text:
\section{Floer theory and Fukaya categories} \label{s:floer-theory}

The goal of this section is to set up the variant of Floer theory that
will be used in this paper. In particular we will show how choose the
auxiliary parameters of this theory so that the Fukaya category
becomes a {\em weakly filtered} $A_{\infty}$-category.

Let $(M, \omega)$ be a symplectic manifold, either closed or convex at
infinity. We always assume $M$ to be connected. Denote by
$\mathcal{L}ag^{we}(M)$ the collection of all closed connected
Lagrangian submanifolds $L \subset M$ that are {\em weakly
  exact}. Recall that $L \subset M$ is weakly exact if for every
$A \in H_2^D(M,L)$ we have $\int_{A} \omega = 0$. (The group of
$H_2^D(M,L) \subset H_2(M,L)$ is by definition the image of the
Hurewicz homomorphism $\pi_2(M,L) \longrightarrow H_2(M,L)$).

Let $\mathcal{C} \subset \mathcal{L}ag^{we}$ be a collection of weakly
exact Lagrangians. Unless explicitly stated otherwise, we henceforth
make the following mild assumption on $\mathcal{C}$, whenever $M$ is
not compact. There exists an open domain $U_0 \subset M$ with compact
closure, such that all Lagrangians $L \subset \mathcal{C}$ lie inside
$U_0$. For further use, also fix another open domain with compact
closure $U_1 \supset \overline{U}_0$ as well as an $\omega$-compatible
almost complex structure $J_{\textnormal{conv}}$ which is compatible
with the convexity of $M$ outside of $\overline{U}_1$.

Fix a base ring $R$ of characteristic $2$ (e.g. $R = \mathbb{Z}_2$)
and let $\Lambda$ be the Novikov ring over $R$ as defined
in~\eqref{eq:Nov-ring}. Denote by $\fuk(\mathcal{C})$ the Fukaya
category, with coefficients in $\Lambda$, whose objects are
$L \in \mathcal{C}$. We mostly follow here the implementation of the
Fukaya category due to Seidel~\cite{Se:book-fukaya-categ} with several
modifications that will be explained shortly.

As in~\cite{Se:book-fukaya-categ}, for every pair of Lagrangians
$L_0, L_1 \in \mathcal{C}$ we choose a Floer datum
$\mathscr{D}_{L_0, L_1} = (H^{L_0, L_1}, J^{L_0, L_1})$ consisting of
a Hamiltonian function
$H^{L_0, L_1}: [0,1] \times M \longrightarrow \mathbb{R}$ and a
time-dependent $\omega$-compatible almost complex structure
$J^{L_0,L_1} = \{J^{L_0,L_1}_t\}_{t \in [0,1]}$. In case $M$ is not
compact we require that outside of $U_1$ we have $H \equiv 0$ and
$J^{L_0,L_1}_t \equiv J_{\textnormal{conv}}$.

Denote by $\mathcal{O}(H^{L_0,L_1})$ the set of orbits
$\gamma:[0,1] \longrightarrow M$ of the Hamiltonian flow
$\phi^{H^{L_0,L_1}}_t$ generated by $H^{L_0,L_1}$ such that
$\gamma(0) \in L_0$ and $\gamma(1) \in L_1$. The Floer complex
$CF(L_0, L_1; \mathscr{D}_{L_0, L_1})$ is a free $\Lambda$-module
generated by the set $\mathcal{O}(H^{L_0,L_1})$:
\begin{equation} \label{eq:CF-1} CF(L_0, L_1; \mathscr{D}_{L_0, L_1})
  = \bigoplus_{\gamma \in \mathcal{O}(H^{L_0,L_1})} \Lambda
  \gamma.
\end{equation}
By adding additional structure it is possible to grade the Floer
complexes but we will not do that in this paper, hence work in an
ungraded setting.  The differential $\mu_1$ on the Floer complex is
defined by counting solutions $u$ of the Floer equation:
\begin{equation} \label{eq:floer-eq-1}
  \begin{aligned}
    & u: \mathbb{R} \times [0,1] \longrightarrow M, \quad u(\mathbb{R}
    \times 0) \subset L_0, \; u(\mathbb{R} \times 1) \subset L_1, \\
    & \partial_s u + J^{H_0,H_1}_t (u) \partial_t u =
    -\nabla H^{L_0,L_1}_t (u), \\
    & E(u) := \int_{-\infty}^{\infty} \int_{0}^{1} \lvert \partial_su
    \rvert^2 dt ds < \infty.
    \end{aligned}
\end{equation}
where $(s,t) \in \mathbb{R} \times [0,1]$. Here,
$H_t^{L_0,L_1}(x) := H^{L_0, L_1}(t,x)$ and $\nabla H_t^{L_0,L_1}$ is
the gradient of the function
$H_t^{L_0,L_1} : M \longrightarrow \mathbb{R}$ with respect to the
Riemannian metric
$g_t(\cdot, \cdot) = \omega(\cdot, J^{L_0, L_1}_t \cdot)$ associated
to $\omega$ and $J^{L_0, L_1}_t$. The quantity $E(u)$ in the last line
of~\eqref{eq:floer-eq-1} is the energy of a solution $u$ and we
consider only finite energy solutions. (Note also that the norm
$\lvert \partial_s u \rvert$ in the definition of $E(u)$ is calculated
with respect to the metric $g_t$.) Solutions $u$
of~\eqref{eq:floer-eq-1} are also called Floer trajectories.

For $\gamma_-, \gamma_+ \in \mathcal{O}(H^{L_0,L_1})$ consider the
space of {\em parametrized} Floer trajectories $u$ connecting
$\gamma_{-}$ to $\gamma_+$:
\begin{equation} \label{eq:floer-traj-space-1} \mathcal{M}(\gamma_-,
  \gamma_+; \mathscr{D}_{L_0,L_1}) = \Big\{ u \mid u \text{
    solves~\eqref{eq:floer-eq-1} and} \lim_{s \to \pm \infty} u(s,t) =
  \gamma_{\pm}(t) \Big\}.
\end{equation}
Note that $\mathbb{R}$ acts on this space by translations along the
$s$-coordinate. This action is generally free, with the only exception
being $\gamma_- = \gamma_+$ and the stationary solution
$u(s,t) = \gamma_-(t)$ at $\gamma_-$.

Whenever, $\gamma_- \neq \gamma_+$ we denote by
\begin{equation}
  \mathcal{M}^*(\gamma_-,
  \gamma_+; \mathscr{D}_{L_0,L_1}) := \mathcal{M}(\gamma_-,
  \gamma_+; \mathscr{D}_{L_0,L_1}) \big/ \mathbb{R}
\end{equation}
the quotient space (i.e. the space of non-parametrized solutions).  In
the case $\gamma_- = \gamma_+$ we define
$\mathcal{M}^*(\gamma_-, \gamma_-; \mathscr{D}_{L_0,L_1})$ in the same
way only that we omit the stationary solution at $\gamma_-$. (Note
that if $\omega$ is exact then
$\mathcal{M}(\gamma_-, \gamma_-; \mathscr{D}_{L_0,L_1})$ contains only
the stationary solution at $\gamma_-$, hence
$\mathcal{M}^* = \emptyset$. But this might not be the case if
$\omega$ is not exact.)

For a generic choice of Floer datum $\mathscr{D}$ the space
$\mathcal{M}^*(\gamma_-, \gamma_+; \mathscr{D}_{L_0,L_1})$ is a smooth
manifold (possibly with several components having different
dimensions). Moreover, its $0$-dimensional component
$\mathcal{M}^*_0(\gamma_{-}, \gamma_{+}; \mathscr{D}_{L_0,L_1})$ is
compact hence a finite set.

Define now
$\mu_1: CF(L_0, L_1; \mathscr{D}) \longrightarrow CF(L_0, L_1;
\mathscr{D})$ by
\begin{equation} \label{eq:mu-1} \mu_1(\gamma_{-}) :=
  \sum_{\gamma_{+}} \sum_{u} T^{\omega(u)} \gamma_+, \quad \forall \;
  \gamma_{-} \in \mathcal{O}(H^{L_0,L_1}),
\end{equation}
and extending linearly over $\Lambda$. Here, the outer sum runs over
all $\gamma_{+} \in \mathcal{O}(H^{L_0,L_1})$ and the inner sum over
all solutions
$u \in \mathcal{M}^*_0(\gamma_{-}, \gamma_{+};
\mathscr{D}_{L_0,L_1})$.  The term $\omega(u)$ is a shorthand notation
for the symplectic area of a Floer trajectory $u$, namely
$\omega(u) := \int_{\mathbb{R} \times [0,1]} u^*\omega$.

It is well known that $\mu_1$ is a differential and we denote the
homology of $CF(L_0,L_1; \mathscr{D}_{L_0,L_1})$ by
$HF(L_0,L_1; \mathscr{D}_{L_0,L_1})$. This homology is independent of
the choice of the Floer datum in the sense that for every two regular
choices of Floer data $\mathscr{D}_{L_0,L_1}$,
$\mathscr{D}'_{L_0,L_1}$ there is a canonical isomorphism
$\psi_{\mathscr{D}, \mathscr{D}'}: HF(L_0,L_1; \mathscr{D}_{L_0,L_1})
\longrightarrow HF(L_0,L_1; \mathscr{D}'_{L_0,L_1})$ which form a
directed system. Therefore we can regard this collection of
$\Lambda$-modules as one and denote it by $HF(L_0,L_1)$. Note however,
that the canonical isomorphisms $\psi_{\mathscr{D}, \mathscr{D}'}$ do
not preserve action-filtrations in general, hence there is no meaning
to $H(CF^{\leq \alpha}(L_0,L_1))$ without specifying the Floer datum.

\medskip

The higher operations $\mu_d$, $d \geq 2$, follow the same scheme as
in~\cite{Se:book-fukaya-categ}, with the main difference being that we
work over the Novikov ring $\Lambda$.

More precisely, we first make a choice of strip-like ends along the
compactification of the moduli-spaces $\mathcal{R}^{d+1}$, $d \geq 2$,
of disks with $(d+1)$-boundary punctures. For every
$r \in \mathcal{R}^{d+1}$ denote by $S_r$ the punctured disk
corresponding to $r$. Denote the punctures by $\zeta_i$,
$i=0, \ldots, d$, going in {\em clockwise} direction. The puncture
$\zeta_0$ will be called the exit and $\zeta_1, \ldots, \zeta_d$ the
entry punctures. We denote the arc along $\partial S_r$ connecting
$\zeta_i$ to $\zeta_{i+1}$ by $C_i$, with the convention that
$\zeta_{d+1} := \zeta_0$. (Of course, $\zeta_i$ and $C_i$ all depend
on $r$ but we suppress this from the notation.)

Next we make a choice of perturbation data
$\mathscr{D}_{L_0, \ldots, L_d} = (K^{L_0, \ldots, L_d}, J^{L_0,
  \ldots, L_d})$ for every tuple of $d+1$ Lagrangians
$L_0, \ldots, L_d \in \mathcal{C}$. The first item in the perturbation
data is a family of $1$-forms
$K^{L_0, \ldots, L_d} = \{ K^{L_0, \ldots, L_r}_r\}_{r \in
  \mathcal{R}^{d+1}}$, parametrized by $r \in \mathcal{R}^{d+1}$, with
values in the space of Hamiltonian functions
$M \longrightarrow \mathbb{R}$. The second one is a family of
$\omega$-compatible domain-dependent almost complex structures on $M$,
$J^{L_0, \ldots, L_d} = \{ J_r^{L_0, \ldots, L_d} \}_{r \in
  \mathcal{R}^{d+1}}$, parametrized by $r \in \mathcal{R}^{d+1}$. In
other words for every $r \in \mathcal{R}^{d+1}$,
$J^{L_0, \ldots, L_d}_r$ is itself a family
$\{ J^{L_0, \ldots, L_d}_{r,z} \}_{z \in S_r}$ of $\omega$-compatible
almost complex structure on $M$, parametrized by $z \in S_r$.

The perturbation data are required to satisfy several additional
conditions. The first one is that along each of the strip-like ends
the perturbation data coincides with the Floer data associated to the
pair of Lagrangians corresponding to that end. More precisely, along
the strip like end corresponding to the puncture $\zeta_i$ of $S_r$ we
have
\begin{equation} \label{eq:pert-floer-da} K^{L_0, \ldots, L_r}_r =
  H_t^{L_{i-1}, L_i} dt, \quad J^{L_0, \ldots, L_d} = J_t^{L_{i-1},
    L_i}, \quad \forall \; 1\leq i \leq d+1,
\end{equation}
where we have used here the convention that $L_{d+1}=L_0$. Here
$(s,t)$ are the conformal coordinates corresponding to the strip-like
ends.

The second condition is that along the arc $C_i$ we have
\begin{equation} \label{eq:K=0-on-del-C_i} K^{L_0, \ldots,
    L_d}(\xi)|_{L_i} = 0, \quad \forall \; \xi \in T(C_i), \; i=0,
  \ldots, d.
\end{equation}

Moreover, the choices of strip-like ends and perturbation data along
$\mathcal{R}^{d+1}$ are required to be compatible with gluing and
splitting, or in the language of~\cite{Se:book-fukaya-categ}
``consistent''.  This means essentially that these choices extend smoothly over
the compactification $\overline{\mathcal{R}}^{d+1}$ of the space of
boundary-punctured disks. In turn, this requires that for every $d$,
the choices of strip-like ends and perturbation data done over
$\mathcal{R}^{d+1}$ are compatible with those that appear on all the
strata of the boundary $\partial \overline{\mathcal{R}}^{d+1}$ of the
compactification $\overline{\mathcal{R}}^{d+1}$ of
$\mathcal{R}^{d+1}$. We refer the reader
to~\cite[Chapter~9]{Se:book-fukaya-categ} for the precise definitions
and implementation.

In case $M$ is not compact we add the following conditions on the
perturbation data. For every $r \in \mathcal{R}^{d+1}$ and
$\xi \in T(S_r)$ the Hamiltonian function
$K_r^{L_0, \ldots, L_d}(\xi)$ is required to vanish outside of $U_1$
and $J_r^{L_0, \ldots, L_d} \equiv J_{\textnormal{conv}}$ outside of
$U_1$.

