Geometric Quantization of Poisson Manifolds Via Symplectic Groupoids

Geometric Quantization of Poisson Manifolds via Symplectic Groupoids

Daniel Felipe Bermudez´ Montana˜

Advisor: Prof. Alexander Cardona Gu´ıo, PhD.

A Dissertation Presented in Candidacy for the Degree of Bachelor of Science in Mathematics

Universidad de los Andes Department of Mathematics Bogota,´ Colombia 2019

Abstract

The theory of Lie algebroids and Lie groupoids is a convenient framework for study- ing properties of Poisson Manifolds. In this work we approach the problem of geometric quantization of Poisson Manifolds using the theory of symplectic groupoids. We work the obstructions of geometric prequantization and show how they can be understood as Lie algebroid’s integrability obstructions. Furthermore, by examples, we explore the complete quantization scheme using polarisations and convolution algebras of Fell line bundles.

ii Acknowledgements

Words fall short for the the graditute I have for people I will mention. First of all I want to thank my advisors Alexander Cardona Guio and Andres Reyes Lega. Alexander Cardona, his advice and insighfull comments guided through this research. I am extremely thankfull and indebted to him for sharing his expertise and knowledge with me. Andres Reyes, as my advisor has tough me more than I could ever ever give him credit for here. By example he has shown me what a good scientist, educator and mentor should be.

To my parents, my ﬁst advisors: Your love, support and caring have provided me with the moral and emotional support to confront every aspect of my life; I owe it all to you. To my brothers: the not that productive time I spent with you have always brought me a lot joy - thank you for allowing me the time to research and write during this work. I am also grateful to my best friend, her encouragement and aﬀection motivated me when times got rough; I am lucky to know her.

At last, a special gratitude for those collegues and friends whose bewteen humor and thoughtful conversations have shaped me and this work.

iii For all those who have guided me through this work, for all those who distracted me from ﬁnishing it, but most of all for those who did both.

Contents

Abstract ...... ii Acknowledgements ...... iii

1 Introduction 1

2 A First Look at Symplectic Groupoids 4 2.1 Lie Groupoids Basics ...... 4 2.2 Lie Algebroids Basics ...... 8 2.3 Integration Theory of Lie Algebroids ...... 14 2.4 Symplectic Groupoids ...... 18

3 Geometric Prequantization of Symplectic Groupoids 24 3.1 Prequantization of Symplectic Spaces ...... 24 3.2 The Groupoid and Algebroid Picture ...... 30 3.3 Quantization of Poisson Manifolds ...... 34

4 Quantization of Poisson Manifolds 39 4.1 Quantization of Symplectic Manifolds ...... 39 4.2 Quantization of Symplectic Groupoids ...... 44 4.3 Examples ...... 51

5 Outlook 52

Bibliography 53

v Chapter 1

Introduction

Loosely speaking quantization is a process for constructing a quantum theory for a given classical system. The generality of the procedure may range, depending on the deﬁnition of classical and quantum systems and the properties one would like to preserve from the former on the quantization process.

A ﬁrst approach would be to deﬁne a classical system as a symplectic manifold, which is manifold together with a non-degenerate, closed 2-form. Within this approach, symplectic manifolds should be regarded as the phase space of some classical system. Similarly, the functions on the symplectic manifolds may be thought of as the observables of the system and the symplectic form as a geometric object that encodes the information of the Hamilton equations of motion[1].

A first generalisation of the classical picture is to focus on the observables of the system rather than on the system itself. It turns out that the space of functions of a symplectic manifold is naturally associated with a bivector field which determines an antisymmetric bracket on differentiable functions that satisfies the Jacobi identity. A manifold with such a bivector field is known as a Poisson Manifold and its space of functions as a Poisson Algebra. Although every symplectic manifold is a Poisson manifold, the converse is not true. It occurs that some Poisson manifolds are not symplectic. In spite of this, Poisson and symplectic structures are intimately related. For example, a Poisson manifold is foliated by immersed manifolds where the restriction of the Poisson bracket actually comes from a symplectic form. One usually refers to this as the foliation by symplectic leaves. Thus, a Poisson manifold can be regarded as a degenerate, or singular, generalisation of a symplectic space [2].

On the quantum counterpart a ﬁrst notion of a Quantum system is given by a Hilbert space. This is a complex inner product vector space that is complete with respect to the metric induced by the inner product. In this description, due primarily to Dirac and von Neumann, the vectors in the Hilbert space are states of the quantum system, the self-adjoint operators are the observables of the system and the inner product assigns an expected value to each of these observables[3]. As in the classical picture, a generalisation of the Hilbert space description of quantum mechanics arises by focusing on the observables of the system. The important algebraic structure of bounded operators on the Hilbert space is that of a C∗-algebra, which is a complete, involutive, normed algebra with the

1 C∗-property, namely kaa∗k = kak2 for all a in the algebra.

A first approximation to a rigorous definition of quantization was proposed by Dirac. From his point of view, the classical system was given by a symplectic manifold, the quantum system by a Hilbert space, and quantization was a map from the space of smooth functions on a symplectic manifold to self-adjoint operators on a Hilbert space, in such a way that the Poisson bracket of two observable becomes the commutator of the corresponding quantum observables. The first procedure for constructing such a map out of the geometric information of the symplectic manifold was due to Konstant[4], Souriau[5] and Kirillov[6] on a process today known as Geometric Quantization. Their approximation focused on realising the symplectic form on a line bundle, in a way that the Poisson algebra could be represented by a subset of the derivations on such line bundle. On some important examples, the description using symplectic manifolds and Hilbert spaces is not suitable to a description of the systems. In these cases a more general theory of quantization aims to assign a C∗-algebra to a Poisson manifold. In this approach to quantization the Poisson manifold is regarded as a geometrical approximation to a noncommutative algebra depending on a parameter ~. Within the rigorousity of this assignment one can consider formal deformation quantization [7] or strict deformation quantization [8].

Symplectic groupoids were independently introduced by Karasëv[9] and Weinstein [10] as a tool to study and generalise diverse quantization procedures. A symplectic groupoid is a Lie groupoid with a compatible symplectic structure; the base manifold of a symplectic groupoid is a Poisson manifold and, if it exists, the symplectic groupoid of such Poisson manifold is unique up to covering. Thus, a symplectic groupoid can be thought as a symplectification of the base Poisson manifold. If such symplectification exists, the geometric quantization procedure developed for symplectic manifolds can be generalized to quantized Poisson manifolds. This generalization, mainly due to Weinstein, Xu [11], Hawkings[12] and Landsman[13], has the aim of constructing a C∗-algebra out of the symplectic groupoid of a Poisson Manifold. This process yields a broad class of C∗-algebras such as commutative C∗-algebras, C∗-algebras of locally compact groups [14], crossed product C∗-algebras [15], non-commutative Tori and foliation C∗-algebras[16].

The process of geometric quantization of Poisson manifolds via symplectic groupoids can be summarised in four steps: first, a symplectification of the Poisson Manifold to a symplectic groupoid; second, geometric prequantization of the symplectic groupoid; third, polarization of the prequantum bundle of the symplectic groupoid; fourth and last, construction of the convolution algebra of the polarized prequantum bundle. In Chapter 2 we will deal with the first step of this process, with this purpose in mind we introduce the reader to the general theory of Lie groups, Lie algebroids and Symplectic groupoids. We discuss the integration theory of Lie algebroids and use it to describe the relationship between Poisson manifolds and symplectic groupoids. The second step of the process is handled in the third chapter, through which we review the prequantization theory of symplectic manifolds and symplectic groupoids. Prequantization is the method of assigning a line bundle to a given symplectic manifold, which within the groupoid scenario will be considered as a central extension of a groupoid. We discuss the obstructions to existence and uniqueness of a prequantization from a cohomological perspective in the symplectic manifold case. Furthermore, we show how in the groupoid framework these obstructions can be reinterpreted as integrability obstructions. At last, we complete an exposition of 2 the quantization process in Chapter 4, where the polarization of symplectic manifolds and groupoids is reviewed and the convolution algebra is described. Due to the complications arising from the polarization process, the main examples of the quantization given in this chapter will refer mainly to nicely behaved objects, namely cotangent groupoids.

Throughout this project, the reader is expected to have a general knowledge on the theory of symplectic manifolds, Poisson Manifolds, Hilbert spaces and C∗-algebras. To obtain an overview of the subject we give some general references. For an introduction to the theory of symplectic manifolds and their relationship with classical mechanics we refer the reader to [1, 17]; for an introduction to the theory of Poisson manifolds refer to [2]; and, for an introduction to the theory of Hilbert spaces and C∗-algebras to [14]. The ﬁrst chapters of any of these references will be more than enough to follow the theory developed in this work. For a general, concise and complete overview on quantization we encourage the reader to look at the prelude of [18].

3 Chapter 2

A First Look at Symplectic Groupoids

In this chapter we introduce the reader to the theory of Lie Symplectic groupoids. For this, we start by introducing the general theory of Lie groupoids and Lie algebroids, following [19]. In a sense, if groups represent abstract symmetries of an object, then groupoids represent abstract symmetry transformations between possibly many objects. A Lie groupoid is a groupoid with a smooth structure. As in the case of a Lie group, the smooth structure lets us work with an inﬁnitesimal version of the object, namely a Lie algebroid. It is worth mentioning that, unlike the Lie group/algebra scenario, to a given Lie algebroid there might not be a groupoid integrating it. We will discuss the obstructions to the integrability of a Lie algebroid. For this we follow [20] and [21].

Given a Poisson manifold P one can associate to it a algebroid A(P ) which will describe many of its infinitesimal properties. For instance, the Poisson cohomology of P will agree with the algebroid cohomology of A(P ) and the foliation described by the anchor of A(P ) will correspond with the foliation by symplectic leaves of P . In case that such an algebroid is integrable, we will be able to find a groupoid with additional structure, that of a symplectic groupoid, which will enlighten the global properties of the Poisson manifold. The symplectic groupoid in discussion is unique up to covering and has as base the original Poisson manifold. Thus, a symplectic groupoid can be regarded as symplectification of a Poisson manifold. For a detailed discussion the reader can consult [2] and [19].

2.1 Lie Groupoids Basics

One of the motivations to use Lie groupoids in diﬀerential geometry is the fact that the smooth category behaves badly with respect to quotients. If M is a smooth manifold with an equivalence relation ∼, then M/ ∼ is usually not a smooth manifold. We introduce the following proposition (Theorem 8.3 of [22]).

Proposition 2.1.1. Let ∼ be an equivalence relation on a manifold M and denote R ⊂ M × M the graph of the equivalence relation. Then, the following are equivalent: i) M/ ∼ has a unique manifold structure such that π : M → M/ ∼ is a submersion. ii) R,→ M × M is a closed embedding and π : R → M is a surjective submersion.

4 In the case of a Lie group G acting smoothly on a manifold M the previous proposition asserts that a sufficient condition for M/G to have a smooth structure is for the action to be free and proper. Proposition 2.1.2. Let G be a Lie group acting smoothly on a manifold M. If the action is free and proper, then the quotient M/G has a unique differentiable structure such that π : M → M/G is a submersion. The above condition is rather strong, hence working with quotients in the smooth category is quite restrictive. A way to avoid non smooth quotients is to enlarge the objects in consideration in a way the information of the equivalence relation can be retrieved in a different setting to avoid working with M/ ∼.

The latter approach leads to the notion of Lie groupoids.

Deﬁnition 2.1.3. A Lie groupoid G ⇒ M is a pair of manifolds (G,M) together with following maps:

i) Surjective submersions s, t : G → M. These maps are called source and target respectively.

ii) A partial multiplication map deﬁned on the set of composable pairs, G(2) = {(g, h) ∈ G × G|s(g) = t(h)} m : G(2) → G (g, h) → gh, which is smooth and associative, i.e. (gh)k = g(hk) whenever one of the sides on the equality makes sense.

iii) An identity embedding 1 : M → G

x → 1x,

which is smooth and satisﬁes s(1x) = t(1x) = x and 1xg = g, g1y = g (whenever the the multipliacation is deﬁned).

iv) A smooth map i : G → G g 7→ g−1,

−1 −1 such that gg = 1t(g) and g g = 1s(g). This map is called the inverse. Remark 2.1.4. Note that G(2) is the ﬁbered product over M of the maps s, t : G → M,

(2) G = Gt ×s G

G G s t M 5 since s and t are submersion the space G(2) has a manifold structure.

The manifold M is called the base of the Groupoid. Sometimes where there is possibility of confusion we will denote the base manifold of a groupoid G by G(0) instead of M. Elements of M are called points, while elements of G are called arrows. In some important examples G is not second-countable and Hausdorff, therefore we will not ask for this in the definition of a groupoid. However we will ask that the base space M and the fibers s−1(m) are Hausdorff-second countable manifolds.

Having defined the main objects of study in this work, it is necessary to define their morphisms. Notice that the definition of a groupoid is actually the definition of a (internal) category in the category of smooth manifolds. Thus, morphisms of a groupoid are functors within the category.

Deﬁnition 2.1.5. A groupoid morphism between G ⇒ M and H ⇒ N is a pair (F, f) of smooth maps F : G → H and f : M → N such that the following holds:

i) The pair commutes with multiplication, i.e. ϕ(mG(g, h)) = mH(ϕ(g), ϕ(h)). Equiva- lently, the following diagram commutes,

ϕ×ϕ G(2) H(2)

mG mH ϕ G H

ii) The pair commutes with the source and target maps, that is sH ◦ ϕ = ϕ0 ◦ sG and tH ◦ ϕ = ϕ0 ◦ tG. Equivalently, the following diagrams commute.

ϕ ϕ G H G H

sG sH tG tH ϕ ϕ M 0 N M 0 N

Proposition 2.1.6. A groupoid morphism preserves identities and inverses, i.e. 1N ◦ ϕ = ϕ ◦ 1M and iH ◦ ϕ = ϕ ◦ iG. Proof. Let x ∈ M and g ∈ G such that s(g) = x. Then

ϕ(g) = ϕ(g1x) = ϕ(g)ϕ(1x),

symmetrically for every h such that t(h) = x it holds ϕ(h) = ϕ(1xh)ϕ(h). The previ- −1 ous equalities imply ϕ(1x) = 1ϕ(1x). Moreover, by the previous argument ϕ(g)ϕ(g ) = ϕ(1 ) = 1 and ϕ(g−1)ϕ(g) = 1 implying ϕ(g−1) = ϕ(g)−1. t(g) ϕ(1t(g)) ϕ(1s(g)) Along with the deﬁnition of a groupoid, there are some important notions which help to characterise its structure, for instance

Deﬁnition 2.1.7.

6 i) The orbit of x ∈ M is the set of points which are related by some arrow to x, i.e. −1 Ox = {t(s ({x}))} ⊂ M.

ii) The isotropy at x ∈ M is set of arrows which ﬁx x, to be precise

−1 −1 Gx = {g ∈ G|s(g) = t(g) = x} = t ({x}) ∩ s ({x})

−1 It is worth noting that Gx is a pullback by a submersion, Gx = t ({x})t ×s G, thus Gx is a Lie group.

In the future, we will be interested in groupoids with some properties on the source ﬁbers. Such properties will be denoted s-properties. For example, a groupoid with simply connected source ﬁbers will be called s-simply connected, or in many occasions simply s-connected.

Examples Groupoids are suﬃciently general objects to encompass a big class of important objects and examples in diﬀerential geometry. We present some of the examples that will be of interest later on. We follow [19], where greater details may be found.

Example 2.1.8. Manifolds: Let M be a manifold, by setting then s = t = Id, the set of multiplicative pairs is ∆(M) := {(m, m) ∈ ×M} and M ⇒ M is a groupoid with multiplicative structure deﬁned by m · m = m.

Example 2.1.9. Lie Groups: Let G be a Lie group, consider the point as a base manifold M = {∗} and set s = t the maps to the point {∗}. Then G ⇒ {∗} is a groupoid with the is multiplicative structure of G.

Example 2.1.10. Pair Groupoid: Let M be a manifold. Consider G = M × M and G(0) = M. Deﬁne:

s(x, y) = y, t(x, y) = x, (x, y)(y, z) = (x, z), (x, y)−1 = (y, x), 1(x) = (x, x).

The previous maps deﬁne a groupoid structure on M × M ⇒ M. Example 2.1.11. Transformation Groupoids: Let α be an action of a Lie group G on a manifold M. Consider G = G n M := G × M and G(0) = M. Deﬁne:

s(g, x) = x t(g, x) = gx, (g, hx)(h, x) = (gh, x), (g, x)−1 = (x−1, gx), 1(x) = (e, x).

