Moduli spaces for quilted surfaces and Poisson structures

0.67 MB·English

by  David Li-Bland

MODULI SPACES FOR QUILTED SURFACES AND POISSON
STRUCTURES
DAVIDLI-BLANDANDPAVOLSˇEVERA
Abstract. LetGbeaLiegroupendowedwithabi-invariantpseudo-Riemannian
metric. Then the moduli space of flat connections on a principal G-bundle,
P → Σ, over a compact oriented surface, Σ, carries a Poisson structure. If
3 we trivialize P over a finite number of points on ∂Σ then the moduli space
1 carries a quasi-Poisson structure instead. Our first result is to describe this
0 quasi-Poisson structure in terms of an intersection form on the fundamental
2 groupoidofthesurface,generalizingresultsofMassuyeauandTuraev[19,27].
Oursecondresultistoextendthisframeworktoquilted surfaces,i.e. sur-
n
faces where the structure group varies from region to region and a reduction
a
(or relation) of structure occurs along the borders of the regions, extending
J
resultsofthesecondauthor[25,23,24].

3 WedescribethePoissonstructureonthemodulispaceforaquiltedsurface
2 intermsofanoperationonspinnetworks,i.e. graphsimmersedinthesurface
whichareendowedwithsomeadditionaldataontheiredgesandvertices. This

] extendstheresultsofvariousauthors[13,12,22,4].
G
D
.
h
t
a
m
[ Contents

2
1. Introduction 2
v
7 1.1. Acknowledgements 4
9 2. Quasi-Poisson manifolds 4
0 3. Reduction and moment maps 5
2
Partial reduction 7
.
2 4. Quasi-Poisson structures on moduli spaces 7
1 5. The homotopy intersection form and quasi-Poisson structures 11
2
6. Surfaces with boundary data 13
1
Description in terms of flat connections 16
:
v 7. Surfaces with domain walls 17
i
X Quilted surfaces with residual marked points 19
8. Spin networks 20
r
a 8.1. Spin networks on a marked surface 20
8.2. Spin networks and functions on the moduli space 21
8.3. The quasi-Poisson bracket on SpinNet (G) 22
Σ,V
8.4. Spin networks on quilted surfaces 25
9. Colorful examples 28
References 32
D.L-B.wassupportedbytheNationalScienceFoundationunderAwardNo. DMS-1204779.
P.Sˇ. was partially supported by the Swiss National Science Foundation (grants 140985 and
141329).
1
2 DAVIDLI-BLANDANDPAVOLSˇEVERA
1. Introduction
If G is a Lie group whose Lie algebra g carries an invariant metric, and Σ is a
closed oriented surface, the corresponding moduli space of flat connections
Hom(π (Σ),G)/G
1
on principal G-bundles P → Σ carries a symplectic form [5]; more generally, if Σ
has a boundary, then the moduli space carries a Poisson structure.
If Σ is connected, and one marks a point on one of the boundary components
Σ=
and trivializes the principal bundle over that point, the moduli space
Hom(π (Σ),G)
1
becomes quasi-Poisson [3, 2, 1]. In a recent paper [19], Massuyeau and Turaev
described this quasi-Poisson structure in terms of an intersection form on the loop
algebra Zπ (Σ), extending a result of Goldman [12, 13]. The first result of our
1
paper is to generalize their result to the case where Σ has multiple marked points
(possibly on the same boundary component):
Σ=
These surfaces allow for more economical description of the moduli spaces — in
particular, we show how to obtain them from a collection of discs with two marked
points each via multiple fusion.
Blowing up at each of the marked points, we obtain a surface which we call a
domain:
Σ=
We refer to the preimage of any marked point as a domain wall (these are the
thickened segments of the boundary in the image above). Our second result is the
following: Supposeonechoosesareductionofstructureseparatelyforeachdomain
wall w, i.e.
