedded vanishing path for $\pi_M$, $c:
[0;1] \rightarrow \C$, with the additional property that $c([0;1])
\cap Fakev(\pi_M) = \emptyset$. Let $\gamma$ be a loop doubling
around $c$ in positive sense. As we know, the monodromy around
$\gamma$ is the Dehn twist along $V_c$. The corresponding statement
for \eqref{eq:braid-monodromy} is that there is an embedded path $d:
[-1;1] \rightarrow \C$, with $d^{-1}(Critv(q_{M_{z_*}})) = \{\pm
1\}$, such that the image of $\gamma$ under the relative monodromy is
the half-twist along $d$. The relation between the two is as follows,
see \cite{seidel04} for the proof:

\begin{prop} \label{th:vanishing-matching}
$d$ is a matching path for $q_{M_{z_*}}$, and the vanishing cycle
$V_c \subset M_{z_*}$ is, up to Lagrangian isotopy, a matching cycle
$\Sigma_d$ for that path. \qed
\end{prop}

\subsection{}
We temporarily digress from K{\"a}hler to general symplectic
geometry. Let $(M^{2n},\o_M)$ be any connected symplectic manifold
with $c_1(M) = 0$. Take an almost complex structure $J_M$ which tames
$\o_M$, and a complex volume form (a nowhere vanishing $(n,0)$-form
with respect to $J_M$) $\eta_M$. Let $\LL_M \rightarrow M$ be the
bundle of (unoriented) Lagrangian Grassmannians. The phase function
associated to $\eta_M$,
\[
\alpha_M: \LL_M \longrightarrow S^1,
\]
is defined as follows. Given $x \in M$ and $\Lambda \in \LL_{M,x}$,
take any (not necessarily orthonormal) basis $e_1,\dots,e_n$ of
$\Lambda$, and set
\begin{equation} \label{eq:squared-phase}
\alpha_M(\Lambda) = \frac{\eta_M(e_1 \wedge \dots \wedge
e_n)^2}{|\eta_M(e_1 \wedge \dots \wedge e_n)|^2}.
\end{equation}
We distinguish the special case where $J_M$ is $\o_M$-compatible by
calling $\alpha_M$ a {\em classical phase function}. These are the
most commonly used ones, and they are much better behaved. General
phase functions will be needed later for technical reasons.

Given $\alpha_M$, one can associate to a Lagrangian submanifold $L
\subset M$ or a symplectic automorphism $\phi: M \rightarrow M$ a
phase function
\[
\begin{cases}
 & \!\!\! \alpha_L: L \longrightarrow S^1, \quad
 \alpha_L(x) = \alpha_M(TL_x), \\
 & \!\!\! \alpha_\phi: \LL_M \longrightarrow S^1, \quad
 \alpha_\phi(\Lambda) = \alpha_M(D\phi(\Lambda))/\alpha_M(\Lambda).
\end{cases}
\]
A grading of $L$ or $\phi$ is a real-valued lift $\alphagr_L$ resp.\
$\alphagr_\phi$, where the convention is that $\R$ covers $S^1$ by $a
\mapsto \exp(2\pi i a)$. Pairs $\Lgr = (L,\alphagr_L)$, $\phigr =
(\phi,\alphagr_\phi)$ are called graded Lagrangian submanifolds
resp.\ graded symplectic automorphisms. The latter form a group, with
a natural action on the set of graded Lagrangian submanifolds. The
composition law and action are defined by
\begin{equation}
\label{eq:compose-gradings}
\begin{cases}
 & \!\!\! \alphagr_{\psi \circ \phi} = \alphagr_\psi \circ D\phi +
 \alphagr_\phi, \\
 & \!\!\! \alphagr_{\phi(L)} = \alphagr_L \circ \phi^{-1} + \alphagr_\phi
 \circ D\phi^{-1}.
\end{cases}
\end{equation}
In particular, for $k \in \Z$ we have the $k$-fold shift $[k] = (\phi
= \id_M,\alphagr_{\phi} = -k)$, which reduces the grading of each
Lagrangian submanifold by $k$. All the ``graded'' notions actually
depend only on the homotopy class of $\eta_M^2$ as a trivialization
of $K_M^2$, or equivalently on the cohomology class $\mu_M =
[\alpha_M] \in H^1(\LL_M)$, called a global Maslov class in
\cite{seidel99}. Namely, suppose that we have two cohomologous phase
functions, which can be written as $\alpha_M' = \alpha_M e^{2\pi i
\chi}$ for some $\chi: \LL_M \rightarrow \R$. Gradings with respect
to the two are related in an obvious way,
\begin{equation} \label{eq:new-phase}
\begin{cases}
 \!\!\! & \alphagr_L'(x) = \alphagr_L(x) + \chi(TL_x), \\
 \!\!\! & \alphagr_\phi'(\Lambda) = \alphagr_\phi(\Lambda) +
 \chi(D\phi(\Lambda)) - \chi(\Lambda).
\end{cases}
\end{equation}
Note that $\chi$ is unique only up to an integer constant. This is
irrelevant for $\alphagr_\phi'$; it affects $\alphagr_L'$ but not in
a really important way, since changing $\chi$ shifts the grading of
all Lagrangians simultaneously.

\begin{remark} \label{th:or}
There is a canonical square root $\alpha_M^{1/2}: \LL^{\orient}
\rightarrow \R$ of $\alpha_M$ on the bundle $\LL^{\orient}$ of
oriented Lagrangian Grassmannians, defined by removing the squares
from \eqref{eq:squared-phase} and using positively oriented bases
$e_1,\dots,e_n$. Any graded Lagrangian submanifold carries a
preferred orientation, characterized by
\[
\exp(\pi i \alphagr_L(x)) = \alpha_M^{1/2}(TL_x).
\]
\end{remark}

\subsection{}
There is an obvious notion of graded symplectic fibre bundle, which
is a locally trivial symplectic fibration $p: E \rightarrow B$ with
an almost complex structure $J_{E_b}$ and complex volume form
$\eta_{E_b}$ on each fibre $E_b$. The monodromy maps of such a
fibration are naturally graded: given a loop $\gamma: [0;l]
\rightarrow B$ and the corresponding parallel transport maps $\phi_t
= h_{\gamma|[0;t]}: E_{\gamma(0)} \rightarrow E_{\gamma(t)}$, one
takes the unique smooth family of functions $\tilde{a}_t$ on the
Lagrangian Grassmannian bundle of $E_{\gamma(0)}$ with $\tilde{a}_0 =
0$ and
\[
\exp(2\pi i \tilde{a}_t) = (\alpha_{E_{\gamma(t)}} \circ
D\phi_t)/\alpha_{E_{\gamma(0)}}.
\]
The graded symplectic monodromy is $\hgr_{\gamma} =
(\phi_l,\alphagr_{\phi_l} = \tilde{a}_l)$. An example which is
relevant for our purpose is where we have a trivial fibre bundle $E =
\C^* \times M$ over $B = \C^*$ with a varying complex volume form
$\eta_{E_z} = z^\mu \eta_M$. The ``invisible singularity'' of this
family at $z = 0$ leads to a monodromy which, for a circle $\gamma$
turning around the origin in clockwise direction, is the shift
\begin{equation} \label{eq:shift-monodromy}
\hgr_\gamma = [2\mu].
\end{equation}

\subsection{}
Gradings make it possible to assign absolute Maslov indices to
Lagrangian intersection points. This depends on the following notion
from symplectic linear algebra: a smooth path of Lagrangian subspaces
$\Lambda_t$ ($0 \leq t \leq 1$) in a fixed symplectic vector space is
{\em crossingless} if it satisfies $\Lambda_0 \cap \Lambda_t =
\Lambda_0 \cap \Lambda_1$ for all $t>0$, and the crossing form
\cite{robbin-salamon93} on $\Lambda_0/\Lambda_0 \cap \Lambda_1$
associated to $(d\Lambda_t/dt)_{t = 0}$ is negative definite.
Nonintersecting paths with given endpoints exist and are unique up to
homotopy, see again \cite{robbin-salamon93}.

\begin{lemma} \label{th:angle-bound}
Suppose that $\alpha_M$ is classical. Let $\Lambda_t$ be a
crossingless path in $TM_x$ for some $x$. Choose $\tilde{a}: [0;1]
\rightarrow \R$ such that $\exp(2\pi i \tilde{a}_t) =
\alpha_M(\Lambda_t)$. Then
\[
\tilde{a}_1 - \tilde{a}_0 \in (-n;0].
\]
\end{lemma}

\proof We can identify $\Phi: TM_x \rightarrow \C^n$ in such a way
that the complex structure and symplectic form become standard,
$\Phi(\Lambda_0) = \R^n$ and $\Phi(\Lambda_1) = e^{i\pi c_1} \R
\times \dots \times e^{i\pi c_n}\R$ with $c_k \in (-1;0]$. For the
obvious crossingless path $\Lambda_t = e^{i\pi t c_1}\R \times \dots
\times e^{i\pi t c_n}\R$ we have $\tilde{a}_1 - \tilde{a}_0 =
c_1+\dots+c_n \in (-n;0]$. Because of homotopy uniqueness, the same
holds for any other crossingless path. \qed

Take two graded Lagrangian submanifolds $\Lgr_0,\Lgr_1$, and a point
$x \in L_0 \cap L_1$. Choose a crossingless path from $\Lambda_0 =
TL_0$ to $\Lambda_1 = TL_1$, and take $\tilde{a}_t \in \R$ such that
$\exp(2\pi i \tilde{a}_t) = \alpha_M(\Lambda_t)$. The absolute Maslov
index is defined as
\begin{equation} \label{eq:absolute-index}
I(x) = (\alphagr_{L_1}(x) - \tilde{a}_1) - (\alphagr_{L_0}(x) -
\tilde{a}_0) \in \Z.
\end{equation}
Lemma \ref{th:angle-bound} implies that this is approximately the
difference of the phases:

\begin{lemma} \label{th:approximate-index}
If $\alpha_M$ is classical, $I(x) - \alphagr_{L_1}(x) +
\alphagr_{L_0}(x) \in [0;n)$. \qed
\end{lemma}

This was pointed out to me by Joyce, but it is known to many other
people in mathematics and physics, see for instance the proof of
\cite[Theorem 4.3]{thomas-yau01}. In fact, ideas of this kind can be
traced back at least to \cite{salamon-zehnder92}. Another easy
consequence of Lemma \ref{th:angle-bound}, this time for symplectic
automorphisms, will be useful later:

\begin{lemma} \label{th:graded-variation}
Suppose that $\alpha_M$ is classical. Let $\phigr =
(\phi,\alphagr_\phi)$ a graded symplectic automorphism of $M$. For
any two Lagrangian subspaces $\Lambda_0,\Lambda_1 \in \LL_{M,x}$ at
the same point,
\[
|\alphagr_\phi(\Lambda_1)-\alphagr_\phi(\Lambda_0)| < n.
\]
\end{lemma}

\proof Choose a crossingless path $\Lambda_t$. Lemma
\ref{th:angle-bound} shows that as we go along this path,
$\alpha_M(\Lambda_t)$ changes by an angle in $(-2\pi n;0]$, and so
does $\alpha_M(D\phi(\Lambda_t))$. The angular change in
$\alpha_M(D\phi(\Lambda_t))/\alpha_M(\Lambda_t)$ is the difference
between the two, hence lies in $(-2\pi n;2\pi n)$. \qed

\subsection{}
We now return to the case of an affine K{\"a}hler manifold $M = X
\setminus X_\infty$. We say that $M$ is {\em affine Calabi-Yau} if it
comes with a preferred isomorphism of holomorphic line bundles,
\begin{equation} \label{eq:beta}
\beta_X : \K_X \stackrel{\iso}{\longrightarrow} o_X^{\otimes -m_X}
\end{equation}
for some $m_X \in \Z$ (called the monotonicity index). On $M$ we have
a preferred trivialization of $o_X$ given by $\sigma_\infty$, hence
through $\beta_X$ a holomorphic volume form $\eta_M$. In the special
case where $m_X = 0$, $X$ itself is Calabi-Yau, and $\eta_M$ extends
to a holomorphic volume form $\eta_X$ on it. We denote by $\Autgr(M)$
the group of exact symplectic automorphisms of $M$ equipped with a
grading. Similarly, if $m_X = 0$, we write $\Autgr(X)$ for the graded
rational automorphisms of $X$.

\begin{remark} \label{th:extend-grading}
In the $m_X = 0$ case, there is a canonical embedding
\[
\Autgr(M) \longrightarrow \Autgr(X).
\]
We already know that every $\phi \in Aut(M)$ extends to a rational
symplectic automorphism of $X$. There are no obstructions to
extending gradings over subsets of codimension $\geq 2$, so the
grading of $\phi$ extends uniquely from $M$ to $X$.
\end{remark}

If $\{X_z \subset X\}$ is a quasi-Lefschetz pencil, we have for $z
\in \C \setminus Critv(\pi_M)$ a canonical isomorphism $\beta_{X_z}:
\K_{X_z} \iso (\K_X \otimes o_X)|X_z \iso o_X^{\otimes -m_X+1}$, so
the $M_z$ are again affine Calabi-Yaus, with monotonicity index
$m_{X_z} = m_X-1$. More explicitly, the complex volume form is
obtained as a quotient $\eta_M/dz$, which means that it is the unique
solution of
\begin{equation} \label{eq:division}
dz \wedge \eta_{M_z} = \eta_M.
\end{equation}
The first consequence of this is that the monodromy maps along loops
$\gamma$ in $\C \setminus Critv(\pi_M)$ have canonical gradings,
$\hgr_\gamma \in \Autgr(M_{\gamma(0)})$. If $c$ is a vanishing path,
the associated vanishing cycle $V_c$ admits a grading; this is
trivial if its dimension is $\geq 2$, and otherwise follows from the
fact that it is bounded by a Lagrangian disc in $M$. Hence, the
associated Dehn twist $\tau_{V_c}$ then has a canonical grading
$\taugr_{V_c}$, which is zero outside a neighbourhood of $V_c$
itself. Taking the loop $\gamma$ which doubles $c$, we have the
following graded version of the Picard-Lefschetz theorem:
\[
\hgr_\gamma \htp \taugr_{V_c} \in \Autgr(M_{c(0)}).
\]
The analogue of \eqref{eq:relation-in-mcg} is

\begin{lemma} \label{th:shift-factor}
Suppose that $\{X_z\}$ is a Lefschetz pencil, and $c_1,\dots,c_m$ a
distinguished basis of vanishing cycles. Then
\[
\taugr_{V_{c_1}} \circ \dots \circ \taugr_{V_{c_r}} \htp [4-2 m_X]
\in \Autgr(M_{z_*}).
\]
\end{lemma}

\proof What we need to show is that the graded monodromy around a
large circle $\gamma_\infty$ is $[4-2 m_X]$. For simplicity, change
variables to $\zeta = z^{-1}$, so that $\gamma_\infty$ becomes a
small circle going clockwise around $\zeta = 0$. Recall that
$\eta_{M_z}$ is obtained by dividing $\eta_M$, which has a zero or
pole of order $-m_X$ along $X_\infty$, by $dz = d\zeta/\zeta^2$,
which has order $-2$. Hence, after introducing the compactified graph
$\bar{\Gamma}$ as in \eqref{eq:bar-gamma}, one has a locally trivial
symplectic fibration near $\zeta = 0$, with a family of complex
volume forms on the fibres that grows like $\zeta^{2-m_X}$. Now we
are in the same situation as in \eqref{eq:shift-monodromy}. \qed

\section{Monodromy and negativity\label{sec:negativity}}

Continuing the discussion above, suppose that $m_X = 1$, so that the
fibres $X_z$ are Calabi-Yau manifolds. In the Lefschetz pencil case,
Lemma \ref{th:shift-factor} tells us that the ``large complex
structure limit monodromy'' is a downward shift. We will need a
generalization of this statement to the case when $X_\infty$ has
normal crossings, where the geometry of the monodromy map becomes
more complicated. Intuition is provided by two aspects of mirror
symmetry: ideas related to the SYZ conjecture suggest that large
complex structure limit monodromy maps should be fibrewise
translations in a Lagrangian torus fibration; and in the context of
homological mirror symmetry we have a formula for the mirror
automorphism of the derived category of sheaves which goes back to
Kontsevich, see \cite{aspinwall01}. Because of technical
difficulties, our result will finally be limited to $K3$ surfaces;
but large parts of the argument are more general, and it seems likely
that with some more effort, the dimensional restriction could be
lifted altogether.

\subsection{}
We start with some generalities. Let $M^{2n}$ be a compact connected
symplectic manifold (possibly with boundary), with a tame almost
complex structure $J_M$, complex volume form $\eta_M$, and associated
phase function $\alpha_M$.

\begin{defn} \label{th:negativity-definition}
A graded symplectic automorphism $\phigr$ of $M$ is {\em negative} if
there is a $d_0>0$ such that
\[
\alphagr_{\phi^{d_0}}(\Lambda) < 0 \quad \text{ for all $\Lambda \in
\LL_M$.}
\]
\end{defn}

Here, the grading of $\phi^{d_0}$ is defined by the composition rule
\eqref{eq:compose-gradings}. The same rule shows that if negativity
holds, the grading of the iterates goes to $-\infty$. Namely, the
grading of $\phi^{d_0}$ is $\leq -\epsilon$ for some $\epsilon>0$,
and the grading of $\phi$ is $\leq C$ for some $C>0$. By writing $d =
k d_0 + l$ with $0 \leq l \leq d_0-1$ one sees that
\begin{equation} \label{eq:growth}
 \alphagr_{\phi^d}(\Lambda) \leq C l - k\epsilon <
 Cd_0 - \epsilon [d/d_0],
\end{equation}
as claimed. A closely related property is that the negativity of a
given graded symplectic automorphism depends only on the global
Maslov class $\mu_M$, and not on the choice of $\alpha_M$ within that
class. For this take two cohomologous phase functions $\alpha_M' =
e^{2 \pi i \chi} \alpha_M$. Since $\chi$ is bounded, it follows from
\eqref{eq:new-phase} that the gradings of $\phi^d$ with respect to
$\alpha_M,\alpha_M'$ differ by a uniformly bounded amount, which in
view of \eqref{eq:growth} gives the desired property.

\begin{lemma} \label{th:maslov-growth}
Let $\phigr$ be a negative graded symplectic automorphism, and
$\Lgr_0,\Lgr_1$ two graded Lagrangian submanifolds. For any $k \in
\Z$ one can find a $d>0$ and a small exact perturbation $L_1'$ of
$L_1$ such that for each $x \in L_0 \cap \phi^d(L_1')$, the absolute
Maslov index is
\[
I(x) < k.
\]
\end{lemma}

\proof Since negativity is independent of the choice of phase
function, we may assume that that function is classical. Suppose that
the grading of $L_1$ is bounded above by $C_1$, and that of $L_0$
bounded below by $C_0$. Choose some $d$ such that $C_1 - C_0 + C d_0
- \epsilon[d/d_0] \leq k-n$. From \eqref{eq:growth} and
\eqref{eq:compose-gradings} it follows that
\begin{equation} \label{eq:image-d}
 \alphagr_{\phi^d(L_1)}(x) - \alphagr_{L_0}(x)
 < k-n
\end{equation}
at every intersection point $x$. This estimate continues to hold if
one perturbs $L_1$ slightly, to make the intersections transverse.
Now apply Lemma \ref{th:approximate-index}. \qed

Negativity is well-behaved with respect to abelian symplectic
reduction (this fact is related to the quotient construction of
special Lagrangian submanifolds, see for instance \cite{gross01b}).
The situation is as follows: $M = M^{2n}$ carries a free Hamiltonian
$T^k$-action, with moment map $\mu: M \rightarrow \R^k$. We have an
almost complex structure $J_M$ which is $\o_M$-compatible and
$T^k$-invariant, and a $T^k$-invariant complex volume form $\eta_M$,
with associated (classical and $T^k$-invariant) phase function
$\alpha_M$. Denote by $K_1,\dots,K_k$ the Killing fields of the
action. Let $\phi$ be a symplectic automorphism which preserves
$\mu$, hence is $T^k$-equivariant; and $\alphagr_\phi$ a grading of
it (necessarily $T^k$-invariant). For every moment value $r$ we have
the symplectic quotient $M^{\red,r} = \mu^{-1}(r)/T^k$. This carries
the reduced symplectic form $\o_{M^{\red,r}}$ and a compatible almost
complex structure $J_{M^{\red,r}}$, the latter of which is obtained
by taking the complexified tangent spaces along the orbits, and
identifying their orthogonal complement with the tangent spaces of
$M^{\red,r}$. Moreover, $\phi$ induces symplectic automorphisms
$\phi^{\red,r}$. In our situation, we also have a reduced complex
volume form $\eta_{M^{\red,r}}$, defined by restricting
$
i_{K_1} \dots i_{K_r} \eta_M
$
to the same orthogonal complement as before. Projection identifies
Lagrangian subspaces $\Lambda \subset TM$ containing the tangent
space to the orbit with Lagrangian subspaces $\Lambda^{\red} \subset
TM^{\red,r}$. The phase functions defined by $\eta_M$,
$\eta_{M^{\red,r}}$ are related by
$\alpha_{M^{\red,r}}(\Lambda^{\red}) = \alpha_M(\Lambda)$. In
particular, we can define a grading of the reduced symplectic
automorphisms by
\begin{equation} \label{eq:lift-lambda}
\alphagr_{\phi^{\red,r}}(\Lambda^{\red}) = \alphagr_\phi(\Lambda).
\end{equation}

\begin{lemma} \label{th:negative-reduction}
Suppose that the $\phigr^{\red,r}$ are all negative. Then $\phigr$ is
negative.
\end{lemma}

\proof Choose some $r$. By assumption there is a ${d_0}$ such that
the grading of $(\phigr^{\red,r})^{d_0}$ is negative, and then the
same thing holds for neighbouring $r$. By a covering argument, one
sees that there is a $d_0$ such that the gradings of the $d_0$-th
iterates of all reduced maps are negative. Applying \eqref{eq:growth}
shows that for some large $d$, the grading of each
$(\phigr^{\red,r})^d$ will be $<-n$ everywhere. By
\eqref{eq:lift-lambda}, this means that at each point of $M$ there
are some Lagrangian tangent subspaces whose $\phigr^d$-grading is
$<-n$. From this and Lemma \ref{th:graded-variation} one gets the
desired conclusion. \qed

\subsection{}
We will now look at the local models for the ``large complex
structure limit'' monodromy, first under the rather unrealistic
assumption that both the symplectic and complex structures are
standard, which allows one to integrate the monodromy vector field
explicitly.

\begin{assumptions} \label{as:1}
Fix $n \geq 2$ and $2 \leq k \leq n$. We take $Y = \C^n$ with the
standard complex structure and symplectic form $\o_Y$, and the map
\[
p: \C^n \longrightarrow \C, \quad y \longmapsto y_1 y_2 \dots y_k.
\]
$Y$ should carry the complex volume form (away from the zero fibre)
\[
\frac{dy_1}{y_1} \wedge \dots \wedge \frac{dy_k}{y_k} \wedge dy_{k+1}
\wedge \dots \wedge dy_n.
\]
\end{assumptions}

The monodromy vector field $Z$ is the unique horizontal lift of the
clockwise rotational field $-i\zeta \partial_\zeta$. Explicitly,
\[
 Z = -i
 \frac{\left(\frac{y_1}{|y_1|^2}, \dots, \frac{y_k}{|y_k|^2},0,\dots,0\right)}
 {\frac{1}{|y_1|^2} + \dots + \frac{1}{|y_k|^2}},
\]
In particular, since each $|y_j|^2$ is invariant, the flow $\psi_t$
of $Z$ is well-defined for all times, in spite of the noncompactness
of the fibres. As in our discussion of Lefschetz pencils, we define
complex volume forms $\eta_{Y_z}$ on the fibres by (formally)
dividing $\eta_Y$ by some complex one-form on the base, which in this
case will be $d\zeta/\zeta^2$. As a consequence, if at some point $y
\in Y_\zeta = p^{-1}(\zeta)$ we have a Lagrangian subspace $\Lambda
\subset TY_y$ with $Dp(\Lambda) = a\R$, $a \in S^1$, then the phase
of the vertical part $\Lambda^v = \Lambda \cap \ker(Dp)$ inside
$Y_\zeta$ is related to the phase of $\Lambda$ by
\begin{equation} \label{eq:divide-degree}
\alpha_{Y_\zeta}(\Lambda^v) = \frac{\zeta^4}{a^2 |\zeta|^4}
\alpha_Y(\Lambda).
\end{equation}

For each point $y$ with $p(y) = \zeta \neq 0$, consider the
Lagrangian tangent subspace $\Lambda_y = \R i y_1 \oplus \dots \oplus
\R i y_k \oplus \R^{n-k} \subset TY_y$. This satisfies
$\alpha_Y(\Lambda_y) = (-1)^k$ and $Dp(\Lambda_y) = i\zeta \R$, hence
\eqref{eq:divide-degree} says that
\begin{equation} \label{eq:lambda-shift}
\alpha_{Y_z}(\Lambda_y^v) = \frac{\zeta^2}{|\zeta|^2} (-1)^{k-1}.
\end{equation}
$\psi_t$ takes the $\Lambda_y$ to each other, and rotates the base
coordinate $\zeta$, hence it also maps the $\Lambda_y^v$ to each
other. Let $h_\zeta = \psi_{2\pi}|Y_z: Y_z \rightarrow Y_z$ be the
monodromy on the fibre over $\zeta$. By definition and
\eqref{eq:lambda-shift}, the canonical grading of this map satisfies
\begin{equation} \label{eq:integrate-shift}
\begin{aligned}
 \alphagr_{h_{\zeta}}(\Lambda_y^v)
 & = \frac{1}{2\pi} \int_0^{2 \pi}
 \frac{d}{dt} \arg(\alpha_{Y_z}(D\psi_t(\Lambda_y^v))) \, dt
 \\ & = \frac{1}{2\pi} \int_0^{2 \pi}
 \frac{d}{dt} \arg( (-1)^{k-1} e^{-2it}) \, dt = -2.
\end{aligned}
\end{equation}
If one passes to a sufficiently large iterate such as $h_\zeta^n$,
the grading on $\Lambda_y^v$ is $-2n$, hence by Lemma
\ref{th:graded-variation} the grading of any $\Lambda$ is $<-n-1$. In
a slight abuse of terminology (since $Y_\zeta$ is not compact), we
conclude that the monodromy is a negative graded symplectic
automorphism, in a way which is uniform for all $\zeta$.

\subsection{}
A different local model appears at the singular points of the base
locus of our pencil. In Ruan's work on the SYZ conjecture, see in
particular \cite{ruan02b}, these are responsible for the
singularities of the Lagrangian torus fibration. For closely related
reasons, the negativity of the monodromy breaks down at such points.

\begin{assumption} \label{as:2}
Fix $n \geq 3$ and $3 \leq k \leq n$. We again take $Y = \C^n$ with
the standard complex and symplectic forms, but now with the rational
map
\[
p: Y \dashrightarrow \C, \qquad p(y) = \frac{y_2 \dots y_k}{y_1},
\]
in the presence of the complex volume form (away from the closure of
the zero fibre)
\[
\eta_Y = dy_1 \wedge \frac{dy_2}{y_2} \wedge \dots \wedge
\frac{dy_k}{y_k} \wedge dy_{k+1} \wedge \dots \wedge dy_n.
\]
\end{assumption}

Denote by $Y_\zeta$ the closure of $p^{-1}(y)$. For $\zeta \neq 0$,
each such fibre can be identified with $\C^{n-1}$ by projecting to
$(y_2,\dots,y_n)$. We will use this identification freely, hoping
that this does not cause too much confusion. For instance, the
symplectic form on $Y_\zeta \iso \C^{n-1}$ is given by
\begin{equation} \label{eq:zeta-symplectic-form}
 \o_{Y_\zeta} = \frac{i}{2} \partial\bar\partial
 \Big(\frac{|y_2\dots y_k|^2}{|\zeta|^2}
 + |y_2|^2 + \dots + |y_n|^2 \Big)
\end{equation}
As in our previous model, we write $Z$ for the clockwise monodromy
vector field (defined by passing to the graph of $p$). In view of our
identification, this becomes a $\zeta$-dependent vector field on
$\C^{n-1}$, namely:
\[
Z = -i
\frac{\left(\frac{y_2}{|y_2|^2},\dots,\frac{y_k}{|y_k|^2},0,\dots,0\right)}
{\frac{1}{|y_1|^2} + \frac{1}{|y_2|^2} + \dots + \frac{1}{|y_k|^2}}
\]
where $y_1 = y_2 \dots y_k/\zeta$. Again, this preserves $|y_2|^2,
\dots, |y_n|^2$, so that the flow $\psi_t$ is well-defined for all
times.

Taking $\eta_Y$ on the total space and $d\zeta/\zeta^2$ on the base,
one finds that the induced volume form $\eta_{Y_\zeta}$ is the
standard form on $\C^{n-1}$. At each point $y \in Y_\zeta$ with $y_2
\dots y_k \neq 0$, consider the Lagrangian subspace $\Lambda_y^v = \R
i y_2 \oplus \dots \oplus \R i y_k \oplus \R^{n-k}$. These subspaces
are preserved by the flow $\psi_t$, and a computation similar to
\eqref{eq:integrate-shift} shows that the canonical grading of the
monodromy map $h_\zeta = \psi_{2\pi}|Y_\zeta$ satisfies
\[
 \alphagr_{h_\zeta}(\Lambda_y^v) = - 2
 \frac{1}{1 + \frac{1}{\frac{|y_1|^2}{|y_2|^2} +
 \cdots + \frac{|y_1|^2}{|y_k|^2}}}.
\]
This is obviously negative, but of course we also need to consider
the grading at the points where $\Lambda_y^v$ is not defined. Suppose
for simplicity that the sizes of the first $k-1$ coordinates of $y
\in Y_\zeta$ are ordered, $|y_2| \leq |y_3| \leq \dots |y_{k-1}|$.
Then
\[
 \alphagr_{h_\zeta}(\Lambda_y) \leq -2
 \frac{1}{1 + \frac{|\zeta|^2}{|y_3|^{2(k-1)}}},
\]
which means that the grading is bounded above by something which
depends on the size of the second smallest coordinate. As a
consequence, if one has a sequence
\[
 y^{(m)} \rightarrow y^{(\infty)}, \quad
 y_2^{(m)} \dots y_k^{(m)} \neq 0 \text{ for all $m$,}
\]
and such that only one of the $(y_2^{(\infty)}, \dots,
y_k^{(\infty)})$ vanishes, the
$\alphagr_{h_\zeta}(\Lambda^v_{y^{(m)}})$ are bounded above by a
negative constant, hence cannot go to zero. Roughly speaking, this
means that $h_\zeta$ is negative away from the subset where at least
two of the $y_2,\dots,y_k$ vanish. In the special case $k = n = 3$,
this subset is a point $0 \in Y_{\zeta}$; one can then apply
\eqref{eq:compose-gradings} and Lemma \ref{th:graded-variation} much
as in our previous local model, to get the following statement:

\begin{lemma} \label{th:bad-points}
Suppose that $k = n = 3$. Let $W \subset Y_\zeta$ be some open ball
around the origin. Then $h_\zeta(W) = W$, and there is a $d>0$ such
that the grading of $h_\zeta^d|(Y_\zeta \setminus W)$ is negative.
\qed
\end{lemma}

\begin{remark}
As mentioned before, ideas associated to the SYZ conjecture suggest
that the monodromy should be a fibrewise translation in a SLAG
fibration. From this point of view, the negativity of the monodromy
is based on the idea that the phases of the tangent spaces to the
fibration should be shifted by $-2$. In the situation of Assumptions
\ref{as:1}, this is indeed the case: $L_r \iso T^{k-1} \times
\R^{n-k} \subset Y_\zeta$, given by
\[
 L_r = \{|y_1|^2 = r_1, \dots, |y_k|^2 = r_k, \im(y_{k+1}) = r_{k+1},
 \dots, \im(y_n) = r_n\}
\]
with $r_1 \dots r_k = |\zeta|^2$, are special Lagrangian with respect
to $\eta_{Y_\zeta}$. The monodromy translates each of them, and the
shift is computed in \eqref{eq:integrate-shift}. For Assumptions
\ref{as:2}, there is a standard SLAG fibration with singularities on
$Y_\zeta \iso \C^{n-1}$ (with respect to
\eqref{eq:zeta-symplectic-form} and the standard complex volume
form), see e.g.\ \cite{gross01b}:
\begin{multline} \label{eq:syz}
 \qquad L_r = \{
 |y_3|^2 - |y_2|^2 =
 r_1, \, \dots \, |y_k|^2 - |y_2|^2 = r_{k-2}, \\
 \im(y_2\dots y_k) = r_{k-1}, \,
 \im(y_{k+1}) = r_k, \dots, \im(y_n) = r_{n-1}\}. \qquad
\end{multline}
However, the monodromy preserves this only asymptotically far away
from the singular locus, and hence the grading is only asymptotically
equal to $-2$. To see more precisely what happens, we apply a
symplectic reduction procedure, which means mapping $Y_\zeta
\rightarrow \C$ by sending $y$ to $w = y_2y_3 \dots y_k$. The fibres
of the SLAG fibration lie over horizontal lines in the base $\C$. The
clockwise monodromy also fibres over a diffeomorphism of $\C$, which
rotates each point $w$ by some $|w|$-dependent angle. For $|w| \gg
0$, this is almost a full clockwise rotation, hence it approximately
preserves the horizontal lines, but near $w = 0$ the angle of
rotation undergoes a full change $0$ to $-2\pi$.