The previous maps deﬁne a groupoid structure on G n M ⇒ M. π Example 2.1.12. Gauge Groupoids: Let G P −→ M denote a principal bundle P with structure group G. Consider G = Gau(P ) := (P × P )/G, where G acts on P × P by the diagonal action and G(0) := M. Deﬁne:

s(g, x) = x t(g, x) = gx, (g, hx)(h, x) = (gh, x), (g, x)−1 = (x−1, gx), 1(x) = (e, x). 7 The previous maps deﬁne a groupoid structure on Gau(M) ⇒ M. Example 2.1.13. Cotangent Bundles of Lie Groups: Let G be a Lie group. Consider ∗ G = T G and G0 = g, where g is the Lie algebra of G. We will describe a groupoid structure ∗ ∗ on T G ⇒ g . Since G acts on itself by left translations Lg there exist a natural map, the pullback map ∗ ∗ µL : T G → g , µL(g, η) = η ◦ De(Lg). Notice the previous map trivialises the cotangent bundle T ∗G =∼ G × g∗. Similarly, for the right action there pullback map is µR(g, Y ) = η ◦ De(Rg). Deﬁne the groupoid structure by

s(g, η) = µL, t(g, η) = µR

(g, η) · (h, ξ) = (gh, η ◦ D(Rg ◦ Lh−1 )), −1 −1 (g, η) = (g , η ◦ DRg).

∗ Proposition 2.1.14. Consider the transformation groupoid Gng deﬁned by the coadjoint action of G on g. The maps id : g → g and

∗ ∗ ϕ : G n g → T G;(g, η) 7→ (g, η ◦ D(Lg−1 )),

∗ ∗ ∗ ∗ deﬁne an isomorphism between the groupoids T G ⇒ g and G n g ⇒ g . Proof. Smoothness of the maps is clear. We will check it is a groupoid morphism. Recall ∗ Adgη = η ◦ D(Lg−1 ◦ Rg), thus

∗ t ◦ ϕ(g, η) = t[(g, η ◦ D(Lg−1 ))] = η ◦ D(Lg−1 ◦ Rg) = Adgη = t[(g, η)].

In the same fashion, s ◦ ϕ = ϕ0 ◦ s follows, proving property ii) of deﬁnition 2.1.5. To check property i):

ϕ[(g, η)(h, ξ)] = ϕ[(gh, η ◦ D(Rg ◦ Lh−1 ))] = [gh, η ◦ D(Rg ◦ Lh−1 ◦ L(gh)−1 )],

by the condition t(h, ξ) = s(g, η) it follows η ◦ D(Rg) = ξ ◦ D(Lh), implying

[gh, η ◦ D(Rg−1 ◦ Lh−1 ◦ L(gh)−1 )] = [gh, ξ ◦ L(gh)−1 )] = ϕ[(gh, ξ)] = ϕ[(g, η)(h, ξ)].

2.2 Lie Algebroids Basics

In Lie theory it is common to work with the Lie algebra of a group instead of the group itself. The Lie algebra is the inﬁnitesimal object associated to a Lie group. This object bears most of the information of Lie group and is much easier to handle. Up to some connectedness and simply connectedness conditions, most of the information of the Lie group can be recovered from its Lie algebra via integration. In the case of Lie groupoids there is a similar procedure generalising that of Lie Groups: to every Lie groupoid one can assign a Lie Algebroid which is the inﬁnitesimal object associated to it. However, this picture is not as well behaved as in the group case, mainly because integrability theory of Lie Algebroids is a more delicate issue here.

8 Deﬁnition 2.2.1. A Lie Algebroid on M is a vector bundle A → M with Lie bracket [·, ·] on Γ(A), together with a vector bundle map, called the anchor, # : A → TM

# A TM s.t [α, fβ] = f[α, β] + #α(f)β, M and the induced morphism in sections # : Γ(A) → X(M) is a Lie algebra morphism.

The anchor encodes a fair amount of geometric information of the Lie algebroid. For instance, the image of the anchor deﬁnes a distribution on M in the sense of Sussman [23].

Proposition 2.2.2. #(A) is in general not a vector bundle. However, it is an integrable distribution, i.e., for each x ∈ M there exists a maximal connected immersed submanifold Lx for which x ∈ Lx and TyLx = #(Ay) for all y ∈ Lx ⊂ M.

Proof. Following Theorem 4.2 of [23], it is enough to prove that for every m ∈ M there exist vector fields X1,...,Xk such that: 1 k (i) #x(Ax) = Span(X (m),...,X (m)), (ii) for every vector field X satisfying X = #α, α ∈ Γ(A), there exists > 0 such that ∞ i there are C functions fj , 1 ≤ i, j ≤ k, defined in (−, ) which satisfy

k i m X i j m [X,X ](ϕX (t)) = fj (t)X (ϕX (t)), j=1 m where ϕX (t) denotes the ﬂow of the vector ﬁeld X starting at m after time t.

To check the existence of such vector ﬁelds, let m ∈ M and consider a trivializing open ∞ set m ∈ Um for both A and TM. Thus Γ(A|U ) is ﬁnitely generated C -module with k ∞ generators α1, . . . , αk, and there exist fij ∈ C (M) such that

k [αi, αj] = fijαk.

We claim #α1,..., #αk are the desired set of vector ﬁelds. By deﬁnition #x(Ax) = Span(#α1,..., #αk) and since the anchor # is a Lie algebra morphism it holds

 k  k X k X k j [#αi, #αj] = #[αi, αj] = #  fijαk = fij#α . j=1 j=1

Proposition 2.2.3. ker #x is a Lie algebra, called isotropy Lie algebra, under the bracket ˜ [αx, βx]ker #x = [˜α, β](x), whereα, ˜ β˜ ∈ Γ(A) satisfyα ˜(x) = αx, β˜(x) = βx.

9 Proof. Since the bracket [α, β] satisﬁes the Leibniz rule it depends just of the germ of α and β. Moreover, if α, β ∈ ker # it follows #α(f) = 0 for every f ∈ C∞(M), thus

[α, fβ] = f[α, β] + #α(f)β = f[α, β].

and the bracket is C∞(M) bi-linear when restricted to ker #. This implies the formula given in the proposition is well deﬁned. It is clear from the Lie Algebroid axioms that this bracket deﬁnes a Lie algebra structure on ker #x.

Examples Just as in the case of Lie groupoids, Lie algebroids encompass a large class of objects and examples important in differential geometry. Example 2.2.4. Tangent Bundles: Let M be a manifold. Consider the tangent bundle TM → M, equipped with anchor # = Id and the vector field bracket the tangent bundle is a Lie Algebroid. Example 2.2.5. Lie Algebra: Let g be a Lie algebra considered as vector space, or equivalently as vector bundle over a point {∗}. By defining the anchor # as the map to the point {∗} and the bracket as the Lie Algebra bracket, the vector bundle g → {∗} is a a Lie algebroid. ρ˜ Example 2.2.6. Infinitesimal Action Algebroid: Let g −→ X(M) be a infinitesimal action (lie algebra homorphism) of a Lie algebra g. Define A = g n M := g × M → M. ∞ Since the bundle is trivial Γ(g n M) = C (M, g) and the bracket is uniquely determined if defined on constant sections α, β : M → g

#(α, x) =ρ ˜(α)|x [α, β](x) = [α(x), β(x)]g

To define the our next example we need a precise definition of Poisson manifold. Definition 2.2.7. A Poisson manifold (P, {·, ·}) is a Manifold together with a R-bilinear map C∞(P ) × C∞(P ) → C∞(P ), such that it satisfies the Jacobi identity, and is a derivation on each component. Example 2.2.8. Cotangent Space of a Poisson Manifold Let (P, {·, ·}) be a Poisson Manifold. The Poisson bivector Π ∈ Λ2TP defined by Π(df, dg) = {f, g}, and the induced map ∗ Π:˜ T P → TP ; df 7→ Π(df, ·) = Xf . Define the algebroid to be the vector bundle T ∗P → P , with anchorπ ˜ and bracket defined on df, dg by [df, dg] = d{f, g}. Many properties of the Poisson manifold P can be understood by means of the algebroid T ∗M. For example, the foliation on P associated to Π:˜ T ∗P → TP of proposition 2.2.2 is the foliation of P by symplectic leaves described in [2]. Another example is that where the Poisson cohomology of a Poisson manifold P corresponds to the algebroid cohomology of T ∗P , described in [2]. 10 Lie Algebroid Cohomology Definition 2.2.9. Let A → M be a Lie algebroid. An A-connection on a vector bundle E → M is a R-bilinear map ∇ : Γ(A) × Γ(E) → Γ(E) satisfying:

i)∇fαs = f∇αs

ii)∇αfs = f∇s + [ρ(α)(f)]s,

for all α ∈ Γ(A), s ∈ Γ(E) and f ∈ C∞(M):

Notice that a usual connection on a vector bundle E → M corresponds to a TM- connection. Just as in the case of usual TM-connections one can deﬁne curvature and torsion.

2 ∗ ∗ Definition 2.2.10. i) The curvature R∇ ∈ Γ(Λ A ⊗ E ⊗ E) of a A-connection ∇ is defined by R∇(α, β)s = ∇α∇βs − ∇β∇αs − ∇[α,β]s. 2 ∗ ii) The torsion of ∇ is T∇ ∈ Γ(Λ A ⊗ A) defined by

T∇(α, β) = ∇αβ − ∇βα − [α, β].

Definition 2.2.11. A representation of a Lie algebroid A → M is a vector bundle E → M together with a flat A-connection ∇, i.e. such that R∇ = 0. Example 2.2.12. Let (A → M) be a Lie algebroid and let L = M × R be the trivial line bundle on M, thus Γ(L) = C∞(M). Define the trivial A-connection ∇triv on L by

triv ∞ ∇α (f) = L#αf f ∈ C (M), α ∈ Γ(A).

The curvature of the trivial connection is [LX , LY ] − L[X,Y ], which is known to vanish. Given a representation (E, ∇) of A we will deﬁne a chain complex with values on (E, ∇). Consider the chain complex Ω(A, E) with diﬀerential

k X d∇ω(α0, ··· , αk) = ∇αi ω(α0, ··· , αˆi, ··· , αk)+ i=0 X i+j + (−1) ω([αi, αj], ω0, ··· , αˆi, ··· , αˆj, ··· , αk). i

In particular for the case of the trivial representation the previous complex is known as the algebroid deRham complex and the corresponding cohomology as the algebroid deRham cohomology of A. In the Poisson manifolds case we have the following result which exhibit relationship between algebroids and Poisson geometry, Lemma 2.1 of [11]:

Theorem 2.2.13. Let P be a Poisson manifold and T ∗P its corresponding algebroid. Then the Poisson cohomology of P is isomorphic to the algebroid deRham cohomology of T ∗P .

11 Lie Algebroid of a Lie Groupoid Let G be a Lie group and consider the set of invariant vector ﬁelds under right translation Rh : G → G deﬁned by Rh(g) = gh

R Xinv(G) = {X ∈ X(G)|(DRh)g(Xg) = Xgh}.

R One can show that Xinv(G) with the bracket [X,Y ] = XY − YX is a Lie algebra, moreover, R Xinv(G) → TeG X → Xe,

is a vector space isomorphism. Thus the vector space TeG inherits the vector ﬁeld bracket, in such a way one obtains the Lie algebra g of G. The previous construction can be generalised to Lie algebroids as follows. Let G be a Lie groupoid, notice that Rg is not a map from G to G but a map between ﬁbers:

−1 −1 Rg : s ({t(g)}) → s ({s(g)}) −1 −1 (right translation between ﬁbers) (DRg)h : Ths ({t(g)}) → Thgs ({s(g)}).

Therefore, we should consider vector fields tangent to the source fibers. Since s : G → M is a submersion, the level set theorem guarantees s−1({x}) is a embedded submanifold with s −1 tangent space Ty G := T s ({x}) = ker Dys. Thus, the space of vector fields tangent to s the fiber is X (G) := Γg(ker Ds), notice Ds is a fiber bundle because s has full rank ([24, Theorem 5.1]). We can define the set of invariant vector field to the fibers as follows

s s Xinv(G) = {X ∈ X (G)|(DRg)Xh = Xgh}.

s Proposition 2.2.14. Xinv(G) is closed under the vector ﬁeld bracket, therefore the set of right invariant vector ﬁelds has a Lie algebra structure.

s In a similar way to the Lie group case, Xinv(G) is determined by its value at the identity. However, in the Lie groupoid case there are many identities, one for each point in M. In some sense we get a smooth collection of vector spaces one for every point in M, that is, a vector bundle. Let us construct an isomorphism analogous to the one of the Lie group case. For this purpose consider the restriction of T sG to M, namely the vector bundle A(G) = 1∗T sG → M, the ﬁbered product of T sG → M over 1 : M → G, and a map s Γ(A) → Xinv(G) given by

∼ s R R Γ(A) −→ Xinv(G); α 7→ αG , αG (g) = (DRg)1t(g) (α(t(g))).

12 s Proposition 2.2.15. The map Γ(A) → Xinv(G) is a vector space isomorphism. Addition- ally, the bracket induced in Γ(A) from the vector ﬁeld bracket satisﬁes the Leibinz rule

[α, fβ] = f[α, β] + (Dt(α)(f))β ,

for all α, β ∈ Γ(A), f ∈ C∞(M).

Proof. Linearity of the maps follows from linearity of the diﬀerential. To prove it is an isomorphism we show an explicit inverse

Xinv(G) → Γ(A); X 7→ X|M , where X|M denotes the restriction of X to M. It is clear this map is linear. Moreover since R R1m |s−1(m) is the identity then (αG|M )(m) = α(DR1m )m(α(m)) = α(m). On the other hand, right invariance implies a vector ﬁeld X ∈ Xinv(G) is completely determined by the values at some ”cross-section” of s, for instance is completely determined by the values at R M (seen as an embedded manifold of G). Thus (X|M )G = X, proving the given map is indeed an inverse.

∞ To prove the second statement we need to extend f ∈ C (M) to Rg-invariant function f˜ ∈ C∞ (G). Since the vector ﬁeld bracket in X (G) satisﬁes the Leibinz rule the desired Rg inv conclusion will not depend of the extension. Consider the extension f˜ = f ◦ t, this is Rg-invariant map since

f ◦ t(1m · g) = f(t(1m · g)) = f(t(1m)) = m.

Now,

R ˜ R ˜ R R R ˜ R R ˜ [α, fβ] = [αG , fβG ]|M = (f[αG , βG ] + (αG f)βG )|M = f[α, β] + (αG f)|M β.

R ˜ To ﬁnish the proof we need to show (αG f)|M = Dt(α)(f). Indeed, for any m ∈ M,

R ˜ [αG f](m) = [(DRg)(α)(f ◦ t)](m) = [α(m)](f ◦ t ◦ Rg) = [α(m)](f ◦ t) = [Dt(α)(f)](m), so the desired equality holds.

Deﬁnition 2.2.16. The Lie algebroid of a Lie groupoid G, denoted by A(G), is the ∗ s vector bundle Ds : 1 T G → M with the bracket induced from the vector ﬁelds Xinv(G) and anchor Dt : 1∗T sG → TM.

Example 2.2.17. Pair Groupoid: For the pair groupoid M × M the source fibers are s−1(x) = {(x, y) ∈ M × M|y ∈ M} which are diffeomorphic to the manifold M. Moreover Dt : T s(M × M) → TM defines an isomorphisms of vector bundles. We will show Dt : A(M × M) → TM is an isomorphisms of Lie algebroids. It is clear the anchor is preserved by the this isomorphism, indeed #TM ◦ Dt = idTM ◦ Dt = #A(M×M). Now, a vector field s on T (M × M) is by definition a collection of vector fields {Xy}y∈M , where Xy ∈ X(M), such a collection defines a right invariant vector field on T s(M × M) if and only if all the Xy are the same. From this description is clear that the bracket of invariant vector fields Xinv(M × M) is isomorphic to the bracket on X (M). Notice that in the case M is simply connected M × M ⇒ M is a s-connected Lie groupoid integrating TM → M. 13 Example 2.2.18. Lie Algebras: For a Lie group G considered a a groupoid over a point {∗}, the contruction of the lie algebroid yield the lie algebra g → {∗} considered as a vector bundle over a point.

2.3 Integration Theory of Lie Algebroids

We described how to any Lie groupoid one can assign a Lie algebroid generalising the construction of the Lie algebra of a Lie group. It is natural to ask under which conditions, the Lie algebroid data integrates to groupoid data, that is, if the classical Lie theorems hold for Lie algebroids/groupoids. There are affirmative answers for generalisation of the first two Lie theorems. Definition 2.3.1. Let be A be a Lie algebroid, then A is integrable if there exists a Lie groupoid G such that A is the Lie algebroid associated to G. A Poisson manifold P is called integrable if the algebroid T ∗P is integrable. Theorem 2.3.2. (Lie I: Proposition 3.3 of [25]) If A is an integrable algebroid, then there exist a unique s-simply connected Lie groupoid integrating A. Theorem 2.3.3. (Lie II: Proposition 3.5 of [25]) Let φ : A → B be a morphism of integrable Lie algebroids, and G and H be the corresponding integrations. If G is s-simply connected, then there exists a unique morphism of Lie groupoids Φ : G → H integrating φ. However, there is a negative answer for the third Lie theorem, i.e. not every Lie algebroid integrates to a Lie groupoid. In this section we will briefly review the integration theory of Lie algebroids, the obstructions to integrability and the construction of the source simply connected Lie groupoid in the integrable case [26].

