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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-83.60898pt}{-62.9221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\mathcal{F}_{\mathcal{A}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\ \ .$ (3.18)

Now let $A$ be an algebra in $\mathcal{A}$. A quantum moment map is an algebra
map $\mu\colon A\longrightarrow\mathcal{F}_{\mathcal{A}}$ in $\mathcal{A}$
such that it fits into the following commutative diagram

=$\theta$$\theta$$\theta$$V\otimes W$$V\otimes
W$$+\text{-linked}$$+\text{-nested}$$+\text{-unlinked}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$L^{+}~{}=$$N^{+}~{}=$$U^{+}~{}=$$\mathcal{F}_{\mathcal{A}}$$V$$V$$\mathcal{F}_{\mathcal{A}}$${A\otimes\mathcal{F}_{\mathcal{A}}}$${A\otimes
A}$${A}$${\mathcal{F}_{\mathcal{A}}\otimes A}$${A\otimes
A}$$\scriptstyle{\tau_{A}^{-1}}$$\scriptstyle{\operatorname{id}\otimes\mu}$$\scriptstyle{m}$$\scriptstyle{\mu\otimes\operatorname{id}}$$\scriptstyle{m}$

It is shown in [BZBJ18b, Corollary 4.7] that algebras
$A\in\int_{\mathbb{S}^{1}\times\mathbb{R}}\mathcal{A}$ amount to the data of a
quantum moment map $\mu\colon\mathcal{F}^{e}_{\mathcal{A}}\longrightarrow A$.

As mentioned in Remark 3.20, braided modules are identified in [BZBJ18b] with
module categories over
$\mathcal{F}_{\mathcal{A}}^{e}\text{-mod}_{\mathcal{A}}$, where the latter is
equipped with the tensor product $\otimes_{\mathbb{R}}$ induced by stacking
annuli in the radial direction. Let now $\mathcal{M}$ be a braided module
category and assume there is a progenerator $m\in\mathcal{M}$ for the induced
$\mathcal{A}$-action. In the situation at hand,
$\mathcal{M}=\int_{(\Sigmait_{o},\varphi_{o})}\mathcal{A}$ and the
progenerator is the distinguished object given by the pointing via the
inclusion of the empty manifold. The following reconstruction result for
$\mathcal{M}$ is proven in [BZBJ18b, Theorem 1.1]: There is an equivalence

$\mathcal{M}\cong
A\text{-mod}_{\int_{\mathbb{S}^{1}\times\mathbb{R}}\mathcal{A}},\quad
A=\underline{\text{End}}_{\mathcal{A}}(m)\ \ ,$

where the endomorphism algebra comes with a canonical quantum moment map
$\mu_{\Sigmait_{o}}\colon\mathcal{F}_{\mathcal{A}}\longrightarrow A$. The
right action of $\mathcal{F}_{\mathcal{A}}\text{-mod}_{\mathcal{A}}$ on
$\mathcal{M}$ is then given by [BZBJ18b, Corollary 4.7] :

$\displaystyle A\text{-mod}\boxtimes\mathcal{F}_{\mathcal{A}}\text{-mod}$
$\displaystyle\longrightarrow A\text{-mod}$ (3.19) $\displaystyle V\boxtimes
X$ $\displaystyle\longmapsto V\otimes_{\mathcal{F}_{\mathcal{A}}}X\ \ ,$
(3.20)

where the algebra homomorphism $\mu_{\Sigmait_{o}}$ and the field goal
transformations are used to form the relative tensor product.

######  Remark 3.22.

Conversely, given an algebra $A\in\mathcal{A}$ and a quantum moment map
$\mu\colon\mathcal{F}_{\mathcal{A}}\longrightarrow A$, the category
$\mathcal{M}=A\text{-mod}_{\mathcal{A}}$ is equipped with the structure of a
braided module category. We refer to [BZBJ18b, Section 4.3] for an explicit
description of the braided module structure that one obtains from the given
quantum moment map $\mu$.

Applying the above reconstruction result to the situation at hand, we get
quantum moment maps

$\displaystyle\mu_{\Sigmait_{o}}\colon\mathcal{F}_{\mathcal{A}}\longrightarrow
a_{P}^{d_{1},\dots d_{n}}\ \ \text{and }\ \
\mu_{\mathbb{D}}\colon\mathcal{F}_{\mathcal{A}}\longrightarrow
1_{\mathcal{A}}\ \ ,$ (3.21)

which endow $a_{P}^{d_{1},\dots d_{n}}$ and $1_{\mathcal{A}}$ with the
structure of algebras in $\mathcal{F}_{\mathcal{A}}\text{-mod}_{\mathcal{A}}$.
Finally, by [BZBJ18b, Corollary 4.8], we get:

######  Proposition 3.23.

The factorisation homology on a closed decorated surface $(\Sigmait,\varphi)$
is given by

$\displaystyle\int\displaylimits_{(\Sigmait,\varphi)}\mathcal{A}\cong(a_{P}^{d_{1},\dots,d_{n}}\text{-}\mathrm{mod}\text{-}1_{\mathcal{A}})_{\mathfrak{F}_{\mathcal{A}}\text{-}\mathrm{mod}_{\mathcal{A}}}\
\ ,$ (3.22)

the category of $a_{P}^{d_{1},\dots,d_{n}}$-$1_{\mathcal{A}}$-bimodules inside
$\mathcal{F}_{\mathcal{A}}\text{-}\mathrm{mod}_{\mathcal{A}}$.

