Moduli Spaces for Quilted Surfaces

and Poisson Structures

David Li-Bland^{1} and Pavol Ševera^{2}

Received: April 2, 2013 Revised: September 21, 2015

Communicated by Eckhard Meinrenken

Abstract. Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P → Σ, over a compact oriented surface with boundary, Σ, carries a Poisson structure. If we trivialize P over a finite number of points on ∂Σ then the moduli space carries a quasi-Poisson structure instead. Our first result is to describe this quasi-Poisson structure in terms of an intersection form on the fundamental groupoid of the surface, generalizing results of Massuyeau and Turaev [27, 38].

Our second result is to extend this framework to quilted surfaces, i.e. surfaces where the structure group varies from region to region and a reduction (or relation) of structure occurs along the borders of the regions, extending results of the second author [33–35].

We describe the Poisson structure on the moduli space for a quilted surface in terms of an operation on spin networks, i.e. graphs immersed in the surface which are endowed with some additional data on their edges and vertices. This extends the results of various authors [16, 17, 31, 32].

2010 Mathematics Subject Classification: Primary 53D30, 53D17 Keywords and Phrases: Moduli Spaces, Flat connections, Flat bundles, Chern Simons, Topological Defects, Poisson Geometry, Symplectic Geometry, Spin Networks, Representation Theory, Lie Group, Lie algebra, Atiyah Bott, Quasi-Poisson geometry, quasi-Poisson reduction, Poisson Lie groups, Poisson Homogeneous spaces Contents 1 Introduction 1072

1D.L-B. was supported by the National Science Foundation under Award No. DMS-1204779.

2P.Š. was partially supported by the Swiss National Science Foundation (grants 140985 and 141329).

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1.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 1076 2 Quasi-Poisson manifolds 1076 3 Reduction and moment maps 1077

3.1 Induction and the Hat construction . . . . . . . . . . . . . . . . 1081 4 Quasi-Poisson structures on moduli spaces 1083 5 The homotopy intersection form and quasi-Poisson struc-

tures 1087 6 Surfaces with boundary data 1092 7 Surfaces with domain walls 1097 8 Spin networks 1102

8.1 Spin networks on a marked surface . . . . . . . . . . . . . . . . 1102 8.2 Spin networks and functions on the moduli space . . . . . . . . 1104 8.3 The quasi-Poisson bracket on SpinNet_{Σ,V} (G) . . . . . . . . . . 1106 8.4 Spin networks on quilted surfaces . . . . . . . . . . . . . . . . . 1109

9 Colorful examples 1113 A Quilted surfaces and moduli spaces of flat bundles 1127 B Some Technical Lemmas 1131

1 Introduction Let G denote a quadratic Lie group, i.e. a Lie group endowed with a bi-invariant pseudo-Riemannian metric. If Σ is a closed oriented surface, the corresponding moduli space of flat G-bundles over Σ (note: we always take our G-bundles to be principal)

Hom(π_{1}(Σ), G)/G

carries a symplectic form [4]; 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_{1}(Σ), G)

becomes quasi-Poisson [1–3]. In a recent paper [27], Massuyeau and Turaev described this quasi-Poisson structure in terms of an intersection form on the loop algebra ℤπ_{1}(Σ), extending a result of Goldman [16,17]. The first result of our 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 iterated fusion. Blowing up at each of the marked points, we obtain a surface which we call a domain:

Σ̂ =

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: Suppose one chooses a reduced structure separately for each domain wall w, i.e. a subgroup L_{w} ⊂ G. If the Lie algebras 𝔩_{w} ⊂ 𝔤 corresponding to L_{w} ⊂ G each satisfy 𝔩_{w}^{⊥} ⊆ 𝔩_{w}, then the moduli space of

• flat G-bundles over Σ̂ equipped with a flat^{3} reduction of the structure group from G to L_{w} over each domain wall w is Poisson. We may think of this as ‘coloring’ each domain wall with a reduced structure group L_{w} ⊆ G, as pictured below:

3Let P |_{w} → w be a flat principal G-bundle. By a a flat reduction of the structure from G to L_{w}, we mean a choice of principal L_{w}-subbundle Q_{w} → w of P |_{w} which is flat with respect to the connection on P |_{w}.

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Σ =

In this way we obtain, in particular, the Poisson structures inverting the symplectic forms carried by the moduli spaces of colored surfaces, introduced in [35] (see also [33, 34]). Suppose now that G′ is a second Lie group whose Lie algebra 𝔤′ carries an invariant metric. Once again, we choose a reduced structure over each domain wall w′ on Σ′, i.e. a subgroup L_{w′} ⊂ G′. If we simultaneously consider flat Gbundles over Σ and G′-bundles over Σ′ which are compatible with the reduced structure on each domain wall, then (as before) the moduli space is Poisson. We picture this as follows:

\begin{aligned}& Σ′ \\ Σ \end{aligned}

However, one might instead wish to choose a common reduction of structure for two domain walls, w and w′ (on Σ and Σ′, resp.). More precisely, to sew the domain walls w and w′ together is to choose an (orientation reversing) identification ϕ : w → w′; Σ ∪_{ϕ} Σ′ is then the sewn surface. We understand the common image of w and w′ to define a domain wall in the sewn surface. A quilted surface Σ_{quilt} is formed by sewing a collection of domains {Σ_{i}}_{i=1}^{n} together along domain walls, and by choosing

• a quadratic Lie group G_{i} for each domain Σ_{i},

• a reduced structure L_{w} ⊂ G_{i} for each unsewn domain wall w ⊂ Σ_{i}; such that 𝔩_{w} ⊆ 𝔩_{w}^{⊥}, and

• a reduced structure L_{w} ⊂ G_{i}×G_{j} for each sewn domain wall w ⊂ Σ_{i}∩Σ_{j}; such that 𝔩_{w} ⊆ 𝔩_{w}^{⊥} as a Lie subalgebra of 𝔤_{i} ⊕ 𝔤̄_{j} (where 𝔤̄_{j} denotes the Lie algebra 𝔤_{j} with the metric negated).

Σ_{quilt} =

Such surfaces (or closely related ones) have played a role in recent developments in both Chern-Simons theory [18–20] and Floer theory [39,40]. Our second main result is to show that the moduli space, M_{Σ_{quilt}}, of

• tuples {P_{i} → Σ_{i}}_{i=1}^{n} of flat G_{i}-bundles, equipped with

1. a flat^{4} reduction of the structure group of P_{i}|_{w} from G_{i} to L_{w} over each unsewn domain wall w ⊂ Σ_{i}, and

2. a flat reduction of the structure group of P_{i} ×_{w} P_{j} from G_{i} × G_{j} to L_{w} over each sewn domain wall w ⊂ Σ_{i} ∩ Σ_{j},

is Poisson. We will often call M_{Σ_{quilt}} a moduli space of flat bundles over a quilted surface. We provide a description of this Poisson structure in terms of spin networks [5, 30], as in [16,17,31,32]. More precisely, we identify functions f ∈ C^{∞}(M_{Σ_{quilt}}) 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},

together with some decoration^{5} 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′, ∗′] is computed as a sum over their intersection points p ∈ Γ ×_{Σ_{quilt}} Γ′,

{[Γ, ∗], [Γ′, ∗′]} = ∑_{p∈Γ×_{Σ_{quilt}} Γ′} ±[Γ ∪_{p} Γ′, ∗′′],

where Γ∪_{p} Γ′ denotes the union of the two graphs with a common vertex added at the intersection point p. This formula generalizes the one found in [32]. The basic technical tool we use is a new type of reduction of quasi-Poisson G-manifolds by subgroups of G. In this paper we study the moduli spaces from the (quasi-)Poisson point of view. An alternative approach via (quasi-)symplectic 2-forms (or more generally, Dirac geometry) appears in [21]; and the equivalence between these two approaches is explained in [36]. It should be mentioned that the approach taken in [21] allows the construction of more general moduli spaces, while the approach taken in the current paper is more easily quantized (cf. [22]).

4i.e. a principal L_{w}-subbundle Q_{w} → w of P_{i}|_{w} which is flat with respect to the connection on P_{i}|_{w}.

5Note, 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 surrounding edges, as in [32].

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Remark 1. Of course, one could also consider sewing k domains together along a single domain wall, together with a compatible reduction of structure along that domain wall. The moduli space of flat bundles over the resulting branched surfaces is also Poisson [21], and there is an analogous description of the Poisson structure in terms of spin networks. To keep our presentation simple, however, we restrict to the case of quilted surfaces (k = 1, 2).

1.1 Acknowledgements

The authors would like to thank Eckhard Meinrenken, Alan Weinstein, Marco Gualtieri, Anton Alekseev, Alejandro Cabrera, Dror Bar-Natan, and Jiang-Hua Lu for helpful discussions, explanations, and advice. We’d also like to thank the various referees for the helpful suggestions we received which improved the overall readability of the paper. David Li-Bland was supported by NSF Grant DMS-1204779. Pavol Severa was supported by the grant MODFLAT of the European Research Council and the NCCR SwissMAP of the Swiss National Science Foundation.

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 [1, 2]. Let G be a Lie group with Lie algebra 𝔤 and with a chosen Ad-invariant symmetric quadratic tensor, s ∈ S^{2}𝔤. Let ϕ ∈ ⋀^{3} 𝔤 be the Ad-invariant element defined by

ϕ(α, β, γ) = \frac{1}{4}α([s^{♯}β, s^{♯}γ]) (α, β, γ ∈ 𝔤^{∗}),  (1)

where s^{♯} : 𝔤^{∗} → 𝔤 is given by β(s^{♯}α) = s(α, β). Suppose ρ : G × M → M is an action of G on a manifold M. Abusing notation slightly, we denote the corresponding Lie algebra action ρ : 𝔤 → Γ(T M), by the same symbol. We extend ρ to a Gerstenhaber algebra morphism ρ : ⋀ 𝔤 → Γ(⋀ T M).

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(⋀^{2} T M) a G-invariant bivector field, satisfying

\frac{1}{2}[π, π] = ρ(ϕ).  (2)

This definition depends on the choice of s. If G_{1}, G_{2} are Lie groups with chosen elements s_{i} ∈ S^{2}𝔤_{i} (i = 1, 2), we set s = s_{1} + s_{2} ∈ S^{2}(𝔤_{1} ⊕ 𝔤_{2}), so that we can speak about quasi-Poisson G_{1} × G_{2}-manifolds. In particular, if (M_{i}, ρ_{i}, π_{i}) is a quasi-Poisson G_{i}-manifold (i = 1, 2) then

(M_{1}, ρ_{1}, π_{1}) × (M_{2}, ρ_{2}, π_{2}) = (M_{1} × M_{2}, ρ_{1} × ρ_{2}, π_{1} + π_{2})

is a quasi-Poisson G_{1} × G_{2}-manifold.

Example 1. G is a quasi-Poisson G×G-manifold, with the action ρ(g_{1}, g_{2})·g = g_{1}gg_{2}^{−1} and with π = 0.

Remark 2. 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_{1} ×G_{2}, s_{1} ⊕s_{2})-manifold is also a quasi-Poisson (G_{1} ×G_{2}, s_{1} ⊕−s_{2})- manifold.

Let ψ ∈ ⋀^{2}(𝔤 ⊕ 𝔤) be given by

ψ = \frac{1}{2} ∑_{i,j} s^{ij} (e_{i}, 0) ∧ (0, e_{j})

where s = ∑_{i,j} s^{ij} e_{i} ⊗ e_{j} in some basis e_{i} of 𝔤.

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).

Fusion is associative (but not commutative): if M is a quasi-Poisson G × G × G × H-manifold then the two G × H-quasi-Poisson structures obtained by the iterated fusions coincide. If M is a quasi-Poisson G^{n} × H-manifold then its (iterated) fusion to a quasi-Poisson G × H-manifold is given by

π^{∗} = π − ∑_{i<j} ρ(ψ_{i,j}),  (3)

where ψ_{i,j} ∈ ⋀^{2}(𝔤^{⊕n}) is the image of ψ under the inclusion 𝔤⊕𝔤 → 𝔤^{⊕n} sending 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^{♯}α, s^{♯}β] ∈ 𝔠 ∀α, β ∈ ann(𝔠).

In particular, if C ⊆ G is coisotropic, i.e. if s^{♯}(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

{f, g} := π(df, dg), f, g ∈ C^{∞}(M)^{C}  (4)

is a Poisson bracket on the space of C-invariant functions. In particular, if the action of C on M induces a regular equivalence relation on M^{6}, then the bivector field π descends to define a Poisson structure on M/C.

Proof. The proof is essentially the same as that of [1, Theorem 4.2.2], but we include 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 (4) satisfies the Jacobi identity, notice that

\{f_{1}, \{f_{2}, f_{3}\}\} + c.p. = \frac{1}{2}[π, π](df_{1}, df_{2}, df_{3}) = ρ(ϕ)(df_{1}, df_{2}, df_{3}),

for any f_{i} ∈ C^{∞}(M)^{C} (i = 1, 2, 3). Now ρ^{∗}df_{i} ∈ ann(𝔠) and C is reducing, hence ρ(ϕ)(df_{1}, df_{2}, df_{3}) = 0.

For ξ ∈ 𝔤 let ξ^{L} and ξ^{R} denote the corresponding left and right invariant vector field on G.

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 · g̃ = τ(g) g̃ g^{−1}  (5)

of G on G, and if the image of π under

μ_{∗} ⊗ id : T M ⊗ T M → T G ⊗ T M

is

−\frac{1}{2} ∑_{i,j} s^{ij}(e_{i}^{L} + τ(e_{i})^{R}) ⊗ ρ(e_{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

𝔠̂:= {(ξ + s^{♯}α, ξ − s^{♯}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^{♯}α, η ± s^{♯}β] = ([ξ, η] + [s^{♯}α, s^{♯}β]) ± s^{♯}(ad_{ξ}^{∗}β − ad_{η}^{∗}α)

and [s^{♯}α, s^{♯}β] ∈ 𝔠, the space 𝔠̂ is closed under the Lie bracket.

6i.e. the orbit space, M/C, is a manifold, and the projection M → M/C is a surjective submersion. Equivalently, by Godement’s criterion, the subset M ×_{M/C} M ⊆ M × M is a closed, embedded submanifold, and the projection onto either factor, M ×_{M/C} M → M, is a submersion.

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Let Ĉ ⊆ G × G be a Lie group with the Lie algebra 𝔠̂; and we suppose that the diagonal inclusion 𝔠 ⊆ 𝔠̂ lifts to an inclusion C ⊆ Ĉ ⊆ G × G. The group G × G acts on G by

(g_{1}, g_{2}) · g = τ(g_{1}) g g_{2}^{−1}.  (6)

The orbits of Ĉ ⊆ 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 action on M induces a regular equivalence relation. Let O ⊆ G be a Ĉ-orbit. If O is in a clean position relative to μ,^{7} then

μ^{−1}(O)/C ⊆ M/C

is a Poisson submanifold. More generally, if S ⊆ G is a Ĉ-stable submanifold and S is in a clean position relative to μ, then μ^{−1}(S)/C ⊆ M/C is a Poisson submanifold.

Proof. This follows from Theorem 1.C when H is trivial.

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 and C ⊆ G is a reducing subgroup.

1. If the C action on M induces a regular equivalence relation, then the bivector field π descends to define a quasi-Poisson H-structure on M/C.

2. Let τ_{G} and τ_{H} be automorphisms of G and H, and let

(μ_{G}, μ_{H}) : M → G × H

be a (τ_{G}, τ_{H})-twisted moment map. If S ⊆ G is a Ĉ-stable submanifold, and the following topological conditions hold:

• S is in a clean position relative to μ_{G}, and

• the action of C on μ_{G}^{−1}(S) induces a regular equivalence relation,

then μ_{G}^{−1}(S)/C is a quasi-Poisson H-manifold, and μ_{H} descends to define a τ_{H}-twisted moment map,

μ_{H} : μ_{G}^{−1}(S)/C → H.

7That is, gr(μ) ∩ (O × M) ⊆ G × M is a submanifold, and T (gr(μ) ∩ (O × M)) = T gr(μ) ∩ (T O × T M), where gr(μ) = {(μ(x), x); x ∈ M}. It happens, in particular, if μ is transverse to O.

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Moreover, if all the stated assumptions hold, then μ_{G}^{−1}(S)/C ⊆ M/C is a quasi-Poisson H-submanifold.

Proof. 1. The proof of this first statement is very similar to that of Theorem 1.A. Let q : M → M/C denote the quotient map (a surjective submersion). π is G × H-invariant, thus it descends to define a bivector field π′ on M/C. Likewise, the H-action on M descends to define an action ρ′ : H × M/C → M/C. Now the Ad-invariant element s ∈ S^{2}(𝔤 ⊕ 𝔥) splits as the sum of s_{G} ∈ S^{2}(𝔤) and s_{H} ∈ S^{2}(𝔥); similarly ϕ ∈ ⋀^{3}(𝔤 ⊕ 𝔥) splits as the sum of ϕ_{G} ∈ ⋀^{3} 𝔤 and ϕ_{H} ∈ ⋀^{3} 𝔥. By definition, we have

\frac{1}{2}[π, π] = ρ(ϕ) = ρ(ϕ_{G}) + ρ(ϕ_{H}).

However, for any α_{i} ∈ Ω^{1}(M/C), (i = 1, 2, 3) the pullbacks ρ^{∗}q^{∗}α_{i} ∈ ann(𝔠), and C is reducing, hence ρ(ϕ_{G})(q^{∗}α_{1}, q^{∗}α_{2}, q^{∗}α_{3}) = 0. Thus

\frac{1}{2}[π′, π′] = ρ′(ϕ_{H}),

as desired.

2. First of all, since S is in a clean position relative to μ_{G}, μ_{G}^{−1}(S) ⊆ M is a submanifold. Moreover, since the action of C induces a regular equivalence relation, μ_{G}^{−1}(S)/C is a manifold and the quotient map is a surjective submersion. Since H acts trivially on S ⊆ G, so H restricts to an action on μ_{G}^{−1}(S)/C ⊆ M/C. For any x ∈ μ_{G}^{−1}(S) and α ∈ T_{x}^{∗}M the moment map condition gives

(μ_{G})_{∗}(π(·, α)) = −\frac{1}{2}((s_{G}^{♯}ρ_{x}^{∗}α)^{L} + τ(s_{G}^{♯}ρ_{x}^{∗}α)^{R}).  (7)

If α is annihilates ρ_{x}(𝔠), then ρ_{x}^{∗}α ∈ ann(𝔠). The vector (7) is thus the action of \frac{1}{2}(s_{G}^{♯}ρ_{x}^{∗}α, −s_{G}^{♯}ρ_{x}^{∗}α) ∈ 𝔠̂ on G. In particular, since S is in a clean position relative to μ_{G}, it follows that π(·, α) is tangent to μ_{G}^{−1}(S). This implies that π descends to μ_{G}^{−1}(S)/C to define a quasi-Poisson Hstructure. Finally, since μ_{H} : μ_{G}^{−1}(S) → H is G×H-equivariant, it descends to a map on μ_{G}^{−1}(S)/C. The image of π under (μ_{H})_{∗} ⊗id : T M ⊗T M → T H ⊗T M is

−\frac{1}{2} ∑_{i,j} s_{H}^{ij}(e_{i}^{L} + τ(e_{i})^{R}) ⊗ ρ(e_{j}),

where s_{H} ∈ S^{2}(𝔥)^{H} denotes the chosen invariant symmetric tensor, and hence this also holds for the reduced bivector field on μ_{G}^{−1}(S)/C, proving that μ_{H} descends to define a moment map.

