INTEGRATING THE NONINTEGRABLE

Alan WEINSTEINDepartment of Mathematics, University of California, Berkeley, USA

Dominique Flament (dir)

Série Documents de travail (Équipe F2DS)

Feuilletages -quantification géométrique : textes des journées d’étude des 16 et 17 octobre 2003,

Paris, Fondation Maison des Sciences de l’Homme, 2004

Integrating the Nonintegrable

Alan Weinstein∗

Department of MathematicsUniversity of California

Berkeley, CA 94720 USA(alanw@math.berkeley.edu)

February 10, 2004

1 Introduction

Foliations occur throughout geometric quantization and related areas ofsymplectic geometry.

Among the symplectic manifolds which represent phase spaces in me-chanics, the simplest are the cotangent bundles of configuration spaces, andthese are quantized in a standard way via wave functions on the configu-ration space. For a general symplectic manifold, one gets something whichplays the role of configuration space by introducing a local structure whichlooks like the decomposition of a cotangent bundle into its fibres (manifoldsof “constant momentum”). This structure is a lagrangian foliation, alsoknown as a (real) polarization; its leaf space is then the natural domain ofquantum “wave functions”. When the foliation has nicely behaved leaves(e.g. when they are the fibres of a fibration), so that the leaf space is amanifold, life is relatively easy, but new methods must be introduced whenthe foliation is more complicated, especially when it has leaves which arenot closed.

Foliations also appear as the characteristic foliations of coisotropic sub-manifolds. These submanifolds may be defined locally by the vanishing ofPoisson commuting “constraint functions”; the leaves of the characteristicfoliation are swept out by the trajectories of the hamiltonian flows of these

∗Research partially supported by NSF Grant DMS-0204100MSC2000 Subject Classification Numbers: xxxxx (Primary), xxxxx (Secondary).Keywords:

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functions. Points on the same characteristic leaf are considered to be phys-ically equivalent, so that the leaf space serves as the space of “true physicaldegrees of freedom”. When the leaf space is a manifold, it has a symplecticstructure, but many constraint manifolds carry more complicated charac-teristic foliations, and one is required once again to introduce new methodsto deal with the leaf space.

Leaf spaces also appear in the solution to a fundamental problem inPoisson geometry going back to Lie [11], that of realizing a given Poissonmanifold P as a quotient of a symplectic manifold. If there are any “nice”(i.e. complete, in a certain sense) realizations, then there is one which isa symplectic groupoid [5]. Cattaneo and Felder [2] construct a universalgroupoid G(P ) by a classical analogue of the Poisson sigma model tech-nique by which they recovered Kontsevich’s [10] deformation quantizationof Poisson manifolds. They start with the symplectic Banach manifold ofC1 paths in T ∗P , restrict attention to a certain coisotropic submanifoldtherein, and then pass to the leaf space of its (finite codimensional) charac-teristic foliation. When this leaf space is not a manifold, P does not haveany nice symplectic realizations. Nevertheless, one would like to work withthe groupoid G(P ). For instance, Xu’s [18] notion of “Morita equivalence”of Poisson manifolds fails to be an equivalence relation because P is Moritaequivalent to itself only when G(P ) is a manifold. In addition, there is amore general notion of “Morita morphism”, but the composition of thesemorphisms is a “geometric tensor product” which is again the leaf space ofa characteristic foliation.

For all the reasons listed above, and many others, it is useful to be ableto treat the leaf space of a foliation as if it were a manifold, and in partic-ular to be able to extend the notions of Lie groupoid, symplectic manifold,and symplectic groupoid to some category including these leaf spaces. Oneapproach to such problems is via noncommutative geometry [3]. In this pa-per, I will describe a different approach, based on the theory of differentiablestacks, especially as it is applied to the problem of integrating Lie algebroidsto Lie groupoids, raised and partially solved by Pradines [14].

