Moduli Spaces of Flat Connections

and Morita Equivalence of Quantum Tori

Pavol Ševera

Received: June 20, 2011 Revised: July 6, 2012

Communicated by Eckhard Meinrenken

Abstract. We study moduli spaces of flat connections on surfaces with boundary, with boundary conditions given by Lagrangian Lie subalgebras. The resulting symplectic manifolds are closely related with Poisson-Lie groups and their algebraic structure (such as symplectic groupoid structure) gets a geometrical explanation via 3dimensional cobordisms. We give a formula for the symplectic form in terms of holonomies, based on a central extension of the gauge group by closed 2-forms. This construction is finally used for a certain extension of the Morita equivalence of quantum tori to the world of Poisson-Lie groups.

2010 Mathematics Subject Classification: 53D30

1 Introduction

Let 𝔤 be a Lie algebra with an invariant inner product ⟨·, ·⟩ (of any signature). It gives rise to two interesting types of symplectic manifolds. The first type are moduli spaces of flat 𝔤-connections on oriented surfaces. The second type are symplectic manifolds connected with Poisson-Lie groups such as the Lu-Weinstein double symplectic groupoid [9] (the symplectic groupoid integrating a Poisson-Lie group) corresponding to a Manin triple

𝔥_{1}, 𝔥_{2} ⊂ 𝔤.

We shall notice that these ”Poisson-Lie type” symplectic manifolds are, in fact, themselves moduli spaces of flat connections, if we allow surfaces with boundary and impose boundary conditions on the flat connections. To get the Lu-Weinstein double groupoid, the surface is a square, with boundary conditions as on the picture:

\begin{matrix}𝔥_{1} \\ 𝔥_{1} \end{matrix} ; \begin{matrix}𝔥_{2} \\ 𝔥_{2} \end{matrix}

We shall provide formulas for symplectic forms using holonomies of the flat connections, in the spirit of Alekseev-Malkin-Meinrenken [1]. The basic idea is that the symplectic form can be interpreted as the integral over the surface of the curvature of a certain connection. The integral is then readily computed in terms of parallel transport. Moreover we shall describe how 3dim bodies give rise to Lagrangian submanifolds; for example, this picture gives one of the two products in Lu-Weinstein double groupoid:

This is a symplectic version of Chern-Simons TQFT in the sense of D. Freed [5], with appropriate boundary conditions. The motivation for this work was to give a symplectic description of Morita equivalence of quantum tori, and moreover, to extend this Morita equivalence from Abelian T-duality to Poisson-Lie T-duality [7] (though just on the symplectic level, without performing geometrical quantization). This is done in the final section.

Aknowledgements

I am grateful to Anton Alekseev, David Li-Bland, Štefan Sakáloš and András Szenes for useful discussion and suggestions. I am also grateful to Albert Schwarz and Alan Weinstein for asking me (independently) about extending the Morita equivalence of non-commutative tori to the case of Poisson-Lie Tduality.

2 Colored surfaces and moduli spaces

Let G be a connected Lie group and ⟨, ⟩ an Ad-invariant inner product (of any signature) on its Lie algebra 𝔤. We shall consider compact oriented surfaces Σ with corners (i.e. locally looking like (ℝ_{≥0})^{2}). We shall assume that none of the components of Σ is closed and that on each component of ∂Σ there is at least one corner.^{1} The boundary of Σ is thus split by the corners into a finite number of arcs. For each arc a we choose a Lie subalgebra 𝔥_{a} ⊂ 𝔤 which is Lagrangian w.r.t. ⟨, ⟩ (i.e. 𝔥_{a}^{⊥} = 𝔥_{a}). Let H_{a} ⊂ G be the corresponding connected Lie subgroup. We demand for every corner x of Σ that if a and b are the arcs meeting at x then 𝔥_{a} ∩ 𝔥_{b} = 0. We shall call such a Σ (together with the choice of subalgebras) a colored surface.

For every colored surface Σ we define a symplectic manifold M_{Σ}. Let us give three equivalent definitions of the manifold M_{Σ}; its symplectic form is defined in Section 3.

2.1 M_{Σ} via cuts of Σ

Let us first describe M_{Σ} in a way which depends on a choice of certain cuts of Σ. We keep cutting Σ along paths connecting corners until we get a polygon. For every side s of the polygon we choose an element g_{s} ∈ G such that:

1. if s is an arc of the boundary of Σ then g_{s} ∈ H_{s}

2. if s and s′ are the two sides which are the result of a cut then g_{s′} = g_{s}^{−1}

3. the product of all g_{s}’s along the boundary of the polygon (in their natural cyclic order) is equal to 1.

An assignment s ↦ g_{s} satisfying these properties is, by definition, a point in M_{Σ}.

r_{1} r_{2} b_{1} b_{2} vgg^{−1} b_{1} g b_{2} v r_{2} g^{−1}r_{1} = 1

1We impose these assumptions only for simplicity reasons, as they imply that the moduli spaces defined below are non-singular.

Documenta Mathematica 17 (2012) 607–625

M_{Σ} can thus be described as the preimage of 1 under a map

∏_{a} H_{a} × G^{#cuts} → G.  (1)

The map is a submersion and thus M_{Σ} is a manifold.