######  Remark 3.24.

Let $\\{x_{1},\dots,x_{r}\\}\subset\Sigmait$ be a collection of marked points
on the surface and
$\varphi\colon\Sigmait\setminus\\{x_{1},\dots,x_{r}\\}\longrightarrow BD$ a
continuous map. Let $\Sigmait_{o}$ be the surface obtained from $\Sigmait$ by
removing a small disk $\mathbb{D}^{d_{i}}$ around each point $x_{i}$, where
the label $d_{i}$ indicates that the holonomy of $\varphi$ around the $i$-th
boundary component $\partial_{i}\Sigmait_{o}$ is given by the group element
$d_{i}\in D$. Let $\mathcal{M}=\bigoplus_{d\in D}\mathcal{M}_{d}$ be an
equivariant balanced right module over $\mathcal{A}$. Applying excision, we
can express factorisation homology over the marked surface $\Sigmait$ via the
following relative tensor product:

$\int_{((\Sigmait,\varphi),\\{x_{1},\dots,x_{r}\\})}(\mathcal{A},\mathcal{M})\cong\int_{(\Sigmait_{o},\varphi)}\mathcal{A}\underset{\big{(}\int_{(\mathbb{S}^{1},d_{1})}\mathcal{A}\boxtimes\dots\boxtimes\int_{(\mathbb{S}^{1},d_{r})}\mathcal{A}\big{)}}{\boxtimes}\Big{(}\mathcal{M}_{d_{1}}\boxtimes\dots\boxtimes\mathcal{M}_{d_{r}}\Big{)}\
\ .$

##  4 Quantisation of flat twisted bundles

In this section we describe the Poisson algebra of functions on the moduli
space of flat $\operatorname{Out}(G)$-twisted $G$-bundles on an oriented
surface $\Sigmait$ and its quantisation via factorisation homology over
$\Sigmait$ with coefficients in the ribbon category ${\mathsf{Rep}}_{q}(G)$
equipped with the $\operatorname{Out}(G)$-action defined in Section 2.3.

###  4.1 The moduli space of flat twisted bundles

We first recollect some background about twisted bundles in the differential
geometric setting, see for example [Mei17] and [Zer21] for more details and
[MSS22] for the non-flat version. We refer to [BY15] for the original
algebraic geometric definition and extension to wild character varieties. Let
$\Sigmait$ be an oriented surface equipped with a principal
$\operatorname{Out}(G)$-bundle $\mathcal{P}\longrightarrow\Sigmait$. The group
homomorphism
$G\rtimes\operatorname{Out}(G)\longrightarrow\operatorname{Out}(G)$, given by
projection onto the second factor, induces a morphism of smooth
groupoids101010Here smooth groupoids can, for example, be modelled as sheaves
of groupoids on the site of Cartisan space as in [BMS21, Section 5.1]. We will
not go into the details here because they will not be important for what
follows. $\operatorname{Bun}^{\text{flat}}_{\operatorname{Out}(G)\rtimes
G}(\Sigmait)\longrightarrow\operatorname{Bun}_{\operatorname{Out}(G)}(\Sigmait)$.
The groupoid of flat $\mathcal{P}$-twisted $G$-bundles is defined as the
homotopy pullback

=$\theta$$\theta$$\theta$$V\otimes W$$V\otimes
W$$+\text{-linked}$$+\text{-nested}$$+\text{-unlinked}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$L^{+}~{}=$$N^{+}~{}=$$U^{+}~{}=$$\mathcal{F}_{\mathcal{A}}$$V$$V$$\mathcal{F}_{\mathcal{A}}$${\operatorname{Bun}_{G\downarrow\mathcal{P}}^{\text{flat}}(\Sigmait)}$${\operatorname{Bun}^{\text{flat}}_{G\rtimes\operatorname{Out}(G)}(\Sigmait)}$${\star}$${\operatorname{Bun}_{\operatorname{Out}(G)}(\Sigmait)\
\ .}$$\scriptstyle{\mathcal{P}}$ (4.1)

The trivial $\mathcal{P}$-twisted $G$-bundle is the bundle
$\mathcal{P}\times_{\operatorname{Out}(G)}(G\rtimes\operatorname{Out}(G))$
associated to $\mathcal{P}$ using the group homomorphism
$\operatorname{Out}(G)\hookrightarrow G\rtimes\operatorname{Out}(G)$,
$\kappa\longmapsto 1\rtimes\kappa$. Note that the automorphisms of the trivial
flat $\mathcal{P}$-twisted $G$-bundle are $G^{\pi_{0}(\Sigmait)}$ and not
$(G\rtimes\operatorname{Out}(G))^{\pi_{0}(\Sigmait)}$ as one might naively
expect.