The body of the paper begins with an explanation of the path space con-struction (inspired by that of [2]) in the context of Poisson sigma models) bywhich Crainic and Fernandes [4] have precisely identified the obstructions tointegration of Lie algebroids. I will then give a brief exposition of the theoryof stacks and its application to leaf spaces. Finally, I will describe workon stacks as groupoids being done by Chenchang Zhu in collaboration withHsian-Hua Tseng, both currently (winter, 2004) PhD students at Berkeley.I would like to thank them for allowing me to present their work at the

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meeting and in this report. I would also like to thank Henrique Bursztyn,Marius Crainic and Rui Fernandes for their comments on this manuscript.

2 Integrating Lie algebroids by path spaces

2.1 Lie algebroids

We recall (see [1] for an exposition) that a Lie algebroid over a manifold Mis a vector bundle π : A → M equipped with a Lie algebra structure [ , ](over R) on its space of sections Γ(A) and a bundle map ρ : A → TM , calledthe anchor, satisfying the Leibniz rule

[X, fY ](x) = f(x)[X, Y ](x) + ((ρX)f)(x)Y (x)

for all X, Y ∈ Γ(A), f ∈ C∞(M), and x ∈ M . It follows from the Leibnizrule and the Jacobi identity that the map from Γ(A) to vector fields on Minduced by ρ is a Lie algebra homomorphism.

Examples of Lie algebroids include tangent bundles and their integrablesubbundles (with the induced bracket, and inclusion as anchor), gauge Liealgebroids TP/H where P is a principal H bundle over M , action Lie al-gebroids h × M determined by homomorphisms from the Lie algebra h tovector fields on M , Lie algebras (when M is a point), and cotangent bun-dles of Poisson manifolds with a bracket on 1-forms originally introducedby Fuchssteiner [8]. It is useful to think of elements of a Lie algebroid A as“generalized tangent vectors” to M , so that the pair (M,A) becomes a newkind of space which we denote by MA; i.e. we may write T (MA) = A (andMTM is just M).

The image of a Lie algebroid anchor is a family of subspaces in the fibresof TM which is integrable in the sense of Stefan and Sussmann; the integralmanifolds form a singular foliation of M whose leaves may be thought ofas the components of MA. In fact, these leaves may be taken as the actualcomponents of a new differentiable structure on M , but with a topologydifferent from the original one, and with (usually) an uncountable numberof connected components. If we look only at the decomposition of MA into itscomponents, though, we lose an important part of the information containedin A, namely the kernel of ρ. Elements of this kernel may be thought of as“internal” tangent vectors to MA; the set of such internal vectors based ata single point inherits a Lie algebra structure from the bracket on sections.

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2.2 Lie groupoids

Lie algebras are the “linearization” of Lie groups. Since linearization involvesdifferentiation, the reverse notion, passing from a Lie algebra to a Lie group,may be thought of as a kind of integration. In these terms, the geometricobjects which integrate Lie algebroids are Lie groupoids (sometimes calleddifferentiable groupoids).

We recall that a groupoid over a set M is a set G whose elements are themorphisms in a category whose objects are the points of M , where all mor-phisms are invertible. M is sometimes called the base of the groupoid. Moreconcretely, there is a target-source map (t, s) from G to M ×M (sometimescalled the anchor of the groupoid, by analogy with the terminology for Liealgebroids), a units map ε : M → G, an inversion map g 7→ g−1 from G toitself, and a multiplication map (g, h) 7→ gh with values in G, defined on theset of G(2) of pairs (g, h) for which s(g) = t(h). (Think of the elements g ∈ Gas arrows going from right to left with head at s(g) and tail at t(g).) Thealgebraic axioms state that multiplication is associative whenever defined,t(gh) = t(g), s(gh) = s(h), ε(t(g))g = g = gε(s(g)), gg−1 = ε(t(g)), andg−1g = ε(s(g)). G is a Lie groupoid if M and G are manifolds, if (t, s) issmooth and transverse to the diagonal so that G(2) is a manifold, and if theunits, inversion, and multiplication maps are smooth.