2.2 M_{Σ} as a space of groupoid morphisms

Let us now describe M_{Σ} without using cuts. Let X ⊂ Σ be the set of corners of Σ and let Π_{1}(Σ, X) be the fundamental groupoid of Σ (the set of objects of Π_{1}(Σ, X) is X and morphisms are homotopy classes of paths between corners). Every arc of the boundary can be seen as a morphism in Π_{1}(Σ, X). Then

M_{Σ} = {F : Π_{1}(Σ, X) → G; F (a) ∈ H_{a} for every arc a}.

2.3 M_{Σ} as a moduli space of flat connections

Finally, let us describe M_{Σ} as a moduli space of flat connections. Let π : P → Σ be a principal G-bundle. For every arc a we choose a reduction of P |_{a} to H_{a} ⊂ G, i.e. a submanifold P_{a} ⊂ π^{−1}(a) which is a principal H_{a}-bundle over a. For every corner x ∈ Σ we choose a point p_{x} ∈ P_{a} ∩ P_{b} where a and b are the arcs meeting at x. Let us call π : P → Σ with its additional structure a colored G-bundle over Σ. Let us consider connections on P which restrict to connections (i.e. to 𝔥_{a}-valued 1-forms) on every P_{a}; we shall call such a connection a colored connection. M_{Σ} can then be described the moduli space of colored flat connections on colored G-bundles over Σ. The groupoid morphism Π_{1}(Σ, X) → G corresponding to a colored flat connection is given by parallel transport (the fiber of P over any corner x is trivialized by the choice of the point p_{x}). In the opposite direction, if F : Π_{1}(Σ, X) → G satisfies F (a) ∈ H_{a} for every arc a then the corresponding flat colored G-bundle is construced as follows. Let p : Σ̂ → Σ be a universal cover of Σ with a chosen corner y_{0} ∈ Σ̂ and let x_{0} = p(y_{0}). Let P̃ = Σ̂ × G → Σ̂ be the trivial G-bundle, with the trivial flat connection. By restriction of F we have a homomorphism π_{1}(Σ, x_{0}) → G and we define the flat G-bundle P → Σ as P = P̃/G. The reduction of P̃ over a corner y ∈ Σ̂ is (y, F ([p ∘ γ_{y_{0}y}])), where γ_{y_{0}y} is a path (unique up to homotopy) from y_{0} to y. These reductions then extend uniquely to a coloring of P̃, and the coloring descends to a coloring of P. It is clear that every flat colored G-bundle P → Σ arises in this way, as the flat connection on p^{∗}P can be used to trivialize it. Notice that M_{Σ} is the disjoint union over the isomorphism classes of colored G-bundles of the moduli spaces with fixed colored G-bundle class. If P is the trivial G-bundle P = Σ × G and its coloring is also trivial (i.e. P_{a} = a × H_{a}, p_{x} = (x, 1)) then a colored connection can be described as a 1form A ∈ Ω^{1}(Σ) ⊗ 𝔤 such that the restriction of A to any arc a is in Ω^{1}(a) ⊗ 𝔥_{a}.

The space of these flat connections modulo the gauge transformations (by maps g : Σ → G such that g(x) = 1 for every corner x and g(a) ⊂ H_{a} for every arc a) is a connected component of M_{Σ}.

3 Symplectic form in terms of holonomies

3.1 Symplectic form on moduli spaces of flat connections

Let P → Σ be a colored G-bundle. Colored connections on P form an affine space A_{col}(P) modeled on Ω_{col}^{1}(Σ, Ad_{P}), where Ω_{col}(Σ, Ad_{P}) ⊂ Ω(Σ, Ad_{P}) is the space of forms that restrict to Ω(a, Ad_{P_{a}}) on every arc a ⊂ ∂Σ. If A is a flat colored connection on P then the covariant differential d_{A} makes Ω_{col}(Σ, Ad_{P}) to a complex and we have a natural isomorphism

T_{[P,A]}M_{Σ} ≅ H^{1}(Ω_{col}(Σ, Ad_{P}), d_{A}),

where [P, A] ∈ M_{Σ} denotes the isomorphism class of (P, A). The antisymmetric pairing

ω([α], [β]) = ∫_{Σ}⟨α ∧ β⟩

on T_{[P,A]}M_{Σ} (α, β ∈ Ω_{col}^{1}(Σ, Ad_{P})) is non-degenerate by Poincaré–Verdier duality. The moduli space M_{Σ} becomes in this way a symplectic manifold. To see that ω is smooth and closed (we already checked that it is non-degenerate), let us choose an open subset U ⊂ M_{Σ} which admits a smooth family of colored flat connections ϕ : U → A_{col}(P), ϕ : x ↦ A_{x}, such that [P, A_{x}] = x. Then

ω = ϕ^{∗}ω_{A}  (2)

where ω_{A} ∈ Ω^{2}(A_{col}(P)) is the constant (hence closed) 2-form

ω_{A}(α, β) = ∫_{Σ}⟨α ∧ β⟩

on the affine space A_{col}(P); ω is therefore closed. The symplectic form ω is a straightforward generalization of the symplectic form of Atiyah-Bott [2] and Goldman [6] who considered closed surfaces.