######  Remark 4.1.

The moduli space of flat $\operatorname{Out}(G)$-twisted bundles on a closed
surface $\Sigmait$ was studied in the differential geometric setting in
[Mei17, Zer21]. In particular, it is shown in loc. cit. that the moduli space
of $\operatorname{Out}(G)$-twisted flat bundles for a compact Lie group $G$
carries a canonical Atiyah-Bott like symplectic structure. Similar symplectic
structures have been constructed in the algebraic geometric setting on of
(wild) twisted character varieties in [BY15].

Since in this paper we obtain our results in the algebraic setting, we will
now give another description of flat twisted bundles that is more suitable for
us, namely the holonomy description of twisted $G$-bundles. We will only
consider surfaces $\Sigmait$ with at least one boundary component and a marked
point $v\in\partial\Sigmait$ on one of the boundary circles. For brevity we
write simply $\pi_{1}(\Sigmait)$ for $\pi_{1}(\Sigmait,v)$. For any group $G$,
we call the space of group homomorphisms
$\operatorname{Hom}(\pi_{1}(\Sigmait),G)$ the $G$-representation variety. It
comes with a natural action of $G$ via conjugation:
$g.\varphi(\gamma)=g\varphi(\gamma)g^{-1}$ for all $g\in G$,
$\gamma\in\pi_{1}(\Sigmait)$ and
$\varphi\in\operatorname{Hom}(\pi_{1}(\Sigmait),G)$. As before, we fix a
principal $\operatorname{Out}(G)$-bundle, here described by a group
homomorphism
$\rho\colon\pi_{1}(\Sigmait)\longrightarrow\operatorname{Out}(G)$. Such a map
$\rho$ is given by picking an element $\kappa\in\operatorname{Out}(G)$ for
every generator in $\pi_{1}(\Sigmait)$. Then, an element in the $\rho$-twisted
$G$-representation variety is a lift

=$\theta$$\theta$$\theta$$V\otimes W$$V\otimes
W$$+\text{-linked}$$+\text{-nested}$$+\text{-unlinked}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{i}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$\mathcal{F}_{\mathcal{A}}^{d_{j}}$$L^{+}~{}=$$N^{+}~{}=$$U^{+}~{}=$$\mathcal{F}_{\mathcal{A}}$$V$$V$$\mathcal{F}_{\mathcal{A}}$${G\rtimes\operatorname{Out}(G)}$${\pi_{1}(\Sigmait)}$${\operatorname{Out}(G)\
\ .}$$\scriptstyle{\rho}$ (4.2)

We write $\operatorname{Hom}_{\rho}(\pi_{1}(\Sigmait),G)$ to denote the space
of lifts. Concretely, elements in
$\operatorname{Hom}_{\rho}(\pi_{1}(\Sigmait),G)$ can be described by maps
$\varphi\colon\pi_{1}(\Sigmait)\longrightarrow G$, which are such that
$\varphi(\gamma_{1}\circ\gamma_{2})=\varphi(\gamma_{1})\rho(\gamma_{1}).\varphi(\gamma_{2})$.
The group $G$ acts via twisted conjugation, i.e. the action of an element
$g\in G$ is given by $\varphi(\gamma)\longmapsto
g\varphi(\gamma)\rho(\gamma).g^{-1}$. Given a set $E$ of free generators of
$\pi_{1}(\Sigmait)$, we get an identification
$\operatorname{Hom}_{\rho}(\pi_{1}(\Sigmait),G)\cong G^{E}$.

There is a bijective correspondence between elements in the twisted
representation variety $\operatorname{Hom}_{\rho}(\pi_{1}(\Sigmait),G)$ and
elements in

$\mathcal{M}^{\circ}_{\rho}(\Sigmait)\coloneqq\\{\text{Isomorphism classes of
flat twisted }G\text{-bundles with trivialisation over }v\in\Sigmait\\}\ \ ,$

which is established via the holonomy map. The group $G$ acts on
$\mathcal{M}_{\rho}^{\circ}(\Sigmait)$ by changing the trivialisation. The
moduli space of flat twisted bundles is then given by the quotient stack

$\mathcal{M}_{\rho}(\Sigmait)=\mathcal{M}_{\rho}^{\circ}(\Sigmait)/^{\rho}G\ \
,$

where the notation $/^{\rho}$ indicates that $G$ acts via twisted conjugation.