Once we have made consistent choices of strip-like ends and
perturbation data we define the higher operations $\mu_d$ for
$L_0, \ldots, L_d \in \mathcal{C}$ as follows. For
$r \in \mathcal{R}^{d+1}$, $z \in S_r$ and $\xi \in T_z(S_r)$ define
$Y_{r,z}(\xi)$ to be the Hamiltonian vector field of the function
$K^{L_0, \ldots, L_d}_{r,z}(\xi) : M \longrightarrow
\mathbb{R}$. Consider now the following Floer equation:
\begin{equation} \label{eq:floer-eq-2}
  \begin{aligned}
    & u: S_r \longrightarrow M, \quad u(C_i) \subset L_i \;\; \forall
    \;
    0 \leq i \leq d, \\
    & Du_z + J^{L_0, \ldots, L_d}_{r,z} (u) \circ Du_z \circ j_r =
    Y_{r,z}(u) + J^{L_0, \ldots, L_d}_{r,z} \circ Y_{r,z}(u) \circ
    j_r, \\
    & E(u) := \int_{S_r} \lvert Du - Y_r \rvert^2_J \sigma < \infty.
  \end{aligned}
\end{equation}
Here $j_r$ stands for the complex structure on $S_r$. The last
quantity in~\eqref{eq:floer-eq-2} is the energy of a solution $u$ and
we consider only solutions of finite energy. The definition of $E(u)$
involves an area form $\sigma$ on $S_r$ and the norm
$\lvert \cdot \rvert_J$ on the space of linear maps
$T_z(S_r) \longrightarrow T_{u(z)}(M)$ which is induced by $j_r$,
$J := J_r^{L_0, \ldots, L_d}$ and $\sigma$. See~\cite[Section~2.2,
Page~20]{McD-Sa:jhol} for the definition. Note that $E(u)$ does not
depend on $\sigma$.

Given orbits $\gamma_{-}^1, \ldots, \gamma_{-}^d, \gamma_+$ with
$\gamma_{-}^i \in \mathcal{O}(H^{L_{i-1}, L_i})$ and
$\gamma_+ \in \mathcal{O}(H^{L_0, L_d})$ define the space of so called
{\em Floer polygons} connecting $\gamma_{-}^1, \ldots, \gamma_{-}^d$
to $\gamma_+$ to be the space of all pairs $(r, u)$ with
$r \in \mathcal{R}^{d+1}$ and $u:S_r \longrightarrow M$ such that:
\begin{enumerate}
\item $u$ is a solution of~\eqref{eq:floer-eq-2}.
\item On the strip-like end corresponding to puncture $\zeta_i$ we
  have $\lim_{s \to \infty} u(s,t) = \gamma^i_{-}(t)$ for
  $1 \leq i \leq d$, where $(s,t) \in (-\infty, 0] \times [0,1]$ are
  the conformal coordinates on the strip-like end of $\zeta_i$.
\item On the strip like end corresponding to puncture $\zeta_0$ we
  have $\lim_{s \to \infty} u(s,t) = \gamma_+(t)$ for
  $1 \leq i \leq d$, where $(s,t) \in [0,\infty) \times [0,1]$ are the
  conformal coordinates on the strip-like end of $\zeta_0$.
\end{enumerate}
We denote this space by
$\mathcal{M}(\gamma^1_-, \ldots, \gamma^d_-, \gamma_+;
\mathscr{D}_{L_0, \ldots, L_d})$. For a generic choice of Floer data
and perturbation data this space is a smooth manifold and its
$0$-dimensional component
$\mathcal{M}_0(\gamma^1_-, \ldots, \gamma^d_-, \gamma_+;
\mathscr{D}_{L_0, \ldots, L_d})$ is compact hence a finite
set. (Recall that $d\geq 2$ hence we do not divide here by any
reparametrization group.)  Define now
\begin{equation} \label{eq:mu_d-def}
  \mu_d(\gamma^1_{-}, \ldots, \gamma^d_{-}) =
  \sum_{\gamma_+} \sum_{(r,u)} T^{\omega(u)} \gamma_+ \in CF(L_0, L_d;
  \mathscr{D}_{L_0, L_d}),
\end{equation}
where the first sum goes over all
$\gamma_+ \in \mathcal{O}(H^{L-0, L_d})$ and the second sum over all
$(r,u) \in \mathcal{M}_0(\gamma^1_-, \ldots, \gamma^d_-, \gamma_+;
\mathscr{D}_{L_0, \ldots, L_d})$. The term $\omega(u)$ stands for the
symplectic area of $u$, namely $\omega(u) = \int_{S_r} u^*\omega$.

Extending $\mu_d$ multi-linearly over $\Lambda$ we obtain an
operation:
$$\mu_d: CF(L_0, L_1; \mathscr{D}_{L_0, L_1}) \otimes \cdots \otimes
CF(L_{d-1}, L_d; \mathscr{D}_{L_{d-1}, L_d}) \longrightarrow CF(L_0,
L_d; \mathscr{D}_{L_0, L_d}).$$

With all the operations above $\fuk(\mathcal{C})$ becomes an
$A_{\infty}$-category. The proof of this is essentially the same as
the one in~\cite{Se:book-fukaya-categ}, the only difference is that
one needs to keep track of the areas appearing as exponents in the
variable $T$ of the Novikov ring.

\subsection{Units} \label{sb:units} We now explain briefly how to
construct homology units in $\fuk(\mathcal{C})$. More details can be
found in~\cite[Chapter~8]{Se:book-fukaya-categ}. Denote by
$S = D \setminus {\zeta_0}$ the unit disk punctured at one boundary
point $\zeta_0 \in \partial D$. Fix a strip-like end around $\zeta_0$
making $\zeta_0$ an exit puncture and let $(s,t)$ be the conformal
coordinates associated to this strip-like end. Let $L \in \mathcal{C}$
and $\mathscr{D}^{L,L}$ be a regular Floer datum for the pair $(L,L)$.
Pick a regular perturbation datum $\mathscr{D}_S = (K, J)$, as
described earlier with the only difference that $K$ and $J$ are are
defined only on $S$ (i.e. there is no dependence on any space like
$\mathcal{R}^{d+1}$). As before, we require that $D_S$ coincides with
the Floer datum $\mathscr{D}_{L,L}$ along the strip-like ends in the
sense of~\eqref{eq:pert-floer-da}. For $z \in S$, $\xi \in T_z(S)$
define $Y_z(\xi)$ as before. Given $\gamma \in \mathcal{O}(H^{L,L})$
consider the space $\mathcal{M}(\gamma; \mathscr{D}_S)$ of solutions
$u:(S, \partial S) \longrightarrow (M, L)$ of the last two lines of
equation~\eqref{eq:floer-eq-2}, with $S_r$, $Y_{r,z}$,
$J^{L_0, \ldots, L_d}_{r,z}$, $j_r$ replaced by $S$, $Y_z$, $J_z$ and
$i$ respectively, and such that along the strip-like end at $\xi_0$ we
have $\lim_{s \to \infty} u(s,t) = \gamma(t)$.  Define now an element
$e_L \in CF(L,L; \mathscr{D}_{L,L})$ by
\begin{equation} \label{eq:e_L} e_L := \sum_{\gamma \in
    \mathcal{O}(H^{L,L})} \sum_{u} T^{\omega(u)} \gamma,
\end{equation}
where the second sum runs over all solutions $u$ in the
$0$-dimensional component $\mathcal{M}_0(\gamma; \mathscr{D}_S)$ of
$\mathcal{M}(\gamma; \mathscr{D}_S)$. By standard theory $e_L$ is a
cycle and its homology class in $HF(L,L)$ is independent of the choice
of the Floer and perturbation data. Moreover, $[e_L] \in HF(L,L)$ is a
unit for the product induced by $\mu_2$.

\subsection{Families of Fukaya categories} \label{sb:fukaya-families}
The Fukaya category $\fuk(\mathcal{C})$ depends on all the choices
made - strip-like ends, Floer and perturbation data. We fix once and
for all a consistent choice of strip-like ends and denote by $E$ the
space of all consistent choices of perturbation data (compatible with
the fixed choice of strip-like ends). The space $E$ can be endowed
with a natural topology (and a structure of a Fr\'{e}chet manifold),
induced from a larger space in which one allows perturbation data in
appropriate Sobolev spaces
(see~\cite[Chapter~9]{Se:book-fukaya-categ}. The subspace
$E_{\textnormal{reg}} \subset E$ of regular perturbation data is
residual hence a dense subset.

The space $E$ contains a distinguished subspace \label{pp:E-reg}
$\mathcal{N} \subset E$ consisting of all consistent choices of
perturbation data $\mathscr{D} = (K, J)$ with $K \equiv 0$. Fix a
subset $E'_{\textnormal{reg}} \subset E_{\textnormal{reg}}$ whose
closure $\overline{E'}_{\textnormal{reg}}$ contains
$\mathcal{N}$. \label{pp:E'}

For $p \in E'_{\textnormal{reg}}$ we denote by $\fuk(\mathcal{C}; p)$
the associated Fukaya category with choice of perturbation data
$p$. We thus obtain a family of $A_{\infty}$-categories
$\{ \fuk(\mathcal{C}; p)\}_{p \in E'_{\textnormal{reg}}}$,
parametrized by $p \in E'_{\textnormal{ref}}$. It is well known that
this is a {\em coherent} system of $A_{\infty}$-categories
(see~\cite[Chapter~10]{Se:book-fukaya-categ}), in particular they are
all mutually quasi-equivalent.

In what follows we will sometimes use the following notation. Given
$L_0, L_1 \in \mathcal{C}$ and $p \in E'_{\textnormal{reg}}$ we write
$CF(L_0,L_1;p)$ for $CF(L_0,L_1; \mathscr{D}_{L_0,L_1})$, where
$\mathscr{D}_{L_0,L_1}$ is the Floer datum prescribed by the choice
$p \in E'_{\textnormal{reg}}$.

\subsection{Weakly filtered structure on Fukaya categories}
\label{sb:wf-fukaya}

We start by defining filtrations on the Floer complexes of pairs of
Lagrangians in $\mathcal{C}$. We follow here the general recipe
from~\S\ref{sbsb:filtration-mixed}.

Let $L_0, L_1 \in \mathcal{C}$ be two Lagrangians and
$\mathscr{D}_{L_0,L_1} = (H^{L_0,L_1}, J^{L_0,L_1})$ a Floer datum.
We define an ``action functional''
$$\mathbf{A}: CF(L_0, L_1; \mathscr{D}_{L_0,L_1}) \longrightarrow
\mathbb{R} \cup \{-\infty\}$$ as follows. Let
$P(T) = \sum_{i=0}^{\infty} a_i T^{\lambda_i} \in \Lambda$ with
$\lambda_0 < \lambda_i$ for all $i\geq 1$, and $a_0 \neq 0$. Let
$\gamma \in \mathcal{O}(H^{L_0,L_1})$ be a Hamiltonian orbit.  We
first define:
$$\mathbf{A}\bigl( P(T)\gamma \bigr) := -a_0 + \int_{0}^{1}
H^{L_0,L_1}_t(\gamma(t)) dt.$$ Now let
$\sum_{k=1}^l P_k(T) \gamma_k \in CF(L_0,L_1;\mathscr{D}_{L_0,L_1})$
be a general non-trivial element, where the $\gamma_k$'s are mutually
distinct.  We extend the definition of $\mathbf{A}$ to such an element
by:
$$\mathbf{A} \bigl( P_1(T)\gamma_1 + \cdots + P_l(T) \gamma_l \bigr)
:= \max \bigl\{ \mathbf{A} \bigl(P_k(T)\gamma_k\bigr) \mid k=1,
\ldots, l \bigr\}.$$ Finally, we put $\mathbf{A}(0) = -\infty$.

We now define a filtration on $CF(L_0,L_1;\mathscr{D}_{L_0,L_1})$
by:
\begin{equation} \label{eq:CF-filtration}
  CF^{\leq \alpha} (L_0,L_1; \mathscr{D}_{L_0,L_1}) :=
  \{x \in CF(L_0,L_1; \mathscr{D}_{L_0,L_1}) \mid \mathbf{A}(x) \leq
  \alpha\}.
\end{equation}
Before we go on, a quick remark regarding the Hamiltonian functions
$H^{L_0,L_1}$ in the Floer data is in order. We do {\em not} assume
that these functions are normalized (e.g. by requiring them to have
zero mean when $M$ is closed, or to be compactly supported when $M$ is
open). This means that if we replace $H^{L_0, L_1}$ by
$H^{L_0, L_1} + c(t)$ for some family of constants $c(t)$, we get the
same chain complex as $CF(L_0,L_1; \mathscr{D}_{L_0,L_1})$ but with a
shifted action-filtration. See Remark~\ref{r:ham-normalization} for
more on this issue and the reason for not insisting on normalizing
these Hamiltonian functions.

Returning to~\eqref{eq:CF-filtration}, it is easy to see that
$CF^{\leq \alpha} (L_0,L_1; \mathscr{D}_{L_0,L_1})$ is a
$\Lambda_0$-module (though {\em NOT} a $\Lambda$-module!).  The fact
that this filtration is preserved by $\mu_1$ and moreover, that it
provides $\fuk(\mathcal{C})$ with a structure of a weakly filtered
$A_{\infty}$-category are the subject of the following proposition.

\begin{prop} \label{p:wf-fukaya} There exists a choice
  $E'_{\textnormal{reg}} \subset E_{\textnormal{reg}} \setminus
  \mathcal{N}$ with
  $\overline{E'}_{\textnormal{reg}} \supset \mathcal{N}$ and such that
  the following holds. Let $p \in E'_{\textnormal{reg}}$ and
  $\fuk(\mathcal{C}; p)$ be the corresponding Fukaya category. Then
  there exist a sequence of non-negative real numbers
  $\bme(p) = (\epsilon_1(p)=0, \epsilon_1(p), \ldots, \epsilon_d(p),
  \ldots)$ and $u(p), \zeta(p), \kappa(p) \in \mathbb{R}_+$, depending
  on $p$, such that:
  \begin{enumerate}
  \item With the filtrations described above on the Floer complexes,
    $\fuk(\mathcal{C};p)$ becomes a weakly filtered
    $A_{\infty}$-category with discrepancy $\leq \bme(p)$.
  \item $\fuk(\mathcal{C}; p)$ is h-unital in the weakly filtered
    sense and there is a choice of homology units with discrepancy
    $\leq u(p)$.
  \item
    $\fuk(\mathcal{C}; p) \in \hyperlink{h:asmp-ue}{\lbue(\zeta(p))}$.
    \label{i:fuk-p-ue}
  \item Let $L \in \mathcal{C}$ and denote by $\mathcal{L}$ its Yoneda
    module. Then
    $\mathcal{L} \in \hyperlink{h:asmp-H}{\lbh_s(\kappa(p))}$.
    \label{i:fuk-p-us}
  \item For every $p_0 \in \mathcal{N} \subset E$ (see
    Page~\pageref{pp:E-reg}) we have:
    $$\lim_{p \to p_0} \epsilon_d(p) = 0, \; \; \forall \; d \geq 2, \quad
    \lim_{p \to p_0} u(p) = \lim_{p \to p_0} \zeta(p) = \lim_{p \to
      p_0} \kappa(p) = 0.$$
    \label{i:u-z-k-to-0}
  \end{enumerate}
\end{prop}

\begin{proof}
  We will only give a sketch of the proof, as most of the ingredients
  are standard in the theory (see e.g.~\cite{Se:book-fukaya-categ}).