On the other hand, as pointed out to me by Mark Gross, in the $K3$
case one can construct a symplectic automorphism which preserves all
the fibres of a SLAG fibration. This is obtained by hyperk{\"a}hler
rotation from a complex automorphism, and it is presumably isotopic
to the monodromy map, even though that has not strictly speaking been
proved. So, it may well be that the failure of our local models to
preserve the fibrations \eqref{eq:syz}, and the corresponding
breakdown in negativity, could be repaired by an isotopy which is
more global in nature.
\end{remark}

\subsection{}
We now relax our previous assumptions on the complex and symplectic
structures, for the first local model. We choose to work in Darboux
coordinates, and with a complex structure and holomorphic function
which are not standard, but which still preserve a $T^k$ symmetry.

\begin{assumptions} \label{as:1prime}
Fix $n \geq 2$ and $2 \leq k \leq n$.
\begin{itemize} \itemsep1em
\item
$Y \subset \C^n$ is an open ball around the origin, carrying the
standard symplectic form $\o_Y$ and the standard diagonal
$T^k$-action
\[
\rho_s(y) = (e^{is_1}y_1,\dots,e^{is_k}y_k,y_{k+1},\dots,y_n),
\]
with moment map $\mu: Y \rightarrow (\Rgeq)^k$. For any regular
moment value $r$, the quotient $Y^{\red,r}$ can be identified with an
open subset of $\C^{n-r}$, with the standard symplectic form
$\o_{Y^{\red,r}}$. We denote points in the reduced spaces by
$y^{\red} = (y_{k+1},\dots,y_n)$.

\item
$J_Y$ is a complex structure which is tamed by $\o_Y$. At the point
$y = 0$ (but not necessarily elsewhere), it is $\o_Y$-compatible and
$T^k$-invariant.

\item
$p: Y \rightarrow \C$ is a $J_Y$-holomorphic function with the
following properties:
\begin{romanlist}
\item \label{item:t-sym}
$p(\rho_s(y)) = e^{i(s_1+\dots+s_k)}p(y)$.
\item \label{item:nondeg}
$\partial_{y_1}\dots \partial_{y_k} p$ is nonzero at the point $y =
0$.
\end{romanlist}

\item
$\eta_Y$ is a $J_Y$-complex volume form on $Y \setminus p^{-1}(0)$,
such that $p(y)\eta_Y$ extends to a smooth form on $Y$, which is
nonzero at $y = 0$. Let $\alpha_Y$ be the corresponding phase
function.
\end{itemize}
\end{assumptions}

\begin{lemma} \label{th:product-function}
One can write
\[
p(y) = 2^{-k/2} y_1 y_2\dots y_k\, q(\half |y_1|^2,\dots,\half
|y_k|^2,y_{k+1},\dots,y_n)
\]
for some smooth function $q$ satisfying $q(0) \neq 0$.
\end{lemma}

\proof Consider the Taylor expansion of $p$ around $y = 0$. From
property \ref{item:t-sym} of $p$ one sees that each monomial which
occurs with nonzero coefficient in that expansion contains one of the
factors $y_l$ for each $1 \leq l \leq k$. As a consequence,
$p(y)/(y_1\dots y_k)$ extends smoothly over $y = 0$. The same
argument can be applied to points where only certain of the
coordinates $y_1 \dots y_k$ vanish. Again applying \ref{item:t-sym},
one sees that $p(y)/(y_1\dots y_k)$ is $T^k$-invariant, which means
that it can be written in the form stated above. Finally, $q(0) \neq
0$ is property \ref{item:nondeg}. \qed

\begin{lemma} \label{th:shrink}
After making $Y$ smaller if necessary, one has
\begin{align*}
 p(y) = 0 & \Longleftrightarrow  y_l = 0
 \text{ for some } l = 1,\dots,k; \\
 Dp(y) = 0 & \Longleftrightarrow y_l = 0
 \text{ for at least two } l = 1,\dots,k.
\end{align*}
\end{lemma}

\proof Suppose that $q(y) \neq 0$ on the whole of $Y$. Then the first
statement is obvious, and so is the $\Leftarrow$ implication in the
second one. Conversely, suppose that $y$ is a critical point.
Property \ref{item:t-sym} implies that $p(y) = 0$, so that $y_l = 0$
for some $l = 1,\dots,k$. From 0 = $Dp_y = y_1 \dots \hat{y}_l \dots
y_k \, q(y)\, dy_l$ one sees that another of the $y_1,\dots, y_k$ has
to vanish too. \qed

We will assume from now on that the conclusions of Lemma
\ref{th:shrink} hold. Consider the function $H(y) = -\half |p(y)|^2$,
its Hamiltonian vector field $X$ and flow $\phi_t$. Since $H$ is
$T^k$-invariant, the flow (where defined) is equivariant and
preserves the level sets of the moment map. For every regular moment
value $r$, we can consider the induced function
\begin{equation} \label{eq:h-red}
H^{\red,r}(y^\red) = -\half r_1 \dots r_k \,
|q(r_1,\dots,r_k,y_{k+1},\dots,y_n)|^2,
\end{equation}
on $Y^{\red,r}$, its vector field $X^{\red,r}$ and flow
$\phi_t^{\red,r}$. If the moment value is small, so are
\eqref{eq:h-red} and all its $y^\red$-derivatives, which means that
the reduced flow moves very slowly. We need a quantitative version of
this:

\begin{lemma} \label{th:small-flow}
Given $\epsilon_1,\epsilon_2>0$, there are
$\delta_1,\delta_2,\delta_3>0$ such that for all
\[
\begin{cases}
 r \in (\R^{\scriptscriptstyle>0})^k &
 \text{with $||r|| < \delta_1$}, \\
 y^\red \in Y^{\red,r} &
 \text{with $||y^\red|| < \delta_2$}, \\
 t \in \R &
 \text{with $|t| < \delta_3 r_1^{-1} \dots r_k^{-1}$},
\end{cases}
\]
the following holds: $\phi_t^{\red,r}$ is well-defined near $y^\red$,
and satisfies
\begin{align}
 \label{eq:c0} &
 ||\phi_t^{\red,r}(y^{\red}) || < \epsilon_1, \\
 \label{eq:c1} &
 ||(D\phi_t^{\red,r})_{y^{\red}} - \mathrm{Id} || < \epsilon_2.
\end{align}
\end{lemma}

\proof By Lemma \ref{th:product-function} there are $\delta_1>0$,
$\delta_2 \in (0;\min(\epsilon_1/2,\epsilon_2/2))$ and a $C>0$, such
that
\[
||X^{\red,r}_{y^\red}||, \; ||DX^{\red,r}_{y^{\red}}|| \leq C
r_1\dots r_k
\]
for each $||r|| < \delta_1$, $||y^{\red}|| < 2\delta_2$ (we use the
Hilbert-Schmidt norm for matrices $DX^{\red,r}$, normalized to
$||\mathrm{Id}|| = 1$). We claim that one can take $\delta_3 =
\delta_2/C $. The first estimate \eqref{eq:c0} is clear: if one
starts at a point $y^\red$ of norm less than $\delta_2$, and moves
with speed at most $C r_1 \dots r_k$ for a time less than $\delta_2
C^{-1} r_1^{-1} \dots r_k^{-1}$, the endpoint will be of norm less
than $2\delta_2 < \epsilon_1$. Next, use the linearized flow to get
an exponential growth bound for the time interval $t$ relevant to us:
$||D\phi_t^{\red,r}|| \leq \exp(C r_1 \dots r_k t) \leq
\exp(\delta_2)$. One may assume that $\exp(\delta_2) \leq 2$. By
plugging this back into the linearized equation, one gets the linear
estimate $||D\phi_t^{\red,r}-\mathrm{Id}|| \leq 2 C r_1 \dots r_k t
\leq 2 \delta_2 < \epsilon_2$, from which \eqref{eq:c1} follows. \qed

We now need to deal with the fact that $J_Y$ and $\eta_Y$ are only
approximately $T^k$-invariant. Let $J_Y'$ be the constant complex
structure on $Y$ which agrees with $J_Y$ at $y = 0$. This is
$\o_Y$-compatible and $T^k$-invariant, so
\[
J_Y' = \begin{pmatrix} i \cdot 1_k & 0 \\ 0 & J_{Y^{\red}}'
\end{pmatrix}.
\]
$J_{Y^{\red}}'$ is the induced complex structure on $Y^{\red,r}$ for
all $r$. Similarly, we consider the $J_Y'$-complex volume form
$\eta'_Y$ which is obtained by taking the value of $p(y)\eta_Y$ at
the point $y = 0$, and dividing that by $y_1\dots y_k$. This can be
written as
\[
\eta_Y' = i^k \frac{dy_1}{y_1} \wedge \dots \wedge \frac{dy_k}{y_k}
\wedge \eta_{Y^\red}',
\]
hence is $T^k$-invariant. $\eta_{Y^\red}'$, which is the induced
complex volume form on the reduced spaces, is again constant and
$r$-independent. Let $\alpha_Y'$ be the phase function associated to
$(J_Y',\eta_Y')$ (unlike $\alpha_Y$, this is a classical phase
function), and $\alpha_{Y^{\red,r}}'$ the induced phase functions on
the quotients.

\begin{lemma} \label{th:phase-difference}
Given $\epsilon_3>0$, there is $\delta_4>0$ such that for all $||y||
< \delta_4$ with $p(y) \neq 0$, and all $\Lambda \in \LL_{Y,y}$,
\[
\Big|\frac{1}{2\pi}\arg(\alpha_Y'(\Lambda)/\alpha_Y(\Lambda))\Big| <
\epsilon_3
\]
(by this we mean that there is a branch of the argument with that
property).
\end{lemma}

\proof The quotient $\alpha_Y'(\Lambda)/\alpha_Y(\Lambda)$ does not
change if we multiply both $\eta_Y$, $\eta_Y'$ with $y_1\dots y_k$.
But by construction,
\begin{align*}
 & y_1 \dots y_k \eta_Y = 2^{-k/2} q(\half |y_1|^2,\dots, \half |y_k|^2,
 y_{k+1},\dots,y_n)^{-1} \sigma_y, \\
 & y_1 \dots y_k \eta_Y' = 2^{-k/2} q(0)^{-1} \sigma_0
\end{align*}
where $\sigma$ is some smooth $n$-form with $\sigma_0 \neq 0$, so the
associated phases also become close as $y \rightarrow 0$. \qed

$\phi_t$ has a canonical grading $\alphagr_{\phi_t}$ with respect to
$\alpha_Y$, obtained by starting with the trivial grading for
$\phi_0$ and extending continuously. Denote by $\alphagr_{\phi_t}'$
the grading obtained in the same way, but with respect to
$\alpha_Y'$. There are corresponding gradings
$\alphagr_{\phi_t^{\red,r}}'$ on the reduced spaces, with respect to
$\alpha_{Y^{\red,r}}'$.

\begin{lemma} \label{th:small-phase}
Given $\epsilon_4>0$, one can find $\delta_5,\delta_6,\delta_7>0$
such that for all
\[
\begin{cases}
 r \in (\R^{\scriptscriptstyle>0})^k &
 \text{with $||r|| < \delta_5$}, \\
 y^\red \in Y^{\red,r} &
 \text{with $||y^\red|| < \delta_6$}, \\
 \Lambda^\red \in \LL_{Y^{\red,r},y^\red} & \text{and} \\
 t \in \R &
 \text{with $|t| < \delta_7 r_1^{-1} \dots r_k^{-1}$},
\end{cases}
\]
$\phi_t^{\red,r}$ is well-defined near $y^\red$, and its grading is
$|\alphagr_{\phi_t^{\red,r}}'(\Lambda^\red)| < \epsilon_4$.
\end{lemma}

\proof This is an easy consequence of Lemma \ref{th:small-flow}: one
can achieve that the derivative $(D\phi_t^{\red,r})_{y^\red}$ remains
very close to the identity in the whole range of $t$'s that we are
considering. Since the volume form $\eta_{Y^{\red,r}}$ is constant
and independent of $r$, it follows that the grading remains small.
\qed

\begin{lemma} \label{th:phi-grading-estimate}
Given $\epsilon_5>0$, one can find $\delta_8,\delta_9,\delta_{10} >
0$ such that for all
\[
\begin{cases}
 y \in Y & \text{with $0 < |p(y)| < \delta_8$ and $||y|| < \delta_9$,} \\
 \Lambda \in \LL_{Y,y}, & \text{and} \\
 t \in \R & \text{with $|t| < \delta_{10} |p(y)|^{-2}$,}
\end{cases}
\]
$\phi_t$ is well-defined near $y$, and its grading satisfies
$
|\alphagr_{\phi_t}(\Lambda)| < n + \epsilon_5.
$
\end{lemma}

\proof Apply Lemma \ref{th:small-flow} to the corresponding point
$y^\red$ in the quotient. In particular, one can choose
$\delta_8,\delta_9,\delta_{10}$ in such a way that the reduced flow
exists near $y^\red$ for $|t| < \delta_{10} |p(y)|^{-2}$. Since the
fibres of the quotient map are compact (tori), it follows that
$\phi_t$ is well-defined near $y$ in the same range of $t$. We now
apply the same basic reasoning as in Lemma
\ref{th:negative-reduction}. After possibly making
$\delta_8,\delta_9$ smaller, Lemma \ref{th:small-phase} ensures that
there is a $\Lambda \in \LL_{Y,y}$ such that
$|\alphagr_{\phi^t}'(\Lambda)| < \epsilon_5/2$. By Lemma
\ref{th:graded-variation}, the grading is $< n + \epsilon_5/2$ for
all other Lagrangian subspaces at the same point. Finally, one uses
Lemma \ref{th:phase-difference} to pass from the grading associated
to $\eta_Y'$ to that for $\eta_Y$, again making the $\delta$s smaller
so that the phase difference becomes $< \epsilon_5/2$. \qed

The next step is to relate $\phi_t$ to the monodromy. As usual, we
take the vector field $-i\zeta \partial_\zeta$ on $\C^*$, and lift it
in the unique way to a horizontal vector field $Z$ on $Y \setminus
p^{-1}(0)$. This is well-defined because $\o_Y$ tames $J_Y$, hence
the regular fibres $Y_\zeta$, $\zeta \neq 0$, are symplectic
submanifolds. We claim that
\begin{equation} \label{eq:f}
Z = f X
\end{equation}
for some positive function $f$ on $Y \setminus p^{-1}(0)$. First,
because $H$ is constant on each fibre, the associated vector field
$X$ is horizontal. Secondly, because $\o_Y(X,Z) = 0$ and the space of
horizontal vectors $TY^h$ is two-dimensional, it is true that $X =
f^{-1}Z$ for some $f$. A little more thought shows that
\begin{equation} \label{eq:explicit-psi}
f = \frac{\o_Y|TY^h}{p^*\o_\C|TY^h} > 0,
\end{equation}
where $TY^h$ is the $\o_Y$-horizontal subspace (which is
two-dimensional, so any two two-forms on it differ by a scalar). Let
$(\psi_s)$ be the flow of $Z$, so that $h_\zeta =
\psi_{2\pi}|Y_\zeta$ is the clockwise monodromy (of course, like
$\phi_s$ this flow is only partially defined). As a consequence of
\eqref{eq:f}, the unparametrized flow lines of $\phi_t$ and $\psi_t$
agree, so
\begin{equation} \label{eq:gt}
\psi_t(y) = \phi_{g_t(y)}(y), \quad \text{where }\;\; g_t(y) =
\int_0^t f(\psi_\tau(y)) d\tau.
\end{equation}

\begin{lemma} \label{th:minimum-speed}
For every $d>0$, $\epsilon_6>0$ there are $\delta_{11},\delta_{12}
> 0$ such that the following holds. For all $0 < |\zeta| < \delta_{11}$ and
all $y \in Y_\zeta$ with $||y|| < \delta_{12}$, the $d$-fold
monodromy $h^d_\zeta$ is well-defined, and \eqref{eq:gt} holds for $t
= 2\pi d$ with
\[
g_{2\pi d}(y) \leq \frac{\epsilon_6}{|\zeta|^2}.
\]
\end{lemma}

\proof Using property \ref{item:t-sym} of $p$ and its
$J_Y$-holomorphicity, one obtains that $Dp_y(J_Y(-iy_1,0,\dots,0)) =
p(y)$ and hence $dH_y(J_Y(-iy_1,0,\dots,0)) = |p(y)|^2 = |\zeta|^2$.
It follows that for small $y$, $||X_y|| \geq c_1 |\zeta|^2/||y||$ for
some $c_1>0$. Hence there is a $C_2>0$ such that
\[
 C_2^{-1} |\zeta|^4/||y||^2 \leq \o_Y(X,J_YX) = f^{-2} \o_Y(Z,J_YZ).
\]
On the other hand, $\o_Y(Z,J_YZ) = |\zeta|^2 f$ by definition of $Z$
and \eqref{eq:explicit-psi}, so finally,
\[
 f(y) \leq C_2 \frac{||y||^2}{|\zeta|^2}.
\]
Integrating gives a similar inequality $g_{2\pi d}(y) \leq C_3
||y||^2/|\zeta|^2$. Clearly, by restricting to smaller $y$ one can
make the right hand side less than $\epsilon_6/|\zeta|^2$ for any
desired $\epsilon_6$ (this, by the way, is the crucial estimate in
our whole computation). \qed

We use $J_Y$-complex volume forms $\eta_{Y_\zeta}$ on the fibres
obtained in the usual way from $\eta_Y$, so that
\eqref{eq:divide-degree} holds. Take $d>0$, $y \in Y_\zeta$, and a
Lagrangian subspace $\Lambda \subset TY_y$ with $Dp(\Lambda) =
i\zeta\R$. This is necessarily of the form $\Lambda = \Lambda^v
\oplus \R Z_y$ for some Lagrangian subspace $\Lambda^v \subset
T_y(Y_\zeta)$. By \eqref{eq:f} and \eqref{eq:gt} one has
\begin{equation} \label{eq:relation-between-diff}
 D(\psi_t)_y = D(\phi_{g_t})_y +
 X_{\psi_t(y)} \otimes dg_t =
 D(\phi_{g_t})_y + Z_{\psi_t(y)} \otimes \frac{dg_t}{f(\psi_t(y))}
\end{equation}
Since $\Lambda$ contains $Z_y$, and $\psi_t$ is the clockwise
parallel transport flow, $D(\psi_t)_y(\Lambda)$ contains
$Z_{\psi_t(y)}$. From this and \eqref{eq:relation-between-diff} it
follows that
\[
 D\psi_t(\Lambda) = D(\phi_{g_t})(\Lambda), \qquad
 D(\psi_t|Y_\zeta)(\Lambda^v) = D(\phi_{g_t})(\Lambda) \cap \ker(Dp),
\]
and hence by \eqref{eq:divide-degree} that
$\alpha_{\psi_t|Y_\zeta}(\Lambda^v) = e^{-4\pi i t}
\alpha_{\phi_{g_t}}(\Lambda)$. By considering the behaviour of this
over the time $t \in [0;2\pi d]$ one finds that the canonical grading
of the monodromy satisfies
\begin{equation} \label{eq:relative-shift}
 \alphagr_{h_\zeta^d}(\Lambda^v) =
 \alphagr_{\phi_{g_{2\pi d}(y)}}(\Lambda)-2d.
\end{equation}

\begin{lemma} \label{th:local-model-1}
For every $d>0$ and $\epsilon_7>0$ there are $\delta_{13},
\delta_{14} >0$ such that the following holds. For every $y \in
Y_\zeta$ with $0 < |\zeta| < \delta_{13}$ and $||y|| < \delta_{14}$,
and every Lagrangian subspace $\Lambda^v \subset T(Y_\zeta)_y$, we
have that $h_\zeta^d$ is well-defined near $y$, and its canonical
grading satisfies
\[
\alphagr_{h_\zeta^d}(\Lambda^v) < n-2d + \epsilon_7.
\]
\end{lemma}

\proof Lemma \ref{th:phi-grading-estimate} says that one can choose
$\delta_{13},\delta_{14}$ so small that there is a $\delta_{15}>0$
with
\begin{equation} \label{eq:epsilon7}
|\alphagr_{\phi_t}(\Lambda)| < n + \epsilon_7
\end{equation}
for all $|t| < \delta_{15}/|p(y)|^2$. On the other hand, after
possibly making $\delta_{13}, \delta_{14}$ smaller, one knows from
Lemma \ref{th:minimum-speed} that $|g_{2\pi d}| <
\delta_{15}/|p(y)|^2$, so \eqref{eq:epsilon7} applies to the values
of $t$ that are relevant for our monodromy map.
\eqref{eq:relative-shift} completes the proof. \qed

This shows that the basic feature of our first local model, the
uniform negativity of the monodromy, survives even if one generalizes
from the idealized situation which we considered first (Assumptions
\ref{as:1}) to a more realistic one (Assumptions \ref{as:1prime}). In
principle, one should carry out a parallel discussion for the second
local model. This would be rather more delicate, since the grading
becomes zero along a submanifold; an arbitrary small perturbation can
potentially make it positive, thereby spoiling the entire argument.
Fortunately, in the $K3$ case the second model occurs only at
isolated points, and one can achieve that the local picture near
those points is standard (satisfies Assumptions \ref{as:2}), so that
the previous discussion suffices after all.

\subsection{}
We will now explain how to patch together the local models to
understand of the ``large complex structure limit'' monodromy. This
is somewhat similar to the strategy used by Ruan to construct
(singular) Lagrangian torus fibrations \cite{ruan02b}. We consider a
Fano threefold $X$ with $o_X = \K_X^{-1}$, and a pair of sections
$\sigma_{X,0},\sigma_{X,\infty}$ which generate a quasi-Lefschetz
pencil of hypersurfaces $\{X_z\}$. From $\sigma_{X,\infty}$ we get a
rational complex volume form $\eta_X$ with a pole along $X_\infty$.
There are canonical induced complex volume forms $\eta_{X_z}$ on the
fibres $X_z$ (if that is smooth and $z \neq \infty$), which are
therefore Calabi-Yau surfaces. In fact, they must be $K3$s, since
$H^1(X) = 0$ for any Fano variety, hence $H^1(X_z) = 0$ by the
Lefschetz hyperplane theorem.

\begin{lemma} \label{th:kaehler-1}
One can find a K{\"a}hler form $\o_X'$ in the same cohomology class
as $\o_X$, with the following properties. (i) Near every point where
three components of $X_\infty$ meet, $(X,\o_X',1/\pi)$ is modelled on
$(Y,\o_Y,p)$ from Assumptions \ref{as:1}, with $n = k = 3$. (ii) Near
every point where $X_0$ meets two components of $X_\infty$,
$(X,\o_X',1/\pi)$ is modelled on $(Y,\o_Y,p)$ from Assumptions
\ref{as:2}, with $n = k = 3$.
\end{lemma}

This is easy because the set of such points (which we will call
``lowest stratum points'' in future) is finite. Near each of them,
there are local holomorphic coordinates in which $(X,1/\pi)$ becomes
$(Y,p)$. One can then locally modify the K{\"a}hler form to make it
standard, see e.g.\ \cite[Lemma 7.3]{ruan02b} or \cite[Lemma
1.7]{seidel01}.

\begin{lemma} \label{th:kaehler-2}
One can find a K{\"a}hler form $\o_X''$ in the same cohomology class
as $\o_X$, which agrees with $\o_X'$ in a neighbourhood of the lowest
stratum points, and such that in addition, any two components of
$X_\infty$ meet orthogonally. \qed
\end{lemma}

Ruan proves a much more general result in \cite[Theorem
7.1]{ruan02b}. The main idea is to apply the techniques from the
previous Lemma to each fibre of the normal bundle $\nu$ to the
intersection of two components of $X_\infty$.
%For instance, if one follows the approach from \cite{ruan02b} then one
%first supposes that the kaehler metric on each fibre is quadratic.
%then one multiplies with \psi(|y|^2) to make it vanish near the
%zero-section, but still \geq 0 everywhere in fibre direction.
%Now one adds some really small multiple of a compactly supported
%form, which is negative only where the other summand is strictly
%positive, and which has the desired behaviour near the zero section.
%A rescaling argument shows that the whole is still Kaehler
%on the total space.
%
%Ruan uses a diagonalization argument, which works in this case only
%because coincidence of eigenvalues of hermitian matrices happens
%generically along a codimension 3 subset, hence never over a Riemann
%surface.

Before proceeding, we need some more notation. Denote by $C$ the
irreducible components of $X_\infty$, and by $L_C = \O(-C)$ the
corresponding line bundles. Let $\gamma_C$ be their canonical
sections, with a zero along $C$. Take two components $C \neq C'$, and
choose a small neighbourhood $U_{C,C'}$ of $C \cap C'$. On this
consider the vector bundle
\[
N_{C,C'} = L_C \oplus L_C^{-1},
\]
We want to equip $L_C$ and hence $N_{C,C'}$ with hermitian metrics
and the corresponding connections, with the following prescribed
behaviour near the lowest stratum pointsin $U_{C,C'}$:
\begin{itemize} \itemsep1em
\item
If $x$ is a point where $C \cap C'$ meets another component $C''$, we
choose local coordinates for $X$ as given by Lemma
\ref{th:kaehler-1}, so that Assumptions \ref{as:1} hold, and a local
trivialization of $L_C$ such that $\gamma_C(y) = y_1$. Take the
trivial metric on $L_C$ with respect to this trivialization.

\item
If $x$ is a point where $C \cap C'$ meets $X_0$, we choose local
coordinates for $X$ as given by Lemma \ref{th:kaehler-1}, so that
Assumptions \ref{as:2} hold, and a local trivialization of $L_C$ such
that $\gamma_C(y) = y_2$. Again, we take the trivial metric.
\end{itemize}

Take the K{\"a}hler metric associated to $\o_X''$, and use its
exponential map to define a retraction $r_{C,C'}: U_{C,C'}
\rightarrow C \cap C'$ of an open neighbourhood onto $C \cap C'$. A
corresponding use of parallel transport on $N_{C,C'}$, with respect
to our chosen connection, yields an isomorphism $R_{C,C'}:
N_{C,C'}|U_{C,C'} \rightarrow r_{C,C'}^*(N_{C,C'}|C \cap C')$. Taking
the two objects together, we have a map
\[
 \nu_{C,C'}: U_{C,C'} \setminus X_0
  \xrightarrow{\xi_{C,C'} = (\gamma_C,\pi_M^{-1}\gamma_C^{-1})}
 N_{C,C'}|U_{C,C'} \xrightarrow{R_{C,C'}} N_{C,C'}|C \cap C'.
\]

\begin{lemma} \label{th:kaehler-3}
After making $U_{C,C'}$ smaller if necessary, it carries a unique
$T^2$-action $\sigma_{C,C'}$ which, under $\nu_{C,C'}$, corresponds
to the obvious fibrewise $T^2$-action on $N_{C,C'}|C \cap C'$. In
particular,
\begin{equation} \label{eq:rotate-base}
\pi_M(\sigma_{C,C',s}(x)) = e^{-i(s_1+s_2)}\pi_M(x).
\end{equation}
Moreover, at each point of $C \cap C'$, the action preserves $\o_X''$
and the complex structure; the same holds in a neighbourhood of the
lowest stratum points.
\end{lemma}

\proof It is sufficient to verify the existence and uniqueness
statement locally. First, take a point $x \in C \cap C'$ which is not
a lowest stratum point. $\gamma_C^{-1}$ has a simple pole along $C$,
and $\pi_M^{-1}$ has a simple zero along $C \cup C'$, hence
$\pi_M^{-1}\gamma_C^{-1}$ has a simple zero along $C'$. It follows
that $\xi_{C,C'}$ is transverse to the zero-section at $x$, so
$\nu_{C,C'}$ is a local diffeomorphism; we simply pull back the
$T^2$-action by it. By definition of the connection on $N_{C,C'}$,
there is a commutative diagram
\[
\xymatrix{
 {U_{C,C'} \setminus X_0} \ar[r]^-{\xi_{C,C'}} \ar[dr]_-{\pi_M^{-1}} &
 {N_{C,C'}|U_{C,C'}} \ar[r]^-{R_{C,C'}} \ar[d] &
 {N_{C,C'}|C \cap C'} \ar[dl] \\
 & {\C} &
}
\]
where the unlabeled arrows are the canonical pairings $L_C \oplus
L_C^{-1} \rightarrow \C$. This shows that our pullback $T^2$-action
satisfies \eqref{eq:rotate-base}. At the point $x$ itself,
\begin{equation} \label{eq:dnu}
D\nu_{C,C'}: TX_x \longrightarrow T(C \cap C')_x \oplus N_x
\end{equation}
is given by orthogonal projection to $C \cap C'$ with respect to the
$o_{X,2}$-metric, together with $D\xi_x: TX_x \rightarrow N_x$. This
is $\C$-linear, which implies that $\sigma_{C,C',x}$ preserves the
complex structure. As for the symplectic structure, observe that by
Lemma \ref{th:kaehler-2} $TX_x$ splits into three complex
one-dimensional pieces, namely $T(C \cap C')_x$ and the normal
directions to that inside $C,C'$. On the right hand side of
\eqref{eq:dnu}, this corresponds to the splitting of $N_x$ into line
bundles. The symplectic structures on the each summand do not
correspond, but they differ only by a scalar, which is enough to
conclude that $\sigma_{C,C'}$ is symplectic at $x$.

It remains to deal with the situation near the lowest stratum points.
For this, we choose local coordinates and the trivialization of
$N_{C,C'}$ as arranged before. Near a point of $C \cap C' \cap C''$,
one then finds that $\nu_{C,C'}(y) = (y_3,y_1,y_2y_3)$, where the
first component is the coordinate on $C \cap C'$. Hence, our
$\sigma_{C,C'}$ near that point is the standard $T^2$-action in the
variables $y_1,y_2$. Similarly, near a point of $C \cap C' \cap X_0$,
$\nu_{C,C'}(y) = (y_1,y_2,y_3/y_1)$, so that one ends up with the
$T^2$-action in the variables $y_2,y_3$. These obviously have the
desired properties. \qed

We now pick an ordering of every pair $\{C,C'\}$ of components, and
will use only the $T^2$-actions $\sigma_{C,C'}$ in that order.

\begin{lemma} \label{th:kaehler-4}
There is a symplectic form $\o_X'''$ which tames the complex
structure, such that for each intersection $C \cap C'$ of two
components of $X_\infty$, there is a neighbourhood $U_{C,C'}$ such
that the local $T^2$-action $\rho_{C,C'}$ leaves $\o_X'''$ invariant.
Moreover, $\o_X'''$ lies in the same cohomology class as $\o_X''$,
and agrees with it at each point $x \in C \cap C'$, as well as in a
neighbourhood of the lowest stratum points.
\end{lemma}

\proof Let $\bar{\o}_{C,C'} \in \Omega^2(U_{C,C'})$ be the result of
averaging $\o_X''$ with respect to $\rho_{C,C'}$. This is a closed
two-form; it agrees with $\o_X''$ along $C cap C'$, and also in a
neighbourhood of the lowest stratum points. Using the Poincar{\'e}
lemma one can write $\bar{\o}_{C,C'} - \o_X'' = d\beta_{C,C'}$, where
$\beta_{C,C'}$ is a one-form that vanishes to second order along $C
\cap C'$, and which is again zero in a neighbourhood of the lowest
stratum point. Take a compactly supported cutoff function
$\psi_{C,C'}$ on $U_{C,C'}$, which is equal to one near $C \cap C'$.
One sees easily that as one rescales the normal directions to make
the support of $\psi_{C,C'}$ smaller, $d(\psi_{C,C'} \beta_{C,C'})$
becomes arbitrarily small in $C^0$-sense. It follows that for
sufficiently small support, $\o_X'' + d(\psi_{C,C'} \beta_{C,C'})$ is
a symplectic form which tames the complex structure. This agrees with
$\o_X''$ near the lowest stratum points, and for that reason, one can
carry out the construction simultaneously for all intersections $C
\cap C'$. \qed

\begin{lemma}
Let $x \in C \cap C'$ be a point where two components of $X_\infty$
meet, which is not a lowest stratum point. Then there are local
$\rho_{C,C'}$-equivariant Darboux coordinates near $x$, in which $X$
with the map $1/\pi$, the symplectic structure $\o_X'''$, the given
complex structure $J_X$, and the meromorphic complex volume form
$\eta_X$, is isomorphic to $(Y,p,\o_Y,J_Y,\eta_Y)$ as in Assumptions
\ref{as:1prime}. \qed
\end{lemma}

This is straightforward checking, which we leave to the reader. The
outcome of the whole construction is this:

\begin{prop} \label{th:negativity}
Let $\gamma_\infty: [0;2\pi] \rightarrow \C$ be a circle of large
radius $R \gg 0$. Then the graded monodromy
\[
\hgr_{\gamma_\infty} \in \Autgr(M_R)
\]
is isotopic (within that group) to a graded symplectic automorphism
$\phigr$ with the following property. There is a finite set $\Sigma
\subset X_z \cap X_\infty$ which admits arbitrarily small open
neighbourhoods $W \subset X_z$, such that $\phi(W) = W$ and
$\phigr|(X_z \setminus W)$.
\end{prop}

\proof The main point is to replace the given K{\"a}hler form $\o_X$
by $\o_X'''$, and we need to convince ourselves that this is
permitted. For each regular $X_z$ there is a symplectic isomorphism
\[
\phi_z: (X_z,\o_{X_z}) \longrightarrow (X_z,\o_X'''|X_z)
\]
which carries $X_{0,\infty}$ to itself. This follows from Moser's
Lemma together with the isotopy theorem for symplectic surfaces with
orthogonal normal crossings, see \cite[Theorem 6.5]{ruan02b} for a
more general statement valid in all dimensions. Next, an inspection
of the construction of monodromy maps for $\pi_M$ shows that they can
be defined using any symplectic form which tames the complex
structure, and in particular $\o_X'''$. The resulting diagram of
symplectic maps
\[
\xymatrix{
 (X_R,\o_{X_R}) \ar[d]^-{h_{\gamma_\infty}}
 \ar[r]^-{\phi_R} &
 (X_R,\o_X'''|X_R) \ar[d]^-{h_{\gamma_\infty}'''} \\
 (X_R,\o_{X_R}) \ar[r]^-{\phi_R} & (X_R,\o_X'''|X_R)
}
\]
commutes up to symplectic isotopy rel $X_{R,\infty}$. Since $X_R$ is
a $K3$ surface, Remark \ref{th:h1} shows that $h_{\gamma_\infty}$ is
isotopic to $\phi_R^{-1} \circ h_{\gamma_\infty}''' \circ \phi_R$
within $Aut(M_R)$. It is no problem to take the gradings into
consideration. Hence, if $h_{\gamma_\infty}'''$ has the desired
negativity property for large $R$, so will $h_{\gamma_\infty}$.