Examples of Lie groupoids (corresponding to the examples of Lie al-gebroids above) include M × M with product (x, y)(y, z) = (x, z), thoseequivalence relations on manifolds which are submanifolds transverse tothe diagonal, gauge groupoids P × P/H of principal bundles, and actionLie groupoids H × M associated with actions of the Lie group H on M .The counterparts of cotangent bundles of Poisson manifolds are symplecticgroupoids, about which we will have say more at the end of this paper. Wemay think of the elements of G as “generalized ordered pairs” in M , so that(M,G) becomes a new kind of space which we could denote by MG.

The image of (t, s) in M×M is an equivalence relation whose equivalenceclasses are (not necessarily connected) submanifolds of M which may bethought of as the components of MG. The extra information in G is foundin the isotropy subgroupoid, defined as the inverse under (t, s) of the diagonalin M × M . This “kernel” of (t, s) is a family of groups Gx lying over thepoints of M . Gx is called the isotropy group of x; its elements may bethought of as “internal symmetries” of x.

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2.3 From Lie groupoids to Lie algebroids

The Lie algebroid of a given Lie groupoid is defined in same way as forLie groups, where the Lie algebra consists of the left invariant vector fields,identified with their values at the unit element. On a groupoid, the lefttranslations by elements of G act between fibres of t, so the left invariantvector fields are necessarily sections of the subbundle ker Tt ⊆ TG tangentto these fibres. As on a group, these vector fields are closed under theusual bracket operation and can be identified with their values along theunit section ε(M). Thus, the pulled back bundle ε∗(ker Tt) becomes a Liealgebroid A(G) over M , with anchor the restriction of Ts. 1

2.4 Development of paths

Integration of left-invariant vector fields yields an “exponential map” fromcompactly supported (and certain other nicely behaved) sections of A(G) tothe group B of submanifolds in G which map under (t, s) to the graphs ofdiffeomorphisms. (These submanifolds are known as bisections, or admissi-ble sections.) There is, though, no exponential map from A(G) itself to Gwhich might allow one to recover G from A(G) by a “Campbell-Hausdorff”construction as in the case of groups. However, there is a “development”bijection between certain paths in A(G) and the paths in G which start atunit elements and remain in a single fibre of t; this map makes it possibleto transport algebraic structure from G to A(G). Defining the analogue ofthis transported structure for an arbitrary Lie algebroid A is not hard, andis a key step in integrating A.

The inverse of the development map is easier to describe. (We follow[4], with slight changes in notation.) Given the groupoid G over M , wedefine the space P (G) of G-paths to consist of the C2 (twice continuouslydifferentiable) paths γ : [0, 1] → G for which γ(0) is a unit element and

1This construction illustrates a persistent headache in the theory of groupoids and Liealgebroids. In addition to the usual question of sign conventions, there is also a questionof ”handedness conventions”. For Lie groups, one defines the Lie algebra by left ratherthan right invariant vector fields in order to have the Lie bracket for GL(n) coincide withthe usual commutator bracket of matrices. But when we adopt this convention for aLie groupoid like the pair groupoid M ×M , the typical Lie algebroid element looks likean infinitesimal arrow pointing from its target to its source, rather than the other wayaround. This problem can be corrected if we think of the groupoid arrows as pointing fromleft to right, defining the anchor as (s, t) rather than (t, s), so that the condition for theproduct gh to exist is that t(g) = s(h). But this does not agree with the usual conventionsfor compositions of functions. If only function evaluation were commonly denoted (x)ginstead of g(x) ! (We could pronounce it “x g’d”, as in ”x squared”.)

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t is constant along γ. The lift (or “logarithm”) L(γ) is the path in A(G)which assigns to each τ ∈ [0, 1] the left-translate of γ′(τ) to the unit elements(γ(τ)). The result is an A(G)-path, i.e. a C1 path σ : [0, 1] → A(G)satisfying the compatibility condition ρ ◦ σ = (π ◦ σ)′, where π : A(G) →M is the vector bundle projection. The inverse map (development, a sortof exponentiation) from the space P (A(G)) of A(G)-paths to P (G) willbe denoted by E . To develop an A(G)-path σ into G, we first extend σto a compactly supported time-dependent section of A(G), which we mayconsider as a time-dependent left-invariant vector field on G. E(σ) is thendefined as the integral curve of this vector field whose initial value is theunit at π(σ(0)). (Existence of the integral curve for all time is guaranteedby compact support of the section and left-invariance of the vector field.)Checking that E inverts L also assures us that the result is independent ofthe choice of extension vector field.

