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Groupoids in Geometric Quantization

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Groupoids in Geometric Quantization Groupoids in geometric quantization Een wetenschappelijke proeve op het gebied van de Natuurwetenschappen, Wiskunde en Informatica Proefschrift Ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. mr. S.C.J.J. Kortmann, volgens besluit van het College van Decanen in het openbaar te verdedigen op vrijdag 28 september 2007 om 15:30 uur precies door Rogier David Bos geboren op 7 augustus 1978 te Ter Aar Promotor: Prof. dr. Klaas Landsman Manuscriptcommissie: Prof. dr. Gert Heckman Prof. dr. Ieke Moerdijk (Universiteit Utrecht) Prof. dr. Jean Renault (Universit´ed’Orl´eans) Prof. dr. Alan Weinstein (University of California) Prof. dr. Ping Xu (Penn State University) This thesis is part of a research program of ‘Stichting voor Fundamenteel Onderzoek der Materie’ (FOM), which is financially supported by ‘Nederlandse Organisatie voor Weten- schappelijk Onderzoek’ (NWO). 2000 Mathematics Subject Classification: 22A22, 22A30, 58H05, 53D50, 53D20, 58B34, 19K35. Bos, Rogier David Groupoids in geometric quantization Proefschrift Radboud Universiteit Nijmegen Met een samenvatting in het Nederlands ISBN 978-90-9022176-2 Printed by Universal Press, Veenendaal. Contents 1 Introduction 7 I Preliminaries 15 2 Groupoids 17 2.1 Groupoids................................. 17 2.2 Liegroupoids ............................... 21 2.3 Morphismsofgroupoids . .. .. .. .. .. .. .. 23 2.4 Liealgebroids............................... 25 2.5 Haarsystems ............................... 28 3 Continuous fields of Banach and Hilbert spaces 31 3.1 Continuous fields of Banach and Hilbert spaces . 31 3.2 Dimension and local pseudo-trivializations . ...... 34 4 C∗-algebras, K-theory and KK-theory 37 4.1 C∗-algebras ................................ 37 4.2 Banach/Hilbert C∗-modules....................... 39 4.3 The convolution algebra of a groupoid . 43 4.4 K-theory of C∗-algebras ......................... 44 4.5 KK-theory ................................ 46 II Continuous representations of groupoids 49 Introduction................................... 51 5 Continuous representations of groupoids 53 5.1 Continuous representations of groupoids . 53 5.2 Continuity of representations in the operator norm . ...... 58 5.3 Example: the regular representations of a groupoid . ...... 63 5.4 Example: continuous families of groups . 64 5.5 Representations of the global bisections group . ...... 67 6 Groupoid representation theory 71 6.1 Decomposability and reducibility . 71 6.2 Schur’slemma .............................. 73 4 CONTENTS 6.3 Square-integrable representations . 75 6.4 ThePeter-WeyltheoremI . 77 6.5 ThePeter-WeyltheoremII . 79 6.6 Representation rings and K-theory of a groupoid . 81 7 The groupoid convolution C∗-category 85 7.1 Fell bundles and continuous C∗-categories . 85 7.2 Representations of G ⇉ M and Lˆ1(G)................. 87 III Geometric quantization of Hamiltonian actions of Lie alge- broids and Lie groupoids 91 Introduction................................... 93 8 Hamiltonian Lie algebroid actions 97 8.1 Actions of groupoids and Lie algebroids . 97 8.2 Internally Hamiltonian actions . 100 8.3 The coadjoint action and internal momentum maps . 102 8.4 Hamiltonian actions and momentum maps . 108 9 Prequantization of Hamiltonian actions 115 9.1 Representations of Lie algebroids . 115 9.2 Longitudinal Cechcohomologyˇ . .. .. .. .. .. .. 118 9.3 Prequantization representations . 121 9.4 Integrating prequantization representations . ....... 126 10 Quantization and symplectic reduction 129 10.1 Quantization through K¨ahler polarization . 129 10.2 Symplecticreduction. 134 10.3 Quantization commutes with reduction . 136 10.4 Theorbitmethod............................. 139 IV Noncommutative geometric quantization 143 Introduction................................... 145 11 Algebraic momentum maps 147 11.1 Hamiltonian actions on Poisson algebras . 147 11.2 Noncommutative differential forms . 151 11.3 Pullbackandpushforwardofforms . 153 11.4 Hamiltonian actions on symplectic algebras . 155 11.5 Algebraic prequantization . 157 12 Noncommutative analytical assembly maps 159 12.1 Properactions .............................. 159 12.2 Example: proper actions on groupoid C∗-algebras . 160 12.3 Hilbert C∗-modules associated to proper actions . 164 12.4 Equivariant KK-theory ......................... 168 CONTENTS 5 12.5 The noncommutative analytical assembly map . 170 12.6 Noncommutative geometric quantization . 