S-Connected Lie Groupoid of G To begin our discussion on the integrability theory of Lie groupoids we begin by reviewing the construction of the s-connected Lie groupoid of a given groupoid G. The space of G-paths is the space

2 P (G) = {γ : I → G|g ∈ C , g(0) = 1x, s(γ(t)) = x, }.

The space P (G) endowed with with the C2-topology is a Banach (sub)manifold with tangent 2 s space at γ ∈ P (A) given by the Banach space Cγ (I,T G) of time dependent (fiber) vector fields over γ ([27] Theorem 1.11 and Proposition 1.13). The Monodromy groupoid Mon(G) of G is Mon(G) = P (G)/ ∼; γ ∼ η, where ∼ is the equivalence of C1-homotopies of path with the same ending and starting points. The groupoid structure is defined by

s([γ]) = γ(0), t([γ]) = γ(1), ( γ(2t) t ∈ [0, 1/2] [η][γ](t) = η(2t − 1)η(1) t ∈ [1/2, 1].

The differential structure of Mon(G) is given in a similar fashion as the differential structure universal covering of a manifold [28]. In the case of a Lie group G the space 14 Mon(G) can be described in terms of homotopies of paths with values on the Lie algebra g, i.e. Mon(G) = Mon(g). Thu,s by starting with the Lie algebra one can describe Mon(g), at least as a topological space, in terms of the Lie algebra alone. One can then use some argument to endow Mon(g) with a differentiable structure coming from the Banach manifold structure on P (g) and show it is actually the s-connected groupoid integrating G ([29, Section 1.14]). The Lie algebroid case follows a similar idea, to begin we require to find the appropriate infinitesimal notion of G-path, this will be given by A-paths; in addition we require some notion of homotopy for A-paths. With these definitions one can define a topological groupoid G(A) of an algebroid A, an analogue of Mon(g). At last, one needs to find necessary and sufficient conditions on A such that G(A) has a differentiable structure, this will be done by considering G(A) as the leaf space of a foliation on a Banach manifold [26].

Deﬁnition 2.3.4. Let π : A → M be an Lie algebroid. A C1-curve a : I → A is an A-path if d #a(t) = γ(t), dt where π(a(t)) = γ is the base path. The space of A-paths is denoted by P (A).

The following proposition shows that P (A) should be thought of as the algebroid analogue of P (G).

Proposition 2.3.5. Let A be an integrable algebroid and G a integration of A. Then the map R R d D : P (G) → P (A); D γ(t) = (DR −1 ) γ(t), γ(t) γ(t) dt deﬁnes a homeomorphism.

Proof. We will prove G-paths correspond to the groupoid morphism I × I → G, where I × I is the pair groupoid, and that A-paths corresponds to algebroid morphism TI → A. Since the map DR is just the derivation of morphism deﬁned by the G-path, and since I × I is the simply connected groupoid integrating TI, then Lie II gives the desired isomorphism.

To check a G-path is a groupoid morphism I × I → G, Let H : I × I → G be a groupoid morphism with base h : I → M, then H(t, ) = H(t, 0)H(0, ) = H(t, 0)H(, 0)−1 thus H is characterized by a path γ(t) = H(t, 0). Note s(H(t, 0)) = h(0) and γ(0) = H(0, 0) = 1h(0), that is γ is a G-path.

To check an A-path is an algebroid morphism TI → A. Since TI has trivial bracket we only need to show the compatibility with the anchor. Since TI is trivializable a vector bundle morphism with base π(a(t)) = γ : I → A can be written as adt : TI → A with d a : I → A. Compatibility with the anchor is equivalent to the equality #a = dt(γ) = dt γ, i.e. the condition for a to be an A-path.

The Weinstein Groupoid of an Lie Algebroid A-paths should be seen as inﬁnitesimal analogues of G-paths. With this in mind let’s see how can we restate the notion of homotopy using just the algebroid data.

15 By a variation of A-paths we mean a family of A-paths

a(t) = a(, t): I × I → A

which is of class C2 on the parameter , and with the property that the base maps γ(t): I → M have ﬁxed end-points.

In the case A is integrable a variation of A-path corresponds to a family of G-paths g(t): I → G such that D(g) = a. In the groupoid scenario a homotopy between G-paths g0, g1 is a map H : I → P (G) with fixed starting and end-points, forgetting the differentiable structure for a second, one verifies that this is the same as a P (G)-path. Thus, heuristically, a homotopy of G-paths can be considered as a groupoid morphism (I × I) × (I × I) → G, where again (I × I) is the pair groupoid, while the outer product (I × I) × (I × I) carries a component-wise groupoid structure. In this picture, an A-path homotopy should be some subset of algebroid morphism

h : TI × TI → A; h = adt + bd a, b : I × I → A.

Now from the G-path picture b = DR(gt), where gt is a path in G deﬁned by gt() = g(, t)([26, Proposition 1.3]). Thus, the condition that the end point are ﬁxed can be restated as b(, 1) = 0.

One can describe the conclusion of the previous discussion without the need of a groupoid G integrating A. If a : I × I → A is a variation of A-paths then the condition for adt + b : TI × TI → A to be an algebroid morphism for some b : I × I → A can be rewritten, by using an auxiliary TM-connection ∇ on A, as a diﬀerential equation on a and b. Proposition 2.3.6. Given a TM-connection in A and a : I × I → A an A-path variation, the condition for adt + bd to be a algebroid morphism is equivalent to the diﬀerential equation

∇∂t b − ∇∂ a = T∇(a, b), b(, 0),

where T∇ denotes the torsion of the connection (see 2.2.10). The solution b of this equation always exist and is an A-path which does not depend on ∆ Proof. The ﬁrst part of the proposition is proposition 1.14 in [30]. The second part is proposition 1.3 in [26].

We summarise the previous discussion in the following deﬁnition of A-path homotopy.

Definition 2.3.7. Two A-paths a0, a1 : I → A are A-homotopic a0 ∼A a1, if there exist a variation a : I × I → A with the property that b given by proposition 2.3.6 satisfies b(, 1) = 0. The A-homotopy relation is an equivalence relation and one can consider the Weinstein groupoid G(A) ⇒ M of a Lie algebroid π : A → M defined by P (A)/ ∼A. This is a topological groupoid over M with the following structure maps:

s([a]) = π(a(0)), t([a]) = π(a(1)), ( 2a2(2t) t ∈ [0, 1/2] [a1][a2] = 2a1(2t − 1) t ∈ [1/2, 1]. 16 Theorem 2.3.8. (Theorem 2.1 of [26]) Let A be Lie algebroid over M, then the We- instein groupoid G(A) is a s-simply connected topological groupoid. Moreover, whenever A is integrable, G(A) admits a diﬀerentiable structure which makes it the unique s-simply connected groupoid integrating A. The previous theorem is an analogue of Lie third theorem in the context of Lie algebroids and Lie groupoids. NOtice the Weinstein groupoid is always a topological groupoids, however it may fail to be a LIe groupoid, In the following section we will explore the obstructions to endow the Weinstein Groupoid witha diﬀerentiable structure.

Obstructions to Integrability: Monodromy We now come to the issue of ﬁnding obstructions to endow G(A) with a diﬀerential structure. The answer to this question is given by the monodromy groups which we will describe in this section. Let A → M be a Lie algebroid and let gx be the isotropy Lie algebra at x ∈ M, i.e. consider G(gx) the Lie group integrating gx, which by construction is subset of G(A)x, where the latter is the isotropy group of G(A) at x.

Deﬁnition 2.3.9. The monodromy group Nx(A) at x ∈ M is the subgroup of G(gx) which consists of the equivalence classes [a] ∈ G(gx) of gx-paths that are trivial as A-paths.

In practice there is a practical way to compute the monodromy groups Nx(A). By proposition 2.2.2 for every x ∈ M there exist leaves Lx such that algebroid A → M restricted to Lx is a Lie algebra bundle

# 0 → gx → A|L −→ TL → 0.

Choosing a linear splitting of the previous exact sequence δ : TL → AL and regarding it as 2 a connection one veriﬁes its curvature Ω (L, gL) is given by

Ωδ(X,Y ) := δ([X,Y ]) − [δ(X), δ(Y )].

Lemma 2.3.10. ([26, Lemma 3.6] ) If there is a splitting δ with the property that its curvature Ωδ is Z(gx)-valued, then Z ∗ 2 Nx(A) = γ Ωδ | (γ : S → L) ∈ π2(L, x) , S2 for all x ∈ L

Remark 2.3.11. The integral is understood as a Z(gx)-valued integral. In general, even if Ωδ is not Z(gx)-valued the set of integrals deﬁned in the proposition is a group P(A, x) containing Nx(A). Theorem 2.3.12. (Theorem 4.1 of [26]) A Lie algebroid A over M is integrable if and only if for all x ∈ M the groups Nx(A) are locally uniformly discrete in x ∈ M. Example 2.3.13. Consider a simply connected symplectic manifold (M, ω). Deﬁne the algebroid Aω = TM ⊕ω R as the algebroid A extended by a trivial line bundle with structure maps:

#ω(a ⊕ λ) = #a, [X ⊕ f, Y ⊕ g]ω = [X,Y ] ⊕ LX g − LY f − ω(X,Y ). 17 This algebroid is transitive i.e. the anchor is surjective at every x ∈ M and there only one leaf, so the only obstruction to integrability is measured by whenever Nx(Aω) is discrete. To measure the obstruction we use lemma 2.3.10, consider the splitting θ(X) = X ⊗ 0, then

Ωθ(X,Y ) = θ([X,Y ]) − [θ(X), θ(Y )]

= [X,Y ] ⊕ 0 − [X,Y ] ⊕ (LX 0 − LY 0 − ω(X,Y )) = ω(X,Y ).

Thus Aω is integrable if and only if the group of spherical periods deﬁned as Z ∗ 2 P(M) = { γ ω | (γ : S → M) ∈ π2(M)}, S2 is discrete. Interesting enough this is the same condition (up to some scalar multiple) for the symplectic manifold (M, ω) to be prequantizable (see the Chapter 3). By the analysis in the previous example, in case of a Poisson manifold P the condition i) of 2.3.12 is equivalent to requiring that the group of periods P(L, x) of every symplectic leaf L of P to be discrete, i.e. every leaf symplectic leaf should be prequantizable. However due to condition ii) this is not a suﬃcient condition, examples of algebroids which fail to be integrable due to condition ii) can be found in [20].

We saw how when Lie algebroid T ∗P of a Poisson manifold P encoded much of the information of P . In the case this algebroid is integrable, the groupoid G(T ∗P ) will have a symplectic structure compatible with the groupoid multiplication in such a way that G(T ∗P ) will represent a symplectiﬁcation of P .

2.4 Symplectic Groupoids

For a groupoid G the space of composable pairs G(2) = {(g, h) ∈ G × G|t(h) = s(g)} has three “face” maps,

pr1, pr2, m : G2 → G, pr1(g, h) = g, pr2(g, h) = h, m(g, h) = gh.

These face maps are actually the face maps of the groupoid seen as a simplicial set by means of its nerve[19]. Deﬁnition 2.4.1. A diﬀerential form ω on G ⇒ M is called multiplicative if

∗ ∗ ∗ pr1ω − m ω + pr2ω = 0.

Deﬁnition 2.4.2. A symplectic groupoid is a pair (G, ω) such that G is a Lie groupoid and ω ∈ Ω2(G) is a multiplicative symplectic form. Remark 2.4.3. Much of the theory to be developed do not require ω to be non-degenerate. A multiplicative closed 2-form will be called multiplicative pre-symplectic form, in this scenario a pair (G, ω) will be called a pre-symplectic groupoid. Lemma 2.4.4. A symplectic form ω in a groupoid G is multiplicative if and only if the graph of the multiplication

(2) Γm = {(g, h, gh) ∈ G × G × G|(g, h) ∈ G }, 18 is a isotropic submanifold of G × G × G¯, where G¯ denotes the symplectic manifold (G, −ω).

Proof. The map (2) G → Γm, (g, h) 7→ (g, h, gh), ∗ ∗ ∗ is a diﬀeormorphism such that ω × ω × ω¯ pulls back to pr1ω + pr2ω − m ω. Then Γm is ∗ ∗ ∗ isotropic if and only if pr1ω + pr2ω − m ω vanishes.

Lemma 2.4.5. Let (G ⇒ M, ω) be a symplectic groupoid. Then:

i) 1(M) is an isotropic submanifold,

ii) Inversion is an anti-symplectomorphism

iii) The s-ﬁbers and t-ﬁbers are symplectically orthogonal, i.e. (ker Ds)ω ⊂ (ker Dt)

Proof. The proof of the 3 statements is very similar, we will only prove iii) and sketch the proof of the other assertions:

−1 −1 −1 iii) Let vs and wt be tangent vector of s (x) and t (x), at some point g ∈ s (x). −1 Let γ be a curve on s (x) with tangent vector vs since (γ(t), 1x, γ(t)) ∈ Γm it follows −1 (vs, 0, vs) ∈ T Γm. In the same fashion considering a curve η ∈ t (x) with tangent vector wt one concludes (0, wt, wt) ∈ T Γm. Multiplicativity implies

ω(0, vs) + ω(wt, 0) − ω(vs, wt) = 0,

thus ω(vs, wt) = 0.

i) Consider the curves γ, η in 1(M) with tangent vectors v, w ∈ Tm(1(M)). Then (γ, γ, γ), (η, η, η) ∈ Γm, inﬁnitesimally multiplicativity yields the desired result.

ii) Consider the curve γ() with tangent vector v at x ∈ M. Then (γ(), γ()−1, t(γ())) ∈ −1 Γm and (γ() , γ(), t(γ())) ∈ Γm, the inﬁnitesimal version together with isotropy of Γm yields the result.

It is clear that M, seen as a submanifold of G by the embedding 1 : M → G, and the source ﬁber s−1(x) are tranversal submanifolds of G implying

−1 ∼ TxG = TxM ⊕ Txs (x) = TxM ⊕ Ax,

for every x ∈ M. Thus T G =∼ TM ⊕ A and ω induces an isomorphism

ω˜ : T G =∼ TM ⊕ A → T ∗G =∼ T ∗M ⊕ A∗.

Since M is an isotropic submanifold, i.e. [˜ω(X)]|TM ≡ 0 for every X ∈ TM, it followsω ˜ restricts to a map ∗ R ω˜|TM : TM → A ; [˜ω(X)](α) = ω(X, αG ).

In a similar way, since target fibers are also tranversal to M it follows TxG = TxM ⊕ −1 Txt (x). By 2.4.5 iii), source fibers and target fibers are symplectically orthogonal, i.e.

19 ω˜(α)|A ≡ 0 for every α ∈ A, thusω ˜ restricts to a map

∗ ω˜|A : Ax → T M.

Lemma 2.4.6. The maps

∗ R ω˜|TM : TM → A ; [˜ω(X)](α) = ω(X, αG ),

∗ R ω˜|A : Am → T M; [˜ω(α)](X) = ω(αG ,X). are pointwise linear isomorphism.

Proof. Consider v ∈ TM such thatω ˜|TM (v) = 0. Again since M is isotropic, it implies that ∗ ω˜(v) ≡ 0 on T G. Then non-degeneracy of ω implies v = 0, i.e.ω ˜|TM is injetive. By a similar argumentω ˜|A is injective. The fact thatω ˜ is an isomorphism from T G to T G together with dimension counting implies both maps are isomorphisms.

Corollary 2.4.7. Let (G ⇒ M, ω) be a symplectic groupoid, then dim G = 2 dim M.

With this results we prove the equivalence of the given deﬁnition of symplectic groupoids and another characterization of symplectic groupoids, historically the original one, which is useful in some occasions.

Corollary 2.4.8. A symplectic form ω in a Lie groupoid G is multiplicative if and only if the graph of the multiplication Γm is a Langrangian submanifold of G × G × G¯.

Proof. We already proved Γ is isotropic, to show Γm is Lagrangian we count dimensions. Recall G(2) is a pulback manifold over M, thus

dim Γm = dim(G × G) − dim(M) = 2 dim(G) − dim(M) = 3 dim(M)

Implying dim(G × G × G) = 2 dim(Γm). Corollary 2.4.9. The s-ﬁbers and t-ﬁbers are each other symplectic complement, i.e. (ker Ds)ω = (ker Dt)

Proof. We already proved (ker Ds)ω ⊂ (ker Dt). Since s−1(x) =∼ t−1(x) via the inversion map, it follows they have the same dimension 1/2 dim(G). Counting dimensions

dim(ker Ds)ω = 1/2 dim(G) = dim(ker Dt),

it follows they are equal.

Symplectic Groupoids and Poisson Manifolds The main property of symplectic groupoids is their close relationship with Poisson manifolds. We will state and prove the theorems that concretely establish the relationship.