Remark 3.1. The symplectic manifold (M_{Σ}, ω) is best described as the symplectic reduction of (A_{col}(P), ω_{A}) by the group of the automorphisms of P preserving the coloring. Making this statement precise is, however, rather technical. Here I present the formal part of the story, ignoring the problems with infinite-dimensional manifolds: To simplify notations, let us discuss the case when the colored G-bundle P → Σ is trivial. Let us recall how symplectic forms appear on moduli spaces of flat connections.

The group of smooth maps Σ → G acts affinely on Ω^{1}(Σ) ⊗ 𝔤 by gauge transformations (g : Σ → G, α ∈ Ω^{1}(Σ) ⊗ 𝔤)

g · α = g^{−1}dg + g^{−1}αg

and this action preserves ω_{A}. The infinitesimal action of a t : Σ → 𝔤 is generated by the Hamiltonian

H_{t}(α) = ∫_{Σ}⟨t, F ⟩ + ∫_{∂Σ}⟨t, α⟩  (3)

where F = dα + α^{2} is the curvature of the 𝔤-connection α; the Poisson bracket of two such Hamiltonians is

{H_{t_{1}}, H_{t_{2}}} = H_{[t_{1},t_{2}]} + c(t_{1}, t_{2})

where

c(t_{1}, t_{2}) = ∫_{∂Σ}⟨t_{1}, dt_{2}⟩.  (4)

Notice that the cocycle (4) vanishes on the Lie algebra

{t : Σ → 𝔤; t(a) ⊂ 𝔥_{a} for every arc a}

of infinitesimal gauge transformations preserving the coloring. The moment map (3) is 0 at α ∈ Ω^{1}(Σ) ⊗ 𝔤 iff

α is flat and α|_{a} ∈ Ω^{1}(a) ⊗ 𝔥_{a} for every arc a.

The symplectic reduction is thus the part of the moduli space M_{Σ} coming from the trivial colored G-bundle.

3.2 Central extension by closed 2-forms

Let M be a manifold and let Ω_{cl}^{2}(M) denote the space of closed 2-forms on M. Let us recall that the Lie algebra 𝔤(M) of smooth maps M → 𝔤 has a central extension 𝔤̃(M) by Ω_{cl}^{2}(M): as a vector space,

𝔤̃(M) = 𝔤(M) ⊕ Ω_{cl}^{2}(M),

and the bracket is

[(t_{1}, ω_{1}), (t_{2}, ω_{2})] = ([t_{1}, t_{2}], ⟨dt_{1} ∧ dt_{2}⟩).  (5)

The corresponding group G̃(M), a central extension of G(M) (the group of smooth maps M → G) by Ω_{cl}^{2}(M), can be described as follows: its elements are pairs

(g, ω), g : M → G, ω ∈ Ω^{2}(M), dω = \frac{1}{2}g^{∗}η  (6)

where the invariant 3-form η on G is given by

η(u, v, w) = ⟨[u, v], w⟩.

The product in the group is

(g_{1}, ω_{1})(g_{2}, ω_{2}) = (g_{1}g_{2}, ω_{1} + ω_{2} + \frac{1}{2}⟨g_{1}^{−1}dg_{1} ∧ dg_{2} g_{2}^{−1}⟩)  (7)

and the inverse

(g, ω)^{−1} = (g^{−1}, −ω).  (8)

Finally, let us also introduce an auxiliary group G̃_{big}(M) ⊃ G̃(M):

G̃_{big}(M) = G(M) × Ω^{2}(M),  (9)

with the product and inverse given by the same formulas (7), (8). The map

G̃_{big}(M) → Ω_{cl}^{3}(M), (g, ω) ↦ dω − \frac{1}{2}g^{∗}η

is a group morphism and G̃(M) is its kernel.

3.3 Symplectic form in terms of holonomies

Let us cut Σ until we get a polygon (as in Section 2). For each side s of the polygon we have a map γ_{s} : M_{Σ} → G (the holonomy along the side).

Theorem 3.1. The symplectic form ω on M_{Σ} is given by

(1, ω) = ∏_{s}(γ_{s}, 0)

where the product is taken in the group G̃_{big}(M_{Σ}) (see Equation (7)) and the sides of the polygon are taken in their natural (cyclic) order.

The idea of the proof is that ω is the integral of the curvature of a 𝔤̃(M_{Σ})-valued connection on Σ, and hence can be expressed in terms of the holonomies g_{s}’s. The proof is in Section 3.6. The formula for ω is a generalization of a similar formula of Alekseev-Malkin-Meinrenken [1] for the case of closed surfaces.

3.4 Integral of curvature

Let

C → K̃ → K

be a central extension of Lie groups. Let P̃ → D be a principal K̃-bundle over a disk D and let P → D be the corresponding K-bundle, P = P̃/C. Suppose that A is a flat connection on P and Ã is a (non-flat) connection on P̃ lifting A. The curvature F̃ of Ã is a 𝔠-valued 2-form on D and its integral is

C ∋ exp ∫_{D} F̃ = hol_{∂D} Ã.  (10)

The proof of this simple claim is obvious: trivialize P̃ → D (and hence P → D) in such a way that the connection A on P = D × K becomes trivial. Such a trivialization can be achieved e.g. by by the parallel transport of Ã along straight lines starting at the center of the disc D. Formula (10) then becomes Stokes theorem. We shall use (10) for the central extension

Ω_{cl}^{2}(M) → G̃(M) → G(M).

