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Blagoje Oblak

BMS Particles in ThreeDimensionsDoctoral Thesis accepted byFree University of Brussels, Belgium

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AuthorDr. Blagoje OblakInstitute for Theoretical PhysicsETH ZürichZürichSwitzerland

SupervisorProf. Glenn BarnichPhysique Théorique et MathématiqueULBBrusselsBelgium

ISSN 2190-5053 ISSN 2190-5061 (electronic)Springer ThesesISBN 978-3-319-61877-7 ISBN 978-3-319-61878-4 (eBook)DOI 10.1007/978-3-319-61878-4

Library of Congress Control Number: 2017944312

© Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof 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 transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe 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 thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Mojoj najmilijoj porodici,s mnogo ljubavi.

Supervisor’s Foreword

Symmetries play a key role in our understanding of Nature. For instance, all ofparticle physics crucially relies on the Poincaré symmetry group of Minkowskispace-time. Once gravity is switched on, however, the isometry group of genericspace-time manifolds is empty and Poincaré symmetry becomes irrelevant. Whatreplaces it are asymptotic symmetries: Those are the symmetries of a space-timemanifold seen by a “faraway” observer.

About fifty years ago, Bondi, Metzner, van der Burg, and Sachs studied theasymptotic symmetries at null infinity of Einstein gravity on a Minkowskian back-ground. What they found, quite surprisingly, was that Poincaré symmetry is extendedinto an infinite-dimensional group now known as the Bondi-Metzner-Sachs(BMS) group. More recent developments confirm that such infinite-dimensionalextensions of exact isometries are actually quite common. In addition, there arereasons to believe that BMS symmetry can be extended further so as to contain localconformal transformations; in that picture, BMS consists of infinite-dimensional“superrotations” and “supertranslations” in the same way that the Poincaré groupconsists of finite-dimensional Lorentz transformations and translations.

Ever since its discovery, the BMS group has been conjectured to play a centralrole in the quest for a quantum theory of gravity. In the last couple of years, excitingnew proposals indicate the existence of hitherto unexplored degrees of freedom,closely connected to and controlled by the BMS group, that may eventually accountfor the Bekenstein-Hawking entropy of realistic black holes in four dimensions.

The complexity of the four-dimensional problem suggests that a good strategy isto turn to a toy model. A natural candidate is provided by three-dimensional gravity,where beautiful asymptotic symmetry groups have been known to exist ever sincethe work of Brown and Henneaux in the eighties. Accordingly, this thesis isdevoted to BMS symmetry in three space-time dimensions. It addresses the specificproblem of classifying the irreducible unitary representations of BMS3 (“BMSparticles”), and relating them to quantum gravity. The material is presented in aself-contained and pedagogical manner, with all the necessary background collectedin the first seven chapters. This allows the reader to fully appreciate the originalresults that have been obtained while learning many fundamental concepts in the

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field along the way. It makes the present work a perfect point of entry into thematter and a most rewarding read for anyone with a serious interest in BMSsymmetry, or asymptotic symmetries in general.

Brussels, BelgiumJune 2017

Prof. Glenn Barnich

viii Supervisor’s Foreword

Abstract

This thesis is devoted to the group-theoretic aspects of three-dimensional quantumgravity on Anti-de Sitter and Minkowskian backgrounds. In particular, we describethe relation between unitary representations of asymptotic symmetry groups andgravitational perturbations around a space-time metric. In the asymptotically flatcase, this leads to BMS particles, representing standard relativistic particles dressedwith gravitational degrees of freedom accounted for by coadjoint orbits of theVirasoro group. Their thermodynamics are described by BMS characters, whichcoincide with gravitational one-loop partition functions. We also extend theseconsiderations to higher spin theories and supergravity.

ix

Preface

This thesis collects thoughts and results that originate from a four-year-longresearch project in theoretical physics. The main topic is representation theory andits application to quantum gravity, in particular in the context of BMS symmetry.The text consists of three parts:

Part I: Group theory;Part II: Virasoro symmetry and AdS3/CFT2;Part III: BMS symmetry in three dimensions.

It is written in such a way that each part can be read more or less independentlyof the others, although the later parts do depend on background material presented inthe earlier ones; the logical flow of chapters is explained in Sect. 1.5. A few sectionsare marked with an asterisk; they contain somewhat more advanced material thatmay be skipped without affecting the reading of the main track.

Brussels, Belgium Dr. Blagoje Oblak

xi

The original contributions of this thesis are based on the followingpublications:

• G. Barnich and B. Oblak, “Holographic positive energy theorems in three-dimensional gravity,” Class. Quant. Grav. 31 (2014) 152001, 1403.3835.