Let $\Sigma$ be the set of those points where two components of
$X_\infty$ intersect each other and $X_0$. This lies in the base
locus of our pencil, hence in every fibre $X_z$. Around each point $x
\in \Sigma$, choose a neighbourhood $U_x \subset X$ which is an open
ball around the origin in local coordinates as in Assumptions
\ref{as:2}, and set $U = \bigcup_x U_x$. We know from our discussion
of the local model that $U$ is invariant under parallel transport
along large circles. The claim is that for sufficiently large $R$,
the grading of
\[
\psigr_R = (\hgr_{\gamma_\infty}''')^2
\]
(with respect to $\o_X'''$, the given complex structure and complex
volume form $\eta_{X_R}$) is negative on $X_R \setminus U$. This is
proved by contradiction: suppose that we have a sequence $R_k
\rightarrow \infty$, and points $x_k \in X_{R_k} \setminus U$ such
that the grading of $\psigr_R$ is not negative on some Lagrangian
subspace in the tangent space at $x_k$. Consider the limit point of a
subsequence, $x_\infty \in X_\infty \setminus U$. If $x_\infty$ is a
smooth point of $X_\infty$, the grading of $\psigr_R$ at $x_{z_k}$
converges to $-2$ for the same reason as in the case of Lefschetz
pencils, see Lemma \ref{th:shift-factor}. If $x_\infty$ is a point
where several components of $X_\infty$ meet, Lemma
\ref{th:local-model-1} says that the grading of $\psigr_R$ at
$x_{z_k}$ will be bounded above by $-1/2$ for large $k$ (when
applying this local result, the reader should keep in mind the change
of coordinates on the base, $\zeta = 1/z$ and $dz = d\zeta/\zeta^2$).

Fix some $R$ such that the claim made above applies. Take a
neighbourhood $W \subset U \cap X_R$ of $\Sigma$. After making $W$
smaller, we may again assume that in the local coordinates of
Assumption \ref{as:2}, this is the intersection of the fibre
$Y_{1/R}$ with an open ball around the origin. Lemma
\ref{th:bad-points} tells us that $h_{\gamma_\infty}'''(W) = W$, and
that $h_{\gamma_\infty}'''$ is negative on $U \setminus W$. \qed

\section{Fukaya categories\label{sec:fukaya}}

There are Fukaya categories for affine and projective Calabi-Yaus, as
well as a relative version which interpolates between the two. All of
them are c-unital $A_\infty$-categories, but with different
coefficient rings:
\begin{center}
\begin{tabular}{l| l | l |l}
 & notation for & & \\
 & category & notation for objects & coefficient ring \\
 \hline
 affine & $\Fuk(M)$ &
 $L^\br = (L,\alphagr_L,\Spin_L) $ & $\C$ \\
 relative & $\Fuk(X,X_\infty)$ &
 $L^\mbr = (L,\alphagr_L,\Spin_L,J_L) $ & $\Lambda_\N$ \\
 projective & $\Fuk(X)$ &
 $L^\rbr= (L,\alphagr_L,\Spin_L,\lambda_L,J_L) $ & $\Lambda_\Q$ \\
\end{tabular}
\end{center}
While our definition of affine Fukaya categories is fairly general,
in the projective and relative setups only Calabi-Yau surfaces ($K3$
and tori) will be considered. This restriction allows us to avoid
using virtual fundamental chains, and it also simplifies the relation
between the different categories: up to quasi-equivalence,
$\Fuk(X,X_\infty)$ reduces to $\Fuk(M)$ if the deformation parameter
$q$ is set to zero, while tensoring with $\Lambda_\Q$ gives a full
subcategory of $\Fuk(X)$ (for the definition of $\Fuk(X)$ in higher
dimensions, see \cite{fooo}; and for its presumed relation with
$\Fuk(M)$, \cite{seidel02}). We emphasize that all properties of
Fukaya categories which we use are quite simple ones, and largely
independent of the finer details of the definition.

\subsection{}
The following is a summary of \cite[Chapter 2]{seidel04}, with minor
changes in notation and degree of generality. Let $M = X \setminus
X_\infty$ be an affine Calabi-Yau. Objects of $\Fuk(M)$ are {\em
exact Lagrangian branes}
\begin{equation} \label{eq:exact-brane}
L^\br = (L,\alphagr_L,\Spin_L),
\end{equation}
where $L \subset M$ is an oriented exact Lagrangian submanifold,
$\alphagr_L$ a grading which is compatible with the orientation (see
Remark \ref{th:or}), and $\Spin_L$ a $Spin$ structure. To define the
morphism spaces, choose for each pair of objects a {\em Floer datum}
\[
(H,J) = (H_{L_0^\br,L_1^\br},J_{L_0^\br,L_1^\br}).
\]
This consists of a function $H \in \smooth_c([0;1] \times M,\R)$, as
well as a family $J = J_t$ of $\o_X$-compatible almost complex
structures parametrized by $t \in [0;1]$, which for all $t$ agree
with the given complex structure in a neighbourhood of $X_\infty$.
The condition on $H$ is that the associated Hamiltonian isotopy
$(\phi^t_H)$ should satisfy $\phi^1_H(L_0) \trans L_1$. We then
define $\CC(L_0^\br,L_1^\br)$ to be the set of trajectories $x: [0;1]
\rightarrow M$, $x(t) = \phi^t_H(x(0))$, with boundary conditions
$x(k) \in L_k$. This is obviously bijective to $\phi^1_H(L_0) \cap
L_1$, hence finite. To each such $x$ one can associate a Maslov index
$I(x)$ and orientation set $or(x)$. For $I(x)$, one observes that the
grading of $L_0$ induces one of $\phi^1_H(L_0)$, and then uses the
index of an intersection point as defined in
\eqref{eq:absolute-index}. $or(x)$ is a set with two elements, the
two possible ``coherent orientations'' of $x$; we refer to
\cite{seidel04,fooo} for its definition, which uses the $Spin$
structures on $L_0,L_1$. The Floer cochain space is the graded
$\C$-vector space
\begin{equation} \label{eq:basic-floer-complex}
CF^j_M(L_0^\br,L_1^\br) = \bigoplus_{\substack{x \in
\CC(L_0^\br,L_1^\br)
\\ I(x) = j}} \C_x,
\end{equation}
where $\C_x$ is defined as the quotient of the two-dimensional space
$\C[or(x)]$ by the relation that the two elements of $or(x)$ must sum
to zero (it is of course isomorphic to $\C$, but not canonically). In
the Fukaya category, $CF^*_M$ is the space of $hom$s from $L_0^\br$
to $L_1^\br$.

Given $x_0,x_1 \in \CC(L_0^\br,L_1^\br)$, we now consider the moduli
spaces $\MM^1_M(x_0,x_1)$ of solutions $u: \R \times [0;1]
\rightarrow M$ to Floer's ``gradient flow'' equation with limits
$x_0,x_1$, divided by the $\R$-action by translation. As part of the
definition of Floer datum, we require that these spaces should all be
regular. Assume from now on that $I(x_1) = I(x_0)-1$. By an index
formula $\MM^1_M(x_0,x_1)$ is zero-dimensional, and a compactness
theorem says that it is a finite set. A few words about this: since
the action functional is exact \eqref{eq:energy} and there is no
bubbling, we only have to worry about sequences of solutions which
become increasingly close to $X_\infty$. To see that this does not
happen, one uses a maximum principle argument based on the function
$-\log\,||\sigma_{X,\infty}||^2$, which goes to $+\infty$ at
$X_\infty$. It is plurisubharmonic since
\begin{equation} \label{eq:log-sigma}
\partial \bar\partial \log\,||\sigma_{X,\infty}||^2 = -F_{\nabla_X} =
2\pi i \o_M.
\end{equation}
Each $u \in \MM^1_M(x_0,x_1)$ comes with a preferred identification
$or(x_1) \iso or(x_0)$, which in turn defines a map $\C_{x_1}
\rightarrow \C_{x_0}$. The sum of these over all $u$ is the Floer
differential $\mu^1_M: CF^j_M(L_0^\br,L_1^\br) \rightarrow
CF^{j+1}_M(L_0^\br,L_1^\br)$, which is the first composition map in
the Fukaya category. Its cohomology is the Floer cohomology
$HF^*_M(L_0^\br,L_1^\br)$, which is therefore the space of morphisms
in the cohomological category $H\Fuk(M)$.

To define the rest of the $A_\infty$-structure, one proceeds as
follows. Take some $d \geq 2$. A {\em $(d+1)$-pointed disc with
Lagrangian labels}
\begin{equation} \label{eq:s}
 (S,\zeta_0,\dots,\zeta_d,L_0^\br,\dots,L_d^\br)
\end{equation}
is a Riemann surface $S$ isomorphic to the closed disc with $d+1$
boundary points removed. Moreover, these points at infinity should
have a preferred numbering $\zeta_0,\dots,\zeta_d$, compatible with
their natural cyclic order. Denote the connected components of
$\partial S$ by $I_0,\dots,I_d$, again in cyclic order and starting
with the component $I_0$ which lies between $\zeta_0$ and $\zeta_1$.
Finally, we want to have exact Lagrangian branes
$L_0^\br,\dots,L_d^\br$ attached to the boundary components. A set of
{\em strip-like ends} for $S$ consists of proper holomorphic
embeddings
\begin{align*}
 \epsilon_{S,0} & : \Rleq \times [0;1] \longrightarrow S, &&
 \textstyle\lim_{s \rightarrow -\infty} \epsilon_{S,0}(s,\cdot) = \zeta_0, \\
 \epsilon_{S,1},\dots,\epsilon_{S,d} & : \Rgeq \times [0;1] \longrightarrow S, &&
 \textstyle\lim_{s \rightarrow +\infty} \epsilon_{S,k}(s,\cdot) = \zeta_k
\end{align*}
taking $\Rleq \times \{0;1\}$, $\Rgeq \times \{0;1\}$ to $\partial S$
and which have disjoint images. A {\em perturbation datum} on $S$ is
a pair $(K_S,J_S)$ consisting of a family $\{J_{S,z}\}_{z \in S}$ of
compatible almost complex structures on $X$, which agree with the
given complex structure in a neighbourhood of $X_\infty$, and a
one-form $K_S \in \Omega^1(S,\smooth_c(M,\R))$ with values in smooth
functions, subject to the condition that $K_S(\xi)|L_k = 0$ for $\xi
\in T(I_k) \subset T(\partial S)$. The behaviour of $(K_S,J_S)$ over
the strip-like ends is fixed by the previously chosen Floer data:
\[
 J_{S,\epsilon_{S,k}(s,t)} = \begin{cases}
 J_{L_0^\br,L_d^\br,t} & k = 0, \\
 J_{L_{k-1}^\br,L_k^\br,t} & k > 0
 \end{cases}
 \quad \text{and} \quad
 \epsilon_{S,k}^*K_S = \begin{cases}
 H_{L_0^\br,L_d^\br,t}\, dt & k = 0, \\
 H_{L_{k-1}^\br,L_k^\br,t} \, dt & k > 0.
 \end{cases}
\]
Suppose that we have made a global choice of strip-like ends and
perturbation data. This means that for every $(d+1)$-pointed disc
with Lagrangian labels we have chosen a set of strip-like ends
$(K_S,J_S)$ (the dependence on the other data in \eqref{eq:s} is
suppressed from the notation for the sake of brevity). This needs to
satisfy two additional kinds of conditions: first of all, it needs to
vary smoothly if we change the complex structure on $S$ in a smooth
way, which means that it is defined on the universal family of such
discs. Secondly, it needs to be well-behaved with respect to the
compactification of that family, which includes degenerations to
discs with nodes. This in fact establishes recursive relations
between the choices of perturbation data for different $d$, see
\cite{seidel04} for details.

Supposing that all these conditions have been met, take $x_0 \in
\CC(L_0^\br,L_d^\br)$ as well as $x_k \in \CC(L_{k-1}^\br,L_k^\br)$
for $k = 1,\dots,d$. Define $\MM^d_M(x_0,\dots,x_d)$ to be the space
of equivalence classes of pairs consisting of a $(d+1)$-marked disc
with labels $L_0^\br,\dots,L_d^\br$, and a solution of the following
generalization of Floer's equation for a map $u: S \rightarrow M$:
\begin{equation} \label{eq:generalized-floer}
\begin{cases}
 \!\!\! & u(I_k) \subset L_k, \\
 \!\!\! & (du(z)-Y_{S,z,u(z)})
 + J_{S,z,u(z)} \circ (du-Y_{S,z,u(z)}) \circ i = 0, \\
 \!\!\! & \lim_{s \rightarrow \pm \infty} u(\epsilon_{S,k}(s,\cdot)) = x_k.
\end{cases}
\end{equation}
Here $Y_S \in \Omega^1(S,\smooth_c(TM))$ is the
Hamiltonian-vector-field valued one-form determined by $K_S$. The
equivalence relation on pairs is isomorphism of the underlying
Riemann surfaces, compatible with all the additional structure, and
composition of maps $u$ with that isomorphism. The resulting quotient
space is (locally) the zero-set of a Fredholm section of a suitably
defined Banach vector bundle. Generically, this will be transverse,
and then $\MM^d_M(x_0,\dots,x_d)$ is smooth of dimension $I(x_0) -
I(x_1) - \dots - I(x_d) + d - 2$. Moreover, the zero-dimensional
spaces are finite. It is maybe worth while stating the basic energy
equality which underlies the compactness argument. Choose primitives
$K_{L_k}$ for $\theta|L_k$. Define the action of the limits $x_k$ by
a generalization of \eqref{eq:action},
\begin{align*}
 A(x_k) & = - \int x_k^*\theta + \\
 + & \begin{cases}
 \int H_{L_0^\br,L_d^\br}(x_0(t))\, dt +
 K_{L_d}(x_0(1)) - K_{L_0}(x_0(0)) & k = 0, \\
 \int H_{L_{k-1}^\br,L_k^\br}(x_k(t))\,dt +
 K_{L_k}(x_k(1)) - K_{L_{k-1}}(x_k(0)) & k = 1,\dots,d.
 \end{cases}
\end{align*}
Then for $u \in \MM^d_M(x_0,\dots,x_d)$,
\begin{equation} \label{eq:polygon-energy}
\begin{aligned}
 E(u) & \stackrel{\mathrm{def}}{=} \half \int_S | du - Y |^2 \\ & =
 A(x_0) - \sum_{k=1}^d A(x_d) - \int_S u^*(-dK_S + \half \{K_S,K_S\}).
\end{aligned}
\end{equation}
The integral on the right hand side can be bounded by a constant
which is independent of $u$, because the curvature term $-dK_S +
\half \{K_S,K_S\}$ vanishes over the strip-like ends of $S$. Of
course \eqref{eq:log-sigma} also plays a role in the proof of
compactness, as was the case for the Floer differential. To continue
with the main story: any element of a zero-dimensional space
$\MM^d_M(x_0,\dots,x_d)$ defines a canonical bijection
\begin{equation} \label{eq:orientation-mechanism}
or(x_0) \iso or(x_1) \times_{\Z/2} \dots \times_{\Z/2} or(x_d).
\end{equation}
The order $d$ composition map $\mu^d_M$ in $\Fuk(M)$ is the sum of
the corresponding maps $\C_{x_d} \otimes \dots \otimes \C_{x_1}
\rightarrow \C_{x_0}$ (the order is reversed according to our
notational conventions for $A_\infty$-categories). There are
additional signs which involve the Maslov indices $I(x_k)$, but since
they depend on the conventions about determinant lines used in
setting up the orientation spaces, we prefer not to spell them out
here. $\Fuk(M)$ is independent of all the choices (Floer data,
strip-like ends, perturbation data) made in its construction up to
quasi-equivalence, and indeed quasi-isomorphism acting trivially on
the objects.

\begin{remark} \label{th:class-of-j}
Suppose that no irreducible component of $X_\infty$ is rational. Then
one can use a wider class of almost complex structures, namely all
those $\o_X$-compatible ones such that each irreducible component $C
\subset X_\infty$ is an almost complex submanifold. Instead of using
\eqref{eq:log-sigma} to prevent solutions of Floer's equation and its
generalization from escaping into $X_\infty$, one now argues as
follows: if the solution stays inside $M$, its intersection number
with $C$ is zero. The limit of a sequence of solutions has the same
property, and nonnegativity of intersection multiplicities shows that
the limit is disjoint from $X_\infty$ (more precisely, one thinks of
a solution of \eqref{eq:generalized-floer} as a map $u: S \rightarrow
S \times X$ which is pseudo-holomorphic for a suitable almost complex
structure, and considers its intersection with $S \times X_\infty$).
Similarly, in the inhomogeneous terms one can use Hamiltonian
functions on $X$ whose values and first derivatives vanish on
$X_\infty$.
\end{remark}

\subsection{\label{subsec:coverings}}
Let $M$ be an affine Calabi-Yau, equipped with a homomorphism
\[
\rho: \pi_1(M) \longrightarrow \Gamma
\]
into some finitely generated abelian group $\Gamma$. Consider the
associated abelian covering $M_\Gamma \rightarrow M$. Denote by
$\Fuk(M)_\rho$ the full subcategory of $\Fuk(M)$ consisting of those
branes $L^\br$ such that the composition $\pi_1(L) \rightarrow
\pi_1(M) \rightarrow \Gamma$ is trivial. For each of these choose a
lift $L_\Gamma \subset M_\Gamma$. Once one has done that, the
$hom$-spaces split as
\[
 CF^*_M(L_0^\br,L_1^\br) =
 \bigoplus_{\gamma \in \Gamma} CF^*_M(L_0^\br,L_1^\br)_\gamma,
\]
where the $\gamma$-summand consists of those $x \in
\CC(L_0^\br,L_1^\br)$ whose unique lift $x_\Gamma: [0;1] \rightarrow
M_\Gamma$ with $x_\Gamma(0) \in L_{0,\Gamma}$ satisfies
$\gamma^{-1}(x_\Gamma(1)) \in L_{1,\Gamma}$. Because each
pseudo-holomorphic polygon can be lifted to $M_\Gamma$, these
splittings are compatible with the compositions, in the sense that
\[
 CF^*_M(L_{d-1}^\br,L_d^\br)_{\gamma_d} \otimes \dots \otimes
 CF^*_M(L_0^\br,L_1^\br)_{\gamma_1} \xrightarrow{\mu^d_M}
 CF^*_M(L_0^\br,L_d^\br)_{\gamma_1\dots \gamma_d}.
\]
This constitutes a $\Gamma$-{\em coaction} on our
$A_\infty$-category. For general nonsense reasons, it defines an {\em
action} of the character group $\Gamma^* = Hom(\Gamma,\C^*)$:
elements $\chi \in \Gamma^*$ act on each
$CF^*_M(L_0^\br,L_1^\br)_\gamma$ by scalar multiplication with
$\chi(\gamma) \in \C^*$.

What we actually need is a parallel but slightly different geometric
setup. Let $M = X \setminus X_\infty$ be an affine Calabi-Yau, and
$\Gamma$ a finite abelian group acting on $X$ by K{\"a}hler
isometries. This should come with a lift of the action to $o_X$,
which preserves $A_X$ and is compatible with the induced action on
$\K_X$ via the isomorphism $\beta_X$ \eqref{eq:beta}. Finally, we
require that the action preserves $\sigma_{X,\infty}$ up to
multiplication with $|m_X|$-th roots of unity (for $m_X = 0$ these
may be arbitrary constants in $S^1$), and that it is free on $M$. As
a consequence of these assumptions, the one-form $\theta_M$ and the
holomorphic volume form $\eta_M$ are $\Gamma$-invariant. Let $\bar{M}
= M/\Gamma$ be the quotient, with its inherited symplectic form
$\o_{\bar{M}}$, one-form $\theta_{\bar{M}}$ and holomorphic volume
form $\eta_{\bar{M}}$. Note that $-\log ||\sigma_{X,\infty}||^2$ also
descends to a proper subharmonic function on $\bar{M}$. One can
therefore define the Fukaya category $\Fuk(\bar{M})$ in the same way
as for $M$ itself, using almost complex structures which agree with
the given complex structure outside a compact subset (this is in
spite of the fact that the obvious compactification $\bar{X} =
X/\Gamma$ may be singular). There is a tautological homomorphism
$\rho: \pi_1(\bar{M}) \rightarrow \Gamma$, whose associated covering
is $\bar{M}_\Gamma = M$. The observation made above applies to this
case as well, so one has a $\Gamma^*$-action on $\Fuk(\bar{M})_\rho$.
Take the semidirect product $A_\infty$-category, as defined in
Section \ref{subsec:ainfty-actions}.

\begin{lemma} \label{th:semidirect-lift}
$\Fuk(\bar{M})_\rho \semidirect \Gamma^*$ is quasi-isomorphic to the
full subcategory of $\Fuk(M)$ consisting of those branes in $M$ which
satisfy $\gamma(L) \cap L = \emptyset$ for all nontrivial $\gamma \in
\Gamma$. \qed
\end{lemma}

The proof is straightforward, and we leave it to the reader. Note
that the $\Gamma^*$-action is not quite unique, since it depends on
choosing lifts for the branes; but different choices lead to group
actions related by (elementary) inner automorphisms as in
\eqref{eq:conjugate-action}, which explains why the semidirect
product is the same for all choices.

\subsection{}
Let $X$ be a projective Calabi-Yau surface, and $L \subset X$ a
Lagrangian submanifold which admits a grading. An $\o_X$-compatible
almost complex structure $J$ is called {\em regular with respect to
$L$} if there are no non-constant $J$-holomorphic spheres, and no
non-constant $J$-holomorphic discs with boundary on $L$.

\begin{lemma}
Almost complex structures which are regular with respect to a given
$L$ are dense within the set of all $\o_X$-compatible almost complex
structures.
\end{lemma}

\proof Since $c_1(X) = 0$, the virtual dimension of the space of
unparametrized $J$-holomorphic spheres at a point $v$ is $4 + 2
\leftsc c_1(X), [v] \rightsc - 6 = -2$. Standard transversality
results tell us that for generic $J$, there are no $J$-holomorphic
spheres that are somewhere injective; and since any non-constant
sphere is a multiple cover of a somewhere injective one, there are no
non-constant spheres at all. The argument for discs goes along the
same lines: because of the grading, the Maslov number $\mu(w)$ of any
disc in $(X,L)$ vanishes, and the virtual dimension is
\begin{equation} \label{eq:minus-one}
2 + \mu(w) - 3 = -1.
\end{equation}
Again, this shows that for generic $J$, there are no somewhere
injective discs. The decomposition theorem of Kwon-Oh
\cite{kwon-oh96} and Lazzarini \cite{lazzarini98} says that if there
is a non-constant $J$-holomorphic disc, there is also a somewhere
injective one, which implies the desired result. \qed

Objects of $\Fuk(X)$ are {\em rational Lagrangian branes}
\begin{equation} \label{eq:rational-brane}
L^\rbr = (L,\alphagr_L,\Spin_L,\lambda_L,J_L)
\end{equation}
where $L$ is an oriented Lagrangian submanifold of $X$ with a grading
$\alphagr_L$ and $Spin$ structure $\Spin_L$; $\lambda_L$ is a
covariantly constant multisection of the circle bundle of $o_X|L$ of
some degree $d_L \geq 1$; and $J_L$ is an $\o_X$-compatible almost
complex structure which is regular with respect to $L$. For any pair
of objects, we take a Floer datum
\[
(H,J) = (H_{L_0^\rbr,L_1^\rbr},J_{L_0^\rbr,L_1^\rbr})
\]
as before, except that the conditions on the behaviour of $H$ and $J$
near $X_\infty$ are omitted, and the following new requirements
imposed instead: $J_t$ must be equal to $J_{L_0}$, $J_{L_1}$ for $t =
0,1$; and there should be no non-constant $J_t$-holomorphic spheres
for any $t$ (this is still true generically, because the virtual
dimension of the parametrized moduli space is $-1$). Define the set
$\CC(L_0^\rbr,L_1^\rbr)$ as before; and for any $x:[0;1] \rightarrow
X$ in that set, write $\bar{A}(x) = \bar{A}(x(1))$ for the mod $\Q$
action \eqref{eq:modq-action} of $x(1)$ as an intersection point of
$\phi^1_H(L_0) \cap L_1$ (where we have used \eqref{eq:deform-lambda}
to get a multisection of $o_X|\phi^1_H(L_0)$ from $\lambda_{L_0}$).
To spell this out, we have
\[
\exp(2\pi i \bar{A}(x)) =
\frac{\lambda_{L_1}(x(1))}{\kappa(\lambda_{L_0}(x(0)))} \exp\Big( 2
\pi i \int_0^1 H_t(x(t))\, dt\Big),
\]
where $\kappa$ is parallel transport on $o_X$ along $x$. For any
number $r \in \R/\Q$, we define a one-dimensional $\Lambda_\Q$-vector
space $q^r\Lambda_{\Q}$ whose elements are formal series $f(q) =
\sum_m a_m q^m$, with $m$ running over all real numbers in $r +
\frac{1}{d}\Z$ for some natural number $d$ which depends on $f$, and
$a_m \in \C$ vanishing for sufficiently negative $m$. Write
$\Lambda_{\Q,x}$ for the quotient of
$q^{\bar{A}(x)}\Lambda_{\Q}[or(x)]$ by the relation that the two
elements of $or(x)$ sum to zero. Then the Floer cochain space is
\[
CF^j_X(L_0^\rbr,L_1^\rbr) = \bigoplus_{\substack{x \in
\CC(L_0^\rbr,L_1^\rbr)
\\ I(x) = j}} \Lambda_{\Q,x}.
\]
Denote by $\MM^1_X(x_0,x_1)$ the space of solutions of Floer's
equation in $X$ with limits $x_0,x_1$ (this means we drop the
previous assumption that the solutions stay in $M$). As before, we
require that this is regular, and in particular zero-dimensional in
the case $I(x_1) = I(x_0)-1$. Take a solution $u$ in such a
zero-dimensional moduli space, and let $E(u) = \int |\partial
u/\partial s|^2 \, ds\, dt
> 0$ be its energy. A version of \eqref{eq:modq-energy} shows that
$E(u) \in \bar{A}(x_0) - \bar{A}(x_1) + \lcm(d_{L_0},d_{L_1})^{-!}
\Z$, so that we get a well-defined $\Lambda_\Q$-module map
\[
\pm q^{E(u)} : \Lambda_{\Q,x_1} \longrightarrow \Lambda_{\Q,x_0}.
\]
The sign of this is determined by the induced bijection of
orientation sets, as before, and summing over all $u$ yields the
Floer differential $\mu^1_X$ in $\Fuk(X)$. The sum is no longer
finite, but it gives a well-defined $\Lambda_\Q$-module map by Gromov
compactness; the basic point being that there is no bubbling off of
holomorphic discs or spheres, by assumption on the Floer datum.
Denote the resulting Floer cohomology by $HF^*_X(L_0^\rbr,L_1^\rbr)$.

The definition of the higher order compositions is largely similar.
One may use only perturbation data $(K_S,J_S)$ where $J_S = J_{L_k}$
over the boundary component $I_k \subset \partial S$. It is no longer
possible to avoid $J_{S,z}$-holomorphic spheres for some values of $z
\in int(S)$, but the argument from \cite{hofer-salamon95} shows that
for a generic choice of $J_S$, these spheres will never occur as
bubbles in the Gromov compactification of the spaces
$\MM^d_X(x_0,\dots,x_d)$ as long as these are of dimension $\leq 1$,
which is all that we will ever use. Equation
\eqref{eq:polygon-energy} still holds if we replace all the actions
by their mod $\Q$ counterparts. We define the contribution of a point
$(S,u)$ in a zero-dimensional moduli space $\MM^d_X(x_0,\dots,x_d)$
to $\mu^d_X$ to be
\begin{equation} \label{eq:qe}
 \pm q^{E(u) + \int_S u^*(-dK_S + \half\{K_S,K_S\})} :
 \Lambda_{\Q,x_d} \otimes \dots \otimes \Lambda_{\Q,x_1}
 \longrightarrow \Lambda_{\Q,x_0},
\end{equation}
with the sign again depending on \eqref{eq:orientation-mechanism}.
There is a uniform lower bound for the $\int_S$ term. This, together
with Gromov compactness, ensures that the sum is a well-defined
$\Lambda_\Q$-multilinear map.

\begin{remark}
The choice of $\lambda_L$ serves only to fix the action: the Floer
cohomology groups for different choices are isomorphic in an obvious
way, by multiplication with $q^m$ for some $m \in \Q$. In particular,
changing $\lambda_L$ does not affect the quasi-isomorphism class of
the resulting object in $\Fuk(X)$.
\end{remark}

\subsection{}
Take $X$ as before, but now equipped with a section
$\sigma_{X,\infty}$ of $o_X$ whose zero set $X_\infty$ is a reduced
divisor with normal crossings, so that $M = X \setminus X_\infty$ is
an affine Calabi-Yau surface. Following Remark \ref{th:class-of-j},
we also assume that $X_\infty$ has no rational components. The
objects of the relative Fukaya category $\Fuk(X,X_\infty)$ are a
mixture of the two previously introduced classes,
\eqref{eq:exact-brane} and \eqref{eq:rational-brane}. One considers
{\em relative Lagrangian branes}
\begin{equation} \label{eq:relative-brane}
L^\mbr = (L,\alphagr_L,\Spin_L,J_L)
\end{equation}
where $(L,\alphagr_L,\Spin_L)$ is an exact Lagrangian brane in $M$;
and $J_L$ is an $\o_X$-compa\-tible almost complex structure on $X$
such that each component of $X_\infty$ is an almost complex
submanifold, and which is regular with respect $L$. The restriction
on $J_L|X_\infty$ does not interfere with the regularity theory for
somewhere injective pseudo-holo\-morphic spheres $v: S^2 \rightarrow
X$, since $v^{-1}(X_\infty)$ is necessarily a finite set, while the
set of somewhere injective points is open; and the same applies to
pseudo-holomorphic discs. For any two objects, choose a Floer datum
\[
(H,J) = (H_{L_0^\mbr,L_1^\mbr},J_{L_0^\mbr,L_1^\mbr})
\]
such that $H \in \smooth([0;1] \times X,\R)$ vanishes along $[0;1]
\times X_\infty$, along with its first derivative; and $J_t|X_\infty$
makes the components of $X_\infty$ almost complex. In addition, we
assume that $J_k = J_{L_k}$ for $k = 0,1$, and that there are no
non-constant $J_t$-holomorphic spheres for any $t$.