• a subgroup L ⊆G, and
w
• a subbundle Q →w of P| →w on which L acts transitively.
w w w
If the Lie algebras l ⊆ g corresponding to L ⊆ G each satisfy l⊥ = l , then
w w w w
the moduli space of flat connections on P which are compatible with the reduced
bundles Q →w is Poisson (this result can be generalized to the case of l⊥ ⊂l ).
w w w
Wemaythinkofthisas‘coloring’eachdomainwallwithareducedstructuregroup
L ⊆G, as pictured below:
w
MODULI SPACES FOR QUILTED SURFACES AND POISSON STRUCTURES 3
Σ=
Inthiswayweobtain,inparticular,thePoissonstructuresinvertingthesymplectic
forms carried by the moduli spaces of colored surfaces, introduced in [25] (see also
[23, 24]).
Suppose now that G(cid:48) is a second Lie group whose Lie algebra g(cid:48) carries an
invariant metric, and P(cid:48) → Σ(cid:48) is a principal G(cid:48)-bundle over a domain Σ(cid:48). Once
again, we choose a reduction of structure for the bundle P| → w(cid:48) over each
w(cid:48)
domain wall w(cid:48) on Σ(cid:48), i.e. a subgroup L ⊆ G(cid:48). If we simultaneously consider
w(cid:48)
flat connections on P and P(cid:48) which are compatible with the reduced structure on
each domain wall, then (as before) the moduli space is Poisson. We picture this as
follows:
Σ Σ(cid:48)
However,onemightinsteadwishtochooseacommonreductionofstructurefortwo
domain walls, w and w(cid:48) (on Σ and Σ(cid:48), resp.). More precisely, to sew the domain
walls w and w(cid:48) together is to choose an identification φ:w→w(cid:48), together with
• a subgroup L ⊆G×G(cid:48), and
φ
• a subbundle Q →w of P| × P(cid:48)| on which L acts transitively,
φ w φ w(cid:48) φ
such that the Lie algebra l ⊆g⊕g¯(cid:48) corresponding to L satisfies l⊥ =l (where g¯(cid:48)
φ φ φ
denotes the Lie algebra g(cid:48) with the metric negated). Quilted surfaces are surfaces
formedbysewingdomainstogetheralongdomainwalls,andbychoosingareduction
of structure on any of the remaining (unsewn) domain walls, as was previously
described.
Σ =
quilt
Such surfaces have played a role in recent developments in both Chern-Simons
theory [15, 16, 14] and Floer theory [29, 28]. Our second main result is to show
that the moduli space of flat connections, M , on a quilted surface is Poisson.
Σquilt
We provide a description of this Poisson structure in terms of spin networks
[21, 6], as in [22, 4, 13, 12]. More precisely, we identify functions f ∈C∞(M )
Σquilt
on the moduli space of flat connections over a quilted surface with spin networks
in the quilted surface. Such a spin network [Γ,∗] consists of an immersed graph
Γ→Σ ,
quilt
4 DAVIDLI-BLANDANDPAVOLSˇEVERA
together with some decoration1 of the edges and vertices of the graph, which (in
the introduction) we will denote abstractly by ∗. The Poisson bracket of two spin
networks [Γ,∗] and [Γ(cid:48),∗(cid:48)] is computed as a sum over their intersection points
p∈Γ× Γ(cid:48),
Σquilt
(cid:8)[Γ,∗],[Γ(cid:48),∗(cid:48)](cid:9)= (cid:88) ±[Γ∪ Γ(cid:48),∗(cid:48)(cid:48)],
p
p∈Γ×ΣquiltΓ(cid:48)
where Γ∪ Γ(cid:48) denotes the union of the two graphs with a common vertex added at
p
the intersection point p. This formula generalizes the one found in [22].
The basic technical tool we use is a new type of reduction of quasi-Poisson G-
manifolds by subgroups of G.
Inthispaperwestudythemodulispacesfromthe(quasi-)Poissonpointofview.
The approach via (quasi-)symplectic 2-forms and a unifying picture using Courant
algebroids will be appear in a future paper.
1.1. Acknowledgements. TheauthorswouldliketothankEckhardMeinrenken,
Alan Weinstein, Marco Gualtieri, Anton Alekseev, Alejandro Cabrera and Dror
Bar-Natan for helpful discussions, explanations, and advice.