• G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions:I. Induced representations,” JHEP 06 (2014) 129, 1403.5803.

• G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions:II. Coadjoint representation,” JHEP 03 (2015) 033, 1502.00010.

• B. Oblak, “Characters of the BMS Group in Three Dimensions,” Commun.Math. Phys. 340 (2015), no. 1, 413–432, 1502.03108.

• G. Barnich, H.A. González, A. Maloney, and B. Oblak, “One-loop partitionfunction of three-dimensional flat gravity,” JHEP 04 (2015) 178, 1502.06185.

• B. Oblak, “From the Lorentz Group to the Celestial Sphere,” Notes de laSeptième BSSM, U.L.B. (2015). 1508.00920.

• A. Campoleoni, H.A. González, B. Oblak, and M. Riegler, “Rotating HigherSpin Partition Functions and Extended BMS Symmetries,” JHEP 04 (2016)034, 1512.03353.

• H. Afshar, S. Detournay, D. Grumiller, and B. Oblak, “Near-Horizon Geometryand Warped Conformal Symmetry,” JHEP 03 (2016) 187, 1512.08233.

• A. Campoleoni, H.A. González, B. Oblak, and M. Riegler, “BMS Modules inThree Dimensions,” Int. J. Mod. Phys. A31 (2016), no. 12, 1650068, 1603.03812.

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Acknowledgements

Professional Community

This thesis could not have been completed without the help and support of a numberof people. First and foremost, I am indebted to my supervisor, Prof. Glenn Barnich,for suggesting the topic in the first place and guiding me through its completion.Working with Glenn has been both a delight and a challenge, the former for hiscontagious passion for physics, and the latter for his sharp critical mind andunwavering skepticism, leading to numerous lively discussions about the nature ofour work and of science altogether. Our complementary approaches have made ourcollaboration all the more fruitful; I am grateful to him for preserving my completefreedom while guiding me along the thesis. He has been a teacher, a friend, and amodel of independence and scientific integrity that I hope to emulate myself.

I also wish to thank my other collaborators. The earliest ones in the history of myPh.D. are Sophie De Buyl, Stéphane Detournay, and Antonin Rovai, with whomI spent a few weeks in Harvard in the Spring of 2013. I am grateful to them for theirfriendship and our many relaxed discussions about physics and other matters,including world economy, Belgian movies, and American food. Later on, my pathcrossed that of Hernán González, starting with several ultra local journal clubs onquantum gravity, and ending with common projects. I am grateful to him forsharing his enthusiasm and almost convincing me that three-dimensional gravity is,in fact, gravity. My gratitude also goes to Hamid Afshar, Daniel Grumiller,Alexander Maloney, Max Riegler, and especially Andrea Campoleoni, for theirfruitful and enjoyable scientific collaborations; I hope there will be many more inthe future.

The friendly environment at the Physics Department of the Université Libre deBruxelles has also been a great help in completing this thesis. In particular, I amgrateful to my office neighbor, Laura Donnay, as well as Pujian Mao and MarcoFazzi, for enjoyable discussions during coffee breaks or lunches. I also thank theother students, postdocs, and professors in the service of mathematical physics formaking my days as a Ph.D. student in Brussels as pleasant as can be; this includes

xv

Riccardo Argurio, Andrés Collinucci, Geoffrey Compère, Laure-AnneDouxchamps, Simone Giacomelli, Gaston Giribet, Paolo Gregori, MarcHenneaux, Victor Lekeu, Arnaud Lepage-Jutier, Andrea Marzolla, Roberto Oliveri,Arash Ranjbar, Waldemar Schulgin, Shyam Sunder Gopalakrishnan, and CélineZwikel. In addition, I wish to thank my colleagues outside of Brussels—HaroldErbin, Lucien Heurtier, Jules Lamers, Ruben Monten, Ali Seraj, and Ellen Van derWoerd—for our numerous delightful interactions. Finally, the logistics and orga-nization of pretty much anything at the service of mathematical physics would beimpossible without the precious help and efficiency of the administrative staff of theservice and the Solvay Institutes—Dominique Bogaerts, Fabienne De Neyn,Marie-France Rogge, Isabelle Van Geet, and Chantal Verrier. To them, thank youfor your unshakeable goodwill in the face of the fiercest of administrativechallenges.

Since October 2015, I have had the chance to meet a number of new colleaguesat the Department of Applied Mathematics and Theoretical Physics of theUniversity of Cambridge. This would not have been possible without the support ofHarvey Reall, whom I wish to thank warmly for this incredible opportunity. I amalso grateful to Siavash Golkar, Shahar Hadar, Sasha Hajnal-Corob, Kai Roehrig,Arnab Rudra, Joshua Schiffrin, and Piotr Tourkine for making my time there a dailyenjoyment.