We have the usual set $\CC(L_0^\mbr,L_1^\mbr)$ of $H$-trajectories
joining $L_0$ to $L_1$, and for $x$ in that set we define
$\Lambda_{\N,x}$ in the same way as $\C_x$, just using $\Lambda_\N$
instead of $\C$ as the ground ring. The Floer cochain space in
$\Fuk(X,X_\infty)$ is
\[
 CF_{X,X_\infty}^j(L_0^\mbr,L_1^\mbr) =
 \bigoplus_{\substack{x \in \CC(L_0^\mbr,L_1^\mbr)
 \\ I(x) = j}} \Lambda_{\N,x}.
\]
For the Floer differential, each $u \in \MM^1_X(x_0,x_1)$ contributes
with $\pm q^{u \cdot X_\infty}: \Lambda_{\N,x_1} \rightarrow
\Lambda_{\N,x_0}$. This makes sense for a single $u$ since the number
$u \cdot X_\infty$ is nonnegative; and Gromov compactness, together
with a version of \eqref{eq:energy}, says that the possibly infinite
sum over all $u$ gives rise to a well-defined map
$\mu^1_{X,X_\infty}$. Denote the resulting Floer cohomology by
$HF^*_{X,X_\infty}(L_0^\mbr,L_1^\mbr)$. The definition of the higher
order compositions works in the same way, and we will therefore omit
the details.

\begin{remark} \label{th:class-of-j-2}
Let $X$, $X'$ be two Calabi-Yau surfaces, with sections
$\sigma_{X,\infty}$, $\sigma_{X',\infty}$ of their ample line bundles
$o_X$, $o_{X'}$ which define reduced divisors with normal crossings,
having no rational components. Suppose that there is a symplectic
isomorphism
\[
\phi: (X,X_\infty) \longrightarrow (X',X'_\infty)
\]
with the following additional properties: $Hom(o_X,\phi^*o_{X'})$ is
the trivial flat line bundle, and over $M = X \setminus X_\infty$,
its covariantly constant sections lie in the same homotopy class of
nowhere zero sections as
$\phi^*\sigma_{X',\infty}/\sigma_{X,\infty}$. Moreover, under the
isomorphism (unique up to homotopy) $\K_X \iso \phi^*\K_{X'}$ induced
by deforming the almost complex structure on $X$ to the pullback of
the one from $X'$, the nowhere zero sections $\eta_X$ and
$\phi^*\eta_{X'}$ are homotopic. Then, by suitably choosing the Floer
and perturbation data, one can define an induced quasi-isomorphism of
$A_\infty$-categories over $\Lambda_\N$,
\[
\phi_*: \Fuk(X,X_\infty) \rightarrow \Fuk(X',X'_\infty).
\]
In the case where $H_1(X) = 0$, all the additional conditions imposed
on $\phi$ are automatically satisfied, for the same reasons as in
Remark \ref{th:h1}.
\end{remark}

\subsection{}
Finally, we need to establish the relation between the different
kinds of Fukaya categories. Let $X$ be a Calabi-Yau surface with a
suitable section $\sigma_{X,\infty}$, so that $\Fuk(M)$, $\Fuk(X)$
and $\Fuk(X,X_\infty)$ are all defined.

\begin{prop} \label{th:q-zero}
Consider $\C$ as a $\Lambda_\N$-module in the obvious way, with $q$
acting trivially. Then the $\C$-linear $A_\infty$-category obtained
from $\Fuk(X,X_\infty)$ by reduction of constants, $\Fuk(X,X_\infty)
\otimes_{\Lambda_\N} \C$, is quasi-equivalent to $\Fuk(M)$.
\end{prop}

\begin{prop} \label{th:invert-q}
Consider $\Lambda_\Q \supset \Lambda_\N$ as a $\Lambda_\N$-module in
the obvious way. Then the $\Lambda_\Q$-linear category obtained from
$\Fuk(X,X_\infty)$ by extension of constants, $\Fuk(X,X_\infty)
\otimes_{\Lambda_\N} \Lambda_\Q$, is quasi-isomorphic to a full
$A_\infty$-subcategory of $\Fuk(X)$. This subcategory consists of all
those $L^\rbr$ such that $L \cap X_\infty = \emptyset$, $o_X|L$ is
the trivial flat bundle, and its covariantly constant sections lie in
the same homotopy class as $\sigma_{X,\infty}|L$.
\end{prop}

Both statements are essentially obvious. For Proposition
\ref{th:q-zero}, one notices that setting $q = 0$ means that we count
only those solutions of Floer's equation or its generalization which
have zero intersection number with, hence are disjoint from,
$X_\infty$. Then, the only difference between $\Fuk(M)$ and
$\Fuk(X,X_\infty) \otimes_{\Lambda_\N} \C$ is that the latter
category has more objects, consisting of the same exact Lagrangian
brane with different choices of $J_L$. However, all these objects are
isomorphic, whence the quasi-equivalence. For Proposition
\ref{th:invert-q} one has to choose a function $K_L$, $dK_L =
\theta_M|L$, for each exact Lagrangian brane in $M$. The associated
object in $\Fuk(X)$ then carries the multisection $\lambda_L$ from
\eqref{eq:k-lambda}, and one identifies
\begin{equation} \label{eq:mq}
 CF^*_{(X,X_\infty)}(L_0^\mbr,L_1^\mbr) \otimes_{\Lambda_\N}
 \Lambda_\Q \stackrel{\iso}{\longrightarrow}
 CF^*_X(L_0^\rbr,L_1^\rbr)
\end{equation}
by using the map $q^{A(x)}: \Lambda_{\N,x} \otimes_{\Lambda_\N}
\Lambda_{\Q} \rightarrow \Lambda_{\Q,x}$ on each one-dimensional
subspace. The compositions in the two categories are based on the
same moduli spaces, and the fact that the weights used to count
solutions correspond under \eqref{eq:mq} is a consequence of
\eqref{eq:energy} and a suitable generalization. Clearly,
$\Fuk(X,X_\infty) \otimes_{\Lambda_\N} \Lambda_\Q$ is not all of
$\Fuk(X)$, since only the objects which are exact Lagrangian
submanifolds in $M$ occur.

\subsection{}
While the basic properties of affine Fukaya categories are familiar
and quite simple, see for instance \cite{seidel04}, the relative and
projective versions reserve some small surprises. These are all
covered by the powerful general theory from \cite{fooo}, but we
prefer a more elementary approach. The relative situation is the more
important one for the arguments later on, and we will consider it
primarily.

Fix an affine Calabi-Yau surface $M = X \setminus X_\infty$, without
rational components in $X_\infty$. Take an exact Lagrangian brane
$(L,\alphagr_L,\Spin_L)$ in $M$, and a one-parameter family $J_t$ ($0
\leq t \leq 1$) of compatible almost complex structures, all of which
make the components of $X_\infty$ almost complex, and are regular
with respect to $L$. These give rise to a family of objects of
$\Fuk(X,X_\infty)$, which we denote by $L_t^\mbr$. By suitably
adapting the construction of Piunikhin-Salamon-Schwarz
\cite{piunikhin-salamon-schwarz94}, one defines a canonical
isomorphism
\begin{equation} \label{eq:pss}
H\Psi_{X,X_\infty} : H^*(L;\C) \otimes_\C \Lambda_\N
\stackrel{\iso}{\longrightarrow}
HF^*_{X,X_\infty}(L_0^\mbr,L_1^\mbr).
\end{equation}
In the special case where $J_t$ is constant, one can use this to
obtain a distinguished element $e = H\Psi_{X,X_\infty}(1) \in
HF^0_{X,X_\infty}(L_0^\mbr,L_0^\mbr)$, which is in fact the
cohomological unit for $L_0^\mbr$ in $H\Fuk(X,X_\infty)$. The more
general isomorphisms \eqref{eq:pss}, for different choices of almost
complex structures, carry the ordinary cup product to the pair of
pants product. This eventually shows that $L_0^\mbr$, $L_1^\mbr$ are
quasi-isomorphic objects. A slight generalization, which in fact is
already implicit in our use of Floer data, then proves the following

\begin{lemma} \label{th:move-object}
Let $L_t$ be a family of Lagrangian submanifolds of $M$, and
$L_t^\mbr$ a family of relative brane structures on them, varying
smoothly with $t$. Then all $L_t^\mbr$ are quasi-isomorphic objects
of $\Fuk(X,X_\infty)$. \qed
\end{lemma}

Note that because all the $L_t$ are supposed to have relative brane
structures, they are in particular exact in $M$, so that the isotopy
is Hamiltonian.

\subsection{}
We will now summarize the definition of $H\Psi_{X,X_\infty}$. Pick a
cohomology class $c \in H^j(L;\Z)$, and represent its Poincar{\'e}
dual by a map $\gamma: C \rightarrow L$ from a closed oriented
manifold $C$ (recall that $L$ is a surface). Take the Riemann surface
$S = \mathbb{H}$, the closed upper half plane, and consider a
generalized Floer equation for pairs $(u,y)$ consisting of a map $u:
S \rightarrow M$ and a point $y \in C$,
\begin{equation} \label{eq:pss-equation}
\begin{cases}
 \!\!\! & u(\partial S) \subset L, \\
 \!\!\! & u(0) = \gamma(c), \\
 \!\!\! & (du - Y_{S,u(z),z}) + J_{S,z,u(z)} \circ (du - Y_{S,u(z),z} \circ i) = 0, \\
 \!\!\! & \lim_{s \rightarrow +\infty} u(e^{\pi(s+it)}) = x_0(1-t),
 \quad x_0 \in \CC(L_0^\mbr,L_1^\mbr).
\end{cases}
\end{equation}
The choice of almost complex structures $J_S$ and inhomogeneous term
$Y_S$ has to obey a whole series of conditions, of which we mention
only the most relevant one: along the boundary $\partial S = \R$,
$J_{S,z} = J_{\psi(z)}$ for some function $\psi: \R \rightarrow
[0;1]$ with $\psi(s) = 0$ for very negative $s$; $\psi(s) = 1$ for
very positive $s$; and $\psi'(s) > 0$ for all $s$ such that $\psi(s)
\in (0;1)$.

Assume that the moduli spaces $\MM^{\Psi}_{X,X_\infty}(x_0;\gamma)$
of solutions of \eqref{eq:pss-equation} are all regular. If $I(x_0) =
j$, the relevant moduli space is zero-dimensional, and every $(u,y)$
in it comes with a canonical choice of point in $or(x_0)$. This will
define an element $\pm q^{u \cdot X_\infty} \in \Lambda_{\N,x_0}$,
and the sum of all these is denoted by
\[
\Psi_{X,X_\infty}(\gamma) \in CF^j_{X,X_\infty}(L_0^\mbr,L_1^\mbr).
\]
The proof that this is a Floer cocycle relies on a standard gluing
argument: one looks at the (one-dimensional) moduli spaces
$\MM^{\Psi}_{X,X_\infty}(x_0;\gamma)$ with $I(x_0) = j+1$, and proves
that points in the boundary of their Gromov compactification are
pairs consisting of an isolated solution which contributes to
$\Psi_{X,X_\infty}(\gamma)$ and a Floer gradient trajectory. The main
point is that due to the choice of $J_S$, there is no bubbling off of
pseudo-holomorphic discs. By definition, the Floer cohomology class
of $\Psi_{X,X_\infty}(\gamma)$ is $H\Psi_{X,X_\infty}(c \otimes 1)$.

\subsection{}
The assumption made up to now, that all the $J_t$ are regular, is not
generic: while a single almost complex structure will typically admit
no nonconstant pseudo-holomorphic discs with boundary on $L$, these
can no longer be avoided in one-parameter families, as one can see
from \eqref{eq:minus-one}. In other words, the space of all almost
complex structures is divided into chambers by codimension one walls
consisting of non-regular ones. Because there is an infinite number
of possible homotopy classes of discs, there is potentially an
infinite number of walls, which might well be everywhere dense. We
will look at the simplest instance of the behaviour of Floer
cohomology under wall-crossing (a complete treatment is beyond our
scope here, since it requires techniques that can deal with the lack
of transversality for multiply-covered discs, see \cite{fooo}).

\begin{assumptions} \label{as:wall-crossing}
Fix an exact Lagrangian brane $(L,\alphagr_L,\Spin_L)$ in $M$, and a
family $(J_t)$ of compatible almost complex structures, each of which
makes the components of $X_\infty$ almost complex. Suppose that this
family has the following properties:
\begin{itemize}
\item
$J_0,J_1$ are regular with respect to $L$;
\item
for all $t$, there are no $J_t$-holomorphic spheres $v: S^2
\rightarrow X$ such that $v \cdot X_\infty = 1$;
\item
there is a $t_0 \in (0;1)$ such that for all $t \neq t_0$, no
$J_t$-holomorphic discs $w: (D,\partial D) \rightarrow (X,L)$ with $w
\cdot X_\infty = 1$ exist. For $J_{t_0}$, there is exactly one such
disc, which we denote by $w_0$. This is parametrized-regular, and
$[w_0|\partial D] \in H_1(L)$ is nonzero.
\end{itemize}
\end{assumptions}

The terminology in the last sentence may require some explanation.
Denote by $D\bar\partial_{w,X}: \Gamma(w^*TX,w^*TL) \rightarrow
\Omega^{0,1}(w^*TX)$ the standard linearized operator associated to a
$J_t$-holomorphic disc $w: (D,\partial D) \rightarrow (X,L)$. Here
$\Gamma(E,F)$ means sections of the vector bundle $E \rightarrow D$
with boundary values in the totally real subbundle $F \subset
E|\partial D$. The parametrized version,
\[
D\bar\partial_{w,X}^{par}: \R \times \Gamma(w^*TX,w^*TL)
\longrightarrow \Omega^{0,1}(w^*TX),
\]
adds a first component $1 \mapsto \half (\partial J_t/\partial t)
\circ dw \circ i$. If this is onto in suitable Sobolev completions,
we say that $w$ is parametrized-regular.

Consider the equation \eqref{eq:pss-equation} as before, choosing
$\psi$ in such a way that $\psi(0) = t_0$. For solutions $(u,y) \in
\MM^{\Psi}_{X,X_\infty}(x_0;\gamma)$ with $I(x_0) = j$, which occur
in zero-dimensional moduli spaces, one can achieve by a suitable
generic choice of $(K_S,J_S)$ that $u(z)$ does not meet the boundary
of $J_{\psi(z)}$-holomorphic discs in $(X,L)$, for any $z \in \R$.
Hence one can define Floer cochains $\Psi_{X,X_\infty}(\gamma)$ as
before. Expand these and the Floer differential on
$CF^*_{X,X_\infty}(L_0^\mbr,L_1^\mbr)$ into powers of $q$,
\[
\begin{aligned}
 & \Psi_{X,X_\infty}(\gamma)
 = \Psi_0(\gamma) + \Psi_1(\gamma) q + \Psi_2(\gamma) q^2 + \dots, \\
 & \mu^1_{X,X_\infty} = \mu_0 + \mu_1 q + \mu_2 q^2 + \dots
\end{aligned}
\]

\begin{claim}
$\mu_0 \Psi_0(\gamma) = 0$; indeed, $\Psi_0$ is an isomorphism
between $H^*(L;\C) \otimes \Lambda_{\N}$ and the cohomology of
$CF^*_{X,X_\infty}(L_0^\mbr,L_1^\mbr)$ with only $\mu_0$ as
differential.
\end{claim}

This is simply the statement of the Piunikhin-Salamon-Schwarz
isomorphism for $L$ as an exact Lagrangian submanifold in the affine
Calabi-Yau $M$: by Proposition \ref{th:q-zero}, $\mu_0 = \mu^1_M$ is
the differential in $\Fuk(M)$, and similarly $\Psi_0$ counts
solutions of \eqref{eq:pss-equation} which stay inside $M$.

\begin{claim}
Suppose that $(C,\gamma) = (L,\id)$ is the whole surface. Then
\[
\mu_1 \Psi_0(\gamma) + \mu_0 \Psi_1(\gamma) = \pm \Psi_0(w_0|\partial
D).
\]
\end{claim}

The $\Psi_0(w_0|\partial D)$ term comes from a new stratum in the
compactification of the one-dimensional moduli spaces, which consists
of solutions of \eqref{eq:pss-equation} with the disc bubble $w_0$
attached to it at the point $z = 0$. Under the regularity condition
on $w_0$ that we have imposed, these are smooth boundary points of
the compactification.

A straightforward computation starting from the two claims above
shows that
\[
HF^0_{X,X_\infty}(L_0^\mbr,L_1^\mbr) = 0.
\]
In particular $L_0^\mbr$, $L_1^\mbr$ are not quasi-isomorphic as
objects of $\Fuk(X,X_\infty)$. Moreover, if we cut off the $q$-series
after the first order, then by the same kind of computation as before
the natural map
\begin{equation} \label{eq:orders}
 H^0(CF^*_{X,X_\infty}(L_0^\mbr,L_1^\mbr)
 \otimes_{\Lambda_\N} \C[q]/q^2) \otimes_{\C[q]/q^2} \C
 \longrightarrow H^0(CF^*_M(L_0^\br,L_1^\br))
\end{equation}
vanishes. More precisely, both sides are isomorphic to $\C$, but the
left hand side is represented by the cocycle $q\Psi_0([L])$, which
maps to zero. As a consequence,

\begin{cor} \label{th:wall-crossing}
Suppose that there is some $(L,\alphagr_L,\Spin_L)$ such that
Assumptions \ref{as:wall-crossing} hold. Then $\Fuk(X,X_\infty)$,
seen as a one-parameter deformation of $A_\infty$-categories, is
nontrivial at first order in $q$. \qed
\end{cor}

\subsection{}
Our assumption that $[w_0|\partial D] \in H_1(L)$ is nontrivial
obviously excludes Lagrangian spheres, and indeed wall-crossing in
trivial in that case:

\begin{lemma} \label{th:spheres}
Up to quasi-isomorphism, a graded Lagrangian sphere $\Lgr$ in $M$
gives rise to a unique object of $\Fuk(X,X_\infty)$.
\end{lemma}

\proof Take two objects $L_0^\mbr,L_1^\mbr$ obtained by various
choices of relative brane structures on $L$ (keeping the same
grading). Clearly, these two objects are quasi-isomorphic in
$\Fuk(X,X_\infty) \otimes_{\Lambda_\N} \C \iso \Fuk(M)$. Because
there is no odd-dimensional cohomology, the quasi-isomorphism between
them survives to an element of
$HF^0_{X,X_\infty}(L_0^\mbr,L_1^\mbr)$, and an obvious spectral
sequence argument shows that this is a quasi-isomorphism in
$\Fuk(X,X_\infty)$. \qed

\begin{remark}
Essentially all of our observations concerning relative Fukaya
categories carry over to the projective case. First of all, we have
an analogue of Lemma \ref{th:move-object}, saying that {\em the
isomorphism class of an object of $\Fuk(X)$ remains constant under
smooth deformation}. Secondly, {\em crossing a wall on which a single
lowest area holo\-morphic disc $w_0$ sits does change the isomorphism
class of an object}, assuming as before that $w_0$ is
parameterized-regular and $[w_0|\partial D]$ is nontrivial. Finally,
for graded Lagrangian spheres in $X$ we have the same statement as in
Lemma \ref{th:spheres}. The proofs are quite similar, except that the
filtration by intersection numbers with $X_\infty$ has to be replaced
by a filtration by values of the action functional. This follows
methods that were pioneered by Oh \cite{oh96} and vastly developed in
\cite{fooo}. In our case, the rationality assumption simplifies the
technical details somewhat, since it makes the filtration discrete.
\end{remark}

\subsection{}
As shown by Corollary \ref{th:wall-crossing}, one can use
wall-crossing phenomena in the relative Fukaya category to prove its
nontriviality in the sense of deformation theory; provided of course
that one can come up with a Lagrangian submanifold for which
wall-crossing actually occurs. To make that easier, we now suppose
that the Calabi-Yau surface under discussion is the fibre of a
quasi-Lefschetz pencil, so that the desired Lagrangian submanifold
can be first constructed in the degenerate fibre at $\infty$.

For this discussion, we change notation as follows: $X$ is a Fano
threefold with $o_X = \K_X^{-1}$, carrying a pair of sections
$\sigma_{X,0},\sigma_{X,\infty}$ which generate a quasi-Lefschetz
pencil of Calabi-Yau surfaces $\{X_z\}$. To be able to define the
relative Fukaya categories, we assume that $X_{0,\infty}$ has no
rational components.

Fix some irreducible component $C$ of $X_\infty$, and denote by $C'$
the union of all the other components. There is an obvious short
exact sequence
\begin{equation} \label{eq:tangent-sequence}
 0 \longrightarrow TC \longrightarrow
 TX|C \xrightarrow{D(\sigma_{X,\infty}/\sigma_{X,0})}
 \O_C(C \cap X_0 - C \cap C') \longrightarrow 0.
\end{equation}
From the resulting isomorphism $\K_C \iso \K_X|C \otimes
\K_X^{-1}(-C')|C \iso \O_C(-C' \cap C)$ one obtains a meromorphic
two-form $\eta_C$ on $C$ with simple poles along $C \cap C'$.
Similarly, from $A_X|C$ and $\sigma_{X,0}|C$ one obtains a one-form
$\theta_C$ on $C \setminus X_0$ with $d\theta_C = \o_C = \o_X|C$.

\begin{assumptions} \label{as:l-zero}
Suppose that we have the following data $L_0$, $w_0$:
\begin{itemize}
\item
$L_0 \subset C$ is a Lagrangian submanifold, disjoint from $C' \cup
X_0$, exact with respect to $\theta_C$, and graded with respect to
$\eta_C$. This implies that the Maslov number and area of any
holo\-morphic disc $w: (D,\partial D) \rightarrow (C,L_0)$ can be
computed as intersection numbers:
\begin{equation} \label{eq:c-index-energy}
\begin{aligned}
 \mu(w) & = 2 (w \cdot C'), \\
 \textstyle \int w^*\o_C & = w \cdot X_0.
\end{aligned}
\end{equation}

\item
$w_0: (D,\partial D) \rightarrow (C,L_0)$ is a specific holo\-morphic
disc with $w_0 \cdot X_0 = w_0 \cdot C' = 1$. The point $w_0(0)$
should lie in $C' \cap X_0$, and $[w_0|\partial D] \in H_1(L_0)$
should be nontrivial. Take the linearized operator of $w_0$, and
restrict its domain to those sections $\xi$ with $\xi(0) = 0$. We ask
that this restriction, denoted by
\begin{equation} \label{eq:restricted-dbar}
 D\bar\partial_{w_0,C}^{res}: \Gamma(w_0^*TC,w_0^*TL_0)^{res}
 \longrightarrow \Omega^{0,1}(w_0^*TC),
\end{equation}
should have one-dimensional cokernel (in suitable Sobolev spaces).

\item
Moreover,
\begin{romanlist}
\item \label{item:no-disc-1}
every holo\-morphic disc $w$ in $(C,L_0)$ with $w \cdot C' = w \cdot
X_0 = 1$ and $w(0) \in C' \cap X_0$ is equal to $w_0$, up to a
possible action of $S^1 \subset Aut(D)$;
\item \label{item:no-disc-2}
there are no holomorphic discs $w$ in $(C,L_0)$ with $w \cdot C' =
0$, $w \cdot X_0 = 1$;
\item \label{item:no-sphere}
there are no holo\-morphic spheres $v$ in $C$ with $\int v^*\o_C =
1$.
\end{romanlist}
\end{itemize}
\end{assumptions}

One can put $L_0$ into a smooth family $\{L_s\}$ (for $s \in \R$,
$|s|$ small) of Lagrangian submanifolds, $L_s \subset X_{1/s}$, such
that the union
$
L = \bigcup_s L_s
$
is Lagrangian (but of course not closed) in $X$. In fact, the
condition on $L$ determines the $L_s$ uniquely, since it means that
they get mapped into each other by symplectic parallel transport in
$\R$-direction.

\begin{lemma}
For each $s \neq 0$, $L_s$ is an exact Lagrangian submanifold in the
affine K{\"a}hler surface $M_{1/s}$, and graded with respect to
$\eta_{X_{1/s}}$.
\end{lemma}

\proof Supposing that $s \neq 0$ remains sufficiently small, one has
$L_s \cap X_0 = \emptyset$, and since $X_{1/s} \cap X_0 = X_{1/s}
\cap X_\infty$, also $L_s \cap X_\infty = \emptyset$. Because
$o_X|L_\infty$ is trivial and $L$ is Lagrangian, $o_X|L_s$ is trivial
for any $s$. Moreover, its covariantly constant sections lie in the
same homotopy class of nowhere zero sections as $\sigma_{X,0}|L_s$,
because that is true for $s = 0$. On the other hand,
$\sigma_{X,\infty}/\sigma_{X,0}$ is constant on $L_s$, so the
covariantly constant sections are also homotopic to
$\sigma_{X,\infty}|L_s$, which is equivalent to exactness. The
argument with the grading is essentially the same. \qed

Exactness of $L_0$ with respect to $\theta_C$ means that the area of
holo\-morphic discs in $(C,L_0)$ is integral. Our disc $w_0$ has
minimal area, since by assumption $\int w_0^*\o_C = w_0 \cdot X_0 =
1$. By \cite{kwon-oh96,lazzarini98}, $w_0$ has a somewhere injective
point. Standard transversality arguments tell us that there is a
family $(J_{C,r})$, for $r \in \R$ with $|r|$ small, of compatible
almost complex structures on $C$ such that $J_{C,0}$ is the given
complex structure, and which makes $w_0$ parametrized-regular. Since
the set of somewhere injective points is open, and $w_0$ intersects
$C' \cup X_0$ only in $w_0(0)$, one can assume that all $J_{C,r}$ are
equal to $J_{C,0}$ near $C \cap (C' \cup X_0)$. Having that, one can
extend them to a family $(J_{X,r})$ of compatible almost complex
structures on $X$, such that again $J_{X,0}$ is the given complex
structure, and with the following two properties for all $r$:
$J_{X,r} = J_{X,0}$ near $X_0$; and the projection $\pi^{-1} =
\sigma_{X,\infty}/\sigma_{X,0}: X \setminus X_0 \rightarrow \C$ is
$J_{X,r}$-holo\-morphic.

Consider the moduli space $\MM^{para}$ of pairs $(r,w)$, where $w:
(D,\partial D) \rightarrow (X,L)$ is a $J_{X,r}$-holo\-morphic disc
such that $w \cdot X_0 = 1$ and $w(0) \in X_0 \cap C'$, divided by
the action of $S^1$ by rotation on $D$.

\begin{lemma} \label{th:is-in-fibre}
For every $(r,w)$ in this moduli space, the image of $w$ is contained
in some fibre $X_{1/s}$ of the Lefschetz pencil, with $s \in \R$.
\end{lemma}

\proof Consider the meromorphic map $\pi^{-1} \circ w: D
\dashrightarrow \C$. $w$ intersects $X_0$ only at $w(0)$ and with
multiplicity one. At the same point it intersects $X_\infty$, and
this implies that $\pi^{-1} \circ w$ is in fact holomorphic. Since
its boundary values are real, it must be constant. \qed

\begin{lemma} \label{th:compactness}
Let $(r_k,w_k)$ be a sequence of pairs, where $w_k: (D,\partial D)
\rightarrow (X,L)$ is a $J_{X,r_k}$-holomorphic disc with $\int
w_k^*\o_X = 1$, whose image is contained in $X_{1/s_k}$ for some $s_k
\neq 0$. Suppose that $r_k,s_k \rightarrow 0$; then the $w_k$, after
a possible reparametrization by automorphisms of $D$ and passing to a
subsequence, converge to $w_0$.
\end{lemma}

\proof Gromov compactness tells us that the limit of a subsequence is
a stable disc in $X_\infty$, but in our case each non-constant
component contributes an integral amount of energy, which means that
there can be only one of them. Furthermore, the case where the
non-constant component is a sphere is excluded by part
\ref{item:no-sphere} of our assumptions. We conclude that after
passing to a subsequence, the convergence is in $\smooth$-sense
towards a limit which is a holomorphic disc $w: (D,\partial D)
\rightarrow (C,L_0)$.

We know that $w \cdot X_0 = 1$, and \ref{item:no-disc-2} says that
$w$ must intersect $C'$. Every $z \in D$ such that $w(z) \in C'$ must
also satisfy $w(z) \in X_0$, otherwise the disc $w$ could not be the
smooth limit of discs in the fibres $X_{1/s_k}$. In fact, a closer
look at the local geometric situation shows that $w \cdot C' \leq w
\cdot X_0$. Hence $w \cdot X_0 = w \cdot C' = 1$, and after
reparametrization we may assume that $w(0) \in X_0 \cap C'$, so an
appeal to \ref{item:no-disc-1} finishes the argument. \qed

For the next step, we need to compare the linearized operators
associated to $w_0$ as a holomorphic disc in $C$ and $X$. From
\eqref{eq:tangent-sequence} one gets a diagram with exact columns,
\begin{equation} \label{eq:e-diagram}
\xymatrix{
 0 \ar[d] & 0 \ar[d] \\
 \Gamma(w_0^*TC,w_0^*TL_0) \ar[r]^-{D\bar\partial_{w_0,C}} \ar[d] &
 \Omega^{0,1}(w_0^*TC) \ar[d] \\
 \Gamma(w_0^*TX,w_0^*TL) \ar[r]^-{D\bar\partial_{w_0,X}} \ar[d] &
 \Omega^{0,1}(w_0^*TX) \ar[d] \\
 \!\!\!\!\! \Gamma(w_0^*\O_C(C \cap X_0 - C \cap C'),w_0^*\Lambda) \ar[d]
 \ar[r]^-{\bar\partial} & \Omega^{0,1}(w_0^*\O_C(C \cap X_0 - C \cap C')) \!\!\!\!
 \ar[d] \\
 0 & 0
}
\end{equation}
where $\Lambda$ is the canonical real subbundle of $\O_C(C \cap X_0 -
C \cap C')|L_0$, whose existence is due to the fact that $L_0$ is
disjoint from $X_0 \cup C'$. As usual, we use suitable Sobolev spaces
to make each row elliptic. The bottom row is a
$\bar\partial$-operator on a line bundle over a disc: in fact, since
$w_0$ intersects both $X_0$ and $C'$ only at $z = 0$ at with
multiplicity one, this is just the standard $\bar\partial$-operator
for functions with real boundary conditions. Hence, its Sobolev
completion is onto with one-dimensional kernel, consisting of nowhere
zero functions. Without losing exactness of the columns, we may
restrict the domains of the top two $D\bar\partial$-operators to
sections with $\xi(0) \in T(X_0 \cap C')$ (this tangent space is zero
for the first row, and two-dimensional for the second one). Kernels
and cokernels of the restrictions will fit into a long exact sequence
\[
 0 \rightarrow \ker\, D\bar\partial_{w_0,C}^{res} \rightarrow
 \ker\, D\bar\partial_{w_0,X}^{res} \rightarrow \R \rightarrow
 \coker\,D\bar\partial_{w_0,C}^{res} \rightarrow \coker\,D\bar\partial_{w_0,X}^{res}
 \rightarrow 0.
\]
From this and \eqref{eq:c-index-energy} one sees that the expected or
virtual dimension of the moduli space $\MM^{para}$ at $(0,w_0)$ is
\[
\text{v.dim}\, \MM^{para} = \ind\,D\bar\partial_{w_0,X}^{res} + 1 - 1
= \ind\,D\bar\partial_{w_0,C} - 3 = 2 + \mu(w_0) - 3 = 1.
\]
(in the second expression, the $+1$ comes from the added parameter
$r$, and the $-1$ from the action of $S^1$ by rotations). By
assumption $\coker\,D\bar\partial_{w_0,C}^{res} \iso \R$, and by
construction of $\{J_{C,r}\}$, this cokernel is spanned by $(\partial
J_{C,r}/\partial r)_{r = 0} \circ dw_0 \circ i$. Hence, the cokernel
of $D\bar\partial_{w_0,X}^{res}$ is spanned by $(\partial
J_{X,r}/\partial r)_{r = 0} \circ dw_0 \circ i$. This means that the
corresponding parametrized linearized operator is onto, so that
$(0,w_0)$ is a regular point of $\MM^{para}$.

\begin{lemma}
There is a family $(r_t,w_t)$ of regular points of $\MM^{para}$, for
$t \in \R$ with $|t|$ small, with $r_0 = 0$ and $w_0$ our given
holomorphic disc. This satisfies $w_t(D) \subset X_{1/s_t}$ for some
$s_t \in \R$ smoothly depending on $t$, and with $s_0 = 0$,
$(ds_t/dt)_{t = 0} \neq 0$.
\end{lemma}

\proof We know that $\MM^{para}$ is locally a smooth one-manifold, so
take a family $(r_t,w_t)$ which parametrizes it locally. The fact
that $w_t \in X_{1/s_t}$ for some $s_t$ comes from Lemma
\ref{th:is-in-fibre}, and the smoothness of $t \mapsto s_t$ is
obvious. Now suppose that $(ds_t/dt)_{t = 0}$ vanishes. Then
$(\partial r_t/\partial t, \partial w_t/\partial t)_{t = 0}$ lies in
the kernel of the parametrized version of
$D\bar\partial_{w_0,C}^{res}$. By construction this is onto, and for
index reasons the only nontrivial kernel elements are those obtained
from infinitesimal rotations of $D$. This contradicts the fact that
we started with a local chart for $\MM^{para}$. \qed

\begin{lemma} \label{th:fibre-regular}
For all small $t \neq 0$, $w_t$ is parametrized-regular as a
$J_{r_t}$-holomor\-phic disc in $(X_{1/s(t)},L_{s(t)})$.
\end{lemma}

The proof is by a comparison argument using a diagram of
$\bar\partial$-operators similar to \eqref{eq:e-diagram}; we leave it
to the reader. Now fix some small $\epsilon>0$ and $0 \neq t \ll
\epsilon$. Then in the family $(J_{X,r}|X_{1/s(t)})$, $r \in
[-\epsilon;\epsilon]$, of almost complex structures there is no
pseudo-holomorphic disc in $(X_{1/s(t)},L_{s(t)})$ with unit area
other than $w_t$. In fact, the contrary assumption for very small
$\epsilon$ and $t$ leads to a contradiction with Lemma
\ref{th:compactness} and the local structure of the moduli space
$\MM^{para}$. By Lemma \ref{th:fibre-regular}, this situation
persists under a small perturbation of our family of almost complex
structures; by such a perturbation, we can make it regular with
respect to $L_{s(t)}$ at the endpoints $r = \pm\epsilon$, and
eliminate pseudo-holomorphic spheres in $X_{1/s(t)}$ for all values
of $r$. We are now in the situation of Corollary
\ref{th:wall-crossing}, which shows that
$\Fuk(X_{1/s(t)},X_{1/s(t),\infty})$ is nontrivial at first order.
Using Remark \ref{th:class-of-j-2} and symplectic parallel transport
one can extend this to other fibres of our pencil, so the conclusion
is:

\begin{cor} \label{th:wall-crossing-2}
Suppose that Assumptions \ref{as:l-zero} hold. Take a smooth fibre
$X_z$ of our pencil of Calabi-Yau surfaces. Then its relative Fukaya
category $\Fuk(X_z,X_{0,\infty})$, seen as a one-parameter
deformation of $A_\infty$-categories, is nontrivial at first order.
\qed
\end{cor}

\section{Computations in Fukaya categories\label{sec:induction}}

To effect the link between symplectic geometry and homological
algebra, we basically rely on two tools. The first is the
correspondence between Dehn twists and algebraic twists, which is a
reformulation of the long exact sequence in Floer cohomology. The
version in the literature \cite{seidel01,seidel04} is for the exact
or affine situation, but as will be outlined below, the proof can be
easily adapted to the projective and relative cases. Our second tool
is a dimensional induction machine based on the notion of matching
cycle, developed in \cite{seidel04}; this is strictly limited to the
affine context, which is the reason why affine varieties appear in
this paper at all. We will treat dimensional induction as a black
box, stating its properties in an axiomatic way only; ultimately it
is again an application of the long exact sequence, but the way in
which this works is too complicated to be summarized briefly.