Theorem 2.4.10. Let (G ⇒ M, ω) be a symplectic groupoid. Then there exists a unique Poisson structure on M such that s : G → M is a Poisson map and t : G → M is an anti-Poisson map 20 Proof. Since s : G → M is a submersion, the pullback

s∗ : C∞(M) → C∞(G); f 7→ f ◦ s,

is an injective map. We will show s∗(C∞(M)) is invariant under the Poisson bracket on C∞(G), thus the restriction deﬁnes a Poisson bracket by

∗ −1 ∗ ∗ {f1, f2}M = (s ) {s f, s f}G.

By deﬁnition the deﬁned bracket makes s into a Poisson map and is clearly unique.

We will call a function f ∈ ι∗(C∞(M)) basic. It is clear that basic fuctions are (locally) constant on fibers, thus its differential lie in the annihilator of the tangent space to the fibers i.e. df ∈ Ann(Ds). Now

ω df ∈ Ann(ker Ds) ⇔ Xf ∈ (ker Ds) ,

where Xf is the Hamiltonian vector ﬁeld of f. Thus lemma 2.4.5 implies

f is basic ⇔ Xf ∈ ker Dt.

Since ker Dt deﬁnes a involutive distribution (it is integrable) then we conclude that ∗ ∞ [Xs∗f ,Ys∗g] = s h for some h ∈ C (M). This fact together with the equality

[Xf ,Xg] = X{f,g}G completes the proof.

Thus, a symplectic groupoid G ⇒ M induces a Poisson structure on M such that the algebroid A(G) =∼ A(T ∗M). On the other hand, if one start with Poisson manifold P one can deﬁne a Lie Algebroid T ∗P , if the Poisson manifold is integrable we will see G(T ∗M) is canonnically a symplectic groupoid. Therefore there is a one to one correspondence between integrable Poisson manifolds and s-connected symplectic groupoids.

To motivate this result we need to recall the construction of G(T ∗M): By definition ∗ ∗ ∗ G(T M) = P (T M)/ ∼T ∗M . We will endow P (T M) with a symplectic structure. Recall P (T ∗M) is Banach manifold with tangent space at a given by time dependent vector fields ∗ ∗ ∗ over a TaP (T M) = {U : I → T (T M)|U(t) ∈ Ta(t)(T M)}[27]. If ω0 is the canonical symplectic form on T ∗M a symplectic form on P (T ∗M) can be defined by

Z 1 ω(U1,U2)a = ω0(U1(t),U2(t))dt. 0 Then, one can show that the equivalence relation of A-paths is actually given by symplectic reduction procedure[31], thus G(T ∗M) has a canonical symplectic structure. Details of this construction can be found in [21, 32], where the following proposition is proved (Theorem 4.6]): Theorem 2.4.11. Let M be a integrable Poisson manifold and let Σ(M) be the s-simply connected groupoid integrating M. Then Σ(M) admits a canonical multiplicative symplectic form ω, so that (Σ(M), ω) is a symplectic groupoid. Moreover, the Poisson bracket induced on M by Σ(M) is the original one. 21 Examples

The following is a general construction for a general class of examples. Let G ⇒ M be a groupoid and let A → M be its associated algebroid. Deﬁne the cotangent groupoid ∗ ∗ T G ⇒ A with the the following structure maps: The source map is deﬁned by pulling G s back the one-forms on T to the algebroid A = T G|M i.e.

[s(α)](Y ) = α(DRg(Y )), [t(α)](Y ) = α(DLg(X − D1(#X))),

∗ ∗ ∗ where α ∈ Tg G, X ∈ At(g) and Y ∈ As(g). For α ∈ Tg G and β ∈ Th G the multiplication is given by α • β(T m(η, ξ)) = α(η) + β(ξ), where η ∈ TgG, ξ ∈ ThG and T m : TgG ×ThG → TghG is the diﬀerential of the multiplication ∗ map. At last the multiplicative symplectic form on T G is ωcan the canonical symplectic form on a cotangent bundle. The details concerning that this is indeed a symplectic groupoid can be found in [33]. Remark 2.4.12. In the previous section we saw a symplectic groupoid induces a Poisson structure on its base Theorem 2.4.10. In the example T ∗G → A∗ this Poisson structure can be described as follows: Let A → M be a Lie algebroid. Then A∗ (dual as a vector bundle) is a Poisson Manifold. Indeed consider the subsets Γ(b A) ⊂ C∞(A∗) and Cb∞(M) ⊂ C∞(A∗) given by, ∗ Γ(A) = {sˆ : A → R|sˆ(a) = aπ(a)(s ◦ π(a)) for s ∈ Γ(A)}, C∞(M) = {fˆ : A∗ → R|fˆ = π∗f for f ∈ C∞(M)}. One can deﬁne a Poisson bracket by,

{f,ˆ gˆ} = 0 {s,ˆ tˆ} = [ds, t] {s,ˆ fˆ} = ρ\(s)f.

It is worth noting that, although we motivated this Poisson structure by theorem 2.4.10, it is independent of the existence of groupoid and actually can be defined for any Lie algebroid A → M (c.f. [33]). In Chapter 4 we will focus in the geometric quantization of this family of Poisson structures, that is, integrable Poisson structures for which the integrating symplectic groupoid is a cotangent groupoid. Some particular examples of the previous construction are: Example 2.4.13. Trivial Poisson Bracket: Let M be a simply connected manifold. Consider M → M as a lie algebroid with zero dimensional fibers, this has integrable ∗ groupoid M ⇒ M. Now the cotangent groupoid is T M ⇒ M where s = t = π and the multiplication is given by addition on fibers. Moreover, the Poisson structure on M is the trivial Poisson structure. Example 2.4.14. Cotangent Bundles: Let N be a simply comnnected manifold. Con- sider the algebroid TN → N, this algebroid has integrable groupoid the pair groupoid ∗ ∗ ∗ N × N ⇒ N. Now the cotangent groupoid is T N × T N ⇒ T N with the pair groupoid structure. Moreover the Poisson structure on T ∗N is the one given by the symplectic structure on T ∗N. Example 2.4.15. Lie Algebras: Consider the algebroid g → {∗}, this algebroid has integrable groupoid G → {∗}, where G is the simply connected groupoid integrating g. The 22 ∗ ∗ contangent groupoid in this example T G ⇒ g is the groupoid of example 2.1.13 with its the natural symplectic form. Moreover, the Poisson structure on g∗ is the Kirillov-Poisson structure of g∗.

23 Chapter 3

Geometric Prequantization of Symplectic Groupoids

The ﬁrst step in the geometric quantization procedure is to construct a prequantization line bundle, or equivalently a principal S1-bundle over the symplectic manifold in consideration, together with a connection whose curvature is the symplectic form. The existence of such a bundle is equivalent to the symplectic form being integral, this is the prequantization condition for a symplectic manifold. If such a bundle exists it describes a Lie groupoid central extension of the pair groupoid of the symplectic manifold. In turn, such groupoid central extension describes an algebroid central extension.

In the general description of prequantization of Lie algebroids and Lie groupoids one starts with the infinitesimal data given by an algebroid extension, and find conditions to integrate such extension to a Lie groupoid extension. In the case such an algebroid extension is the one of the geometric quantization for symplectic manifolds, the integrability obstruction is the same as the prequantization condition. Our main interest in this description will be when the considered algebroid A(P ) comes from a Poisson manifold P , if the Poisson manifold is integrable, then one would seek to prequantize the symplectic groupoid G(P ) integrating A(P ) by finding a suitable extension.

3.1 Prequantization of Symplectic Spaces

Principal S1-Bundles We begin this section by introducing some of main objects in geometric quantization.

Deﬁnition 3.1.1. A principal S1-bundle is a ﬁber bundle L → M together with a smooth (left) action S1 × L → L such that:

i) The action of S1 preserves the ﬁbers of L, i.e. S1 ·π−1(m) ⊂ π−1(m) for every m ∈ M,

ii) S1 acts freely and transitively on ﬁbers, and

iii) L → M is locally trivial, i.e. there exists an open covering {Uα}α∈Λ together with a −1 1 collection of equivariant diﬀeomorphisms ηα : π (Uα) → Uα × S such that the following

24 diagram commutes −1 ηα π (Uα) Uα × G

π pr1 Uα

Deﬁnition 3.1.2. A base preserving S1-bundle isomorphism between (L → M) and (L0 → M) is a S1-equivariant diﬀeomorphism from L to L0.

1 Let L → M be a S bundle and {Uα, ηα}α∈Λ a trivialization for L → M, if Uβ is another trivializing open set such that Uα ∩ Uβ 6= ∅ we can consider the following diagram:

η−1 1 β −1 ηα 1 (Uα ∩ Uβ) × S π (Uα ∩ Uβ) (Uα ∩ Uβ) × S pr 1 π pr1 Uα ∩ Uβ

−1 1 1 The mapg ˜αβ := ηα ◦ηβ : Uα ∩Uβ ×S → Uα ∩Uβ ×S should preserve the ﬁrst component, 0 1 thusg ˜αβ(m, λ) = (m, λ ). Moreover, since the action of S is transitive and free on ﬁbers 1 0 there exist a unique gαβ(m) ∈ S such that λ = [gαβ(m)]λ. Therefore, the covering (Uα, ηα) 1 induces a set of transition functions gαβ : Uα ∩ Uβ → S which satisfy:

i)gαα = 1, ii)gαβgβα = 1, iii)gαβgβγgγα = 1. (3.1)

For a good covering {Uα}, a covering such that every intersection Uα1 ∩ · · · ∩ Uαn is contractible, the transition functions are actually an example of a element the first Chechˇ 1 1 1 1 cohomology Hˇ (M,S ). Actually for a good covering {Uα} an element in Hˇ (M,S ) can 1 be defined as a collection of functions cαβ : Uα ∩Uβ → S satisfying (3.1). For the definition of higher Chechˇ cohomology groups Hˆ •(M,A) of a manifold M with coefficients on an abelian group A, and some of its properties we refer to the reader to [34]. On the other ˇ 1 1 1 hand given a element [c] ∈ H (M,S ) one can find a S -bundle L[c] such that the transition functions defines the same element [c]. Fixing a good covering {Uα} the element [c] can be 1 represented by a collection of functions cαβ : Uα ∩Uβ → S satisfying (3.1), then the bundle L[c] can be constructed as a 1 L[c] = ( Uα × S )/ ∼, α∈

where (p, λ) ∼ (q, µ) if and only if p = q and [cαβ(p)] · λ = µ. The conditions imposed in (3.1) assure the quotient is a manifold with a principal S1-bundle structure. Thus we have sketched the proof pf Theorem 1.1.1 of [35]:

1 Proposition 3.1.3. Let M be a manifold and denote by PicS1 (M) the group of S -Bundles over M, known as the Picard group, then

∼ ˇ 1 1 PicS1 (M) = H (M,S ).

Connections and Curvature on S1-Bundles For our purposes we will require more geometrical insight of S1-bundles, namely the notion of connection and curvature. Since the action of S1 on fibers is free and transitive it estab- 25 1 −1 lishes a diffeomorphism between S and the fiber Lm = π (m) for every fiber. Moreover 1 ∼ we have an identification of the tangent spaces TeS = R and TpV := ker dπ, given by the derivative of the action, explicitly d #: R → T V ; #(X, p) = (eiXt · p)| = X#. p dt t=0 p With this identification we have the following exact sequence of vector bundles,

TV TL Dπ TM.

In the above sequence the vertical space TV is naturally embedded in TL, however there is not a canonical way of choosing a “horizontal” complement for TV inside TL. Such a complement can be chosen with help of a connection form.

Deﬁnition 3.1.4. A connection form θ on a principal S1-bundle L → M is a element of Ω1(L) such that:

∗ ∗ i) Rλθ = θ, where Rλ denotes the action of λ on L,

# 1 ii) θ(X ) = X, for every X ∈ TeS . Proposition 3.1.5. The map # ◦ θ : TL → TV is a splitting of the exact sequence

TV TL Dπ TM.

Proof. We just need to check (# ◦ θ) ◦ d(ιp) = IdTV , where ιp is the inclusion of the ﬁber π−1(p) in L. Take some Y˜ ∈ TV represented by Y˜ = Y # with Y ∈ g, then

# # # (# ◦ θ) ◦ d(ιp)(Y ) = #[θ(Y )] = #(Y ) = Y .

Now, one important use of the identification TL = TV ⊕ TM is that it lets us lift (uniquely) vector fields X ∈ X(M) to vector fields H(X) ∈ X(L) by the requirement that the vertical component of H(X) be zero.

Proposition 3.1.6. Let θ be a connection form and X ∈ X(M) then there exists a unique vector ﬁeld H(X) ∈ X(L) characterised by the following properties properties: i) dπ(H(X)) = X, ii) θ(H(X)) = 0. This vector ﬁeld is called the horizontal lift of X by θ.

Proof. For the existence, consider some Y ∈ L such that dπ(Y ) = X (such a Y exists because π is a submersion). One checks by direct computation Y − θ(Y )# satisﬁes the given conditions. To prove it is unique notice that, by the splitting TL = TV ⊕ TH, a vector Y ∈ TL that vanishing both on θ and dπ should be zero. Since the diﬀerence of two vectors H1(X), H2(H) satisfying the previous conditions vanishes both under θ and under dπ, it follows H1(X) = H2(H).

26 Deﬁnition 3.1.7. Let L → M be a principal S1-bundle and θ ∈ Ω(L, g) be a connection one form on L. The curvature Ω of θ is element of Ω2(L) deﬁned by

Ω = dθ.

Having introduced our basic objects of study we are ready start our discussion on prequantization.

Prequantum Bundles and Prequantum Hilbert Spaces The idea of geometric quantization (at least in its beginning) is to construct a Hilbert space together with a representation of the Poisson algebra of smooth functions on a symplectic manifold on this Hilbert space. The Hilbert space will turn out to be some completition of a space of the sections of a line bundle. The ﬁrst step in a geometric quantization procedure is to construct such a prequantization line bundle, or equivalently a principal S1-bundle, with a connection whose curvature is the symplectic form. In this section we review facts about the existence and uniqueness of such a bundle.

Deﬁnition 3.1.8. A symplectic manifold (M, ω) is prequantizable if there exists a principal S1-bundle L −→π M with connection form θ such that dθ = π∗ω. Such a pair (L → M, θ) is called a prequantum bundle.

Example 3.1.9. Let M be a manifold and T ∗M its the cotangent bundle. The cotangent bundle is always a symplectic manifold with symplectic form ω = dθ0, where θ0 is the Liouville form. Now, (T ∗M, ω) is prequantizable by a trivial prequantum bundle (S1 × ∗ π 1 T M −→ M, dθ0 + πS1 ∗ θL), where θL is the Maurer-Cartan form on S defined by θL(Y ) = 1 DRλ−1 (Y ) for Y ∈ TλS . The above example enlightens the role of the connection. In a physical system the cotangent bundle T ∗M is the phase space for a configuration space M, the coordinates on this space can be classified on position coordinates qi, given by coordinates in M, and ∗ (local) momentum coordinates pj, given by coordinates along the fibers of T M. The i Liouville form with respect to those coordinates is θL = pidq , thus the Liouville form can be used to differentiate along position coordinates. In the general case of a symplectic manifold the connection one form plays the role of the Liouville form, that is, a first approach of choosing how to differentiate along the position coordinates.

Back to our main topic, given a prequantum bundle (L → M, θ) for (M, ω) we can consider the vector space of smooth S1-invariant functions

∞ S1 C ¯ C (L)0 := {h : L → |f(λ · p) = λf(p) and π(Suppf) is compact.}.

This set will be called the space of compactly supported sections and will be denoted by Γ(L). It is called the space of sections because it is isomorphic to the space of section of the associated vector bundle to L[4]. We will study further the relationship between principal bundle and associated line bundles later in the chapter.

Since the pullback π∗f of a function f ∈ C∞(M) is S1 invariant it follows that (π∗f)·h ∈ ∞ G ∞ G ∞ ∞ G C (L)0 , i.e. the space C (L)0 is naturally a C (M)-module. We can endow C (L)0

27 with an inner product, Z hϕ, ψi = ϕ(m)ψ(m)ωm. M Remark 3.1.10. Notice ϕ(λ · p)ψ(λ · p) = λλϕ¯ (p)ψ(p) = ϕ(p)ψ(p), thus ϕ(·)ψ(·) deﬁnes a function on M, which can be integrated by using the symplectic volume form ωn = ω∧· · ·∧ω.

∞ G The completion of the inner product space (C (L)0 , h·, ·i) is the desired prequantum Hilbert space Hω. Although not entirely important for the future discussions, for sake of completeness, we turn to the task of constructing a representation of the Poisson algebra ∞ ∞ C (M), i.e. a linear map Q : C (M) → End(H), such that [Qf ,Qg] = Q{f,g}. Deﬁne

∞ Q : C (M) → End(Hω)

Q(f)(ϕ) = H(Xf )ϕ − f · ϕ,

where Xf is the Hamiltonian vector ﬁeld of f.