  The precise definition of the set of choices of perturbation data
  $E'_{\textnormal{reg}}$ will be given in the course of the proof.

  We begin by showing that the filtration~\eqref{eq:CF-filtration} is
  preserved by $\mu_1$. Let $L_0, L_1 \in \mathcal{C}$ and
  $\mathscr{D}_{L_0,L_1} = (H^{L_0,L_1}, J^{L_0,L_1})$ be a Floer
  datum. Let $\gamma_{-},\gamma_{+} \in \mathcal{O}(H^{L_0,L_1})$ be
  two generators of $CF(L_0,L_1; \mathscr{D}_{L_0,L_1})$ and let
  $u \in \mathcal{M}_0(\gamma_{-}, \gamma_{+}; \mathscr{D}_{L_0,L_1})$
  be an element of the $0$-dimensional component of Floer trajectories
  connecting $\gamma_{-}$ to $\gamma_{+}$. By~\eqref{eq:mu-1}, the
  contribution of $u$ to $\mu_1(\gamma_{-})$ is
  $T^{\omega(u)} \gamma_{+}$.  We now have the following standard
  energy-area identity for solutions
  $u \in \mathcal{M}(\gamma_{-}, \gamma_{+}; \mathscr{D}_{L_0,L_1})$
  of the Floer equation:
  \begin{equation} \label{eq:E-omega-strips} E(u) = \omega(u) +
    \int_{0}^{1} H^{L_0,L_1}_t(\gamma_{-}(t)) dt - \int_{0}^{1}
    H^{L_0,L_1}_t(\gamma_{+}(t)) dt.
  \end{equation}
  It immediately follows that
  $$\mathbf{A}(T^{\omega(u)} \gamma_{+}) = -\omega(u) + \int_{0}^{1}
  H^{L_0,L_1}_t(\gamma_{+}(t)) dt \leq \int_{0}^{1}
  H^{L_0,L_1}_t(\gamma_{-}(t)) dt = \mathbf{A}(\gamma_{-}).$$ This
  shows that $\mu_1$ preserves the filtration~\eqref{eq:CF-filtration}
  on $CF(L_0,L_1; \mathscr{D}_{L_0,L_1})$.

  We move on to analyze the behavior of the higher operations $\mu_d$,
  $d \geq 2$, with respect to our filtration. Let
  $L_0, \ldots, L_d \in \mathcal{C}$ and
  $\mathscr{D}_{L_0, \ldots, L_d}$ be the corresponding perturbation
  data. Let $\gamma_{-}^i \in \mathcal{O}(H^{L_{i-1}, L_i})$,
  $\gamma_+ \in \mathcal{O}(H^{L_0, L_d})$, and
  $(r,u) \in \mathcal{M}_0(\gamma_{-}^1, \ldots, \gamma_{-}^d,
  \gamma_+; \mathscr{D}_{L_0, \ldots, L_d})$. The contribution of $u$
  to $\mu_d(\gamma_{-}^1, \ldots, \gamma_{-}^d)$ is
  $T^{\omega(u)} \gamma_{+}$.

  Similarly to~\eqref{eq:E-omega-strips} we have the following
  energy-area identity for solutions $u$ of~\eqref{eq:floer-eq-2}:
  \begin{equation} \label{eq:E-omega-polygons} E(u) = \omega(u) -
    \int_0^1 H^{L_0,L_d}_t(\gamma_+(t))dt + \sum_{j=1}^d \int_0^1
    H_t^{L_{j-1},L_j}(\gamma_{-}^{j}(t))dt - \int_{S_r} R^{K^{L_0,
        \ldots, L_d}}(u),
  \end{equation}
  where $R^{K^{L_0, \ldots, L_d}}$ is the curvature $2$-form on $S_r$
  associated to the perturbation form $K^{L_0,\ldots, L_d}$. In local
  conformal coordinates $(s,t) \in S_r$ it can be written as follows.
  Write $K^{L_0, \ldots, L_d} = F_{s,t}ds + G_{s,t} dt$ for some
  functions $F_{s,t}, G_{s,t}: M \longrightarrow \mathbb{R}$. Then
  \begin{equation} \label{eq:curvature-term} R_{s,t}^{K^{L_0, \ldots,
        L_d}} = \bigl( -\tfrac{\partial F_{s,t}}{\partial t} +
    \tfrac{\partial G_{s,t}}{\partial s} - \{F_{s,t}, G_{s,t}\} \bigr)
    ds \wedge dt,
  \end{equation}
  where $\{F_{s,t}, G_{s,t}\} := -\omega(X^{F_{s,t}}, X^{G_{s,t}})$ is
  the Poisson bracket of the functions $F_{s,t}$, $G_{s,t}$.

  We now need to bound the term
  $\int_{S_r} R^{K^{L_0, \ldots, L_d}}(u)$
  from~\eqref{eq:E-omega-polygons} independently of $(r,u)$. To this
  end, first note that for any given $r \in \mathcal{R}^{d+1},$ the curvature
  $R^{K^{L_0, \ldots, L_d}}$ vanishes identically along the strip-like
  ends of $S_r$ by assumption on the perturbation $1$-form. Next, let $\mathcal{S}^{d+1}$ be the universal
  family of disks with $d+1$ boundary punctures
  (see~\cite[Chapter~9]{Se:book-fukaya-categ},see
  also~\cite[Section~3.1]{Bi-Co:lcob-fuk}). This is a fiber bundle
  over $\mathcal{R}^{d+1}$ whose fiber over $r \in \mathcal{R}^{d+1}$
  is the surface $S_r$. The space $\mathcal{S}^{d+1}$ admits a partial
  compactification $\overline{\mathcal{S}}^{d+1}$ over
  $\overline{\mathcal{R}}^{d+1}$ and can be endowed with a smooth
  structure. Since the perturbation data
  $\mathscr{D}_{L_0, \ldots, L_d}$ was chosen consistently, the forms
  $K^{L_0, \ldots, L_d}$ extend to the partial compactification
  $\overline{S}^{d+1}$ over $\overline{\mathcal{R}}^{d+1}$. Now let
  $\mathcal{W} \subset \overline{S}^{d+1}$ be the union of all the
  strip-like ends corresponding to all the surfaces parametrized by
  $r \in \overline{\mathcal{R}}^{d+1}$. Then
  $\overline{\mathcal{S}}^{d+1} \setminus
  \textnormal{Int\,}\mathcal{W}$ is compact.  It follows that for all
  $(r,u) \in \mathcal{M}_0(\gamma_{-}^1, \ldots, \gamma_{-}^d,
  \gamma_+; \mathscr{D}_{L_0, \ldots, L_d})$ we have:
  \begin{equation} \label{eq:bound-R} \Bigl \lvert \int_{S_r} R^{K^{L_0,
        \ldots, L_d}}(u) \Bigr \rvert \leq \epsilon_d(K^{L_0, \ldots,
      L_d}),
  \end{equation}
  where $\epsilon_d(K^{L_0, \ldots, L_d})$ depends only on the
  $C^1$-norm of $K^{L_0, \ldots, L_d}$ (defined in the
  $\mathcal{S}^{d+1}$ as well as $M$ directions). Moreover, we have
  $\epsilon_d(K^{L_0, \ldots, L_d}) \longrightarrow 0$ as
  $K^{L_0, \ldots, L_d} \longrightarrow 0$ in the $C^1$-topology
  (along
  $\overline{\mathcal{S}}^{d+1} \setminus
  \textnormal{Int\,}\mathcal{W}$ and $M$).

  A few words are in order for the case when $M$ is not compact. In
  that case the arguments above continue to work due to our choice of
  perturbation data. More precisely, recall that we had two open
  domains $U_0, U_1 \subset M$ with compact closure, with
  $\overline{U}_0 \subset U_1$, and with the following properties: all
  Lagrangians $L \in \mathcal{L}$ lie in $U_0$ and outside of $U_1$ we
  have $\mathscr{D}^{L_0, \ldots, L_d} = (0, J_{\textnormal{conv}})$
  for all $r \in \mathcal{R}^{d+1}$. This implies that the Floer
  equations~\eqref{eq:floer-eq-1} and~\eqref{eq:floer-eq-2} become
  homogeneous at the points where $u(z) \in M \setminus U_1$. Since
  $(M, \omega, J_{\textnormal{conv}})$ is convex at infinity, the
  maximum principle implies that all solutions $u$ lie within one
  compact domain of $M$.  Thus the estimate~\eqref{eq:bound-R} follows
  by bounding the $C^1$-norm of $K^{L_0, \ldots, L_d}$ only along that
  compact domain.

  Coming back to the estimate~\eqref{eq:bound-R}, it follows
  from~\eqref{eq:E-omega-polygons} that:
  \begin{equation} \label{eq:eps-d-K} \mathbf{A}(T^{\omega(u)} \gamma_+)
    \leq \mathbf{A}(\gamma_{-}^1) + \cdots + \mathbf{A}(\gamma_{-}^d) +
    \epsilon_d(K^{L_0, \ldots, L_d}).
  \end{equation}
  In order to obtain a weakly filtered structure on
  $\fuk(\mathcal{C};p)$ we need to bound from above
  $\epsilon_d(K^{L_0, \ldots, L_d})$ uniformly in
  $L_0, \ldots, L_d \in \mathcal{C}$, so that the ultimate discrepancy
  $\epsilon_d(p)$ depends only on the choice of
  $p \in E'_{\textnormal{reg}}$. This is easily done by restricting
  the set $E'_{\textnormal{reg}}$ to choices of perturbation data
  $p = \{ \mathscr{D}_{L_0, \ldots, L_d} \}_{L_0, \ldots, L_d \in
    \mathcal{C}}$ for which the $C^1$-norms of the forms
  $K^{L_0, \ldots, L_d}$ are uniformly bounded (in
  $L_0, \ldots, L_d$). Since $E_{\textnormal{reg}} \subset E$ is
  residual it follows that the restricted set of choices
  $E'_{\textnormal{reg}}$ still has $\mathcal{N}$ in its closure.

  This concludes the proof that
  $\fuk(\mathcal{C}; p)_{p \in E'_{\textnormal{reg}}}$ is a family of
  weakly filtered $A_{\infty}$-categories, and that the bounds on
  their discrepancies $\bm{\epsilon}(p)$ have the property that for
  all $p_0 \in \mathcal{N}$ we have
  $\lim_{p \to p_0} \epsilon_d(p) = 0$ for every $d \geq 2$.

  We now turn to the statements about the unitality of the categories
  $\fuk(\mathcal{C}; p)$ and their Yoneda modules. Let
  $p \in E'_{\textnormal{reg}}$. Fix $L \in \mathcal{C}$ and let
  $\mathscr{D}_{L,L} = (H^{L,L}, J^{L,L})$ be the Floer datum of
  $(L,L)$ prescribed by $p$. Recall that a homology unit
  $e_L \in CF(L,L;\mathscr{D}_{L,L})$ can be defined
  by~\eqref{eq:e_L}. Let $S = D \setminus \{\zeta_0\}$ and
  $\mathscr{D}_{S} = (K,J)$ as in~\S\ref{sb:units}.  Let
  $\gamma \in \mathcal{O}(H^{L,L})$ and
  $u \in \mathcal{M}_0(\gamma; \mathscr{D}_S)$. According
  to~\eqref{eq:e_L} the contribution of $\gamma$ and $u$ to $e_L$ is
  $T^{\omega(u)}\gamma$. The energy-area identity for $u$ gives:
  $$E(u) = \omega(u) - \int_0^1 H_t^{L,L}(\gamma(t))dt  -
  \int_S R^{K}(u),$$ where $R^{K}(u)$ is the curvature associated to
  the $1$-form $K$ from the perturbation datum $\mathscr{D}_S$ and is
  defined in a similar way as in~\eqref{eq:curvature-term}. Note that
  we can choose the perturbation datum $\mathscr{D}_S = (K, J)$ such
  that the $C^1$-norm of the $1$-form $K$ is of the same order size as
  the $C^1$-norm of $H^{L,L}$ (i.e.
  $\lVert K \rVert_{C^1} \leq C \lVert H^{L,L} \rVert_{C^1}$ for some
  constant $C$). By doing that we obtain
  $\lvert \int_{S}R^{K}(u) \rvert \leq C' \lVert H^{L,L} \rVert_{C^1}$
  for some constant $C'$. It follows that
  $\mathbf{A}(e_L) \leq C' \lVert H^{L,L} \rVert_{C^1}$. By
  restricting all the Hamiltonians $H^{L,L}$, for all
  $L \in \mathcal{C}$, to have a uniformly bounded $C^1$-norm we
  obtain one constant $u(p)$ (that depends on the choice $p$) such
  that for all every $L \in \mathcal{C}$ we have
  $\mathbf{A}(e_L) \leq u(p)$. Moreover, $u(p) \longrightarrow 0$ as
  $p \longrightarrow p_0 \in \mathcal{N}$ in the $C^1$-topology.  This
  proves the statement about the discrepancy of the units in
  $\fuk(\mathcal{C}; p)$.

  We now turn to proving statements~(\ref{i:fuk-p-ue})
  and~(\ref{i:fuk-p-us}) of the proposition and the corresponding
  claims on $\zeta(p)$ and $\kappa(p)$ from
  statement~(\ref{i:u-z-k-to-0}).