When G is a group and σ is a constant path in its Lie algebra A(G),E(σ) is in fact the initial segment of the one-parameter group generated bythe image of σ; i.e. it really is an exponential. But when G is a generalLie groupoid, a constant path in A(G) is an A(G)-path only when its valuebelongs to the kernel of the anchor, so that the exponentiation of individualelements of A(G) only reaches the isotropy subgroupoid of G.

2.5 Product structure on the path space

The space G can be recovered from P (G): we identify two paths if theyhave the same value at τ = 1. To recover the multiplication on G, we needa product structure on paths. When G is a group, there are two such struc-tures. The first is pointwise multiplication, and the second is concatenation,modified to be consistent with the condition that all paths start at the unitelement, and so that the concatenated path is smooth. (The usual homo-topy theory of Lie groups teaches us that these two products are equivalentup to homotopy.) Pointwise multiplication was used by Duistermaat andKolk [6] to recover groups from their Lie algebras, but it is not available forpaths in general groupoids, since the pointwise products generally do notexist. Therefore, we must use concatenation.

When G is a groupoid over M , the concatenation product will makeP (G) into something which resembles a groupoid over the same M . Thecompatibility condition for forming the product γ1 ? γ2 of two G-paths isthat s(γ1(1)) = t(γ2(0)). When this condition is satisfied, we can replace γ2

by its left translate2 γ2 = γ1(1)γ2, which has the property that, since γ2(0)2This construction suggests that it might also be useful to think of P (G) as the space

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is a unit element, γ2(0) = γ1(1). We can now define γ1 ? γ2 to be the usualconcatenation of γ1 and γ2. Notice that (γ1 ? γ2)(1) = γ2(1) = γ1(1)γ2(1),so evaluation at τ = 1 recovers the product on G from that on P (G). Ifwe define the units of P (G) to be the constant curves with values in theunits of G, and inversion of a path to be the same path traversed in theopposite direction and left-translated to begin at a unit, these operations,too, project to the corresponding operations in G.

Although they project to the groupoid operations in G, the operations inP (G) do not satisfy the groupoid identities. The product is not associative,the units are not units, and the inversion of a path is not its inverse. We willdeal with this issue later. An additional technical problem is that, althoughwe have arranged via the left translations to make the concatenation of twopaths continuous, there is no reason for it to be differentiable. To remedythe latter problem, we must either restrict to paths which have second-ordercontact with constant paths at their endpoints, or enlarge the path space toinclude piecewise smooth paths. The former approach seems to be the moreeffective one, and we will from now on use the notation P (G) to denote thisrestricted class of paths.

The operations on P (G) which project to the groupoid structure on Gcan now be transfered by the development map to P (A(G)). The trans-ferred product on P (A(G)) is simply concatenation, without any remnantof the left-translations. Inversion becomes “time” reversal combined withmultiplication by −1, and the units are the constant paths with zero value.The boundary condition which allows for smooth concatenation is that thepaths in P (A(G)) should vanish at their endpoints, along with their firstderivatives.

There is one more structure on P (G) which we must transfer to P (A(G)):the equivalence relation determined by the projection from P (G) to G. Wefirst note that, although there may be many groupoids with the same A(G)(with different connectivity of their s and t fibres), they all have isomorphicP (G)’s. In order to have G determined uniquely by P (G), we will assumethat G has simply connected (i.e. connected with trivial fundamental group)s and t fibres. In this case, the fibres of the projection from P (G) to G maybe thought of as “homotopy classes of paths in the restructured space MG”.They are the leaves of a foliation which, when transferred to P (A), may bedescribed strictly in terms of a notion of homotopy of A-paths which makes

of all paths in G on which t is constant, modulo left translation, instead of requiring thecondition that the paths have unit initial values. The projection from P (G) to G wouldthen be given by γ 7→ γ(0)−1γ(1).