If the action of C on M induces a regular equivalence relation, then μ_{G}^{−1}(S)/C ⊆ M/C is certainly an H-invariant submanifold, and the previous argument shows that the bivector field is tangent to μ_{G}^{−1}(S)/C. That is, μ_{G}^{−1}(S)/C ⊆ M/C is a quasi-Poisson H-submanifold.

Example 2. Suppose that (M, ρ, π) is a quasi-Poisson G × G × H manifold. Let

G_{Δ} := {(ξ, ξ) ∈ G × G} ⊆ G × G

denote the diagonal subgroup. One may first fuse the two G factors of the quasi-Poisson structure on M (in either order), yielding a quasi-Poisson G × H structure on M, and then reduce by G, to obtain a quasi-Poisson H structure on M/G_{Δ}. Alternatively, notice that G_{Δ} ⊆ G × G is a reducing subgroup, so one may reduce M directly by G_{Δ}, as in Theorem 1.C, obtaining a quasi-Poisson H structure on M/G_{Δ}. Both these quasi-Poisson H structures on M/G_{Δ} are identical.

3.1 Induction and the Hat construction

When C ⊆ Ĉ is closed, and O ⊂ G is a Ĉ-orbit, the reduced space μ_{G}^{−1}(O)/C can be conveniently described in the following way. Let M̂ be the manifold obtained from M by induction from C to Ĉ (using the diagonal embedding C ⊂ Ĉ). Concretely,

M̂ = (Ĉ × M)/C  (8)

where the C-action on Ĉ × M is c · (ĉ, m) = (ĉc^{−1}, c · m). Now suppose O = Ĉ · g_{0} for some g_{0} ∈ G, then

μ_{G}^{−1}(O)/C ≅ μ̂_{G}^{−1}(g_{0})/ Stab(g_{0}),  (9)

where

μ̂_{G} : M̂ ⟶^{(ĉ,m)→ĉ·μ_{G}(m)} G,

is the induced Ĉ-equivariant moment map and

Stab(g_{0}) = {ĉ ∈ Ĉ; ĉ · g_{0} = g_{0}} ⊆ Ĉ.

More generally, suppose that S ⊆ G is a Ĉ-invariant submanifold. Then for any submanifold X ⊆ G such that S = Ĉ · X, we have natural bijections

μ_{G}^{−1}(S)/C ≅ μ̂_{G}^{−1}(S)/Ĉ ≅ μ̂_{G}^{−1}(X)/ Stab(X),  (10)

where the stabilizer groupoid, Stab(X) ⇉ X, is defined as the restriction of the action groupoid Ĉ ⋉ G to X ⊆ G.

Remark 3 (The stabilizer groupoid). Recall that the stabilizer groupoid is defined as Stab(X) := {(ĉ, x) ∈ Ĉ ×X | ĉ·x ∈ X} with source and target maps given by s(ĉ, x) = x, and t(ĉ, x) = ĉ · x, respectively, and multiplication given by (ĉ′, ĉ · x) ∘ (ĉ, x) := (ĉ′ · ĉ, x). The action of Stab(X) on μ̂_{G}^{−1}(X) is the obvious one: the moment map is μ̂_{G}, and for any m̂ ∈ μ̂_{G}^{−1}(X) and (ĉ, μ̂_{G}(m̂)) ∈ Stab(X), we have (ĉ, μ̂_{G}(m̂)) · m̂ = ĉ·m̂. In particular, two elements m̂, m̂′ ∈ μ̂_{G}^{−1}(X) are in the same Stab(X)-orbit if and only if there exists a ĉ ∈ Ĉ such that m̂′ = ĉ · m̂.

The following proposition gives sufficient conditions on X ⊆ G for the quotient (10) to be a smooth quasi-Poisson manifold.

Proposition 1. Suppose that C ⊆ G is a reducing subgroup, and C ⊆ Ĉ is closed. Suppose further that (M, ρ, π) is a quasi-Poisson G × H-manifold with a (τ_{G}, τ_{H})-twisted moment map,

(μ_{G}, μ_{H}) : M → G × H.

If X ⊆ G, Stab(X) ⊆ Ĉ ⋉ G, and S := Ĉ · X ⊆ G are embedded submanifolds, and the following topological conditions hold:

• X is in a clean position relative to μ̂_{G}, and

• the action of Stab(X) on μ̂_{G}^{−1}(X) induces a regular equivalence relation,

then μ̂_{G}^{−1}(X)/ Stab(X) is a quasi-Poisson H-manifold, and μ_{H} descends to define a τ_{H}-twisted moment map,

μ_{H} : μ̂_{G}^{−1}(X)/ Stab(X) → H.

Proof. Since μ_{G}^{−1}(S)/C ≅ μ̂_{G}^{−1}(X)/ Stab(X), it suffices to show that the assumptions of Theorem 1.C.2 hold. We begin by showing that S is in a clean position relative to μ_{G}. First, consider the diagonal action map

A : Ĉ ⋉ (G × M̂) ⟶^{(ĉ;g,m̂)→(ĉ·g,ĉ·m̂)} (G × M̂).

By assumption, U := Ĉ × (X × M̂) ⊆ Ĉ ⋉ (G × M̂) and V := Ĉ × gr(μ̂_{G}) ⊆ Ĉ ⋉(G×M̂) intersect cleanly, with U ∩V ≅ Ĉ ×μ̂_{G}^{−1}(X). Since V is a union of fibres of A, it follows that A(U) = S × M̂ and A(V) = gr(μ̂_{G}) intersect cleanly (cf. Lemma 6). That is, S is in a clean position relative to μ̂_{G}. Moreover, the restriction A : U ∩ V → A(U) ∩ A(V) defines a surjective submersion

Ĉ × μ̂_{G}^{−1}(X) → μ̂_{G}^{−1}(S).

Next, consider the Ĉ-equivariant map B : G × M̂ ⟶^{(g,[ĉ,m])→[ĉ]} Ĉ/C. The restrictions of B to both S × M̂ and gr(μ̂_{G}) are surjective submersions (since both submanifolds are Ĉ-invariant). Thus S × M = (B|_{S×M̂})^{−1} (C) and gr(μ_{G}) = (B|_{gr(μ̂_{G})})^{−1} (C) intersect cleanly. That is, S is in a clean position relative to μ_{G}. Since the action Lie groupoids C ⋉μ_{G}^{−1}(S) and Stab(X)⋉_{X} μ̂_{G}^{−1}(X) are Morita equivalent, and the equivalence relation defined by the action of Stab(X) on μ̂_{G}^{−1}(X) is regular, it follows that the equivalence relation corresponding to the action of C on μ_{G}^{−1}(S) is also regular (cf. Lemma 7).

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_{1}(Σ, V) denote the fundamental groupoid of Σ with the base set V. The composition in Π_{1}(Σ, V) is from right to left: ab means path b followed by path a. For a ∈ Π_{1}(Σ, V) let s(a) denote the source and t(a) the target of a; ab is defined if t(b) = s(a). Let

M_{Σ,V} (G) = Hom(Π_{1}(Σ, V), G).

M_{Σ,V} (G) can be seen as the moduli space of pairs:

(P → Σ, {v̂ ∈ P |_{v}}_{v∈V}),  (11)

consisting of a flat (principal) G-bundle over Σ together with a framing over V.^{8} For any arrow a ∈ Π_{1}(Σ, V) let

hol_{a} : M_{Σ,V} (G) → G

denote evaluation at a (in terms of flat connections it is the holonomy along a). There is a natural action ρ = ρ_{Σ,V} of the group G^{V} on M_{Σ,V} (G) which is defined by

hol_{a}(ρ(g)x) = g_{t(a)} hol_{a}(x)g_{s(a)}^{−1}.  (12)

Infinitesimally,

(hol_{a})_{∗}(ρ(ξ)) = −ξ_{t(a)}^{R} + ξ_{s(a)}^{L}

for any ξ ∈ 𝔤^{V}, where ξ^{L/R} denotes the left/right invariant vector field on G corresponding to ξ ∈ 𝔤. By a skeleton of (Σ, V) we mean a graph Γ ⊂ Σ with the vertex set V, such that there is a deformation retraction of Σ to Γ.^{9} If we choose an orientation of every edge of Γ then M_{Σ,V} (G) gets identified (via (hol_{a}, a ∈ E_{Γ})) with G^{E_{Γ}}, where E_{Γ} is the set of edges of Γ. In particular, if Σ is a disc and V has two elements then we get M_{Σ,V} (G) = G. Our convention in this paper is to orient the ∂Σ against the induced boundary orientation. Cutting the boundary of Σ at the marked points, V, splits it into oriented arcs (the components of ∂Σ that don’t contain a marked point^{10} are not considered to be arcs). If we choose an ordered pair (v_{s}, v_{t}) of marked points (v_{s} ≠ v_{t} ∈ V) then the corresponding fused surface Σ^{∗} is obtained by gluing a short piece of the arc ending at v_{s} with a short piece of the arc starting at v_{t} (so that v_{s} and v_{t} get identified). The subset V^{∗} ⊂ ∂Σ^{∗} is obtained from V by identifying v_{s} and v_{t}. Notice that the map

8By a framing over V, we mean a choice of a lift v̂ ∈ P_{v} of each marked point v ∈ V to the fibre over v.

9It is a simple exercise to show that there exists a skeleton for every marked surface.

However, it should be emphasized that this skeleton is not unique! The edges of a skeleton determine generators of Π_{1}(Σ, V), but the converse fails (generators for Π_{1}(Σ, V) may be chosen which intersect in an essential way).

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M_{Σ^{∗},V^{∗}}(G) → M_{Σ,V} (G),

coming from the map (Σ, V) → (Σ^{∗}, V^{∗}), is a diffeomorphism: if Σ retracts to a skeletal graph Γ then Σ^{∗} retracts to its image Γ^{∗}, and the two graphs have the same number of edges. We can thus identify the manifolds M_{Σ^{∗},V^{∗}}(G) and M_{Σ,V} (G).

v v_{s} v_{t}  v^{∗} Σ  Σ^{∗} v_{s}^{∗} = v_{t}^{∗}

Figure 1: Fusion. The marked points v_{s}, v_{t} ∈ V are identified after fusion, while every other marked point v ∈ V (v_{s} ≠ v ≠ v_{t}) is unaffected. Notice that v_{s} remains the source of the same edge after fusion, and v_{t} remains the target of the same edge after 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_{Γ} is a disc D_{e}, and Σ is obtained from D_{e}’s by repeated fusion.

Theorem 2. There is a natural bivector field π_{Σ,V} on M_{Σ,V} (G) such that

(M_{Σ,V} (G), ρ_{Σ,V} , π_{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_{Σ,V} = 0 Figure 2: A surface with a skeleton, fused from four discs. The discs have been assigned colors in the picture, but this figure should not be confused with a quilted surface.

10that is, an element of V

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2. if (Σ, V) = (Σ_{1}, V_{1}) ⊔ (Σ_{2}, V_{2}) then π_{Σ,V} = π_{Σ_{1},V_{1}} + π_{Σ_{2},V_{2}}

3. if (Σ^{∗}, V^{∗}) is obtained from (Σ, V) by fusion, then (M_{Σ^{∗},V^{∗}}(G), ρ_{Σ^{∗},V^{∗}}, π_{Σ^{∗},V^{∗}}) is obtained from (M_{Σ,V} (G), ρ_{Σ,V} , π_{Σ,V}) by the corresponding fusion.

If there is no danger of confusion, we shall denote π_{Σ,V} simply by π.

Remark 4. Alejandro Cabrera has independently studied quasi-Hamiltonian G^{V} -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_{Σ,V}, as (Σ, V) is a fusion of a collection of discs with two marked points. Let us denote the resulting bivector field on M_{Σ,V} (G) by π_{Γ}. Theorem 2 follows from the following Lemma:

Lemma 2. The bivector field π_{Γ} on M_{Σ,V} (G) is independent of the choice of Γ.

Remark 5. The lemma follows from the special case where (Σ, V) is a disc with 3 marked points (see Example 4 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_{Σ,V} (G). Properties (1)–(3) of the theorem are satisfied by the construction of π_{Γ}.

Let us now describe the calculation of π = π_{Γ} in more detail. Notice that for 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 [15]. We choose an orientation of every edge of Γ to get an identification M_{Σ,V} (G) = G^{E_{Γ}}. First we see it as a G^{E_{Γ}} × G^{E_{Γ}}-quasi-Poisson space with zero bivector

v_{s}  v_{t}

Figure 3: The annulus with one marked point is obtained by fusion from the disc with two marked points.

(i.e. as M_{Σ′,V′}(G), where (Σ′, V′) is a disjoint union of discs with two marked points each). Then, fusing at each vertex using the reversed^{11} linear order, we obtain a G^{V} -quasi-Poisson space.

Example 3. 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′, V′) with two marked points, as in Fig. 3. Now M_{Σ′,V′} = G with the the quasi-Poisson G × G-structure described in Example 1: the bivector field is trivial and G × G acts by (g_{1}, g_{2}) · g = g_{1}gg_{2}^{−1}. Thus M_{Σ,V} = G, the G-action is by conjugation, and

π = \frac{1}{2} ∑_{i,j} s^{ij} e_{i}^{R} ∧ e_{j}^{L}.

Example 4. Let Σ be a triangle and V is the set of its vertices.

\begin{matrix}a \\ b \\ c \end{matrix}

We can identify M_{Σ,V} with G^{2} via (hol_{a^{−1}}, hol_{b}), i.e. Γ is the graph with the oriented edges a^{−1}, b. In this case

π = −\frac{1}{2} ∑_{i,j} s^{ij} e_{i}^{L}(1) ∧ e_{j}^{L}(2)  (13)

11We reverse the linear order when fusing to account for the minus sign appearing in (3)

(cf. [2]).

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(where e_{i}^{L}(k) denotes the left-invariant vector field which is tangent to the k^{th} factor of G^{2} (k = 1, 2)). Equivalently,

π(hol_{a^{−1}}^{∗} θ^{L}, hol_{b}^{∗} θ^{L}) = −s ∈ 𝔤 ⊗ 𝔤,

where θ^{L} ∈ Ω^{1}(G, 𝔤) is the left-invariant Maurer-Cartan form. An easy calculation shows

π(hol_{b^{−1}}^{∗} θ^{L}, hol_{c}^{∗} θ^{L}) = −s ∈ 𝔤 ⊗ 𝔤,

confirming that π is independent of the choice of Γ.

For a general surface (Σ, V) with a choice of a skeleton Γ, we get an identification M_{Σ,V} = G^{E_{Γ}}. Applying (3) we obtain

π_{Σ,V} = \frac{1}{2} ∑_{v∈V} ∑_{a<b} ∑_{i,j} s^{ij} e_{i}(a, v) ∧ e_{j}(b, v)  (14)

where a, b run over the (half)edges adjacent to v,

e_{i}(a, v) = \begin{cases}−e_{i}^{R}(a) & a goes into v \\ e_{i}^{L}(a) & a goes out of v \end{cases}

and for a ∈ E_{Γ}, e_{i}^{R,L}(a) denotes the right/left-invariant vector field on G^{E_{Γ}} equal to

(0, …, 0, \overbrace{e_{i}}^{a}, 0, …, 0) ∈ 𝔤^{E_{Γ}}

at the identity element.

Remark 6. Essentially the same formula was discovered by Fock and Rosly [15], for Poisson structures on M_{Σ,V} obtained by a choice of a classical r-matrix. More precisely, if one considers the bivector described in Equation (16) of [15], but replacing the r-matrix r^{ij} with it’s symmetrization s^{ij}, then one arrives at the same formula as (14). Meanwhile, Skovborg studied the corresponding formula in the absence of an r-matrix for invariant functions [37].

5 The homotopy intersection form and quasi-Poisson structures

Massuyeau and Turaev [27] made a beautiful observation that, in the case of one marked point and G = GL_{n}, the quasi-Poisson structure on M_{Σ,V} (G) can be expressed in terms of the homotopy intersection form on π_{1}(Σ), introduced by Turaev in [38]. Here we extend their result to the case of arbitrary (G, s) and arbitrary V. Let us first extend (a skew-symmetrized version of) Turaev’s homotopy intersection form to fundamental groupoids. If a, b ∈ Π_{1}(Σ, V), let us represent them by transverse smooth paths α, β. For any point A in their intersection, let

λ(A) = \begin{cases}1 & if A ∈ ∂Σ \\ 2 & otherwise \end{cases}

sign(A) := \begin{cases}1 & if (α̇|_{A}, β̇|_{A}) is positively oriented \\ −1 & otherwise. \end{cases}

as in Fig. 4.

A α  β +  A β  α −

Figure 4: sign(A) = ±1 is determined by comparing the orientation of α and β with that of Σ.

Let α_{A} denote the portion of α parametrized from the source of α up to the point A. Finally, let

(a, b) := ∑_{A} λ(A) sign(A)[α_{A}^{−1}β_{A}] ∈ ℤΠ_{1}(Σ, V),  (15)

where ℤΠ_{1}(Σ, V) denotes the groupoid ring^{12} generated by Π_{1}(Σ, V) over ℤ. As in [38] one can check that (a, b) is well defined, i.e. independent of the choice of α and β. Let us list the properties of (a, b). For x ∈ ℤΠ_{1}(Σ, V), x = ∑ n_{i}a_{i}, let x̄ = ∑ n_{i}a_{i}^{−1}.

Proposition 2. The pairing

(·, ·) : Π_{1}(Σ, V) × Π_{1}(Σ, V) → ℤΠ_{1}(Σ, V)

satisfies 1. (a, b) is a linear combination of paths from the source of b to the source

12As a ℤ-module, ℤΠ_{1}(Σ, V) is freely generated by Π_{1}(Σ, V). For generators a, b ∈ Π_{1}(Σ, V), their product in ℤΠ_{1}(Σ, V) is

ab := \begin{cases}a ∘ b & if a and b are composable, i.e. s(a) = t(b) \\ 0 & otherwise. \end{cases}

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of a

2. (b, a) = −\overline{(a, b)}

3. (a, bc) = (a, c) + (a, b)c

4. if (Σ^{∗}, V^{∗}) is obtained from (Σ, V) by fusing Σ at v_{s}, v_{t} ∈ V and a^{∗}, b^{∗} denotes the image of a, b in Π_{1}(Σ^{∗}, V^{∗}), then

(a^{∗}, b^{∗}) = (a, b)^{∗} − (δ_{t(a),v_{s}} a^{−1} − δ_{s(a),v_{s}}1_{v_{s}^{∗}})(δ_{t(b),v_{t}} b − δ_{s(b),v_{t}}1_{v_{s}^{∗}})+

+ (δ_{t(a),v_{t}} a^{−1} − δ_{s(a),v_{t}}1_{v_{s}^{∗}})(δ_{t(b),v_{s}} b − δ_{s(b),v_{s}}1_{v_{s}^{∗}}).

It is the only pairing satisfying these properties.

Proof. The fact that (15) satisfies these properties is readily verified. We now show that these properties uniquely determine a pairing

(·, ·) : Π_{1}(Σ, V) × Π_{1}(Σ, V) → ℤΠ_{1}(Σ, V).  (16)

First suppose (Σ, V) is a disjoint union of disks each of which is marked with two points, then any pairing which satisfies the first three properties must be trivial: From property 1, we see that (a, b) = 0 unless a and b lie in the same connected component of Σ. This leaves the following cases to check:

a = 1_{v} is a unit, for some v ∈ V  (17a)
b = 1_{v} is a unit, for some v ∈ V  (17b)
b = a  (17c)
b = a^{−1}  (17d)

Now in case (17b) property 3 (with c = 1_{v}) implies that (a, 1_{v}) = 0. Now in case (17a), property 2 implies (a, b) = (1_{v}, b) = −\overline{(b, 1_{v})} = 0. In case (17c), (a, a) = k1_{s(a)} (by property 1), for some k ∈ ℤ. Then property 2 implies that k1_{s(a)} = −\overline{k1_{s(a)}} := −k1_{s(a)}, which forces k = 0, and hence (a, b) = (a, a) = 0. Finally, in case (17d), property 3 implies (a, 1_{t(a)}) = (a, a^{−1}) + (a, a)a^{−1}. Since (a, 1_{t(a)}) = 0 = (a, a), we have (a, b) = (a, a^{−1}) = 0. Next, notice that properties 3 and 4 determine the pairing for a fused surface (Σ^{∗}, V^{∗}) in terms of the pairing for (Σ, V). Since any marked surface can be obtained by fusing a collection of disks (with two marked points each), a finite number of times, this shows that (16) is uniquely determined by properties 1-4.