####  4.1.1 The twisted Fock-Rosly Poisson structure

For the remainder of this section, $\Sigmait$ is a connected surface with at
least one boundary component. We will give an explicit description of the
Poisson structure on $\mathcal{M}^{\circ}_{\rho}(\Sigmait)$, following the
strategy of Fock and Rosly [FR98] using lattice gauge theory.

We choose a ciliated fat graph model for $\Sigmait$ with one vertex and edges
$E=\\{e_{1},\dots,e_{n}\\}$, constructed from a gluing pattern for $\Sigmait$
as defined in Section 3.2. Furthermore, we choose an
$\operatorname{Out}(G)$-labeling $\\{\kappa_{1},\dots,\kappa_{n}\\}$ of the
gluing pattern describing the twisting principal
$\operatorname{Out}(G)$-bundle $\rho$. The fundamental group of $\Sigmait$ is
freely generated by the edges $E$ of the graph model, as depicted in Figure
14. Using the holonomy description from the previous section, we can
characterise a $\rho$-twisted bundle on $\Sigmait$ by a graph connection, that
is a labeling of every edge $e_{i}\in E$ with a group element $g_{i}\in G$:

$\text{hol}\colon\mathcal{M}_{\rho}^{\circ}(\Sigmait)\xrightarrow{\cong}\operatorname{Hom}_{\rho}(\pi_{1}(\Sigmait,v),G)=G^{E}\
\ .$

This identification chooses an orientation for every edge in the fat graph
model which we choose to agree with the natural orientation coming from the
gluing pattern. Hence, we get an identification

$\mathcal{M}_{\rho}(\Sigmait)\cong G^{E}/^{\rho}G\ \ ,$

where $h\in G$ acts via twisted conjugation

$(g_{e_{1}},\dots
g_{e_{n}})\longmapsto(hg_{e_{1}}\kappa_{1}(h)^{-1},\dots,hg_{e_{n}}\kappa_{n}(h)^{-1})\
\ .$ (4.3)

In this way, we consider the algebraic functions on $G^{E}$ as an element of
${\mathsf{Rep}}(G)$ and we denote this algebra by $\mathcal{O}^{\rho}(G^{E})$.
Quasi-coherent sheaves on $\mathcal{M}_{\rho}(\Sigmait)$ can now be identified
with modules over $\mathcal{O}^{\rho}(G^{E})$ in ${\mathsf{Rep}}(G)$.

######  Proposition 4.2.

Let $\Sigmait$ be a surface of genus $g$ and with $r\geq 1$ boundary
components. Given a principal $\operatorname{Out}(G)$-bundle
$\rho\colon\pi_{1}(\Sigmait)\longrightarrow\operatorname{Out}(G)$, described
by the elements $\kappa_{1},\dots,\kappa_{2g+r-1}\in\operatorname{Out}(G)$,
and a gluing pattern $P$ for $\Sigmait$, there is an isomorphism
$\mathcal{O}^{\rho}(G^{2g+r-1})\cong a_{P}^{\kappa_{1},\dots,\kappa_{2g+r-1}}$
of algebras in ${\mathsf{Rep}}(G)$.

###### Proof.

To establish the isomorphism on the level of vector spaces, we use the
algebraic Peter-Weyl theorem:

$\mathcal{O}(G)\cong\bigoplus_{V}V^{\vee}\otimes V\ \ ,$

where the sum on the right hand side is over all irreducible representations
of $G$ and $\mathcal{O}(G)$ is the Hopf algebra of matrix coefficients of
irreducible $G$-representations. Next we take into account the twist by a
given automorphism $\kappa\in\operatorname{Out}(G)$: a group element $h\in G$
acts on $\phi\in\mathcal{O}^{\kappa}(G)$ via
$h\triangleright\phi=\phi(h^{-1}(-)\kappa(h))$. As explained in Example 3.2,
we thus get an isomorphism
$\mathcal{O}^{\kappa}(G)\cong\bigoplus_{V}V^{\vee}\otimes\kappa^{*}V=\mathcal{F}_{{\mathsf{Rep}}(G)}^{\kappa}$
compatible with the $G$-action. ∎

In combination with Theorem 3.5, the above result shows that
$\int_{(\Sigmait,\rho)}{\mathsf{Rep}}(G)$ agrees with the category of quasi-
coherent sheaves on the moduli space $\mathcal{M}_{\rho}(\Sigmait)$ of twisted
bundles. Note that $G^{E}$ is a finite dimensional smooth algebraic variety
and independent of the concrete form of the gluing pattern or topology of
$\Sigmait$. However, we will see shortly that the Poisson structure is
sensitive to the topology.

\begin{overpic}[scale={0.4},tics=10]{graphsurface.pdf} \end{overpic} Figure
14: Generators of the fundamental group for an $r$-punctured genus $g$
surface.