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no reference to G.

2.6 From Lie algebroids to Lie groupoids

We are now ready to go in the other direction. Starting with an arbitraryLie algebroid A over M , we will attempt to realize it as (isomorphic to)A(G) for a groupoid G.

Start with the space P (A) of A-paths, defined as above by the conditionρ ◦ γ = (π ◦ γ)′ and the boundary condition of vanishing at least to secondorder at the endpoints. Multiplication is defined by concatenation, inversionby reversal of direction and multiplication by −1; the units are the constantzero paths. We think of P (A) as diffeomorphic via a development map toP (G(A)) for a groupoid G(A) which is still to be defined. If G(A) exists, itmust be the quotient of P (G(A)), and hence of P (A), by the foliation definedin the previous section. In other words, we define on P (A) the “homotopy”foliation H mentioned above.

We now define G(A) as the leaf space P (A)/H. One can show that thealgebraic structure on P (A) descends to G(A) and that all the “anomalies”vanish, so that G(A) becomes a true groupoid. What is harder is to put ageometric structure on this leaf space. When H is a simple foliation, thenG(A) has a (unique) structure of smooth manifold for which the projectionfrom P (A) is a smooth submersion. In this case (which is precisely thecase when A is integrable), G(A) is a Lie groupoid whose Lie algebroid isnaturally isomorphic to A. In the general case, G(A) has been describedin [4] simply as a topological groupoid, but this is not enough geometricstructure from which to recover the Lie algebroid A. It turns out that asuitable structure to put on G(A) is that of a differentiable stack. This kindof structure will be described in the next section.

3 Differentiable stacks

Stacks originated in algebraic geometry, and their definition in that field isthe basis for the differentiable version. We refer to the articles of Pronk [16]and Metzler [12] for careful treatments of the theory of differentiable stacks.What follows is a very rough, “cartoon” version of that theory.

The most abstract definition of a differentiable stack is as a category Xwith a covariant functor F from X to the category C of smooth manifoldsand smooth mappings. We will not state the axioms which must be satisfiedby these objects, but we merely mention that the most important of them isa “gluing property”. From here, we proceed immediately to some examples.

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First of all, for each ordinary manifold M , there is a stack XM for whichthe objects are smooth maps N → M , where N ranges over the objects ofC. The morphisms from N1 → M to N2 → M are smooth maps N1 → N2

forming a commutative triangle with the other two, and the functor F isthe one which forgets about M and looks only at the domains N and themaps between them.

Another class of examples are the “classifying stacks”. For each Liegroup H, BH is the stack whose objects are smooth principal bundles withstructure group H and whose morphisms are H-equivariant maps. Thefunctor F assigns to each bundle its base. In fact, there is a notion ofprincipal bundle with structure groupoid, and BG can be defined in thesame way when G is any Lie groupoid. In particular, when G is a groupoidover M consisting entirely of unit morphisms, BG is isomorphic to XM .

Care must taken with respect to morphisms of stacks. The most obviousdefinition is as a functor between the categories, forming a commutativediagram with the functors to C. But two functors between stacks should beconsidered as giving the same morphism when they are related by a naturalequivalence. Strictly speaking, the stacks should be regarded as objects ina 2-category in which these natural equivalences are the 2-morphisms. Werefer to the references cited above for a careful discussion of this point.

3.1 Atlases as groupoids

A somewhat less abstract description of stacks is via their presentations bygroupoids. These presentations are known as atlases–to see why, we willexplain how to view ordinary smooth atlases in terms of groupoids.