∞ Proposition 3.1.11. (Lemma 3.9 of [36]) The assignment Q : C (M) → End(Hω) is a morphism of Lie algebras.

Example 3.1.12. In example 3.1.9 we saw that T ∗R was prequantizable by a trivial bundle 1 ∗ π ∗ ∞ ∗ (S × T R −→ M, dθ0 + π θL). Thus the space of sections is Γ(L) = C (T R, C) and the inner product is the usual inner product of C-valued functions. We conclude the prequantum Hilbert space is L2(R2).

Prequantum Line Bundles: Existence and Uniqueness In the previous section we constructed a Hilbert space together with a representation of the Poisson algebra C∞(M) of a prequantizable symplectic manifold (M, ω). In this section we will deal with the problem of existence and uniqueness of prequantum S1-bundles.

To begin with we will introduce the classification problem of S1-bundles over a fixed manifold M. By the discussion in section 3.1, S1-bundles over a manifold M are characterized by their transition functions, i.e. by an element in the first Cechˇ cohomolgy group exp Hˇ 1(M,S1). The exact sequence of groups 2πiZ → R −−→ S1, induces a long exact sequence in Chechˇ cohomology,

· · · → Hˇ 1(M, C) → Hˇ 1(M,S1) → Hˇ 2(M, 2πiZ) → Hˇ 2(M, C) → Hˇ 2(M,S1) ···

Moreover, since H•(M, C) admits partition of unity it follows Hn(M, C) = 0 for n > 0 [34]. These two facts imply:

Proposition 3.1.13. Let M be a manifold, then

Hˆ 1(M,S1) =∼ Hˆ 2(M, Z).

Given a connection on a S1-principal bundle, and considering the equivalence between the deRham cohomology and the real Chechˇ cohomology H2dR(M) =∼ Hˇ (M, R), one can realize the isomorphism map as the chern class of the curvature. In the case of a prequantizable space this implies the necessary condition ω ∈ H2(M, 2πiZ), where the latter

28 cohomology group is the image in the deRham cohomology of Hˇ 2(M, 2πi) under the afore- mentioned isomorphism. A detailed discussion of the previous paragraph can be found on [34], an explicit construction of the objects and isomorphisms mentioned can be found in [4].

On the other hand, if ω ∈ H2(M, 2πiZ) then one can associate to it a S1-bundle π : L → M via the isomorphism of proposition 3.1.13. A theorem by Weil (Proposition 2.1.1 of [4]) asserts such a bundle can always be endowed with a connection θ such that dθ = π∗ω.

Theorem 3.1.14. A symplectic manifold (M, ω) is prequatizable if and only if ω ∈ H2(M, 2πiZ).

In practice, given the pairing between deRham cohomology and singular (co)homology given by Z n R n R n ∗ H (M, ) ⊗ HdR(M) → ; [(δ : ∆ → M), θ] 7→ δ θ, ∆n one can check ω ∈ H2(M, 2πiZ) if and only if the integral above has values in 2πZ([37, Theorem 18.14]). Then the theorem 3.1.14 can be restated:

Proposition 3.1.15. A is prequatizable if and only if

P(M) ⊂ 2πZ.

This is, up to a constant, exactly the condition for the algebroid Aω of example §2.3.13 to be integrable.

The previous theorem concludes the discussion concerning existence of prequantum bundles over a symplectic manifold (M, ω). The discussion of uniqueness is a bit more subtle, and goes as follows. We already saw that given the curvature ω ∈ HdR(M, 2πiZ) there is one S1-bundle, up to S1-bundle isomorphism, with the curvature ω. However if we want to distinguish between prequantization (L → M, θ) and (L0 → M, θ0) we also require to distinguish the S1-bundle up to connection preserving S1-bundle isomorphism, i.e. a bundle diffeomorphism τ : L → L0 such that τ ∗θ0 = θ. Given a loop γ : S1 → M, using the connection one can lift the vector field it defines to a vector field in L over γ, then the integral curve of this vector field defines a horizotal lift ofγ ˜θ : L → M. Sinceγ ˜(0) andγ ˜(1) belongs to the same fiber their difference is an element Q(γ) ∈ S1 called the holonomy of γ. It turns out this holonomy classifies S1-bundle with connection ([4, 1.12.3]). Now, for null-homotopic γ it holds Z Z Q(γ) = exp(−i θ) = exp(i ω) = 1, γ ∆ where ∆ is a surface with boundary γ. Since ω is integral then null-homotopic curves have fixed holonomy. We arrive to the conclusion:

Proposition 3.1.16. Let (M, ω) be a simply connected symplectic manifold. Then there exists, up to connection preserving S1-bundle morphism, a unique prequantization (L → M, θ) of (M, ω).

29 The Kostant Picture of Prequantization We end this section with a remark about the vector bundle picture of prequantization, also known as the Kostant picture. What we have described so far is the principal bundle description of prequantization, also known as the Souriau picture. There is another description making use of line bundles instead of S1-principal bundles, in this description the prequantization bundle is given by a Hermitian, line bundle with an compatible affine connection. A Hermitian line bundle (L → M, h·, ·i) is a one dimensional complex vector bundle with and fiberwise Hermitian inner product, and an affine connection is a TM-connection ∇ on L → M, such that X(hs, ti) = hs, ∇xt, i + h∇xs, ti, for all s, t ∈ γ(L) and X ∈ X (M). The two pictures are related by the relationship between principal bundles and associated bundles. Given an S1-bundle L → M and the natural representation χ : S1 → GL(C) = C×, λ 7→ λ, 1 one can compose the transition functions {gαβ : Uα → S } of L → M to obtain transition function {χ ◦ gαβ : Uα → C}. In the same way as described in section 3.1, these transition functions can be used to construct a fiber bundle with typical fiber C. The described bundle is known as the associated bundle A(L, C, χ) of L with respect to χ. This bundle can 1 also be described as the quotient L × C/ ∼, where (s, λ) ∼ (s · µ, χµ−1 λ) for s ∈ L, µ ∈ S , λ ∈ C. It is worth mentioning that in this description the set of sections of the line bundle is isomorphic to the space we called the sections of the prequantum S1-bundle. For a detailed discussion about the equivalence of objects in the principal bundle description and in the associated bundle description we refer to the reader to [4].

3.2 The Groupoid and Algebroid Picture

In this section we will generalise the previous discussion from prequantization of symplectic manifolds to prequantization of symplectic groupoids. By using the integration theory of Chapter 2, we will recover the prequantization uniqueness and existence conditions showing how prequantizability can be understood as an integrability problem. Together with the sympliﬁcation of Poisson manifolds discussed in Chapter 2 this will allow us to generalise the prequantization procedure to (integrable) Poisson manifolds.

Prequantum Symplectic Groupoid Extension We now describe how groupoids and algebroids appear in the picture. Suppose (L → M, θ) is the prequantization of a simply connected symplectic manifold M. We will show how the prequantum bundle induces a central Lie groupoid extension.

1 Deﬁnition 3.2.1. Let G ⇒ M be a groupoid and let S := S ×M endowed with a ﬁberwise groupoid structure. A central S1-Lie groupoid extension of G is a exact sequence of groupoids morphism over M, S −→ι G˜ −→Gπ , such that ι is an embedding, π a submersion and ι(γ)g = gι(γ) for every g ∈ G and γ ∈ S.

30 The sequence is exact in the sense that the kernel of the morphism π

ker π = {g ∈ G|˜ π(g) = 1x for some x ∈ M}

is SM It is worth noting that for every g ∈ ker π sG˜(g) = tG˜(g) = sG(π(g)) = x for some x ∈ M, therefore the kernel of a groupoid morphism has the structure of an Lie group bundle with base M.

Proposition 3.2.2. Let S → G˜ −→Gπ , be a central extension of Lie groupoid then π : G˜ → G is a S1-bundle over G .

1 Proof. By denoting λm the element (λ, m) of S. One can deﬁne the action of S on G˜ by

λ · g = ι(λt(g))g,

where the latter is the multiplication in the groupoid G˜. This action preserves ﬁbers since

π(λ) = [π ◦ ι(λt(g))]π(g) = 1t(g)π(g) = π(g).

0 ˜ 0 0−1 0 −1 Furthermore, if g, g ∈ G satisfy π(g) = π(g ) then π(gg ) = π(g)π(g ) = 1t(g). Hence, 0−1 0−1 0 gg ∈ ker π and there exist some λt(g) such that gg = λt(g), i.e. g = λ · g . Since ι is injective such λ is unique if we fix the fiber. We conclude the action of G is free and transitive on fibers.

Back to the prequantization picture, let (L → M, θ) be a prequantization of a simply connected symplectic manifold (M, ω). Associated to the S1-bundle (L → M) there is a gauge groupoid Gau(L) §2.1.12. Since L projects to M in an S1-invariant fashion there is a map to the pair grupoid,

π × π : Gau(L) → M × M, π × π([p, q]) = (π(q), π(q)).

This map is a groupoid morphism and its kernel is given by equivalence classes [p, q] such that π(p) = π(q). An equivalence class is uniquely determined by a π(p) ∈ M and a 1 λ[p,q] ∈ S such that p = λ[p,q] · q. For another pair of representatives it holds

λ[t·p,t·q]t · q = t · p = tλ[p,q] · q = λ[p,q]t · q,

then effectiveness of the action on the fibers implies λ[p,q] = λ[t·p,t·q], i.e. λ[p,q] is well defined. We conclude ker π =∼ S1 ×M, and that the embedding into Gau(L) is given by λ 7→ [λ·p, q]. Moreover, since [λ · p, p][p, q] = [λ · p, q] = [λ · p, λ · q][λ · q, q] = [p, q][λ · p, p] we have proved:

Proposition 3.2.3. The sequence

S −→ι Gau(P ) −−−→π×π M × M, is central S1-extension of Lie groupoids.

The correspondence between S1-extensions of the pair groupoid and S1-bundles is even more profound, for instance the geometric information of the symplectic form ω and the connection θ can be incorporated in the previous S1-extension as multiplicative forms.

31 Proposition 3.2.4. There is a one to one correspondence between S1−bundles over M and S1 extensions of M × M, S −→ι Gau(L) −−−→π×π M × M. Moreover, given a S1 bundle L there are one to one correspondences between: i) Closed 2-forms ω on M and closed multiplicative 2 formsω ˜ on M × M ii) Connections on L and multiplicative 1-forms θ˜ on Gau(L), ∗ ∗ ˜ iii) the relation πP ω = dθ is equivalent to π ω˜ = dθ. Proof. In this case multiplicativity of 1-forms on Ω(M × M) can be rewritten as

ωm[(X,Y )] = ωm[(X,Z)]. − ωm[(Y,Z)]

For higher forms a similar relation holds component-wise. In what follows we will prove statement for 1-forms which generalize directly to k-forms. Recall s(x, y) = x and t(x, y) = y. Deﬁne a bijection between k-forms Ωk(M) and multiplicative k-forms Ωm(M × M) by

k ∗ ∗ Ω (M) 7→ Ωm(M × M), ω 7→ s ω − t ω

k ∗ Ωm(M × M) 7→ Ω(M), ωm 7→ ιxωm, ∗ where ιx : M → M × M is deﬁned by ιx(y) = (y, x) for x ∈ M. Notice that ιxωm(X) = ∗ ∗ ωm(X, 0) = ιyωm(X), so the above map does ιxω not depend on x. Now since (s ◦ ιx)∗(X,Y ) = (X, 0) and (t ◦ ιx)∗(X,Y ) = (Y, 0) it follows

∗ ∗ (s ◦ ιx) ωm[(X,Y )] − (t ◦ ιx) ωm[(X,Y )] = ωm[(X, 0)] − ωm[(Y, 0)],

∗ ∗ multiplicativity implies (s ◦ ιx) ωm − (t ◦ ιx) ωm = ωm. Likewise, since (s ◦ ιx)∗ = IdTM and (t ◦ ιx)∗ = 0, then ∗ ∗ ∗ ∗ ι (s ω − t ω) = (s ◦ ιx) ω = ω. Concluding the proof that the given maps establish a bijection.

i) This is the previous discussion in the particular case of 2-forms.

ii) Multiplicative 1-forms on L × L are in bijection with 1-forms on L. Moreover a one form θ ∈ Ω(L) is S1-invariant if and only if factors through the quotient to a one form θ˜ in Gau(L) = (L × L)/S1.

iii) It is clear from the discussions in ii) and the given map between multiplicative forms on M × M and forms on M. One can generalize the previous picture to other symplectic groupoids leading to the general definition of prequantization. Definition 3.2.5. Let G ⇒ M be a groupoid and ω a multiplicative 2-form on G.A prequantization of (G, ω) is a central extension of Lie groupoids (R → G ⇒ M) together with a multiplicative 1-form which is a connection form for the principal bundle G˜ → G that satisfies π∗ω = dθ In the case of integrable Poisson manifolds one says one has a definition for prequantizability via its s-connected symplectic groupoid. 32 Definition 3.2.6. A integrable Poisson manifold is called prequantizable if its symplectic groupoid G(P ) is prequantizable.

Algebroid Extension Associated to a Groupoid Extension Having described the relationship between the usual prequantization and the groupoid picture we now turn to describe the inﬁnitesimal counterpart of the situation. Just as in the case of groupoids the description is focused around extensions.

Deﬁnition 3.2.7. A Lie algebroid extension of A by R is short exact sequence of algebroids ι π RM −→ A˜ −→ A, where RM is the trivial line bundle. Such an extension is called central if

[˜α, ι(f)] = L#αf,

∞ for allα ˜ ∈ A˜, f ∈ Γ(RM ) = C (M). Just as in the case of Lie algebras, extensions of Lie algebroids are classiﬁed by the a second cohomology class of the deRham algebroid cohomology. Given a cocycle c ∈ H2(A) one can extend the algebroid A using the cocycle, explicitly,

R → Ac → A, where Ac = A ⊕ R is endowed with the bracket,

[α ⊕ f, β ⊕ g]c = [α, β] ⊕ (L#αg − L#βf + c(α, β)),

and anchor #c = # ◦ prA. Proposition 3.2.8. The Lie algebroid central extensions of A by R are classiﬁed by the deRham algebroid cohomology H2(A).

2 Proof. Let c ∈ H (A), we will prove Ac is indeed and algebroid. The only non clear step in this assertion is checking that the bracket [·, ·]c satisﬁes the Leibniz rule:

[α ⊕ f, h(β ⊕ g)]c = [α, hβ] ⊕ (L#αhg − L#hβf + c(α, hβ))

= (h[α, β] + (#α)hβ) ⊕ (hL#αg + gL#αh − hL#βf + hc(α, β))

= h ([α, β] ⊕ (L#αg + L#βf + c(α, β))) + (#αh)β ⊕ gL#αh

= h([α ⊕ f, β ⊕ g]) + [#c(α ⊕ f)h](β ⊕ g).

2 Now, we check the extension depends on the cohomology class. Let dAb ∈ H (A) be a boundary term, we will show it induces an isomorphic extension. Indeed if dAb is a boundary term then,

dAb(α, β) = L#αb(β) − L#βb(α) − b([α, β]),

for some b ∈ H1(A). A straightforward (but tedious) calculation shows the map ϕ(α⊕f) = α ⊕ (f − b(α)) is an isomorphism of extensions.

33 At last, we will show that an extension R → A˜ → A induces a cocycle c ∈ H2(A). Since the previous extension is in particular a vector bundle extension, there exist a linear splitting σ : A → A˜. Again a straighforward (and also tedious) computation shows,

c(α, β) = σ([α, β]) − [σ(α), σ(β)], is a 2-cocycle.

Since the Lie algebroid A(G) is a sub-bundle of the tangent bundle T G any closed multilplicative 2-form on G induces (by restriction) a 2-cocycle cω on A

cω(α, β) = ω(α, β).

Thus given a closed multiplicative 2-form on a groupoid G, one can consider the algebroid

Acω given by the extension of A(G) by the cocycle cω. In the case the groupoid is prequan-

tizable, we will show the algebroid Acω is integrable. Proposition 3.2.9. Let (G, ω) be a symplectic groupoid, then any prequantization ˜ groupoid (G → G ⇒ M) of (G, ω) integrates Acω . Proof. Differentiating the extension given by the prequantization (G˜, θ˜) we obtain an extension R → A˜ −−→Dπ A, where A˜ = A(G˜). Moreover, the one form θ˜ defines (by restriction) ˜ ∗ ∗ a one cocycle lθ on A, and the condition π ω = dθ restricts to π cω = dA˜lθ . ˜ ∼ ˜ We claim A = Acω , implying G integrates Acω . Define the map

Θ: A˜ → A ⊕ R, α 7→ (dπ(α), l(α)).