2. Quasi-Poisson manifolds
In this section we recall the basic definitions from the theory of quasi-Poisson
manifolds, as introduced by Alekseev, Kosmann-Schwarzbach, and Meinrenken [2,
1].
Let G be a Lie group with Lie algebra g and with a chosen Ad-invariant sym-
metricquadratictensor,s∈S2g. Letφ∈(cid:86)3gbetheAd-invariantelementdefined
by
(1) φ(α,β,γ)= 1α(cid:0)[s(cid:93)β,s(cid:93)γ](cid:1) (α,β,γ ∈g∗),
4
where s(cid:93) :g∗ →g is given by β(s(cid:93)α)=s(α,β).
Suppose ρ : G×M → M is an action of G on a manifold M. Abusing nota-
tion slightly, we denote the corresponding Lie algebra action ρ : g → Γ(TM), by
(cid:86)
the same symbol. We extend ρ to a Gerstenhaber algebra morphism ρ : g →
(cid:86)
Γ( TM).
Definition 1. A quasi-Poisson G-manifold is a triple (M,ρ,π), where M is a
manifold, ρ an action of G on M, and π ∈Γ((cid:86)2TM) a G-invariant bivector field,
satisfying
1
[π,π]=ρ(φ).
2
This definition depends on the choice of s. If G , G are Lie groups with chosen
1 2
elements s ∈ S2g (i = 1,2), we set s = s +s ∈ S2(g ⊕g ), so that we can
i i 1 2 1 2
speak about quasi-Poisson G × G -manifolds. In particular, if (M ,ρ ,π ) is a
1 2 i i i
quasi-Poisson G -manifold (i=1,2) then
i
(M ,ρ ,π )×(M ,ρ ,π )=(M ×M ,ρ ×ρ ,π +π )
1 1 1 2 2 2 1 2 1 2 1 2
is a quasi-Poisson G ×G -manifold.
1 2
Example 1. G is a quasi-Poisson G×G-manifold, with the action ρ(g ,g )·g =
1 2
g gg−1 and with π =0.
1 2
1Note, when our structure groups are compact, the graph is decorated with a representation
on each edge, and each vertex is decorated with an intertwinor of the representations on the
surroundingedges,asin[22].
MODULI SPACES FOR QUILTED SURFACES AND POISSON STRUCTURES 5
Remark 1. Since s appears twice in Eq. (1), it follows that any quasi-Poisson
(G,s)-manifold is also a quasi-Poisson (G,−s)-manifold. Likewise, any quasi-
Poisson (G ×G ,s ⊕s )-manifold is also a quasi-Poisson (G ×G ,s ⊕−s )-
1 2 1 2 1 2 1 2
manifold.
Let ψ ∈(cid:86)2(g⊕g) be given by
1(cid:88)
ψ = sij(e ,0)∧(0,e )
2 i j
i,j
where s=(cid:80) sije ⊗e in some basis e of g.
i,j i j i
Definition 2. If (M,ρ,π) is a quasi-Poisson G×G×H-manifold then its fusion
is the quasi-Poisson G×H-manifold (M,ρ∗,π∗), where ρ∗(g,h)=ρ(g,g,h) and
π∗ =π−ρ(ψ).
Fusionisassociative(butnotcommutative): ifM isaquasi-PoissonG×G×G×
H-manifold then the two G×H-quasi-Poisson structures obtained by the double
fusioncoincide. IfM isaquasi-PoissonGn×H-manifoldthenits(multiple)fusion
to a quasi-Poisson G×H-manifold is given by
(cid:88)
(2) π∗ =π− ρ(ψ ),
i,j
i<j
where ψ ∈(cid:86)2(n×g) is the image of ψ under the inclusion g⊕g→n×g sending
i,j
the two g’s to i’th and j’th place respectively.
3. Reduction and moment maps
A Lie subgroup C ⊆G will be called reducing if its Lie algebra c⊆g satisfies
φ(α,β,γ)=0 ∀α,β,γ ∈ann(c)
where ann(c)⊆g∗ is the annihilator of c. Equivalently,
[s(cid:93)α,s(cid:93)β]∈c ∀α,β ∈ann(c).