\subsection{}
Let $M$ be an affine Calabi-Yau of dimension $n>1$, and $L \subset M$
a Lagrangian sphere. Equip it with a grading and the unique $Spin$
structure, making it into an object $L^\br$ of $\Fuk(M)$. Let
$\tau_L$ be the Dehn twist along $L$, and $\taugr_L$ its canonical
grading (again, there is the small matter of framing, which we
ignore). Consider also the algebraic twist $T_{L^\br}$ along $L^\br$,
which is a map of objects of $D^b\Fuk(M)$. Let $L_1^\br$ be any exact
Lagrangian brane, with underlying Lagrangian submanifold $L_1$.
Clearly $\tau_L^{-1}(L_1)$ carries an induced brane structure, making
it into an object $\taugr_L^{-1}(L_1^\br)$ of the Fukaya category.
What \cite{seidel04} says is that

\begin{prop} \label{th:exact-sequence-1}
$T_{L^\br}(\taugr_L^{-1}(L_1^\br))$ is isomorphic to $L_1^\br$ in
$D^b\Fuk(M)$.
\end{prop}

On the algebraic side, this is an application of Lemma
\ref{th:recognize-twist}, where $W$ is some other exact Lagrangian
brane $L_0^\br$; we need to explain how a suitable moduli space setup
provides the necessary pair $(k,h)$. This is a variation of the
construction in \cite{seidel01}, based on pseudo-holomorphic sections
of exact symplectic Lefschetz fibrations with Lagrangian boundary
conditions. Specifically, taking $S = \mathbb{H}$ the upper half
plane, there is such a fibration $E_L \rightarrow S$, which has a
single critical point somewhere over $int(S)$. The regular fibres of
$E_L$ are isomorphic to $M$; the monodromy around the singularity is
of course $\tau_L$; and there is a canonical trivialization away from
a compact subset of $S$. Given any exact Lagrangian brane in $M$ such
as $L_1^\br$, one can define a subbundle $F_{L_1} \subset
E_L|\partial S$ which, in the canonical trivialization, satisfies
\[
 F_{L_1} \cap E_s = \begin{cases}
 \tau_L^{-1}(L_1) & s \ll 0, \\
 L_1 & s \gg 0.
 \end{cases}
\]
By considering an equation similar to \eqref{eq:generalized-floer}
for sections of $E_L$ with boundary values on $F_{L_1}$, one then
obtains a ``relative invariant'' in the style of
\cite{piunikhin-salamon-schwarz94} taking values in
$HF^0(\taugr_L^{-1}(L_1^\br),L_1^\br)$. Take $k$ to be the cocycle
representing it, which of course depends on the specific choices of
almost complex structures etc.\ used in the definition.
%
\FIGURE{fig:homotopy}{h.eps}{hb}%

For $h$ one uses a parametrized version of the same construction,
applied to a family of Lefschetz fibrations $E^t$ with boundary
conditions $F^t$, $t \in \R$. Figure \ref{fig:homotopy} shows the
asymptotic behaviour of this family: for $t \ll 0$, $(E^t,F^t)$ is
the fibre connected sum of $(E_L,F_{L_1})$ and a trivial fibration
over the three-punctured disc, with boundary conditions over the
different boundary components given by the Lagrangian submanifolds
$L$, $\tau_L^{-1}(L_1)$ and $L_1$. The invariant associated to the
degenerate limit $t = -\infty$ is therefore the composition
\begin{equation} \label{eq:t-plus}
\begin{aligned}
 CF_M^*(L^\br,\taugr_L^{-1}(L_1^\br)) & \longrightarrow CF_M^*(L^\br,L_1^\br), \\
 a & \longmapsto \mu^2_M(a,k).
\end{aligned}
\end{equation}
For $t \gg 0$, $(E^t,F^t)$ again decomposes as a connected sum. Its
first component is $E_L$ as before, but now with boundary condition
$F_L$; and the second component is trivial, with boundary conditions
given by $L$, $\tau_L^{-1}(L_1)$ and again $L$. In principle, the
associated invariant is
\begin{equation} \label{eq:t-minus}
\begin{aligned}
 CF_M^*(L^\br,\taugr_L^{-1}(L_1^\br)) \iso
 CF_M^*(\taugr_L(L^\br),L_1^\br) &
 \longrightarrow CF_M^*(L^\br,L_1^\br), \\
 a & \longmapsto \mu^2_M(l,a),
\end{aligned}
\end{equation}
where $l \in CF^0(L^\br,\taugr_L(L^\br)) \iso
CF^0(\taugr^{-1}(L^\br),L^\br)$ is obtained from $(E_L,F_L)$.
However, since $\taugr_L(\Lgr) = \Lgr[1-n]$ with $n>1$, we have
$HF^0(L^\br,\taugr_L(L^\br)) = HF^{1-n}(L^\br,L^\br) \iso
H^{1-n}(S^n;\C) = 0$, and by suitably choosing the Floer data, one
can achieve that $CF^0(L^\br,\taugr_L(L^\br))$ vanishes too. Hence
$l$ and the expression \eqref{eq:t-minus} are both zero, and from
this one deduces that the parametrized moduli space of sections
associated to the deformation from $t = -\infty$ to $t = +\infty$
yields a chain homotopy $h$ between \eqref{eq:t-plus} and zero, as
desired (this vanishing argument, which relies on the grading, is
different from the one in \cite[Proposition 2.13]{seidel01}; we use
it because it extends easily to the projective and relative cases).

Since all Lagrangian submanifolds involved are exact, there are
real-valued action functionals for all the Floer cochain groups
involved, so that in the relevant complex \eqref{eq:total-cone} one
can introduce an $\R$-filtration by the values of the action. Its
acyclicity is then proved by a spectral sequence argument. Briefly,
one replaces the $\R$-filtration by a $\Z$-subfiltration, which
divides the action values into intervals of some suitably chosen
small size $\epsilon>0$. After passing to the associated graded space
of the filtration, one finds that the Floer differentials $\mu^1$ and
the terms $\mu^2(h(a_2),a_1)$, $\mu^3(k,a_2,a_1)$ disappear, while
the remaining terms $\mu^2(a_2,a_1)$ and $\mu^2(k,b)$ make the
complex exact, hence acyclic. For this to work, some care needs to be
observed when choosing the map $\tau_L$ and the cochain level
representatives for $k$ and $h$, see \cite[Chapter 3]{seidel01} for
details.

\subsection{}
We now turn to the consequences of Proposition
\ref{th:exact-sequence-1}, combining arguments from \cite{seidel04}
with the notion of negative graded symplectic automorphisms
introduced in Section \ref{sec:negativity}.

\begin{lemma} \label{th:negative-exact}
Suppose that $m_X = 0$, so $M$ is the affine part of a projective
Calabi-Yau $X$. Let $L_1,\dots,L_m \subset M$ be Lagrangian spheres,
and suppose that
\[
\taugr_{L_1}\dots \taugr_{L_m} \in \Autgr(M)
\]
is isotopic, within that group, to a graded symplectic automorphism
$\phigr$ whose extension to $X$ is negative (see Remark
\ref{th:extend-grading}). Then $L_1^\br,\dots,L_m^\br$ are
split-generators for $D^\pi\Fuk(M)$.
\end{lemma}

\proof Let $L^\br$ be any exact Lagrangian brane. From Lemma
\ref{th:maslov-growth} we know that if $d$ is sufficiently large, a
small exact perturbation $L'$ of $L$ will cause the Maslov index of
any intersection point $x \in L' \cap \phi^d(L)$ to be strictly
negative. By definition of the Floer cochain space
\eqref{eq:basic-floer-complex}, it follows that
\[
Hom_{H^0\Fuk(M)}(L^\br,\phigr^d(L^\br)) =
HF^0_M(L^\br,\phigr^d(L^\br)) = 0.
\]
On the other hand, $\phigr^d(L^\br)$ is isomorphic in $D^b\Fuk(M)$ to
$(T_{L_1^\br} \dots T_{L_m^\br})^d(L^\br)$. Lemma
\ref{th:twist-generators} says that $L^\br$ lies in the subcategory
of $D^\pi\Fuk(M)$ split-generated by the $L_k^\br$; by construction
of that category, this implies that these objects are
split-generators. \qed

\begin{remark} \label{th:codimension-n}
In Lemma \ref{th:negative-exact}, one can replace the negativity of
$\phigr$ by the following slightly weaker assumption. There is a
closed subset $\Sigma \subset X$ which is a finite union of real
submanifolds of dimension $<n$, which admits arbitrarily small open
neighbourhoods $W$ satisfying $\phi(W) = W$, and such that $\phigr$
is negative on $X \setminus W$. The reason is that by a generic
perturbation, one can make $\Lambda$ disjoint from $\Sigma$. Then it
is disjoint from a sufficiently small $W$, and the same will hold for
all iterates $\phi^d(\Lambda)$, so that the breakdown of negativity
on $\Sigma$ can be disregarded.
\end{remark}

\subsection{}
To adapt Proposition \ref{th:exact-sequence-1} to the case of a
projective Calabi-Yau surface $X$, with $L$ a Lagrangian sphere and
$L_1^\rbr$ a rational brane, two points have to be taken into
account. Firstly, more care needs to be exercised when defining the
invariants derived from pseudo-holomorphic sections, to avoid
bubbling off of holomorphic discs. This is not really new, since we
have dealt with the same problem in the construction of $\Fuk(X)$.
Note that up to shifts, $L$ gives rise to a unique object $L^\rbr$ of
the Fukaya category, by Lemma \ref{th:spheres}. For
$\taugr_L^{-1}(L_1^\rbr)$, one needs to choose the regular almost
complex structure which is the $\tau_L$-pullback of that associated
to the $L_1$. The second modification concerns the filtration
argument. Our ground ring is now $\Lambda_\Q$, and each
pseudo-holomorphic section contributes $\pm q^e$ as in \eqref{eq:qe}.
In the $\R$-filtration of the Floer cochain groups, one now uses
powers of $q$ as a replacement for values of the action functional.
Since only a finite number of rational branes are involved ($L^\rbr$,
$L_0^\rbr$, $L_1^\rbr$), one can actually work over a subfield
$\Lambda_{(1/d)\Z} \subset \Lambda_\Q$ containing only roots of $q$
of some fixed order $d$. An essentially equivalent and more familiar
formulation is to say that all action functionals together have only
a discrete set of periods. This allows one to carry over the previous
argument, with its use of a $\Z$-subfiltration: it suffices to make
sure that $\epsilon \ll 1/d$ (actually, this is the only point in the
entire paper that would have to be modified substantially if one
wished to include non-rational branes in $\Fuk(X)$). The outcome is

\begin{prop} \label{th:exact-sequence-2}
$T_{L^\br}(\taugr_L^{-1}(L_1^\br))$ is isomorphic to $L_1^\br$ in
$D^b\Fuk(X)$. \qed
\end{prop}

There is also a version for the relative Fukaya category, where $L$
is again a Lagrangian sphere in $M$ and $L_1^\mbr$ an object of that
category. The definition of the relevant version of $(k,h)$ is as in
the projective case, except sections $u$ now contribute with integer
powers of $q$ according to their intersection with $X_\infty$. As for
the acyclicity of \eqref{eq:total-cone}, one filters it by powers of
$q$, and then taking the associated graded space brings one back to
the affine situation, where we already know that acyclicity holds.

\begin{prop} \label{th:exact-equence-3}
$T_{L^\mbr}(\taugr_L^{-1}(L_1^\mbr))$ is isomorphic to $L_1^\mbr$ in
$D^b\Fuk(X,X_\infty)$. \qed
\end{prop}


The argument from Lemma \ref{th:negative-exact} carries over to the
projective and relative cases without any modifications, and so does
Remark \ref{th:codimension-n}. By combining this with Proposition
\ref{th:negativity}, one arrives at the following conclusion:

\begin{cor} \label{th:generates-everything}
Let $X$ be a projective K{\"a}hler threefold which is Fano, with $o_X
= \K_X^{-1}$, equipped with sections $\sigma_{X,0}$,
$\sigma_{X,\infty}$ which generate an almost Lefschetz pencil of
Calabi-Yau surfaces $\{X_z\}$. Assume that $X_0 \cap X_\infty$ has no
rational components. Pick some base point $z_* \in \C \setminus
Critv(\pi_M)$, base path $c_*$, and a distinguished basis of
vanishing cycles $V_{c_1},\dots,V_{c_r} \subset M_{z_*}$. Then the
$V_{c_i}$, equipped with some gradings, are split-generators for all
versions of the split-closed derived Fukaya category:
$D^\pi\Fuk(M_{z_*})$, $D^\pi\Fuk(X_{z_*})$ and $D^\pi\Fuk(X_{z_*},
X_{z_*,\infty})$. \qed
\end{cor}

\subsection{\label{subsec:induction}}
Let $M = X \setminus X_\infty$ be an affine Calabi-Yau of dimension
$n > 1$, equipped with a quasi-Lefschetz pencil $\{X_z\}$ and the
associated holomorphic function $\pi_M: M \rightarrow \C$. Choose a
base point $z_*$ and base path $c_*$. Let $c_1,\dots,c_r$ be a basis
of vanishing paths, and $V_{c_1},\dots,V_{c_r} \subset M_{z_*}$ the
associated vanishing cycles. Make them into objects $V_{c_i}^\br$ of
$\Fuk(M_{z_*})$; any choice of grading is allowed, and the $Spin$
structure is unique except for $n = 2$, where one needs to take the
{\em nontrivial} one. Let
\begin{equation} \label{eq:directed}
\F^\rightarrow = \Fuk^\rightarrow(V_{c_1}^\br,\dots,V_{c_r}^\br)
\end{equation}
be the directed $A_\infty$-subcategory of $\Fuk(M_{z_*})$ consisting
of these objects. We associate to each embedded vanishing path $c$,
subject to the conditions $im(c) \cap im(c_*) = \{z_*\}$ and $\Rgeq
c'(0) \neq \Rgeq c_*'(0)$, an object $D_c$ of $D^b(\F^\rightarrow)$,
unique up to isomorphism and shifts, according to the following
rules:
\begin{itemize}
\item
to $c_i$ belongs the object $D_{c_i} = V_{c_i}^\br$ of
$\F^\rightarrow$ itself;
\item
an isotopy of $c$ within the class of such paths does not affect
$D_c$;
\item if we have vanishing paths $c,c',c''$ as in Figure
\ref{fig:hurwitz}, then $D_{c'} \iso T_{D_c}(D_{c''})$ and $D_{c''}
\iso T^\vee_{D_c}(D_{c'})$.
\end{itemize}
From this and the elements of the theory of mutations
\cite{rudakov90}, one sees that all the $D_c$ are exceptional
objects, which means that $Hom^*(D_c,D_c)$ is one-dimensional in
degree zero, and zero in all other degrees.

Next, we associate objects $S_d$ of $D^b(\Fuk^\rightarrow)$ to
matching paths which are disjoint from $c_*$. For this, take paths
$c',c'',d$ as in Figure \ref{fig:triangle}. Suppose that the
vanishing cycles associated to $c',c''$ are isotopic, so that $d$ is
a matching paths. The objects associated to $c',c''$ will satisfy
\begin{equation} \label{eq:matching-objects}
 Hom^k_{D^b(\Fuk^\rightarrow)}(D_{c'},D_{c''}) =
 \begin{cases} \C & k = l,l+n-1, \\
 0 & \text{otherwise} \end{cases}
\end{equation}
for some $l \in \Z$. After a shift, we may assume that $l = 0$, and
then the rule is:
\begin{itemize}
\item
$S_d$ is the mapping cone $\{D_{c'} \rightarrow D_{c''}\}$ over some
nontrivial morphism.
\end{itemize}
An easy computation shows that $Hom^*(S_d,S_d)$ is one-dimensional in
degrees $0$ and $n$, and zero otherwise. This resemblance to the
cohomology of an $n$-sphere is not accidental:
%
\FIGURE{fig:hurwitz}{hurwitz.eps}{ht}
\FIGURE{fig:triangle}{triangle.eps}{ht} %

\begin{theorem} \label{th:induction}
Let $d_1,\dots,d_m$ be matching paths which are disjoint from our
base path $c_*$. Assign objects $S_{d_1},\dots,S_{d_m}$ in
$D^b(\Fuk^\rightarrow)$ as explained above. Then the full
$A_\infty$-subcategory of $Tw\Fuk^\rightarrow$ consisting of these
objects is quasi-isomorphic to the full subcategory of the Fukay
category of the total space, $\Fuk(M)$, whose objects are the
matching cycles $\Sigma_{d_1},\dots,\Sigma_{d_m}$, with some choice
of gradings. \qed
\end{theorem}

This way of stating the result leaves several things implicit. Since
one can get any vanishing path by starting with $c_1,\dots,c_r$ and
applying the moves shown in Figure \ref{fig:triangle}, our rules
determine the objects $D_c$. However, it is not a priori clear that
the assignment $c \mapsto D_c$ is unambiguously defined, since one
can get the same path by different sequences of moves. Similarly,
\eqref{eq:matching-objects} and the well-definedness of $S_d$ are not
obvious. The fact that all of this works is a nontrivial part of the
statement. For proofs see \cite{seidel04}.

\subsection{\label{subsec:combinatorial}}
The usefulness of Theorem \ref{th:induction} is particularly obvious
for $n = 2$, when the vanishing cycles are curves on a surface, since
then $\Fuk^\rightarrow$ can be computed in a purely combinatorial
way, up to quasi-isomorphism (versions of this fact have been
discovered independently in various contexts, see e.g.\
\cite{desilva98,polishchuk-zaslow98,chekanov99}). We will now outline
how to do this computation, referring to the papers just quoted and
to \cite{seidel04} for further details and justification.

{\em Exactness:} For simplicity, we use a compact piece $N \subset
M_{z_*}$ which is a surface with boundary, capturing all of the
topology of $M_{z_*}$ (the inclusion is a homotopy equivalence). Draw
simple closed curves $\nu_k$ on $N$ which are isotopic to $V_{c_k}$
and in general position (any two intersect transversally, and there
are no triple intersections). One should then verify that these
curves are exact Lagrangian submanifolds for some choice of one-form
$\theta_N$ with $d\theta_N > 0$. For this, consider the decomposition
of $N$ given by the graph $\Gamma = \bigcup_i \nu_i$. Suppose that
one can associate to each component $Z \subset N \setminus \Gamma$
which is disjoint from $\partial N$ a positive weight $w(Z) > 0$,
with the following property: if $\sum_k m_k \bar{Z}_k$, $m_k \in \Z$,
is a two-chain whose boundary is a linear combination of the $\nu_k$,
then
\begin{equation} \label{eq:zero-energy}
\sum_k m_k w(Z_k) = 0.
\end{equation}
One needs to check this only for a finite number of chains, which
form a basis for the homology relations between the $\nu_k$. If this
works out, one can find a symplectic form $\o_N$ with $\int_Z \o_N =
w(Z)$, and that can be written as $\o_N = d\theta_N$ with
$\int_{\nu_k} \theta_N = 0$ for all $k$.

\begin{remark} \label{th:painting-by-numbers}
This ``painting by numbers'' step is rather tedious, so we will
mention a way of reducing the amount of work. Relations in homology
between the vanishing cycles correspond to elements of $H_2(M)$
(because $H_2(M_{z_*}) = 0$, and $M$ is obtained topologically by
attaching two-cells to the vanishing cycles). It is sufficient to
check \eqref{eq:zero-energy} for relations corresponding to elements
in $im(\pi_2(M) \rightarrow H_2(M))$. Equivalently, these are
relations obtained from maps $\Sigma \rightarrow M_{z_*}$ where
$\Sigma$ is a genus zero surface with boundary, each boundary
component getting mapped to some $\nu_i$. This weaker condition does
not guarantee that the vanishing cycles can be made exact, but it
allows one to define the directed Fukaya category, and moving the
$\nu_i$ within that class does not affect the quasi-isomorphism type
of the category. Hence, after a (possibly non-Hamiltonian) isotopy,
one sees that the directed Fukaya category is the same as for the
exact choice of vanishing cycles. The appearance of $\pi_2(M)$ should
remind the reader of the assumption $\o|\pi_2(M) = 0$ used frequently
in symplectic topology, and indeed it addresses the same issue
(getting a bound on the energy of pseudo-holomorphic polygons).
\end{remark}

{\em Gradings:} Recall that the holomorphic one-form $\eta_{M_{z_*}}$
defines a phase function $\alpha_{M_{z_*}}$. One can associate to
this an unoriented foliation of $M_{z_*}$, consisting of those
tangent lines $\Lambda$ for which $\alpha_{M_{z_*}}(\Lambda) = 1$. We
draw some unoriented foliation $\Fol_N$ on $N$ which is homotopic to
the one determined by $\alpha_{M_{z_*}}$. To equip the $\nu_k$ with
gradings, one has to choose a point $b_k \in \nu_k$ which should not
be an intersection point with any other $\nu_l$, and a homotopy
between $\Fol_N$ and $T\nu_k$ at that point (within the $\RP{1}$ of
lines in $TN_{b_k}$). This determines the Maslov indices $I(x)$ of
all intersection points $x \in \nu_i \cap \nu_j$, $i < j$, and also
orientations of the $\nu_i$. Note that $(-1)^{I(x)+1}$ is the local
intersection number $(\nu_i \cdot \nu_j)_x$.

{\em $Spin$ structures:} For every $x$ as before with $(\nu_i \cdot
\nu_j)_x = +1$, we choose a nonzero tangent vector $\xi_x$ to $\nu_j$
at $x$. Of course, one can take these to be the vectors pointing
along the orientation, but that is not strictly speaking necessary.
Next, recall that we have to equip the $\nu_i$ with nontrivial $Spin$
structures, or what is the same, with nontrivial double covers.
Having already chosen points $b_k$, the simplest way to do this is to
take the double cover which is trivial on $\nu_k \setminus b_k$, and
whose two sheets get exchanged when passing over that point.

{\em Counting polygons:} We now introduce a directed
$A_\infty$-category $\Cat^\rightarrow$ with objects
$\nu_1,\dots,\nu_r$. Because of directedness, it is sufficient to
define the $hom_\B(\nu_i,\nu_j)$ for $i<j$, and the composition maps
between those spaces. Set as usual
\[
 hom_{\Cat^\rightarrow}^k(\nu_i,\nu_j) = \bigoplus_{I(x) = k} \C_x,
\]
but where now $\C_x$ is identified with $\C$ in a fixed way. Given
$i_0 < \dots < i_d$ in $\{1,\dots,r\}$ and intersection points $x_0
\in \nu_{i_0} \cap \nu_{i_d}$, $x_1 \in \nu_{i_0} \cap \nu_{i_1}$,
\dots, $x_d \in \nu_{i_{d-1}} \cap \nu_{i_d}$, one considers oriented
immersions $u: \Delta \rightarrow N$, where $\Delta$ is some model
$(d+1)$-gon in the plane with numbered sides and corners. Under $u$,
the sides should go to the $\nu_{i_k}$, and the corners to the $x_k$;
moreover, the immersion should be a local diffeomorphism up to the
corners, so that the image has locally an angle $<\pi$. One considers
two such $u$ to be equivalent if they differ by a diffeomorphism of
$\Delta$. Each equivalence class contributes to
$\mu^k_{\Cat^\rightarrow}$ by
\[
 \pm 1: \C_{x_d} \otimes \dots \otimes \C_{x_1} \rightarrow \C_{x_0},
\]
and the sign is the product of the following factors: (i) for each $k
\geq 1$ such that $(\nu_{i_{k-1}} \cdot \nu_{i_k})_{x_k} = 1$, we get
a $-1$ iff the tangent vector $\xi_{x_k}$ points away from
$u(\Delta)$. (ii) We get another $-1$ if $(\nu_{i_0} \cap
\nu_{i_d})_{x_0} = 1$ and the tangent vector $\xi_{x_0}$ points
towards $u(\Delta)$. (iii) Every time that $u|\partial\Delta$ passes
over a point $b_k$, this adds a further $-1$.

\begin{lemma}
$\Cat^{\rightarrow}$ is quasi-isomorphic to the directed subcategory
$\Fuk^{\rightarrow}$ from \eqref{eq:directed}, at least up to a shift
in the gradings of the $V_{c_i}^\br$. \qed
\end{lemma}

Starting from this, Theorem \ref{th:induction} allows one to
determine the full $A_\infty$-subcategory of $\Fuk(M)$ consisting of
any finite number of matching cycles, and in particular the Floer
cohomology of any two such cycles. Note that matching cycles are
Lagrangian two-spheres in a four-manifold, so their Floer cohomology
is not amenable to computation in any direct way.

\subsection{\label{subsec:addition}}
We will need a minor addition to the previously explained
computational method, which takes into account the presence of a
homomorphism $\rho: \pi_1(M) \rightarrow \Gamma$ into a finitely
generated abelian group $\Gamma$. Restriction yields a homomorphism
from the fundamental group of the fibre $M_{z_*}$, and all vanishing
cycles $V_{c_i}$ bound balls in $M$, hence the composition
$\pi_1(V_{c_i}) \rightarrow \pi_1(M_{z_*}) \rightarrow \Gamma$ is
zero. As discussed in Section \ref{subsec:coverings}, after choosing
lifts $V_{c_i,\Gamma}$ to the associated $\Gamma$-cover
$M_{z_*,\Gamma}$, one has a $\Gamma^*$-action on \eqref{eq:directed}.
We can use the same rules as before to associate to each vanishing
path $c$ an object $D_c$ of the equivariant derived category
$D^b_{\Gamma^*}(\A)$, see Remark \ref{th:twist-is-equivariant}. For
$c',c'',d$ are as in Figure \ref{fig:triangle}, the objects $D_{c'}$,
the objects $D_{c''}$ will now satisfy
\[
 Hom^k_{D^b(\A)}(D_{c'},D_{c''}) \iso \begin{cases} W & k = l,l+n-1, \\
 0 & \text{otherwise,} \end{cases}
\]
where $W$ is some one-dimensional $\Gamma^*$-representation. After
shifting $D_{c'}$ and tensoring it with $W^*$, one may assume that $l
= 0$ and $W$ is trivial. Then the cone $S_d = \{D_{c'} \rightarrow
D_{c''}\}$ over a nontrivial degree zero morphism is again an object
of the equivariant derived category. The equivariant version of
Theorem \ref{th:induction} says that the full $A_\infty$-subcategory
of $Tw_{\Gamma^*}\Fuk^\rightarrow$ with objects
$S_{d_1},\dots,S_{d_m}$ is $\Gamma^*$-equivariantly quasi-isomorphic
to the full subcategory of $\Fuk(M)$ with objects
$\Sigma_{d_1},\dots,\Sigma_{d_m}$, where we assume that gradings and
lifts to $M_\Gamma$ have been chosen in some appropriate way for
these matching cycles. For the proof, which is not particularly
difficult, there are two essentially equivalent viewpoints: one
either goes through the relevant material in \cite{seidel04}, making
sure that everything can be made compatible with the
$\Gamma^*$-actions; or one works directly with the covering
$M_\Gamma$ and its holomorphic function $M_\Gamma \rightarrow M
\rightarrow \C$.

\section{The algebras $Q_4$ and $Q_{64}$\label{sec:4-64}}

Using split-generators, the categories which occur in homological
mirror symmetry can be described in terms of $A_\infty$-structures on
finite-dimensional algebras. This section introduces the algebra
relevant for our computation, as well as a simpler version which
serves as a stepping-stone. The main point is to use the general
theory from Section \ref{sec:deformations} to partially classify
$A_\infty$-structures on those algebras. We end by applying the
results to coherent sheaves on the mirror of the quartic surface:
this finishes work on the algebro-geometric side of mirror symmetry,
and it also helps to clarify the amount of things still to be proved
on the symplectic side.

\subsection{\label{subsec:q-preliminary}}
We begin by recalling a general ``doubling'' construction. Let
$C^\rightarrow$ be a directed ($\C$-linear graded) category, with
objects $(X_1,\dots,X_m)$. The {\em trivial extension category} of
degree $d > 0$ is a category $C$ with the same objects, and it
satisfies:
\begin{itemize}
\item
The directed subcategory of $C$ is the given $C^\rightarrow$.
\item
$C$ is a Frobenius category of degree $d$, in the sense that there
are linear maps $\int_{X_j}: Hom_C^d(X_j,X_j) \rightarrow \C$, such
that
\begin{equation} \label{eq:int-pairing}
\leftsc a,b \rightsc = \int_{X_j} ab
\end{equation}
for $a \in Hom_C^*(X_k,X_j)$, $b \in Hom_C^*(X_j,X_k)$ is a
nondegenerate pairing. These pairings are graded symmetric, $\leftsc
a,b \rightsc = (-1)^{deg(a)deg(b)} \leftsc b,a \rightsc$.
\item
One can split $Hom_{C}^*(X_j,X_k) = Hom_{C^\rightarrow}^*(X_j,X_k)
\oplus J_{jk}$ in such a way that the product of two morphisms
belonging to the $J$ spaces is zero (in view of the previous
requirements, we have $J_{jk} = 0$ for $j<k$, $J_{jk} =
Hom_C^*(X_j,X_k)$ for $j>k$; and since it is an isotropic subspace,
$J_{jj}$ must be one-dimensional and concentrated in degree $d$).
\end{itemize}
These properties characterize $C$ up to isomorphism. Alternatively,
one can define it explicitly by
\begin{equation} \label{eq:double}
 Hom_C(X_j,X_k) = Hom_{C^\rightarrow}(X_j,X_k)
 \oplus Hom_{C^\rightarrow}(X_k,X_j)^\vee[-d]
\end{equation}
and $(a,a^\vee)(b,b^\vee) = (ab,a^\vee(b\,\cdot) +
(-1)^{deg(a)(deg(b)+deg(b^\vee))} b^\vee(\cdot\, a))$.

The example relevant to us as is follows: Let $V$ be a
four-dimensional complex vector space. Consider the (linear graded)
directed category $C^{\rightarrow}_4$ having four objects
$X_1,\dots,X_4$ and morphisms
\[
 Hom_{C^\rightarrow_4}(X_j,X_k) = \begin{cases}
 \Lambda^{k-j}V & j \leq k, \\
 0 & \text{otherwise,} \end{cases}
\]
with the obvious grading and (wedge product) composition. We define
$C_4$ to be the trivial extension category of $C_4^\rightarrow$ in
degree $d = 2$. Its total morphism algebra, linear over $R_4 \iso
\C^4$, will be denoted by $Q_4$. Once one has fixed a choice of
volume element in $\Lambda^4 V^\vee$, the duals in the general
formula \eqref{eq:double} can be removed, so that then
\begin{equation} \label{eq:c4-morphism-revised}
 Hom_{C_4}(X_j,X_k) = \begin{cases}
 \Lambda^{k-j}V & j < k, \\
 \Lambda^0V \oplus (\Lambda^4V)[2], & j = k, \\
 (\Lambda^{k-j+4}V)[2] & j > k.
 \end{cases}
\end{equation}

From a different viewpoint, take the exterior algebra $\Lambda V$,
and consider the action of the center of $SL(V)$, $\Gamma_4 = \{\One,
i\One, -\One, -i\One\}$, on it. For explicitness, pick the generator
$\gamma = i\One$. From our general discussion of semidirect products,
we know that $\Lambda V \semidirect \Gamma_4$ is an algebra over $\C
\Gamma_4 \iso R_4$, with a basis of idempotents given by $u_j =
\quarter\chi_j = \quarter(e + i^{-j} \gamma + i^{-2j} \gamma^2 +
i^{-3j} \gamma^3)$ for $j = 1,\dots,4$. Concretely,
\begin{equation} \label{eq:c4-chi}
 u_k (\Lambda V \semidirect \Gamma_4) u_j =
 \begin{cases}
 \Lambda^{k-j} V \otimes \C u_j & j<k, \\
 (\Lambda^0 V \oplus \Lambda^4 V) \otimes \C u_j & j = k, \\
 \Lambda^{k-j+4} V \otimes \C u_j & j>k. \\
 \end{cases}
\end{equation}
So $\Lambda V \semidirect \Gamma_4$ is the total morphism algebra of
a category with four objects. If we view it in this way, the directed
subcategory is isomorphic to $C_4^\rightarrow$. Moreover, after
choosing a volume element on $V$, the resulting integration maps make
$\Lambda V \semidirect \Gamma_4$ into a Frobenius category of degree
4. One can split the morphism spaces into those of the directed
subcategory and a complementary summand $J$, consisting of those
$\Lambda^jV \otimes \C u_k$ with $j+k > 4$. This satisfies the
properties stated above, so we conclude that $\Lambda V \semidirect
\Gamma_4$ is the trivial extension category of $C_4^\rightarrow$ in
degree 4. Hence there is an isomorphism $\Lambda V \semidirect
\Gamma_4 \iso Q_4$ of algebras, which is $R_4$-linear but only
$\Z/2$-graded.

