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Geometric Quantization of Poisson Manifolds Via Symplectic Groupoids

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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. iv 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 Karas¨ev[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].
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