If M is a manifold with an atlas of coordinate charts Uα → Vα, wherethe Uα’s are open subsets of M and the Vα’s are open subsets of Rn, we candefine a groupoid GM over the disjoint union Q of the Vα’s whose morphismspace is the disjoint union of the pairwise intersections Uα ∩Uβ, with sourceand target maps given by the charts. As a set, M is isomorphic to theorbit space of this groupoid; its differentiable structure is the unique onefor which the quotient map from Q to M is a local diffeomorphism. (Theclassifying stack BGM is isomorphic to XM .) If we have a second atlas forthe same manifold, the corresponding groupoid G′

M over a manifold Q′ isnot isomorphic to GM , but the two groupoids are equivalent by a (GM , G′

M )-“bibundle”, a notion which we now explain.

In general, an equivalence bibundle for Lie groupoids G and G′ overmanifolds Q and Q′ is a manifold B on which G and G′ act from the leftand right respectively in such a way that, if we think of the elements of B as

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morphisms from objects in Q′ to objects in Q and introduce another copyB of B to represent the inverses of these morphisms, then the disjoint unionof G, G′, B, and B becomes a groupoid over the disjoint union of Q and Q′

in such a way that its restrictions to Q and Q′ are G and G′.Returning to our two atlases, we define the bibundle B to be the disjoint

union of the intersections Uα∩U ′β, with projections to Q and Q′ again givenby the charts. Note that this is essentially the same as a common definitionof compatibility between smooth atlases: their union should also be a smoothatlas.

We can also define mappings between different manifolds in terms of amore general kind of bibundle. If GM and G′

M ′ are groupoids associated withatlases for M and M ′, and if f : M ′ → M is a smooth map, then we definethe bibundle Bf to be the disjoint union of the open subsets f−1(Uα) ∩ U ′β,with projection to Q given by the charts for M composed with f , whileprojection to Q′ is given by the charts for M ′. In other words, this bibundleencodes the “coordinate representation” of the map f . In this case, we donot introduce inverses for the “morphisms” in B, and the resulting objectover the union of Q and Q′ is not a groupoid but a category having certain“divisibility” properties. Of course, even having chosen atlases for M andM ′, we should identify the map f not with a particular bibundle, but withan isomorphism class of bibundles.

3.2 Groupoids as atlases

Following the example above, we may consider any differentiable groupoidas a presentation of (or “atlas for”) some stack, with the stack itself definedby all presentations which are related by equivalence bibundles. Morphismsbetween stacks are then defined, with respect to given presentations, as iso-morphism classes of bibundles. (Again we have a 2-category, with morphismsbetween bibundles as the 2-morphisms.)

3.3 Etale stacks and foliations

A stack is called etale if it has a presentation by an etale groupoid (i.e. onewhose source and target maps are etale). Such stacks arise naturally fromfoliations. If F is a foliation of a manifold P , there is an associated holonomygroupoid. (One may also use the monodromy groupoid.) Restricting to aglobal (not necessarily connected, of course) cross section Σ to the foliationgives an equivalent etale groupoid. Thus, the etale groupoids attached todifferent cross sections are all equivalent and are therefore presentations of

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a single differential stack known as the leaf stack of the foliation. If thefoliation is simple, the leaf space P/F is a manifold, and the leaf stack isjust XP/F . (If we use the monodromy groupoid instead of the holonomygroupoid, the leaf stack incorporates the fundamental groups of the leaves.)

The ideas in the paragraph above were already clearly expressed in workof Haefliger [9], van Est [7], and Pradines [15] from the mid 1980’s, andfurther developed by Moerdijk [13].

4 Stacks as groupoids

We now resume the discussion which was temporarily abandoned at the endof Section 2. Starting with a Lie algebroid A over M , we have on the Banachmanifold P (A) of A-paths a collection of operations resembling those of agroupoid, and a foliation H such that these operations project to a groupoidstructure on G(A) = P (A)/H. We now consider G(A) as a differentiablestack, the leaf stack of H. (Notice that, since H has finite codimension, theetale groupoids representing G(A) are purely finite-dimensional objects.)Tseng and Zhu [17] show that the groupoid operations on G(A) are mor-phisms of differentiable stacks. This is not the end of the story though.When we check the groupoid identities, they are not satisfied exactly, butonly “up to equivalence.” A further analysis of the situation shows that allthe “equivalences” involved here are consistent with one another. Finally,one defines the Lie algebroid for a general “stack groupoid” and shows thatthe Lie algebroid of G(A) is isomorphic to A, thus completing the integra-tion of A. The main result of [4] is the establishment of computable criteriawhich determine precisely when G(A) is an ordinary manifold.