In particular, if C ⊆G is coisotropic, i.e. if s(cid:93)(ann(c))⊆c, then C is reducing.
Theorem 1.A. Suppose that (M,ρ,π) is a quasi-Poisson G-manifold and that
C ⊆G is a reducing subgroup. Then
(3) {f,g}:=π(df,dg), f,g ∈C∞(M)C
is a Poisson bracket on the space of C-invariant functions. In particular, if the
C-orbits of M form a regular foliation2, then the bivector field π descends to define
a Poisson structure on M/C.
Proof. Theproofisessentiallythesameasthatof[1,Theorem4.2.2],butweinclude
it for completeness.
First, we observe that {f,g}∈C∞(M)C, since f,g and π are each C-invariant.
To see that the bracket (3) satisfies the Jacobi identity, notice that
1
{f ,{f ,f }}+c.p.= [π,π](df ,df ,df )=ρ(φ)(df ,df ,df ),
1 2 3 2 1 2 3 1 2 3
for any f ∈ C∞(M)C (i = 1,2,3). Now ρ∗df ∈ ann(c) and C is reducing, hence
i i
ρ(φ)(df ,df ,df )=0. (cid:3)
1 2 3
For ξ ∈g let ξL and ξR denote the corresponding left and right invariant vector
field on G.
2i.e. the orbit space, M/C, is a manifold, and the projection M → M/C is a surjective
submersion
6 DAVIDLI-BLANDANDPAVOLSˇEVERA
Definition 3. Let (M,ρ,π) be a quasi-Poisson G-manifold and let τ : G → G be
an s-preserving automorphism. A map µ : M → G is a (τ-twisted) moment map
if it is equivariant for the action g·g˜=τ(g)g˜g−1 of G on G, and if the image of
π under
µ ⊗id:TM ⊗TM →TG⊗TM
∗
is
−1sij(cid:0)eL+τ(e )R(cid:1)⊗ρ(e ).
2 i i j
We shall use moment maps to get Poisson submanifolds of M/C, in analogy
with Marsden-Weinstein reduction (under certain non-degeneracy conditions these
submanifolds will be the symplectic leaves of M/C). First we need an analogue of
coadjoint orbits.
Lemma 1. If C ⊆G is a reducing subgroup then
ˆc:={(ξ+s(cid:93)α,ξ−s(cid:93)α); ξ ∈c,α∈ann(c)}
is a Lie subalgebra of g⊕g.
Proof. Let ξ,η ∈c, α,β ∈ann(c). Since
[ξ±s(cid:93)α,η±s(cid:93)β]=(cid:0)[ξ,η]+[s(cid:93)α,s(cid:93)β](cid:1)±s(cid:93)(ad∗β−ad∗α)
ξ η
and [s(cid:93)α,s(cid:93)β]∈c, the spaceˆc is closed under the Lie bracket. (cid:3)
Let Cˆ ⊆G×G be a Lie group with the Lie algebraˆc. The group G×G acts on
G by
(4) (g ,g )·g =τ(g )gg−1.
1 2 1 2
The orbits of Cˆ ⊆G×G on G will serve as analogues of coadjoint orbits.
Theorem 1.B. Let (M,ρ,π) be a quasi-Poisson G-manifold with a moment map
µ:M →G and C ⊆G a reducing subgroup. Suppose that the C-orbits of M form
a regular foliation. Let O ⊆G be a Cˆ-orbit. If the graph of µ intersects O cleanly3,
then
µ−1(O)/C ⊆M/C
is a Poisson submanifold. More generally, if K ⊆ G is a Cˆ-stable submanifold
and the graph of µ intersects K cleanly, then µ−1(K)/C ⊆ M/C is a Poisson
submanifold.
Proof. First of all notice that µ−1(K)⊆M is stable under the action of C ⊆G.