Still on the international side, I would like to thank Matthias Gaberdiel andBianca Dittrich for giving me the opportunity to give seminars at their respectiveinstitutions. Both of these visits were a delight, and I had great pleasure in sharingsome thoughts about my research with them and their colleagues.

Finally, I am grateful to Guillaume Bossard, Axel Kleinschmidt, and PetrTinyakov (in addition to Glenn Barnich, Stéphane Detournay, and Marc Henneauxwho have already been cited) for accepting to be part of my thesis jury, and for theirmany questions and suggestions at the private and public defences.

Family and Friends

On a more private side, I must mention the help and unconditional support providedby my family, and in particular my parents, Tijana and Dušan Oblak. Their con-tribution to this work is invisible to the naked eye, but it is actually so all-pervasivethat it is hard to tell what would have become of me if I hadn’t had such an amazingteam behind my back. I am grateful to them for all the love they have given me and inparticular for their support (both moral and practical) during the last year. I also wishto thank my cousins Sofija and Milena Stevanović, my aunt Ksenija Stevanović, andmy grandmother Vera Vujadinović for always being there for me when it counts andfor the many beautiful moments we have had the chance to share.

My friends have also contributed greatly to this thesis, often unknowingly. Aboveall, I wish to thank David Alaluf, Mitia Duerinckx, Jihane Elyahyioui, GeoffreyMullier, Olmo Nieto-Silleras, and Roxane Verdikt for our many adventures together,

xvi Acknowledgements

including holidays and unforgettable parties (in the Balkans and elsewhere). As amatter of fact, Mitia Duerinckx has been a great help in understanding some of themathematics used in this thesis, always patiently replying to my many e-mails withquestions about Hilbert spaces, measures, and the like. More generally, I am gratefulto my friends from the Mathematics Department of ULB for their friendship andsometimes professional interactions, including Charel Antony, Cédric De Groote,Julien Meyer, and Patrick Weber. Finally, I am indebted to Thierry Maerschalk forhelping me out long ago with some cute LATEX tricks, many of which were used inthis thesis.

On the other side of the channel, I wish to thank the flurry of people whomI have had the chance to meet during the last year of my Ph.D. and who haveallowed me to take helpful breaks away from physics. This includes AbhimanyuChandra, Robert Cochrane, Marius Leonhardt, Dušan Perović, Frank Schindler,Bianca Schor, and Aaron Wienkers. I hope to meet them again in the future, despiteour living in different corners of the world.

Finally, I wish to thank Vanessa Drianne, who has been in an entangled statebetween Brussels and Cambridge for most of the past year and whose support hasbeen critical for my well-being in the last stages of writing the thesis.

Physics and Art

At this point, I would like to thank two professors who have been most importantfor my development, both as a scientist and as a person. The first is my high schoolphysics teacher, Emmanuel Thiran, who first managed to show me a glimpse of thebeauty of Nature and the thrill of lifting its veil. While indirect, his influencepervades the entirety of my approach to physics and guides me to this very day. Thesecond is Michel Laurent, my piano teacher. For more than a decade, he has beenshowing me the subtleties of music, but in truth, his teaching extends far beyondthat. While difficult to express in words, the conceptions of art and beauty that hehas conveyed to me have had a great influence on me, and hence on this thesis.

Financial and Logistic Support

To conclude, I wish to thank the institutions who have provided me with financialand logistic support throughout my Ph.D. Above all, I am grateful to the Fonds dela Recherche Scientifique—F.N.R.S. for the grant (number FC-95570) that hasallowed me to make a living out of the most delightful of occupations. On the otherhand, the financial support allowing me to work at the University of Cambridgeduring the last academic year has been granted to me by the FondationWiener-Anspach . I gratefully acknowledge their support in this marvelous expe-rience, and wish to thank particularly Nicole Bosmans for her help with logistics.