In order to describe the Poisson structure on the representation variety
$\mathcal{M}_{\rho}^{\circ}(\Sigmait)$, we notice that there is an equivariant
embedding

$\displaystyle\iota\colon G^{E}$
$\displaystyle\longrightarrow(G\rtimes\operatorname{Out}(G))^{E}$ (4.4)
$\displaystyle(g_{e_{1}},\dots g_{e_{n}})$
$\displaystyle\longmapsto(g_{e_{1}}\rtimes\kappa_{1},\dots
g_{e_{n}}\rtimes\kappa_{n})$ (4.5)

which identifies $G^{E}$ with a connected component of
$(G\rtimes\operatorname{Out}(G))^{E}$ since $\operatorname{Out}(G)$ is
discrete. The $G$-action on the right side is via the embedding
$G\longrightarrow G\rtimes\operatorname{Out}(G)$ and conjugation inside
$G\rtimes\operatorname{Out}(G)$. Using the gluing pattern for $\Sigmait$,
together with the choice of an $\operatorname{Out}(G)$-invariant classical
r-matrix111111For example, the semi-classical limit of the quantum R-matrix
$\mathcal{R}$ of $U_{\hbar}(\mathfrak{g})$ is
$\operatorname{Out}(G)$-invariant, see Proposition 2.13.
$r\in\left(\mathfrak{g}\otimes\mathfrak{g}\right)^{\operatorname{Out}(G)}$,
Fock and Rosly’s construction [FR98] gives a Poisson structure
$\pi_{\text{FR}}$ on $(G\rtimes\operatorname{Out}(G))^{E}$, such that the
action of $G\rtimes\operatorname{Out}(G)$ is Poisson-Lie. Pulling back
$\pi_{\text{FR}}$ along $\iota$, we get the desired Poisson structure on
$\mathcal{M}_{\rho}^{\circ}(\Sigmait)$, which is compatible with the twisted
$G$-action. In Proposition 4.3 below we give an explicit formula for the
Poisson structure $\pi_{\mathcal{M}_{\rho}^{\circ}(\Sigmait)}$ we just
described on $\mathcal{M}_{\rho}^{\circ}(\Sigmait)$, which is a twisted
version of the Fock-Rosly Poisson structure on $G^{E}$ given in [FR98,
Proposition 3].

######  Proposition 4.3.

Let the surface $\Sigmait$ be represented by a ciliated fat graph with one
vertex $v$ and a set $E$ of edges. Let
$(x_{i})_{i=1,\dots,\text{dim}(\mathfrak{g})}$ be a basis of $\mathfrak{g}$.
Then for a given choice $r=r^{ij}x_{i}\otimes
x_{j}\in\left(\mathfrak{g}\otimes\mathfrak{g}\right)^{\operatorname{Out}(G)}$
of $\operatorname{Out}(G)$-invariant classical $r$-matrix there is a Poisson
structure on $\mathcal{M}_{\rho}^{\circ}(\Sigmait)$ given by the bivector

$\pi_{\mathcal{M}_{\rho}(\Sigmait)}=\sum_{\alpha\prec\beta}r^{ij}x_{i}(\alpha)\wedge
x_{j}(\beta)+\frac{1}{2}\sum_{\alpha}r^{ij}x_{i}(\alpha)\wedge x_{j}(\alpha)$

where $\alpha$ and $\beta$ run over the set of half-edges121212We break up the
edges of the graph, so that from each edge we get an incoming and an outgoing
half-edge at the vertex $v$. Since the chosen graph is ciliated, we get an
ordering $\prec$ on the set of half-edges. and

$x_{i}(\alpha)\coloneqq\begin{cases}-x_{i}^{R}(\alpha),&\alpha\text{ is
incoming at v}\\\ (\kappa_{\alpha})_{*}x_{i}^{L}(\alpha),&\alpha\text{ is
outgoing at v}\end{cases}$

where $x^{R/L}_{i}(\alpha)$ denotes the right/left-invariant vector field of
$x_{i}$ acting on the $\alpha$-copy of $G^{E}$. Furthermore, the induced
Poisson structure on the subalgebra of $G$-invariant functions is independent
of the chosen fat graph model for $\Sigmait$.

###  4.2 Quantisation

In Section 3.2 we constructed an algebra
$a_{P}^{\kappa_{1},\dots,\kappa_{n}}$, $n=2g+r-1$, from a combinatorial
presentation of the decorated surface $\Sigmait$. We now explain how these
algebras provide a deformation quantisation of the twisted Fock-Rosly Poisson
structure on $\mathcal{M}_{\rho}^{\circ}(\Sigmait)$. To that end, we consider
$a_{P}^{\kappa_{1},\dots,\kappa_{n}}$ as an object in the representation
category ${\mathsf{Rep}}_{\hbar}(G)$ of the formal quantum group. It is the
tensor product $\bigotimes_{i=1}^{n}\mathcal{O}_{\hbar}^{\kappa_{i}}(G)$,
where each $\mathcal{O}_{\hbar}^{\kappa_{i}}(G)$ is a $\kappa_{i}$-twisted REA
of quantised coordinate functions. The multiplication on the tensor product is
defined in terms of the crossing morphisms depicted in Figure 6. We will show
in Theorem 4.4 that for all elements
$f_{\hbar}^{\kappa_{i}}\in\mathcal{O}_{\hbar}^{\kappa_{i}}$ and
$g_{\hbar}^{\kappa_{j}}\in\mathcal{O}_{\hbar}^{\kappa_{j}}$ we have