  Let $L, L' \in \mathcal{C}$. Choose $S$, $\mathscr{D}_S$ and define
  $e_L$ as explained above. Denote by $\mathscr{D}_{L,L,L'}$ be the
  perturbation datum of the triple $(L,L,L')$ as prescribed by
  $p$. Consider also a disk
  $S' = D \setminus \{\zeta_0', \zeta_1', \zeta_2'\}$ with three
  boundary punctures, ordered clockwise along $\partial D$. We fix
  strip-like ends near these three punctures such that
  $\zeta_0', \zeta_1'$ are entries and $\zeta_2'$ is an exit.
  Consider a $1$-parametric family
  $(\{S''_{\tau}\}_{\tau \in (0,1]}, j_{\tau})$ of surfaces (endowed
  with complex structures) obtained by performing gluing $S$ and $S'$
  at the points $\zeta_0$, $\zeta_0'$ respectively. We construct this
  family so that $S''_{\tau} \longrightarrow S \coprod S'$ as
  $\tau \longrightarrow 0$ and $S''_1 = \mathbb{R} \times [0,1]$ is
  the standard strip. Next, we choose a generic family
  $\{\mathscr{D}_{\tau}\}_{\tau \in (0,1]}$ of perturbation data over
  the family $\{S''_{\tau}\}_{\tau \in (0,1]}$ such that:
  \begin{enumerate}
  \item for $\tau \longrightarrow 0$, $\mathscr{D}_{\tau}$ converges
    to $\mathscr{D}_{S}$ on the $S$ component and
    $\mathscr{D}_{L,L,L'}$ on the $S''$ component.
  \item $\mathscr{D}_1 = \mathscr{D}_{L,L}$.
  \end{enumerate}
  As the family $\{\mathscr{D}_{\tau}\}_{\tau \in (0,1]}$ is generic,
  none of the elements in the $\mathscr{D}_{\tau}$, $\tau <1$ is
  invariant under reparametrization by any non-trivial automorphism
  $\sigma \in Aut(S_{\tau})$.

  Let $\gamma, \lambda \in \mathcal{O}(H^{L,L'})$ and consider the
  space $\mathcal{M}(\gamma, \lambda ; \{ \mathscr{D}_{\tau} \})$ of
  all pairs $(\tau, u)$, with $\tau \in (0,1]$ and
  $u:S_{\tau} \longrightarrow M$ a solution of the Floer
  equation~\eqref{eq:floer-eq-2} with the obvious modifications:
  namely, the lower part of $\partial S_{\tau}$ is mapped by $u$ to
  $L$ and the upper one to $L$', $u$ converges to $\gamma$ at the
  entry $\zeta_1'$ and to $\lambda$ at the exit $\zeta_2'$,
  $(S_r, j_r)$ is replaced by $(S_{\tau}, j_{\tau})$, and $J_{r,z}$
  and $Y_{r,z}$ are replaced by the corresponding structures from
  $\mathscr{D}_{\tau}$.

  Assume that $\gamma \neq \lambda$, and consider the $0$-dimensional
  component
  $\mathcal{M}_0(\gamma, \lambda ; \{ \mathscr{D}_{\tau} \})$. This is
  compact $0$-dimensional manifold hence a finite set. It gives rise
  to a map
  \begin{equation} \label{eq:ch-homotopy-phi-1}
    \begin{aligned}
      & \Phi: CF(L,L'; \mathscr{D}_{L,L'}) \longrightarrow CF(L,L';
      \mathscr{D}_{L,L'}), \\
      & \Phi(\gamma) := \sum_{\lambda} \sum_{(\tau,u)} T^{\omega(u)}
      \lambda, \;\; \forall \gamma \in \mathcal{O}(H^{L,L'}),
    \end{aligned}
  \end{equation}
  where the outer sum is over all $\lambda \in \mathcal{O}(H^{L,L'})$
  with $\lambda \neq \gamma$ and the second sum is over all
  $(\tau, u) \in \mathcal{M}_0(\gamma, \lambda ; \{ \mathscr{D}_{\tau}
  \})$. We extend the formula in the 2'nd line
  of~\eqref{eq:ch-homotopy-phi-1} linearly over $\Lambda$.

  We claim that the following formula holds:
  \begin{equation} \label{eq:ch-homotopy-phi-2} \mu_2(e_L, x) = x +
    \mu_1 \circ \Phi(x) + \Phi \circ \mu_1(x), \; \; \forall \; x \in
    CF(L,L'; \mathscr{D}_{L,L'}),
  \end{equation}
  i.e. $\Phi$ is a chain homotopy between the map $\mu_2(e_L, \cdot)$
  and the identity.

  To prove this, let $\gamma, \gamma_+ \in \mathcal{O}(H^{L,L'})$ and
  consider the $1$-dimensional component
  $\mathcal{M}_1(\gamma, \gamma_+ ; \{ \mathscr{D}_{\tau} \})$ of the
  space $\mathcal{M}(\gamma, \gamma_+ ; \{ \mathscr{D}_{\tau}
  \})$. This space admits a compactification
  $\overline{\mathcal{M}}_1(\gamma, \gamma_+ ; \{ \mathscr{D}_{\tau}
  \})$ which is a $1$-dimensional manifold with boundary. The elements
  in the boundary of this space fall into four types:
  \begin{enumerate}
  \item Elements corresponding to the splitting of $S''$ into $S$ and
    $S'$ at $\tau=0$. These can be written as pairs $(u_S, u_{S'})$
    with $u_S \in \mathcal{M}_0(\gamma'; \mathscr{D}_S)$ for some
    $\gamma' \in \mathcal{O}(H^{L,L})$ and
    $u_{S'} \in \mathcal{M}_0(\gamma', \gamma, \gamma_+;
    \mathscr{D}_{L,L,L'})$. \label{i:split-1}
  \item Elements corresponding to splitting of $S_{\tau}$ at some
    $0 < \tau_0<1$ into a Floer strip $u_0$ followed by a solution
    $u_1:S_{\tau_0} \longrightarrow M$ of the Floer equation for the
    perturbation datum $\mathscr{D}_{\tau_0}$. More precisely, these
    can be written as $(\tau_0,u_0, u_1)$ with $0<\tau_0<1$,
    $u_0 \in \mathcal{M}^*(\gamma, \gamma'; \mathscr{D}_{L,L'})$ and
    $u_1 \in \mathcal{M}(\gamma', \gamma_+; \mathscr{D}_{\tau_0})$ for
    some $\gamma' \in \mathcal{O}(H^{L,L'})$. \label{i:split-2}
  \item The same as~(\ref{i:split-2}) only that the splitting occurs
    in reverse order, namely first an element of
    $\mathcal{M}(\gamma, \gamma'; \mathscr{D}_{\tau_0})$ followed by
    an element of
    $\mathcal{M}^*(\gamma', \gamma_+;
    \mathscr{D}_{L,L'})$. \label{i:split-3}
  \item Elements corresponding to $\tau=1$. These are
    $u: \mathbb{R} \times [0,1] \longrightarrow M$ that belong to the
    space $\mathcal{M}_0(\gamma, \gamma_+; \mathscr{D}_{L,L'})$ or in
    other words elements of the $0$-dimensional component of the space
    $\mathcal{M}(\gamma, \gamma_+; \mathscr{D}_{L,L'})$ of {\em
      parametrized} Floer trajectories connecting $\gamma$ to
    $\gamma_+$. The latter space has a $0$-dimensional component if
    and only if $\gamma=\gamma_+$ in which case that component
    contains only the stationary trajectory at $\gamma$. Summing up,
    this type of boundary point occurs if and only if
    $\gamma = \gamma_+$ and $u$ is the stationary solution at
    $\gamma$.
    \label{i:split-4}
  \end{enumerate}
  Summing up over all these four possibilities (for every given area
  of solutions $u$) yields formula~\eqref{eq:ch-homotopy-phi-2}. Note
  that the first term (i.e. the summand $x$) on the right-hand side
  of~\eqref{eq:ch-homotopy-phi-2} comes exactly from the boundary
  points of type~(\ref{i:split-4}).

  To conclude the proof we only need to estimate the shift in action
  (or filtration) of the chain homotopy $\Phi$. This is done in a
  similar way to the argument used above to estimate $\bme(p)$, namely
  by using an energy-area identity as
  in~\eqref{eq:E-omega-polygons}. Indeed we can choose the
  perturbation data $\mathscr{D}_S$ and $\mathscr{D}_{\tau}$,
  $0< \tau < 1$, to be of the same size order (in the $C^1$-norm) as
  the Hamiltonian $H^{L,L'}$, hence the curvature term in the
  energy-area identity can be bounded by a constant $C(H^{L,L'})$ that
  depends on $H^{L,L'}$ and such that $C(H^{L,L'}) \longrightarrow 0$
  as $H^{L,L'} \longrightarrow 0$ in the $C^1$-topology. By taking all
  the Hamiltonians $H^{L,L'}$ for all $L,L' \in \mathcal{C}$ to be
  uniformly bounded in the $C^1$-topology we obtain a uniform bound
  $\kappa(p)$ on the action shift of the chain homotopy $\Phi$ that
  holds for all pairs $L,L' \in \mathcal{C}$ and such that
  $\kappa(p) \longrightarrow 0$ as
  $p \longrightarrow p_0 \in \mathcal{N}$.  This shows that the Yoneda
  module $\mathcal{L}$ satisfies
  Assumption~$\hyperlink{h:asmp-H}{\lbh_s(\kappa(p))}$. By taking
  $L'=L$ it also follows immediately that
  $\mu_2(e_L,e_L) = e_L + \mu_1(c)$ for some chain $c$ with
  $\mathbf{A}(c) \leq \kappa(p)$, hence
  $\fuk(\mathcal{C}; p) \in \hyperlink{h:asmp-ue}{\lbue(\kappa(p))}$
  (so we can actually take $\zeta(p) = \kappa(p)$).
\end{proof}

\begin{rem} \label{r:ham-normalization} In some variants of Floer
  theory it is common to normalize the Hamiltonian functions involved
  in the definition of the Floer complexes. For example, when the
  ambient manifold is closed one often normalizes the Hamiltonian
  functions to have zero mean, and for open manifolds one requires the
  Hamiltonian functions to have compact support. This solves the
  ambiguity of adding constants to the Hamiltonian functions and
  consequently provides a ``canonical'' way to define action
  filtrations. This especially makes sense when one aims to construct
  invariants of Hamiltonian diffeomorphisms (or flows) by means of
  filtered Floer homology. See e.g. the theory of spectral
  numbers~\cite{Sc:action-spectrum, Oh:spec-inv-1, Oh:spec-inv-2,
    Oh:spec-inv-3, En-Po:calqm}, see
  also~\cite{Vi:generating-functions-1} for an earlier approach via
  generating functions.

  While we could have normalized the Hamiltonian functions in the
  Floer and perturbation data, we have opted not to do so. At first
  glance, this might seem to have odd implications. For example,
  suppose that $p_1, p_2 \in E'_{\text{reg}}$ are two choices of
  perturbation data such that $p_2$ is obtained by adding a
  (different) constant to each of the Hamiltonian functions (or forms)
  in the perturbation data from $p_1$. Clearly, the Fukaya categories
  $\fuk(\mathcal{C}; p_1)$ and $\fuk(\mathcal{C}; p_2)$ are precisely
  the same, but they have completely different (and generally
  unrelated) weakly filtered structures.

  Our justification for not imposing any normalization on the
  Hamiltonian functions is that their role is purely auxiliary, and
  moreover, ideally we would like to make them arbitrarily small. More
  specifically, the focus of our study is the collections of
  Lagrangians $\mathcal{C}$ and its Fukaya category, whereas the
  Hamiltonian functions in the Floer data serve only as perturbations
  whose sole purpose is technical, namely to set up the Floer theory
  so that it fits into a (infinite dimensional) Morse theoretic
  framework. In reality, we view the Hamiltonian perturbations as
  quantities that can be taken arbitrarily small and consider families
  of Fukaya categories parametrized by choices of perturbations that
  tend to $0$. (See e.g. Proposition~\ref{p:wf-fukaya}.)

  In fact, our theory would become simpler and cleaner if we
  could set up the Fukaya category without appealing to any
  perturbations at all. If this were possible (which means that all
  the Floer trajectories and polygons are unperturbed
  pseudo-holomorphic curves) our Fukaya categories would be
  genuinely filtered rather than only ``weakly
  filtered''. (See~\cite{FO3:book-vol1, FO3:book-vol2} for a
  ``perturbation-less'' construction of an $A_{\infty}$-algebra
  associated to a single Lagrangian.)

  Another point related to the matter of normalization is that when
  extending our theory to Lagrangian cobordisms
  (see~\S\ref{sb:ext-lcob}) we are forced to work with non-compactly
  supported Hamiltonian perturbations. While one could have attempted
  a different sort of normalization in that case (suited for the class
  of non-compactly supported perturbations used for cobordisms), we
  will not do that for the very same reasons as those for not doing it for
  $\fuk(\mathcal{C};p)$.
\end{rem}


text:
\begin{document}

\title{Quantum geometry from $2+1$ AdS quantum gravity on the torus}

\author{J. E. Nelson \and R. F. Picken}

\institute{J. E. Nelson (corresponding author) \at Dipartimento di Fisica Teorica, Universit\`a
degli Studi di Torino\\
and\\
INFN, Sezione di Torino\\
via Pietro Giuria 1, 10125 Torino, Italy\\
  \email{nelson@to.infn.it}
\and  R. F. Picken \at Departamento de Matem\'{a}tica\\
and\\
CAMGSD - Centro de An\'{a}lise Matem\'{a}tica, Geometria e Sistemas Din\^{a}micos\\
Instituto Superior T\'{e}cnico, Technical University of Lisbon\\
Avenida Rovisco Pais, 1049-001 Lisboa, Portugal\\
\email{rpicken@math.ist.utl.pt}
}

\date{Received: date / Accepted: date}

\maketitle

\begin{abstract}

Wilson observables for $2+1$ quantum gravity with negative cosmological constant, when
the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and
their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our
study of this bracket, which can be interpreted as a $q$-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.