This map is an isomorphism. Indeed, since G˜ is a S1-bunde over G then T G˜ = T G ⊕ R, where the isomorphism is given by the splitting deﬁned by the connection one form. Recall A˜ = T sG˜, because the groupoid multiplication commutes with the S1-action it implies the splitting restricts to T sG =∼ T sG ⊕ R, this restriction is Θ.

3.3 Quantization of Poisson Manifolds

Lets recall the panorama, given a symplectic manifold (G, ω) one can associate to it the symplectic groupoid (M × M, ω˜ = ω × ω), which in turn describes a Lie algebroid extension R → Acω → TM. In the case (M, ω) is prequantizable, say by (P, θ), the previous algebroid extension can be integrated to a groupoid extension S1 → Gau(P ) −→π M ×M, which carries an invariant multiplicative form θ˜ such that π∗ω˜ = dθ˜. Moreover, we saw that the necessary and suﬃcient condition for (M, ω) to be prequantizable is equivalent to the integrability condition of the algebroid extension Aω. We will see how we can recover the prequantum data from the integration of the Lie algebroid extension concluding that, in the general scenario, prequantization can be seen as an integration problem.

Integrability of Algebroid Extensions In this section we will use the theory of integration of Lie algebroids introduced in chapter 2 to solve the integrability problem of extensions of integrable Lie algebroids by 2 cocycles. 34 For this, let A be a algebroid and c ∈ H2(A) be a 2-cocycle, we will ﬁnd a explicit description for Ac-paths and Ac-homotopies in terms of A-paths and A-homotopies. By deﬁnition a Ac-path over γ : I → M is a path

ac : I → A ⊕ R; ac = (a, f); for a : I → A, f : I → R,

d satisfying dt γ = #ac = #a. That is a Ac-path is a couple (a, f) where a : I → A is an A-path and f : I → R is a function. Now, the condition for a variation of Ac-paths

(a, f): I × I → A ⊕ R,

to be a Ac-homotopy is the condition that unique (b, h): I × I → A ⊕ R making

(a, f)dt + (b, h)d : TI × TI → A ⊕ R into a algebroid morphism satisfies (b(, 1), h(, 1)) = (0, 0). Thus it breaks into two conditions: first b(, 1) = 1, i.e. a should be an A-homotopy; and second, h(, 1) = 0 for every ∈ [0, 1]. The second condition can be computed explicitly, recall that given a TM- connection on an algebroid the condition that (a, f)dt + (b, h)d : TI × TI → A ⊕ R is an algebroid morphism can be rewritten as a differential equation. In this scenario considering the connection (∇, ∂), where ∂ is the canonical flat connection in R, the condition in this component becomes d d h − f = c(a, b), h(, 0) = 0. dt d Integrating we find 0 Z t df h(, t0) = − c(f, h)dt; 0 d Z 1 df Z 1 h(, 1) = 0 ⇒ dt = c(a, b)dt. 0 d 0

Thus, the condition for (a, f) to be an Ac-homotopy is equivalent to a being a A-homotopy and f : I × I → R to satisfy the previous equation.

Proposition 3.3.1. Any Ac-path (a, f) is Ac-homotopic to a path (a, r) where a is an A-path and r ∈ R. Moreover, any two such paths (a0, ro) and (a1, r1) are homotopic if a ∼ a as A-paths and 0 1 Z r1 − r0 = c(a, b)ddt. I2 R 1 Proof. Consider the variation of Ac-path between (a, f) and (a, 0 f) given by Z 1 f = f + (1 − ) fdt. 0 This variation is a A-homotopy, indeed

Z 1 df Z 1 Z 1 Z 1 Z 1 dt = (f − fdt)dt = fdt − fdt = 0, 0 d 0 0 0 0 since c(a, a) = 0 the above equality is the condition to be a Ac-homotopy.

35 R Now if r1 − r0 = I2 c(a, b) then it is clear f = r0 + (1 − )r1 is a homotopy between the paths.

Now, if A → M is the Lie algebroid of a s-connected groupoid G we will describe a 2 ∗ s topological extension of G by the Weinstein groupoid of Ac. Since c ∈ Γ(Λ (1 T G)) one x 2 −1 can extend this to a 2-form ωc ∈ Ω (s (x)) by

x ωc (Xg,Yg) = ct(g)(DRg−1 (Xg),DRg−1 (Yg)).

And deﬁne the period bundle by Z ∗ x 2 −1 −1 P(c) → M, Px(c) := γ ωc | (γ : S → s (x)) ∈ π2(s (x)) . S2 One can also deﬁne the bundle

c R S(c) → M, Sx = /Px(c).

Notice that the period bundle is fiberwise countable, thus if it is a smooth embedded sub-bundle of R then necessarily the fiber is a discrete subgroup. Moreover this implies S is a smooth (trivial) bundle of abelian groups with fiber S1.

Example 3.3.2. Let (M, ω) be a simply connected symplectic manifold. The algebroid extension corresponding to the symplectic groupoid M × M is the algebroid AωTM ⊕ω R described in example §1.4.13. Since TM is integrable by M × M then TM-homotopies are given by differentiating homotopies h : I × I → M × M which are characterised by homotopies on M. Because M is simply connected any two paths with the same starting point and ending point are homotopic. Summarizing, G(TM) is the the set of differentials of paths I → M where two paths are homotopic if they have the same starting and ending points. Now Ac-paths (γ, r0), (γ, r1), where γ ∼ η i.e. γ(0) = η(0) = x and γ(1) = η(1) = x, are Ac-homotopic if and only if Z ω = r0 − r1, r0 − r1 ∈ P(M), ∇ where ∂∇ = γ − η. If the periods satisfies P(ω) ⊂ 2πZ then they describe a line bundle M × R/P(ω).

The previous example can be taken to a general setting. In general by the from of the A-paths on A ⊕ R the monodromy groups of G will be given by the a direct product the monodromy groups of G and the periods bundle. Since the groupoid G is integrable the condition of A⊕R being integrable is equivalent to the monodromy groups being uniformly discrete. Now if the bundle of periods is an embedded sub-bundle of R × M it would imply the monodromy groups are, indeed, uniformly discrete and 2.3.8 would imply A ⊕ R is integrable. We have sketched:

Proposition 3.3.3. Let G be a s-connected Lie groupoid with algebroid A. Suppose c ∈ 2 H (A) is a 2-cocycle and let R → Ac → A be the extension induced by such cocycle. Then there exist a extension of topological groupoids

S(c) → G(Ac) → G.

36 This extension is a central extension of Lie groupoids if and only if P(c) is a smooth sub- bundle of R.

From Integrability to Prequantizability: Existence and Uniqueness Back to the prequantizability problem, just as in the case of symplectic manifolds, the existence of S1-extensions implies the existence of a 1-form turning it into a prequantum extension. For this we use the auxiliary result by Crainic (Theorem 4.2 of [38]):

Theorem 3.3.4. Let G be a s-simply connected Le groupoid, let ω ∈ Ω2(G) be a closed 2 multiplicative 2-form, and let cω ∈ H (A) be the algebroid 2-cocycle induced by restriction of ω. Then there exist a multiplicative 1-form θ satisfying dθ = ω if and only if cω is an exact algebroid cocycle.

To show the extension of proposition 3.3.3 is a prequantum extension we will prove ∗ 2 ∗ the 2-cocycle π (c) ∈ H (Ac) coming from π ω ∈ G(Ac) is exact. Consider the algebroid Aω = A ⊕ R, then the projection on the second component lc(α, λ) = λ is a one form on A ⊕ R which satisﬁes

dA(lc)((α, f), (β, g)) = L#αlc(β, g) + L#βlc(α, f) − lc([(α, f), (β, g)])

= L#αg + L#βf − L#αlcg − L#βf + c(α, β) = π∗c((α, f), (β, g))

Theorem 3.3.5. Let (G, ω) be a s-connected symplectic groupoid with algebroid A. Then G is prequantizable if and only if the bundle of periods P(c) is a smooth sub-bundle of R, where C ∈ H2(A) is the 2-cocycle induced by ω. Moreover the prequantization is unique.

Proof. One on hand by proposition 3.2.9 the algebroid extension R → Ac → A is integrable, thus theorem 3.3.3 implies the bundle of periods is smooth. On the other hand, if the bundle of periods is smooth then theorem 3.3.3 implies the extension A → Ac → R is integrable to ∗ S(c) → G(Ac) → G. Moreover, since π (c) = dA(l) theorem 3.3.4 assures the existence of a ∗ 1-form θ such that π ω = dθ. Notice the groupoid G(Ac) extending G is unique given G is s-simply connected. In addition the 1-form θ described is unique up to boundary terms.

For a integrable Poisson manifold P , we consider the s-simply connected symplectic groupoid integrating P and carry the discussion of prequantizability with this groupoid. We end the prequantization discussion with an example.

∗ Example 3.3.6. Cotangent groupoid: Consider the cotangent groupoid (T G, ωcan). Since there exist θ ∈ Ω(G) such that dθ = ω then the algebroid cocycle cω is trivial, up to boundary terms. Proposition 3.2 implies the algebroid extension A0(G) = A(G) ⊕ R has no twist in the bracket. From example ?? the groupoid T ∗G × S1 integrates the algebroid A0(G). It worth noting that in this case Z Z ω = dθ = 0, S2 S2 ∗ and the periods of bundle is trivial. Thus the set of A0-paths is isomorphic to T G × R which is exactly the s-connected groupoid of G ×S1. This implies that in the case the period bundle is trivial one should consider the groupoid T ∗G × S1 instead of the one given in the theorem 3.3.5. Now following the symplectic manifold case the prequantization of T ∗G is 37 ∗ 1 ∗ 1 (T G × S , dθ0 + π θL), where θL is the Maurer-cartan form on S . The multiplicativity of this connection follows from the multiplicativity of dθ0 and the multiplicativity of θL.

Fell Line bundles We end this chapter by discussing the Line bundle picture on groupoid S1-extensions. We saw groupoids S1-extension are in particular S1-bundle over G, which itself had a multiplicative structure. The concept that formalises the multiplicative structure in the Kostant picture is that of a Fell line bundle:

Deﬁnition 3.3.7. Suppose π : L → G is a line bundle over a groupoid G. Let

(2) ∗ ∗ L := pr1L ⊗ pr2L, thus a generic ﬁber of L(2) is ﬁnite linear combination of elements of the set {s ⊗ t ∈ L ⊗ L|(π(s), π(t)) ∈ G(2)}. We say L → G is a Fell line bundle over G if there is an continuous multiplication bundle morphism m : L(2) → L, and an involution s 7→ s∗ such that: i) π(m(s, t)) = π(s)π(t), ii) π(b∗) = π(s)−1, iii) m(s, t)∗ = m(s∗, t∗), for all s, t ∈ L. We will normally suppress the map m, and simply write st in place of m(s, t).

We will show the associated bundle A = (L × C)/ ∼ to a S1-central extension L → G of G is a Fell line bundle. For this consider the multiplication and involution

[s, λ][t, µ] := [st, λµ], [s, λ]∗ = [s−1, λ¯],

for all s, t ∈ L and λ, µ ∈ C. It is easy to check that these operation endow A → G with a Fell line bundle structure, in fact all the properties of a Fell line bundle follow from the properties of groupoid structure on L. We only have to check the maps are well deﬁned. Let [sz, zλ¯ ] be another representative of the class [s, λ], where z ∈ S1, then

[sz, zλ¯ ][t, µ] = [szt, zλµ¯ ] = [stz, zλµ¯ ] = [st, λµ],

where we used the fact that L → G is a central S1-extension. A similar calculation shows the involution is also well deﬁned.

Now the multiplicativity of the connection on a S1 extension is translated in an aﬃne connection ∇ on L → M satisfying the Leibinz rule

∇Dm(X)st = [∇Dpr1(X)s]t + s[∇Dpr2(X)t],

for any X ∈ T G(2) and s ⊗ t ∈ Γ(L(2)).

38 Chapter 4

Quantization of Poisson Manifolds

The last step in a geometric quantization procedure is to cut down the space of sections of the prequantum line bundle constructed in the chapter 3. This is done by using a polarisation on the symplectic manifold, in the good case scenario the reduced space of sections can be described as the space of sections of another line bundle. Afterwards, with the use of half-densities we will deﬁne an inner product in this new space of sections in order to complete it to the desired quantum Hilbert space of the symplectic manifold. Through the general discussion the example of the quantization of the cotangent bundle will motivate and illustrate the main concepts and constructions.

4.1 Quantization of Symplectic Manifolds

Again, as in the previous chapter, we begin our discussion for the case of symplectic manifolds and work our way to the general notion for symplectic groupoids. Until now we have constructed a prequantum Hilbert space out of a prequantum line bundle. However this Hilbert space is not the right one; in some sense it is to big to represent the quantization of the symplectic manifold (M, ω). Considering the symplectic manifold (M, ω) as the phase space of some physical system the quantum Hilbert space has information of both the configuration variables and the momenta variables. Since in quantum mechanics the Hilbert space has information just of the configuration variables the prequantum Hilbert space is twice as big as the desired Hilbert space. Thus we need a way of choosing configuration and momenta variables on the manifold (M, ω), this is done though a polarisation. In the case of a good enough polarisation we will discuss how to construct the quantum Hilbert space of a symplectic manifold (M, ω) out of the prequantum data and the polarisation data. polarisation of Symplectic Manifolds The need of a polarisation can be motivated by the quantization of the prototypical symplectic space T ∗Rn =∼ R2n of example §2.3.13. The prequantum Hilbert space of this symplectic manifold is L2(R2). However, T ∗R is the phase space of a free particle in one dimension, and it is known that the corresponding quantum space of this space should be L2(R), not L2(R2). As mentioned before, the intuitive idea is that the quantum Hilbert space should only contain information of the configuration variables, however the prequantum space has the information of both configuration and momenta variables. Thus we need to halve the variables of the prequantum Line bundle in order to get the right quantum Hilbert space.

39 Definition 4.1.1. A Real polarisation of a symplectic manifold (M, ω) is an involutive Lagrangian subbundle F of TM. Since a real polarisation is involutive, Frobenius theorem asserts that there exists a foliation FM of M such that T FM = Fx, thus a real polarisation induces a foliation by Lagrangian manifolds. On the other hand, it is easily seen that the tangent bundle to the leafs of a foliation by Lagrangian submanifolds is a involutive lagrangian subbundle, that is, a real polarisation. Thus we have proved, Proposition 4.1.2. There is a one to one correspondence between real polarisations and foliations by Lagrangian submanifolds. ∗ Example 4.1.3. Let M be a manifold and let (T M, ωcan) be the cotangent space with the canonical symplectic structure. Now T ∗M −→π M has a natural Lagrangian foliation by the fibers of the cotangent bundle, this foliation induces a real polarisation known as the −1 vertical polarisation. Since the tangent space to the fiber (x, p) ∈ π (x) is ker(Dπ)(x,p) the polarisation can be described as

F = ker(Dπ): T (T ∗M → TM).

It is worth mentioning that real polarisations might not exist for a general symplectic manifold. There is a more general definition of complex polarisation, this is a complex involutive Lagrangian sub-bundle of the complexified tangent bundle TCM. Although this kind of polarisations are more general there and allow to consider more examples even in this case there examples of symplectic manifolds that admit no polarisations whatsoever, real or complex, see [39]. We will be interested in well behaved examples where there exist real polarisations, thus we will not be interested in complex polarisation and will not discuss them further. Definition 4.1.4. A section s ∈ Γ(L) of a line bundle (L → M, ∇) is said to be polarised if ∇X s = 0 for all X ∈ ΓF. Remark 4.1.5. In the Souriau picture a section ϕ ∈ Γ(L) = C∞(L, C)S1 of a S1-bundle with connection (L → M, θ) is said to be polarised if Hθ(X)s = 0 for all X ∈ Γ(F). The space of polarised sections will be denoted by ΓF (L). ∗ Example 4.1.6. Consider (T M, ωcan) the cotangent space with canonical symplectic 1 ∗ π ∗ structure, (S × T M −→ M, dθ0 + π θL) the prequantum bundle, and F the vertical polarisation of T ∗M. In this scenario the space of sections is C∞(T ∗M) and H(X)ϕ = Xϕ. Since a vector field on the vertical polarisation is a vector field tangent to the fibers, we conclude that the space of polarised sections is the same as the space of functions constant on fibers

π∗C∞(T ∗M) = {ϕ ∈ C∞(T ∗M)|ϕ = π∗f, for some f ∈ C∞(M)}.

Fibrating polarisations and Line Bundles on Leaf Spaces

Naively the desired quantum Hilbert space would be a completition of ΓF (L) with respect to some inner product. However there is no clear description space of polarised sections, thus there is no clear way of how to endow ΓF (L) with a natural inner product. In what follows we will deal with a good polarisation scenario. In this scenario the space of polarised sections can be realised as the space of sections of some other line bundle. 40 Deﬁnition 4.1.7. A real polarisation is said to be ﬁbrating if:

i) the leaf space M/F of the foliaton deﬁned by the real polarisation is a manifold and that the quotient map π : M → M/F is a submersion,

ii) each leaf of the polarisation is simply connected.