For any f ∈C∞(M) the moment map condition gives
(5) µ (cid:0)π(·,df)(cid:1)=−1(cid:0)(s(cid:93)ρ∗df)L+τ(s(cid:93)ρ∗df)R(cid:1).
∗ 2
If f is C-invariant then α := ρ∗df belongs to ann(c). The vector (5) is thus the
action of 1(s(cid:93)α,−s(cid:93)α)∈ˆc on G. In particular, π(·,df) is tangent to µ−1(K). This
2
implies that µ−1(K)/C ⊆M/C is a Poisson submanifold. (cid:3)
Thespaceµ−1(O)/C canbeconvenientlydescribedinthefollowingway. LetMˆ
be the Cˆ-manifold obtained from M by induction from C to Cˆ (using the diagonal
embedding C ⊂Cˆ). Concretely,
(6) Mˆ =(Cˆ×M)/C
3That is, gr(µ)∩O ⊆ G×M is a submanifold, and T(gr(µ)∩O) = Tgr(µ)∩TO, where
gr(µ)={(µ(x),x); x∈M}. Ithappens,inparticular,ifµistransversetoO.
MODULI SPACES FOR QUILTED SURFACES AND POISSON STRUCTURES 7
where the C-action on Cˆ × M is c · (cˆ,m) = (cˆc−1,c · m). The moment map
µ:M →G induces a Cˆ-equivariant map
µˆ :Mˆ →G, µˆ(cˆ,m)=cˆ·µ(m).
Let now O =Cˆ·g for some g ∈G. Since µ−1(O)/C ∼=µˆ−1(O)/Cˆ, we have
0 0
(7) µ−1(O)/C ∼=µˆ−1(g )/Stab(g )
0 0
where
Stab(g )={cˆ∈Cˆ; cˆ·g =g }⊆Cˆ.
0 0 0
Partial reduction. One can generalize both Theorem 1.A and Theorem 1.B in
order to reduce quasi-Poisson G×H-manifolds to quasi-Poisson H-manifolds:
Theorem 1.C. Suppose that (M,ρ,π) is a quasi-Poisson G×H-manifold, C ⊆G
is a reducing subgroup, and the C-orbits of M form a regular foliation.
(1) The bivector field π descends to define a quasi-Poisson H-structure on
M/C.
(2) Let τ and τ be automorphisms of G and H, and let
G H
(µ ,µ ):M →G×H
G H
bea(τ ,τ )-twistedmomentmap. IfK ⊆GisaCˆ-stablesubmanifoldand
G H
the graph of µ intersects K cleanly, then µ−1(K)/C ⊆ M/C is a quasi-
G G
Poisson H-submanifold, and µ descends to define a τ -twisted moment
H H
map,
µ :µ−1(K)/C →H.
H G
Proof. The proof of the first statement is only superficially different from that of
Theorem 1.A and so we omit it.
Likewise, we omit the proof that µ−1(K)/C ⊆ M/C is a quasi-Poisson H-
G
submanifold, as it differs only superficially from that of Theorem 1.B.
Finally, since µ :µ−1(K)→H is G×H-equivariant, it descends to a map on
H G
µ−1(K)/C. The image of π under (µ ) ⊗id:TM ⊗TM →TH ⊗TM is
G H ∗
−1sij(cid:0)eL+τ(e )R(cid:1)⊗ρ(e ),
2 H i i j
where s ∈S2(h)H denotes the chosen invariant symmetric tensor, and hence this
H
alsoholdsforthereducedbivectorfieldonbothM/C andµ−1(K)/C,provingthat
G
µ descends to define a moment map. (cid:3)
H
4. Quasi-Poisson structures on moduli spaces
Let Σ be a compact oriented surface with boundary, and let V ⊂ ∂Σ be a
finite collection of marked points such that every component of Σ intersects V.
Let Π (Σ,V) denote the fundamental groupoid of Σ with the base set V. The
1
composition in Π (Σ,V) is from right to left: ab means path b followed by path
1
a. For a ∈ Π (Σ,V) let out(a) denote the source and in(a) the target of a; ab is
1
defined if in(b)=out(a).
Let
M (G)=Hom(Π (Σ,V),G).