\keywords{Wilson observables \and $2+1$ quantum gravity \and Goldman bracket \and area phases}
\PACS{02.40.Gh \and 04.60.Kz }
\subclass{81R50 \and 83C45 }
\end{abstract}

\section{Introduction} \label{intro} \noindent Interest in the
construction of a
commutator for Wilson loops associated to intersecting loops
on surfaces arose from the study of quantum gravity for spacetimes of
dimension $2+1$ with negative cosmological constant \cite{NR1,NRZ}.
When the spatial manifold is a torus, piecewise linear (PL) paths in
the covering space
${\IR}^2$ were used to represent loops on the torus, and a quantum
connection with noncommuting components was introduced to construct quantum
holonomies whose traces quantize the Poisson bracket structure
of $2+1$ gravity \cite{NP1,qmp,goldman}. This may be regarded as a
quantum version of the Goldman bracket (see Section \ref{sec3})
for loops on a surface. The purpose of this paper is to describe
progress in the study of this bracket

The classical action of $2+1$ gravity with negative cosmological
constant $\Lambda$ is related to Chern-Simons theory for the gauge
group $SL(2,\IR)\times SL(2,\IR)$ \cite{wit}. The corresponding classical phase
space is the moduli space of flat $SL(2,\IR)\times SL(2,\IR)$
connections on the spatial manifold, which here is the torus
$\IT^2=\IR^2/\IZ^2$. This moduli space represents the fundamental
group of the torus, and has two generators, or generating cycles,
say $\gamma_1,\, \gamma_2$ which satisfy the single relation
\be
\ga_1^{\vphantom{-1}}\cdot\ga_2^{\vphantom{-1}}\cdot\ga_1^{-1}\cdot\ga_2^{-1}
  = {\II}.
\label{gp}
\ee
i.e. the generators $\gamma_1,\, \gamma_2$ commute.  The moduli space
therefore consists of pairs of commuting $SL(2,\IR)$ matrices (up to
simultaneous conjugation by
an element of $SL(2,\IR)$), representing the holonomies of the flat
connection along the two cycles of the torus (up to gauge
transformation). It was studied in \cite{mod}.

The diagonal, hyperbolic--hyperbolic sector of this moduli space is
the relevant sector for $2+1$ gravity \cite{Ezawa1,Ezawa2}, and each
sector can be parametrised by constant connections
\cite{goldman,mikpic} on the torus of the form
\be
A= (r_1 dx + r_2 dy)
\left(\begin{array}{clcr} 1& 0 \\0& -1
\end{array}\right)
\label{conn}
\ee
where $x,\,y$ are coordinates on the torus (with periodicity $1$), or
on its covering space $\IR^2$. The connections are constant in that
$r_1,\,r_2$ are global constants (independent of $x$ and $y$). After integration,
this yields the holonomies:
\be
U_i = \exp \int_{\ga_i} A = \left(\begin{array}{clcr}e^{{r_i}}& 0 \\0& e^{-
{r_i}}\end{array}\right) \quad i=1,2.
\label{hol1}
\ee
The traces of the holonomies \rref{hol1} are precisely the variables used in \cite{NR1}
to quantize this model of $2+1$ gravity. The relation between them  and the ADM variables
was described in \cite{CN}. Therefore our present approach can be related to standard descriptions, but a richer picture is obtained by including more general Wilson loops, derived from the quantum connection \rref{conn}.

In Chern-Simons theory components of the connection $A$ (equation \rref{conn}) satisfy
non-trivial Poisson brackets. For the global parameters
$r_i,\,i=1,2$ this implies the Poisson bracket
\be
\left\{ r_1, r_2\right\} = -\frac{\sqrt {-\Lambda}}{4},
\ee
which is quantized by the commutator
\begin{equation}
[\hat{r}_1, \hat{r}_2] = \hat{r}_1\hat{r}_2 - \hat{r}_2\hat{r}_1 = \frac{i\hbar \sqrt{-\Lambda}}{4}.
\label{rcommrel}
\end{equation}

Further, the commutator \rref{rcommrel} implies that the {(now \it
quantum matrices}) $\hat U_1, \hat
U_2$ \rref{hol1} must satisfy {\it by both matrix and operator multiplication}, the
$q$--commutation relation
\beq
\hat U_1 \hat U_2 = q \hat U_2 \hat U_1
\label{fund2}
\eeq
where the $q$ parameter\footnote{Note that in $q$ the exponent of
$\exp$ is dimensionless, when all physical constants are taken into account.} is $q=\exp (- \frac {i \hbar \sqrt{-\Lambda}}{4})$ i.e. the matrices $\hat U_1, \hat U_2$ form a matrix--valued Weyl pair. Equation \rref{fund2} can be understood as a deformation of equation \rref{gp}. Analogous commutators for a general class of Wilson loops give rise to the quantum Goldman bracket, our central theme.

The plan of the paper is as follows. In Section \ref{sec2} some interesting features of the emerging quantum geometry are reviewed. These are the representation of loops on the torus by piecewise--linear (PL) paths between integer points in $\IR^2$, constant matrix--valued connections applied to a much larger class of loops, and a definition of a $q$--deformed representation of the fundamental group where signed area phases relate the quantum matrices assigned to homotopic loops. In Section \ref{sec3} intersections and reroutings, and two quantizations of the Poisson bracket between paths are described: the `direct' quantization and the `refined' quantization. In Section \ref{sec4} we describe new results which represent progress in understanding the quantum nature of intersections (expressed as commutators) of straight paths (i.e. straight in $\IR^2$) and `crooked' paths (those resulting from previous reroutings) by using the concepts of integer points and relative phases for a crooked rerouting.  As a consequence we obtain a quantum version of a formula for the area of a polygon with integer
vertices. We also show the equivalence of the refined and direct quantization of the bracket for straight paths, and the antisymmetry of the refined bracket.

\section{Piecewise linear paths and quantum holonomy matrices}\label{sec2}
\subsection{Piecewise linear paths}\label{sub21}
Loops (closed paths) on the torus $\IT^2=\IR^2/\IZ^2$ can be conveniently identified with paths on its covering space ${\IR}^2$, i.e. we represent all loops on the torus by piecewise-linear (PL) paths on ${\IR}^2$ i.e. the $(x,y)$ plane, between integer points $(m,n)\in \IZ^2$. All these integer points are identified, and correspond to the same point on the torus. It follows that a path in ${\IR}^2$ representing a loop on the torus can be replaced by any parallel path starting at a different integer point, e.g. the path from $O=(0,0)$ to $(1,1)$ represents the same loop as the path from $(2,0)$ to $(3,1)$, as shown in Figure \ref{fund}.

\begin{figure}
\begin{center}
\includegraphics[width=16pc]{fund.eps}
\end{center}
\caption{\label{fund} The path from $O$ = $(0,0)$ to $(1,1)$ and the path from $(2,0)$ to $(3,1)$ represent the same loop on the torus}
\end{figure}
A natural subclass of paths in ${\IR}^2$ are those straight paths denoted $p=(m,n)$ that start at the origin $O =(0,0)$ and end at an integer point $(m,n)\in \IZ^2$. They generalise the cycles $\gamma_1,\, \gamma_2$ (corresponding to the paths $(1,0)$ and $(0,1)$ respectively). When the path $(m,n)$ is a multiple of another integer path, i.e. $(m,n)=c(m',n')$, where $m,n,c,m',n'$ are all integers, with $c\geq 2$, we say it is {\it reducible}. Otherwise it is irreducible.

The identification between loops on the torus $\IT^2=\IR^2/\IZ^2$ and PL paths in $\IR^2$ can be  further understood by considering the concept of fundamental reduction introduced in \cite{goldman}. There we introduced this concept in order to better study the intersections between paths $p_1$ and $p_2$. It consists of reducing one or more paths to a fundamental domain of
$\IR^2$, namely the unit square with vertices $(0,0), (1,0)$ and $(1,1), (0,1)$. This works as follows: a path that passes through more than one cell (a unit square in $\IR^2$) consists of ordered segments, each of which passes through only one cell. The fundamental reduction is obtained by superimposing each of these cells, in the order of the segments, on the fundamental domain (the square, or cell, with vertices $(0,0), (1,0)$ and $(1,1), (0,1)$). In this fundamental domain the left and right edges should be identified, and similarly for the top and bottom edges.

Two examples of fundamental reduction for straight paths are shown in Figures \ref{21} and \ref{-12}. Figure \ref{21} shows a straight path in the first quadrant, namely the path $(2,1)$, with its two segments labelled $s_1$ and $s_2$ (in that order), and its reduction to the fundamental domain

\begin{figure}[hbtp]
\centering
\includegraphics[height=2cm]{21j.eps}

\caption{The path $(2,1)$ and its fundamental reduction}
\label{21}
\end{figure}
\noindent whereas Figure \ref{-12} shows a straight path in the second quadrant, namely $(-1,2)$, and its two segments $s_1$ and $s_2$ (in that order). Note that, when fundamentally reduced, this path starts at $(1,0)$ (not the origin $O=(0,0)$) and ends at $(0,1)$. This is because its first cell does not coincide with the fundamental domain, and is true for non--reduced paths in all quadrants except the first.

\begin{figure}[hbtp]
\centering
\includegraphics[height=3cm]{12j.eps}

\caption{The path $(-1,2)$ and its fundamental reduction}
\label{-12}
\end{figure}
It is important that in the fundamental domain diagrams (the diagrams on the right) of Figures \ref{21} and \ref{-12} the left and right edges should be identified, and similarly for the top and bottom edges, i.e. these diagrams depict a loop on the torus.

\subsection{Quantum holonomy matrices and signed area}\label{sub22}
Using the connection \rref{conn} a quantum matrix is assigned to any straight path $(m.n)$ by
\be
\hat{U}_{(m,n)}= \exp \int_{(m,n)} \hat{A} =
\left(\begin{array}{ll} e^{m\hat{r}_1+ n\hat{r}_2} & 0
\\ 0 & e^{-m\hat{r}_1 -n\hat{r}_2}
\end{array}\right).
\label{Umn}
\ee
Clearly the assignment \rref{Umn} extends straightforwardly to any PL path between integer points by assigning a quantum matrix to each linear segment of the path, as in \rref{Umn}, and multiplying the matrices in the same order as the segments along the path. This prescription obviously coincides with the general relation:
\be
p\mapsto \hat{U}_p = {\cal P }\exp \int_{p} \hat{A}.
\label{hol}
\ee
where $\cal P$ denotes path-ordering.

In the covering space ${\IR}^2$, two homotopic loops on the torus are represented by two PL paths on the plane, $p_1,\, p_2$, with the same integer starting point and the same integer endpoint.  It was shown in \cite{goldman} that the following relationship holds for the respective quantum matrices:
\begin{equation}
\hat{U}_{p_1}=q^{S(p_1,p_2)}\hat{U}_{p_2},
\label{areaphase}
\end{equation}
where $S(p_1,p_2)$ denotes the signed area enclosed between the paths
$p_1$ and $p_2$. Equation \rref {areaphase} generalises equation \rref{fund2}, for which $S(p_1,p_2)$ is the area of the square with vertices $(0,0), \, (0,1), \, (1,0)$ and $(1,1)$, i.e. 1. For the general case, the signed area between two PL paths is defined as follows: for any finite region $R$ enclosed by $p_1$ and $p_2$, if the boundary of $R$ consists of oriented segments of $p_1$ and $p_2^{-1}$ (the path $p_2$ followed in the opposite direction), and is globally oriented in the positive (anticlockwise), or negative (clockwise) sense, this gives a contribution of $+{\rm area}(R)$ , or $-{\rm area}(R)$ respectively, to the signed sum $S(p_1,p_2)$ (otherwise the contribution is zero). See Figure \ref{signedarea}.
\begin{figure}
\begin{center}
\includegraphics[width=16pc]{signedareaj.eps}
\end{center}
\caption{\label{signedarea} The signed area between $p_1$ and $p_2$ is the area of the the horizontally shaded region minus the area of the vertically shaded region}
\end{figure}

In \cite{goldman} this relation between the holonomy matrices for homotopic loops/ paths was interpreted as a $q$-deformed surface group representation, i.e. a deformation of a representation of the fundamental group in terms of matrices. There we used the fact that signed area between paths has the following two properties:
\begin{equation}
 S(p_1,p_3) =  S(p_1,p_2) + S(p_2,p_3), \quad S(p_1p_2,p_3p_4) = S(p_1,p_3) + S(p_2,p_4)
\label{signedareaprops}
\end{equation}
where $p_1p_2$ denotes the concatenation of paths $p_1$ and $p_2$.

Another perspective on this comes from \cite{FMP}, where the concept of $2$-dimensional holonomy was explored. In the present context, $2$-dimensional holonomy may be described as the assignment of an element of a group $G$ to each of a pair of homotopic loops, and simultaneously the assignment of an element of a group $E$ to the homotopy between the two loops, subject to certain consistency relations. Amongst these relations, there is the requirement that the elements assigned to the homotopies behave well under vertical and horizontal composition, which translates precisely to the properties of signed area as in \rref{signedareaprops}. We note also that a crucial ingredient in the construction of \cite{FMP} is a connection $2$-form with values in the Lie algebra of $E$. Indeed, in the present context there is a natural $2$-form, namely the non-vanishing curvature of the connection \rref{conn} (non-vanishing because of the non-commutativity of the components $r_1$ and $r_2$, so that $A\wedge A$ is non-zero, as pointed out in \cite{goldman}). Thus, there are strong hints that the quantum holonomies of $2+1$ gravity may be interpreted in terms of $2$-dimensional holonomy.

\section{Intersecting paths}\label{sec3}
That Wilson loops associated to intersecting paths on surfaces have non--zero Poisson brackets was noted in \cite{gol}. These brackets and the corresponding commutators, for $2+1$--dimensional AdS gravity, were studied in \cite{NRZ}. In all cases paths which are {\it rerouted} at the intersection points appear on the r.h.s. In this Section we review some results about intersections and reroutings, and their classification by integer points. We also show how these concepts appear in the Poisson brackets of Wilson loops, and give some explicit examples.

\subsection{Intersections and reroutings}\label{sub31}
The intersection number between two paths $p_1$ and $p_2$, corresponding to loops on the torus, at an intersection point $S$, is defined to be (for single, not multiple, intersections) $+1$ if the angle between the tangent vector of $p_1$ at $S$ and the tangent vector of $p_2$ at $S$ is between $0$ and $180$ degrees, and $-1$ if it is between $180$ and $360$ degrees.

A rerouted path is denoted $p_1Sp_2$, or $p_1Sp_2^{-1}$, and is the path that follows $p_1$ as far as the intersection point $S$, then follows $p_2$ (or the inverse loop $p_2^{-1}$) from $S$ back to $S$, and finally proceeds along $p_1$ from $S$ back to the starting point of $p_1$. This can be thought of as `inserting' the path $p_2$ into the path $p_1$ at the intersection point $S$. Note that the point $S$ may occur at the origin of $\IR^2$, in which case we follow $p_2$ straight away. An example of the intersections at the points $P$ (the origin), $R$ and $Q$ for the paths $p_1=(1,2)$ and $p_2=(2,1)$ (and its inverse $(-2,-1)$) is shown in Figure \ref{p20a}.

\begin{figure}[hbpt]
\centering
\includegraphics[height=4cm]{p20aj.eps}
\hspace{2cm}
\includegraphics[height=4cm]{p20bj.eps}
\caption{ The reroutings $(1,2)S(2,1)$ and $(1,2)S(-2,-1)$ for $S=P,R,Q$ }
\label{p20a}
\end{figure}

The concept of fundamental reduction described in Section \ref{sub21} can be used to simultaneously show the reduction of the paths $(1,2)$ and $(2,1)$ and, more importantly, their three intersections $P,Q,R$, as shown in Figure \ref{ga4}. Note that the point $S=(1,1)$ does not contribute since it coincides with the point $P=(0,0)$. Figure \ref{ga4} should be compared with Figure \ref{p20a}.

\begin{figure}[hbtp]
\centering
\includegraphics[width=3cm]{ga4.eps}
\caption{The intersections $P,Q,R$ of the straight paths $(1,2)$ and $(2,1)$ }
\label{ga4}
\end{figure}

Here we shall work directly in $\IR^2$. Even though all integer points are the same when projected down to the torus a very clear picture of where the intersection point $R$ is located along both paths is obtained by fixing $p_1$ and parallel translating $p_2$ to start at a new integer point, denoted $\alpha$, in such a way that it intersects $p_1$ at $R$ (see Figure
\ref{rerouting}).