Later on, we will also need some understanding of the automorphisms
of $Q_4$, as a graded algebra over $R_4$. To get the trivial thing
out of the way, the group $I = (\C^*)^4$ acts by (elementary) inner
automorphisms $\rho_I$, with $\rho_I(r_1,\dots,r_4)$ being
multiplication with $r_k/r_j$ on $u_k Q_4 u_j$. Next, the canonical
action of $G = GL(V)$ on $C_4^\rightarrow$ (leaving the objects
fixed, and being the obvious representation on $\Lambda^{k-j}V$)
extends to an action $\rho_G$ on $C_4$, determined by asking that
\begin{equation} \label{eq:det}
\int_{X_j} \rho_G(A)(a) = det(A) \int_{X_j} a.
\end{equation}
With respect to the isomorphism introduced above, this is the same as
the $G$-action on $\Lambda V \semidirect \Gamma_4$ induced by the
natural action on $V$. It is a general feature of semidirect products
that $\Gamma_4 \subset G$ acts by inner automorphisms, more precisely
\[
 \rho_G(i\One) = \rho_I(1,i,i^2,i^3).
\]
This is the only overlap between $\rho_I$ and $\rho_G$, which
together make up the whole automorphism group. To see this, note that
any automorphism of $Q_4$ determines linear automorphisms of the
spaces $u_{j+1}Q_4 u_j \iso V$ for all $j$ (in this case, the index
is cyclic, so we are thinking of $u_1 Q_4 u_4$ as well). Moreover,
the product of two elements for successive $j$ is zero iff they are
linearly dependent in $V$, and this shows that the actions for
various $j$ are by linear maps that differ only by multiplicative
constants. By composing the given automorphism with an inner one,
these constants can be made equal to 1, and then the remaining
automorphism is easily seen to lie in the image of $\rho_G$. To
summarize, we have
\[
\begin{aligned}
 & Inn(Q_4/R_4) \iso (\C^*)^4/\C^*, \\
 & Out(Q_4/R_4) \iso GL(V)/\Gamma_4, \\
 & Aut(Q_4/R_4) \iso (\C^*)^4/\C^* \times_{\Z/4} GL(V).
\end{aligned}
\]

At this point, fix a maximal torus $H \subset SL(V)$, and let $T =
H/\Gamma_4$ be its image in $PSL(V)$. Choose a splitting $\Sigma: T
\rightarrow H$ of the projection, and denote by $\rho_T$ the
$T$-action obtained by $\Sigma$ from $\rho_G$ (this should not be
confused with $\rho_I$). Let $\Gamma_{64}$ be the subgroup of those
$A \in H$ such that $A^4 = \One$, and $\Gamma_{16} =
\Gamma_{64}/\Gamma_4$ its image in $T$. We use the restriction of
$\rho_T$ to that finite group to form the semidirect product
\[
 Q_{64} \stackrel{\text{def}}{=} Q_4 \semidirect \Gamma_{16}.
\]
This is an algebra linear over $\C\Gamma_4 \otimes \C\Gamma_{16} =
\C\Gamma_{64}$, so one can also view it as a category $C_{64}$ with
64 objects. The actions $\rho_T|\Gamma_{16}$ obtained from any two
choices of $s$ differ by homomorphisms $\Gamma_{16} \rightarrow
Inn(Q_4/R_4)$, which can always be lifted to $I$. This implies that
$Q_{64}$ is independent of $\Sigma$ up to ($\C\Gamma_{64}$-linear)
isomorphism, see \eqref{eq:conjugate-action}. Moreover, by combining
our previous discussion with \eqref{eq:in-stages} one finds that
there is a $\Z/2$-graded isomorphism
\[
 Q_{64} \iso (\Lambda V \semidirect \Gamma_4) \semidirect \Gamma_{16}
 \iso \Lambda V \semidirect \Gamma_{64}.
\]

\subsection{}
We want to use the interpretation of $Q_4$ as a twisted exterior
algebra to compute its Hochschild cohomology, and that requires
paying careful attention to the different gradings. It is convenient
to temporarily introduce a bigrading indexed by $\half \Z \times
\Z/2$ on $A = \Lambda V \semidirect \Gamma_4$, with elements of
$\Lambda^k V \otimes \C \Gamma_4$ given bidegree $(k/2, [k])$. One
defines the Hochschild cohomology of such bigraded algebras by using
the $\Z/2$-grading for parities and hence the signs in the Hochschild
differential, and the $\half\Z$-grading to determine the degree of
Hochschild cochains, hence the $t$-part of $HH^{s+t}(A,A)^t$. It is
straightforward to adapt the proof of Proposition
\ref{th:twisted-hkr} to the new gradings, so we have
\begin{equation} \label{eq:fractional-hkr}
 HH^{s+t}(A,A)^t \iso
 \bigoplus_{\gamma \in \Gamma_4} \Big( S^s(V^{\gamma,\vee}) \otimes
 \Lambda^{s+2t-codim(V^\gamma)}(V^\gamma) \Big)^{\Gamma_4}.
\end{equation}
Having done that, we modify the bigrading by adding $((k-j)/2,0)$ to
the bidegree of each piece $(1 \otimes u_k)A(1 \otimes u_j)$, and
denote the outcome by $B$. If we view our algebras as categories,
this corresponds to a fractional shift or translation of the $j$-th
object downwards by $j/2$. As a consequence, restriction to the
quasi-isomorphic subcomplex of $R_4$-linear cochains shows that the
Hochschild cohomologies of $A$ and $B$ are the same. On the other
hand, a direct check using \eqref{eq:c4-chi} shows that $B$ is
nontrivial only in bidegrees $(k,[k])$ with $k \in \Z$, so that we
can consider it as an ordinary graded algebra. As such, it becomes
isomorphic to $Q_4$, see \eqref{eq:c4-morphism-revised}; which shows
that \eqref{eq:fractional-hkr} also computes the Hochschild
cohomology of that algebra. In particular, the $\gamma \neq e$
summands only contribute to $(s,t) = (0,2)$.

Let $\QQ_4$ be an $R_4$-linear and $\rho_T$-equivariant
$A_\infty$-algebra whose underlying vector space is $Q_4$, which
satisfies \eqref{eq:mu1zero}, and which is nontrivial in the sense of
not being gauge equivalent to the trivial $A_\infty$-structure. As an
aside, note that because the elementary inner automorphisms of $Q_4$
are also automorphisms of $\QQ_4$, $\rho_T$-equivariance is actually
the same as $\rho_G|H$-equivariance, hence independent of $\Sigma$.
We have no need of proving the existence of such an
$A_\infty$-algebra abstractly, since it will be constructed
geometrically later on from coherent sheaves or as a Fukaya category.
The uniqueness statement is:

\begin{lemma} \label{th:recognize-q4}
$\QQ_4$ is unique up to ($R_4$-linear, $T$-equivariant)
$A_\infty$-isomorphism.
\end{lemma}

\proof We will use the equivariant version of the deformation theory
from Section \ref{sec:deformations}, which was discussed extensively
in Section \ref{sec:group-actions} in the case of finite groups. We
need it for $T$, but that is not a problem since the $T$-action on
all relevant Hochschild cochain spaces splits them into direct
products of subspaces with given weights, which is all we needed.
Recall that the HKR isomorphism is $G$-equivariant. From this and
\eqref{eq:fractional-hkr}, one gets
\[
 \big(HH^2(Q_4,Q_4)^{2-d}\big)^T =
 \big(S^dV^\vee \otimes \Lambda^{4-d}V \big)^H =
 \begin{cases}
 \C & d = 4, \\
 0 & \text{for all other $d>2$}.
 \end{cases}
\]
More explicitly, choose an identification $V = \C^4$ making the
maximal torus $H$ standard. Denote the standard basis of $V$ by
$v_0,\dots,v_3$, and the dual basis by $y_0,\dots,y_3$. Then the only
nontrivial summand consists of multiples of the $H$-invariant
polynomial $y_0y_1y_2y_3$. Since $u_k Q_4 u_j$ is nonzero only in
degrees $\equiv k-j$ mod 2, any odd degree $R_4$-bimodule map
$
Q_4 \otimes_{R_4} \cdots \otimes_{R_4} Q_4 \longrightarrow Q_4
$
is necessarily zero. That applies to all the odd order
$\mu^d_{\QQ_4}$ and in particular to $d = 3$, which shows that
$\QQ_4$ has a well-defined order 4 deformation class \[m^4_{\QQ_4}
\in \big(HH^2(Q_4,Q_4)^{-2}\big)^T.\] This is nonzero because
otherwise, since all the higher obstruction groups vanish, $\QQ_4$
would be gauge equivalent to the trivial $A_\infty$-structure, which
is contrary to assumption. Using the equivariant version of Lemma
\ref{th:versal-1}, we find that versality applies: any other
$A_\infty$-structure with the same properties as $\QQ_4$ is gauge
equivalent to $\epsilon^*\QQ_4$ for some $\epsilon \in \C^*$, and in
particular isomorphic to $\QQ_4$. \qed

\begin{remark} \label{th:more-recognize-q4}
One can formulate the previous Lemma as a more general recognition
principle. Suppose that we have an $A_\infty$-algebra $\A$ linear
over $R_4$ and carrying a $T$-action, whose cohomology algebra is
equivariantly isomorphic to $Q_4$. Choose $\lambda$, $\pi$ and $h$ as
in Remark \ref{th:explicit-hpt} (and $R_4$-linearly), so that we can
run the explicit form of the Homological Perturbation Lemma to
produce an $A_\infty$-structure $\B$ on $H(\A)$. This will satisfy
$\mu^3_\B = 0$ for the same reasons as in the proof above. Take
classes $a_4 \in u_1 H(\A) u_4$, $a_3 \in u_4 H(\A) u_3$, $a_2 \in
u_3 H(\A) u_2$, $a_1 \in u_2 H(\A) u_1$, such that the product of any
two successive ones is zero. Plug the $a_k$ into the expression
\eqref{eq:mu4-hpt} for $\mu^4_\B$; if the outcome is nonzero, then
$\A$ is quasi-isomorphic to $\QQ_4$. To see this, note that by the
explicit formula \eqref{eq:reverse-hkr} for the HKR isomorphism, or
rather a slight variation of it for semidirect products,
$\mu^4_\B(a_4,a_3,a_2,a_1) = p(v,v,v,v) \otimes u_1$, where $p \in
S^4V^\vee$ is the polynomial corresponding to the obstruction class
$m^4_\B \in HH^2(Q_4,Q_4)^{-2}$, and $v \in V$ reflects the choice of
the $a_k$. As soon as one knows that $m^4_\B$ is nonzero, Lemma
\ref{th:recognize-q4} does the rest.
\end{remark}

\subsection{\label{subsec:u4}}
Since $\QQ_4$ is $\rho_T$-invariant, we can form the semidirect
product $A_\infty$-algebra $\QQ_{64} = \QQ_4 \semidirect
\Gamma_{16}$, whose underlying algebra we denote by $Q_{64}$.

\begin{lemma}
The truncated Hochschild cohomology of $\QQ_{64}$ satisfies
\begin{equation} \label{eq:hh12}
 HH^1(\QQ_{64},\QQ_{64})^{\leq 0} \iso \C^3, \quad
 HH^2(\QQ_{64},\QQ_{64})^{\leq 0} \iso \C^7.
\end{equation}
\end{lemma}

\proof We use the spectral sequence \eqref{eq:truncated-ss}. The same
trick of passing through an appropriate bigrading which we used to
compute the Hochschild cohomology of $Q_4$ also works for $Q_{64}$,
yielding
\begin{multline} \label{eq:hh-q64}
 HH^{s+t}(Q_{64},Q_{64})^t \iso \\ \iso
 \bigoplus_{\gamma \in \Gamma_{64}} \Big( S^s(V^{\gamma,\vee}) \otimes
 \Lambda^{s+2t-codim(V^\gamma)}(V^\gamma) \otimes
 \Lambda^{codim(V^\gamma)}(V/V^\gamma) \Big)^{\Gamma_{64}}.
\end{multline}
In particular, our spectral sequence satisfies $E_2^{s,t} = 0$ for $s
\leq 0$ or $s + 2t < 0$, which ensures its convergence. Case-by-case
study of the conjugacy classes in $\Gamma_{64}$ shows that the
summands with nontrivial $\gamma$ contribute only to $E^{s,t}_2$ with
$s+t \geq 3$. Since the differentials in the spectral sequence
preserve each summand, we can concentrate on $\gamma = e$. Here is
the relevant part of the $E_2$ term ($S^j\Lambda^k$ is shorthand for
$(S^j V^\vee \otimes \Lambda^k V )^{\Gamma_{64}}$, and the crossed
out summands vanish because the $\Gamma_4$-fixed point sets are
already trivial):
\[
\begin{array}{r|ccccccc}
 & s = 0 & s = 1 & s = 2 & s = 3 & s = 4 & s = 5 & s = 6
 \\[0.25em]
 \hline
 t = 1 &
 0
 & \xout{S^1\Lambda^3}
 & \xout{S^2\Lambda^4}
 & 0
 & 0
 & 0
 & 0
 \\[0.25em]
 t = 0 &
 0
 & S^1\Lambda^1
 & S^2\Lambda^2
 & S^3\Lambda^3
 & S^4\Lambda^4
 & 0
 & 0
 \\[0.25em]
 t = -1 &
 0
 & 0
 & \xout{S^2\Lambda^0}
 & \xout{S^3\Lambda^1}
 & \xout{S^4\Lambda^2}
 & \xout{S^5\Lambda^3}
 & \xout{S^6\Lambda^4}
 \\[0.25em]
 t = -2 &
 0
 & 0
 & 0
 & 0
 & S^4\Lambda^0
 & S^5\Lambda^1
 & S^6\Lambda^2
 \\[0.25em]
 t = -3 &
 0
 & 0
 & 0
 & 0
 & 0
 & 0
 & \xout{S^6\Lambda^0}
\end{array}
\]
The first nonzero differential is $\delta_3$, which is the Schouten
bracket with the order four deformation class. We identify $V = \C^4$
as before, so that the deformation class is $y_0 y_1 y_2 y_3$ (up to
a multiple, which we may safely ignore since it can be absorbed into
a rescaling of $V$). In total degree $s+t = 1$, we have the
$\Gamma_{64}$-invariant part of $V^\vee \otimes V$ which is spanned
by elements $y_k \otimes v_k$, and these satisfy
\begin{equation} \label{eq:d311}
 \delta_3^{1,0}(y_k \otimes v_k) = y_0 y_1 y_2 y_3
\end{equation}
for all $k$. Note that $\half \sum_k y_k \otimes v_k$ is what we
called the Euler element in Section \ref{sec:deformations}, and so
the fact that its image under $\delta_3$ is twice the deformation
class is an instance of the general property
\eqref{eq:euler-element}. For $s+t = 2$ we have firstly the
$\Gamma_{64}$-invariant part of $S^2V^\vee \otimes \Lambda^2V$,
generated by the six elements $y_j y_k \otimes v_j \wedge v_k$ which
satisfy
\[
 \delta_3^{2,0}(y_j y_k \otimes v_j \wedge v_k) =
 (y_0 y_1 y_2 y_3) y_k \otimes v_k -
 (y_0 y_1 y_2 y_3) y_j \otimes v_j;
\]
and secondly the invariant part of $S^4V^*$, whose basis consists of
the $y_k^4$ together with $y_0y_1y_2y_3$, the latter being
annihilated by being in the image of \eqref{eq:d311}. The structure
of the $E_2$ term shows that the differentials of order $d>3$ all
vanish, so our computation is complete. To summarize, explicit
generators of $E_2$ which represent the Hochschild cohomology groups
in \eqref{eq:hh12} are (with some slight redundancy): for $HH^1$,
$y_0 \otimes v_0 - y_1 \otimes v_1$ and its three cyclic permutations
(the sum of all four being zero). And for $HH^2$, $y_0y_1 \otimes v_0
\wedge v_1 + y_12y_2 \otimes v_1 \wedge v_2 + y_2y_0 \otimes v_2
\wedge v_0$ and its cyclic permutations (the sum being again zero),
together with the monomials $y_k^4$, $k = 0,\dots,3$. \qed

Consider $U_4 \in Aut(Q_4/R_4)$ of the form $U_4 = \rho_I(r)
\rho_G(A)$, where $Av_k = \lambda_kv_{k+1}$, $Av_3 = \lambda_4 v_0$
with $\lambda_0\lambda_1\lambda_2\lambda_3 = 1$. Slightly more
intrinsically, this class of automorphisms can be characterized by
the following properties:
\begin{itemize}
\item
$U_4$ acts on the second summand in $u_j Q_4 u_j \iso \C \oplus
(\Lambda^4 V)[2]$ by $-1$;
\item
Let $\xi_H: H \rightarrow H$ be the shift $diag(t_0,\dots,t_3)
\mapsto diag(t_1,t_2,t_3,t_0)$, and $\xi_T$ the induced automorphism
of $T$. Then there is a homomorphism $\iota: \Gamma_{16} \rightarrow
I$, such that
\begin{equation} \label{eq:skew-commute}
\rho_T(\xi_T(\gamma)) \circ U_4 = \rho_I(\iota(\gamma)) \circ U_4
\circ \rho_T(\gamma).
\end{equation}
$\iota$ is necessary here because the splitting $\Sigma$ breaks some
natural symmetries: essentially, it measures the difference between
$\xi_H(\Sigma(\gamma))$ and $\Sigma(\xi_T(\gamma))$.
\end{itemize}
This is the situation from \eqref{eq:inner-twist}, so one can use
$\iota$ to lift $U_4$ to an automorphism $U_{64} = U_4 \semidirect
(\xi|\Gamma_{16})$ as in \eqref{eq:twisted-u}. Suppose that $\U_{64}$
is an $A_\infty$-isomorphism from $\QQ_{64}$ to itself whose
underlying algebra automorphism is $U_{64}$, and let $\M_{64}$ be its
graph, which is an $A_\infty$-bimodule over $\QQ_{64}$. Consider a
one-parameter deformation $\QQ_{64,q}$ of $\QQ_{64}$ which is
nontrivial at first order in $q$, and has the property that $\U_{64}$
can be simultaneously deformed to an $A_\infty$-homomorphism
$\U_{64,q}$ from $\QQ_{64,q}$ to itself. The existence of these
structures is again not an issue, but uniqueness is:

\begin{lemma} \label{th:recognize-q64}
Up to the pullback action of $End(\Lambda_\N)^\times$ combined with
equivalence of one-parameter deformations, $\QQ_{64,q}$ is unique.
\end{lemma}

\proof We use coupled deformation theory for the pair
$(\QQ_{64},\M_{64})$. The same parity issues as in Lemma
\ref{th:recognize-q4} show that $\mu^1$ remains zero during the
deformation to $\QQ_{64,q}$, and that also holds for the differential
$\mu^{0,0}$ of the $A_\infty$-bimodule. Here is the relevant piece of
the long exact sequence \eqref{eq:coupled-sequence-2}, with the
information from Lemma \ref{th:graph} (or rather its analogue for
truncated Hochschild cohomology) already built in:
\begin{multline} \label{eq:dull-sequence}
 \qquad HH^1(\QQ_{64},\QQ_{64})^{\scriptscriptstyle\leq 0} \xrightarrow{1-\U_{64}^*}
 HH^1(\QQ_{64},\QQ_{64})^{\scriptscriptstyle\leq 0} \longrightarrow \\ \longrightarrow
 HH^2(\QQ_{64} \& \M_{64},\QQ_{64} \& \M_{64})^{\scriptscriptstyle\leq 0} \longrightarrow \qquad \qquad \\
 \longrightarrow HH^2(\QQ_{64},\QQ_{64})^{\leq 0} \xrightarrow{1-\U_{64}^*}
 HH^2(\QQ_{64},\QQ_{64})^{\leq 0}.
\end{multline}
At the end of the proof of the previous Lemma, we wrote down four
generators which give an $H$-equivariant isomorphism
$HH^1(\QQ_{64},\QQ_{64})^{\scriptscriptstyle\leq 0} \iso
\C^4/\C(1,1,1,1)$; since the coordinates are permuted cyclically by
$U_{64}$, the first arrow in \eqref{eq:dull-sequence} is an
isomorphism. $HH^2$ requires slightly more care, since two different
pieces of the $E_\infty$ term contribute to it. What the spectral
sequence shows is that there is a short exact sequence with an
endomorphism,
\[
\xymatrix{
 0 \ar[r]
 & {\C^4} \ar[r] \ar[d]
 & HH^2(\QQ_{64},\QQ_{64})^{\scriptscriptstyle\leq 0} \ar[r] \ar[d]^{\U_{64}^*}
 & {\C^4/\C(1,1,1,1)} \ar[r] \ar[d]
 & 0
 \\
 0 \ar[r]
 & {\C^4} \ar[r]
 & HH^2(\QQ_{64},\QQ_{64})^{\scriptscriptstyle\leq 0} \ar[r]
 & {\C^4/\C(1,1,1,1)} \ar[r]
 & 0
}
\]
where both the left and right $\downarrow$ are cyclic permutations of
the coordinates. This implies that the last arrow in
\eqref{eq:dull-sequence} has one-dimensional kernel, hence the
coupled truncated Hochschild cohomology group is one-dimen\-si\-onal.
On the other hand, since $\QQ_{64,q}$ is nontrivial at first order,
the primary deformation class of any pair $(\QQ_{64,q},\M_{64,q})$ in
that group must be nonzero; at which point Lemma \ref{th:versal-4}
takes over. \qed

It is maybe not entirely clear from the statement, but the proof of
the Lemma above shows that we were talking about uniqueness of
$\QQ_{64,q}$ with respect to a fixed choice of $U_4$, $\iota$ and
$\U_{64}$. The next results removes the apparent dependence of those
data:

\begin{lemma}
Up to isomorphism of $A_\infty$-algebras linear over $\Lambda_\N$,
combined with the action of $End(\Lambda_\N)^\times$, $\QQ_{64,q}$ is
independent of all choices made in its construction.
\end{lemma}

\proof It is convenient to take on the choices in reverse order. (1)
Concerning the $A_\infty$-bimodule structure $\M_{64}$ on the fixed
bimodule $M_{64} = Graph(U_{64})$, we know from \eqref{eq:hh-q64} and
classical homological algebra that
\[
 Ext^k(M_{64},M_{64}[1-k]) \iso HH^1(Q_{64},Q_{64})^{1-k} = 0
\]
for all $k \geq 2$, and then Lemma \ref{th:formal-m} implies that
$\M_{64}$ is unique up to isomorphism. (2) Any two choices of $\iota$
differ by a $\chi \in \Gamma_{16}^*$, and that defines an
automorphism of $\QQ_{64}$. Since all the Hochschild cohomology in
\eqref{eq:hh12} lies in the $\Gamma_{16}^*$-invariant subspace, one
can use the standard quasi-isomorphism theorem of abstract
deformation theory \cite[Theorem 2.4]{goldman-millson88} to transform
a given $\QQ_{64,q}$ into an equivalent deformation which is
$\Gamma_{16}^*$-equivariant. Having done that, it is easy to see that
if $\U_{64}$ deforms compatibly to an $\U_{64,q}$ for one choice of
$\iota$, then it does for all the others. (3) Changing $U_4$ by an
inner automorphism does not affect its graph (up to bimodule
isomorphism), so we may assume that $U_4 = \rho_G(A)$. The remaining
freedom is the choice of the $\lambda_k$, but one can relate
different choices by an automorphism of $Q_4$ which rescales the
different coordinates of $V$ in a suitable way. \qed

\subsection{}
Take the Beilinson basis of the derived category of coherent sheaves
on $P = {\mathbb P}(V)$, which consists of $F_k =
\Omega^{4-k}(4-k)[4-k]$ for $k = 1,\dots,4$. The $\Omega^k$ are
naturally $PGL(V)$-equivariant sheaves, and if we give $\O(-1)$ the
$GL(V)$-action obtained by embedding it into $\O \otimes V$, the
$\Omega^{4-k}(4-k) = \Omega^{4-k} \otimes \O(-1)^{\otimes k-4}$
become $GL(V)$-equivariant. A straightforward computation, which is
the basis of the argument in \cite{beilinson78}, shows that

\begin{lemma} \label{th:beilinson}
The linear graded category with objects $F_1,\dots,F_4$, and where
the morphisms are homomorphisms of all degrees in the derived
category, is $GL(V)$-equiva\-ri\-antly isomorphic to
$C_4^\rightarrow$. \qed
\end{lemma}

Let $\iota_0: Y_0 \hookrightarrow P$ be the unique $H$-invariant
quartic hypersurface, in appropriate coordinates $Y_0 =
\{y_0y_1y_2y_3 = 0\}$. Write $E_{0,k} = \iota_0^*F_k$. Repeating an
argument from \cite{seidel-thomas99}, we observe that
\begin{align*}
 Hom^*_{Y_0}(E_{0,j},E_{0,k})
 & = Hom^*_P(F_j,(\iota_0)_*\iota_0^*F_k)
 = Hom^*_P(F_j,\O_{Y_0} \stackrel{{\bf L}}{\otimes} F_k) \\
 & = Hom^*_P(F_j,\{\K_P \otimes F_k \rightarrow \O_P \otimes F_k\})
\end{align*}
sits in a long exact sequence which one can write, using Serre
duality on $P$, as
\[
 \cdots \rightarrow Hom^d_P(F_j,F_k) \stackrel{\iota_0^*}{\longrightarrow}
 Hom^d_{Y_0}(E_{0,j},E_{0,k}) \stackrel{(\iota_0^*)^\vee}\longrightarrow
 Hom^{2-d}_P(F_k,F_j)^\vee \rightarrow \cdots
\]
Since the $F_k$ form an exceptional collection, we have unique
splittings
\begin{equation} \label{eq:doubling}
 Hom^*_{Y_0}(E_{0,j},E_{0,k}) = Hom^*_P(F_j,F_k) \oplus Hom^*_P(F_k,F_j)^\vee[-2].
\end{equation}
$Y_0$ has trivial dualizing sheaf, so Serre duality makes
$D^bCoh(Y_0)$ into a Frobenius category of degree $d = 2$. If we
denote by $J_{jk}$ the second summand in \eqref{eq:doubling}, then
the composition of two morphisms in $J$ spaces is zero for dimension
reasons (these morphisms are $Ext^2$'s of locally free sheaves on
$Y_0$). We have now checked off the properties of a trivial extension
category. Moreover, the induced $GL(V)$-action on
$Hom^*_{Y_0}(E_{0,j},E_{0,k})$ satisfies \eqref{eq:det}, so

\begin{lemma}
The linear graded category with objects $E_{0,k}$, and where the
morphisms are maps of arbitrary degree in $D^bCoh(Y_0)$, is
$GL(V)$-equiva\-riantly isomorphic to $C_4$. \qed
\end{lemma}

Denote by $S_4$ the total morphism algebra of that subcategory, so
$S_4 \iso Q_4$. This is the cohomology algebra of a dg algebra
$\SS_4$, canonical up to quasi-isomorphism, which can be defined as
total dg algebra of the same objects in the {\v C}ech dg category
$\cech{\SS}(Y_0)$. The next step comes from
\cite{douglas-govindarajan-jayaraman-tomasiello01}:

\begin{lemma} \label{th:massey}
$\SS_4$ is not formal, meaning not quasi-isomorphic to its
cohomology.
\end{lemma}

\proof The skyscraper sheaf at any point $x = \C v \in P$ admits a
locally free Koszul resolution
\[
 \big\{\Omega^3(3) \longrightarrow \Omega^2(2)
 \longrightarrow \Omega^1(1) \longrightarrow \O\big\} \iso \O_x
\]
where each differential corresponds to $v \in V$ under the
isomorphism from Lemma \ref{th:beilinson}. After restricting, it
follows that there is a spectral sequence converging to
$Hom^*_{Y_0}(\Omega^3(3)[3], L\iota_0^*\O_x)$ whose $E_1$ term is
\[
\begin{array}{r|ccccccc}
 & s = 0 & s = 1 & s = 2 & s = 3 & s = 4
 \\[0.25em]
 \hline
 t = 2 & \C & 0 & 0 & 0 & 0 \\
 t = 1 & 0  & 0 & 0 & 0 & 0 \\
 t = 0 & \C & V & \Lambda^2V & \Lambda^3V & 0
\end{array}
\]
The differential $\delta_1: E_1^{s,t} \rightarrow E_1^{s+1,t}$ is
wedge product with $v$ in the bottom row, which leaves just two
nonzero terms $E_2^{0,2} \iso \C$, $E_2^{3,0} \iso \C$. If $x \notin
Y_0$ then $L\iota_0^*\O_x = 0$, so these two terms must kill each
other through the only remaining differential $\delta_3^{0,2}$. On
the other hand, if $x \in Y_0$ then $L\iota_0^*\O_x = \O_x \oplus
\O_x[1]$ has nonzero sections, so the spectral sequence degenerates
at $E_2$.

The connection with the question we are interested in comes about as
follows. If $\SS_4$ was formal, the triangulated subcategory of
$D^bCoh(Y_0)$ generated by the $E_{0,k}$ could be modelled by twisted
complexes defined on the category $C_4$ with its trivial
$A_\infty$-structure. We know that $L\iota_0^*\O_x$ is a twisted
complex of the form $(E_{0,1} \oplus E_{0,2} \oplus E_{0,3} \oplus
E_{0,4},\delta)$, and the behaviour of first differential in the
spectral sequence means that $\delta$ must be (up to irrelevant
scalar multiples)
\[
\delta = \begin{pmatrix}
 0 & & & \\
 v & 0 & & \\
 \ast & v & 0 & \\
 \ast & \ast & v & 0
\end{pmatrix},
\]
But for degree reasons the terms $\ast$ are necessarily equal to
zero. Having identified the object, we can check that in $D^b(C_4)$
the analogue of the spectral sequence above degenerates at $E_2$ for
any nonzero $v$, which is a contradiction. \qed

By choosing the {\v C}ech covering to be $H$-invariant, one can
arrange that $\SS_4$ has a natural $H$-action. Take the same
splitting $\Sigma: T \rightarrow H$ as before, and consider the
induced $T$-action. The $T$-equivariant version of the Homological
Perturbation Theorem (which applies because the action on
$hom_{\SS_4}$ is well-behaved), together with Lemma
\ref{th:recognize-q4}, show that $\SS_4$ is
$A_\infty$-quasi-isomorphic to $\QQ_4$, and moreover that
quasi-isomorphism is $R_4$-linear and $T$-equivariant. As a
consequence, the semidirect product $\SS_{64} = \SS_4 \semidirect
\Gamma_{16}$ defined using the restriction of the $T$-action to
$\Gamma_{16}$ is quasi-isomorphic to $\QQ_{64}$. On the geometric
side, consider each $E_{0,k}$ as a $\Gamma_{16}$-equivariant sheaf in
all 16 possible ways (mutually twisted by characters). This gives a
total of 64 objects in the derived category
$D^bCoh_{\Gamma_{16}}(Y_0)$, and Remark \ref{th:equivariant-sheaves}
tells us that $\SS_{64}$ is the dg algebra underlying their total
morphism algebra. The flat formal deformation $Y_q = \{p_q(v) = 0\}$
of $Y_0$ given by \eqref{eq:fermat} still admits a
$\Lambda_\N$-linear $\Gamma_{16}$-action. Since the $E_{0,k}$ are
restricted from projective space, they have obvious equivariant
extensions $E_{q,k}$ to locally free coherent sheaves over $Y_q$. By
the discussion in Section \ref{subsec:formal-schemes}, or rather its
equivariant analogue, this gives rise to a $\Gamma_{16}$-equivariant
one-parameter deformation $\SS_{64,q}$ of $\SS_{64}$, which satisfies
\begin{equation} \label{eq:stable-h}
 H^*(\SS_{64,q}) \iso H^*(\SS_{64}) \otimes_\C \Lambda_\N.
\end{equation}
One can check \eqref{eq:stable-h} either by repeating the duality
argument above for $Y_q$, or more simply by observing that the
morphisms between any two $E_{0,k}$ in $D^bCoh(Y_0)$ are concentrated
in a single degree, which precludes the possibility of any
deformation of the differential affecting the cohomology.

\begin{lemma} \label{th:64-nontrivial}
${\SS}_{64,q}$ is nontrivial at first order in $q$.
\end{lemma}

\proof This is quite similar to the previous Lemma. Let $x_q$ be a
$\Lambda_\N$-point of $P \times_\C \Lambda_{\N}$, which is now given
by a family $v_q$ of nonzero vectors in $V$ varying with the formal
parameter $q$. We have the same spectral sequence as before for the
derived restriction of its structure sheaf $\O_{x_q}$ to $\iota_q:
Y_q \hookrightarrow P \times_\C \Lambda_\N$, and therefore
$Hom^*_{Y_q}(\Omega^3(3)[3],L\iota_q^*\O_{x_q})$ is the cohomology of
\[
 \Lambda_\N \xrightarrow{q \mapsto p_q(v_q)} \Lambda_\N.
\]
$L\iota_q^*\O_{x_q}$ is not an equivariant sheaf, but one can remedy
this by summing over the $\Gamma_{16}$-orbit of $x_q$. By taking
$x_q$ in such a way that $p_q(v_q)$ has precisely order $O(q)$, one
obtains examples of twisted complexes over $\SS_{64,q}$ (considered
here as an $A_\infty$-category with 64 objects) such that the map
\eqref{eq:q2} is not an isomorphism. Now apply Lemma
\ref{th:deformation-generators} to $Tw\,\SS_{64,q}$. \qed

The automorphism $A \in GL(V)$, $A v_k = v_{k+1}$, $A v_3 = Av_0$
preserves $Y_0$ and the objects $E_{0,k}$, hence defines an
automorphism of $\SS_4$, unique up to elementary inner automorphisms.
Since $det(A) = -1$, this automorphism acts by $-1$ on the space of
sections of the canonical sheaf of $Y_0$, hence by Serre duality on
$Hom^2(E_{0,k},E_{0,k})$ for any $k$. Moreover, it skew-commutes with
the $T$-action on $Y_0$ in the obvious sense, so it can be identified
on the level of cohomology with one of the automorphisms $U_4$ of
$Q_4$ introduced before. Because of the twisted equivariance, one can
define an induced automorphism of $\SS_{64}$, and since $A$ survives
the deformation to $Y_q$, this can be deformed to an automorphism of
$\SS_{64,q}$. Together with the previous Lemma, this allows us to
apply Lemma \ref{th:recognize-q64} which shows that $\SS_{64,q}$ is
quasi-isomorphic to $\QQ_{64,q}$, at least in the sense in which the
latter object is unique, which means up to reparametrization by $\psi
\in End(\Lambda_\N)^\times$.