Σ,V 1
M (G)canbeseenasthemodulispaceofflatconnectionsonprincipalG-bundles
Σ,V
over Σ which are trivialized over V.
For any arrow a∈Π (Σ,V) let
1
hol :M (G)→G
a Σ,V
8 DAVIDLI-BLANDANDPAVOLSˇEVERA
denote evaluation at a (in terms of flat connections it is the holonomy along a).
There is a natural action ρ = ρ of the group GV on M (G) which is defined
Σ,V Σ,V
by
(8) hol (ρ(g)x)=g hol (x)g−1 .
a in(a) a out(a)
Infinitesimally,
(hol ) (ρ(ξ))=−ξR +ξL
a ∗ in(a) out(a)
for any ξ ∈ gV, where ξL/R denotes the left/right invariant vector field on G
corresponding to ξ ∈g.
Byaskeleton of(Σ,V)wemeanagraphΓ⊂ΣwiththevertexsetV, suchthat
there is a deformation retraction of Σ to Γ.4 If we choose an orientation of every
edge of Γ then MΣ,V(G) gets identified (via (hola,a∈EΓ)) with GEΓ, where EΓ is
the set of edges of Γ. In particular, if Σ is a disc and V has two elements then we
get M (G)=G.
Σ,V
The boundary of Σ is split by V to arcs (the components of ∂Σ that don’t
contain a marked point5 are not considered to be arcs). If we choose an ordered
pair (P,Q) of marked points (P (cid:54)= Q ∈ V) then the corresponding fused surface
Σ∗ is obtained by gluing a short piece of the arc starting at P with a short piece
of the arc ending at Q (so that P and Q get identified). The subset V∗ ⊂ ∂Σ∗ is
obtained from V by identifying P and Q. Notice that the map
M (G)→M (G),
Σ∗,V∗ Σ,V
coming from the map (Σ,V) → (Σ,V∗), is a diffeomorphism: if Σ retracts to a
skeletalgraphΓthenΣ∗ retractstoitsimageΓ∗,andthetwographshavethesame
number of edges. We can thus identify the manifolds M (G) and M (G).
Σ∗,V∗ Σ,V
A C A∗ C∗
P Q
P∗ =Q∗
B D B∗ D∗
Σ Σ∗
Figure 1. Fusion
Every (Σ,V) can be obtained by fusion from a collection of discs, each with two
marked points: If Γ ⊂ Σ is a skeleton then the subset of Σ that retracts onto an
edge e∈E is a disc D , and Σ is obtained from D ’s by repeated fusion.
Γ e e
Theorem 2. There is a natural bivector field π on M (G) such that
Σ,V Σ,V
(M (G),ρ ,π )
Σ,V Σ,V Σ,V
is a quasi-Poisson manifold, uniquely determined by the properties
(1) if Σ is a disc and V has two elements then π =0
Σ,V
(2) if (Σ,V)=(Σ ,V )(cid:116)(Σ ,V ) then π =π +π
1 1 2 2 Σ,V Σ1,V1 Σ2,V2
(3) if(Σ∗,V∗)isobtainedfrom(Σ,V)byfusion,then(M (G),ρ ,π )
Σ∗,V∗ Σ∗,V∗ Σ∗,V∗
is obtained from (M (G),ρ ,π ) by the corresponding fusion.
Σ,V Σ,V Σ,V
If there is no danger of confusion, we shall denote π simply by π.
Σ,V
4Itisasimpleexercisetoshowthatthereexistsaskeletonforeverymarkedsurface. However,
itshouldbeemphasizedthatthisskeletonisnotunique!
5thatis,anelementofV
MODULI SPACES FOR QUILTED SURFACES AND POISSON STRUCTURES 9
Figure 2. A surface with a skeleton, fused from four discs. The
discshavebeenassignedcolorsinthepicture,butthisfigureshould
not be confused with a quilted surface.
Remark 2. Alejandro Cabrera has independantly studied quasi-Hamiltonian GV-
structures for the marked surfaces described above.