$\frac{[f_{\hbar}^{\kappa_{i}},g_{\hbar}^{\kappa_{j}}]}{\hbar}\text{
mod}(\hbar)=\\{f^{\kappa_{i}},g^{\kappa_{j}}\\}\ \ ,$

where $\\{\cdot,\cdot\\}$ is the twisted Fock-Rosly Poisson structure from
Proposition 4.3, and $f^{\kappa_{i}}=f_{\hbar}^{\kappa_{i}}\text{
mod}(\hbar)\in\mathcal{O}^{\kappa_{i}}(G)$, and similarly for
$g^{\kappa_{j}}$.

We present a reformulation of the Poisson structure on
$\mathcal{M}_{\rho}^{\circ}(\Sigmait)$ that will prove useful for what
follows. Let $r=\omega+t$ be the decomposition of the classical r-matrix into
an anti-symmetric part $\omega$ and an invariant symmetric element $t$. For a
given automorphism $\kappa\in\operatorname{Out}(G)$, define the bivector field

$\displaystyle\pi^{\kappa}_{\text{STS}}\coloneqq\omega^{\operatorname{ad}(\kappa),\operatorname{ad}(\kappa)}+t^{R,L(\kappa)}-t^{L(\kappa),R}\
\ ,$ (4.6)

where the superscripts indicate that the action by left-invariant vector
fields is twisted by the automorphism $\kappa$, and we used the notation
$x^{\operatorname{ad}(\kappa)}=x^{R}-\kappa_{*}x^{L}$ for the vector field
generated by the element $x\in\mathfrak{g}$ via the twisted adjoint action
$h\longmapsto gh\kappa(g^{-1})$ of $G$ on itself. In the case $\kappa=e$, the
bivector field $\pi^{e}_{\text{STS}}$ agrees with the Semenov-Tian-Shansky
(STS) Poisson structure on $G$, see [STS94]. Using the decorated gluing
pattern $(P,\\{\kappa_{1},\dots,\kappa_{2g+n-1}\\})$ for $\Sigmait$, we define
the bivector

$\displaystyle\pi=\sum_{\alpha\in
E}\pi_{\text{STS}}^{\kappa_{\alpha}}+\sum_{\alpha<\beta}\pi_{\alpha,\beta}-\pi_{\beta,\alpha}\
\ ,$ (4.7)

where $\pi_{\alpha,\beta}$ is a 2-tensor, acting on the $\alpha$-component of
the first factor and on the $\beta$-component of the second factor of
$G^{E}\times G^{E}$, and is defined by

$\displaystyle\pi_{\alpha,\beta}\coloneqq\begin{cases}-r_{2,1}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}&\text{,
if $\alpha$ and $\beta$ are positively unlinked}\\\
-r_{2,1}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}-2t^{L(\kappa_{\alpha}),R}&\text{,
if $\alpha$ and $\beta$ are positively linked}\\\
-r_{2,1}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}-2t^{L(\kappa_{\alpha}),R}+2t^{L(\kappa_{\alpha}),L(\kappa_{\beta})}&\text{,
if $\alpha$ and $\beta$ are positively nested}\end{cases}$ (4.8)

And similarly, the 2-tensor $\pi_{\beta,\alpha}$ acts on the $\beta$-component
of the first factor and on the $\alpha$-component of the second factor of
$G^{E}\times G^{E}$ and is defined as
$\pi_{\beta,\alpha}=\tau(\pi_{\alpha,\beta})$, where $\tau$ swaps the two
tensor factors. Similarly, for the remaining three cases, we define

$\displaystyle\pi_{\alpha,\beta}\coloneqq\begin{cases}r_{1,2}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}&\text{,
if $\alpha$ and $\beta$ are negatively unlinked}\\\
r_{1,2}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}+2t^{R,L(\kappa_{\beta})}&\text{,
if $\alpha$ and $\beta$ are negatively linked}\\\
r_{1,2}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}+2t^{R,L(\kappa_{\beta})}-2t^{L(\kappa_{\alpha}),L(\kappa_{\beta})}&\text{,
if $\alpha$ and $\beta$ are negatively nested}\end{cases}$ (4.9)

and set again $\pi_{\beta,\alpha}=\tau(\pi_{\alpha,\beta})$. A direct
computation shows that $\pi$ agrees with the twisted Fock-Rosly Poisson
structure defined in Proposition 4.3.