The disk will be the result of cutting Σ and M will run over certain open subsets of M_{Σ}.

3.5 Symplectic form as integral of curvature

Let Σ be a colored surface. Let U ⊂ M_{Σ} be an open subset, P → Σ a colored G-bundle and A_{x} a smooth family of colored flat connections on P parametrized by x ∈ U, such that the class of (P, A_{x}) is x. M_{Σ} can be covered by such open subsets U. Using the inclusion G → G̃(U), g ↦ (g, 0), we lift P to a principal G̃(U)-bundle P̃_{U} → Σ. Similarly, the inclusion G → G(U) lifts P to a principal G(U)-bundle P_{U} → Σ and P_{U} = P̃_{U}/Ω_{closed}^{2}(U). The family A_{x} can be seen as a flat connection A on P_{U}, and Ã = (A, 0) as a (non-flat) connection on P̃_{U}. The curvature F̃ of Ã is a Ω_{cl}^{2}(U)-valued 2form on Σ, and the integral of F̃ is (using (2) and (5)) the symplectic form on U ⊂ M_{Σ}:

ω = ∫_{Σ} F̃.  (11)

Remark 3.2. To speak properly about principal G(U)- and G̃(U) bundles, we should understand in what sense G(U) and G̃(U) are Lie groups. However, we don’t need to do it. A principal G(U)-bundle ober Σ is given by an open cover {V_{α}} of Σ and by a cocycle V_{α} ∩ V_{β} → G(U), i.e. by a cocycle of smooth maps (V_{α} ∩ V_{β}) × U → G. In our case the maps are constant on U. A connection on such a bundle is given by 1-forms A_{α} ∈ Ω^{1}(V_{α}, 𝔤(U)), compatible on the overlaps V_{α} ∩ V_{β}. A 1-form A_{α} ∈ Ω^{1}(V_{α}, 𝔤(U)) is by definition a family of 1-forms in Ω^{1}(V_{α}, 𝔤), smoothly parametrized by U. That is how the family A_{x} is seen as a flat connection A on P_{U}. Notice also that since P can be trivialized when we cut Σ to a dics, P (U) and P̃(U) are also trivialized. Let us also remark that we are not constructing G(M_{Σ})- and G̃(M_{Σ})-bundles over Σ. Formula (11) gives the symplectic form only on our open subsets U ⊂ M_{Σ}, but these open subsets cover M_{Σ}.

3.6 Symplectic form in terms of holonomies (proof)

Proof of Theorem 3.1. It follows immediately from (11) and (10). We cut Σ to a polygon. If s ⊂ ∂Σ then (γ_{s}, 0) is the holonomy of Ã along s (since 𝔥_{s} is isotropic and thus the cocycle in (5) vanishes). If s comes from a cut then the holonomy of Ã along s is (γ_{s}, β) for some 2-form β. However, the holonomy along the other side coming from the same cut is its inverse; we can thus replace β with 0 and the product of holonomies will not change. This proves Theorem 3.1 for open U ⊂ M_{Σ} satisfying the condition of Section 3.5, and since they cover M_{Σ}, it proves it for entire M_{Σ}

3.7 Examples

Example 3.1. Let Σ be a square colored by a Manin triple 𝔯, 𝔟 ⊂ 𝔤:

\begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix}

Let us denote the holonomies as on the picture, i.e.

M_{Σ} = {(r_{1}, r_{2}, b_{1}, b_{2}) ∈ R × R × B × B; r_{1}b_{1} = b_{2}r_{2}} :

\begin{matrix}r_{1} \\ r_{2} \\ b_{2} \\ b_{1} \end{matrix}

In G̃(M_{Σ}) we have (r_{1}, 0)(b_{1}, 0) = (r_{1}b_{1}, ⟨r_{1}^{−1}dr_{1} ∧ db_{1}b_{1}^{−1}⟩/2) and (b_{2}, 0)(r_{2}, 0) = (b_{2}r_{2}, ⟨b_{2}^{−1}db_{2} ∧ dr_{2}r_{2}^{−1}⟩/2). The symplectic form on M_{Σ} is thus

ω = \frac{1}{2}⟨r_{1}^{−1}dr_{1} ∧ db_{1}b_{1}^{−1}⟩ − \frac{1}{2}⟨b_{2}^{−1}db_{2} ∧ dr_{2}r_{2}^{−1}⟩.

This symplectic manifold (M_{Σ}, ω) is the Lu-Weinstein double symplectic groupoid [9] corresponding to the triple R, B ⊂ G. This fact was already noticed by the author in [12]. A similar interpretation of the Lu-Weinstein double groupoid was found by P. Boalch in [3] using irregular connections. In one of the groupoid structures, the space of objects is B and a point in M_{Σ} is an arrow from b_{2} to b_{1}; in the other groupoid structure, the roles of R and B are exchanged. The groupoid products are given by concatenation of squares (either horizontal or vertical); they will be explained more properly in the following section.

Example 3.2. Let now Σ be a square colored as follows:

𝔯 ; 𝔳 ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; r ; v ; b_{2} ; b_{1}

The symplectic form is

ω = \frac{1}{2}⟨r^{−1}dr ∧ db_{1}b_{1}^{−1}⟩ − \frac{1}{2}⟨b_{2}^{−1}db_{2} ∧ dv v^{−1}⟩.