Let $Y_q^* = Y_q \times_{\Lambda_\N} \Lambda_\Q$ be the general fibre
of our formal scheme, and $E_{q,k}^*$ the restrictions of the
$E_{q,k}$ to it. Let $Z_q^*$ be the minimal resolution of the
quotient $Y_q^*/\Gamma_{16}$. We need two more basic facts:

\begin{lemma}
The 64 equivariant versions of the $E_{q,k}^*$ are split-generators
for $D^bCoh_{\Gamma_{16}}(Y_q^*)$.
\end{lemma}

\proof Beilinson's resolution of the diagonal \cite{beilinson78}
shows that the $F_{q,k}^*$ are generators for the derived category of
$P \times_\C \Lambda_\Q$. His argument carries over to the
equivariant case, which means that the 64 objects obtained by making
the $F_{q,k}^*$ equivariant in all possible ways generate the derived
category of $\Gamma_{16}$-equivariant sheaves. The equivariant
analogue of Lemma \ref{th:kontsevich} shows that the restrictions of
these objects to $Y_q^*$ split-generate its derived category of
equivariant sheaves. \qed

\begin{lemma} \label{th:kapranov-vasserot00}
$D^bCoh(Z_q^*) \iso D^bCoh_{\Gamma_{16}}(Y_q^*)$, as triangulated
categories linear over $\Lambda_\Q$.
\end{lemma}

This is a theorem of Kapranov and Vasserot
\cite{kapranov-vasserot00}, valid for any finite group action on a
smooth $K3$ surface which preserves the holomorphic two-form. We now
have the situation where we know that
\[
D^bCoh_{\Gamma_{16}}(Y_q^*) \iso D^\pi(\SS(Y_q^*))
\]
is split-generated by certain objects, and where we know that the dg
subcategory of $\SS_{\Gamma_{16}}(Y_q^*)$ consisting of those
objects, denoted by $\SS_{64,q}$ above, is quasi-isomorphic to
$\QQ_{64,q}$. Lemma \ref{th:derived-equivalence} then says that

\begin{cor} \label{th:mirror-1}
There is some $\psi \in End(\Lambda_\N)^\times$, such that
$D^bCoh(Z_q^*) \iso D^\pi(\psi^*\QQ_{64,q} \otimes_{\Lambda_\N}
\Lambda_\Q)$. \qed
\end{cor}


\section{Counting polygons\label{sec:computation}}

We begin by showing that for a quartic surface in $\CP{3}$ whose part
at infinity is the intersection with the ``simplex'' $x_0x_1x_2x_3 =
0$, the deformation from affine to relative Fukaya category is
nontrivial. This is based on Corollary \ref{th:wall-crossing}, but
the construction of the relevant Lagrangian torus involves geometric
ideas related to the SYZ conjecture \cite{ruan02a}, in particular the
appearance of amoebas in that context \cite{ruan02c,mikhalkin01}.
Having done that, the next step is to apply Corollary
\ref{th:generates-everything} to prove that the 64 vanishing cycles
of the Fermat pencil of quartics are split-generators for the derived
Fukaya category. This reduces us to studying the full
$A_\infty$-subcategory formed by these cycles, and finally, the
dimensional induction machinery from Section
\ref{subsec:combinatorial} allows us to compute that subcategory
combinatorially.

\subsection{}
Let $X = \CP{3}$ with $o_X = \K_X^{-1} = \O(4)$, and
$\sigma_{X,\infty} = x_0x_1x_2x_3$. Choose some section
$\sigma_{X,0}$ which with $\sigma_{X,\infty}$ generates a
quasi-Lefschetz pencil, and such that $X_0$ is regular.

\begin{lemma} \label{th:quartic-deformation}
$\Fuk(X_0,X_{0,\infty})$, as a one-parameter deformation of
$A_\infty$-cate\-go\-ries, is nontrivial at first order in $q$.
\end{lemma}

The validity of the statement is independent of the particular choice
of $\sigma_{X,0}$. Any two different choices $\sigma_{X,0}^0$,
$\sigma_{X,0}^1$ can be joined by a smooth family $\sigma^t_{X,0}$,
of sections of $o_X$, such that for all $t$,
\[
X^t_0 = (\sigma^t_{X,0})^{-1}(0)
\]
is a smooth quartic and intersects each stratum of $X_{\infty}$
transversally. For the same reasons as in the proof of Proposition
\ref{th:negativity}, there is a family of symplectic isomorphisms
$\phi^t: X^0_0 \rightarrow X^t_0$ which maps $X^0_0 \cap X_\infty$ to
$X^t_0 \cap X_\infty$. Since quartic surfaces are simply-connected,
one sees from Remark \ref{th:class-of-j-2} that $\phi^1$ defines an
equivalence of relative Fukaya categories.

It remains to prove Lemma \ref{th:quartic-deformation} in a single
case. Let $C = \{x_3 = 0\} \iso \CP{2}$ be one of the irreducible
components of $X_\infty$. The intersection with the other components
is $\Delta = C \cap C' \iso \{x_0x_1x_2 = 0\} \subset \CP{2}$. By
construction, $\CP{2}$ carries the Fubini-Study symplectic form
$\o_C$, rescaled in such a way that a line has area $4$. We also have
the meromorphic complex volume form $\eta_C$ with poles along
$\Delta$. Take (this will not quite be our final word) a section of
the form
\begin{equation} \label{eq:prelim-choice}
\sigma_{X,0} = x_0^4 + x_1^4 + \quarter x_2^4 + 2 x_0 x_1 x_2^2 +
\text{(monomials containing $x_3$)}.
\end{equation}
$X_0 \cap C$ is the smooth quartic curve $Q = \{x_0^4 + x_1^4 +
\quarter x_2^4 + 2 x_0 x_1 x_2^2 = 0\} \subset \CP{2}$, which
intersects all the strata of $\Delta$ transversally. Hence, by
suitably choosing the terms not spelled out in
\eqref{eq:prelim-choice}, one can achieve that $\sigma_{X,0}$
generates a quasi-Lefschetz pencil. Consider the family of Lagrangian
torus orbits
\[
 T_\lambda = \{
 \lambda_2 |x_0|^2 = \lambda_0 |x_2|^2, \;
 \lambda_2 |x_1|^2 = \lambda_1 |x_2|^2
 \} \subset \CP{2} \setminus \Delta,
\]
for $\lambda = (\lambda_0,\lambda_1,\lambda_2)$ with $\lambda_k >0$
and $\sum_k \lambda_k = 1$. We take
\[
\textstyle L_0 = T_{\frac{1}{4},\frac{1}{4},\frac{1}{2}},
\]
which by an elementary estimate is disjoint from $Q$. The next
question is how far this satisfies Assumptions \ref{as:l-zero}. Each
$T_\lambda$ is graded with respect to $\eta_C$, and $O(4)|T_\lambda$
is the trivial flat line bundle iff $\lambda_k \in \quarter\Z$ for
all $k$, which is the case for $L_0$. Look at the holomorphic discs
$w: (D,\partial D) \rightarrow (C,L_0)$ given by
\begin{equation} \label{eq:8-discs}
 w(z) = \begin{cases} [\zeta: z: \sqrt{2}], \\
 [z: \zeta: \sqrt{2}] \end{cases}
 \text{for some $\zeta \in \C$ with $\zeta^4 = -1$}.
\end{equation}
They satisfy $\int w^*\o_C = w \cdot Q = 1$, and their boundary
circles generate $H_1(L_0)$. By Lemma \ref{th:check-exactness}, this
implies that $L_0$ is exact with respect to $\theta_C$.

\begin{lemma} \label{th:regular-p2}
Every holomorphic disc $w: (D,\partial D) \longrightarrow
(\CP{2},T_\lambda)$ is regular.
\end{lemma}

\proof Take the Euler sequence on $\CP{2}$,
\begin{equation} \label{eq:euler}
 0 \longrightarrow \O \longrightarrow \O(1) \otimes H^0(\O(1))^\vee
 \longrightarrow T\CP{2} \longrightarrow 0.
\end{equation}
Choose a basis for $H^0(\O(1))^\vee$ dual to the three coordinate
functions $x_0,x_1,x_2$. Then the first map is just given by these
functions, and the second map takes $i(x_k \otimes x_k^\vee)$ to the
vector field which infinitesimally rotates the $k$-th coordinate. It
follows that over $T_\lambda$, we have a sequence of real subbundles
of \eqref{eq:euler},
\[
 0 \longrightarrow i\R \longrightarrow
 i\R x_0 \oplus i\R x_1 \oplus i\R x_2
 \longrightarrow T(T_\lambda) \longrightarrow 0.
\]
After pulling this back to the disc via $w$ and introducing the
associated Cauchy-Riemann operators, one finds that there is a
surjection
\[
 \coker(D_0 \oplus D_1 \oplus D_2) \twoheadrightarrow \coker\, D_{w,\CP{2}},
\]
where $D_k$ is the $\bar\partial$-operator on $w^*\O(1)$ with totally
real boundary conditions $w^*(i\R x_k)$. The Maslov number of
$(w^*\O(1),w^*(i\R x_k))$ is twice the intersection number of $w$
with the line $\{x_k = 0\}$, hence nonnegative, which means that
$D_k$ has trivial cokernel. \qed

\begin{lemma} \label{th:maslov-2}
Up to the $PSL(2,\R)$-action on $D$, any holomorphic disc $w:
(D,\partial D) \rightarrow (\CP{2},T_\lambda)$ with Maslov number 2
is either of the form
\[
 w(z) = [\lambda_0 : \zeta\lambda_1 : \lambda_2 z] \;\; \text{ or }
 \;\; w(z) = [\lambda_0 : \zeta\lambda_1 : \lambda_2 z^{-1}]
\]
for some $|\zeta| \in S^1$, or else the same with the roles of the
coordinates exchanged.
\end{lemma}

\proof The virtual dimension of the moduli space of unparametrized
discs is equal to one. By Lemma \ref{th:regular-p2}, this moduli
space is indeed a smooth one-dimensional manifold. Since on the other
hand, $T^2 \subset PU(3)$ acts on it, we find that there must be a
sub-circle of $T^2$ which leaves $w$ invariant as an unparametrized
disc. In other words, after applying an initial $PSL(2,\R)$-action
one can assume that there are weights $d_0,d_1,d_2 \in \Z^3$, not all
of them equal, such that
\[
w(e^{it}z) = \mathrm{diag}(e^{it d_0},e^{it d_1}, e^{it d_2})\, w(z).
\]
By holomorphicity $w(z) =
\mathrm{diag}(z^{d_0},z^{d_1},z^{d_2})\,w(1)$. However, at $z = 0$
our disc may intersect $\Delta$ only with multiplicity one, and that
implies $d = (0,0,\pm 1)$ or else the same with the coordinates
permuted. \qed

\begin{lemma} \label{th:8-discs}
Up to the action of $S^1$ on $D$ by rotations, there are precisely
eight holomorphic discs $w: (D,\partial D) \rightarrow (\CP{2},L_0)$
with $w \cdot \Delta = w \cdot Q = 1$ and $w(0) \in \Delta \cap Q$.
Moreover, for each of them $[w|\partial D] \in H_1(L_0)$ is
nontrivial, and the restricted linearized operator
$D\bar\partial_{w,\CP{2}}^{res}$ as in \eqref{eq:restricted-dbar} has
one-dimensional cokernel.
\end{lemma}

\proof Because $L_0$ is graded with respect to $\eta_C$, $w \cdot
\Delta = 1$ implies that the Maslov number is 2, so we can apply
Lemma \ref{th:maslov-2}. Out of the possible six choices of weights,
only $d = (1,0,0)$ or $(0,1,0)$ lead to discs with $\int w^*\o_C = w
\cdot Q = 1$. The condition that $w(0) \in \Delta \cap Q$ further
restricts us to the eight discs from \eqref{eq:8-discs}. The
statement about $[w|\partial D]$ is obvious, and that about the
cokernel is a straightforward computation.
% in fact, since we are in a clifford torus like situation,
% the dbar operator is O(0) \oplus O(1), so restricting to a point = 0
% yields precisely the desired cokernel \R on the O(0) factor.
\qed

Now perturb the equation defining $Q$ slightly, in such a way that
out of the eight points of $Q \cap \{x_0x_1 = 0\}$, one remains fixed
while the others move in a generic way. Then if we replace $Q$ by its
perturbed version $Q'$ in Lemma \ref{th:8-discs}, only one of the 8
discs persists, which we call $w_0$. Take a corresponding slight
perturbation $\sigma_{X,0}'$ of $\sigma_{X,0}$ which still generates
a quasi-Lefschetz pencil. Then Assumptions \ref{as:l-zero} are
satisfied: we already verified those concerning $L_0$ and $w_0$, and
as for those excluding other pseudo-holomorphic maps, part
\ref{item:no-disc-1} comes out of Lemma \ref{th:8-discs};
\ref{item:no-disc-2} follows from Lemma \ref{th:regular-p2} because a
disc $w$ which does not intersect $\Delta$ would have zero Maslov
number, hence negative virtual dimension; and \ref{item:no-sphere} is
obvious, since the $\o_C$-area of any $v: S^2 \rightarrow \CP{2}$ is
a multiple of 4. An appeal to Corollary \ref{th:wall-crossing-2}
completes the proof of Lemma \ref{th:quartic-deformation}.

\subsection{}
As before take $X = \CP{3}$ with $\sigma_{X,\infty}(x) =
x_0x_1x_2x_3$, so that the complement of $X_\infty$ can be written as
a quotient by the action of $\Gamma_4^* = \{\pm \One, \pm i \One\}
\subset SL_4(\C)$,
\begin{equation} \label{eq:m-coordinates}
 M = \{x \in \C^4 \suchthat x_0 x_1 x_2 x_3 = 1\}/\Gamma_4^*.
\end{equation}
We now specify $X_0$ to be the Fermat quartic $\sigma_{X,0} =
\quarter( x_0^4+x_1^4+x_2^4+x_3^4)$. $X_0$ intersects the strata of
$X_\infty$ transversally, and $\pi_M =
\sigma_{X,0}/\sigma_{X,\infty}$ has 64 nondegenerate critical points,
coming in groups of 16 which lie in the same fibre (this is not a
problem, see Remark \ref{th:multiple-fibres}). We take $z_* = 0$ as
our base point, and use the same embedded vanishing path for all
critical points which lie in a given fibre. This yields a collection
of 64 vanishing cycles in $M_0 = X_0 \setminus X_{0,\infty}$, divided
into four groups of 16 mutually disjoint ones. We make them into
branes by choosing gradings and, for the projective and relative
versions of that notion, the other additional data. Notation:
$\Fuk_{64}$ will be the full $A_\infty$-subcategory of $\Fuk(M_0)$
whose objects are these vanishing cycles; and $\Fuk_{64,q}$,
$\Fuk_{64,q}^*$ the corresponding $A_\infty$-subcategories of
$\Fuk(X_0)$, $\Fuk(X_0,X_{0,\infty})$. From Propositions
\ref{th:q-zero} and \ref{th:invert-q} we know that $\Fuk_{64,q}$ is a
one-parameter deformation of $\Fuk_{64}$, and $\Fuk_{64,q}^* \iso
\Fuk_{64,q} \otimes_{\Lambda_\N} \Lambda_\Q$.

\begin{lemma} \label{th:f64-generates}
$D^\pi\Fuk(X_0) \iso D^\pi(\Fuk_{64,q}^*)$.
\end{lemma}

\proof Corollary \ref{th:generates-everything} shows that the
vanishing cycles are split-generators for $D^\pi\Fuk(X_0)$, and Lemma
\ref{th:generating-subcategory} yields the desired conclusion. \qed

\begin{lemma} \label{th:f64-nontrivial}
$\Fuk_{64,q}$, seen as a one-parameter deformation of $\Fuk_{64}$, is
nontrivial at first order.
\end{lemma}

\proof We know from Lemma \ref{th:quartic-deformation} that
nontriviality is true for the whole of $\Fuk(X_0,X_{0,\infty})$. A
closer look at the argument, and in particular at the proof of
Corollary \ref{th:wall-crossing}, reveals that we have in fact
established the assumptions of Lemma \ref{th:deformation-generators},
which shows that the deformation remains nontrivial when restricted
to any subcategory which has enough objects to split-generate the
whole. \qed

To simplify the geometry of our pencil, consider the action of the
group $\Gamma_{16}^* \subset PSL_4(\C)$ of diagonal matrices $A$
satisfying $A^4 = \One$ on $X$. This action is free on $M$; it can be
lifted to $o_X$ and preserves the Fubini-Study connection, as well as
$\sigma_{X,0},\sigma_{X,\infty}$; and it is compatible with the
isomorphism $o_X \iso \K_X^{-1}$. Hence, on the quotient $\bar{M} =
M/\Gamma_{16}^*$ we have an induced symplectic form, a one-form
primitive for it, a holomorphic volume form, and a function
$\pi_{\bar{M}}: \bar{M} \rightarrow \C$. Explicitly,
\[
 \bar{M} = \{ u_0 u_1 u_2 u_3 = 1\}, \quad
 \pi_{\bar{M}}(u) = \quarter(u_0 + u_1 + u_2 + u_3)
\]
where the coordinates are related to the previous ones by $u_k =
x_k^4$. Since $\Gamma_{16}^*$ permutes the critical points in any
given fibre transitively, $\pi_{\bar{M}}$ has one nondegenerate
critical point in each each of the four singular fibres. In other
words, the 64 vanishing cycles in $M_0$ are all possible lifts of the
corresponding 4 vanishing cycles in the quotient $\bar{M}_0$. We need
some basic information about the topology of this space:
\begin{equation} \label{eq:pi12}
\begin{split}
 & \pi_1(\bar{M}_0) = \{ h \in \Z^4 \suchthat h_0 + \dots + h_3 = 0\}, \\
 & im(\pi_2(\bar{M}_0) \otimes \Q \rightarrow H_2(\bar{M}_0;\Q)) \iso \Q^3
\end{split}
\end{equation}
where the first group maps isomorphically to $\pi_1(\bar{M})$, with
the generators $h_k$ becoming loops around the coordinates axes $u_k
= 0$ at infinity; and the second is generated by the 4 vanishing
cycles. Both things follow from elementary Morse and Lefschetz
theory, since $\bar{M} \htp T^3$ is obtained by attaching four
3-cells to $\bar{M}_0$, which is itself homotopy equivalent to a
2-complex. As explained in Section \ref{subsec:coverings}, this means
that we have an action of $\pi_1(\bar{M}_0)^*$ on a subcategory of
$\Fuk(\bar{M}_0)$ which consists of branes with a certain condition
on their fundamental groups, and in particular contains our vanishing
cycles since they are spheres. Recall that in Section
\ref{subsec:q-preliminary} we considered the group $T = H/\Gamma_4
\subset PSL(V)$. If one chooses a basis of $V$ making the maximal
torus $H \subset SL(V)$ standard, $T$ can be written as
$(\C^*)^4/\C^*$ (the quotient of the group of all diagonal matrices
by the multiples of the identity) and hence can be identified with
$\pi_1(\bar{M}_0)^*$. The following is our main computational result,
and will be proved later:

\begin{prop} \label{th:computation}
For a suitable choice of the lifts to $M_0$ which define the
$T$-action, full $A_\infty$-subcategory $\Fuk_4 \subset
\Fuk(\bar{M}_0)$ whose objects are the 4 vanishing cycles of
$\pi_{\bar{M}}$ is $T$-equivariantly quasi-isomorphic to $\QQ_4$.
\end{prop}

The covering $M_0 \rightarrow \bar{M}_0$ provides an epimorphism
$\pi_1(\bar{M}_0) \rightarrow \Gamma_{16}^*$, and the dual of that is
the inclusion of the subgroup $\Gamma_{16} \hookrightarrow T$
considered in Section \ref{subsec:q-preliminary}. Combining the
statement above with Lemma \ref{th:semidirect-lift}, one finds that
\[
 \Fuk_{64} \iso \Fuk_4 \semidirect \Gamma_{16} \iso
 \QQ_4 \semidirect \Gamma_{16} \iso \QQ_{64}.
\]
Since each morphism space $u_i Q_{64} u_j$ is concentrated in a
single degree, the one-parameter deformation $\Fuk_{64,q}$
necessarily satisfies $H^*(\Fuk_{64,q}) \iso H^*(\Fuk_{64})
\otimes_\C \Lambda_\N$, so we can use the Perturbation Lemma to find
a quasi-isomorphic deformation of $\QQ_{64}$. To complete the
argument, consider the symplectic automorphism $\bar\phi$ of
$\bar{M}$ given by $\bar\phi(u_0,u_1,u_2,u_3) = (u_1,u_2,u_3,u_0)$.
This acts fibrewise with respect to $\pi_{\bar{M}}$, hence takes each
vanishing cycle to itself, but it reverses their orientation since it
acts with determinant $-1$ on the tangent space to each critical
point. If we equip $\bar\phi_0 = \bar\phi|\bar{M}_0$ with a suitable
``odd'' grading $\alphagr_{\bar\phi_0}$, it will map each vanishing
cycle to itself as a graded Lagrangian submanifold. Since it
preserves the complex structure, it is no problem to make
$\bar\phi_0$ act on the Fukaya category, yielding an
$A_\infty$-homomorphism $W_4: \Fuk_4 \rightarrow \Fuk_4$ which acts
as $-1$ on $HF^2(V,V)$ for each of the four vanishing cycles. This
also skew-commutes with the $T$-action up to an automorphism $\xi$ of
$T$, which is simply the dual of the action of $\bar\phi$ on
$\pi_1(\bar{M})$, hence which permutes the coordinates cyclically.
Under the isomorphism $H(\Fuk_4) \iso Q_4$, we can therefore identify
$H(W_4)$ with one of the automorphisms $U_4$ considered in Section
\ref{subsec:u4}. $\bar\phi_0$ lifts to an automorphism of $M_0$,
$\phi_0(x_0,\dots,x_3) = (x_1,x_2,x_3,x_0)$, and this extends to the
compactification $X_0$. This means that $W_{64} = W_4 \semidirect
(\xi|\Gamma_{16})$ which can be identified with the action of
$\phi_0$ on $\Fuk_{64}$, admits a deformation $W_{64,q}$ to an
$A_\infty$-homomorphism from $\Fuk_{64,q}$ to itself. Together with
Lemma \ref{th:f64-nontrivial}, this means that we have verified all
the conditions of Lemma \ref{th:recognize-q64}, and therefore can
conclude that $\Fuk_{64,q}$ is quasi-isomorphic to $\QQ_{64,q}$ up to
some change of parameter in $End(\Lambda_\N)^\times$. Using Lemma
\ref{th:f64-generates}, we obtain:

\begin{cor} \label{th:mirror-2}
There is a $\psi \in End(\Lambda_\N)^\times$ such that
\[ D^\pi\Fuk(X_0) \iso D^\pi(\Fuk_{64,q}^*) \iso
\psi^*D^\pi(\QQ_{64,q} \otimes_{\Lambda_\N} \Lambda_\Q).\qed \]
\end{cor}

This is the mirror statement to Corollary \ref{th:mirror-1}, and by
comparing the two, Theorem \ref{th:main} follows.


\subsection{}
It remains to explain Proposition \ref{th:computation}. Following the
strategy from Section \ref{subsec:braid-monodromy}, the first step in
understanding the geometry of the vanishing cycles is to represent
them as matching cycles, which involves the choice of a generic
auxiliary section. Let us momentarily work ``upstairs'' in $M$. An
ansatz for the section is
\begin{equation} \label{eq:ansatz}
 \sigma_{X,0}' = \quarter(\xi_0 x_0^4 + \xi_1 x_1^4 + \xi_2 x_2^4 + \xi_3 x_3^4)
\end{equation}
with pairwise distinct constants $\xi_k \in \C^*$. This always
satisfies the first of the genericity assumptions, namely
$(\sigma_{X,0}')^{-1}(0)$ intersects each stratum of $X_{0,\infty}$
transversally. The critical point set of the associated map
\eqref{eq:bm-map} is always smooth,
\[
 Crit(b_M) = \Big\{ x_k^4 = (s+\xi_k t)^{-1},
 \text{ where } \prod_k (s+t \xi_k) = 1 \Big\}.
\]
The parameter value $t = 0$ corresponds precisely to the 64 critical
points of $\pi_M$. At those points, $b_M: Crit(b_M) \rightarrow \C^2$
is an immersion, and its second component
$\sigma'_{X,0}/\sigma_{X,\infty} : Crit(b_M) \rightarrow \C$ a simple
branched cover, iff the coefficients $\xi_k$ satisfy
\begin{equation} \label{eq:nondegeneracy}
 \sum_k \xi_k^2 \neq \quarter \Big(\sum_k \xi_k\Big)^2.
\end{equation}
Next, $b_M|Crit(b_M)$ cannot be a generic embedding because it is
$\Gamma_{16}^*$-invariant, but after passing to the quotient we do
get a generic embedding $b_{\bar{M}}: Crit(b_{\bar{M}}) \rightarrow
\C^2$, for elementary reasons: the image $\bar{C}$ of this map is an
irreducible algebraic curve in $\C^2$ having four distinct points at
infinity, and since $Crit(b_{\bar{M}})$ has Euler characteristic -2,
the map cannot be a multiple branch cover. The upshot is that the
dimensional induction technique from Section \ref{subsec:induction}
can be applied ``downstairs'', meaning to the vanishing cycles of
$\pi_{\bar{M}}$.

Let's get down to concrete numbers. Our choice of weights is $\xi_0 =
5/4$, $\xi_1 = i$, $\xi_2 = -1$, $\xi_3 = -i$, which satisfies
\eqref{eq:nondegeneracy}. The branch curve (determined by computer
using elimination theory) is
\[
\begin{split}
 \bar{C} = \{ & 128000 z^{12} - 76800 z^{11}w
 + 92160 z^{10}w^2 - 108544 z^9w^3 - 258048 z^8w^4 \\ & + 86016z^7w^5
 - 112640 z^6w^6 + 141312 z^5w^7 + 211968 z^4w^8 - 25600z^3w^9 \\
 & + 36864 z^2 w^{10} - 49152 zw^{11} - 65536 w^{12}
 - 391307 z^8 + 180632 z^7w \\ & + 1940 z^6w^2 - 503112 z^5w^3
 - 2102866 z^4w^4 + 457512 z^3w^5 - 122764 z^2w^6 \\ & - 102904 zw^7
 - 251483 w^8 + 399430 z^4 - 106312 z^3w - 155028z^2w^2 \\ & - 53752 zw^3
 - 320858 w^4 - 136161 = 0 \}.
\end{split}
\]
Figure \ref{fig:paths1} shows the $z$-plane $\C$ with the 4 points of
$Critv(\pi_{\bar{M}}) = \{\pm 1, \pm i\}$, as well the 40 points of
$Fakev(\pi_{\bar{M}})$. Luckily, none of the latter lie on $[-1;1]$
or $i[-1;1]$, so we can take the basis of embedded critical paths
which are straight lines $b_k(t) = i^{2-k}t$, $k = 1,\dots,4$. Denote
by $V_{b_k} \subset \bar{M}_0$ the resulting vanishing cycles. We now
turn to the function
\begin{equation} \label{eq:q0}
q_{\bar{M}_0} = \quarter \sum_k \xi_k u_k: \bar{M}_0 =
\pi_{\bar{M}}^{-1}(0) \longrightarrow \C,
\end{equation}
which is also a quasi-Lefschetz fibration (or more accurately, the
$\Gamma_{16}^*$-quotient of such a fibration on $M_0$). Figure
\ref{fig:paths2} shows its basis $\C$ with the 12 critical values,
which are also the points of $\bar{C} \cap \{z = 0\}$, and the
matching paths $d_k$ obtained from the braid monodromy of $\bar{C}$
over $b_k$. The associated matching cycles are our vanishing cycles:
$V_{b_k} \htp \Sigma_{d_k}$.

Taking $0$ as a base point, choose a base path $c_*$ and a basis of
vanishing paths $c_1,\dots,c_{12}$ for \eqref{eq:q0} as indicated in
Figure \ref{fig:paths3}. This gives rise to vanishing cycles $V_{c_k}
\subset \bar{M}_{0,0} = q_{\bar{M}_0}^{-1}(0)$, which are simple
closed loops on a four-pointed genus three Riemann surface.

\begin{remark}
If one took $\xi_k = i^k$ instead, condition \eqref{eq:nondegeneracy}
would be violated but the structure of $q_{\bar{M}_0}$ would be much
simpler: it would have 4 degenerate critical points of type $A_3$.
The actually chosen $\xi_k$ are a small perturbation of these values,
in the sense that one can still recognize the groups of 3 critical
points obtained by ``morsifying'' the degenerate singularities. This
allows one to choose the vanishing paths $c_k$ in such a way as to
minimize the intersections between the vanishing cycles in each
group.
\end{remark}

Let $\Fuk^\rightarrow_{12}$ be the directed $A_\infty$-subcategory of
the Fukaya category of $\bar{M}_{0,0}$ whose objects are the
$V_{c_k}$. From the restriction of the universal cover of
$\bar{M}_0$, we get an action of $\pi_1(\bar{M}_0)^* \iso T$ on this
$A_\infty$-category. The equivariant version of Theorem
\ref{th:induction} shows that $\Fuk_4$ is equivariantly
quasi-isomorphic to the full $A_\infty$-subcategory of
$Tw_T(\Fuk_{12}^\rightarrow)$ whose objects are the following four
equivariant twisted complexes, obtained by applying the rules from
Section \ref{subsec:induction} to the paths from Figures
\ref{fig:paths2}, \ref{fig:paths3}:
\begin{equation} \label{eq:s-objects}
\begin{split}
 &
 S_1 = \big\{ T'_{V_{c_7}}T'_{V_{c_5}}(V_{c_3})
 \longrightarrow T_{V_{c_{10}}}T_{V_{c_{11}}}(V_{c_{12}}) \big\}, \\
 &
 S_2 = \big\{ T_{V_{c_1}}T_{V_{c_2}}(V_{c_3})
 \longrightarrow T'_{V_{c_{10}}}T'_{V_{c_8}}(V_{c_6}) \big\}, \\
 &
 S_3 = \big\{ T_{V_{c_1}}T_{V_{c_4}}T_{V_{c_5}}(V_{c_6})
 \longrightarrow T_{V_{c_{11}}}'(V_{c_9}) \big\}, \\
 &
 S_4 = \big\{ T_{V_{c_2}}T_{V_{c_4}}T_{V_{c_7}}T_{V_{c_8}}(V_{c_9})
 \longrightarrow V_{c_{12}} \big\}
\end{split}
\end{equation}
where $\{\cdots \rightarrow \cdots\}$ is the cone over the lowest
degree nonzero morphism, which is unique up to a $\C^*$ multiple.
Actually, that morphism may not have degree zero, or it may not be
$T$-invariant, so one needs to change one of the two objects forming
the cone by a translation and tensoring with a suitable
$T$-character. In that sense, the formulae given \eqref{eq:s-objects}
are not entirely precise, but the appropriate corrections can not yet
be determined at this stage in the computation.