######  Theorem 4.4.

The algebra $a_{P}^{\kappa_{1},\dots,\kappa_{2g+r-1}}$ is a quantisation of
the twisted Fock-Rosly Poisson structure on
$\mathcal{M}^{\circ}_{\rho}(\Sigmait)\cong G^{2g+r-1}$. Its subalgebra of
$U_{\hbar}(\mathfrak{g})$-invariants does not depend on the choice of the
gluing pattern $P$ and is a quantisation of the Poisson structure on the
affine quotient $\mathcal{M}_{\rho}^{\circ}(\Sigmait)\text{//}G$.

###### Proof.

First, we show that the quasi-classical limit of the commutator of two
quantised functions in $\mathcal{O}_{\hbar}^{\kappa}(G)$ agrees with the
$\kappa$-twisted STS Poisson structure $\pi_{\text{STS}}^{\kappa}$. We recall
from Example 3.2 that the multiplication in the $\kappa$-twisted REA
$\mathcal{O}_{\hbar}^{\kappa}(G)$ is related to the multiplication in the FRT-
algebra via a twisting cocycle given in terms of R-matrices. The commutator in
the (untwisted) FRT-algebra $H^{\circ}$, $H=U_{\hbar}(\mathfrak{g})$, can be
computed by acting with

$(1\otimes^{\operatorname{rev}}1)\boxtimes(1\otimes
1)-(\mathcal{R}_{2}^{-1}\otimes^{\operatorname{rev}}\mathcal{R}^{-1}_{1})\boxtimes(\mathcal{R}^{\prime}_{2}\otimes\mathcal{R}^{\prime}_{1})$

on the components $V^{\vee}\otimes^{\operatorname{rev}}W^{\vee}\boxtimes
V\otimes W$, for $V,W\in{\mathsf{Rep}}_{\hbar}(G)$, since the multiplication
in the FRT-algebra is given by the Hopf pairing $\langle-,-\rangle$ between
$H^{\circ}$ and $H$:

$\langle
m_{\text{FRT}}(\phi\psi),h\rangle=\langle\phi\otimes\psi,\Delta(h)\rangle,\quad\phi,\psi\in
H^{\circ},h\in H$

and $\Delta(-)=\mathcal{R}^{-1}\Delta^{\text{op}}(-)\mathcal{R}$. Now we take
into account the twist by $\kappa$, as well as the twisting cocycle
$\mathcal{R}^{\prime}_{1}\otimes\kappa.\mathcal{R}_{1}\otimes\mathcal{R}^{\prime}_{2}\mathcal{R}_{2}\otimes
1$, to compute the commutator in $\mathcal{O}_{\hbar}^{\kappa}(G)$ component-
wise by acting with

$\displaystyle(\mathcal{R}^{\prime}_{1}\otimes^{\operatorname{rev}}\mathcal{R}^{\prime}_{2}\mathcal{R}_{2})\boxtimes(\kappa.\mathcal{R}_{1}\otimes
1)-C\circ(\mathcal{R}^{\prime}_{2}\mathcal{R}_{2}\otimes^{\operatorname{rev}}\mathcal{R}^{\prime}_{1})\boxtimes(1\otimes\kappa.\mathcal{R}_{1})$
(4.10) $\displaystyle\text{where
}C=(\mathcal{R}_{2}^{-1}\otimes^{\operatorname{rev}}\mathcal{R}^{-1}_{1})\boxtimes(\kappa.\mathcal{R}^{\prime}_{2}\otimes\kappa.\mathcal{R}^{\prime}_{1})$
(4.11)

on $V^{\vee}\otimes^{\operatorname{rev}}W^{\vee}\boxtimes V\otimes W$. To
compute the quasi-classical limit of the action (4.10), we use that in the
limit $\exp(\hbar)\longrightarrow 1$, the R-matrix has the following
expansion: $\mathcal{R}=1+\hbar r+\mathcal{O}(\hbar^{2})$, where
$r=r_{1}\otimes r_{2}\in\mathfrak{g}^{\otimes 2}$ is the classical r-matrix.
Explicitly, the quasi-classical limit of (4.10) is

$r^{3(\kappa),2}+r^{1,2}-r^{4(\kappa),1}-r^{2,1}+r^{2,1}-r^{4,3}\in
U(\mathfrak{g})^{\otimes 4}\ \ ,$

where for instance $r^{3(\kappa),2}=1\otimes r_{2}\otimes
r_{1}^{\kappa}\otimes 1\in U(\mathfrak{g})^{\otimes 4}$ and the superscript
$\kappa$ means that the respective action will be twisted by $\kappa$. More
explicitly, the first two copies of $U(\mathfrak{g})^{\otimes 4}$ act on
$\mathcal{O}^{\kappa}(G)$ via $x\longmapsto x^{r}$, for $x\in\mathfrak{g}$,
and the last two copies act via $x\longmapsto-\kappa_{*}x^{L}$. Thus, we find
that the quasi-classical limit of the commutator is the bivector field on $G$
given by