Let $(p)_{A,B}$ denote the subpath of the path $p$ from the point $A$ on $p$ to the point $B$ on $p$. Also, for any path $p$, let $\overline{p}$ denote the integer endpoint of $p$. For straight paths $p=(m,n)$, the notation for the endpoint and for the path itself coincide, i.e. $\overline{p}=p=(m,n)$,  but when the path is non-straight, there is a distinction between $p$ and
$\overline{p}$. This latter situation occurs when we discuss the extension of the refined bracket (to be discussed in the next subsection) to `crooked' paths, i.e. non-straight paths, resulting from a former rerouting, like, for example, those of Figure \ref{p20a}.

Write $p_1= (p_1)_{OR}(p_1)_{R\overline{p_1}}$, to indicate how $p_1$ is divided into two segments by the intersection point $R$. Here $p_1$ starts at the origin $O$, but the parallel translated $p_2$ starts at the integer point $\alpha$. Let $p^\alpha$ denote the path $p$ parallel translated to start at $\alpha$ instead of its original starting point. (Of course, projected down to the torus, the original $p$ and the shifted $p^\alpha$ give rise to identical loops). In this way, the explicit algebraic expression for the rerouting $p_1Rp_2$ represented as a path in the plane (the four dark segments of Figure \ref{rerouting}) is:
\be
p_1Rp_2 = (p_1)_{O,R}\, (p_2^\alpha)_{R,\beta}\, (p_2^\beta)_{\beta, R+\overline{p_2}} \,
(p_1^{\overline{p_2}})_{R+ \overline{p_2},\overline{p_1} +\overline{p_2}},
\label{reroute1}
\ee
where the start and endpoints $\alpha$ and $\beta$ of $p_2^\alpha$ are related by: $\beta =
\alpha + \overline{p_2}$.

\begin{figure}
\begin{center}
\includegraphics[width=12pc]{reroutingj1.eps}
\end{center}
\caption{\label{rerouting} The parallelogram with edges $p_1$ and $p_2$, and the rerouting $p_1Rp_2$ displayed as dark segments. The segment between integer points $\alpha$ and $\beta$ is a parallel copy of $p_2$, translated from starting at the origin to starting at $\alpha$; the grid of all other integer points is not displayed. }
\end{figure}

It is therefore clear why intersections occur and how they give rise to reroutings, i.e. for an intersection to occur, it is necessary that either $p_2^\alpha$ (the path $p_2$ parallel translated to start at $\alpha$) intersects $p_1$ in a point $R$ (which may be the origin, but not the endpoint of $p_1$), or equivalently, the endpoint $\beta$ is such that the appropriate $p_2^\alpha$ ending in $\beta$ intersects $p_1$ in a point $R$ as before.

The possible starting points $\alpha$ are the integer points lying in a `pre - parallelogram' with vertices $-\overline{p_2}$, $-\overline{p_2} +\overline{p_1}$, $\overline{p_1}$ and the origin $O$. See Figure \ref{preparallelogram}. Here any integer points lying on the edge between $-\overline{p_2} +\overline{p_1}$ and $\overline{p_1}$ are excluded, since the corresponding paths $p_2^\alpha$ intersect $p_1$ at its endpoint. Similarly, any integer points lying on the edge between
$-\overline{p_2}$ and $-\overline{p_2} +\overline{p_1}$ are excluded, to avoid double counting, since the integer points lying along the edge between $O$ and $\overline{p_1}$ are included.

\begin{figure}
\begin{center}
\includegraphics[width=12pc]{preparallelogramj.eps}
\end{center}
\caption{\label{preparallelogram} The parallelogram and pre-parallelogram (dotted line) for $p_1$ and $p_2$, showing some of the integer points (black dots) and the corresponding parallel-translated copies of $p_2$}
\end{figure}

Equivalently, consider the endpoints $\beta$ lying inside the parallelogram generated by $p_1$ and $p_2$, i.e. with vertices $O$, $\overline{p_1}$, $\overline{p_1} + \overline{p_2}$ and $\overline{p_2}$. Here the integer points lying on the edge between  $\overline{p_1}$ and $\overline{p_1} +\overline{p_2}$ and those lying along the edge between $O$ and $\overline{p_1}$ are excluded (they correspond to the starting points we excluded from the pre-parallelogram).

From both perspectives it can be checked that the number of intersections is correct, since the total intersection number is:
\be
\epsilon (p_1,p_2) = \det \left(p_1 p_2\right).
\label{int}\ee
The total intersection number (counting multiplicities), all of whose contributions have the same sign, is therefore the modulus of $\epsilon (p_1,p_2)$, i.e. in geometric terms, the area of the parallelogram or the pre-parallelogram. In turn, this area is given by a classical theorem of Pick \cite{pick}, which states that the area $A(P)$ of a planar polygon $P$ with
vertices at integer points of the plane is given in terms of the number of
interior integer points $I(P)$ and the number of boundary integer points $B(P)$
as follows:
\be
A(P) = I(P) + \frac{B(P)}{2} -1.
\label{pick}\ee
This value is evidently the same as the number of integer starting points $\alpha$ in the pre-parallelogram, or equivalently the number of integer end points $\beta$ in the parallelogram generated by $p_1$ and $p_2$, since we have excluded two edges, i.e. half the integer points lying along the interior of the edges and three out of four of the vertices (this is accounted for by the $-1$ in the formula, since $\frac{4}{2} -1 =1$, the single remaining vertex).

This approach clearly allows us to handle reducible paths in a natural manner. If $p_2$ is reducible, this means that there are integer points along $p_2$ other than the endpoints. In terms of the pre-parallelogram analysis, there will be $c$ integer starting points $\alpha$ for each intersection point $R$ along $p_1$. The intersection point $R$ therefore has intersection number $c$ and the rerouting at $R$ will appear with multiplicity $c$. These issues were briefly touched upon at the end of Section 5 of \cite{goldman}.

Since any rerouting must pass through one of the integer points inside the parallelogram generated by $p_1$ and $p_2$ (namely the appropriate endpoint $\beta$), we may label all intersections by the integer points that the rerouted paths (rerouted at each intersection) must pass through, rather than by their corresponding intersection points.

\subsection{Two quantizations of the Goldman bracket}\label{sub32}
When analysing the behaviour of the Wilson loops $\hat{T}(p)= {\rm tr}~\hat{U}_{p}$ for the quantum connection \rref{conn} in \cite{goldman} a link with the Goldman bracket \cite{gol} emerged. This bracket is a Poisson bracket for the traces $T(\gamma)= {\rm tr}\, U_\ga$, defined on homotopy classes of loops on a surface, which for $U_\ga \in SL(2,\IR)$ takes the following form (see
\cite{gol} Thm. 3.14, 3.15 and Remark (2), p. 284):
\be
\{T(\ga_1), T(\ga_2)\} = \sum_{S \in \ga_1 \sharp \ga_2}
\epsilon(\ga_1,\ga_2,S)(T(\ga_1S\ga_2) -
T(\ga_1S\ga_2^{-1})).
\label{gold}
\ee
Here $\ga_1 \sharp \ga_2$ denotes the set of (transversal) intersection points
of $\ga_1$ and $\ga_2$, and $\epsilon(\ga_1,\ga_2,S)$ is the intersection number
of $\ga_1$ and $\ga_2$ at the intersection point $S$. One should note the rerouted paths $\ga_1S\ga_2$ and
$\ga_1S\ga_2^{-1}$ on the r.h.s. of \rref{gold}.

We found that the Wilson loops $\hat{T}(p)$, for $p=(m,n)$ a straight path corresponding to a loop on the torus, satisfied a quantum version of the Goldman relation \rref{gold}. In fact two different quantizations emerged, a direct one, and a refined one, which we will now describe.

First, for straight paths $p_1=(m,n)$ and $p_2=(s,t)$, the Goldman bracket takes the form:
\be
\left\{T(m,n), T(s,t)\right\} = (mt-ns)
(T(m+s,n+t)- T(m-s,n-t)).
\label{straightforward}
\ee
Here $mt-ns$ is the total intersection number between $p_1$ and $p_2$, and this factor appears because there are effectively $mt-ns$ simple intersection points and the corresponding reroutings  $p_1Sp_2$ are all homotopic to the straight path $(m+s, n+t)$, with an analogous statement for the negative reroutings $p_1Sp_2^{-1}$.

By a simple direct calculation, it was shown in \cite{goldman} that the Wilson loops satisfy:
\begin{equation}
[\hat{T}(m,n), \hat{T}(s,t)]=
(q^{\frac{mt-ns}{2}}-q^{-\frac{mt-ns}{2}}) \left(\hat{T}(m+s,n+t) - \hat{T}(m-s,n-t)\right)
\label{qgb1}
\end{equation}
i.e. a quantization of \rref{straightforward}, with the total intersection number $mt-ns$ replaced by a quantum total intersection number (the first factor on the r.h.s. of \rref{qgb1}). We call this the direct quantization of the Goldman bracket (previously referred to as preliminary, or straightforward, or unrefined).

A refined quantization was also obtained in \cite{goldman}, where each rerouting appears as a separate term, and the relative area phases of these different but homotopic reroutings are taken into acccount. This takes the following form:
\be
[\hat{T}(p_1), \hat{T}(p_2)] = \sum_{ {S} \in p_1 \sharp p_2}
(q^{\epsilon(p_1,p_2,{S})} - 1)\hat{T}(p_1{S}p_2)
+ (q^{-\epsilon(p_1,p_2,{S})} - 1)\hat{T}(p_1{S}p_2^{-1})
\label{qgold}
\ee
which quantizes the bracket \rref{gold} (with loops $\ga$ substituted by paths $p$) by replacing the intersection numbers $\epsilon(p_1,p_2,{S})$ by quantum intersection numbers $(q^{\epsilon(p_1,p_2,{S})} - 1)$.

An example of equation \rref{qgold} is given by the choice $p_1=(1,2)$, $p_2=(2,1)$
\be
[\hat{T}(1,2),\hat{T}(2,1)] =\sum_{S=P,R,Q}
(q^{-1} -1) \hat{T}((1,2)S(2,1)) + (q-1) \hat{T}((1,2)S(-2,-1)),
\label{qgoldexp}
\ee
where the terms on the r.h.s. come from the positive and negative reroutings at three intersection points denoted $P,\,R,\,Q$, see Figure \ref{p20a}. To relate \rref{qgoldexp} to the direct form \rref{qgb1} of this commutator we calculate the area phases of each rerouting on the r.h.s. of
\rref{qgoldexp} relative to the corresponding straight paths $(3,3)$ or $(-1,1)$, and using
\begin{equation}
\hat{T}({p_1})=q^{S(p_1,p_2)}\hat{T}({p_2}),
\label{areaphasewilson}
\end{equation}
which follows from \rref{areaphase} we obtain:
\begin{eqnarray}
[\hat{T}(1,2),\hat{T}(2,1)] &=& (q^{-1}-1)(q^{\frac{3}{2}} + q^{\frac{1}{2}} + q^{-\frac{1}{2}}) \hat{T}(3,3)\nonumber \\
&+& (q-1) (q^{-\frac{3}{2}} + q^{-\frac{1}{2}} + q^{\frac{1}{2}}) \hat{T}(-1,1) \nonumber \\
&=& (q^{-\frac{3}{2}} - q^{\frac{3}{2}}) (\hat{T}(3,3) - \hat{T}(-1,1)).
\end{eqnarray}
as required.

\section{Properties of commutators of Wilson loops}\label{sec4}
To obtain a fully consistent theory for intersecting loops, based on the refined bracket \rref{qgold}, we need to address the following issue: the paths on the l.h.s. are straight, whereas on the r.h.s. they are crooked, so for the bracket to close, we need to understand the refined bracket with crooked paths on the l.h.s.

When attempting to extend the refined bracket to crooked paths, we encountered unexpected extra phases when the second path $p_2$ in the rerouting $p_1Rp_2$ was crooked, and the first path $p_1$ was straight. These difficulties did not occur when the first path was crooked and the second straight, i.e. the discrepancy occurs when the middle part of the rerouting (the inserted path) is crooked. To better understand these `crooked reroutings' and the extra phases they generate, we derive a formula for the relative phase of such reroutings compared to the `first' rerouting, i.e. the rerouting that occurs straight away at the origin (the origin is always an intersection point for any two paths).

\subsection{The relative phase for a crooked rerouting}\label{sub41}
We refer to Figure \ref{crookedrerouting} for the following discussion and calculations. There we have written $p_2$ as a double arrow, to indicate that it is crooked, i.e. it comes from one or more previous rerouting processes. The endpoint of the double arrow is the endpoint of $p_2$, written $\overline{p_2}$ using the notation of Section \ref{sub31}.
We derive an expression for the signed area, or relative phase, between the rerouting at the intersection point $R$, namely $p_1Rp_2$ and the rerouting at the origin (i.e. the first rerouting that occurs when starting along $p_1$).

\begin{figure}
\begin{center}
\includegraphics[width=10pc]{crookedreroutingj.eps}
\end{center}
\caption{\label{crookedrerouting} Rerouting along a crooked path $p_2$, represented as a double line.}
\end{figure}

The `crooked rerouting' may be expressed exactly like \rref{reroute1} as:
\be
p_1Rp_2 = (p_1)_{O,R}\, (p_2^\alpha)_{R,\beta}\, (p_2^\beta)_{\beta, R+\overline{p_2}} \,
(p_1^{\overline{p_2}})_{R+ \overline{p_2},\overline{p_1} +\overline{p_2}},
\label{reroute2}\ee
where the start and endpoints $\alpha$ and $\beta$ of $p_2^\alpha$ are related by: $\beta =
\alpha + \overline{p_2}$, whereas the rerouting at the origin is:
\be
 p_1Op_2 = (p_2)_{O, \overline{p_2}}\, (p_1^{\overline{p_2}})_{\overline{p_2}, R+ \overline{p_2}}
\label{reroute3}\ee
This is all that is needed since the relative phase of $p_1Op_2$ compared to the natural reference path, i.e. the straight path from the origin to $\overline{p_1}+ \overline{p_2}$ is easily calculated.