The symplectic manifold (M_{Σ}, ω) is again a well-known object: it is the symplectic groupoid integrating the homogeneous Poisson space given by R, B, V ⊂ G via Drinfeld’s classification [4]. This symplectic groupoid was discovered by Jiang-Hua Lu [8].

Example 3.3. Now let Σ be a triangle.

\begin{matrix}𝔯 \\ 𝔳 \\ 𝔟 \\ r \\ v \\ b \end{matrix}

In this case

M_{Σ} = {(r, b, v) ∈ R × B × V; rbv = 1}.

The symplectic form is

ω = \frac{1}{2}⟨v^{−1}dv ∧ db b^{−1}⟩.

This symplectic manifold is, up to covering, the big symplectic leaf in the homogeneous Poisson space given by R, B, V ⊂ G. It will play a role when we discuss Morita equivalence.

Example 3.4. Finally, let us discuss the simplest Σ that requires a cut.

\begin{matrix}𝔯 \\ 𝔟 \\ 𝔯 \\ 𝔟 \\ r_{1} \\ b_{1} \\ r_{2} \\ b_{2} \\ g \end{matrix}

M_{Σ} = {(r_{1}, r_{2}, b_{1}, b_{2}, g) ∈ R^{2} × B^{2} × G; r_{1}b_{1}g = g r_{2}b_{2}}
(1, ω) = (r_{1}, 0)(b_{1}, 0)(g, 0) ((g, 0)(r_{2}, 0)(b_{2}, 0))^{−1}

The symplectic manifold (M_{Σ}, ω) is the double symplectic groupoid integrating the Drinfeld double given by the triple R, B ⊂ G, i.e. the Lu-Weinstein groupoid of the triple R × B, G_{diag} ⊂ G × G.

4 Painted bodies and Lagrangian submanifolds

In this section we shall discuss how cobordisms of painted surfaces give rise to Lagrangian submanifolds in the moduli spaces. These Lagrangian submanifolds will turn the moduli spaces into interesting algebraic objects, such as (double) groupoids, modules, etc. The Lagrangian submanifold will consist of those flat connection on the surface that can be extended to flat connections on the 3dim manifold (cobordism). This construction is a straightforward generalization of the symplectic Chern-Simons theory of D. Freed [5], who considered closed surfaces.

4.1 Painted bodies

Let us consider a compact oriented 3dim manifold with corners (i.e. locally looking as (ℝ_{≥0})^{3}). Its boundary is divided to vertices (corners), edges and faces. For some of the faces we choose a Lagrangian Lie subalgebra of 𝔤 (we shall call such a face painted). We shall require the following. Whenever two faces meet along an edge then at least one of them is painted, and if both are painted, then the two subalgebras are transverse. At each vertex should meet two painted and one unpainted face. Finally, the unpainted part of the boundary should be a colored surface, i.e. each of its components should have boundary and on each of the boundary circles there should be a corner. We shall call such a manifold a painted body. The unpainted part of the boundary of a body X will be denoted Σ_{X}.

4.2 Flat connections on a painted body

Let X be a painted body. We shall consider principal G-bundles P → X with a reduction to H_{a} over every painted face a and with a section over every edge between painted faces. We then consider flat connections compatible with the reductions. Let L_{X} ⊂ M_{Σ_{X}} denote the set of equivalence classes of flat colored connections on Σ_{X} that are extensible to X. We shall call L_{X} smooth if it is a submanifold and moreover it can be locally lifted to a smooth family of flat connections on X.

Theorem 4.1. If L_{X} ⊂ M_{Σ_{X}} is smooth, it is a Lagrangian submanifold.

Proof. We shall prove that the formal tangent spaces to L_{X} are Lagrangian in the tangent spaces of M_{Σ_{X}}. If L_{X} is smooth then these formal tangent spaces are the actual tangent spaces. Let P be a painted G-bundle over X, Ad_{P} → X the vector bundle associated to the adjoint representation of G on 𝔤, and let

Ω_{col}(X, Ad_{P}) ⊂ Ω(X, Ad_{P})

be the space of Ad_{P} -valued differential forms that take values in the corresponding subalgebra of 𝔤 when restricted to a painted face of X. Let A be a flat colored connection; then d_{A} makes Ω_{col}(X, Ad_{P}) into a complex. Let us denote this complex Ω_{A}(X) and its cohomology H_{A}(X). Let P′ = P |_{Σ_{X}} and A′ = A|_{Σ_{X}}. Let us consider the short exact sequence

0 → Ω_{A,0}(X) → Ω_{A}(X) → Ω_{A′}(Σ_{X}) → 0

(where Ω_{A,0}(X) are the forms vanishing at Σ_{X}) and the following piece of the resulting long exact sequence:

H_{A}^{1}(X) → H_{A′}^{1}(Σ_{X}) → H_{A,0}^{2}(X).  (12)

We have

H_{A′}^{1}(Σ_{X}) = T_{[P′,A′]}M_{Σ_{X}} ,

and the image of the first arrow is the formal T_{[P′,A′]}L_{X}. By Poincaré duality the dual of (12) is obtained just by reversing the arrows:

H_{A,0}^{2}(X) ← H_{A′}^{1}(Σ_{X}) ← H_{A}^{1}(X)

(in particular, the identification of H_{A′}^{1}(Σ_{X}) with its dual is via the symplectic form). As a consequence, the image of the first arrow in (12) is a Lagrangian subspace.