For clearness of the discussion in what follows we will deal with the vector bundle description of prequantization, that is, Hermitian line bundles. Suppose (M, ω) is a prequantizable symplectic manifold and (L → M, θ) is a prequantum Line bundle of M, and suppose F is a ﬁbrating real polarisation. Let Λ be a leaf of the polarisation, since Λ is Lagrangian and the curvature of θ is ω it follows that (L|Λ → Λ, θ|Λ) is a ﬂat line bundle over a simply connected space.

One can define parallel section, i.e. H(X)ϕ = 0, on the L|Λ as follows: given any path γ on Λ and a starting point l ∈ L|Λ one can use the connection θ to horizontaly lift γ to a path on L|Λ. Since Λ is simply connected the lifting does not depend on the path, the difference between two paths will be given by the holonomy of the bundle which in this case is trivial, the parallel section ϕ is well defined. Notice that the set of parallel sections is parametrized by the initial value at a point, thus it is a one dimensional complex vector space. One can consider a such a one dimensional space for each leaf moreover the local triviality of the foliation implies such an assignment is locally trivial. Thus one can consider a line bundle LF → M/F over the leaf space.

Proposition 4.1.8. The space of section on Γ(LF ) is isomorphic to the space ΓF (L) of polarised sections on L.

Proof. A section s ∈ Γ(LF ) is by deﬁnition a transversaly smooth collection {sy}y ∈ M/F of parallel sections each over the restriction of L on Λx. The smoothness of such a collection imply that they deﬁne a smooth section on L.

On the other hand, the restriction of a polarised section s ∈ ΓF (L) on a leaf Λx is by deﬁnition a parallel section on L|Λx . Thus the restriction of s to each leaf is a transversally smooth family of parallel sections on {Λx}, that is, a section of Γ(LF ) Remark 4.1.9. Notice that the property that each leaf of the polarisation was a simply connected manifold is not necessary for the previous discussion. One could simply ask that the holonomy of the connection restricted to each leaf be trivial in order to carry with the construction presented here.

Intermezzo: Half-Densities

We will associate a Hilbert space to LF , however as a preparatory for this purpose we will quickly discuss about densities and half-densities. Densities are geometric objects that under change of coordinates change as the absolute value of the Jacobian matrix of the change of variables. Thus densities satisfy the change of coordinates rule necessary to deﬁne integration over them. On an oriented manifold densities can be identiﬁed with positive top forms on M, thus densities can be regarded as generalisations of volume forms.

41 Let FC(M) → M be the complexiﬁed frame bundle over a n-dimensional manifold M, this is the principal GL(n, C) bundle over M associated to the complexiﬁed tangent bundle T CM → M. Since there is a natural GL(n, C) representation on C by

GL(N, C) → C∗, α(T ) = | det T |−α,

one can consider an associated bundle with respect to this representation.

Deﬁnition 4.1.10. For α > 0, the bundle of α-densities |Λ|α(M) is the line bundle over M associated to (F (M), C, | det |α). An α-density is a section of |Λ|α(M).

We will be interested in 1-densities, which will be simply called densities, and 1/2- densities which will be referred as half-densities. In the principal bundle description a α-density is a map ρ : TM → C satisfying the equivariance condition

−α ρx(T v1, . . . T vn) = | det(T ) |ρx(V1,...,Vn),

n for any T ∈ GL(n, R) and any frame v1, . . . vn ∈ (TxM) . In particular under a change of coordinates from (Uα, φα) to (Uβ, φβ) given by gαβ :(Uα ∩ Uβ) → (Uα ∩ Uβ) a density changes as ρα(Jv1, . . . Jvn) = | det J| · ρβ(v1, . . . vn)

where J = Dgαβ is the Jacobi matrix of the change of coordinates gαβ.

Due to the transformation rule of densities the integral of a density ρ ∈ |Λ|M supported on the chart (Uα, φα) Z Z −1 ρ = ψα ◦ ρ ◦ φα dµ, U φα(Uα) R is well deﬁned. In the above expression tα : |Λ|M|Uα → φ(Uα)× is the trivialisation of the density bundle induced from the chart (Uα, φα) (recall the density bundle is an associated bundle of F (M)) and dµ is the Lebesgue measure. Thus, using partitions of unity {bα} subject to chart covering {Uα, φα} of M one can deﬁne an integral for any compactly supported density ρ ∈ |Λ|cM by

Z X Z ρ = bα ρ. Uα By the Riezs representation theorem (c.f [40, Section 7.2]) such an integral deﬁnes a measure dρ on M which is locally absolutely continuous with respect to the Lebesgue measure. By construction, any other measure η arising from another density is absolutely continuous with respect to the one described above, thus Z Z dρ ρ = η dη

dρ where dη is the Radon-Nikodym derivative. The local form of the integral and the fun- dρ damental theorem of calculus shows that locally dη is given by a smooth function. Since smoothness is a local property it implies that ﬁxing a a density ρ any other density can be written as ρ = dρ η for some smooth function dρ ∈ C∞(M). The previous discussion yields dη η. c the following proposition. 42 Proposition 4.1.11. The bundle |Λ|cM is a trivialisable bundle, with the trivialisation dρ given by η 7→ dη for some ﬁxed density ρ. Given that a density ρ induces a measure on M one can talk about Lp(M, |Λ|) the space of p-integrable functions with respect to the measure dρ. Notice this space does not depend on the density since any two densities are absolutely continuous with respect to each other. It is worth noting that in the case the manifold M has just one chart the measure dρ is mutually absolutely continuous to the Lebesgue measure, implying L2(M, |Λ|) =∼ L2(M).

To end this section we will discuss how half-densities can be multiplied to deﬁne a densities.

Proposition 4.1.12. The multiplication map

α β α+β |Λ| M ⊗ |Λ| M → |Λ| ρ1 ⊗ ρ2 7→ ρ1ρ¯2

induces a bundle isomorphism.

α β Proof. If ρ1 ∈ |Λ| M and ρ2 ∈ |Λ| M, then

implying the image is indeed a α + β-density. It is clear the map is differentiable and by definition it preserve fibers. To check it is a bundle isomorphism we only need to check it is an isomorphism on fibers. However this is clear since fiberwise it is a linear non-zero map bewteen one dimensional spaces.

Summarising, two half-densities M can be multiplied to give a density on M. We will use this fact to construct a Hilbert space out of the line bundle on the leaf space of a ﬁbrating polarisation.

Quantum Hilbert Space of a Symplectic Manifold With the notion of Half-densities we are ready to describe the quantum Hilbert space of a symplectic manifold. Let (M, ω) be a simply connected prequantizable symplectic manifold, (L → M, θ) be the prequatum line bundle of (M, ω), F a ﬁbrating real polarisation of M, L/F → M/F the line bundle described in section 4.1, and |Λ|1/2(M/F) the line bundle of half-densities on M/F. Let (L/F)1/2 denote the tensor product

(L/F)1/2 := L/F ⊗ |Λ|1/2(M/F).

1/2 Given two sections s1 ⊗ ρ1 , s2 ⊗ ρ2 of (L/F) one can deﬁne the density r(x) := hs1(x), s2(x)iρ1ρ¯2. If the half-densities ρ1, ρ2 have compact support one can integrate the density r to deﬁne an inner product Z hs1 ⊗ ρ1, s2 ⊗ ρ2i = hs1(x), s2(x)iρ1ρ¯2. M/F

43 1/2 The completition of (L/F) under the previous inner product HM is the Half- Density Quantum Hilbert Space we desired.

There is another construction similar to the construction by half-densities using half- forms. The bundle of half-forms Ω1/2(M) is the square root to the determinant bundle, i.e. Ωn(M) = Ω1/2(M) ⊗ Ω1/2(M). The half-form bundle might not exist, thus in order to find a prequantum line bundle such that it exists, one needs to discuss metaplectic structures. However in the case M/F is orientable, the density bundle and the determinant bundle are isomorphic and the half-form and half-density Quantum Hilbert spaces are isomorphic. For a detailed discussion on half-forms and the relationship with quantization via half-densities can be found on [41, Chapter 5] and [42]. ∗ Example 4.1.13. Consider (T M, ωcan) with the vertical polarisation. The leaf space of the polarisation is M and the leafs of the polarisation are vector spaces which are contractible, implying the vertical polarisation is a fibrating polarisation. Since the prequantum line bundle is trivial, then the induced bundle on M is also trivial. The quantum Hilbert space is then the completition of the space of square integrable functions with respect to the 2 1/2 measure defined by the half-densities, which we will denote by HT ∗M := L (M, Λ ). In ∗ ∼ 2 the case of T R the discussion on section 4.1 implies HT ∗R = L (R), thus we have arrived to the right quantum Hilbert space.

4.2 Quantization of Symplectic Groupoids

After brieﬂy reviewing the quantization of symplectic manifolds with a good enough polarisation we come back to the symplectic groupoid picture, in particular we explore the notion of good enough polarisation in this framework. We will discuss how to construct a C∗-algebra of a groupoid, and in general, of a Fell line bundle over a groupoid. Furthermore, we will use this construction to ﬁnd the C∗-algebra of a good enough polarised prequantum Fell line bundle, which will be our quantization of the symplectic groupoid.

Convolution C∗ Algebra of a Groupoid In this section we will deal with the construction of a C∗-algebra of a groupoid, or in general, the C∗-algebra of a Fell line bundle over a groupoid. To begin our discussion, following [16, Chapter 2.8] , [43, §3.3] and [44] we will generalise the half-density line bundle over a manifold to the half-bidensity Fell line bundle over a groupoid.

∗ Consider the half-density bundle on the tangent space of the s-fiber over m, s |Λ|m(G) := |Λ|1/2T (s−1(x)). The collection of all such bundles form a bundle s∗|Λ|1/2(G) over G. One ∗ 1/2 −1 can make and analogous construction with r-fibers and define r |Λ|m(G) := |Λ| T (r (x)) and r∗|Λ|1/2(G). Definition 4.2.1. The bundle of bi-densities is the tensor product

A(G) := s∗|Λ|1/2(G) ⊗ r∗|Λ|1/2(G).

There are canonical actions of G on s∗|Λ|1/2(G) and r∗|Λ|1/2(G) given by the pullback right translation and left translation maps

∗ ∗ 1/2 ∗ 1/2 ∗ Rg−1 : s |Λ|h (G) → s |Λ|hg (G),Rg−1 ρ(v1, . . . , vn) = ρ(DRg−1 v1,...,DRg−1 vn), 44 ∗ ∗ 1/2 ∗ 1/2 ∗ Lg−1 : r |Λ|h (G) → r |Λ|gh (G),Lg−1 η(v1, . . . , vn) = η(DLg−1 v1,...,DLg−1 vn), where DRg−1 and DLg−1 are the derivatives of the right translation and left translation maps respectively. A bi-section is called invariant if

∗ ∗ ρ(hg) ⊗ η(gh) = Rg−1 ρ(h) ⊗ Lg−1 η(h). Just as in the case of left-invariant vector ﬁelds, invariant bi-densities are characterised by ∼ densities over M, that is, the space of invariant bi-densities is isomorphic to Ainv(G) = 1∗A(G) → M. Notice that for an invariant bi-section

∗ ∗ 1/2 1/2 ρ ⊗ η(g) = Rg−1 ρ(t(g)) ⊗ Lg−1 η(s(g)), for ρ(t(g)) ⊗ η(s(g)) ∈ |Λ|t(g)(G) ⊗ |Λ|s(g)(G).

Thus (Ainv(G))g, the ﬁber at g of the bundle of invariant bi-densitites, can be identiﬁed 1/2 1/2 (2) with the space |Λ|t(g)(G) ⊗ |Λ|s(g)(G). For a composable pair (g, h) ∈ G , i.e. such that s(g) = t(h), we can consider the tensor product

∼ 1/2 1/2 1/2 1/2 (Ainv(G))g ⊗ (Ainv(G))h = (|Λ|t(g)(G) ⊗ |Λ|t(h)(G)) ⊗ (|Λ|t(h)(G) ⊗ |Λ|s(h)(G)) ∼ 1/2 1/2 = |Λ|t(h)(G) ⊗ (|Λ|t(g)(G) ⊗ |Λ|s(h)(G)) ∼ = |Λ|t(h)(G) ⊗ (Ainv(G))gh)

On sections ρ1 ⊗ η1, ρ2 ⊗ η2 ∈ Γ(Ainv(G)) the previous isomorphism, from the ﬁrst row to the last, is given by

∗ ∗ (ρ1(g) ⊗ η1(g)) ⊗ (ρ2(h) ⊗ η2(h)) 7→ (η1 · ρ2(t(h))) ⊗ (R(gh)−1 ρ1 ⊗ L(gh)−1 η2).

The element (η1 ·ρ2(t(h))) ∈ |Λ|t(h)(G) can be extended by right translations to a density on the whole t-fiber t−1(t(h)) ⊂ G, and further integrated to a invariant bi-density on the gh fiber (Ainv(G))gh. Thus is for two sections invariant bi-section ρ1, the following expression makes sense Z ! ∗ ∗ ∗ (ρ1 ⊗ η1 ∗ ρ2 ⊗ η2)(k) = Rg−1 (η1 · ρ2(t(k))dg (Rk−1 ρ1 ⊗ Lk−1 η2), g∈t−1(t(k)) and defines another invariant bi-section.

Deﬁnition 4.2.2. The convolution algebra of the Lie groupoid G is the space Γc(Ainv(G)) of smooth compactly supported sections of the bundle Ainv(G) with product Z (ϕ ∗ ψ)(k) = ϕ(g)ψ(g−1k), t−1(t(k)) and involution ϕ∗(g) = τ(ϕ(g−1)), where τ is the pullback of the inversion map.

We will not carry on with the task of showing these map indeed deﬁnes a ∗-algebra. For an exposition of this fact we refer the reader to [44] or [43, §3.3]. Then one may consider the enveloping C∗-algebra of a Banach algebra, that is the completition of the previous algebra 45 under the norm kϕk = sup kπ(ϕ)k, π∈R where π ranges over al ∗-representations of Γc(Ainv(G)). For details in this construction we refer to the reader to [14, Chapter 2.7] or [45, Chapter 2].

In a similar way to the case of half-densities, the bundle of half-densities over each t-fiber is trivialisable, and a trivialisation is given by fixing a density on each fiber. In fact, by our previous discussions, a transversaly smooth family of such trivialisation is equivalent ∗ to choosing a section ρ0 ⊗ η0 ∈ 1 Ainv(G), for details see [44]. This family of trivialisations induces a Haar system in the sense of [45, Definition 1.2.2] on the groupoid G.

Example 4.2.3. Manifolds: For a manifold M considered as a groupoid M → M the t−fibers are just points, thus the fibers of the bundle of bi-densities are 0 dimensional. Then the bundle of half-denstities to the s-fibers and t-fibers are the associated bundles to the trivial representation of GL(0, R) =∼ {e}, which are just trivial bundles over the fibers. Thus C ∞ Ainv(M) = M × and the the space of sections is just Cc (M) with the pointwise product

ϕ ∗ ψ(m) = ϕ(m)ψ(m)

and involution ϕ∗(m) = ϕ(m).

This is a commutative ∗-algebra and its completition is C0(M) the space of continuous functions on M vanishing at inﬁnity.

Example 4.2.4. Groups: For a group G ⇒ {∗} there is only one source and target fiber, namely G. Given a Haar measure dµ on G the space of compactly supported invariant ∞ densities on G can be trivialized, and the space of section can be identified with Cc (G). Implying 1/2 s 1/2 t ∼ 1/2 1/2 Ac(G) = |Λ| T G ⊗ |Λ| T G = |Λ|c TG ⊗ |Λ|c TG thus a Haar measure on G induces a trivialisation on the bundle of bi-densities. With this ∼ ∞ identification Γ(Ainv(G)) = Cc (G) and the convolution product and the involution can be written as Z −1 ϕ ∗ ψ(k) = ϕ(g)ψ(g k)dµg, G ϕ∗(g) = ϕ(g−1), which are the usual product and involution of the convolution algebra C∗(G) of a locally compact group G.

Example 4.2.5. Pair Groupoids: For the Pair groupoid M × M ⇒ M the t-fibers of the groupod are diffeomorphic to M, therefore a trivialisation of the bundle of invariant bi-densities is a density over M. Fixing such density induces a measure ds on each t-fiber ∗ and the space of compactly supported sections Γc(1 Ainv(N × N)) can be identified with Cc(N × N) with product and involution Z φ ∗ ψ(m, n) = ϕ(m, s)ψ(s, n)ds, ϕ∗(m, n) = (ϕ(n, m)). M

46 Notice that each ϕ ∈ Cc(N × N) can be considered as the kernel of an integral operator on L2(N), moreover the composition and involution of such integral operators is given by the formulas above. The completition of the space of integral operators with compact support under the given norm is the algebra K(L2(N)) of compact operators on L2(N).