The signed area $S(p_1Rp_2, p_1Op_2)$ between \rref{reroute2} and \rref{reroute3} is found in two steps, by geometric adjustments of the area we are considering. First, notice that we can omit the last part of the two paths $(p_1^{\overline{p_2}})_{R+ \overline{p_2},\overline{p_1} +\overline{p_2}}$, which is the same for both reroutings, and can add and subtract a `triangle' with two straight edges, $(p_1)_{O,R}$ and $(\beta)_{O,\beta}$ (the path going from the origin to the integer point $\beta$), and one crooked edge $(p_2^\alpha)_{R,\beta}$. This gives
\be
S(p_1Rp_2, p_1Op_2) = S( (\beta)_{O,\beta}\, (p_2^\beta)_{\beta, \beta+\overline{p_2}},
(p_2)_{O, \overline{p_2}} (\beta)_{\overline{p_2},\beta + \overline{p_2}}).
\label{s1}\ee
Then we simply replace the two parallel crooked edges of the parallelogram, i.e. $(p_2^\beta)_{\beta, \beta+\overline{p_2}}$ and $(p_2)_{O, \overline{p_2}}$, by the corresponding straight edges, to obtain:
\bea
 && S( (\beta)_{O,\beta}\, (p_2^\beta)_{\beta, \beta+\overline{p_2}},
(p_2)_{O, \overline{p_2}} (\beta)_{\overline{p_2},\beta + \overline{p_2}})\nonumber\\
 && = S( (\beta)_{O,\beta}\, (\overline{p_2}^\beta)_{\beta, \beta+\overline{p_2}},
(\overline{p_2})_{O, \overline{p_2}} (\beta)_{\overline{p_2},\beta + \overline{p_2}}) =
 \det (\beta \overline{p_2}).
\label{s2}\eea
That is, the signed area, or relative phase, between the rerouting $p_1Rp_2$ and the rerouting at the origin $p_1Op_2$ is
\be
 S(p_1Rp_2, p_1Op_2) = \det (\beta \overline{p_2})
\label{s3}\ee
where $\beta$ is the integer endpoint associated to the intersection point $R$. This
result will be used in Section \ref{sub43}.

\subsection{A geometric formula}\label{sub42}
Here we use the classification by integer points of Section \ref{sub31} to obtain a quantum version of equation
\rref{pick}, namely Pick's formula \cite{pick}.

We define a set $V$ of valid integer points associated to the parallelogram $P(p_1,p_2)$ generated by the straight paths $p_1$ and $p_2$. This set consists of all interior integer points, all non-vertex integer points lying along $p_2$ and $p_1^{\overline{p_2}}$, as well as the vertex integer point $p_2$ itself. The significance of the points of $V$ in relation to the previous sections is that these are endpoints of parallel copies of $p_2$ with starting points in the pre-parallelogram of $P(p_1,p_2)$.

The number of points $\bf V$, in $V$ is equal to the area $A(P)$ of $P(p_1,p_2)$ because of Pick's theorem \rref{pick}:
\be
{\bf V} =  \sum_{\beta \in V} 1 = I(P) + B(P)/2 -1 = A(P).
\label{pick2}\ee
This is clear since the number of non-vertex integer points lying along $p_2$ and $p_1^{\overline{p_2}}$ is half the total number of non-vertex integer points on the boundary, i.e.  $(B(P) -4)/2= B(P)/2-2$. Adding the single vertex point $p_2$ gives the r.h.s. of Pick's formula.

Consider first the case where $p_1$ and $p_2$ have a positive intersection number $\det(p_1,p_2)>0$, and $p_2$ is irreducible (see Figure \ref{geomformulafig}). We have the following formula:
\be
\sum_{\beta \in V}q^{\det(\beta,p_2)} = \frac{q^{\det(p_1,p_2)}-1}{q-1}
\label{geomformula}
\ee
To see this, note that any line parallel to $p_2$ in the parallelogram has, since $p_2$ is irreducible, at most one point of $V$ lying in it. It follows that there are precisely $N=\det(p_1,p_2)$ such lines passing through a point of $V$ (including $p_2$ itself), and these lines divide the parallelogram into $N$ thin parallelograms, each of area $1$ (see Figure \ref{geomformulafig}). The area of all the thin parallelograms between the point $\beta$ and the edge $p_2$ is equal to $\det(\beta,p_2)$. Therefore
\be
\sum_{\beta \in V}q^{\det(\beta,p_2)}=\sum_{i=0}^{N-1}q^{i}= \frac{q^{N}-1}{q-1}.
\label{det}\ee

\begin{figure}
\begin{center}
\includegraphics[width=25pc]{geomformulafigr.eps}
\end{center}
\caption{\label{geomformulafig} Elements of V (black dots) for $p_2$ irreducible (left) and reducible (right).}
\end{figure}

For the case $\det(p_1,p_2)<0$, and $p_2$ irreducible, the geometric formula \rref{geomformula} is modified (equivalent to replacing $q$ with $q^{-1}$) to:
\be
\sum_{\beta \in V}q^{\det(\beta,p_2)} = \frac{q^{\det(p_1,p_2)}-1}{q^{-1}-1}
\label{geomformula_negint}
\ee
This follows by analogous arguments starting from the geometric series formula \rref{geomformula}
with $q$ replaced by $q^{-1}$.

Suppose now that again $p_1$ and $p_2$ have a positive intersection number, i.e.
$\det(p_1,p_2)>0$, but that $p_2$ is reducible, in other words $p_2$ has $M>1$ points of $V$ lying along it. The area of $P(p_1,p_2)$ is an integer multiple of $M$, say $\det(p_1,p_2)=NM$, and the $NM$ points of $V$ lie along $N$ lines parallel to $p_2$ (see Figure \ref{geomformulafig}), with $M$ points in each line. These lines divide the parallelogram into $N$ thin parallelograms, each of area $M$. We have a new geometric formula:
\be
\frac{1}{M}\sum_{\beta \in V}q^{\det(\beta,p_2)} = \frac{q^{\det(p_1,p_2)}-1}{q^M-1}
\label{geomformula_red}
\ee
which follows by grouping together the $M$ equal terms coming from points $\beta$ lying on the same parallel line and using the geometric series formula:
 $$
 \sum_{i=0}^{N-1}q^{Mi}= \frac{q^{NM}-1}{q^{M}-1}
$$

Finally the analogous formula for the case $\det(p_1,p_2)<0$ and $p_2$ reducible is as follows:
\be
\frac{1}{M}\sum_{\beta \in V}q^{\det(\beta,p_2)} = \frac{q^{\det(p_1,p_2)}-1}{q^{-M}-1}
\label{geomformula_red_negint}
\ee

We remark that the geometric formula \rref{geomformula} can be interpreted as a quantum version of \rref{pick2}. That is, the r.h.s. of \rref{geomformula} is the quantum area of the parallelogram $P(p_1,p_2)$, is expressed as a sum over terms labelled by $\beta \in V$, each of which equals $1$ in the classical limit $q\rightarrow 1$.  A simple example is given by $p_1=(2,1)$ and $p_2=(1,2)$,  with a corresponding quantum area of $1 + q + q^2$, instead of, classically, 3. We conjecture that this interpretation can be generalised in some way to an arbitrary polygon $P$.

\subsection{Equivalence of the refined and direct brackets for straight paths}\label{sub43}
We now show, for the general case, how the direct and refined formulas \rref{straightforward} and \rref{qgold} for $[\hat{T}(p_1),\hat{T}(p_2)]$ are equivalent, when $p_1$ and $p_2$ are
straight.  When $p_2$ is irreducible and $\det(p_1,p_2)>0$, the refined formula \rref{qgold} gives:
\begin{eqnarray*}
[\hat{T}(p_1), \hat{T}(p_2)]& = &(q-1) \sum_{\beta \in V(p_1,p_2)}
q^{\det(\beta,p_2)} \hat{T}(p_2 p_1^{\overline{p_2}})\\& + &
 (q^{-1}-1) \sum_{\gamma \in V(p_1,p_2^{-1}) }
q^{\det(\gamma,p_2^{-1})} \hat{T}(p_2^{-1} p_1^{-\overline{p_2}})
\end{eqnarray*}
where we have used \rref{s3} to obtain the geometric factors
of the reroutings associated to an integer point $\beta$ (or $\gamma$)
relative to the respective `first reroutings' (i.e. reroutings at
the origin). With the geometric formulae  \rref{geomformula} and
\rref{geomformula_negint} we obtain:
\begin{eqnarray}
[\hat{T}(p_1), \hat{T}(p_2)] & = & (q^{\det(p_1,p_2)}-1)  \hat{T}(p_2
p_1^{\overline{p_2}}) +  (q^{\det(p_1,p_2^{-1})}-1) \hat{T}(p_2^{-1} p_1^{-\overline{p_2}})\nonumber \\
& = & (q^{\det(p_1,p_2)}-1) q^{-\det(p_1,p_2)/2} \hat{T}(p_1+p_2) + \nonumber \\
& & \quad (q^{\det(p_1,p_2^{-1})}-1) q^{-\det(p_1,p_2^{-1})/2} \hat{T}(p_1-p_2) \nonumber \\
& = & (q^{\det(p_1,p_2)/2}- q^{-\det(p_1,p_2)/2}) (\hat{T}(p_1+p_2) - \hat{T}(p_1-p_2))\nonumber \\
\label{refined-unref}
\end{eqnarray}
i.e. the direct formula  \rref{straightforward}, where we have used
the relative geometric factors for the `first' reroutings relative
to the corresponding straight diagonal paths $p_1 \pm p_2$, as well as
the property $\det(p_1, p_2^{-1}) = -\det(p_1,p_2)$.

As discussed briefly in \cite{goldman}, when $p_2$ is
reducible, the refined formula must be modified by the introduction of
quantum {\em multiple} intersection numbers. Again let
$\det(p_1,p_2)>0$, and suppose $p_2$ has $M>1$ points of $V$ along it. The intersection number of the intersection corresponding to $M$ integer points lying on a line in $P(p_1,p_2)$ parallel to $p_2$, is $M$ classically, and the quantum multiple intersection number is $q^M-1$. The refined formula for $[\hat{T}(p_1), \hat{T}(p_2)]$ when $p_2$ is reducible of multiplicity $M$, is given by:
\be
[\hat{T}(p_1), \hat{T}(p_2)] = \sum_{ {S} \in p_1 \sharp p_2}
(q^{M} - 1)\hat{T}(p_1{S}p_2)
+ (q^{-M} - 1)\hat{T}(p_1{S}p_2^{-1})
\ee
Each intersection point $S$ is associated to $M$ integer points, so this can be rewritten as
\begin{eqnarray*}
[\hat{T}(p_1), \hat{T}(p_2)] &=& \frac{(q^{M} - 1)}{M}
 \sum_{ \beta \in V(p_1,p_2)} q^{\det(\beta,p_2)} \hat{T}(p_2 p_1^{\overline{p_2}}) \\
&+&\frac{(q^{-M} - 1)}{M} \sum_{ \gamma \in V(p_1,p_2^{-1})} q^{\det(\gamma,p_2^{-1})}
\hat{T}(p_2^{-1} p_1^{-\overline{p_2}})
\end{eqnarray*}
(recall that the factors $q^{\det(\beta,p_2)}$ are the same for all points $\beta$ lying on a line parallel to $p_2$).

Now, using the corresponding geometrical formula \rref{geomformula_red}, we obtain:
\begin{eqnarray*}
[\hat{T}(p_1), \hat{T}(p_2)]& = & (q^{\det(p_1,p_2)}-1)
\hat{T}(p_2 p_1^{\overline{p_2}}) +  (q^{-\det(p_1,p_2)}-1)
\hat{T}(p_2^{-1} p_1^{-\overline{p_2}}) \\
& = & (q^{\det(p_1,p_2)/2}- q^{-\det(p_1,p_2)/2}) (\hat{T}(p_1+p_2) - \hat{T}(p_1-p_2)
\end{eqnarray*}
where the steps in the final equality are the same as those in \rref{refined-unref}.

Finally the cases where $\det(p_1,p_2)<0$ are dealt with by using the geometric formulae \rref{geomformula_negint} and \rref{geomformula_red_negint} respectively, since the terms associated with a positive rerouting $p_1Sp_2$ now appear with a factor $q^{-1}-1$, and the terms associated with a negative rerouting $p_1Sp_2^{-1}$ appear with a factor $q-1$. In all cases the direct form \rref{straightforward} of the bracket $[\hat{T}(p_1), \hat{T}(p_2)]$ is found.