173 References 179 Index 185 Samenvatting in het Nederlands 191 Dankwoord/Acknowledgements 201 Curriculum vitae 203 Chapter 1 Introduction Geometric quantization The aim of this thesis is to introduce various new approaches to geometric quantiza- tion, with a particular rˆole for groupoids. This introduction gives a general outline of the thesis and intends to explain its coherence. A more technical introduction explaining the results more precisely is found in separate introductions preceding each part. Geometric quantization is a mathematical method to relate classical physics to quantum physics. In classical mechanics the information of the system under investigation is assembled in the concept of a phase space. Each point in this space corresponds to a possible state of the system. The observable quantities of the system correspond to functions on the phase space. Typical for classical mechanics is the existence of a Poisson bracket on the algebra of functions. An important source of commutative Poisson algebras are algebras of functions on symplectic manifolds. The dynamics of the system is determined by a special function, the Hamiltonian, and by the Poisson bracket. The possible states of a quantum system correspond to unit vectors in a Hilbert space. The observable quantities correspond to certain operators on this Hilbert space. These operators form an operator algebra. Analogously to classical mechan- ics, there is a Lie bracket on this algebra, namely the commutator. The dynamics of a quantum system is determined by a special operator (the Hamiltonian) and the commutator. The program of geometric quantization was initiated around 1965 by I. E. Segal, J. M. Souriau, B. Kostant, partly based on ideas of A. A. Kirillov going back to 1962 (cf. e.g. [38, 92] and references therein). Its aim is to construct a Hilbert space from a symplectic manifold, and to construct operators on that Hilbert space from functions on the symplectic manifold. The purpose is to relate the Poisson algebra to the operator algebra in such a way that the Poisson bracket relates to the commutator bracket. In this way one relates the two ways in which the dynamics of the classical and quantum systems are described. 8 Chapter 1: Introduction Symmetry and geometric quantization If a classical system has some symmetry, then one would hope that this symmetry is preserved under quantization. This idea does not work in general. Instead, it applies to a particular type of symmetries called Hamiltonian group actions. Geometric quantization turns a Hamiltonian group action on a symplectic manifold into a unitary representation on the quantizing Hilbert space. In this sense geometric quantization can be seen as a way to construct representations of groups. In the case of a Lie group G, the associated dual g∗ of the Lie algebra g can serve as a rich source of symplectic manifolds endowed with a Hamiltonian action. Indeed, the orbits of the coadjoint action of G on g∗ are symplectic manifolds and the coadjoint action is actually Hamiltonian. For simply connected solvable Lie groups all representations are obtained from geometric quantization of the coadjoint orbits. The idea that representations correspond to coadjoint orbits is called the “orbit philosophy” (cf. [38, 39]). Actually, the orbit philosophy goes much further in linking the representations of a Lie group to the coadjoint orbits, but we shall not discuss this. For compact Lie groups and non-compact semisimple Lie groups, geometric quantization is still a good method to obtain (some of) the representations, but the tight bond as described in the orbit philosophy has to be relaxed. The advantage of the occurrence of a symmetry in a physical system is that the number of parameters used to describe the system can be reduced. For classical me- chanics this mathematically boils down to a symplectic reduction of the symplectic manifold under the Hamiltonian action (cf. [50]). The obtained reduced symplectic manifold is called the Marsden-Weinstein quotient. On the other hand, if a quan- tum system has a symmetry one can perform a so-called quantum reduction. An important statement, the Guillemin-Sternberg conjecture, explains the relationship between geometric quantization of the Hamiltonian action on a symplectic manifold and the geometric quantization of the associated Marsden-Weinstein quotient (cf. [28]). The Guillemin-Sternberg conjecture states that quantization commutes with reduction or, more precisely, that the so-called quantum reduction of the geometric quantization should be isomorphic to the quantization of the Marsden-Weinstein quotient. It has been formulated and proved by Guillemin and Sternberg for com- pact Lie groups in the 80’s of the previous century (cf. [28]). Other proofs were found in the 90’s (cf. [52, 53, 62, 76]). But there are various new developments now ([47, 36, 35]) and this thesis is part of those developments. Groupoids in geometric quantization Suppose (M, ) is a foliated manifold. This means that M is partioned by immersed F submanifolds F . One could
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