Once we choose a skeleton Γ of (Σ,V), Theorem 2 gives us a formula for the
quasi-Poisson structure on M , as (Σ,V) is a fusion of a collection of discs with
Σ,V
two marked points. Let us denote the resulting bivector field on M (G) by π .
Σ,V Γ
Theorem 2 follows from the following Lemma:
Lemma 2. The bivector field π on M (G) is independent of the choice of Γ.
Γ Σ,V
Remark 3. The lemma follows from the special case where (Σ,V) is a disc with 3
marked points (see Example 3 below). However, we shall give a different proof in
the next section.
Proof of Theorem 2. By the lemma we have a well-defined quasi-Poisson structure
on M (G). Properties (1)–(3) of the theorem are satisfied by the construction of
Σ,V
π . (cid:3)
Γ
Let us now describe the calculation of π = π in more detail. Notice that for
Γ
any vertex v of Γ, the (half)edges adjacent to v are linearly ordered: a cyclic order
is given by the orientation of Σ. Since v is on the boundary, the cyclic order is
actually a linear order. Γ is a ciliated graph in the terminology of Fock and Rosly
[11].
We choose an orientation of every edge of Γ to get an identification M (G)=
Σ,V
GEΓ. Firstweseeitasa GEΓ×GEΓ-quasi-Poissonspacewithzerobivector(i.e.as
M (G),where(Σ(cid:48),V(cid:48))isadisjointunionofdiscswithtwomarkedpointseach).
Σ(cid:48),V(cid:48)
Then, fusing at each vertex using the linear order, we obtain a GV-quasi-Poisson
space.
Example 2. As the simplest example, suppose (Σ,V) is an annulus with a single
marked point (on one of the boundary circles). Then (Σ,V) may be obtained by
fusion from a disc (Σ(cid:48),V(cid:48)) with two marked points, as in Fig. 3. Now M =G
Σ(cid:48),V(cid:48)
with the the quasi-Poisson G×G-structure described in Example 1: the bivector
fieldistrivialandG×Gactsby(g ,g )·g =g gg−1. ThusM =G,theG-action
1 2 1 2 Σ,V
is by conjugation, and
1
π = sijeR∧eL.
2 i j
10 DAVIDLI-BLANDANDPAVOLSˇEVERA
P Q
Figure 3. The annulus with one marked point is obtained by
fusion from the disc with two marked points.
Example 3. Let Σ be a triangle and V is the set of its vertices.
c b
a
WecanidentifyM withG2via(hol ,hol ),i.e.Γisthegraphwiththeoriented
Σ,V a−1 b
edges a−1, b. In this case
1
(9) π =− sijeL(1)∧eL(2)
2 i j
(whereeL(k)denotestheleft-invariantvectorfieldwhichistangenttothekthfactor
i
of G2 (k =1,2)). Equivalently,
π(hol∗ θL,hol∗θL)=−s∈g⊗g,
a−1 b
where θL ∈Ω1(G,g) is the left-invariant Maurer-Cartan form. An easy calculation
shows
π(hol∗ θL,hol∗θL)=−s∈g⊗g,
b−1 c
confirming that π is independent of the choice of Γ.
Forageneralsurface(Σ,V)withachoiceofaskeletonΓ,wegetanidentification
MΣ,V =GEΓ. Applying (2) we obtain
1 (cid:88)(cid:88)
(10) π =− sije (a,v)∧e (b,v)
Σ,V 2 i j
v∈V a<b
where a,b run over the (half)edges adjacent to v,
(cid:40)
eR(a) a goes into v
e (a,v)= i
i −eL(a) a goes out of v
i
and for a ∈ EΓ, eRi ,L(a) denotes the right/left-invariant vector field on GEΓ equal
to
a
(0,...,0,(cid:122)(cid:125)e(cid:124)(cid:123),0,...,0)∈gEΓ
i
at the identity element. Essentially the same formula was discovered by Fock and
Rosly [11], for Poisson structures on M obtained by a choice of a classical r-
Σ,V
matrix. Meanwhile, Skovborg studied the corresponding formula in the absence of
an r-matrix for invariant functions [26].