In all the examples that we consider below, L_{X} is easily seen to be smooth.

4.3 Examples

As we noticed above, Lu-Weinstein’s double symplectic groupoid corresponding to a Manin triple B, R ⊂ G is the moduli space for the surface

\begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix}

The graph of one of the products in this double groupoid is L_{X} where X is

In other words, the product is given by gluing squares along the adjacent sides on the picture

\begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; \begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; 𝔟

The other product is obtained when we exchange the colors. The moduli space of

𝔯 ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; 𝔳

is a symplectic groupoid via the gluing

𝔯 ; 𝔳 ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; 𝔯 ; 𝔳 ; 𝔟

It is a symplectic groupoid integrating the Poisson B-homogeneous space corresponding to the quadruple R, B, V ⊂ G. It is also a module of

\begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix}

via the gluing

𝔯 ; 𝔳 ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; 𝔯 ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix}

Similar pictures can be drawn for the double symplectic groupoid integrating the Drinfeld double and also for its R-matrix in the sense of Weinstein and Xu; see [12] for details.

5 Morita equivalence of quantum tori and beyond

This last section is a bit speculative. On the other hand, it describes the motivation for the constructions described above, so it is included anyway.

5.1 Morita equivalence of quantum tori

Recall that two algebras A and B are said to be Morita equivalent if their categories of modules are linearly equivalent. Equivalently, there exist a A ⊗ B^{op}-module M and a B⊗A^{op} module N such that M ⊗_{B} N ≅ A and N ⊗_{A}M ≅ B. Let θ_{ij} = −θ_{ji}, 1 ≤ i, j ≤ n, be a skew-symmetric matrix with real elements. We suppose that the graph of the corresponding linear map ℝ^{n} → ℝ^{n} intersects ℤ^{2n} ⊂ ℝ^{2n} only in 0 ∈ ℤ^{2n}. To the matrix θ we associate the algebra 𝕋_{θ}^{n} (a quantum torus) generated by elements u_{i} (1 ≤ i ≤ n) and their inverses, modulo relations u_{i}u_{j} = exp(2π\sqrt{−1}θ_{ij})u_{j}u_{i}. A famous result of Rieffel and Schwarz [10] says that the algebra 𝕋_{θ}^{n} is Morita equivalent to 𝕋_{θ^{−1}}^{n}.^{2} The quantum torus 𝕋_{θ}^{n} can be seen as a quantization of the n-dimensional torus 𝕋^{n} with the constant Poisson structure given by θ. The following natural questions are due to A. Schwarz and A. Weinstein (motivated by an extension of T -duality [11] to Poisson-Lie T -duality [7]).

1. Is there a generalization of Morita equivalence when 𝕋^{n} is replaced by a quantum group H and 𝕋_{θ}^{n} by a torsor of H?

2. Is there a symplectic/Poisson version of Morita equivalence for tori with constant Poisson structure? Can it be extended to Poisson-Lie groups, giving a symplectic/Poisson analog of Question 1?

We shall give an answer to Question 2. It will provide a conjectural answer to Question 1.

5.2 H-Morita equivalence

Let H be a Hopf algebra. Let A be an associative algebra in the (monoidal) category H-mod of left H-modules. In other words, A is an H-module and the product A ⊗ A → A is a morphism of H-modules. Let A-mod_{H} be the category of A-modules in H-mod, i.e. the category of vector spaces V which are modules of both A and H, such that A ⊗ V → V is a morphism of H-modules. Let A-mod be the category of A-modules in the category of vector spaces. We have the forgetful functor res : A-mod_{H} → A-mod and its left adjoint ind : A-mod → A-mod_{H}. Let now B be an algebra in the (monoidal) category H-comod of right Hcomodules. We have the category B-mod^{H} of B-modules in H-comod and the category B-mod. Now we have the forgetful functor cores : B-mod^{H} → B-mod and its right adjoint coind : B-mod → B-mod^{H}.

Definition 5.1. We shall say that A and B are H-Morita equivalent if there are equivalences of linear categories A-mod → B-mod^{H} and A-mod_{H} → B-mod such that the diagram

2it implies a Morita equivalence of 𝕋_{θ}^{n} with 𝕋_{A·θ}^{n} for any A ∈ SO(n, n; ℤ), where SO(n, n; ℤ) acts on the graph of θ in ℝ^{2n}, provided the transformed graph is again a graph

Documenta Mathematica 17 (2012) 607–625

A-mod  B-mod^{H}

A-mod_{H}  B-mod

ind cores

commutes up to a natural isomorphism, or equivalently, such that

A-mod  B-mod^{H}

A-mod_{H}  B-mod

res coind

commutes up to a natural isomorphism.

The simplest example is when A = k is trivial (k is the base field) and B = H. The category H-mod^{H} is called the category of Hopf modules of H. A linear equivalence F : k-mod → H-mod^{H} making the diagram

k-mod  H-mod^{H}

H-mod  H-mod

F ind = cores

commutative (up to a natural isomorphism) is due to Sweedler [14]; F is given simply by F (V) = H ⊗ V.