$\displaystyle-r^{L(\kappa),R}+r^{R,R}+r_{2,1}^{R,L(\kappa)}-r_{2,1}^{L,L}$
$\displaystyle=\omega^{\operatorname{ad}(\kappa),\operatorname{ad}(\kappa)}+t^{R,L(\kappa)}-t^{L(\kappa),R}$
$\displaystyle=\pi_{\text{STS}}^{\kappa}\ \ ,$

where we used that $r^{R,R}-r_{2,1}^{L,L}=\omega^{R,R}+\omega^{L,L}$.

Next, we prove the claim for two positively unlinked edges $\alpha<\beta$. We
recall that the crossing morphism for two unlinked edges $\alpha<\beta$ is
given by acting on
$\mathcal{O}^{\kappa_{\beta}}_{\hbar}(G)\otimes\mathcal{O}^{\kappa_{\alpha}}_{\hbar}(G)$
with

$\displaystyle U^{+}$ $\displaystyle=\tau_{12,34}\circ(\mathcal{R}_{1}\otimes
1\otimes
1\otimes\kappa_{\alpha}.\mathcal{R}_{2})(1\otimes\kappa_{\beta}.\mathcal{R}_{1}\otimes
1\otimes\kappa_{\alpha}.\mathcal{R}_{2})(\mathcal{R}_{1}\otimes
1\otimes\mathcal{R}_{2}\otimes
1)(1\otimes\kappa_{\beta}.\mathcal{R}_{1}\otimes\mathcal{R}_{2}\otimes 1)$
$\displaystyle\coloneqq\tau_{12,34}\circ\widetilde{U}^{+}$

Hence, the commutator on components
$\phi\otimes\kappa_{\alpha}^{*}v\in\mathcal{O}^{\kappa_{\alpha}}_{\hbar}(G)$
and $\psi\otimes\kappa_{\beta}^{*}w\in\mathcal{O}^{\kappa_{\beta}}_{\hbar}(G)$
can be computed via

$(m_{\mathcal{O}^{\kappa_{\alpha}}_{\hbar}(G)}\otimes
m_{\mathcal{O}^{\kappa_{\beta}}_{\hbar}(G)})\circ(1-(U^{+})^{7,8,1,2})(\phi\otimes\kappa_{\alpha}^{*}v\otimes
1^{\otimes 4}\otimes\psi\otimes\kappa_{\beta}^{*}w)\ \ .$

Taking the quasi-classical limit of this action thus amounts to

$\frac{1-\tau(\widetilde{U}^{+})}{\hbar}~{}\text{mod}(\hbar)=-r^{3,2(\kappa_{\alpha})}-r^{4(\kappa_{\beta}),2(\kappa_{\alpha})}-r^{3,1}-r^{4(\kappa_{\beta}),1}\in
U(\mathfrak{g})^{\otimes 4}\ \,$ (4.12)

where this time the first and third copy in $U(\mathfrak{g})^{\otimes 4}$ act
via $x\longmapsto x^{R}$ and the second and the forth copy via
$x\longmapsto-\kappa_{*}x^{L}$, so that the right hand side of (4.12) acts on
$\mathcal{O}^{\kappa_{\alpha}}(G)\otimes\mathcal{O}^{\kappa_{\beta}}(G)$ via
$-r_{2,1}^{\operatorname{ad}(\kappa_{\alpha}),\operatorname{ad}(\kappa_{\beta})}$,
which agrees with $\pi_{\alpha,\beta}$ as claimed. Similarly, for two
positively linked edges we have

$\frac{1-\tau(\widetilde{L}^{+})}{\hbar}\text{
mod}(\hbar)=r^{2(\kappa_{\alpha}),3}-r^{4(\kappa_{\beta}),2(\kappa_{\alpha})}-r^{3,1}-r^{4(\kappa_{\beta}),1}\
\,$

and we see that in the positively linked case the 2-tensor
$\pi_{\alpha,\beta}$ differs from the unlinked case by adding a term
$-2t^{L(\kappa_{\alpha}),R}$. Lastly, for two positively nested edges we find

$\frac{1-\tau(\widetilde{N}^{+})}{\hbar}\text{
mod}(\hbar)=r^{2(\kappa_{\alpha}),3}+r^{2(\kappa_{\alpha}),4(\kappa_{\beta})}-r^{3,1}-r^{4(\kappa_{\beta}),1}\
\,$

which differs from the linked case by adding the term
$2t^{L(\kappa_{\alpha}),L(\kappa_{\beta})}$, which ends the proof for the
positively unlinked, linked and nested case. The remaining three cases can be
worked out analogously. ∎

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