When these constructions are applied in Poisson geometry to the Lie al-gebroid T ∗P over a Poisson manifold P , one recovers the universal groupoidG(P ) of [2], which now becomes a differentiable stack. The notion of sym-plectic structure on a differentiable stack is easy to define, and G(P ) nowbecomes a symplectic stack groupoid. In particular, it provides a Moritaequivalence from P to itself. To see that we have not gone too far by intro-ducing these exotic objects, we note that, if P and Q are integrable Poissonmanifolds which are Morita equivalent via a symplectic bibundle which isa differentiable stack, then this stack is actually isomorphic to a manifold,so the Poisson manifolds were already Morita equivalent in the usual sense.Thus, we have accomplished the goal of extending Xu’s Morita equivalenceto an equivalence relation among all Poisson manifolds.

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References

[1] Cannas da Silva, A., and Weinstein, A., Geometric Models for Noncom-mutative Algebras, Berkeley Math. Lecture Notes, Amer. Math. Soc.,Providence, 1999.

[2] Cattaneo, A.S., and Felder, G., Poisson sigma models and sym-plectic groupoids, in Quantization of Singular Symplectic Quotients,(ed. N. P. Landsman, M. Pflaum, M. Schlichenmaier), Progress inMathematics 198 (2001), 41–73.

[3] Connes, A., Noncommutative Geometry, Academic Press, San Diego,1994.

[4] Crainic, M., and Fernandes, R., Integrability of Lie brackets, Ann.Math. 157 (2003), 575–620.

[5] Crainic, M., and Fernandes, R., Integrability of Poisson brackets,preprint math.DG/0210152.

[6] Duistermaat, J.J., and Kolk, J.A.C., Lie Groups, Springer-Verlag,Berlin-Heidelberg-New York, 1999.

[7] Est, E.T. van, Rapport sur les S-atlas, in Structure Transverse desFeuilletages, Toulouse 17-19 fev 1982, Asterisque 116 (1982), 235-292.

[8] Fuchssteiner, B., The Lie algebra structure of degenerate Hamiltonianand bi-Hamiltonian systems, Progr. Theoret. Phys. 68 (1982), 1082-1104.

[9] Haefliger, A., Groupoıdes d’holonomie et classifiants, in StructureTransverse des Feuilletages, Toulouse 17-19 fev 1982, Asterisque 116(1984), 70–97.

[10] Kontsevich, M., Deformation quantization of Poisson manifolds, I,preprint q-alg/9709040.

[11] Lie, S., Theorie der Transformationsgruppen, (Zweiter Abschnitt, unterMitwirkung von Prof. Dr. Friedrich Engel), Teubner, Leipzig, 1890.

[12] Metzler, D., Topological and smooth stacks, preprintmath.DG/0306176.

[13] Moerdijk, I., Foliations, groupoids and Grothendieck etenndues, Rev.Acad. Cienc. Zaragoza (2) 48 (1993), 5–33.

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[14] Pradines, J., Troisieme theoreme de Lie sur les groupoıdes differen-tiables, C. R. Acad. Sc. Paris, 267 (1968), 21-23.

[15] Pradines, J., How to define the differentiable graph of a singular folia-tion, Cahiers de Top. et Geom. Diff. 26 (1985), 339-380.

[16] Pronk, D.A., Etendues and stacks as bicategories of fractions, Compo-sitio Math. 102 (1996), 243–303.

[17] Tseng, H.-H., and Zhu, C., Integrating Lie algebroids via stacks, inpreparation.

[18] Xu, P. Morita equivalence of Poisson manifolds, Comm. Math. Phys.142 (1991), 493-509.

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