For a, b ∈ Π_{1}(Σ, V) let us consider the 𝔤 ⊗ 𝔤-valued function on M_{Σ,V},

π(hol_{a}^{∗} θ^{L}, hol_{b}^{∗} θ^{L}).

These functions, in turn, specify π completely. We can now state our version of the result of Massuyeau and Turaev [27].

Theorem 3. For any a, b ∈ Π_{1}(Σ, V) we have

π(hol_{a}^{∗} θ^{L}, hol_{b}^{∗} θ^{L}) = \frac{1}{2}(Ad_{hol_{(a,b)}} ⊗ 1) s.  (18)

Remark 7. Essentially the same formula was discovered independently by Xin Nie [29].

Proof of Theorem 3 and of Lemma 2. To prove both Theorem 3 and Lemma 2 (and thus finish the proof of Theorem 2) we need to check that

π_{Γ}(hol_{a}^{∗} θ^{L}, hol_{b}^{∗} θ^{L}) = \frac{1}{2}(Ad_{hol_{(a,b)}} ⊗ 1) s  (19)

for every a, b ∈ Π_{1}(Σ, V) and any skeleton Γ of (Σ, V). Notice (via hol_{bc}^{∗} θ^{L} = hol_{c}^{∗} θ^{L} + Ad_{(hol_{c})^{−1}} hol_{b}^{∗} θ^{L}) that

π_{Γ}(hol_{a}^{∗} θ^{L}, hol_{bc}^{∗} θ^{L}) = π_{Γ}(hol_{a}^{∗} θ^{L}, hol_{c}^{∗} θ^{L})+

(1 ⊗ Ad_{(hol_{c})^{−1}})π_{Γ}(hol_{a}^{∗} θ^{L}, hol_{b}^{∗} θ^{L}).

As a result, if (19) is true for all a, b in a set of generators of Π_{1}(Σ, V), it is then true (by Proposition 2 Part 3) for all elements of Π_{1}(Σ, V). Equation (19) is true if (Σ, V) is a disc with two marked points, as both sides of the equation vanish. The same is true for the disjoint union of a collection of such discs. As any (Σ, V, Γ) can be obtained from such a collection by a repeated fusion, it remains to check that (19) is preserved under fusion. Suppose that (19) is satisfied for some (Σ, V) and its skeleton Γ. Let (Σ^{∗}, V^{∗}) be a fusion of (Σ, V) and let Γ^{∗} be the image of Γ in Σ^{∗}. Then π_{Γ^{∗}} is obtained from π_{Γ} by the corresponding quasi-Poisson fusion. By Proposition 2 Part 4 we then get

π_{Γ^{∗}}(hol_{a}^{∗} θ^{L}, hol_{b}^{∗} θ^{L}) = \frac{1}{2}(Ad_{hol_{(a^{∗},b^{∗})}} ⊗ 1) s.

In other words, (19) is satisfied also for (Σ^{∗}, V^{∗}, Γ^{∗}) for the elements of Π_{1}(Σ^{∗}, V^{∗}) in the image of Π_{1}(Σ, V). As the image generates Π_{1}(Σ^{∗}, V^{∗}), we conclude that (19) is satisfied for (Σ^{∗}, V^{∗}, Γ^{∗}).

Remark 8. Theorem 3 can be used as an alternative definition of π. Properties 1–3 of the homotopy intersection form (cf. Proposition 2) mean that there is a unique G^{V} -invariant bivector field π satisfying (18). Property 4 means that π is compatible with fusion.

If f : (Σ′, V′) → (Σ, V) is an embedding then clearly

(f_{∗}a, f_{∗}b) = f_{∗}(a, b)

for every a, b ∈ Π_{1}(Σ′, V′). From Theorem 3 we thus get the following result.

S_{i}^{1} = v_{3} v_{n} v_{2} v_{1}

Figure 5: Pictured, is a connected component of the boundary, S_{i}^{1} ⊆ ∂Σ. S_{i}^{1} is oriented against the induced boundary orientation. The marked points {v_{1}, …, v_{n}} = V ∩ S_{i}^{1} inherit a cyclic order, and the permutation is defined as σ(v_{i}) = v_{i+1} (where the index i + 1 is computed modulo n).

Corollary 1. If f : (Σ′, V′) → (Σ, V) is an embedding then

f^{∗} : M_{Σ,V} (G) → M_{Σ′,V′}(G)

is a quasi-Poisson map, i.e. π_{Σ,V} and π_{Σ′,V′} are f^{∗}-related.

Let us now consider the special case Σ′ = Σ, V′ ⊂ V. Recall that if (M, ρ, π) is a G × H-quasi-Poisson manifold and if M/G is a manifold (e.g. if the action of G is free and proper) then π descends to a bivector field π′ on M/G such that p_{∗}π = π′, where p : M → M/G is the projection, and that M/G thus becomes a H-quasi-Poisson manifold, called the quasi-Poisson reduction of M by G (cf. [1, 2] or Theorem 1.C). Using this terminology, Corollary 1 becomes

Corollary 2. If V′ ⊂ V then M_{Σ,V′}(G) is the quasi-Poisson reduction of M_{Σ,V} (G) by G^{V \V′}.

Finally, again following Massuyeau and Turaev [27], we can define a moment map for the quasi-Poisson G^{V} -manifold M_{Σ,V} (G). Let us orient ∂Σ against the orientation induced from Σ, and let S_{1}^{1} ⊔ ⋯ ⊔ S_{n}^{1} = ∂Σ be the decomposition into connected components. The subset of marked points V ∩S_{i}^{1} which all lie on a given component of the boundary inherit a cyclic order from the orientation of ∂Σ. Let σ : V → V be the corresponding permutation (as in Figure 5). Let τ : G^{V} → G^{V} be the automorphism defined for any g ∈ G^{V} and v ∈ V by

τ(g)_{v} = g_{σ(v)},

so that the τ-twisted action of G^{V} × G^{V} on G^{V} is

((g_{1}, g_{2}) · g)_{v} = (g_{1})_{σ(v)} g_{v} (g_{2})_{v}^{−1}  (20)

(cf. Eq. (6)). For every v ∈ V let μ_{v} : M_{Σ,V} (G) → G be

μ_{v} = hol_{a_{v}}

where a_{v} is the boundary arc from v to σ(v). Let us combine the maps μ_{v} to a single map μ : M_{Σ,V} (G) → G^{V}.

Theorem 4. The map μ : M_{Σ,V} (G) → G^{V} is a τ-twisted moment map.

Proof. The equivariance of μ is obvious. If v ∈ V and b ∈ Π_{1}(Σ, V) then by the definition of (a_{v}, b) we have

(a_{v}, b) = −δ_{s(b),v} 1_{v} − δ_{s(b),σ(v)} a_{v}^{−1} + δ_{t(b),v} b + δ_{t(b),σ(V)} a_{v}^{−1}b.

Theorem 3 then implies that μ is a moment map, as the 1-forms hol_{b}^{∗} θ^{L} span the cotangent bundle of M_{Σ,V} (G).

Remark 9. An alternative way of proving Theorem 4 is to verify it for the case of a disc with two marked points on the boundary, and then to use fusion.

6 Surfaces with boundary data

For every point v ∈ V let us choose a coisotropic subgroup C_{v} ⊆ G.^{13} Let C = ∏_{v∈V} C_{v} ⊆ G^{V}. Suppose the action of C on M_{Σ,V} (G) defines a regular equivalence relation, and consider the orbit space

M_{Σ,V} (G, (C_{v})_{v∈V}) := M_{Σ,V} (G)/C.

Geometrically, M_{Σ,V} (G, (C_{v})_{v∈V}) is the moduli space of flat (principal) Gbundles P → Σ equipped with a reduction to C_{v} over v for every v ∈ V. Since C ⊆ G^{V} is coisotropic, by Theorem 1.A we know that M_{Σ,V} (G, (C_{v})_{v∈V}) is Poisson. More generally, if V′ ⊆ V then Theorem 1.C implies that

M_{Σ,V} (G)/ ∏_{v∈V′} C_{v}

is a quasi-Poisson G^{V ∖V′}-manifold.

Example 5. Let Σ be a triangle and V the set of its vertices. Suppose that s ∈ S^{2}𝔤 is non-degenerate, and that (𝔤, 𝔞, 𝔟) is a Manin triple. Let A, B ⊂ G be the corresponding subgroups, and let us suppose that A ∩ B = {1} and

BA = G, i.e. B × A ⟶^{(b,a)→ba} G is a diffeomorphism. Let us choose the subgroup, C_{v}, at two of the vertices to be B and the remaining vertex to be A, as in the picture below

13If G is semisimple then any parabolic subgroup is coisotropic; if G is simple then these are all the coisotropic subgroups

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g_{1} g_{2} B  A B

Now the holonomies g_{1}, g_{2} ∈ G along the edges pictured above identify M_{Σ,V} (G) with G × G, where the bivector field was described in Eq. (13). The diffeomorphism

((b, a) → ba) : B × A → G

identifies M_{Σ,V} with B × A × B × A, and the action of C = B × A × B on M_{Σ,V} (G) becomes

(b, a, b′) · (b_{1}, a_{1}, b_{2}, a_{2}) = (bb_{1}, a_{1}a^{−1}, b′b_{2}, a_{2}a^{−1}).

Thus the map (b_{1}, a_{1}, b_{2}, a_{2}) → a_{2}a_{1}^{−1} identifies M_{Σ,V} (G)/C with the Lie group A. Using (13) we see that the resulting bivector field on A ≅ B\G is the pushforward of

π = \frac{1}{2} ∑_{i} ξ_{i}^{L} ∧ η^{i}^{L},

where {η^{i}} ⊂ 𝔟 and {ξ_{i}} ⊂ 𝔞 are bases in duality. A comparison shows that the Poisson structure on M_{Σ,V} (G)/C ≅ A is the Poisson-Lie structure on A described in [23]. The assumption that G = BA is rather strong. In Example 8, we will identify the Poisson Lie group with a moduli space in the absence of this assumption.

Example 6. If Σ, V is a disc with two marked points and C ⊆ G is a coisotropic subgroup which we embed as a subgroup of the second factor of G × G, then M_{Σ,V} (G)/C = G/C is a quasi-Poisson G-manifold, with π = 0.

Since, according to Theorem 4, the holonomies along the boundary arcs define a moment map μ : M_{Σ,V} (G) → G^{V}, we can apply the moment map reduction (Theorem 1.B) to get Poisson submanifolds of M_{Σ,V} (G, (C_{v})_{v∈V}) = M_{Σ,V} (G)/C.^{14} To give geometric descriptions of these Poisson submanifolds it will be convenient to use the “hat”-construction from Section 3.1. We begin be describing M̂, μ̂ and Ĉ. Recall that σ : V → V is the permutation obtained by walking along ∂Σ against the orientation induced from Σ, and that a_{v} is the boundary arc from v to σ(v).

14If s ∈ S^{2}𝔤 is nondegenerate and if every component of ∂Σ contains a marked point then these submanifolds are in fact the symplectic leaves. See [21] for more details.

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We now describe the group Ĉ. Let 𝔠_{v}^{⊥} = s^{♯} ann(𝔠_{v}); it is an ideal in 𝔠_{v}. Let C_{v}^{⊥} ⊆ C_{v} denote the corresponding connected Lie group.^{15} Then C_{v} normalizes C_{v}^{⊥}. Hence

Ĉ_{v} = {(g_{1}, g_{2}) ∈ C_{v} × C_{v} | g_{1}g_{2}^{−1} ∈ C_{v}^{⊥}} ⊂ G × G

is a subgroup. We take

Ĉ = ∏ Ĉ_{v} ⊂ G^{V} × G^{V} .

a_{v} v a_{v′}  a_{v}  a_{v′} v_{+} v_{−} w_{v} Σ  Σ̂

Figure 6: Σ̂ is obtained from Σ by blowing up at each point v ∈ V. We denote the exceptional divisor (a segment on the boundary of Σ) by w_{v}, its initial point by v_{−} and its endpoint by v_{+}. Note that v = σ(v′), and the orientation of the arcs is opposite to the induced boundary orientation.

Let us now give a geometric description of the induced Ĉ-manifold M̂ (see Eq. (8)) and of the induced Ĉ-equivariant moment map μ̂ : M̂ → G^{V} in the case of M = M_{Σ,V} (G). First, let Σ̂ be the surface obtained from Σ by blowing up at each point v ∈ V, as in Fig. 6. We let w_{v} denote the exceptional divisor obtained by blowing up at v. We label the initial and end points of the segment w_{v} by v_{−} and v_{+}, respectively. We let Wall denote the set of w_{v}’s, and we let V_{−} and V_{+} denote the set of initial and end points of the w_{v}’s. Thus (Σ̂, V_{−} ∪ V_{+}) is a marked surface. Then

M̂ = {f ∈ M_{Σ̂,V_{−}∪V_{+}}(G); f(w_{v}) ∈ C_{v}^{⊥} (∀v ∈ V)}.  (21)

The group G^{V} × G^{V} ≅ G^{V_{−}∪V_{+}} acts naturally on M_{Σ̂,V_{−}∪V_{+}}(G) (cf. Eq. (12)) and the subgroup Ĉ preserves M̂ ⊂ M_{Σ̂,V_{−}∪V_{+}}(G). The elements of Ĉ are (g_{v_{−}}, g_{v_{+}} ∈ C_{v})_{v∈V} such that g_{v_{+}}g_{v_{−}}^{−1} ∈ C_{v}^{⊥}. The map μ̂ : M̂ → G^{V} is given by

μ̂(f) = (f(a_{v}))_{v∈V} .

Suppose we now choose an element h = (h_{v})_{v∈V} ∈ G^{V}. Recall from Eq. (20) that the action of Ĉ ⊂ G^{V_{−}∪V_{+}} on G^{V} is by

(g · h)_{v} = g_{σ(v)_{−}}h_{v}g_{v_{+}}^{−1}.

Let O ⊆ G^{V} denote the Ĉ-orbit containing h ∈ G^{V}. Using Eq. (9) we get

μ^{−1}(O)/C ≅ μ̂^{−1}(O)/Ĉ ≅ {f ∈ M̂; f(a_{v}) = h_{v} (∀v ∈ V)}/ Stab(h).

15If G is simple, so that C_{v} is parabolic, then C_{v}^{⊥} ⊆ C_{v} is the nilpotent radical.

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Theorem 5. Suppose that the graph of μ̂ composes cleanly with h ∈ G^{V}, and the Stab(h)-orbits of μ̂^{−1}(h) form a regular foliation, then the quotient space

μ̂^{−1}(h)/ Stab(h)

= {f ∈ M_{Σ̂,V_{−}∪V_{+}}(G); f(w_{v}) ∈ C_{v}^{⊥}, f(a_{v}) = h_{v} (∀v ∈ V)}/ Stab(h),

is a Poisson manifold. Moreover, if the C-orbits of M_{Σ,V} (G) form a regular foliation, then μ̂^{−1}(h)/ Stab(h) is naturally isomorphic to a Poisson submanifold of M_{Σ,V} (G)/C.

Proof. We apply Proposition 1 to prove the first statement. To do so, we only need to show that C ⊆ Ĉ is closed. But for each v ∈ V, we have Ĉ_{v} ≅ C_{v}^{⊥} ⋊ C_{v} via the map (g_{1}, g_{2}) → (g_{1}g_{2}^{−1}, g_{2}); hence C_{v} ⊆ Ĉ_{v} is closed, which yields the result. The proof of Proposition 1 shows that the assumptions of Theorem 1.C.2 hold, and if the C-orbits of M_{Σ,V} (G) form a regular foliation, then the assumptions of Theorem 1.C.1 also hold. Hence μ̂^{−1}(h)/ Stab(h) is naturally isomorphic to a Poisson submanifold of M_{Σ,V} (G)/C.

Theorem 5 is particularly interesting in the case when h = 1 is the unit. Since this constrains the holonomies along the paths a_{v} to be trivial, we can contract these paths to points.

v  v′′ v′  v_{+} v_{−}  v′′_{+} v′′_{−} v′_{+} v′_{−} w_{v}  w_{v′′} w_{v′} a_{v} a_{v′′} a_{v′}  w_{v}  w_{v′′} w_{v′} Σ  Σ̂  Σ̂^{c}

Figure 7: Blow up at the marked points, followed by contraction of a_{v}’s

Example 7. [ [33–35]] Once again, let (𝔤, 𝔞, 𝔟) be a Manin triple, let Σ, V be a disk with four marked points, and let us choose the subgroups C_{v} as in the picture,

B B A A Σ

where A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 with A ∩ B = {1}. Setting h = 1 (which is a regular value for μ̂), we get the constraint on the holonomies pictured in Fig. 8.

B B A A  B B A A Σ̂  b_{1} b_{2} a_{1} a_{2}

Figure 8: Setting h = 1 is effectively the same as contracting the uncolored portions of the boundary in the first picture; this results in the second picture. We label the holonomies as in the third picture. Since Σ̂ is contractible, the product of the holonomies around the boundary must be trivial, that is: a_{1}b_{1} = b_{2}a_{2}.

In this case Stab(h) = 1 and μ̂^{−1}(1)/ Stab(1) is

μ̂^{−1}(1) = {(a_{1}, b_{1}, a_{2}, b_{2}) ∈ A × B × A × B; a_{1}b_{1} = b_{2}a_{2}},

where the holonomies are as pictured in Fig. 8.^{16} M_{Σ,V} (G)/C is the Lu-Weinstein double symplectic groupoid (cf. [23,25]). The moduli space for this surface was first identified with the Lu-Weinstein double symplectic groupoid in the work of the second author [33–35]. In the case that G = AB, we can identify the moduli space with G via

g = a_{1}b_{1} = b_{2}a_{2}.

The resulting Poisson structure on G is the so-called Heisenberg double.

Description in terms of flat bundles

We now give an alternative description of M_{Σ,V} /C and its Poisson submanifolds in terms of flat bundles. Given a domain wall w_{v} ∈ Wall obtained by blowing up the marked point v ∈ V, we define C_{w_{v}} := C_{v}.

Definition 4. Suppose that w ⊆ ∂Σ̂ is a domain wall. A flat reduction of the structure from G to C_{w} over w, is a choice of principal C_{w}-subbundle Q_{w} ⊆ P |_{w} which is flat with respect to the connection on P |_{w}.^{17}

Theorem 6. Suppose the C-orbits of M_{Σ,V} form a regular foliation. Then the moduli space of pairs

(P → Σ̂, {Q_{w} → w}_{w∈Wall})  (22)

16In the case that the discrete group A ∩ B is non-trivial, μ̂^{−1}(1) is a covering space for M_{Σ,V} (G)/C, and thus inherits a Poisson structure.

17Equivalently, this is a choice of a flat principal C_{w}-bundle Q_{w} → w together with an isomorphism of flat bundles P |_{w} → Q_{w} ×_{C_{w}} G.

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of flat G-bundles over Σ̂ equipped with a flat reduction of the structure group from G to C_{w} over each domain wall w ∈ Wall, carries a natural Poisson structure and can be canonically identified with M_{Σ,V} /C (as a Poisson manifold).