Proposition 5.1. An H-Morita equivalence is equivalent to a vector space M which is a right A-module and left B-module, satisfying the compatibility relation b · (m · a) = (b_{(1)} · m) · (b_{(2)} · a) for all a ⊗ m ⊗ b ∈ A ⊗ M ⊗ B, where b ↦ b_{(1)} ⊗ b_{(2)} ∈ B ⊗ H is the H-comodule structure of B.

Proof. If M is given then M ⊗H is a B-A-bimodule via b·(m⊗h) = (b_{(1)} ·m)⊗ (b_{(2)} · h), (m ⊗ h) · a = (m · (h_{(2)} · a)) ⊗ h_{(1)}. It induces an H-Morita equivalence as follows. The functor A-mod → B-mod^{H} is given by V ↦ V ⊗_{A}(M ⊗H), and A-mod_{H} → B-mod by W ↦ (W ⊗_{A} (M ⊗ H))^{H}, where X^{H} (for a H-module X) is {x ∈ X; (∀h ∈ H) h · x = ϵ(h)x}. If an H-Morita equivalence is given then M = F (A)^{H}, where F is the functor A-mod → B-mod^{H}.

5.3 Conjectural answer to Question 1

Suppose again that G is a connected Lie group and its Lie algebra 𝔤 has an invariant inner product ⟨, ⟩. Let 𝔯, 𝔟 ⊂ 𝔤 be a Manin triple and let R and B be the corresponding Poisson-Lie groups. Suppose also that 𝔳 is another Lagrangian subalgebra with the property that 𝔳 ∩ 𝔯 is the Lie algebra of a closed connected subgroup R_{𝔳} ⊂ R and similarly, 𝔳 ∩ 𝔟 is the Lie algebra of a closed connected subgroup B_{𝔳} ⊂ B. By Drinfeld’s classification of Poisson homogeneous spaces [4] the homogeneous space R/R_{𝔳} has a Poisson structure such that the map R × (R/R_{𝔳}) → R/R_{𝔳} is Poisson, and similarly for B/B_{𝔳}. Below we shall prove a symplectic version of the following loosely stated conjecture: If Hopf algebra H is a (suitable) quantization of the Lie bialgebra corresponding to the Manin triple 𝔯, 𝔟 ⊂ 𝔤 and algebras A and B are (suitable) quantizations of the Poisson manifolds R_{𝔳} and B_{𝔳} respectively, then A and B are H-Morita equivalent. In particular, if 𝔳 = 𝔯 we get A = k and B = H, i.e. Sweedler’s example of H-Morita equivalence. Let us notice that we prove the symplectic version of the conjecture only under the additional assumption that 𝔳 is transverse to both 𝔯 and 𝔟. The general case would require colored surfaces where we allow adjacent subalgebras to have non-trivial intersection. The corresponding colored G-bundles would have a reduction over the corresponding corner to the group exponentiating the intersection. We shall treat this generalization elsewhere.

5.4 Symplectic H-Morita equivalence

In this final section we provide a symplectic analogue of our conjectural answer to Question 1. Vector spaces are replaced by symplectic manifolds of the form M_{Σ} as follows:

𝔯 ; 𝔳 ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; A ⇝ ; 𝔟 ; 𝔳 ; \begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; B ⇝ ; 𝔟 ; 𝔟 ; \begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; H ⇝ ; 𝔯 ; 𝔳 ; 𝔟 ; M ⇝

In other words, A is replaced by the symplectic groupoid integrating the Poisson homogeneous space R_{𝔳}, B by the symplectic groupoid integrating B_{𝔳}, H by the double symplectic groupoid integrating both R and B, and M by the moduli space of the displayed triangle.

Let us recall the definition of Morita equivalence of symplectic groupoids [15]. Let I denote the groupoid with two objects 0 and 1 and with a unique morphism 0 → 1. Let Γ be a Lie groupoid with a groupoid morphism Γ → I. Γ splits naturally to 4 components: Γ_{ij} (i, j ∈ {0, 1}) is the space of arrows lying over the unique morphism i → j. Let X_{i} denote the space of objects of Γ lying over i ∈ {0, 1}.

Definition 5.2. If the maps Γ_{01} → X_{1} and Γ_{10} → X_{0} are surjective then the groupoids Γ_{00} ⇉ X_{0} and Γ_{11} ⇉ X_{1} are said to be Morita equivalent via the bimodules Γ_{01} and Γ_{10}. If the groupoid Γ is symplectic then we have a Morita equivalence of symplectic groupoids.