\subsection{Antisymmetry of the refined bracket}\label{sub44}

Finally, we show how the analysis of the previous section permits the antisymmetry of the refined bracket
\be
[\hat{T}(p_1), \hat{T}(p_2)] = - [\hat{T}(p_2), \hat{T}(p_1)]
\label{antisymm}
\ee
to be deduced directly from its expression \rref{qgold}, rather than indirectly from the equality of the refined and direct bracket, together with the patent antisymmetry of the latter. Instead of a general argument, we will focus on the particular case where $p_1$ and $p_2$ are both irreducible, $\det(p_1,p_2)>0$, and will only consider the positive rerouting terms, illustrating the arguments with the example of Figure \ref{antisymmetry} where $p_1=(2,1)$ and $p_2=(1,3)$.
\begin{figure}
\begin{center}
\includegraphics[width=10pc]{antisymmetryr.eps}
\end{center}
\caption{\label{antisymmetry} V($p_1,p_2$) contains the four internal points and $p_2$, V($p_2,p_1$) contains the four internal points and $p_1$.}
\end{figure}

The contribution of the positive rerouting terms to (\ref{antisymm}), using the results from the previous section, is:
$$
(q-1) \sum_{\beta \in V(p_1,p_2) } q^{\det(\beta,p_2)} \hat{T}(p_2 p_1^{\overline{p_2}}) =
- (q^{-1}-1) \sum_{\beta' \in V(p_2,p_1) } q^{\det(\beta',p_1)} \hat{T}(p_1 p_2^{\overline{p_1}}).
$$
By factoring out  $q^{-1}$ on the r.h.s., and using
$$
\hat{T}(p_1 p_2^{\overline{p_1}}) = q^{\det(p_1,p_2)}  \hat{T}(p_2 p_1^{\overline{p_2}}),
$$
it remains to prove the equality:
\be
 \sum_{ \beta \in V(p_1,p_2)} q^{\det(\beta,p_2)} = q^{-1} q^{\det(p_1,p_2)} \sum_{ \beta'' \in V(p_2,p_1)} q^{\det(\beta',p_1)}
\label{sumformula}
\ee
Note that the sets $V(p_1,p_2)$ and $V(p_2,p_1)$ share the same internal integer points, and differ only in that $V(p_1,p_2)$ contains $p_2$ and not $p_1$, whereas $V(p_2,p_1)$ contains $p_1$ and not $p_2$. The exponents $\det(\beta',p_1)$ on the r.h.s. are disjoint non-positive integers, running from $0$ to $-\det(p_1,p_2) + 1$, and represent the negative of the areas of the parallelograms bounded on the left and right by $p_2$ and $p_2^{\overline{p_1}}$, bounded below by $p_1$ and above by a line parallel to $p_1$ through $\beta'$.  Multiplying the sum on the r.h.s. by $q^{-1}$ corresponds to shifting the range of negative areas to run from $-1$ to $-\det(p_1,p_2)$, and taking the complementary areas $-\det(p_1,p_2)| -(\det(\beta',p_1)-1)$ (corresponding to multiplying on the r.h.s. with $q^{\det(p_1,p_2)}$), which are positive, these will run from $0$ to $\det(p_1,p_2)-1$. These latter exponents are exactly the exponents $\det(\beta,p_2)$ that appear on the l.h.s. of (\ref{sumformula}), i.e. corresponding to the (positive) areas of parallelograms bounded above and below by $p_1$ and $p_1^{\overline{p_2}}$, on the left by $p_2$ and on the right by the line parallel to $p_2$ through $\beta$.

This geometric argument establishes the equality (\ref{sumformula}) and hence the antisymmetry (\ref{antisymm}) for the case considered.

\section{Conclusions}\label{concl}
We have argued that this model of $2+1$ quantum gravity implies considering Wilson variables
for a large class of loops, related by area phases. Our study has substantially clarified the nature of the Wilson observables for crooked paths, namely paths that come from intersecting straight paths and rerouting along one of them. It has prepared us for a fuller description of the refined Goldman bracket for this larger class of loops, i.e. the class that contains loops coming from both straight paths and crooked paths.

In particular,
\begin{itemize}
\item we have given a full proof showing that, for straight paths, the refined bracket and the direct bracket are equivalent.
\item This proof also deals satisfactorily with reducible paths, i.e. paths such that $p=(m,n)=c(m',n')$, where $m,n,c,m',n'$ are all integers, with $c\geq 2$.
\item Our methods are a substantial step forwards towards our ultimate goal of defining the refined bracket which closes on a suitable class of straight and crooked paths.
\item We have achieved an independent understanding of the antisymmetry of the refined bracket for two straight paths, which is not manifest.
\end{itemize}

Apart from the main goal mentioned above, there are a couple of other questions which should be addressed in a fully consistent intersection and rerouting theory based on the refined bracket:

\begin{itemize}
\item The quantum intersection numbers which naturally appear in the two quantizations \rref{qgb1} and \rref{qgold} have a different appearance, and in particular they are symmetric, for the direct bracket, under the interchange $q \leftrightarrow q^{-1}$. This is not the case for the refined bracket.

\item We must prove the Jacobi identity for the refined bracket extended to a wider class of paths.
\end{itemize}

In future work we hope to explore in more detail the link with $2$-dimensional holonomy \cite{FMP}, as discussed at the end of Section 2, and gain further understanding of the elegant quantum geometry that emerges through the quantized Goldman bracket, relating it e.g. to noncommutative geometry \cite{con}, or to other quantizations of the Goldman bracket \cite{Turaev}, or the BTZ black hole \cite{vazwit}.

\begin{acknowledgements}
This work was supported by the Istituto Nazionale di Fisica Nucleare (INFN) of Italy, Iniziativa Specifica FI41, the Italian Ministero dell' Universit\`a e della Ricerca Scientifica e Tecnologica (MIUR), and the projects Quantum Topology, POCI/MAT /60352/2004, and New Geometry and Topology, PTDC/MAT/101503/2008, financed by the {\em Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia} (FCT) and cofinanced by the European Community fund FEDER.
\end{acknowledgements}

config: arxiv


TITLE: Describe comparison matrix algorithm as equation
QUESTION [1 upvotes]: I'm developing a statistical function for name matching. A core component of the function is an algorithm that determines which individual names from a full name correspond to eachother. For the explanation of my function, i'd like to capture this algorithm in a mathematical equation.
Take for example the input string "JN SMITH T" and the reference string "JOHN ALEXANDER SMITH TAYLOR".
I want to determine how alike the input is to the reference, but i want to account for the possibility of missing names. So i generate a comparison matrix like so:

Each combination of names (like "JN" and "JOHN") gets a comparison score from some other function. Now i want to obtain the best possible total score, where each name can only match one other name, and ordering must be preserved.
In my current algorithm, i generate a list of permutations that can be added together, of which the maximum scoring one is then selected. The best mathematical description i can come up with is this:
Let the comparison matrix be $X$. Each name comparison gets a score at position $X_{ij}$. The input string has 3 names in it (which i'll describe as $n=3$), so we get at most 3 comparisons. That means (i think) that we're looking for the maximum of $\sum_{k=1}^{k=3} X_{i_{k}j_{k}}$, where $i_k < i_{k+1}$ and $j_k < j_{k+1}$ because order is preserved. If i assign the maximum score to the symbol $\Theta$, i'd finally get:
$$\Theta = max\left(\sum_{k=1}^{k=n} X_{i_{k}j_{k}}|i_k < i_{k+1}, j_k < j_{k+1}\right)$$
However, this equation looks like you'll always match all the input names. If my reference name is just "JOHN TAYLOR", you could only match two of the input names, not all three. The max sum would become $X_{11} + X_{23}$. Is it possible to reflect that into the equation for $\Theta$?

REPLY [1 votes]: I think i've found the answer:
instead of defining $n$ to be the number of names in the input string, i'll define $m$ to be the number of names in the input string.
Then i'll define $n=(1,2,...,m)$
and the total equation becomes
$$\Theta = max\left(\sum_{k=1}^{k=n} X_{i_{k}j_{k}}|i_k < i_{k+1}, j_k < j_{k+1},n=(1,2,...,m)\right)$$


TITLE: Units of vector differential operator del ($\nabla$)
QUESTION [2 upvotes]: My book says that $\left[\nabla \cdot (\vec E \times \vec H)\right] = \mathrm{W/m^3}$. I see that $\vec E$ is in $\mathrm{V/m}$ and $\vec H$ is $\mathrm{A/m}$, so these multiplied is $\mathrm{W/m^2}$, but how does dotting with $\nabla$ give another $\mathrm{m^{-1}}$?

REPLY [9 votes]: Note on notation: I use $[\cdot]$ do denote the units of the quantity in brackets.
Derivatives always have units of $1/[\text{differentiation variable}]$. This can be clearly seen from the definition of the derivative in terms of difference qutionts:
$$ \partial_x f(x) := \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$$
So if $x$ has some unit $[x]$ then $\partial_x f$ will have units $[f]/[x]$. (As the limit does obviously not change the units.)
As
$$\nabla = \begin{pmatrix} \partial_x \\ \partial_y \\ \partial_z \end{pmatrix}$$
and the coordinates in space carry the unit $\mathrm{m}$, you have that
$$[\nabla \cdot \vec V] = [\partial_x V_x + \partial_y V_y + \partial_z V_z] = [\partial_x V_x] = [V_x]/[x] = [V_x]/\mathrm{m}.$$
(Where I arbitrarily chose the $x$-component of the vector, as all components have the same units.)

REPLY [3 votes]: The $\nabla$ operator is a spatial derivative of the $\frac{\delta}{\delta x}$ etc kind. This has units of $1/m$


TITLE: What does Addition distribute over?
QUESTION [1 upvotes]: I was considering I’m trying to find (hopefully continuous) functions $f: \mathbb{C}^2 \rightarrow \mathbb{C}$ that obey
$$ K + f(x,y) = f(x+K, y+K) \forall K \in \mathbb{C} $$

REPLY [1 votes]: So $f(x,y)=x+f(0,y-x)$, thus the general solution will be $f(x,y)=x+g(y-x)$ where $g$ is an arbitrary function.


TITLE: Is there a "topology" of functions?
QUESTION [1 upvotes]: A pair of real variables could be the surface coordinates for a 2 dimensional surface such as a sphere or a torus or a genus n surface for example.
Going from the space of real numbers to the space of functions is there an equivalent "topology" possible? One might imagine a functional of a pair of functions with certain properties defining a "topological surface" paramaterised by pairs of functions.
I.e. do function spaces also have a notion of topology that is a higher order version of topology?

REPLY [1 votes]: I don't know if this answer will be satisfying to you, but there is some general topology context that might change how you look at this question.
Firstly, if you want to handle pairs of anything-with-topologies, you can use the product topology. This applies equally well to anything you have topologies on, and often gives nice answers. For example, if $\mathbb R$ has the standard topology and the circle $S^1$ has its standard topology (induced from its embedding in the plane, say), then:

The product topology on $\mathbb R\times\mathbb R$ is indeed the standard topology on $\mathbb R^2$
The product topology on $S^1\times\mathbb R$ corresponds exactly to the standard topology on an infinite cylinder living in $\mathbb R^3$.
The product topology on $S^1\times S^1$ corresponds exactly to the standard topology on a torus living in $\mathbb R^3$.

So if we had a reasonable topology on a collection of functions, then we could get a reasonable topology on pairs of functions via this same product-topology construction.
Now the question is: what is a reasonable topology on a bunch of functions $f:X\to Y$? Well, suppose the functions are (some of) the continuous maps between two (possibly identical) topological spaces (maybe nice ones like metric spaces or something).
One candidate would be to treat the space of these functions just like (infinite if $X$ is infinite) products of the space $Y$, but this is ignoring the topology on $X$. Another  common one which takes that into account is called the compact-open topology. For a number of reasons (one is mentioned at this MSE answer), the compact-open topology is usually nicer/more appropriate.
Since you used the tag functional analysis, if we're looking at the continuous (aka bounded) linear maps on a Hilbert space $H$, then there are lots of topologies in use. And this MO answer mentions that on norm-bounded subsets, the "strong operator topology" agrees with the compact-open topology I mentioned above.


TITLE: Partial Derivative of $x^y$
QUESTION [2 upvotes]: $$\frac{\partial}{\partial x}x^y$$

$x^y=e^{ln(x)^y}=e^{y*ln(x)}$
$$\frac{\partial}{\partial x}e^{y\cdot ln(x)}=e^{y\cdot ln(x)}\cdot\frac{y}{x}=x^y\cdot\frac{y}{x}=x^{y-1}\cdot y$$
But the answer is different , where did I get it wrong?

REPLY [3 votes]: You have the correct answer. WolframAlpha thinks $y$ is a function of $x$, which is why it gives the answer it does. Try using another variable like $t$ instead of $y$ and you'll see that WolframAlpha gives the right answer.
As a side note, I noticed that the query you typed into WolframAlpha was "derive x^y with respect to x". The process of finding a derivative is differentiation, not derivation, so it would be more correct to ask "WolframAlpha to "differentiate x^y with respect to x". The two terms are (incorrectly) used interchangeably so often that WolframAlpha knows you mean "differentiate" when you say "derive".

REPLY [2 votes]: You didn't get it wrong: Wolfram Alpha just doesn't know that $y$ doesn't depend on $x$.
If it did, then the term that comes from using the product rule would be $$x \log(x) y'=0$$


TITLE: How to see that $F$ $F$ dual is a surface term?
QUESTION [4 upvotes]: The renormalisable 'theta term' that one can add to a Lagrangian describing Yang-Mills fields is often neglected on the grounds that it contributes a surface term. For QED, this is easy to see:
$$\theta \int F \wedge F = \theta \int \mathrm{d}(A \wedge F) $$
But for a non-Abelian field with $F = \mathrm{d}A + A \wedge A$, $F \wedge F$ contains an $A^4$ term which isn't obviously an exact form. Either I'm missing something obvious here, or perhaps $F \wedge F$ is not the correct way to write the theta term for a non-Abelian field?
EDIT (problem solved): I found a proof that
$$ \mathrm{tr}(F \wedge F) = \mathrm{d}\, \mathrm{tr}\left(A \wedge \mathrm{d}A + \frac{2}{3}A^3 \right)$$
In section 10.5.5 (lemma 10.3) of Nakahara. As pointed out by ACuriousMind below, the proof isn't very enlightening, but one crucial step I was missing is that
$$ \mathrm{tr}(A^4) = 0$$
On account of the cyclicity of the trace and antisymmetry of the wedge product.

REPLY [3 votes]: So as to complement Nakahara's and Nogeira's proofs, when I did this calculation the most puzzling part was the origin of the $\frac{2}{3}$ in front of $A\wedge A\wedge A$, but it is easy to find it out:
\begin{align*}
   Tr[F\wedge F] =&\ Tr\left[dA\wedge dA+dA\wedge A\wedge A
                    +A\wedge A\wedge dA+A\wedge A\wedge A\wedge A\right] \\[0.5em]
                 =&\ Tr\left[dA\wedge dA+2dA\wedge A\wedge A\right]           \\
                 =&\ Tr\left[dA\wedge dA+\frac23(dA\wedge A\wedge A
                    -A\wedge dA\wedge A+A\wedge A\wedge dA)\right]      \\
                 =&\ Tr\left[d(A\wedge dA)+\frac23d(A\wedge A\wedge A)\right] \\
                 =&\ d\ Tr\left[A\wedge dA+\frac23A\wedge A\wedge A\right]
\end{align*}
where the key point is that $Tr[dA\wedge A\wedge A]=-Tr[A\wedge dA\wedge A]=Tr[A\wedge A\wedge dA]$, as can be easily checked from the cyclicity of the trace and  the antisymmetry of the wedge product (recall that $A$ are matrix-valued, not $\mathbb{R}$-valued 1-forms).
This shows that the Pontryagin density $Tr[F\wedge F]$ is exact and that its generating form is a Chern-Simons term.