We now describe the Poisson submanifold μ̂^{−1}(1)/ Stab(1) ⊆ M_{Σ,V} /C as a moduli space. For simplicity, we assume^{18} that s ∈ S^{2}(𝔤) is non-degenerate, and each C_{v} ⊆ G is Lagrangian (i.e. C_{v}^{⊥} = C_{v} for each v ∈ V). Let Σ̂^{c} be the surface Σ̂ with each arc a_{v} contracted to a point, as in Figure 7. The surface Σ̂^{c} is thus obtained from Σ by blowing up the marked points and contracting the original boundary arcs. We identify every domain wall w ⊆ ∂Σ̂ with its image in ∂Σ̂^{c}. Note that adjacent domain walls intersect non-trivially in Σ̂^{c}.

Definition 5. Suppose P → Σ̂^{c} is a principal G-bundle, and that w, w′ ⊆ ∂Σ̂^{c} are two domain walls. We say that a reduction of the structure group from G to C_{w} over w is compatible with a reduction of the structure group from G to C_{w′} over w′ if there exists a common reduction of the structure group from G to C_{w} ∩ C_{w′} over the intersection w ∩ w′ (if it exists, the common reduction of structure is unique up to a unique isomorphism).^{19}

We have:

Theorem 7. Suppose that the graph of μ̂ composes cleanly with 1 ∈ G^{V}, and the Stab(1)-orbits of μ̂^{−1}(1) form a regular foliation, then the moduli space of pairs

(P → Σ̂^{c}, {Q_{w} → w}_{w∈Wall}),  (23)

consisting of a flat G-bundles over Σ̂^{c} equipped with mutually compatible flat reductions of the structure group from G to C_{w} over each domain wall w ∈ Wall, is a Poisson manifold canonically isomorphic to μ̂^{−1}(1)/ Stab(1) ⊆ M_{Σ,V} /C.

Remark 10. The moduli space described in Theorem 7 is a symplectic manifold (cf. [21]).

7 Surfaces with domain walls

Consider a finite family (Σ_{d}, V_{d}) indexed by a set, Dom. Once again, we construct Σ̂_{d} by blowing up Σ_{d} at every v ∈ V_{d}. We let w_{v} ⊂ Σ̂_{d} denote the preimage of v ∈ V_{d} and denote the boundary points of w_{v} by v_{−} and v_{+} (see Fig. 6). We let

18These are not necessary assumptions, and we encourage the interested reader to describe the generalization.

19Equivalently, the corresponding C_{w}-subbundle Q_{w} ⊂ P |_{w} and the corresponding C_{w′}- subbundle Q_{w′} ⊂ P |_{w′} intersect non-trivially over every point in the intersection w ∩ w′.

Moreover, in the special case that w = w_{v} = w′ and the image of w_{v} in Σ̂^{c} forms a closed loop, we insist (naturally) that the two reductions of the structure from G to C_{w_{v}} over the common endpoint q(v_{+}) = q(v_{−}) agree.

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{ \atop Wall} = ⨆_{d∈Dom,v∈V_{d}}w_{v},

and choose an orientation for each connected component of { \atop Wall}, defining the map Sign : π_{0}({ \atop Wall}) → \{−, +\} by

Sign(w) := \begin{cases}+ & if w ⊆ Σ_{d}, and the inclusion \\ & w → ∂Σ_{d} is orientation preserving, \\ − & otherwise, \end{cases}

for any w ∈ π_{0}({ \atop Wall}). Now choose an orientation preserving action of ℤ_{2} on { \atop Wall}, which we extend trivially to an action on ⊔_{d∈Dom}Σ̂_{d}. We let

Σ = (⊔_{d∈Dom}Σ̂_{d}) /ℤ_{2},

and Wall = π_{0}({ \atop Wall})/ℤ_{2}. As before, we refer to each connected component w ∈ Wall as a domain wall. For each d ∈ Dom choose a Lie group G_{d} and an element s_{d} ∈ (S^{2}𝔤_{d})^{G_{d}} - we describe this choice as coloring the domain Σ_{d} with (G_{d}, s_{d}). Then

M = ∏_{d} M_{Σ_{d},V_{d}}(G_{d})

is a quasi-Poisson G-manifold, where

G = ∏_{d} G_{d}^{V_{d}}.

Let us now construct a reducing subgroup of G in the following way. Taking V = ⊔_{d∈Dom}V_{d}, we have a natural identification V ≅ π_{0}({ \atop Wall}). The orbits of the induced action of ℤ_{2} on V are either pairs (orbit of cardinality 2), or singletons (orbits of cardinality 1), and the orbit space is naturally identified with the set of domain walls V/ℤ_{2} ≅ Wall. For every singleton-orbit {v}, for which v ∈ V_{d} ⊆ V, we choose a subgroup

C_{{v}} ⊆ G_{d}

which is coisotropic with respect to s_{d}. The singleton-orbit, {v} ∈ V/ℤ_{2} ≅ Wall, is naturally identified with a domain wall w ∈ Wall, and we describe the choice of C_{{v}} as coloring the domain wall w with C_{w} := C_{{v}}. For future use we define s_{w} = Sign(v)s_{d}. For every pair-orbit O = {v, v′}, for which v ∈ V_{d}, v′ ∈ V_{d′} we choose a subgroup

C_{{v,v′}} ⊆ G_{d} × G_{d′}

which is coisotropic with respect to Sign(v)s_{d} + Sign(v′)s_{d′}. The pair-orbit, {v, v′} ∈ V/ℤ_{2} ≅ Wall, is naturally identified with a domain wall w ∈ Wall, and we describe the choice of C_{{v,v′}} as coloring the domain wall w with C_{w} := C_{{v,v′}}. As before, we define s_{w} = Sign(v)s_{d} + Sign(v′)s_{d′} Following Wehrheim and Woodward [39,40] we shall call the resulting surface, Σ, together with the chosen coloring of each domain Σ_{d} with (G_{d}, s_{d}), and each domain wall w ∈ Wall with the coisotropic subgroup C_{w}, a quilted surface. The product

C = ∏_{w∈Wall} C_{w} = [( ∏_{{v}∈V/ℤ_{2}} C_{v}) × ( ∏_{{v,v′}∈V/ℤ_{2}} C_{{v,v′}})] ⊆ ∏_{d∈Dom} G_{d}^{V_{d}}

is a reducing subgroup of G. Indeed, it is coisotropic with respect to

s := ∑_{d∈Dom,v∈V_{d}} Sign(v)s_{d},

as in Remark 2. As a result (by Theorem 1.A),

M/C = ( ∏_{d} M_{Σ_{d},V_{d}}(G_{d}))/C

is a Poisson manifold (provided the equivalence relation is regular). The manifold M/C can be interpreted as a moduli space classifying compatible flat bundles over the quilted surface; for brevity, we will not describe the corresponding moduli problem in detail (as we did in Theorem 6 for a single domain), though we encourage the interested reader to do so. Instead, we will now use the “hat” construction from Section 3.1 to give an explicit geometric description of M/C in terms of holonomy representations of the fundamental groupoid (as would be obtained from such flat bundles). We begin by describing the induced space M̂ corresponding to M = ∏_{d} M_{Σ_{d},V_{d}}(G_{d}), the action of Ĉ on M̂, and the map μ̂ : M̂ → G in more detail. As before, we let 𝔠_{w}^{⊥} = s_{w}^{♯} ann(𝔠_{w}), where 𝔠_{w} is the Lie algebra corresponding to C_{w}. Once again, 𝔠_{w}^{⊥} ⊆ 𝔠_{w} is an ideal, and we let C_{w}^{⊥} ⊆ C_{w} be the corresponding connected Lie group (which is normalized by C_{w}). We take

Ĉ_{w} = {(g_{w_{−}}, g_{w_{+}}) ∈ C_{w} × C_{w} | g_{w_{+}}g_{w_{−}}^{−1} ∈ C_{w}^{⊥}}.

We have

Ĉ = ∏_{w∈Wall} Ĉ_{w} ⊆ ∏_{w∈Wall} C_{w} × C_{w} = ∏_{d∈Dom} G^{V_{d}} × G^{V_{d}}.

Now we relate the induced space M̂ to the the quilted surface. An element f ∈ M̂ is a collection of elements f_{d} ∈ M_{Σ̂_{d},V_{d}_{−}∪V_{d}_{+}}(G_{d}) (d ∈ Dom), satisfying the condition:

f(w) ∈ C_{w}^{⊥},

Figure 9: A quilted surface with 2 domains and 4 domain walls

for every domain wall w ∈ Wall. Here f(w) is the holonomy (if w is on the boundary of Σ) or the pair of holonomies (if w is inside Σ) along w. Every element (g_{w_{−}}, g_{w_{+}}) ∈ Ĉ_{w} acts on M̂ by acting at the initial and final boundary points of w, (w_{−}, and w_{+}, respectively).^{20} This defines the action of Ĉ on M̂. Finally, the map

μ̂ : M̂ → G = ∏_{d∈Dom} G_{d}^{V_{d}}

is the collection of holonomies along the arcs a_{v}, v ∈ V_{d} (i.e. along the boundary arcs of Σ which are not domain walls). The isomorphism

M̂/Ĉ ≅ M/C

can be seen via the embedding M ↪ M̂ given by the constraint f(w) = 1 for all w ∈ Wall (effectively contracting every w to a point). We write

M_{Σ} := M̂/Ĉ  (24)

Theorem 1.B and Proposition 1 now give the following result.

Theorem 8.A. Let h ∈ G. If h is in a clean position relative to μ̂ and the Stab(h)-orbits of μ̂^{−1}(h) form a regular equivalence relation, then the quotient space

N := {f ∈ M̂; f(a_{v}) = h_{v}}/ Stab(h)  (25)

is a Poisson manifold. Moreover, if the C-orbits of M form a regular equivalence relation, then N is naturally isomorphic to a Poisson submanifold of M/C.

(w_{−}, w_{+}) = \begin{cases}(v_{−}, v_{+}) & if Sign(v) = +, and \\ (v_{+}, v_{−}) & if Sign(v) = −. \end{cases}

20Recall that the orientation of w as a domain wall was chosen arbitrarily, and may not coincide with the orientation of the boundaries of the domains. For example, if w ∈ Wall ≅ V/ℤ_{2} corresponds to the singleton-orbit {v} ∈ V_{d} ⊆ V, then we have

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The theorem is particularly interesting when h = 1, as then we can contract the arcs a_{v}. More generally we might want to only contract a subset of the arcs a_{v}. For instance, suppose we choose a subset V_{triv} ⊆ V and want to contract those arcs a_{v} with v ∈ V_{triv}. We can do this as follows: Let

H = {h ∈ ∏_{d∈Dom} G_{d}^{V_{d}}; h_{v} = 1 for all v ∈ V_{triv}},

and

Ĥ = {g ∈ ∏_{d∈Dom} G_{d}^{V_{d}_{−}∪V_{d}_{+}}; g_{v_{+}} = g_{σ(v)_{−}} for all v ∈ V_{triv}}.

Then for any g ∈ ∏_{d} G_{d}^{V_{d}_{−}∪V_{d}_{+}}, and h ∈ H,

g · h ∈ H ⇒ g ∈ Ĥ,  (26)

where the action on the left hand side is the twisted action (6). Thus the stabilizer groupoid for H is Stab(H) = (Ĉ ∩ Ĥ) ⋉ H (cf. Remark 3). Let Σ_{V_{triv}} be the surface obtained from Σ by contracting each arc a_{v} ⊆ Σ (for v ∈ V_{triv}) to a point. Thus Proposition 1 implies:

Theorem 8.B. If H is in a clean position relative to μ̂, S = Ĉ · H is a submanifold of ∏_{d} G_{d}^{V_{d}}, and the Ĉ ∩ Ĥ-action on μ̂^{−1}(H) defines a regular equivalence relation, then the moduli space

M_{Σ_{V_{triv}}} := μ̂^{−1}(H)/(Ĉ ∩ Ĥ)

is a Poisson manifold. Moreover, if the action of C on M defines a regular equivalence relation, then M_{Σ_{V_{triv}}} is naturally isomorphic to a Poisson submanifold of M/C.

Explicitly, we have

μ̂^{−1}(H) = {f ∈ M̂; f(a_{v}) = 1 for all v ∈ V_{triv}}

= \{f ∈ ∏_{d∈Dom} M_{Σ̂_{d},V_{d}_{−}∪V_{d}_{+}}(G_{d})| \begin{matrix}f(w) & ∈ C_{w}^{⊥} for every w ∈ Wall, and \\ f(a_{v}) & = 1 for all v ∈ V_{triv} \end{matrix} \}

In the sequel, we will refer to

Ĉ ∩ Ĥ = \{g ∈ ∏_{d} G_{d}^{V_{d}_{−}∪V_{d}_{+}} | \begin{matrix}g_{w_{−}}, g_{w_{+}} & ∈ C_{w} \\ g_{w_{+}}g_{w_{−}}^{−1} & ∈ C_{w}^{⊥} \\ g_{v_{+}} & = g_{σ(v)_{−}} for all v ∈ V_{triv} \end{matrix} \}

as the group of residual gauge transformations. Here g_{w} := g_{v} when the domain wall w ∈ Wall ≅ V/ℤ_{2} corresponds to the singleton-orbit {v} ⊂ V, and g_{w} := (g_{v}, g_{v′}) when the domain wall w corresponds to the pair-orbit {v, v′} ⊂ V. We leave the reformulation of Theorem 8 in terms of moduli spaces of flat bundles (in the spirit of Theorems 6 and 7) to the reader.

Remark 11 (Quilted surfaces with residual marked points). One may also leave the marked points on certain boundary components of the domains Σ_{d} (d ∈ Dom) unreduced (or unsewn); the resulting moduli spaces are quasi-Poisson rather than Poisson. For example, suppose that we decompose the set B := ⊔_{d}∂Σ_{d} of all boundary components into two closed subsets B = B^{sew} ⊔ B^{res}. We will leave those marked points V^{res} := V ∩ B^{res} in B^{res} unreduced (here V = ⊔_{d}V_{d}). As before, le Σ̂_{d} denote the surface obtained by blowing up Σ_{d} at V_{d}^{sew} := V_{d} ∩ B^{sew}, { \atop Wall} the union of the exceptional divisors and { \atop Wall} → Wall the quotient obtained by sewing along the domain walls. We color each domain, Σ_{d} with a pair (G_{d}, s_{d}), and each domain wall, w ∈ Wall, with a coisotropic subgroup C_{w}. We take

G = ∏_{d∈Dom} G_{d}^{V_{d}^{sew}} H = ∏_{d∈Dom} G_{d}^{V_{d}∩V^{res}}, and C = ∏_{w∈Wall} C_{w} ⊆ G

Once again, we let M̂ denote the induced manifold,

M̂ = {f ∈ ∏_{d} M_{Σ̂_{d},V_{d}^{res}∪V_{d}_{−}^{sew}∪V_{d}_{+}^{sew}}(G_{d})|f(w) ∈ C_{w}^{⊥} for each w ∈ Wall}

Then Proposition 1 implies that the moduli space

{f ∈ M̂; f(a_{v}) = 1 for each v ∈ V^{sew}}/ Stab(1)

is a quasi-Poisson H-manifold, where

Stab(1) = {ĉ ∈ Ĉ; ĉ · 1 = 1}.

8 Spin networks

In this section, we reinterpret the quasi-Poisson structure on marked surfaces as well as the Poisson structure on quilted surfaces in terms of spin networks.

Remark 12. Spin networks were first introduced by Penrose [30] (see also [5]). Poisson brackets of spin networks were studied by Roche and Szenes [32], following work by Goldman [16, 17], Andersen, Mattes and Reshetikhin [31].

8.1 Spin networks on a marked surface

Let (Σ, V) denote a marked surface, as in Section 4.

Definition 6. A graph diagram in (Σ, V) consists of the following data:

• a directed graph, Γ with edges E_{Γ} and vertices V_{Γ},

• a subset of vertices V_{Γ}^{anch} ⊆ V_{Γ}, called ‘anchor points’, and • a smooth map ι : Γ → Σ (it is understood that the domain of this map is

really the geometric realization, |Γ|, of Γ) such that

ι^{−1}(V) = V_{Γ}^{anch}.

Often we will abuse notation and denote the graph diagram simply by Γ. A morphism between graph diagrams ι : Γ → Σ and ι′ : Γ′ → Σ is a map between the geometric realizations, μ : |Γ| → |Γ′|, such that ι = ι′ ∘ μ and μ(V_{Γ}) ⊆ V_{Γ′}. A homotopy between graph diagrams ι_{0} : Γ → Σ and ι_{1} : Γ → Σ is a one parameter family of graph diagrams ι_{t} : Γ → Σ, t ∈ [0, 1].

Suppose Γ is a graph diagram in (Σ, V). The Lie group G^{V_{Γ}} acts on G^{E_{Γ}} by

(g · g′)_{e} = g_{t(e)}g′_{e}g_{s(e)}^{−1},

where g ∈ G^{V_{Γ}}, g′ ∈ G^{E_{Γ}}, and e ∈ E_{Γ}. We define

A_{Γ}(G) := C^{∞}(G^{E_{Γ}})^{G^{V{int \atop Γ}}} ,

where V_{Γ}^{int} = V_{Γ} ∖ V_{Γ}^{anch}. Note that there is a residual action of G^{V_{Γ}^{anch}} on A_{Γ}(G). For any v ∈ V_{Γ}^{anch}, we let ρ_{v} : G × A_{Γ}(G) → A_{Γ}(G) denote the action of the v^{th}-factor.

Remark 13. Since Γ is a graph, G^{E_{Γ}} = Hom (Π_{1}(|Γ|, V_{Γ}), G). Therefore

A_{Γ}(G) = C^{∞}( Hom (Π_{1}(|Γ|, V_{Γ}), G))^{G^{V{int \atop Γ}}} .

By functoriality, a morphism μ : |Γ| → |Γ′| of graph diagrams defines a morphism

μ^{!} : Hom (Π_{1}(|Γ′|, V_{Γ′}), G) → Hom (Π_{1}(|Γ|, V_{Γ}), G),

which is equivariant with respect to the map

μ_{V}^{!} : G^{V_{Γ′}} → G^{V_{Γ}}

defined by μ^{!}(g)_{v} = g_{μ(v)}, for any g ∈ G^{V_{Γ′}}. Since μ_{V}^{!} maps the normal subgroup G^{V_{Γ′}^{int}} into G^{V_{Γ}^{int}}, pull-back along μ^{!} defines an equivariant morphism of algebras μ_{∗} : A_{Γ} → A_{Γ′}. We may summarize this as:

Lemma 3. The assignment (ι : Γ → Σ) ↦ A_{Γ} is a functor from graph diagrams to algebras endowed with an action of G^{V}, where g ∈ G^{V} acts via

g · f = ( ∏_{v∈V_{Γ}^{anch}} ρ_{v}(g_{ι(v)}))f.

Definition 7. A spin network in (Σ, V) is a pair (Γ, f), where Γ → Σ is a graph diagram, and f ∈ A_{Γ}(G). We say that spin networks (Γ_{0}, f_{0}) and (Γ_{1}, f_{1}) are homotopic if the underlying graph diagrams are homotopic and f_{0} = f_{1} ∈ A_{Γ_{0}} ≡ A_{Γ_{1}} (to identify the algebras, we use the fact that the definition of A_{Γ} only depends on the underlying graph, and not the map Γ → Σ). A morphism of spin networks μ : (Γ, f) → (Γ′, f′) is a morphism of graph diagrams μ : Γ → Γ′ such that f′ = μ_{∗}f.