The next step is to determine $\Fuk^\rightarrow_{12}$, or rather a
directed $A_\infty$-category $\Cat^\rightarrow_{12}$ quasi-isomorphic
to it, using the combinatorial description of directed Fukaya
categories on Riemann surfaces. Following the procedure described in
Section \ref{subsec:combinatorial}, we take a compact piece $N
\subset \bar{M}_{0,0}$ which is a surface with boundary and a
deformation retract. Figure \ref{fig:curve1} shows a decomposition of
$N$ into eight polygons (I)--(VIII), with the solid lines the actual
$\partial N$, and the dashed lines to be glued to the boundaries of
other polygons as indicated. $\bar{M}_{0,0}$ inherits the structure
of an affine Calabi-Yau from $\bar{M}$, and Figure \ref{fig:curve2}
shows an unoriented foliation on $N$ lying in the same homotopy class
as the distinguished trivialization of the squared canonical bundle.
Moreover, restriction
\[
\Z^4/\Z(1,1,1,1) \iso H^1(\bar{M}) \longrightarrow H^1(\bar{M}_{0,0})
\]
gives four canonical elements of $H^1(N)$ whose sum is zero; the
Poincar{\'e} duals of the first three elements are represented by the
one-cycles $x,y,z$ in $(N,\partial N)$ shown in Figure
\ref{fig:curve3}. Figures \ref{fig:vanishing1}--\ref{fig:vanishing8}
show simple closed curves $\nu_k$ isotopic to the vanishing cycles
$V_{c_k}$, drawn on each of the eight polygons which make up $N$.

\begin{remark}
These pictures were arrived at by an application of braid monodromy
(in one dimension lower than before), which means that we introduce
another auxiliary holomorphic function on $\bar{M}_0$, whose
restriction to any generic fibre of $q_{\bar{M}_0}$ represents that
surface as a branched cover with only simple branch points. The
function we use is simply a projection
\[
 r_{\bar{M}_0}(u) = u_2 \in \C^*,
\]
making $\bar{M}_{0,0}$ into a fourfold branched cover of $\C^*$ with
eight simple branch points. Cutting the base of that into suitable
pieces gives rise to our decomposition of $N$ into polygons.
Elimination theory is used to compute the branch curve of
\[
(q_{\bar{M}_0},r_{\bar{M}_0}): \bar{M}_{0,0} \longrightarrow \C
\times \C^*,
\]
and from that one obtains the vanishing cycles
$V_{c_1},\dots,V_{c_{12}}$. We have omitted the details of this step,
since it is essentially topological, and completely parallel to the
procedure which led to the paths $b_1,\dots,b_4$ in Figure
\ref{fig:paths2}.
\end{remark}

When drawing the $\nu_k$, there is some freedom in their relative
position with respect to each other. The only constraint, which comes
from exactness, says that we should be able to assign positive areas
to the connected components of $N \setminus \bigcup_k \nu_k$ such
that the two-chains which represent relations between the $\nu_k$
have signed area zero. In fact, as explained in Remark
\ref{th:painting-by-numbers}, not all relations need to be
considered; in our case \eqref{eq:pi12} shows that there are three
essential ones. These can be read off from \eqref{eq:s-objects} and
the intersection numbers of the $\nu_k$, and are
\begin{equation}
\begin{split}
 & \nu_5 + \nu_8 + \nu_{10} - \nu_1 - \nu_2 - \nu_3 \sim 0, \\
 & \nu_3 + \nu_5 + \nu_7 - \nu_{10} - \nu_{11} - \nu_{12} \sim 0, \\
 & \nu_8 + \nu_2 + \nu_4 - \nu_7 - \nu_8 - \nu_0 \sim 0.
\end{split}
\end{equation}
After having drawn the relevant two-chains explicitly in Figures
\ref{fig:vanishing1}--\ref{fig:vanishing8}, one realizes that for
each of them, there are pieces of $N \setminus \bigcup_k \nu_k$ which
appear in it with either sign, and which do not appear in the other
two-chains. This means that the configuration of curves $\nu_k$ can
indeed be used as basis for the computation.

Choose gradings and $Spin$ structures for the $\nu_k$, lifts to the
abelian covering of $N$ induced from $\pi_1(N) \iso
\pi_1(\bar{M}_{0,0}) \rightarrow \pi_1(\bar{M}_0)$, and
identifications $\C_x \iso \C$ for each intersection point of $\nu_k
\cap \nu_l$ with $k<l$. One can then list all intersection points
with their (Maslov index) degrees as well as their ($T$-action)
weights. This is done in Tables \ref{table:intersections1} and
\ref{table:intersections2}, where the notation is that
\begin{equation} \label{eq:abcd}
 m^d x^a y^b z^c
\end{equation}
is an intersection point with Maslov index $d$, and on which
$[x,y,z,1] \in T$ acts as $x^ay^bz^c$, respectively. By inspecting
the table, one sees that different intersection points of the same
$\nu_k \cap \nu_l$ are always distinguished by $(a,b,c,d)$. We may
therefore denote the generator of
$hom_{\Cat^\rightarrow_{12}}(\nu_k,\nu_l)$ given by an intersection
point \eqref{eq:abcd} by
\[
 \nu_k \cap \nu_l: m^d x^a y^b z^c.
\]
The next step is to count all immersed holomorphic polygons which
have sides on the $\nu_k$ (in the correct order), with the
appropriate signs. The outcome of that are the composition maps
$\mu^d$ of $\Cat^{\rightarrow}_{12}$. It turns out that only
$\mu^2,\mu^3$ are nonzero; their coefficients are listed in Tables
\ref{table:mu2} and \ref{table:mu3}.

\begin{remark}
The $T$-action is actually a big bonus in this computation, since the
fact that the $\mu^d$ have to be homogeneous means that a lot of
their coefficients have to vanish. For instance, the fact that $\mu^d
= 0$ for $d > 3$ follows by inspection of the list of $T$-weights of
the intersection points, so one does not need to verify geometrically
that there are no immersed pentagons, hexagons, and so forth.
\end{remark}

At this point, we can write down the objects $S_k$ from
\eqref{eq:s-objects}, or rather the corresponding objects $C_k$ in
$Tw_T(\Cat^{\rightarrow}_{12})$, more explicitly. They are
\begin{align*}
 &
 C_1 = \nu_3[1]\twist{x} \oplus \nu_5\twist{y} \oplus \nu_7\twist{yz}
 \oplus \nu_{10}\twist{x^{-1}y} \oplus \nu_{11}[-1]\twist{y} \oplus
 \nu_{12}[-1], \\
 & \delta_1 \!=\!
 {\arraycolsep2.5pt \begin{pmatrix}
 0 & 0 & 0 & 0 & 0 & 0 \\
 \nu_3 \cap \nu_5\!:\!x^{-1}y \!\! & 0 & 0 & 0 & 0 & 0 \\
 0 & \!\! \nu_5 \cap \nu_7\!:\!mz & 0 & 0 & 0 & 0 \\
 0 & \!\! \nu_5 \cap \nu_{10}\!:\!mx^{-1} & 0 & 0 & 0 & 0 \\
 0 & 0 & \nu_7 \cap \nu_{11}\!:\!z^{-1} & \nu_{10} \cap \nu_{11}\!:\!x & 0 & 0 \\
 0 & 0 & 0 & 0 & \!\!\!\! \nu_{11} \cap \nu_{12}\!:\!my^{-1} & 0
 \end{pmatrix}},
 \\
 &
 C_2 = \nu_1[1]\twist{x^{-1}y} \oplus \nu_2\twist{y} \oplus \nu_3
 \oplus \nu_6\twist{z^{-1}} \oplus \nu_8[-1]\twist{z^{-1}}
 \oplus \nu_{10}[-1]\twist{x^{-1}}, \\
 & \delta_2 =
 {\arraycolsep2.5pt \begin{pmatrix}
 0 & 0 & 0 & 0 & 0 & 0 \\
 \nu_1 \cap \nu_2\!:\!x\!\! & 0 & 0 & 0 & 0 & 0 \\
 0 & \nu_2 \cap \nu_3\!:\!my^{-1} & 0 & 0 & 0 & 0 \\
 0 & \nu_2 \cap \nu_6\!:\!my^{-1}z^{-1} & 0 & 0 & 0 & 0 \\
 0 & 0 & \!\!\!\!\!\! -(\nu_3 \cap \nu_8\!:\!z^{-1}) & \nu_6 \cap \nu_8\!:\!1 & 0 & 0 \\
 0 & 0 & 0 & 0 & \!\! \nu_{8} \cap \nu_{10}\!:\!mx^{-1}z & 0
 \end{pmatrix}},
 \\
 &
 C_3 = \nu_1\twist{x^{-1}} \oplus \nu_4\twist{x^{-1}y} \oplus
 \nu_5[-1]\twist{x^{-1}y} \oplus \nu_6[-1]\twist{x^{-1}z^{-1}} \\
 & \qquad \qquad
 \oplus \nu_9[-1]\twist{y^{-1}z^{-1}} \oplus \nu_{11}[-2]\twist{z^{-1}}, \\
 & \delta_3 =
 {\arraycolsep2.5pt \begin{pmatrix}
 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 \\
 \nu_1 \cap \nu_5\!:\!y & \nu_4 \cap \nu_5\!:\!1 & 0 & 0 & 0 & 0 \\
 0 & 0 & \nu_5 \cap \nu_6\!:\!my^{-1}z^{-1} & 0 & 0 & 0 \\
 0 & 0 & \!\!\nu_5 \cap \nu_9\!:\!mxy^{-2}z^{-1}\!\! & 0 & 0 & 0 \\
 0 & 0 & 0 & \nu_6 \cap \nu_{11}\!:\!x & \nu_9 \cap \nu_{11}\!:\!y & 0
 \end{pmatrix}}, \displaybreak[0]
 \\
 &
 C_4 = \nu_2[1]\twist{z^{-1}} \oplus \nu_4[1]\twist{x^{-1}} \oplus \nu_7[1]\twist{x^{-1}y}
 \oplus \nu_8\twist{x^{-1}z^{-1}} \\ & \qquad \qquad \oplus \nu_9\twist{x^{-1}y^{-1}z^{-1}}
 \oplus \nu_{12}\twist{y^{-1}z^{-1}}, \displaybreak[0] \\
 & \delta_4 =
 {\arraycolsep2.5pt \begin{pmatrix}
 0 & 0 & 0 & 0 & 0 & 0 \\
 \nu_2 \cap \nu_4 \!:\! mx^{-1}z & 0 & 0 & 0 & 0 & 0 \\
 \nu_2 \cap \nu_7 \!:\! mx^{-1}yz & 0 & 0 & 0 & 0 & 0 \\
 0 & \nu_4 \cap \nu_8\!:\!z^{-1} & \nu_7 \cap \nu_8\!:\!y^{-1}z^{-1} & 0 & 0 & 0 \\
 0 & 0 & 0 & \nu_8 \cap \nu_9\!:\!my^{-1} & 0 & 0 \\
 0 & 0 & 0 & \nu_8 \cap \nu_{12}\!:\!mxy^{-1} & 0 & 0
 \end{pmatrix}}.
\end{align*}
where $\nu_k[d]\twist{x^ay^bz^c}$ means the object $\nu_k$ shifted
down by $d$, and tensored with the one-dimensional representation of
$T$ on which $[x,y,z,1]$ acts as $x^{-a}y^{-b}z^{-c}$.

It is now easy to compute explicitly the full $A_\infty$-subcategory
of $Tw_T\Cat^{\rightarrow}_{12}$ consisting of these four equivariant
twisted complexes $C_k$. More precisely, one verifies (by computer)
first that the cohomology of this subcategory is $T$-equivariantly
isomorphic to $Q_4$ and then, following the indications in Remark
\ref{th:more-recognize-q4}, that a suitable fourth order Massey
product is nonzero, which implies that our full subcategory is
$T$-equivariantly quasi-isomorphic to $\QQ_4$. By Theorem
\ref{th:induction}, or rather the equivariant version of it explained
in Section \ref{subsec:addition}, we have a $T$-equivariant
quasi-isomorphism of our subcategory with $\Fuk_4$, and this
completes the proof of Proposition \ref{th:computation}.

\begin{figure}[H]
\begin{centering}
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\includegraphics{paths1.pstex}%
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\end{picture}
\end{centering}
\caption{\label{fig:paths1} The actual ($\bullet$) and fake ($\circ$)
critical points of $\pi_{\bar{M}}$, together with our choice of
vanishing paths $b_1,\dots,b_4$. The image has been distorted for
better legibility.}
\end{figure}

\begin{figure}[H]
\begin{center}
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\includegraphics{paths2.pstex}%
\end{picture}%
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\end{center}

\caption{\label{fig:paths2}The four matching paths obtained from the
vanishing paths in Figure \ref{fig:paths1} by braid monodromy. These
were drawn using a computer to watch how the points of $\bar{C} \cap
\{z = b_k(t)\}$ move in $\C$ as one goes along the vanishing paths.
The picture would be $\Z/4$-symmetric if we hadn't moved the paths by
an isotopy, so as to avoid the future base path (the dashed line).}
\end{figure}

\begin{figure}[H]
\begin{center}
 \epsfig{file=paths3.eps}

 \caption{\label{fig:paths3}
 The base path and the 12 vanishing paths $c_1,\dots,c_{12}$
 (ordered as they must be, clockwise with respect to
 the derivatives at the base point).
 }
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
 \epsfig{file=curve1.eps}

 \caption{\label{fig:curve1}}
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
 \epsfig{file=curve2.eps}

 \caption{\label{fig:curve2}}
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
 \epsfig{file=curve3.eps}
 \caption{\label{fig:curve3}}
\end{center}
\end{figure}

\begin{figure}[H]
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\caption{\label{fig:vanishing1}}
\end{figure}

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\caption{\label{fig:vanishing2}}
\end{figure}

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\caption{\label{fig:vanishing3}}
\end{figure}

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\end{picture}

\caption{\label{fig:vanishing4}}
\end{figure}

\begin{figure}[H]
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\caption{\label{fig:vanishing5}}
\end{figure}

\begin{figure}[H]
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\caption{\label{fig:vanishing6}}
\end{figure}

\begin{figure}[H]
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\caption{\label{fig:vanishing7}}
\end{figure}

\begin{figure}[H]
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\caption{\label{fig:vanishing8}}
\end{figure}

\begin{table}[H] \extrarowheight0.2em
\begin{equation*}
\begin{array}{l||l|l|l|l|l|l|}
 & 1 & 2 & 3 & 4 & 5 & 6 \\
 \hhline{=#======} 1  \\
 \hhline{--} 2 & x  \\
 \hhline{---} 3 & & my^{-1}  \\
 \hhline{----} 4 & 1 & mx^{-1}z & \\
 & x && \\
 \hhline{-----} 5 & x & x^{-1}y & x^{-1}y & 1  \\
 & 1      & 1 && \\
 & y &&& \\
 \hhline{------} 6 & & my^{-1}z^{-1} & x^{-1}yz^{-1} && my^{-1}z^{-1}  \\
 & & mx^{-1}z^{-1} & x^{-1} && \\
 & & mx^{-1}y^{-1} &&& \\
 \hline 7 & myz & mx^{-1}z && 1 & mz & \\
 & x & mx^{-1}yz && x && \\
 && mz &&&& \\
 \hline 8 & xz^{-1} & z^{-1} & x^{-1} & xy^{-1}z^{-1} & z^{-1} & 1 \\
 & xy^{-1}z^{-1} & mx^{-1}y^{-1} & z^{-1} & y^{-1}z^{-1} & xy^{-1}z^{-1} & \\
 & z^{-1} && x^{-1}yz^{-1} & z^{-1} && \\
 \hline 9 && my^{-1}z^{-1} & x^{-1} && mxy^{-2}z^{-1} & 1 \\
 && mx^{-1}y^{-1} & my^{-1}z^{-1} && my^{-1}z^{-1} & y^{-1}z \\
 && my^{-2} &&& my^{-2} & \\
 \hline 10 & m & mx^{-2} && mx^{-1} & mx^{-1}y^{-1} & \\
 & mx^{-1} & mx^{-1} && y^{-1}z^{-1} & mx^{-1} & \\
 & & mx^{-1}y^{-1} & & & my^{-1} & \\
 \hline 11 & xz^{-1} & my^{-1} & x^{-1}y & xz^{-1} & xz^{-1} & z \\
 && mx^{-1} & 1 & xy^{-1}z^{-1} & my^{-1} & x \\
 && & yz^{-1} & z^{-1} && y \\
 \hline 12 && my^{-1} & my^{-1} && mxy^{-1}z^{-1} & z \\
 &&& mz^{-1} && my^{-1} & mxy^{-1} \\
 &&&&& mxy^{-2} & \\ \hline
\end{array}
\]

\caption{\label{table:intersections1} The points of $\nu_k \cap
\nu_l$ for $1 \leq k \leq 6$ and $l > k$, with their Maslov indices
and $T$-action weights.}
\end{table}

\begin{table}[H] \extrarowheight0.2em
\begin{equation*}
\begin{array}{l||l|l|l|l|l|l|}
 & 7 & 8 & 9 & 10 & 11 & 12 \\
 \hhline{=#======} 7  \\
 \hhline{--} 8 & y^{-1}z^{-1}  \\
 \hhline{---} 9 & & my^{-1}  \\
 \hhline{----} 10 & x^{-1}y^{-1}z^{-1} & mx^{-1}z & \\
 & y^{-1}z^{-1} && \\
 \hhline{-----} 11 & xy^{-1}z^{-1} & y & y & x \\
 & y^{-1}z^{-1}      & x && \\
 & z^{-1} &&& \\
 \hhline{------} 12 & & mxy^{-1} & y && my^{-1}  \\
 & & m & z && \\
 & & my^{-1}z &&& \\
 \hline
\end{array}
\]

\caption{\label{table:intersections2} The points of $\nu_k \cap
\nu_l$ for $7 \leq k \leq 12$ and $l > k$, with their Maslov indices
and $T$-action weights.}
\end{table}

{ \smaller\smaller
\begin{longtable}[c]{ll|l|l}
 $a_1$ & $a_2$ & $b$ & $\pm$ \\
 \hline
 $\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_5:1$
 & $\nu_1 \cap \nu_5:x$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_1 \cap \nu_5:y$
 & $-$ \\
$\nu_1 \cap \nu_4:x$
 & $\nu_4 \cap \nu_5:1$
 & $\nu_1 \cap \nu_5:x$
 & $+$ \\
$\nu_1 \cap \nu_4:1$
 & $\nu_4 \cap \nu_5:1$
 & $\nu_1 \cap \nu_5:1$
 & $-$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_6:x^{-1}yz^{-1}$
 & $\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_6:x^{-1}$
 & $\nu_2 \cap \nu_6:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_4 \cap \nu_8:z^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_7:x$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_7:1$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_4 \cap \nu_8:y^{-1}z^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_9:1$
 & $\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_9:y^{-1}z$
 & $\nu_5 \cap \nu_9:my^{-2}$
 & $-$ \\
$\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $+$ \\
$\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $-$ \\
$\nu_7 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_7 \cap \nu_{10}:x^{-1}y^{-1}z^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_7 \cap \nu_{11}:y^{-1}z^{-1}$
 & $-$ \\
$\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_9 \cap \nu_{12}:y$
 & $\nu_8 \cap \nu_{12}:m$
 & $-$ \\
$\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_9 \cap \nu_{12}:z$
 & $\nu_8 \cap \nu_{12}:my^{-1}z$
 & $-$ \\
$\nu_8 \cap \nu_{11}:y$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_8 \cap \nu_{12}:m$
 & $-$ \\
$\nu_8 \cap \nu_{11}:x$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_7:1$
 & $\nu_2 \cap \nu_7:mx^{-1}z$
 & $-$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_7:x$
 & $\nu_2 \cap \nu_7:mz$
 & $+$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_2 \cap \nu_7:mz$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_2 \cap \nu_7:mx^{-1}yz$
 & $-$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_3 \cap \nu_8:z^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_6:x^{-1}yz^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_6:x^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_3 \cap \nu_8:x^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_1 \cap \nu_7:myz$
 & $+$ \\
$\nu_1 \cap \nu_4:x$
 & $\nu_4 \cap \nu_7:1$
 & $\nu_1 \cap \nu_7:x$
 & $-$ \\
$\nu_1 \cap \nu_4:1$
 & $\nu_4 \cap \nu_7:x$
 & $\nu_1 \cap \nu_7:x$
 & $+$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_1 \cap \nu_7:myz$
 & $+$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_8:x^{-1}$
 & $\nu_2 \cap \nu_8:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_8:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_2 \cap \nu_8:z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_2 \cap \nu_8:z^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_6:mx^{-1}y^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_2 \cap \nu_8:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_7:mx^{-1}z$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_8:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_3 \cap \nu_9:my^{-1}z^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_6:x^{-1}$
 & $\nu_6 \cap \nu_9:1$
 & $\nu_3 \cap \nu_9:x^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_6:x^{-1}yz^{-1}$
 & $\nu_6 \cap \nu_9:y^{-1}z$
 & $\nu_3 \cap \nu_9:x^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_3 \cap \nu_9:my^{-1}z^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_8:z^{-1}$
 & $\nu_1 \cap \nu_8:xz^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_4:x$
 & $\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_1 \cap \nu_8:xz^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_4:x$
 & $\nu_4 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_4:1$
 & $\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_1 \cap \nu_8:z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_4:1$
 & $\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:x$
 & $\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_1 \cap \nu_8:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_1 \cap \nu_8:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:1$
 & $\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_1 \cap \nu_8:z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:1$
 & $\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_7:x$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_9:x^{-1}$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_9:my^{-2}$
 & $\nu_2 \cap \nu_9:my^{-2}$
 & $+$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_2 \cap \nu_9:my^{-1}z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_2 \cap \nu_9:my^{-1}z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_9:my^{-2}$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_9:1$
 & $\nu_2 \cap \nu_9:my^{-1}z^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $\nu_6 \cap \nu_9:y^{-1}z$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_6:mx^{-1}y^{-1}$
 & $\nu_6 \cap \nu_9:1$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_9:y^{-1}z$
 & $\nu_2 \cap \nu_9:my^{-2}$
 & $-$ \\
$\nu_2 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_2 \cap \nu_9:my^{-1}z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{10}:x^{-1}y^{-1}z^{-1}$
 & $\nu_5 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_5 \cap \nu_{10}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_5 \cap \nu_{10}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $-$ \\
$\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_6 \cap \nu_{11}:x$
 & $-$ \\
$\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_6 \cap \nu_{11}:y$
 & $-$ \\
$\nu_6 \cap \nu_9:1$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_6 \cap \nu_{11}:y$
 & $-$ \\
$\nu_6 \cap \nu_9:y^{-1}z$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_6 \cap \nu_{11}:z$
 & $+$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_4 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_7:x$
 & $\nu_7 \cap \nu_{10}:x^{-1}y^{-1}z^{-1}$
 & $\nu_4 \cap \nu_{10}:y^{-1}z^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_7:1$
 & $\nu_7 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_4 \cap \nu_{10}:y^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_4 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:z$
 & $\nu_5 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:y^{-1}z^{-1}$
 & $\nu_5 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_5 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_5 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_9:my^{-2}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_5 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_5 \cap \nu_{11}:my^{-1}$
 & $-$ \\
$\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_6 \cap \nu_{12}:mxy^{-1}$
 & $+$ \\
$\nu_6 \cap \nu_9:y^{-1}z$
 & $\nu_9 \cap \nu_{12}:y$
 & $\nu_6 \cap \nu_{12}:z$
 & $-$ \\
$\nu_6 \cap \nu_9:1$
 & $\nu_9 \cap \nu_{12}:z$
 & $\nu_6 \cap \nu_{12}:z$
 & $+$ \\
$\nu_6 \cap \nu_{11}:x$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_6 \cap \nu_{12}:mxy^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{11}:xz^{-1}$
 & $\nu_4 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_7:x$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_4 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_7:x$
 & $\nu_7 \cap \nu_{11}:y^{-1}z^{-1}$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_7:1$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_7:1$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_4 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{12}:z$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:my^{-1}z$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $+$ \\
$\nu_5 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:m$
 & $\nu_5 \cap \nu_{12}:mxy^{-1}z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_5 \cap \nu_{12}:mxy^{-1}z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{12}:my^{-1}z$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_9 \cap \nu_{12}:y$
 & $\nu_5 \cap \nu_{12}:mxy^{-1}z^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{12}:z$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_9:my^{-2}$
 & $\nu_9 \cap \nu_{12}:y$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_9 \cap \nu_{12}:z$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $-$ \\
$\nu_5 \cap \nu_{11}:xz^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_5 \cap \nu_{12}:mxy^{-1}z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{10}:my^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-2}$
 & $-$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_7 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_7 \cap \nu_{10}:x^{-1}y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-2}$
 & $-$ \\
$\nu_2 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{10}:x^{-1}y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_7:mx^{-1}z$
 & $\nu_7 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_2 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{11}:xz^{-1}$
 & $\nu_3 \cap \nu_{11}:yz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_6:x^{-1}yz^{-1}$
 & $\nu_6 \cap \nu_{11}:z$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $+$ \\
$\nu_3 \cap \nu_6:x^{-1}yz^{-1}$
 & $\nu_6 \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:yz^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_6:x^{-1}$
 & $\nu_6 \cap \nu_{11}:y$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $+$ \\
$\nu_3 \cap \nu_6:x^{-1}$
 & $\nu_6 \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:1$
 & $-$ \\
$\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:yz^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_8:x^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:1$
 & $+$ \\
$\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_3 \cap \nu_{11}:yz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_8:x^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $-$ \\
$\nu_3 \cap \nu_9:x^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_{10}:mx^{-1}$
 & $\nu_1 \cap \nu_{10}:m$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_{10}:mx^{-2}$
 & $\nu_1 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_4:x$
 & $\nu_4 \cap \nu_{10}:mx^{-1}$
 & $\nu_1 \cap \nu_{10}:m$
 & $+$ \\
$\nu_1 \cap \nu_4:1$
 & $\nu_4 \cap \nu_{10}:mx^{-1}$
 & $\nu_1 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $\nu_1 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_1 \cap \nu_{10}:mx^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_{10}:my^{-1}$
 & $\nu_1 \cap \nu_{10}:m$
 & $-$ \\
$\nu_1 \cap \nu_5:x$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_1 \cap \nu_{10}:m$
 & $+$ \\
$\nu_1 \cap \nu_7:myz$
 & $\nu_7 \cap \nu_{10}:x^{-1}y^{-1}z^{-1}$
 & $\nu_1 \cap \nu_{10}:mx^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_7:myz$
 & $\nu_7 \cap \nu_{10}:y^{-1}z^{-1}$
 & $\nu_1 \cap \nu_{10}:m$
 & $-$ \\
$\nu_1 \cap \nu_8:xz^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_1 \cap \nu_{10}:m$
 & $-$ \\
$\nu_1 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_1 \cap \nu_{10}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_{11}:1$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{11}:my^{-1}$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{11}:my^{-1}$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:z$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_6:mx^{-1}y^{-1}$
 & $\nu_6 \cap \nu_{11}:x$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:z$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_6:mx^{-1}y^{-1}$
 & $\nu_6 \cap \nu_{11}:y$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_7:mx^{-1}z$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_7 \cap \nu_{11}:y^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_7:mx^{-1}z$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_8:mx^{-1}y^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_8:mx^{-1}y^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_9:my^{-2}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_2 \cap \nu_{11}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_{10}:mx^{-2}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_2 \cap \nu_{11}:mx^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{12}:mxy^{-1}z^{-1}$
 & $\nu_3 \cap \nu_{12}:mz^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $\nu_3 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_6:x^{-1}$
 & $\nu_6 \cap \nu_{12}:mxy^{-1}$
 & $\nu_3 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_6:x^{-1}yz^{-1}$
 & $\nu_6 \cap \nu_{12}:mxy^{-1}$
 & $\nu_3 \cap \nu_{12}:mz^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{12}:my^{-1}z$
 & $\nu_3 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_8:x^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_3 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{12}:m$
 & $\nu_3 \cap \nu_{12}:mz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_3 \cap \nu_{12}:mz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{12}:y$
 & $\nu_3 \cap \nu_{12}:mz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{12}:z$
 & $\nu_3 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_{11}:yz^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_3 \cap \nu_{12}:mz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_{11}:1$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_3 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_4:1$
 & $\nu_4 \cap \nu_{11}:xz^{-1}$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_4:x$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{11}:xz^{-1}$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_7:x$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{11}:x$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{12}:z$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{12}:my^{-1}z$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_9:my^{-2}$
 & $\nu_9 \cap \nu_{12}:y$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{12}:z$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\[1em]

\caption*{{\normalsize {\sc Table} \thetable. Products $\mu^2$ in the
directed $A_\infty$-category $\Cat_{12}^\rightarrow$. The notation
means that $\mu^2(a_2,a_1) = \pm b$. \label{table:mu2}}}
\\
\end{longtable}
}

{ \smaller\smaller
\begin{longtable}[l]{lll|l|l}
 $a_1$ & $a_2$ & $a_3$ & $b$ & $\pm$ \\
 \hline
 $\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_1 \cap \nu_5:1$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_2 \cap \nu_6:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_4 \cap \nu_8:y^{-1}z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_5 \cap \nu_9:my^{-2}$
 & $-$ \\
$\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_7 \cap \nu_{11}:y^{-1}z^{-1}$
 & $+$ \\
$\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_8 \cap \nu_{12}:my^{-1}z$
 & $-$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $\nu_1 \cap \nu_8:z^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_2 \cap \nu_7:mx^{-1}z$
 & $+$ \\
$\nu_2 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_2 \cap \nu_9:my^{-2}$
 & $-$ \\
$\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_1 \cap \nu_8:z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:x$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_1 \cap \nu_8:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_1 \cap \nu_8:z^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_2 \cap \nu_9:my^{-2}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_2 \cap \nu_9:my^{-2}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_2 \cap \nu_9:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_3 \cap \nu_8:x^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $-$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:x$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_5 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_{10}:my^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $-$ \\
$\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $+$ \\
$\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_4 \cap \nu_{11}:z^{-1}$
 & $+$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:m$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $+$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_5 \cap \nu_{12}:mxy^{-2}$
 & $+$ \\
$\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_5 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_6 \cap \nu_{11}:z$
 & $+$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_{11}:yz^{-1}$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_{10}:mx^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:x$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_2:x$
 & $\nu_2 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:x$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_1 \cap \nu_5:x$
 & $\nu_5 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-2}$
 & $-$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{10}:my^{-1}$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_3:my^{-1}$
 & $\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_2 \cap \nu_{10}:mx^{-2}$
 & $-$ \\
$\nu_2 \cap \nu_6:my^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_2 \cap \nu_{10}:mx^{-1}y^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_6:mx^{-1}z^{-1}$
 & $\nu_6 \cap \nu_8:1$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_2 \cap \nu_{10}:mx^{-2}$
 & $+$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_8:xy^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:m$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $+$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $\nu_3 \cap \nu_{11}:1$
 & $+$ \\
$\nu_1 \cap \nu_8:xz^{-1}$
 & $\nu_8 \cap \nu_9:my^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{10}:mx^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $-$ \\
$\nu_3 \cap \nu_5:x^{-1}y$
 & $\nu_5 \cap \nu_{10}:my^{-1}$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:1$
 & $-$ \\
$\nu_3 \cap \nu_8:x^{-1}yz^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:x^{-1}y$
 & $-$ \\
$\nu_3 \cap \nu_8:z^{-1}$
 & $\nu_8 \cap \nu_{10}:mx^{-1}z$
 & $\nu_{10} \cap \nu_{11}:x$
 & $\nu_3 \cap \nu_{11}:1$
 & $+$ \\
$\nu_1 \cap \nu_5:x$
 & $\nu_5 \cap \nu_9:my^{-1}z^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $+$ \\
$\nu_1 \cap \nu_5:y$
 & $\nu_5 \cap \nu_9:mxy^{-2}z^{-1}$
 & $\nu_9 \cap \nu_{11}:y$
 & $\nu_1 \cap \nu_{11}:xz^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_5:1$
 & $\nu_5 \cap \nu_{12}:mxy^{-1}z^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_4:mx^{-1}z$
 & $\nu_4 \cap \nu_{11}:xz^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:mxy^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\
$\nu_2 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_8:y^{-1}z^{-1}$
 & $\nu_8 \cap \nu_{12}:m$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_7:mz$
 & $\nu_7 \cap \nu_{11}:z^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $+$ \\
$\nu_2 \cap \nu_7:mx^{-1}yz$
 & $\nu_7 \cap \nu_{11}:xy^{-1}z^{-1}$
 & $\nu_{11} \cap \nu_{12}:my^{-1}$
 & $\nu_2 \cap \nu_{12}:my^{-1}$
 & $-$ \\[1em]

\caption*{{\normalsize {\sc Table} \thetable. Products $\mu^3$ in
$\Cat_{12}^\rightarrow$, written in the same way as in the previous
table. \label{table:mu3}}}
\\
\end{longtable}
}

%\bibliographystyle{amsplain}
%\bibliography{../../../bib/all,../../../bib/new}
\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
} \providecommand{\href}[2]{#2}
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