Let 𝔯, 𝔟, 𝔳 ⊂ 𝔤 be as above and let R, B, V ⊂ G be the corresponding groups. Let X_{0} = R × B and let the arrows (r_{1}, b_{1}) → (r_{2}, b_{2}) in Γ_{00} be (v, r) ∈ V × R such that r_{1}b_{1}v = b r_{2}b_{2}; composition of arrows is by (v, r)(v′, r′) = (vv′, rr′). The groupoid Γ_{00} can be seen as M_{Σ} for the surface

𝔟 ; 𝔳 ; \begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; \begin{matrix}𝔟 \\ 𝔟 \end{matrix} ; Γ_{00} ∼

which makes it to a symplectic groupoid. The groupoid composition is by (horizontal) gluing of rectangles. The symplectic groupoid Γ_{00} integrates the following Poisson structure on R × B. We have the Poisson action B × B_{𝔳} → B_{𝔳}. The forgetful functor

Poisson manifolds with a moment map to R → Poisson 𝔟-manifolds

has a right adjoint F (see [13]), and F (B_{𝔳}) = R×B is our Poisson manifold. It is a semi-classical analog of the crossed product A⋊H, where A is a quantization of B_{𝔳} (an associative algebra) and H a quantization of the Lie bialgebra 𝔟 (a Hopf algebra). Notice that A ⋊ H-mod is equivalent to A-mod_{H}. Let us suppose for simplicity that the map R × B → G, (r, b) ↦ rb, is a diffeomorphism. The symplectic groupoid Γ_{00} is Morita equivalent to the symplectic groupoid Γ_{11} integrating the Poisson manifold R_{𝔳}, i.e. M_{Σ} for the surface

𝔟 ; 𝔳 ; \begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; Γ_{11} ∼

The bimodule Γ_{01} is M_{Σ} for the surface

𝔟 ; 𝔳 ; \begin{matrix}𝔯 \\ 𝔯 \end{matrix} ; 𝔟 ; Γ_{01} ∼

with the bimodule structure given by gluing along the vertical sides. This Morita equivalence of symplectic groupoids is analogous to a Morita equivalence A-mod_{H} → B-mod. If we exchange 𝔯 and 𝔟 then we get a similar Morita equivalence, analogous to A-mod → B-mod^{H}, and these two Morita equivalences of symplectic groupoids are easily seen to form a commutative square analogous to H-Morita equivalence of A and B. Finally, the moduli space for the triangle

\begin{matrix}𝔯 \\ 𝔳 \\ 𝔟 \end{matrix}

is analogous to the vector space M; the A- and B-module structure of M is given by gluing squares to the triangle along the edge of the corresponding color. In the case when R, B = 𝕋^{n}, G = R × B, all these symplectic manifolds are of the form ℝ^{2m}/ℤ^{k}, with constant symplectic structure. They can be therefore easily geometrically quantized (provided the symplectic form is integral on ℤ^{k}) and these quantizations are compatible with the groupoid/module structures, so we get an H-Morita equivalence. If we choose the Planck constant so that the geometrical quantization H of the double symplectic groupoid R × B is trivial, we get the standard proof of Morita equivalence of quantum tori [10]: In the simplest case of 2-dimensional tori, A is given by the relation uv = e^{2πiθ}vu, B by u′v′ = e^{2πi/θ}v′u′, and the bimodule is M = L^{2}(ℝ) given by (u · f)(x) = f(x+1), (v ·f)(x) = e^{2πiθx}f(x), (u′ ·f)(x) = f(x+1/θ), (v′ ·f)(x) = e^{2πix}f(x).

References

[1] A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps. J. Differential Geom. 48 (1998), no. 3, 445–495.

[2] M. F. Atiyah, R. Bott: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. London 308 (1982), 523–615.

[3] P. Boalch, Quasi-Hamiltonian geometry of meromorphic connections. Duke Math. J. 139, No. 2 (2007), 369–405 .

[4] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups. Teoret. Mat. Fiz. 95 (1993), no. 2, 226–227.

[5] D.S. Freed, Classical Chern-Simons theory I, Adv. Math. 113 (1995) 237.

[6] W. M. Goldman, The symplectic nature of fundamental groups of surfaces. Adv. in Math. 54 (1984), no. 2, 200–225.

[7] C. Klimčík, P. Ševera, Dual non-Abelian duality and the Drinfeld double, Phys. Lett. B351 (1995) 455.

[8] J.-H. Lu, Multiplicative and Affine Poisson Structures on Lie groups, PhD thesis, UC Berkeley, 1990.

[9] J.-H. Lu, A. Weinstein, Groupoïdes symplectiques doubles des groupes de Lie-Poisson, C. R. Acad. Sci. Paris, Sér. I Math. 309 (1989), No. 18.

[10] M. Rieffel, A. Schwarz, Morita equivalence of multidimensional noncommutative tori. Internat. J. Math. 10 (1999), no. 2, 289–299.

[11] A. Schwarz, Morita equivalence and duality. Nuclear Phys. B 534 (1998), no. 3, 720738.

[12] P. Ševera, (Non-)abelian Kramers-Wannier duality and topological field theory. J. High Energy Phys. 2002, no. 5, no. 49, 10 pp.

[13] P. Ševera, Poisson actions up to homotopy and their quantization. Lett. Math. Phys. 77 (2006), no. 2, 199–208.

[14] M. E. Sweedler, Hopf algebras. Mathematics Lecture Note Series, W. A. Benjamin, New York 1969.

[15] P. Xu, Morita equivalent symplectic groupoids. Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), 291–311, Math. Sci. Res. Inst. Publ., 20, Springer, New York, 1991.

Pavol Ševera

Section of Mathematics University of Geneva 2-4 rue du Lièvre 1211 Genève, Switzerland On leave from FMFI UK Bratislava, Slovakia pavol.severa@gmail.com 626

Documenta Mathematica 17 (2012)