We consider two spin networks to be equivalent if they are related by a chain of homotopies and morphisms (or the formal inverses of morphisms). For a spin network (Γ, f), we let [Γ, f] denote the corresponding equivalence class. Define SpinNet_{(Σ,V)}(G) to be the set of equivalence classes of spin networks.

Lemma 4. SpinNet_{(Σ,V)}(G) is a G^{V} -algebra where scalar multiplication, addition, and multiplication are defined as

λ · [Γ, f] = [Γ, λf]  (27a)
[Γ, f] + [Γ′, f′] = [Γ ∪ Γ′, f ⊕ f′]  (27b)
[Γ, f] · [Γ′, f′] = [Γ ∪ Γ′, f ⊗ f′]  (27c)

Lemma 4 will follow from Proposition 3.

8.2 Spin networks and functions on the moduli space

Suppose (Γ, f) is a spin network in (Σ, V). Then the we may push f along the map ι : Γ → Σ to define a function ev(Γ, f) on the moduli space M_{Σ,V} (G), as we shall now describe. For a finite set of points X = {x_{i}} ∈ Σ ∖ V, we let

M_{Σ,V,X}(G) := Hom (Π_{1}(Σ, V ∪ X), G)

Notice that

M_{Σ,V} (G) = M_{Σ,V,X}(G)/G^{X}.

The map

ι : (|Γ|, V_{Γ}) → (Σ, V ∪ V_{Γ}^{int})

yields a G^{V} × G^{V_{Γ}^{int}}-equivariant map

ι^{!} : M_{Σ,V,V_{Γ}^{int}}(G) → Hom (Π_{1}(|Γ|, V_{Γ}), G).  (28)

Therefore, the function

ev(Γ, f) := ι^{!}^{∗}f ∈ C^{∞}(M_{Σ,V,V_{Γ}^{int}}(G))

is G^{V_{Γ}^{int}}-invariant. Hence it descends to define a function on the moduli space

M_{Σ,V} (G) ≅ M_{Σ,V,V_{Γ}^{int}}(G)/G^{V_{Γ}^{int}}.

Moreover, the map

ev(Γ, ·) : A_{Γ}(G) → C^{∞}(M_{Σ,V} (G))

is G^{V} -equivariant. Notice that ev(Γ_{0}, f_{0}) = ev(Γ_{1}, f_{1}) whenever (Γ_{0}, f_{0}) and (Γ_{1}, f_{1}) are homotopic. Moreover, if

μ : (ι : |Γ| → Σ) → (ι′ : |Γ′| → Σ)

is a morphism of graph diagrams, then the following diagram of equivariant morphisms commute M_{Σ,V,V_{Γ′}^{int}}(G)

Hom (Π_{1}(|Γ′|, V_{Γ′}), G)  Hom (Π_{1}(|Γ|, V_{Γ}), G) ι^{′!} ι^{!} μ^{!}

Therefore, ι^{!}^{∗}(f) = ι^{′!}^{∗} ∘ μ_{∗}(f) for any f ∈ A_{Γ}, i.e. ev(Γ′, μ_{∗}f) = ev(Γ, f). So ev descends to a map on the set of equivalence classes of spin networks, SpinNet_{Σ,V} (G).

Proposition 3. The map

ev : SpinNet_{Σ,V} (G) → C^{∞}(M_{Σ,V} (G))

is an isomorphism of G^{V} -algebras.

Proof. With only a slight modification, the statement follows from the proofs of the corresponding statements for (unmarked) surfaces in [5,32], but we outline it here for completeness. Let SpinNet_{Σ,V} (G) be the set of all spin networks in (Σ, V) (i.e. we do not identify equivalent spin networks). We may define the operations in Eqs. 27 directly on SpinNet_{Σ,V} (G) (we do not claim that they satisfy the axioms of an algebra on this set). Since (Γ, f) → ev(Γ, f) is induced by the map of spaces (28), it follows that ev intertwines the operations of scalar multiplication, addition, and multiplication. Next we show that ev : SpinNet_{Σ,V} (G) → C^{∞}(M_{Σ,V} (G)) is surjective. Let ι_{skel} : |Γ_{skel}| → Σ be an embedded graph diagram with a single anchor point at every marked point, for which there exists a deformation retract r : Σ → ι_{skel}(|Γ_{skel}|). Then the map defined in Eq. (28),

Hom (Π_{1}(Σ, V ∪ V_{Γ}^{int}), G) → Hom (Π_{1}(|Γ_{skel}|, V_{Γ_{skel}}), G)

is a diffeomorphism. It follows that ev : A_{Γ_{skel}} → C^{∞}(M_{Σ,V} (G)) is an isomorphism.

It remains to show that that ev(Γ, f) = ev(Γ′, f′) only if the two spin networks are equivalent. We may assume that there exist maps between the geometric realizations μ : |Γ| → |Γ_{skel}| (just compose the map |Γ| → Σ with the retract Σ → |Γ_{skel}|) and likewise μ′ : |Γ′| → |Γ_{skel}|. Since ev : A_{Γ_{skel}} → C^{∞}(M_{Σ,V} (G)) is an isomorphism, ev(Γ, f) = ev(Γ′, f′) only if μ_{∗}f = μ′_{∗}f′, i.e. (Γ, f) and (Γ′, f′) are equivalent.

8.3 The quasi-Poisson bracket on SpinNet_{Σ,V} (G)

e e′ A

(a) The edge e ∈ E_{Γ} and e′ ∈ E_{Γ′} intersect at A.

ẽ_{A}^{ẽ′_{A}} A e_{A} e′_{A}

(b) After subdividing the edges e and e′ at A and then the newly created vertices, the newly created edges are labelled as depicted.

Figure 10

Suppose that ι : Γ → Σ and ι′ : Γ′ → Σ are graph diagrams such that the maps ι and ι′ are transverse to each other.^{21} Let (Γ×_{Σ} Γ′)^{anch} = (V_{Γ}^{anch})×_{Σ} (V_{Γ′}^{anch}) and

(Γ ×_{Σ} Γ′)^{int} = (Γ ×_{Σ} Γ′) ∖ (Γ ×_{Σ} Γ′)^{anch}.

Given A ∈ (Γ ×_{Σ} Γ′)^{int}, let e ∈ E_{Γ} and e′ ∈ E_{Γ′} be the two edges intersecting at A. Let Γ ∪_{A} Γ′ be the graph obtained from Γ ∪ Γ′ by subdividing the edges e and e′ at A and then merging the newly created vertices. It is clear that

ι ∪ ι′ : Γ ∪_{A} Γ′ → Σ

is a graph diagram. We let e_{A} and e′_{A} denote the newly created edges running from s(e) and s(e′) (respectively) to A and ẽ_{A} and ẽ′_{A} denote the newly created edges running from A to t(e) and t(e′) (respectively), as depicted in Fig. 10. Note that the orientations of the new edges are inherited from the orientations of the original edge. We define the map

ϱ : (𝔤 × 𝔤) × G^{E_{(Γ∪_{A}Γ′)}} → T (G^{E_{Γ}}) × T (G^{E_{Γ′}} )

21That is, the restriction of ι and ι′ to any two edges are transverse, and ι(V_{Γ}^{int})∩ι′(V_{Γ′}^{int}) = ∅.

Documenta Mathematica 20 (2015) 1071–1135

by

ϱ(ξ, η, g)_{e_{0}} := \begin{cases}(L_{gẽ_{A}} R_{g_{e_{A}}})_{∗}ξ & if e_{0} = e, \\ (L_{g_{}_{ẽ′_{A}}} R_{g_{e′_{A}}})_{∗}η & if e_{0} = e′, \\ g_{e_{0}} & otherwise. \end{cases}

Where L_{g}, R_{g} : G → G denote Left,Right multiplication by g ∈ G, and we have identified 𝔤 with the tangent space at the identity. The map ϱ is G^{V_{(Γ∪_{A}Γ′)}}- equivariant, where the action is defined on 𝔤 × 𝔤 by

g · (ξ, η) := (Ad_{g_{A}}ξ, Ad_{g_{A}}η), g ∈ G^{V_{(Γ∪_{A}Γ′)}}, ξ, η ∈ 𝔤.

The universal property of the tensor product implies that ϱ extends to a map

ϱ : (𝔤 ⊗ 𝔤) × G^{E_{(Γ∪_{A}Γ′)}} → T (G^{E_{Γ}}) ⊗ T (G^{E_{Γ′}} ) ⊆ ⊗^{2}T (G^{E_{(Γ∪Γ′)}})

We define

Ψ_{s} := ϱ(s, ·) : G^{E_{(Γ∪_{A}Γ′)}} → ⊗^{2}T (G^{E_{(Γ∪Γ′)}}).

Since s is G invariant, it follows that Ψ_{s} is G^{V_{(Γ∪_{A}Γ′)}}-equivariant. Therefore

(f ⊗ f′ → Ψ_{s}^{∗}(df ⊗ df′)) : A_{Γ}(G) ⊗ A_{Γ′}(G) → A_{(Γ∪_{A}Γ′)}(G)

is a G^{V_{Γ}^{anch}} × G^{V_{Γ′}^{anch}}-equivariant linear map.

Proposition 4. Let SpinNet_{Σ,V} (G) be endowed with the bracket

{[Γ, f], [Γ′, f′]} := ∑_{A∈(Γ×_{Σ}Γ′)^{int}} sign(A)[Γ ∪_{A} Γ′, Ψ_{s}^{∗}(df ⊗ df′)]

+ ∑_{A=(v,v′)∈(Γ×_{Σ}Γ′)^{anch}} \frac{1}{2} sign(A)[Γ ∪ Γ′, (df ⊗ df′)((ρ_{v} ⊗ ρ_{v′})s)], (29)

where sign(·) = ±1 is computed as pictured in Fig. 4, and Γ and Γ′ are assumed to be transverse graph diagrams. Then

π(d ev(Γ, f), d ev(Γ′, f′)) = ev {[Γ, f], [Γ′, f′]}.  (30)

Proof. Suppose first that V_{Γ}^{int} = ∅ = V_{Γ′}^{int}. Then A_{Γ}(G) = C^{∞}(G^{E_{Γ}}), and d ev(Γ, f) = ∏_{e∈E_{Γ}} hol_{e}^{∗} df. Thus

d ev(Γ, f) = ∑_{e∈E_{Γ}} ((L_{hol_{e}})_{e}^{∗}df)(hol_{e}^{∗} θ^{L})  (31)

where (L_{g})_{e}, (R_{g})_{e} : G^{E_{Γ}} → G^{E_{Γ}} denotes the left,right multiplication by g ∈ G on the e ∈ E_{Γ}-th factor of G^{E_{Γ}}. Substituting Eq. (31) (and the corresponding expression for d ev(Γ′, f′)) into Eq. (18) results in the equality

π(d ev(Γ, f), d ev(Γ′, f′))

= \frac{1}{2} ∑_{e∈E_{Γ},e′∈E_{Γ′}} ((L_{hol_{e}})_{e}^{∗}df) ⊗ ((L_{hol_{e′}})_{e′}^{∗}df′)(Ad_{hol_{(e,e′)}} ⊗ 1)s

Expanding (e, e′) in the last line using Eq. (15), and simplifying the resulting expression using the equalities hol_{e} = holẽ_{A} hol_{e_{A}} and hol_{e′} = hol_{}_{ẽ′_{A}} hol_{e′_{A}}, yields

∑_{A∈e∩e′,e∈E_{Γ},e′∈E_{Γ′}} \frac{1}{2}λ(A) sign(A)(df ⊗ df′)

(((L_{holẽ_{A}})_{e}(R_{hol_{e_{A}}})_{e} ⊗ (L_{hol_{ẽ′_{A}}})_{e′}(R_{hol_{e′_{A}}})_{e′})s).

If A ∈ ∂Σ, then λ(A) = 1, and

(L_{holẽ_{A}})_{e}(R_{hol_{e_{A}}})_{e} = ρ_{v}, (L_{hol_{ẽ′_{A}}})_{e}(R_{hol_{e′_{A}}})_{e} = ρ_{v′},

where A = (v, v′) ∈ (Γ ×_{Σ} Γ′)^{anch}. On the other hand, if A ∉ ∂Σ, then λ(A) = 2, and

(df ⊗ df′)(((L_{holẽ_{A}})_{e}(R_{hol_{e_{A}}})_{e} ⊗ (L_{hol_{ẽ′_{A}}})_{e′}(R_{hol_{e′_{A}}})_{e′})s) = Ψ_{s}^{∗}(df ⊗ df′).

Therefore Eq. (30) holds in this case.

Σ  Σ^{c}  Σ^{C}^{v}  ^{v}  ^{v}^{Dv}  ^{Dv}

Figure 11: For each vertex v, we delete a small open disk, D_{v}, from Σ such that its boundary ∂D_{v} intersects the graph precisely at v.

Suppose now that Γ and Γ′ are arbitrary graph diagrams (which are transverse to each other). Let Σ^{c} be the marked surface obtained from Σ as follows: for each v ∈ V_{Γ}^{int} ∪ V_{Γ′}^{int} delete a small open disk D_{v} ⊂ Σ ∖ Γ ∪ Γ′ such that

∂D_{v} ∩ (Γ ∪ Γ′) = v.

Let Σ^{C} be the marked surface obtained by Σ^{c} by adding a marked point at v (cf. Fig. 11). If we view (Γ, f) and (Γ′, f′) as spin networks in Σ^{C}, all their vertices are anchor points. Therefore Eq. (29) holds for these spin networks in Σ^{C}. Finally, applying Corollary 1 to the embeddings

(Σ^{c}, V) → (Σ, V)

and

(Σ^{c}, V) → (Σ^{C}, V ∪ V_{Γ}^{int} ∪ V_{Γ′}^{int})

shows that Eq. (29) holds for these spin networks in Σ.

8.4 Spin networks on quilted surfaces

Let Σ be a quilted surface.

Definition 8. A graph diagram in Σ consists of the following data:

• a directed graph, Γ with edges E_{Γ} and vertices V_{Γ},

• a subset of vertices V_{Γ}^{anch} ⊆ V_{Γ}, called ‘anchor points’, and

• a smooth map ι : Γ → Σ (it is understood that this map is really from the geometric realization) such that

ι^{−1}(Wall) = V_{Γ}^{anch}.

Often we will abuse notation and denote the graph diagram simply by Γ. A morphism between graph diagrams ι : Γ → Σ and ι′ : Γ′ → Σ is a map between the geometric realizations of the graphs, μ : |Γ| → |Γ′|, such that ι = ι′ ∘ μ and μ(V_{Γ}) ⊆ V_{Γ′}. A homotopy between graph diagrams ι_{0} : Γ → Σ and ι_{1} : Γ → Σ is one parameter family of graph diagrams ι_{t} : Γ → Σ, t ∈ [0, 1].

Suppose ι : Γ → Σ is a graph diagram. We denote by ι̃ the natural map V_{Γ}^{anch} → Wall. Let Γ_{d} = ι^{−1}(Σ_{d}),^{22} then Γ_{d} → Σ_{d} is a graph diagram in the marked surface (Σ_{d}, V_{d}), (here we have composed with the blow down map Σ̂_{d} → Σ_{d}). Define

A_{(ι:Γ→Σ)} := C^{∞}( ∏_{d∈Dom} G_{d}^{E_{Γ_{d}}} )^{C_{Γ}^{int}×C_{Γ}^{anch}},

where

C_{Γ}^{int} := ∏_{d∈Dom} G_{d}^{V_{Γ_{d}}^{int}}, C_{Γ}^{anch} := ∏_{v∈V_{Γ}^{anch}} C_{ι̃(v)},

with V_{Γ_{d}}^{int} := V_{Γ_{d}} ∖ V_{Γ_{d}}^{anch}. Notice that

A_{(ι:Γ→Σ)} = ( ˆ⊗_{d∈Dom}A_{Γ_{d}}(G_{d}))^{C_{Γ}^{anch}}.  (32)

To simplify notation, we will often abbreviate A_{(ι:Γ→Σ)} to A_{Γ}. As in the case of spin networks in marked surfaces, we have the following lemma.

Lemma 5. • The algebras A_{(ι_{0}:Γ→Σ)} and A_{(ι_{1}:Γ→Σ)} are canonically identified for homotopic graph diagrams ι_{0} : Γ → Σ and ι_{1} : Γ → Σ.

• The assignment (ι : Γ → Σ) ↦ A_{(ι:Γ→Σ)} is a functor from graph diagrams to algebras.

22I.e. the subgraph of Γ consisting of edges mapped into Σ_{d}

Documenta Mathematica 20 (2015) 1071–1135

Definition 9. A spin network in Σ is a pair (Γ, f), where Γ → Σ is a graph diagram, and f ∈ A_{Γ}. We say that spin networks (Γ_{0}, f_{0}) and (Γ_{1}, f_{1}) are homotopic if the underlying graph diagrams are homotopic and f_{0} = f_{1} ∈ A_{Γ_{0}} ≡ A_{Γ_{1}}. A morphism of spin networks μ : (Γ, f) → (Γ′, f′) is a morphism of graph diagrams μ : |Γ| → |Γ′| such that f′ = μ_{∗}f.

Let

M_{Σ} := ( ∏_{d∈Dom} M_{Σ_{d},V_{d}}(G_{d}))/C

denote the moduli space for the quilted surface, Σ. Then

C^{∞}(M_{Σ}) = C^{∞}( ∏_{d∈Dom} M_{Σ_{d},V_{d}}(G_{d}))^{C} = ( ˆ⊗_{d∈Dom}C^{∞}(M_{Σ_{d},V_{d}}(G_{d})))^{C}

Since the map

ev : ˆ⊗_{d∈Dom}A_{Γ_{d}}(G_{d}) → ˆ⊗_{d∈Dom}C^{∞}(M_{Σ_{d},V_{d}}(G_{d}))

is ∏_{d∈Dom} G_{d}^{V_{d}}-equivariant it restricts to a map of algebras

ev : A_{Γ} → C^{∞}(M_{Σ}).

As before, we consider two spin networks to be equivalent if they are related by a chain of homotopies and morphisms, and we define SpinNet_{Σ} to be the set of equivalence classes of spin networks in the quilted surface, Σ. The same arguments as in Section 8.2 shows that ev descends to a map of equivalence classes:

ev : SpinNet_{Σ} → C^{∞}(M_{Σ}).

Moreover, the analogues of Lemma 4, Proposition 3, and Proposition 4 hold for SpinNet_{Σ}. To be precise, suppose that ι : Γ → Σ and ι′ : Γ′ → Σ are two graph diagrams which are transverse. That is, ι(V_{Γ})∩ι′(V_{Γ′}) = ∅ and the restrictions of ι and ι′ to the edges are transverse. Suppose A ∈ Γ×_{Σ} Γ′ and let d_{A} ∈ Dom denote the domain such that A ∈ Σ_{d_{A}}. We define the graph diagram ι ∪_{A} ι′ : Γ ∪_{A} Γ′ → Σ as in Section 8.3, and we define

Ψ : ∏_{d∈Dom} G_{d}^{E_{(Γ∪pΓ′)_{d}}} → ⊗^{2}T ( ∏_{d∈Dom} G_{d}^{E_{(Γ∪Γ′)_{d}}} )

to be the sum of Ψ_{s_{d_{A}}} on the d_{A}-th factor with the zero sections on the other factors.

Theorem 9. The map

ev : SpinNet_{Σ} → C^{∞}(M_{Σ})

is an isomorphism of Poisson algebras, where scalar multiplication, addition, and multiplication are defined on SpinNet_{Σ} by Eqs. 27, and the Poisson bracket is defined by

{[Γ, f], [Γ′, f′]} := ∑_{A∈Γ×_{Σ}Γ′} sign(A)[Γ ∪_{A} Γ′, Ψ^{∗}(df ⊗ df′)],  (33)

where Γ and Γ′ are assumed to be transverse graph diagrams.

