SPRINGER BRIEFS IN MATHEMATICS Shubham Dwivedi Jonathan Herman Lisa C. Jeffrey Theo van den Hurk Hamiltonian Group Actions and Equivariant Cohomology SpringerBriefs in Mathematics Series Editors Palle Jorgensen, Iowa, USA Roderick Melnik, Waterloo, Canada Lothar Reichel, Kent, USA George Yin, Detroit, USA Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Tatsien Li, Shanghai, China Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, USA Yuri Tschinkel, New York, USA Ping Zhang, Kalamazoo, USA [email protected] SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030 [email protected] Shubham Dwivedi • Jonathan Herman • Lisa C. Jeffrey • Theo van den Hurk Hamiltonian Group Actions and Equivariant Cohomology 123 [email protected] Shubham Dwivedi Jonathan Herman Department of Pure Mathematics Department of Mathematical and University of Waterloo Computational Sciences Waterloo, ON, Canada University of Toronto at Mississauga Mississauga, ON, Canada Lisa C. Jeffrey Theo van den Hurk Department of Mathematics Department of Mathematics University of Toronto University of Toronto Toronto, ON, Canada Toronto, ON, Canada ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-030-27226-5 ISBN 978-3-030-27227-2 (eBook) https://doi.org/10.1007/978-3-030-27227-2 Mathematics Subject Classification (2010): 57R17 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland [email protected] Preface This monograph could serve as the textbook for a graduate course on symplectic geometry. It has been used for this purpose in graduate courses taught in the Mathematics Department at the University of Toronto. This book evolved from the lecture notes for the graduate course on symplectic geometry taught in fall 2016. Three of the coauthors (Shubham Dwivedi, Jonathan Herman and Theo van den Hurk) were students in this course. Alternatively, it could be used for independent study. A course on differential topology is an essential prerequisite for this course (at the level of the texts by Boothby [1] or Lee [2]). Some of the later material will be more accessible to readers who have had a basic course on algebraic topology, at the level of the book by Hatcher [3]. For some of the later chapters (such as the chapter on geometric quantization and the chapter on flat connections on 2-manifolds), it would be helpful to have some background on representation theory (such as the book by Bröcker and tom Dieck [4]) and complex geometry (such as the first chapter of the book by Griffiths and Harris [5]). The layout of this monograph is as follows. The first chapter introduces sym- plectic vector spaces, followed by symplectic manifolds. The second chapter treats Hamiltonian group actions. The Darboux theorem comes in Chap. 3. Chapter 4 treats moment maps. Orbits of the coadjoint action are introduced in Chap. 5. Chapter 6 treats symplectic quotients. The convexity theorem (Chap. 7) and toric manifolds (Chap. 8) come next. Equivariant cohomology is introduced in Chap. 9. The Duistermaat–Heckman theorem follows in Chap. 10, geometric quantiza- tion in Chap. 11 and flat connections on 2-manifolds in Chap. 12. Finally, the appendix provides the background material on Lie groups. v [email protected] vi Preface Exercises related to the material presented here may be found at Chaps. 1–5 http://www.math.toronto.edu/*jeffrey/mat1312/exerc1.pdf Chaps. 6–8, 11 http://www.math.toronto.edu/*jeffrey/mat1312/exerc2rev.pdf Chaps. 9, 10, 12 http://www.math.toronto.edu/*jeffrey/mat1312/exerc3rev.pdf The authors thank Yucong Jiang and Caleb Jonker for carefully reading and commenting on several chapters of the manuscript. Waterloo, Canada Shubham Dwivedi Mississauga, Canada Jonathan Herman Toronto, Canada Lisa C. Jeffrey Toronto, Canada Theo van den Hurk References 1. W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry. Pure and Applied Mathematics, vol. 120 (Academic Press, New York, 1986) 2. J. Lee, Introduction to Smooth Manifolds. GTM (Springer, New York, 2006) 3. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2001) 4. T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups. GTM (Springer, New York, 1985) 5. P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New Jersey, 1994) [email protected] Contents 1 Symplectic Vector Spaces ................................ 1 1.1 Properties of Symplectic Vector Spaces .................. 1 1.2 Review of Results From Differential Topology ............. 4 1.3 Symplectic Manifolds ............................... 5 1.4 Examples ........................................ 6 References ............................................ 7 2 Hamiltonian Group Actions .............................. 9 2.1 Hamilton’s Equations ............................... 9 2.2 Hamiltonian Flow of a Function ....................... 10 2.3 Poisson Bracket ................................... 11 2.4 Uniqueness ....................................... 14 References ............................................ 15 3 The Darboux–Weinstein Theorem.......................... 17 References ............................................ 20 4 Elementary Properties of Moment Maps..................... 21 4.1 Introduction ...................................... 21 4.2 Examples of Moment Maps ........................... 22 4.3 The Normal Form Theorem ........................... 25 References ............................................ 26 5 The Symplectic Structure on Coadjoint Orbits ................ 27 Reference ............................................. 29 6 Symplectic Reduction ................................... 31 6.1 Introduction ...................................... 31 6.2 Symplectic Quotients ............................... 34 6.3 Reduction at Coadjoint Orbits and the Shifting Trick ........ 37 vii [email protected] viii Contents 6.4 Reduction in Stages ................................ 41 6.5 Symplectic Cutting ................................. 45 References ............................................ 46 7 Convexity ............................................ 47 7.1 Introduction ...................................... 47 7.2 Digression on Morse Theory .......................... 49 7.3 Almost Periodic Hamiltonians ......................... 51 7.4 Proof of the Convexity Theorem ....................... 54 7.5 Applications and Examples ........................... 57 References ............................................ 60 8 Toric Manifolds ....................................... 61 8.1 Introduction ...................................... 61 8.2 Integrable Systems ................................. 65 8.3 Primitive Polytopes ................................. 66 8.4 Delzant Correspondence ............................. 68 References ............................................ 70 9 Equivariant Cohomology ................................ 71 9.1 Introduction ...................................... 71 9.2 Homotopy Quotients ................................ 71 9.3 The Cartan Model .................................. 73 9.4 Characteristic Classes of Bundles over BUð1Þ and BT ....... 76 9.5 Characteristic Classes in Terms of the Cartan Model......... 78 9.6 Equivariant First Chern Class of a Prequantum Line Bundle ... 79 9.7 Euler Classes and Equivariant Euler Classes ............... 80 9.8 Localization Formula for Torus Actions .................. 81 9.9 Equivariant Characteristic Classes ...................... 84 9.10 The Localization Theorem in Equivariant Cohomology ....... 86 References ............................................ 88 10 The Duistermaat–Heckman Theorem ....................... 89 10.1 Introduction .....................................