Example 4.2.6. Transformation Groupoids: For a transformation groupoid G n M ⇒ M each fiber is diffeomorphic to the group G, thus choosing a trivialisation on each fiber is −1 equivalent to choosing a Haar measure dµm on each fiber t (m). Choosing such trivialisa- ∗ tion the set of section of compactly supported sections Γc(1 Ainv(G n M) can be identified with Cc(G × M) with product and involution Z −1 −1 ∗ −1 ϕ ∗ ψ(g, m) = ϕ(gh, h · m)ψ(h , m)dµh, ϕ (g, m) = ϕ(g , g · m). G The completition of this ∗-algebra is usually known as the transformation groupoid C∗- algebra. Now, we want to generalise the previous construction to the case of a Fell line bundle L → G over a groupoid G since this will be the scenario in latter section. Realising that the previous discussion is the particular case when L is a trivial bundle the generalisation to a Fell line bundle is given by Definition 4.2.7. The C∗-algebra of a Fell line bundle L → M, also known as the ∗ ∗ twisted C -algebra of G, is enveloping C algebra of Γ(L ⊗ Ainv(G)) with product Z (s ⊗ ϕ ∗ t ⊗ ψ)(k) = (s(g)t(g−1k)) ⊗ (ϕ(g)ψ(g−1k)), t−1(t(k))

s, t ∈ Γ(L), ϕ, ψ ∈ Γ(Ainv(G)) and the product of s and t is the Fell line bundle product. And involution (s ⊗ ϕ)∗(g) = s∗ ⊗ τ(ϕ(g−1)),

where s ∈ Γ(L), ϕ ∈ Γ(Ainv(G)) and s∗ is the involution on the Fell line bundle. In future examples the line bundle we will consider will turn out to be trivial thus we will on be concerned with twisted C∗-algebras. We refer the interested reader to [46] for a detailed discussion of this algebras and their structure.

polarisation of Groupoids The notion of right polarisation of symplectic groupoid is still a unexplored question, in this work we will use the definition of Hawkings [12]. The motivation behind this definition is that in order to define a convolution algebra of a polarised groupoid the definition of the polarisation should be compatible with the groupoid structure maps, that is, multiplication and inverse. Definition 4.2.8. Let G be a Lie groupoid and F ⊂ T G a real distribution on G, and (2) denote F2 := (F × F) ∩ T G . The distribution F is multiplicative if for any (g, h) ∈ G2:

T m(F2(g,h)) = Fgh.

The definition of multiplicativity is the compatibility condition of a distribution with the multiplication of the groupoid. We also require a compatibility with inversion 47 Definition 4.2.9. A real distribution F ⊂ T G is Hermitian if Di(F) = F, where Di is the differential of the inversion map on the groupoid. Definition 4.2.10. A real polarisation of a Lie grupoid is a multiplicative, hermitian polarisation. Definition 4.2.11. A real polarisation of a symplectic Lie groupoid is a Lie groupoid polarisation which is also a symplectic manifold polarisation. In the symplectic manifold case we handled polarised sections by considering them as sections of a line bundle over the leaf space. In the groupoid case we would like to proceed in an analogous way, however in this case we require that the leaf space to have a groupoid structure.

Leaf Spaces of Lie groupoids In the general the quotient of a Lie groupoid may not be a manifold, neither a groupoid, let alone a Lie groupoid. The notion that allows to consider quotients in the groupoid framework is the one of a smooth congruence and ﬁbration developed by Higgins and Mackenzie [47].

Definition 4.2.12. Let G ⇒ M be Lie groupoid. A differentiable congruence in G is a pair (S, R), where S ⊂ G × G and R ⊂ M × M satisfy the following conditions: i) S is a closed, embedded, differentiable subgroupoid of the pair groupoid G × G with base G. R is a closed, embedded, differentiable subgroupoid of the pair groupoid M × M with base M, ii) S ⇒ R is a Lie subgroupoid of G × G ⇒ M × M, where the latter has a pointwise groupoid structure, iii) the pullback pr S 1 G

s×s s pr R 1 M exist and the induced map S → Rpr1 ×s G is a submersion.

Deﬁnition 4.2.13. A ﬁbration of Lie groupoids is a smooth homorphism F :(G ⇒ M) → (H ⇒ N) such that the base map F0 : M → N and the map

(F, t): G → Ht ×F0 M,

are submersions. The above structures are equivalent in the sense that one determines the other. We will briefly sketch such correspondence: on one hand the kernel of a fibration determines a equivalence relation both on G and M, that is subsets S ⊂ G × G and R ⊂ M × M. One can check the definition of S and R being equivalence relations is equivalent to being subgroupoids of the pair groupoid. Moreover, since the quotient is a submersion proposition 2.1.1 implies that S, M are closed embedded subgroupoids of G × G and M × M respectively. By definition of the kernel S → R is a groupoid satisfying ii) and the fact that the fibration is a groupoid morphism implies iii).

48 On the other hand, the fact that S ⊂ G × G and R ⊂ M × M are embedded, closed, subgroupoids implies, again by proposition 2.1.1 and the previous discussion, that S and R determine equivalence relations such that G/S, M/R are manifolds and the quotient maps F : G → G/S, f : M → M/R are submersions. Deﬁne a groupoid structure on G/S → M/R by s¯(F (g)) = f ◦ s(g), t¯(F (g)) = f ◦ t(g),

F (g)F (h) = F (gh), 1F (g) = F (1g). Some comments are in place: condition iii) implies that for every composable pair (¯g, h¯) ∈ G/S(2) there exist a composable pair (g, h) ∈ G(2) such that the product can be defined as before. Moreover condition ii) implies the definition ofs ¯, t¯ and the multiplication are well defined. At last one can show that the described maps are smooth and the map (F, f):(G ⇒ M) → (G/S ⇒ M/R) is a fibration of Lie groupoids. We have sketched Theorem 2.3 of [47]:

Theorem 4.2.14. If (F, f):(G ⇒ M) → (H ⇒ N) is a fibration of Lie groupoids , then the kernel of the morphism determines a differentiable congruence on G. Conversely, given a Lie groupoid G ⇒ M and a differentiable congruence (S, R) on G, there is a unique differentiable groupoid structure on the quotient sets G/S ⇒ M/R such that the quotient maps F : G → G/S and M → M/R determines a fibration (F, f):(G ⇒ M) → (G/S ⇒ M/R).

Definition 4.2.15. A fibrating polarisation of a symplectic Lie groupoid G is a real groupoid polarisation such that as a symplectic polarisation is fibrating and the foliation of the polarisation by Lagrangian submanifolds determines a congruence on G.

Proposition 4.2.16. Let F :(G ⇒ M) → (H ⇒ N) be a ﬁbration of Lie groupoids, then the foliation by preimages {(F −1(h), f −1(h))} determines a Lie groupoid polarisation L.

Proof. Since {(F −1(g), f −1(g))} is a foliation it determines a real distribution denoted TF of T G. We will prove this polarisation is both multiplicative and hermitian. To begin notice −1 0 Tg0 (F (h)) = ker(DF )g, thus if (X,Y ) ∈ F2(g,h) it follows TF (X) = 0 and TF (Y ) = 0. Since F is a groupoid morphism, then

TghF (T(g,h)m(X,Y )) = T(g,h)(F ◦ m)(X,Y ) = T(g,h)(m ◦ F )(X,Y ),

implying that TFgh(T(g,h)m(X,Y )) = 0 and T m(F2(g,h)) ⊂ Fgh. Now if γ(t) is a curve in G such that F (γ(t)) = h, then in the quotient G/S (where S is the congruence associated with (F, f) ) the curve isγ ¯(t) is a constant h. Consider a decompositionγ ¯(t) = h0h00, , by the 0 remarks above theorem 4.2.14 there exist curves γ1(t), γ2(t) in G such that F (γ1(t)) = h , 00 F (γ2(t)) = h and the pair (γ1(t), γ2(t)) is a composable pair for every t. This implies that for every Z ∈ ker DghF there exist (X,Y ) ∈ F2(g,h) such that T(g,h)(X,Y ) = Z proving T m(F2(g,h)) = Fgh, i.e. the polarisation is multiplicative The proof that the polarisation is hermitian is similar to the proof that it is multiplicative.

Example 4.2.17. Let A → M be an integrable algebroid, G be the s-connected groupoid ∗ integrating A → M and (G ⇒ A , ωcan) be the associated cotangent groupoid. Then the vertical polarisation is a ﬁbrating symplectic groupoid polarisation. Indeed, we have already seen that as a symplectic manifold it is a ﬁbrating real polarisation. On the other hand,

49 ∗ ∗ the bundle projection (T G ⇒ A ) → (G ⇒ M) is a ﬁbration [33], thus by proposition the previous proposition the polarisation is multiplicative and the foliation integrates to a congruence.

Turning back to the quantization on symplectic groupoids. Let (G ⇒ M, ω) be a s- connected prequantizable symplectic groupoid, (S → G ⇒ M, θ) the prequantization of G and let F be a real ﬁbrating polarisation of symplectic groupoid. Then using the same arguments as in section 4.1 one can construct a line bundle (L/F → G/S) over the groupoid G/S ⇒ M/R. Proposition 4.2.18. The line bundle L/F → G/F is a Fell line bundle over G/F.

Proof. For g ∈ G let Og denote the leaf of the foliation G/F such that g ∈ Og. By

construction, an element s ∈ (L/F)O} over the leaf Og ∈ G/F is a parallel section of the line bundle L → Og- Recall this is completely characterised by it value at a point, thus we may consider that s is given by a point s(g) ∈ Lg. Given a composable pair of leaves Og and (2) Oh, i.e. such that (g, h) ∈ G , define the product of s ∈ (L/F)Og and t ∈ (L/F)Og as the Fell product on L and the involution s 7→ s∗ as the Fell involution on L. The compatibility of the connection with the Fell product implies the resulting section is also parallel. This implies the product is well defined and the properties of the Fell multiplication factors through the quotient defining a Fell product on L/F.

Quantum C∗-algebra of Symplectic Groupoid ∗ At last we will describe the quantum C -algebra of the symplectic groupoid.Let (G ⇒ M, ω) be a s-connected prequantizable symplectic groupoid, (S → G ⇒ M, θ) the prequantization of G, F be a real ﬁbrating symplectic groupoid polarisation of G,(L → G/S) the Fell line bundle described in proposition 4.2.18. Then the quantum C∗-algebra of (G, ω) denoted by C∗(G) is the twisted C∗-algebra of (L → G/S) as described in section 4.2. We will be mainly interested in the case where the Line bundle of proposition 4.2.18 turn out to be trivial. This case arises naturally in the quantization of the groupoid T ∗G and is an analogue of the well behaved case of the quantization of the cotangent bundle T ∗M.

Deﬁnition 4.2.19. Let (L → G, ω) be a groupoid prequantum line bundle. A multiplicative symplectic potential is a multiplicative 1-form θ ∈ Ω1(G) such that dθ = ω.A symplectic potential is said to be adapted to a polarisation F if θ(X) = 0 for all X ∈ F.

Exercise 4.2.20. Let A → M be an integrable algebroid, G be the s-connected groupoid ∗ ∗ 1 integrating A → M,(G ⇒ A , ωcan) be the associated cotangent groupoid, (T G × S , dθ0 + ∗ ∗ π π θL) the associated prequantum line bundle and F the vertical polarisation on G −→G. ∗ We claim that the Liuville 1-form θ0 on T G is an adapted symplectic potential. To begin with, by definition dθ = ωcan and by construction of θ is multiplicative [33]. Now to show it is adapted, notice vector field X ∈ F is by definition a vector field such that DπX = 0, thus for every X ∈ F and p = (m, ξ) ∈ T ∗G it holds

θp(X) = ξ(Dπp(X)) = ξ(0) = 0.

Proposition 4.2.21. ([12]) Let (L → G, ω) be a groupoid prequantum line bundle, F be real ﬁbrating polarisation and let p : G → G/F andp ˜ : L → L/F be the quotients map

50 deﬁned in proposition 4.2.18. If there exist an adapted symplectic potential θ then the Fell line bundle L/F → G/F is trivial.

Proof. The existence of a symplectic potential θ implies the prequantization of G is given by 1 π ∗ a (§ × G −→G, θ + π θL), where θL is the Maurer-Cartan form (compare with the cotangent groupoid example). In the vector bundle picture this is a trivial Fell bundle with connection ∇ = d + θ, where d is the exterior diﬀerential. Now adaptability implies that on F the connection is just ∇|F = d and covariantly constant section are just constant sections. Thus L/F is a trivial bundle.

4.3 Examples

The Cotangent Groupoid To end this chapter, and this monograph, we will present the quantization of a fairly general family of Poisson Manifolds, namely integrable Poisson structures for which the integrating symplectic groupoid is a cotangent groupoid. The example we will develop in this section where the ones presented at the end of Chapter 2, and the C∗−algebras obtained where the ones worked in the examples of section 4.2. Let A → M be an integrable algebroid, G be the s-connected groupoid integrating A → M. In chapter 2 we discussed how the dual bundle of a Lie algebroid has a natural Poisson structure such that the s-connected ∗ ∗ symplectic groupoid integrating A was the associated cotangent groupoid (G ⇒ A , ωcan). In the chapter 3 we worked the prequantization of such symplectic groupoids, namely it ∗ 1 ∗ was given by the trivial extension (T G × S , dθ0 + π θL). At last in this chapter we worked the vertical polarisation of T ∗G −→Gπ , by proposition 4.2.21 and the discussion on section 4.2 the quantization of T ∗G is given by the groupoid convolution algebra of the groupoid T ∗G/F = G i.e C∗(G). This gives a fair amount of examples of quantization depending in the initial algebroid A → M.

Example 4.3.1. Trivial Poisson Bracket: Let N be a manifold. Consider the algebroid N → N with 0-dimensional ﬁbers, then N is a Poisson manifold with the trivial Poisson structure. The s-connected groupoid integrating N → N is N ⇒ N, thus the quantization ∗ ∼ of N is C (N) = C0(N) the algebra of function over N vanishing at inﬁnity. Example 4.3.2. Cotangent Bundles: Let N be a simply comnnected manifold. Consider the algebroid TN → N, then T ∗N is a Poisson manifold with the Poisson structure given by N the symplectic structure on T . The s-connected groupoid integrating TN is N × N ⇒ N, thus the quantization of T ∗N is C∗(N × N) =∼ K(L2(N)) the algebra of compact operators on the Hilbert space L2(N).

Example 4.3.3. Lie Algebras: Let g∗ be a Lie algebra and G the simply connected group integrating g. Consider the algebroid g → {∗}, then g∗ is a Poisson manifold with the Kirillov-Poisson structure. The s-connected groupoid integrating g is G ⇒ {∗}, thus the quantization of g∗ is C∗(G) the convolution algebra of a group.

51 Chapter 5

Outlook

In this work we presented an approach to quantization whose aim was to construct a C∗-algebra out of an integrable Poissson Algebra. In particular, for a Groupoid G we realised its groupoid C∗-algebra C∗(G) as the quantization of the Lie algebroid A(G) considered as a Poisson Manifold. Implying that this approach is capable of producing a broad class of important examples in noncommutative geometry. As exposed through this work, most of the questions of existence and uniqueness of the objects required to prequantize a Poisson manifolds are well understood and there are clear descriptions to the obstructions of this process. However, there are still many issues needed to address in the polarization step. On one hand, the question of existence and uniqueness of polarizations of symplectic groupoids is still an open question. Furthermore, the dependence of the ﬁnal quantized C∗-algebra on the polarization has been hardly addressed, Hawkings [12] conjectured that the quantization should not depend on the polarization and worked some examples of this fact, however the general question of whenever this is true is still unresolved.

On the other hand, the quantization exposed in this monograph was in a sense a weak one, we did not ask for the C∗-algebra to be a deformation quantization. Regarding this issue Landsman showed that that the C∗-algebra of a groupoid is a strict deformation quantization [48] of its algebroid considered as a Poisson manifold, therefore the described prescription yields a strict deformation quantization in the case of a cotangent groupoid. In the author’s knowledge there still not a characterisation of the Poisson manifolds for which the quantization, as described in this work, yields a strict deformation quantization.

One of the main applications of the geometric quantization of symplectic manifolds is the orbit method, developed by Kirillov and Konstant [6, 4]. This method allows to construct all the irreducible representations of certain Lie groups. To the author’s knowledge there is no precise generalization of this procedure in the groupoid setting. A possible generalisation would construct a representation of the quantization C∗-algebra on some Hilbert out of an integrable Poisson manifold and a symplectic realisation of such Poisson manifold. Work towards such a theory has been developed by Xu and Weinstein [49]. From a diﬀerent context, Williams and Brown have worked on C∗-algebra representation theory by using S1- extensions of locally compact groupoids, which resembles many of the objects appearing in the quantization of symplectic groupoids [46, 50]. However, a concrete method generalising the Kirillov orbit method seems to be missing in the symplectic groupoid case.

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