Proof. The proof that ev : SpinNet_{Σ} → C^{∞}(M_{Σ}) is an isomorphism of algebras is entirely analogous to that of Proposition 3, and so we omit it. As explained in Section 8.3, Ψ is equivariant with respect to the action of ∏_{d∈Dom} G_{d}^{V_{(Γ∪_{A}Γ′)_{d}}} . In particular, Ψ^{∗}(df ⊗ df′) ∈ A_{(Γ∪_{A}Γ′)}. Equation (32) identifies SpinNet_{Σ} with the subalgebra of C-invariant elements of

ˆ⊗_{d∈Dom} SpinNet_{Σ_{d},V_{d}}(G_{d}).

Moreover, the following diagram commutes:

SpinNet_{Σ}  C^{∞}(M_{Σ})

ˆ⊗_{d∈Dom} SpinNet_{Σ_{d},V_{d}}(G_{d})  C^{∞}( ∏_{d∈Dom} M_{Σ_{d},V_{d}}(G_{d}))

ev

ev

Hence Theorem 1.A and Proposition 4 imply that Eq. (29) defines a Poisson bracket on SpinNet_{Σ} for which ev is a morphism of Poisson algebras. Explicitly, this bracket is

{[Γ, f], [Γ′, f′]} := ∑_{d∈Dom} ( ∑_{A∈(Γ×_{Σ}Γ′)_{d}^{int}} sign(A)[Γ ∪_{A} Γ′, Ψ^{∗}(df ⊗ df′)]

+ ∑_{A=(p,p′)∈(Γ_{d}×_{Σ_{d}} Γ′_{d})^{anch}} \frac{1}{2} sign(A)[Γ ∪ Γ′, (df ⊗ df′)((ρ_{p} ⊗ ρ_{p′})s_{d})]).

The first term simplifies to yield Eq. (33), so we need only show that the second term vanishes. In terms of graph diagrams in the quilted surface, Σ, the second term can be rewritten as

\frac{1}{2} ∑_{w∈Wall} ∑_{p∈w∩Γ,p′∈w∩Γ′} ∑_{d∈Dom(w)} ϵ(p, p′, w, d)[Γ∪Γ′, (df ⊗df′)((ρ_{p} ⊗ρ_{p′})s_{d})]

where

Dom(w) = {d ∈ Dom; w ⊂ ∂Σ̂_{d}}.

+Σ̂_{d} p w  −Σ̂_{d} p w

(a) Sign(p, d) = ±1 depends on the direction of the edge in Γ leaving p.

+ Σ̂_{d} p′ p w  − Σ̂_{d} p p′ w Γ′  Γ Γ  Γ′

(b) Sign(p, p′, w, d) = ±1 depends on the order of p and p′, where we give w ⊂ ∂Σ̂_{d} the boundary orientation.

+ Σ̂_{d} A  − Σ̂_{d} A Γ′  Γ Γ  Γ′

(c) The blow down map Σ̂_{d} → Σ_{d} contracts w to a point. Thus the blow down equates ϵ(p, p′, w, d) with sign(A), where the latter is computed as in Fig. 4

Figure 12

Both

ϵ(p, p′, w, d) = sign(p, d) sign(p′, d) sign(p, p′, w, d),

and sign(p, d) = ±1 and sign(p, p′, w, d) = ±1 are computed as in Fig. 12. Note that ϵ(p, p′, w, d) = sign(A), where the left hand side is computed in the blow up, Σ̂_{d}, and the right hand side is computed for A = (p, p′) in the blow down, Σ_{d}. Now both f and f′ are C_{w}-invariant, that is

⨁_{d∈Dom(w)} ρ_{p}^{∗}df ∈ 𝔠_{w}^{⊥}, ⨁_{d∈Dom(w)} ρ_{p′}^{∗}df′ ∈ 𝔠_{w}^{⊥}

Since C_{w} was chosen to be coisotropic, ∑_{d∈Dom(w)} ϵ(p, p′, w, d)s_{d} vanishes on 𝔠_{w}^{⊥}. This completes the proof.

Suppose Σ and Σ′ are two quilted surfaces. A map Σ → Σ′ is called an embedding of quilted surfaces if

• it is an embedding of surfaces Σ → Σ′,

• it maps each domain, Σ_{d}, of Σ into a domain, Σ′_{d′}, of Σ′ which is colored with the same structure group (i.e. G_{d} ≡ G_{d′}), and • it maps each domain wall, w, of Σ into a domain wall, w′, of Σ′ which is

colored with the same coisotropic relation (i.e. C_{w} ≡ C_{w′}).

Corollary 3. An embedding of quilted surfaces Σ → Σ′ induces a Poisson morphism

M_{Σ′} → M_{Σ}

of the corresponding moduli spaces.

Proof. This follows directly from the functoriality of the assignment Σ → SpinNet_{Σ} of the Poisson algebra of spin networks to quilted surfaces and Theorem 9. Alternatively, it follows from Corollary 1 and the definition of the Poisson structure on M_{Σ} given in Eq. (24).

9 Colorful examples

Example 8 (Poisson Lie groups). Suppose that (𝔤, 𝔞, 𝔟) is a Manin triple and consider the quilted surface Σ pictured below

B A 1 B 1

where A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 such that A ∩ B = 1. Constraining the holonomies along the uncolored boundary arcs marked by 1 to be trivial is effectively the same as contracting those arcs to points, which results in the first image below.

b_{1} b_{2}  g a B  G A  B

Following Theorem 8.B, the moduli space is

{(b_{1}, g, b_{2}, a) ∈ B × G × B × A; b_{2}gb_{1} = a}/(Ĥ ∩ Ĉ),

where the holonomies are as pictured above. Since A ∩ B = 1, the group (Ĥ ∩ Ĉ) ≅ B × B of residual gauge transformations acts non-trivially only at the right two vertices by

(b, b′) · (b_{1}, g, b_{2}, a) = (bb_{1}, b′gb^{−1}, b_{2}b^{′−1}, a).

Thus the moduli space is identified with A. The Poisson structure on A is the Poisson Lie structure. This example should be compared with Example 5. Consider the embedding of quilted surfaces pictured below:

t  s

As explained above, the moduli space corresponding to the quilted surfaces depicted on the left and right is the Poisson Lie group, A. Meanwhile, as explained in Example 7, the moduli space for the quilted surface depicted in the middle is the symplectic groupoid

{(a_{1}, b_{1}, a_{2}, b_{2}) ∈ A × B × A × B; a_{1}b_{1} = b_{2}a_{2}},

integrating the Poisson Lie structure on A. The embedding of quilted surfaces induces the map

t × s : (a_{1}, b_{1}, a_{2}, b_{2}) → (a_{2}, a_{1}).

The first component of the map is the Lie groupoid target map, while the second component is the Lie groupoid source map. Corollary 3 shows that this map is a Poisson morphism. The multiplication for the Poisson Lie group A ≅ M_{Σ} can be understood by considering the following sequence of embeddings:

Σ ⊔ Σ  Σ̃ ⊔ Σ̃  Σ̃′  =  Σ̃′  Σ

which induces the following sequence of Poisson morphisms:

M_{Σ⊔Σ} ⟶^{ι} M_{Σ̃⊔Σ̃} ⟵^{φ} MΣ̃′ ⟶^{=} MΣ̃′ ⟶^{m} M_{Σ}

The map φ is an embedding which contains the image of ι. Composing this sequence, μ := m ∘ φ^{−1} ∘ ι, yields a multiplication map

μ : M_{Σ} × M_{Σ} = M_{Σ⊔Σ} → M_{Σ}.

Associativity can be verified via another sequence of embeddings of quilted surfaces (we omit the details). Under the canonical identification M_{Σ} ≅ A with the Lie group A, the map μ becomes the usual multiplication. However, as a consequence of this construction, we see that multiplication is a Poisson map.

Example 9 (Double Poisson Lie group). Suppose once again that (𝔤, 𝔞, 𝔟) is a Manin triple, and that A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 such that A ∩ B = 1. Then, G is a Poisson Lie group (cf. [11], see also [23], for example), we may describe this Poisson structure in terms of a moduli space as follows: Let Σ be the quilted surface pictured below left,

B A  g  B A 1 1B A  B A

Contracting the uncolored boundary arcs marked by 1 results in the image on the right. Since A∩B = 1, the group (Ĥ ∩Ĉ) of residual gauge transformations is trivial (cf. Theorem 8.B). Consequently, computing the holonomy along the dotted arc pictured in the second image, we may identify the moduli space with G. The resulting Poisson structure on G is Drinfel’d’s Double Poisson Lie structure. Applying Corollary 3 to the following embedding of quilted surfaces

yields the morphism of Poisson Lie groups A → G.

Example 10 (Symplectic double groupoid integrating Drinfel’d’s double [33, 35]). Suppose once again that (𝔤, 𝔞, 𝔟) is a Manin triple, and that A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 such that the product map A × B → G is a diffeomorphism. We may identify the symplectic double groupoid integrating the Poisson Lie structure on G (described in the previous example) with a moduli space, as follows: Let Σ be the quilted surface pictured below, where we have already contracted all the uncolored boundary arcs.

B A B A  b_{1} a_{1} b_{2} a_{2} g

The holonomies along the arcs pictured above identify the moduli space with

{(a_{1}, b_{1}, a_{2}, b_{2}, g) ∈ A × B × A × B × G; ga_{1}b_{1} = a_{2}b_{2}g},

where once again, the group of residual gauge transformations is trivial. Applying Corollary 3 to the following embedding of quilted surfaces

t  s

yields the Poisson morphism

t × s : (a_{1}, b_{1}, a_{2}, b_{2}, g) → (g, b_{2}gb_{1}^{−1})

whose components are the target/source map (respectively) onto G endowed with the Poisson Lie structure described in Example 9. This moduli space was first studied in [33, 35].

Example 11 (Lu-Yakimov Poisson homogeneous spaces). Suppose once again that (𝔤, 𝔞, 𝔟) is a Manin triple, and that A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 such that A ∩ B = 1. Let C ⊆ G be a closed subgroup whose Lie subalgebra 𝔠 ⊆ 𝔤 is coisotropic. Lu and Yakimov [26] describe a Poisson structure on G/C, which we may identify with the moduli space for the following quilted surface:

B A  C g

Computing the holonomy along the dotted arc yields an element g ∈ G, but the group C of residual gauge transformations acts by right multiplication on this element. Thus, following Theorem 8.B, the moduli space G/C carries a Poisson structure. The symplectic groupoid integrating the Poisson structure on G/C is the moduli space corresponding to the quilted surface pictured below:

B A  ba c g C

The holonomies along the arcs pictured above identify the moduli space with

{(a, b, c, g) ∈ A × B × C^{⊥} × G; abg = gc}/C

where c′ ∈ C acts by

c′ · (a, b, c, g) = (a, b, c′cc^{′−1}, gc^{′−1}).

Applying Corollary 3 to the embedding of quilted surfaces pictured below

yields the following source and target maps onto G/C:

t × s : (a, b, c, g) → (g, bg).

That the Lu-Yakimov space, G/C, carries a Poisson action of the Drinfeld Double, G, can be seen from the following sequence of embeddings (cf. Example 8):

=

Example 12 (Poisson Homogeneous spaces). Suppose once again that (𝔤, 𝔞, 𝔟) is a Manin triple, and that A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 such that A ∩ B = 1. As explained in Example 8, A carries the structure of a Poisson Lie group. Following Drinfel’d’s classification [12], pointed Poisson homogeneous spaces for A with connected stabilizers are classified, up to isomorphism, by Lagrangian subalgebras 𝔠 ⊆ 𝔤.^{23} Let C ⊂ G be a subgroup whose corresponding subalgebra 𝔠 ⊆ 𝔤 is Lagrangian. Let K = A ∩ C. Consider the quilted surface pictured below:

cb  g a B  G A  C

Following Theorem 8.B, the moduli space is

{(c, g, b, a) ∈ C × G × B × A; bgc = a}/ ∼,

where the holonomies are as pictured above, and the quotient is by the group of residual gauge transformations. Since A∩B = 1, the group of residual gauge transformations is K × C × B acting by

(k, c′, b′) · (c, g, b, a) = (c′ck^{−1}, b′gc^{′−1}, bb^{′−1}, ak^{−1}).

Thus the moduli space is identified with A/K. The Poisson Lie group A acts on A/K via the following sequence of embeddings:

=

To compute the bivector field on A/K, we work with the following skeleton:

B  G A  Cg_{1} g_{2}

23Note that not every Lagrangian subalgebra 𝔠 corresponds to a Poisson homogeneous space, but the correspondences become one-to-one when one works with local lie groups and homogeneous spaces (cf. [12–14]).

Alternatively (if one doesn’t wish to work with pointed spaces), Poisson homogeneous spaces for A with connected stabilizers are classified, up to isomorphism, by A-orbits of Lagrangian subalgebras 𝔠 ⊆ 𝔤.

Documenta Mathematica 20 (2015) 1071–1135

Let g_{1} = b_{1}a_{1} ∈ BA, and g_{2} = a_{2}c_{2} ∈ AC denote the holonomies along the corresponding edges, then we may choose the following (skeletal) coordinates for A/K:

{(ba_{1}, a_{2}c) ∈ BA × AC}/ ∼ ⟶^{(ba_{1},a_{2}c)→(a_{1}a_{2})K} A/K,

where the equivalence relation on the left hand side is induced by the action

(a′, b′, c′) · (ba_{1}, a_{2}c) = (b′ba_{1}a^{′−1}, a′a_{2}cc′−1), (a′, b′, c′) ∈ A × B × C (34)

In these coordinates, the bivector is the restriction of (14):

π = −\frac{1}{2} ∑_{i} ξ_{i}^{L}(1) ∧ η^{i}^{R}(2) + η^{i}^{L}(1) ∧ ξ_{i}^{R}(2)

to BA×AC ⊂ G^{2}, where, for ζ ∈ 𝔤, ζ^{L}(k) and ζ^{R}(k) denote the corresponding left and right-invariant vector fields which are tangent to the k^{th} factor of G^{2} (k = 1, 2), and {ξ_{i}} ⊂ 𝔞, and {η^{i}} ⊂ 𝔟 are dual bases. At the point (1, 1) ∈ G^{2},

π_{(1,1)} ∼ −\frac{1}{2} ∑_{i} ξ_{i}(2) ∧ η^{i}(2)

modulo the equivalence relation (34) (since at (1, 1), η^{i} ∈ 𝔟 ⇒ η^{i}^{L}(1) ∼ 0 and ξ_{i} ∈ 𝔞 ⇒ ξ_{i}^{L}(1) ∼ ξ_{i}^{R}(2)). Now, choose a vector space complement 𝔭 ⊆ 𝔟 to ann(𝔨) ⊆ 𝔟 ≅ 𝔞^{∗} (where 𝔨 ⊆ 𝔞 is the Lie algebra of K ⊆ A). Then 𝔞 = 𝔨 ⊕ ann(𝔭), and 𝔟 = 𝔭 ⊕ ann(𝔨). There exists a unique ϖ ∈ ∧^{2}(𝔞/𝔨) ≅ ∧^{2} ann(𝔭) such that

𝔠 = {(ξ + ϖ^{♯}(η) + η) ∈ 𝔤 | ξ ∈ 𝔨, η ∈ ann(𝔨)},

where ϖ^{♯}(η) = (η ⊗ id)(ϖ) and we have used the identification ϖ ∈ ∧^{2}(𝔞/𝔨) ≅ ∧^{2}( ann(𝔨)^{∗}). Let {ξ_{i}^{1}} ⊂ 𝔨 and {η_{1}^{i}} ⊂ 𝔭 be bases in duality, and similarly with {ξ_{i}^{0}} ⊂ ann(𝔭) and {η_{0}^{i}} ⊂ ann(𝔨). Then

π_{(1,1)} ∼ −\frac{1}{2} ∑_{i} ξ_{i}(2) ∧ η^{i}(2) = −\frac{1}{2} ∑_{i} ξ_{i}^{0}(2) ∧ η_{0}^{i}(2) + ξ_{i}^{1}(2) ∧ η_{1}^{i}(2)

Modulo (34), we have

π_{(1,1)} ∼ \frac{1}{2} ∑_{i} ξ_{i}^{0}(2) ∧ ϖ^{♯}(η_{0}^{i})(2)

(since ξ_{i}^{1} ∈ 𝔠 ⇒ ξ_{i}^{1}(2) ∼ 0 and ϖ^{♯}(η_{0}^{i}) + η_{0}^{i} ∈ 𝔠 ⇒ η_{0}^{i}(2) ∼ −ϖ^{♯}(η_{0}^{i})(2)). Thus, at the point K ∈ A/K, the bivector is just

ϖ ∈ ∧^{2}(𝔞/𝔨) ≅ ∧^{2}T_{K}(A/K),

which shows that the pointed Poisson homogeneous space (A/K, ∗ = K) corresponds to the Lagrangian Lie subalgebra 𝔠 ⊂ 𝔤 under Drinfel’d’s classification.

Example 13 (Drinfel’d Polyubles [15]). Suppose once again that (𝔤, 𝔞, 𝔟) is a Manin triple, and that A, B ⊂ G are Lie groups integrating 𝔞 and 𝔟 such that A ∩ B = 1. The n^{th} Drinfel’d polyuble of A (introduced in [15] for factorizable Poisson Lie groups, A, and generalized to arbitrary Poisson Lie groups by Jiang Hua-Lu and Victor Mouquin [24]) is a Lie-Poisson structure on

P_{n}(A) := \begin{cases}A × G^{k} & when n = 2k + 1 is odd, and \\ G^{k+1} & when n = 2k + 2 is even. \end{cases}

The Poisson manifold P_{n}(A) is naturally identified with the moduli space M_{Σ_{n}} for a quilted surface Σ_{n}, as we now describe.^{24} When n is even, the quilted surface Σ_{n} is the k = \frac{n−2}{2}-fold punctured disk pictured below left,

\begin{aligned}1 & B & B & 1 & B & B & k & G & A & g_{0} & g_{1} & g_{k} & A & A & A \\ & & Figure 13 \end{aligned}

contracting the uncolored boundary arcs marked by 1 results in the surface pictured above right. Since A ∩ B = 1, the group (Ĥ ∩ Ĉ) of residual gauge transformations is trivial (cf. Theorem 8.B). Thus, computing the holonomies along the dotted arcs canonically identifies M_{Σ_{n}} with

{(g_{0}, g_{1}, …, g_{k}) ∈ G^{k+1}} = P_{n}(A).

The resulting Poisson structure on P_{n}(A) is the Drinfel’d Polyuble Poisson Lie structure, as explained in [15] (cf. Remark 6). In particular, for n = 2, the quilted surface pictured in Figure 13 is just the one from Example 9 corresponding to the Drinfel’d Double. The double symplectic groupoid integrating P_{n}(A) is the moduli space for the quilted surface from Example 10, modified by attaching k handles in the interior. On the other hand, when n is odd, then the quilted surface Σ_{n} is the k = \frac{n−1}{2}- fold punctured disk pictured below left,

24For the factorizable case, this is essentially explained in [15], but using a slightly different language.

Documenta Mathematica 20 (2015) 1071–1135

1B B 1  B  B k  G Aa g_{1} g_{k} A

contracting the uncolored boundary arcs marked by 1 results in the surface pictured above right. As before, the group of residual gauge transformations is trivial, and computing the holonomies along the dotted arcs canonically identifies M_{Σ_{n}} with

{(a_{0}, g_{1}, …, g_{k}) ∈ A × G^{k}} = P_{n}(A).

The resulting Poisson structure on P_{n}(A) is the Drinfel’d Polyuble Poisson Lie structure, as explained in [15] (cf. Remark 6). The double symplectic groupoid integrating P_{n}(A) is the moduli space for the quilted surface pictured in Figure 8, modified by attaching k handles in the interior. As mentioned in [15], for n either even or odd, the natural embedding of Σ_{n} into the (unpunctured) disk from Example 8 yields a Poisson morphism A → P_{n}(A) between the corresponding moduli spaces (cf. Example 9 for the case n = 2). To take a different perspective on this example, we recall that the n^{th} Drinfeld’ polyuble is the Poisson Lie group whose corresponding Manin triple is (𝔤_{n}, 𝔞_{n}, 𝔟_{n}), where

𝔤_{n} := \begin{cases}𝔤 ⊕ (𝔤̄ ⊕ 𝔤)^{\frac{n−1}{2}} , & if n is odd, \\ (𝔤 ⊕ 𝔤̄)^{\frac{n}{2}} , & if n is even, \end{cases}  (35)

𝔞_{n} := \begin{cases}𝔞 ⊕ (𝔤_{Δ})^{\frac{n−1}{2}} , & if n is odd, \\ (𝔤_{Δ})^{\frac{n}{2}} , & if n is even, \end{cases}

𝔟_{n} := \begin{cases}(𝔤_{Δ})^{\frac{n−1}{2}} ⊕ 𝔟, & if n is odd, \\ 𝔞 ⊕ (𝔤_{Δ})^{\frac{n−2}{2}} ⊕ 𝔟, & if n is even, \end{cases}

Where 𝔤_{Δ} = {(ξ, ξ)} ⊆ 𝔤 ⊕ 𝔤̄ is the diagonal subalgebra, as in Example 2. Thus, after first replacing the Manin triple (𝔤, 𝔞, 𝔟) with (𝔤_{n}, 𝔞_{n}, 𝔟_{n}), we may identify the n^{th} Drinfel’d polyuble with the moduli space for the quilted surface from Example 8:

B_{n} A_{n} 1 B_{n} 1 G_{n}

G_{n} := G^{n}
A_{n} := \begin{cases}A × (G_{Δ})^{\frac{n−1}{2}} , & if n is odd, \\ (G_{Δ})^{\frac{n}{2}} , & if n is even, \end{cases}

B_{n} := \begin{cases}(G_{Δ})^{\frac{n−1}{2}} × B, & if n is odd, \\ A × (G_{Δ})^{\frac{n−2}{2}} × B, & if n is even, \end{cases}

Figure 14

where G_{Δ} is the diagonal subgroup, as in Example 2. In particular, using the multiplicative structure described in Example 8, this shows that P_{n} is a Poisson Lie group, as claimed. We now relate the G_{n}-colored quilted surface pictured in Figure 14 to the Gcolored quilted surface pictured in Figure 13. For brevity, we will consider only the case where n is even (the odd case is completely analogous). Let (Σ, V) be the underlying domain for the quilted surface pictured in Figure 14 (i.e. the disk with three marked points on the boundary). First, notice that coloring a domain Σ with G_{n} = G^{n} is equivalent to coloring n copies of the domain with G.

B_{n}  A_{n} G  B_{n}  n layers of G-colored sheets

Next, we use the fact (cf. Example 2) that coloring a pair of marked points by G_{Δ} corresponds to fusing those two marked points and then reducing by G (effectively erasing the marked point); that is, coloring a domain wall by G_{Δ} corresponds to sewing the surface together along that domain wall:

G_{Δ}-colored Domain wall.  =  No domain wall.

Since A_{n} = GΔ^{\frac{n}{2}} is just \frac{n}{2} copies of G_{Δ}, reducing by A_{n} corresponds to sewing the alternating pairs of sheets together at that marked point, as pictured below (for the case n = 6):

6 layers of G-colored sheets G_{Δ}  G_{Δ} G_{Δ} G^{Δ}  B G_{Δ} A B G_{Δ} A  G_{Δ}

Figure 15: We sew the sheets together along the domain walls coloured by G_{Δ}, so that the orientations coincide when one crosses the G_{Δ}- coloured domain wall. Note that the orientations of alternating layers are reversed - this corresponds to the fact that alternating layers (domains) are coloured with 𝔤̄ rather than 𝔤 (cf. (35)). Similarly, since B_{n} = A × (G_{Δ})^{\frac{n−2}{2}} × B is just \frac{n−2}{2} copies of G_{Δ} (bounded on one end by a copy of A, and on the other end by a copy of B) reducing by B_{n} corresponds to sewing alternating pairs of the interior sheets together at that marked point (as pictured above), and then coloring the marked points on the initial (respectively, final) sheet by A (respectively B). Unfolding the resulting quilted surface to lay it flat yields the quilted surface pictured in Figure 13.

Example 14 (Fission spaces [10]). Suppose that s ∈ S^{2}(𝔤)^{𝔤} is non-degenerate, 𝔤 = 𝔲_{+} ⊕ 𝔥 ⊕ 𝔲_{−} as a vector space (but not as a Lie algebra), where 𝔟_{±} := 𝔥⊕𝔲_{±} ⊆ 𝔤 are coisotropic subalgebras satisfying 𝔟_{±}^{⊥} = 𝔲_{±}. Suppose further that the Lie subalgebras 𝔲_{±}, 𝔟_{±}, 𝔥 all integrate to closed subgroups U_{±}, B_{±}, H ⊆ G such that H = B_{+} ∩ B_{−}. The metric on 𝔤 descends to a non-degenerate invariant metric on 𝔥 ⊆ 𝔤, and

C_{±} := {(c, c̃) ∈ H × G; c̃c^{−1} ∈ U_{±}}  (36)

is a coisotropic subgroup of H × G. Consider the quilted surface pictured in Fig. 16. Computing holonomies along the dashed lines yields

{(h, h_{0}, …, h_{2r−1}; C_{0}, C_{1}, …, C_{2r}) ∈ H^{2r+1} × G^{2r+1}

; h_{2i+1}^{−1}C_{2i+1}C_{2i}^{−1}h_{2i} ∈ U_{+} and h_{2i}^{−1}C_{2i}C_{2i−1}^{−1}h_{2i−1} ∈ U_{−}, }.

Meanwhile, since B_{+} ∩ B_{−} = G, the group of residual gauge transformations is ∏_{x_{1},...,x_{2r}} H, acting at the appropriate points on the quilted surface. Thus, up to a gauge transformation, we may assume that h_{0} = h_{1} = ⋯ = h_{2r−1} = 1. Setting S_{i} = C_{i}C_{i−1}^{−1}, we see that that the moduli space can be identified with

GA_{H}^{r} := {(h; S_{2r}, …, S_{1}; C_{0}) ∈ H × (U_{−} × U_{+})^{r} × G}.

As explained in Remark 11, _{G}A_{H}^{r} is a quasi-Poisson G × H manifold, where g ∈ G and k ∈ H act at the marked points v_{G} and v_{H}, respectively:

(g, k) · (h; S_{2r}, …, S_{1}; C_{0}) = (khk^{−1}, kS_{2r}k^{−1}, …, kS_{1}k^{−1}, kC_{0}g^{−1}),

x_{3}  x_{2r−3}  C_{0} C_{1} C_{2} hC_{2r}C_{2r−1}C_{2r−2} HG  C_{0} Ad_{h}h_{0} hh_{0} h_{2r−1} h_{2r−2} h_{2} h_{1} Ad_{h}h_{0} h x_{2r} x_{1} x_{2}  x_{2r−2} x_{2r−1} v_{G} v_{H}  v_{G} v_{H}

Figure 16: On the surface pictured above, the structure group in the yellow domain is G while the structure group in the blue domain is H. Along the boundary of the two domains, blue edges are colored with C_{+} while the red edges are colored with C_{−}. The dotted segment of the boundary between the two annuli is meant to indicate an alternating sequence of blue and red edges. Cutting along the vertical dotted line in the first picture yields the second picture. Acting by H at the points x_{1}, …, x_{2r} allows us to set the holonomies h_{0}, …, h_{2r−1} to the identity.

The holonomy along the boundary components,

(h; S_{2r}, …, S_{1}; C_{0}) → (C_{0}^{−1}hS_{2r} ⋯ S_{1}C_{0}, h^{−1}),

defines a moment map on _{G}A_{H}^{r}. This quasi-Hamiltonian G×H-space was first discovered by Boalch [8–10], who used it to study meromorphic connections on Riemann surfaces.

Example 15 (Bott-Samelson variety [6]). Suppose that s_{𝔤} ∈ S^{2}(𝔤)^{𝔤} is nondegenerate, 𝔤 = 𝔲_{+} ⊕ 𝔥 ⊕ 𝔲_{−} as a vector space (but not as a Lie algebra), where 𝔟_{±} := 𝔲_{±} ⊕ 𝔥 ⊆ 𝔤 are coisotropic subalgebras satisfying 𝔟_{±}^{⊥} = 𝔲_{±}. Let s_{𝔥} ∈ S^{2}(𝔥)^{𝔥} be the projection of s_{𝔤} along

𝔤 = 𝔥 ⊕ (𝔲_{+} ⊕ 𝔲_{−}) → 𝔥.

Suppose further that the Lie subalgebras 𝔟_{±}, 𝔥 integrate to closed subgroups B_{±}, H ⊆ G such that H = B_{+} ∩ B_{−}. We take 𝔤̃ = 𝔤 ⊕ 𝔥, and define

𝔟̂_{±} = {(ξ ± η, η) | ξ ∈ 𝔲_{±} and η ∈ 𝔥} ⊆ 𝔤̃

Then (𝔤̃, 𝔟̂_{+}, 𝔟̂_{−}) forms a Manin triple with respect to the non-degenerate copairing s := s_{𝔤} ⊕ −s_{𝔥} ∈ S^{2}(𝔤̃)𝔤̃.

Suppose that the Lie subalgebras 𝔟̂_{±} integrate to closed subgroups B̂_{±} ⊆ G̃ := G × H which project isomorphically to B_{±} along the map G × H → G; and suppose for simplicity t that B̂_{+} ∩ B̂_{−} = 1. Let (𝔤̃_{n}, 𝔟_{n}^{+}, 𝔟_{n}^{−}) denote the Manin triple corresponding to the n-th Drinfel’d Polyuble, P_{n}(B̂_{+}), as described in Example 13, and let (G̃_{n}, B_{n}^{+}, B_{n}^{−}) denote the corresponding triple of Lie groups (cf. Figure 14). Notice that B̃_{+} = B_{+} × H ⊆ G̃ is a coisotropic subgroup. We consider the Lu-Yakimov Poisson homogeneous space, M_{big} = G̃^{n}/B̃_{+}^{n} ≅ G^{n}/B_{+}^{n}, seen as the moduli space for the following quilted surface with G̃^{n}-colored domain (cf. Example 11):

B_{n}^{−} B_{n}^{+}  B̃_{+}^{n} G̃^{n}

Recall that a G̃^{n}-colored domain Σ is equivalent to n copies of Σ, colored by G̃. ‘Unfolding’ the n-layered quilted surface pictured above (as in Example 13), we obtain the following G̃-colored quilted surface with n B̃_{+}-colored domain walls:

\begin{aligned}B̂_{−} & B̂_{+} & B̃_{+} & B̃_{+} & B̃_{+} & B̃_{+} & g_{1} & g_{2} & g_{n} \\ & Figure 17 \end{aligned}

Computing the holonomies along the dotted arcs identifies the corresponding moduli space with

M_{big} := \overbrace{G̃ ×B̃_{+} ⋯ ×B̃_{+} G̃}^{n−1} ×B̃_{+} G̃/B̃_{+}.

As in Example 11, M_{big} has a Poisson action of the Drinfel’d Polyuble P_{2n}(B̂_{+}). We now take a closer look at this moduli space. Let Σ′ denote the disk with n + 2 marked points V = {−1, 0, …, n}:

−1 0  n n − 1 1 2 p_{1} p_{0} p_{−1}  p_{n} p_{2}

Computing holonomies along the boundary arcs identifies

M_{Σ′,V} (G̃) ≅ {(p_{0}, …, p_{n}) ∈ G^{n+1}},

a quasi-Poisson G̃^{n+2} space with the G̃^{n+2} action given by

(g′_{−1}, g′_{0}, …, g′_{n}) · (p_{0}, …, p_{n}) = (g′_{−1}p_{0}g′_{0}^{−1}, …, g′_{n−1}p_{n}g′_{n}^{−1}),

and moment map μ : M_{Σ′,V} (G̃) → G̃^{n+2} given by

μ(p_{0}, …, p_{n}) = (p_{−1}, p_{0}, …, p_{n}) (where p_{−1} := (p_{0} ⋯ p_{n})^{−1}).

Let

S_{big} = G̃ × (B̂_{−} · B̂_{+}) × G̃^{n} ⊆ G̃^{n+2},

and

C = B̂_{−} × B̂_{+} × B̃_{+}^{n} ⊆ G̃^{n+2}.

The moduli space M_{big} for the quilted surface picture in Figure 17 is the moment map quotient

M_{big} = μ^{−1}(S_{big})/C

Now choose subgroups B_{+} ⊆ P_{1}, …, P_{n} ⊆ G between B_{+} and the ambient group G. With P̃_{i} := P_{i} × H ⊆ G̃, we define

S_{small} = G̃ × (B̂_{−} · B̂_{+}) × P̃_{1} × ⋯ × P̃_{n} ⊆ S_{big}

(note that S_{small} is a Ĉ-stable submanifold of G̃^{n+2}). Computing the holonomies^{25} along the dotted arcs in Figure 17 identifies the moduli space M_{small} := μ^{−1}(S_{small})/C with the Bott-Samelson variety

P_{1} ×_{B_{+}} P_{2} ×_{B_{+}} ⋯ ×_{B_{+}} P_{n}/B_{+} ≅ P̃_{1} ×B̃_{+} ⋯ ×B̃_{+} P̃_{n}/B̃_{+} ⊆ M_{big}.

25More specifically

μ^{−1}(S_{small}) = {(p_{0}, p_{1}, …, p_{n}) | p_{0} = b_{−}b_{+} ∈ B̂_{−} · B̂_{+}, and p_{i} ∈ P̃_{i}} and (b̂_{−}, b̂_{+}, b̃_{1}, …, b̃_{n}) ∈ C acts by (b̂_{−}, b̂_{+}, b̃_{1}, …, b̃_{n}) · (b_{−}b_{+}; p_{1}, …, p_{n}) = (b̂_{−}b_{−}b_{+}b̂_{+}^{−1}; b̂_{+}p_{1}b̃_{1}^{−1}, b̃_{1}p_{2}b̃_{2}^{−1}, …, b̃_{n−1}p_{n}b̃_{n}^{−1}).

Thus the map (which computes the holonomies pictured in Figure 17)

(b_{−}b_{+}, p_{1}, …, p_{n}) → [g_{1}, …, g_{n}] := [b_{+}p_{1}, p_{2}, p_{3}, …, p_{n}] ∈ P̃_{1} ×B̃_{+} ⋯ ×B̃_{+} P̃_{n}/B̃_{+} descends to an isomorphism on μ^{−1}(S_{small})/C.

Documenta Mathematica 20 (2015) 1071–1135

In particular, we recover the result of Jiang-Hua Lu [24] that the Bott-Samelson variety M_{small} is a Poisson submanifold of M_{big}. There are n and distinct Poisson morphisms from the Bott-Samelson variety M_{small} to the Shubert variety G/B_{+} ≅ G̃/B̃_{+} (which carries the Lu-Yakimov Poisson structure, cf. Example 11),

q_{k} : [g_{1}, …, g_{n}] → [g_{1} ⋯ g_{k}], 1 ≤ k ≤ n,

corresponding to the embeddings (we picture the case k = 2):

A Quilted surfaces and moduli spaces of flat bundles

In this section we prove Theorems 6 and 7, and identify the more general Poisson manifolds μ̂^{−1}(h)/ Stab(h) ⊆ M_{Σ,V} /C with moduli spaces.

Proof of Theorem 6. First, we recall that M_{Σ,V} is the moduli space of pairs (11)

(P → Σ; {v̂ ∈ P_{v}}_{v∈V}),

of flat G-bundles^{26} over Σ equipped with a framing over V ⊂ ∂Σ. Of course this moduli space only depends on the homotopy type of the pair (Σ, V). Since (Σ, V) has the same homotopy type as (Σ̂, ∪_{v∈V} w_{v}), M_{Σ,V} is equivalently described as the moduli space of pairs

(P → Σ̂; {ŵ ∈ C_{flat}^{∞}(w, P |_{w})}_{w∈Wall}),

of flat G-bundles over Σ̂ which are equipped with a flat framing over each domain wall w ∈ Wall. Likewise, M_{Σ,V} /C is canonically isomorphic to the moduli space of pairs

(P → Σ, {Q_{v} → v}_{v∈V})

of flat G-bundles over Σ which are equipped with reduction of the structure group from G to C_{v} over each marked point v ∈ V.^{27} As above, we can also canonically identify M_{Σ,V} /C with the moduli space of pairs

(P → Σ̂, {Q_{w} → w}_{w∈Wall})

26Recall that we always take our G-bundles to be principal.

27That is, a choice of a principal C_{v}-subbundle Q_{v} ⊆ P_{v} over each marked point v ∈ V.

Documenta Mathematica 20 (2015) 1071–1135

of flat G-bundles over Σ̂ equipped with a flat reduction of the structure group from G to C_{w} over each domain wall w ∈ Wall (as in the description we gave in the introduction). Indeed the isomorphism between these two moduli spaces is given by sending (P → Σ, {Q_{v} → v}_{v∈V}) to its pullback (p^{∗}P → Σ̂, {p^{∗}Q_{v} → w_{v}}_{v∈V}) along the blowdown map p : Σ̂ → Σ. Thus we have proven Theorem 6.

Proof of Theorem 7. To describe the Poisson submanifolds μ̂^{−1}(1)/ Stab(1) ⊆ M_{Σ,V} /C as moduli spaces, we need to first identify M̂ with a moduli space. By assumption, s ∈ S^{2}(𝔤) is non-degenerate, and each C_{v} ⊆ G is Lagrangian (i.e. C_{v}^{⊥} = C_{v} for each v ∈ V). Thus Ĉ_{v} = C_{v} × C_{v} ⊆ G × G. (We will treat the general case later). As a first step, consider the moduli space of triples

(P → Σ̂, {Q_{w} → w}_{w∈Wall}, {x_{v} ∈ Q_{w_{v}}|_{v}}_{v∈V_{−}∪V_{+}}),  (37)

i.e a pair of the form (22), but equipped with an additional framing of the C_{v}-bundle Q_{w_{v}} at either end point v_{−}, v_{+} ⊂ w_{v}. Since x_{v} ∈ Q_{w_{v}} ⊂ P, we have a fully faithful forgetful functor which sends a triple (37) to the pair ((P → Σ̂, {x_{v} ∈ P |_{v}}_{v∈V_{−}∪V_{+}}). Thus, the moduli space of triples (37) imbeds as a subspace of the moduli space M_{Σ̂,V_{−}∪V_{+}} of pairs (11). Now the holonomy of a given connection along the domain wall w_{v}, as measured with respect to the framing x_{v_{−}}, x_{v_{+}}, is an element of C_{v}. This is the only restriction placed on the holonomies by the structure of the triples (37), and it identifies the moduli space of triples with the imbedded submanifold M̂ ⊆ M_{Σ̂,V_{−}∪V_{+}} (cf. (21)). The forgetful functor sending a triple (37) to the corresponding pair (22) coincides with the quotient map M̂ → M̂/Ĉ ≅ M_{Σ,V} /C. Meanwhile, μ̂^{−1}(1) ⊂ M̂ consists of those triples (37) for which parallel transport along a_{v} takes x_{v_{+}} to x_{σ(v)_{−}}. Let q : Σ̂ → Σ̂^{c} denote the quotient map which collapses each respective segment a_{v} to a point, as in Figure 7. We denote the image of each domain wall w in Σ̂^{c} by the same symbol. The flat connection along each a_{v} canonically trivializes P |_{a_{v}}, allowing us to identify P → Σ̂ with the pullback of a flat bundle P^{c} → Σ̂^{c}, i.e. P = q^{∗}P^{c}. Moreover, if the parallel transport along a_{v} takes x_{v_{+}} to x_{σ(v)_{−}}, then q(x_{v_{+}}) = q(x_{σ(v)_{−}}) ∈ (Q_{w_{v}} ∩ Q_{w_{σ(v)}})|_{q(v)} is a common framing. Thus the moduli space μ̂^{−1}(1) of triples (37) is canonically isomorphic to the moduli space of triples over Σ̂^{c}:

(P^{c} → Σ̂^{c}, {Q_{w} → w}_{w∈Wall}, {x_{v} ∈ (Q_{w_{v}} ∩ Q_{w_{σ(v)}})|_{q(v)}}_{v∈V}) (38)

where each x_{v} is a common framing of Q_{w_{v}} and Q_{w_{σ(v)}} over q(v). Now Ĉ acts on the moduli space M̂ of triples (38) by forgetting the framing. The essential image of this forgetful functor consists of those pairs

(P^{c} → Σ̂^{c}, {Q_{w} → w}_{w∈Wall})

of flat G-bundles over Σ̂^{c} equipped with reductions of the structure group from G to C_{w} over each domain wall w such that ∅ ≠ (Q_{w_{v}} ∩ Q_{w_{σ(v)}})|_{q(v)}. Equivalently, for any two domain walls w, w′ ∈ Wall, the fibres of Q_{w} and Q_{w′} intersect non-trivially over every point in the intersection of their bases w ∩ w′. That is, the reductions of structure along w and w′ are compatible (cf. Definition 5). This proves Theorem 7

We now lift the restrictions that s ∈ S^{2}(𝔤) be non-degenerate, and each C_{v} ⊆ G be Lagrangian; instead we suppose only that C_{v}^{⊥} ⊂ C_{v} is closed, for each v ∈ V. To describe the Poisson submanifolds μ̂^{−1}(h)/ Stab(h) ⊆ M_{Σ,V} /C as moduli spaces, we proceed as above, by first identifying M̂ with a moduli space. Consider the moduli space of triples

(P → Σ̂, {Q_{w} → w}_{w∈Wall}, {x_{v} ∈ Q_{w_{v}}|_{v}}_{v∈V_{−}∪V_{+}}),  (39)

i.e a pair of the form (22), but equipped with an additional framing of the C_{v}-bundle Q_{w_{v}} at either end point v_{−}, v_{+} ⊂ w_{v} such that parallel transport in the quotient bundle Q_{w_{v}}/C_{w_{v}}^{⊥} identifies their images x̃_{v_{−}}, x̃_{v_{+}} ∈ Q_{w_{v}}/C_{w_{v}}^{⊥}. As in the proof of Theorem 7, the forgetful functor which sends a triple (39) to the pair

((P → Σ̂, {x_{v} ∈ P |_{v}}_{v∈V_{−}∪V_{+}})

identifies the moduli space of triples with the imbedded submanifold M̂ ⊆ M_{Σ̂,V_{−}∪V_{+}} (cf. (21)). The forgetful functor sending a triple (39) to the corresponding pair (22) coincides with the quotient map M̂ → M̂/Ĉ ≅ M_{Σ,V} /C; and the moment map μ̂ : M̂ → G^{V} coincides with the map which sends an isomorphism class of triples (39) to the collection of holonomies (hol_{a_{v}})_{v∈V} ∈ G^{V} (these holonomies are measured with respect to the chosen framings). Let q : Σ̂ → Σ̂^{c} denote the quotient map which collapses each respective segment a_{v} to a point, as in Figure 7. As before, P → Σ̂ is pullback of a flat bundle P^{c} → Σ̂^{c}. Thus the moduli space M̂ of triples (39) is canonically isomorphic to the moduli space of triples over Σ̂^{c}:

(P^{c} → Σ̂^{c}, {Q_{w} → w}_{w∈Wall}, {x_{v} ∈ Q_{w_{v}}|_{q(v)}}_{v∈V_{−}∪V_{+}}),

and the moment map μ̂ sends a given triple to the element (g_{v})_{v∈V} ∈ G^{V} such that x_{σ(v)_{−}} = g_{v} · x_{v_{+}} for each v ∈ V. In particular, the moment map factors through the isomorphism classes of triples over the boundary:

(P′ → ∂Σ̂^{c}, {Q′_{w} → w}_{w∈Wall}, {x_{v} ∈ Q′_{w_{v}}|_{q(v)}}_{v∈V_{−}∪V_{+}}),  (40)

where P′ → ∂Σ̂^{c} is a G-bundle, Q′_{w} ⊆ P′|_{w} is a C_{w}-subbundle equipped with

1. a flat connection on the quotient bundle Q_{w}/C_{w}^{⊥}, and

2. a framing at either endpoint of w such that parallel transport in the quotient bundle, Q_{w}/C_{w}^{⊥}, identifies the images of the framings.

An element (g_{v_{−}}, g_{v_{+}})_{v∈V} ∈ Ĉ ⊆ G^{V_{−}∪V_{+}} acts on a triple (40) by x_{v} → g_{v} · x_{v} for each v ∈ V_{−} ∪ V_{+}. Notice that neither P′ nor Q′_{w} carry a distinguished flat structure, so the Ĉequivariant map which sends an isomorphism class of such quadruples (40) to the unique element (g_{v})_{v∈V} ∈ G^{V} such that x_{σ(v)_{−}} = g_{v} · x_{v_{+}} for each v ∈ V is injective. In particular, a given isomorphism class of quadruples (40) is naturally identified with an element of G^{V}, and under this identification, the moment map μ̂ : M̂ → G^{V} coincides with the functor which sends a quadruple of the form (39) to a quadruple of the form (40) by restricting the ambient principal bundle P^{c} to ∂Σ̂^{c}, and forgetting the superfluous flat connections.

Definition 10. Boundary data for the surface Σ̂^{c} is a pair

(P → ∂Σ̂^{c}, {Q_{w} → w}_{w∈Wall}),

where P → ∂Σ̂^{c} is a G-bundle, and each Q_{w} ⊆ P |_{w} is a C_{w}-subbundle equipped with a choice of flat connection on the quotient bundle Q_{w}/C_{w}^{⊥}.

Thus the isomorphism classes of boundary data for Σ̂^{c} correspond precisely to the Ĉ-orbits of isomorphism classes of quadruples (40). In particular, a given isomorphism class of boundary data is naturally identified with a Ĉorbit O ⊆ G^{V}. Of course, given a flat G-bundle over Σ̂^{c} equipped with a flat reduction of the structure group from G to C_{w} over each domain wall {w}_{w∈Wall}:

(P → Σ̂^{c}, {Q_{w} → w}_{w∈Wall}),

restricting P to ∂Σ̂^{c}, and forgetting the superfluous flat connections defines a boundary data,

(P |_{∂Σ̂^{c}}, {Q_{w}}_{w∈Wall}),

for Σ̂^{c}. Summarizing, we have:

Theorem 10. Let O ⊆ G^{V} be a Ĉ-orbit corresponding to a given isomorphism class of boundary data for Σ̂^{c}. Suppose that O is in a clean position relative to μ̂, and the Ĉ-orbits of μ̂^{−1}(O) form a regular foliation. Then the moduli space of pairs

(P → Σ̂^{c}, {Q_{w} → w}_{w∈Wall}),

a flat G-bundles over Σ̂^{c} equipped with a flat reduction of the structure group from G to C_{w} over each domain wall {w}_{w∈Wall} and such that (P |_{∂Σ̂^{c}}, {Q_{w}}_{w∈Wall}), lies in the given isomorphism class of boundary data, is a Poisson manifold which is canonically isomorphic to μ̂^{−1}(O)/Ĉ as a Poisson manifold.

Remark 14. When s ∈ S^{2}(𝔤) is non-degenerate, and each C_{v} ⊆ G is Lagrangian (i.e. C_{v}^{⊥} = C_{v} for each v ∈ V), each moduli space described in Theorem 7 is a symplectic manifold (cf. [21]).

B Some Technical Lemmas

We expect the following lemmas are common knowledge, but we were unable to find any proofs for them, so we have included these proofs for completeness.

Lemma 6. Suppose that embedded submanifolds U, V ⊆ W intersect cleanly, q : W → Q is a surjective submersion, and V is a union of q-fibres, i.e V = q^{−1}(q(V)). If q(U), q(V) ⊆ Q are embedded submanifolds, and q|_{U} : U → q(U) is a submersion, then q(U) and q(V) intersect cleanly.

Proof. We claim that q(U) ∩ q(V) ⊆ Q is an embedded manifold. Assuming this, we need only show that

T (q(U) ∩ q(V)) = T q(U) ∩ T q(V).

It is clear that the right hand side contains the left hand side. Suppose that

ξ ∈ T q(U) ∩ T q(V) = q_{∗}(T U) ∩ q_{∗}(T V).

By assumption, T V contains the kernel of q_{∗}, so there exists a lift ξ̃ ∈ T U ∩T V such that ξ = q_{∗}(ξ̃). Since U and V intersect cleanly, ξ̃ ∈ T (U ∩ V), and hence ξ = q_{∗}(ξ̃) ∈ q_{∗}(T (U ∩ V)) ⊆ T (q(U) ∩ q(V)). We now prove the claim that q(U) ∩ q(V) ⊆ Q is an embedded submanifold. Consider the groupoid U ×_{q(U)} U over U corresponding to the surjective submersion q : U → q(U). Then U ×_{q(U)} U is Morita equivalent to q(U), thus it is a proper groupoid with trivial isotropy groups [28, § 5.6]. Since U ∩ V ⊆ U is an embedded submanifold, it follows that

U ×_{q(U)} (U ∩ V) ⊆ U ×_{q(U)} U

is an embedded submanifold. Now V is a union of q-fibres, therefore U ∩V ⊆ U is a union of q|_{U}-fibres, and (U ∩ V) ×_{q(U)} (U ∩ V) = U ×_{q(U)} (U ∩ V). In summary, we have shown that

H := (U ∩ V) ×_{q(U)} (U ∩ V) ⊆ U ×_{q(U)} U

is an embedded subgroupoid over base U ∩V. Since H is a proper Lie groupoid with trivial isotropy groups (it inherits these properties from the inclusion H ⊆ U ×_{q(U)} U), it follows that H is Morita equivalent to a manifold [28, § 5.6]. That is to say, the orbit space of H, q(U ∩V), is a manifold. Since V is a union of q-fibres, q(U) ∩ q(V) = q(U ∩ V) is also a manifold. By construction, the inclusion q(U)∩q(V) → Q is an injective immersion. Now U∩V ⊂ M has the subset topology, while q(U∩V) and Q both have the quotient topology. Since q is a surjective submersion, q(U) ∩ q(V) = q(U ∩ V) ⊆ Q has the subset topology. Thus q(U) ∩ q(V) ⊆ Q is an embedded submanifold.

Lemma 7. Suppose that G and H are Morita equivalent Lie groupoids, and the orbit-equivalence relation on H is regular, then the orbit-equivalence relation on G is regular too.

Proof. By Godement’s criterion, the image R_{H} = t × s(H_{1}) ⊆ H_{0} × H_{0} is a closed embedded submanifold. Let G ← K → H be a Morita equivalence.^{28} Then K_{0} → H_{0} is a surjective submersion, and the pullback R_{K} = R_{H} ×_{H_{0}×H_{0}} (K_{0} × K_{0}) is a closed embedded submanifold (since K_{0} → H_{0} is a surjective submersion). But R_{K} = t × s(K_{1}), so the projection from from R_{K} onto either factor of K_{0} is a surjective submersion. It follows from Godement’s criterion that R_{K} is a regular equivalence relation. Now form Q = K_{0}/R_{K}. The quotient map q : K_{0} → Q is a surjective submersion, which factors through the surjective submersion f : K_{0} → G_{0}, i.e. there exists a map p : G_{0} → Q such that q = p ∘ f. Since f and q are smooth, surjective submersions, it follows that q is a smooth, surjective submersion. Since Q = G_{0}/G_{1} is the orbit space for G, this proves the lemma.

References

[1] Anton Yu Alekseev and Yvette Kosmann-Schwarzbach. Manin pairs and moment maps. Journal of Differential Geometry, 56(1):133–165, 2000.

[2] Anton Yu Alekseev, Yvette Kosmann-Schwarzbach, and Eckhard Meinrenken. Quasi-Poisson manifolds. Canadian Journal of Mathematics, 54(1):3–29, 2002.

[3] Anton Yu Alekseev, Anton Z. Malkin, and Eckhard Meinrenken. Lie group valued moment maps. Journal of Differential Geometry, 48(3):445–495, 1998.

[4] Michael Atiyah and Raoul Bott. The Yang-Mills Equations over Riemann Surfaces. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 308(1505):523–615, March 1983.

[5] John C. Baez. Spin networks in gauge theory. Advances in Mathematics, 117(2):253–272, 1996.

[6] Elek Balázs. Computing the Standard Poisson Structure on Bott-Samelson Varieties in Coordinates. Master’s thesis, The University of Hong Kong, June 2012.

[7] Kai Behrend and Ping Xu. Differentiable stacks and gerbes. J. Symplectic Geom., 9(3):285-341, 2011.

[8] Philip Paul Boalch. Quasi-Hamiltonian geometry of meromorphic connections. Duke Mathematical Journal, 139(2):369–405, August 2007.

[9] Philip Paul Boalch. Through the analytic Halo: fission via irregular singularities. Université de Grenoble. Annales de l’Institut Fourier, 59(7):2669– 2684, 2009.

[10] Philip Paul Boalch. Geometry and braiding of Stokes data; fission and wild character varieties. Annals of Mathematics. Second Series, 179(1):301–365, 2014.

28i.e. K is a Lie groupoid, and the two maps are Morita morphisms (cf. [7, 28]).

Documenta Mathematica 20 (2015) 1071–1135

[11] Vladimir Gershonovich Drinfel’d. Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations. Doklady Akademii Nauk SSSR, 268(2):285–287, 1983.

[12] Vladimir Gershonovich Drinfel’d. On Poisson homogeneous spaces of Poisson-Lie groups. Theoretical and Mathematical Physics, 95(2):524–525, May 1993.

[13] Sam Evens and Jiang-Hua Lu. On the variety of Lagrangian subalgebras. I. Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 34(5):631–668, 2001.

[14] Sam Evens and Jiang-Hua Lu. On the variety of Lagrangian subalgebras. II. Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 39(2):347–379, 2006.

[15] V. V. Fock and A. A. Rosly. Poisson structure on moduli of flat connections on Riemann surfaces and the r-matrix. In Amer. Math. Soc. Transl. Ser. 2, pages 67–86. Amer. Math. Soc., Providence, RI, 1999.

[16] William M. Goldman. The symplectic nature of fundamental groups of surfaces. Advances in Mathematics, 54(2):200–225, 1984.

[17] William M. Goldman. Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Inventiones Mathematicae, 85(2):263–302, 1986.

[18] Anton Kapustin. Topological field theory, higher categories, and their applications. In Proceedings of the International Congress of Mathematicians. Volume III, pages 2021–2043, New Delhi, 2010. Hindustan Book Agency.

[19] Anton Kapustin and Natalia Saulina. Surface operators in 3d topological field theory and 2d rational conformal field theory. In Proc. Sympos. Pure Math., pages 175–198. Amer. Math. Soc., Providence, RI, 2011.

[20] Anton Kapustin and Natalia Saulina. Topological boundary conditions in abelian Chern-Simons theory. Nuclear Physics. B, 845(3):393–435, 2011.

[21] David Li-Bland and Pavol Ševera. Symplectic and Poisson geometry of the moduli spaces of flat connections over quilted surfaces. In Damien Calaque and Thomas Strobl, editors, Mathematical Aspects of Quantum Field Theories (Mathematical Physics Studies). Springer, December 2014.

[22] David Li-Bland and Pavol Ševera. On Deformation Quantization of Poisson Lie Groups and Moduli Spaces of Flat Connections. International Mathematics Research Notices (to appear), 19, 2014.

[23] Jiang-Hua Lu. Multiplicative and affine Poisson structures on Lie groups. PhD thesis, University of California, Berkeley, 1990.

[24] Jiang-Hua Lu and Victor Mouquin. Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras. preprint availble at:

http://arxiv.org/abs/1504.06843

[25] Jiang-Hua Lu and Alan Weinstein. Groupoides symplectiques doubles des groupes de Lie-Poisson. Comptes Rendus de l’Academie des Sciences - Series I: Mathematics, 309:951–954, 1989.

[26] Jiang-Hua Lu and Milen Yakimov. Group orbits and regular partitions of Poisson manifolds. Communications in Mathematical Physics, 283(3):729– 748, 2008.

[27] Gwenael Massuyeau and Vladimir G Turaev. Quasi-Poisson structures on representation spaces of surfaces. International Mathematics Research Notices, (1):1–64, 2014.

[28] Ieke Moerdijk and Janez Mrčun. Introduction to foliations and Lie groupoids. Cambridge University Press, 2003.

[29] Xin Nie. The quasi-Poisson Goldman formula. Journal of Geometry and Physics, 74:1–17, 2013.

[30] Roger Penrose. Applications of negative dimensional tensors. In Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pages 221–244. Academic Press, London, 1971.

[31] Nicolai Yu Reshetikhin, Jørgen Ellegaard Andersen, and J. Mattes. The poisson structure on the moduli space of flat connections and chord diagrams. Topology, 35(4):1069–1083, 1996.

[32] Philippe Roche and Andras Szenes. Trace functionals on noncommutative deformations of moduli spaces of flat connections. Advances in Mathematics, 168(2):133–192, 2002.

[33] Pavol Ševera. (Non-)abelian Kramers-Wannier duality and topological field theory. Journal of High Energy Physics. A SISSA Journal, (5):no. 49–10, 2002.

[34] Pavol Ševera. Some title containing the words “homotopy” and “symplectic”, e.g. this one. In Travaux mathématiques. Fasc. XVI, pages 121–137. Univ. Luxemb., Luxembourg, 2005.

[35] Pavol Ševera. Moduli spaces of flat connections and Morita equivalence of quantum tori. Documenta Math, 17:607–625, 2012.

[36] Ševera, Pavol. Left and right centers in quasi-Poisson geometry of moduli spaces. Advances in Mathematics, 279:263-290, 2015.

[37] Anders Reiter Skovborg. The Moduli Space of Flat Connections on a Surface, Poisson Structures and Quantization. PhD thesis, Aarhus University, July 2006.

[38] Vladimir G Turaev. Intersections of loops in two-dimensional manifolds. Matematicheskiĭ Sbornik. Novaya Seriya, 106(148)(4):566–588, 1978.

[39] Katrin Wehrheim and Christopher Thomas Woodward. Functoriality for Lagrangian correspondences in Floer theory. Quantum Topology, 1(2):129– 170, 2010.

[40] Katrin Wehrheim and Christopher Thomas Woodward. Quilted Floer cohomology. Geometry & Topology, 14(2):833–902, 2010.

David Li-Bland Dept. of Math. U.C. Berkeley Berkeley California, USA. libland@math.berkeley.edu Pavol Ševera Dept. of Math. Université de Genève Geneva, Switzerland on leave from: Department of Theoretical Physics FMFI UK Bratislava, Slovakia pavol.severa@gmail.com 1136

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