Acknowledgements xvii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Asymptotic BMS Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Global BMS and Extended BMS . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Holography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 BMS Particles and Soft Gravitons . . . . . . . . . . . . . . . . . . . . . . . 91.5 Plan of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Part I Quantum Symmetries

2 Quantum Mechanics and Central Extensions . . . . . . . . . . . . . . . . . . 172.1 Symmetries and Projective Representations . . . . . . . . . . . . . . . . 17

2.1.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Symmetry Representation Theorem . . . . . . . . . . . . . . . . 192.1.3 Projective Representations . . . . . . . . . . . . . . . . . . . . . . . 202.1.4 Central Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.5 Topological Central Extensions . . . . . . . . . . . . . . . . . . . 222.1.6 Classifying Projective Representations. . . . . . . . . . . . . . 24

2.2 Lie Algebra Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Central Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Central Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Wavefunctions and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Hilbert Spaces of Wavefunctions. . . . . . . . . . . . . . . . . . 383.1.3 Equivalent Measures and Radon–Nikodym

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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3.2 Quasi-regular Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.1 Quasi-invariant Measures on Homogeneous Spaces . . . 423.2.2 The Simplest Induced Representations. . . . . . . . . . . . . . 453.2.3 Radon–Nikodym Is a Cocycle* . . . . . . . . . . . . . . . . . . . 47

3.3 Defining Induced Representations. . . . . . . . . . . . . . . . . . . . . . . . 483.3.1 Standard Boosts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.2 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . 503.3.3 Properties of Induced Representations . . . . . . . . . . . . . . 513.3.4 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Characters Are Partition Functions . . . . . . . . . . . . . . . . 573.4.2 The Frobenius Formula . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.3 Characters and Fixed Points . . . . . . . . . . . . . . . . . . . . . 60

3.5 Systems of Imprimitivity*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5.1 Projections and Imprimitivity . . . . . . . . . . . . . . . . . . . . 613.5.2 Imprimitivity Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 63

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Semi-direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Representations and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Semi-direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1.3 Orbits and Little Groups . . . . . . . . . . . . . . . . . . . . . . . . 704.1.4 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.1.5 Exhaustivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Poincaré Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.1 Poincaré Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2.2 Orbits and Little Groups . . . . . . . . . . . . . . . . . . . . . . . . 804.2.3 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.4 Massive Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2.5 Massless Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2.6 Wigner Rotations and Entanglement* . . . . . . . . . . . . . . 90

4.3 Poincaré Particles in Three Dimensions . . . . . . . . . . . . . . . . . . . 934.3.1 Poincaré Group in Three Dimensions . . . . . . . . . . . . . . 934.3.2 Particles in Three Dimensions . . . . . . . . . . . . . . . . . . . . 964.3.3 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Galilean Particles* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.1 Bargmann Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.2 Orbits and Little Groups . . . . . . . . . . . . . . . . . . . . . . . . 1004.4.3 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4.4 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xx Contents

5 Coadjoint Orbits and Geometric Quantization . . . . . . . . . . . . . . . . . 1095.1 Symmetric Phase Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.1.2 Adjoint and Coadjoint Representations . . . . . . . . . . . . . 1115.1.3 Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.1.4 Symplectic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.1.5 Kirillov–Kostant Structures . . . . . . . . . . . . . . . . . . . . . . 1165.1.6 Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2 Geometric Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2.1 Line Bundles and Wavefunctions . . . . . . . . . . . . . . . . . 1245.2.2 Quantization of Cotangent Bundles . . . . . . . . . . . . . . . . 1255.2.3 Quantization of Arbitrary Symplectic Manifolds* . . . . . 1285.2.4 Symmetries and Representations . . . . . . . . . . . . . . . . . . 130

5.3 World Lines on Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.1 World Lines and Quantization Conditions . . . . . . . . . . . 1315.3.2 Interlude: The Maurer–Cartan Form . . . . . . . . . . . . . . . 1335.3.3 Coadjoint Orbits and Sigma Models . . . . . . . . . . . . . . . 1365.3.4 Coadjoint Orbits and Characters of SLð2;RÞ*. . . . . . . . 138

5.4 Coadjoint Orbits of Semi-direct Products . . . . . . . . . . . . . . . . . . 1415.4.1 Adjoint Representation of GnA . . . . . . . . . . . . . . . . . . 1425.4.2 Coadjoint Representation of GnA . . . . . . . . . . . . . . . . 1435.4.3 Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.4.4 Geometric Quantization and Particles . . . . . . . . . . . . . . 1495.4.5 World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.5 Relativistic World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.5.1 Coadjoint Orbits of Poincaré . . . . . . . . . . . . . . . . . . . . . 1545.5.2 Scalar World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.5.3 Galilean World Lines* . . . . . . . . . . . . . . . . . . . . . . . . . 156

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Part II Virasoro Symmetry and AdS3 Gravity

6 The Virasoro Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.1 Diffeomorphisms of the Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.1.1 Infinite-Dimensional Lie Groups . . . . . . . . . . . . . . . . . . 1646.1.2 The Group of Diffeomorphisms of the Circle . . . . . . . . 1656.1.3 Topology of DiffðS1Þ . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.1.4 Adjoint Representation and Vector Fields . . . . . . . . . . . 1696.1.5 Primary Fields on the Circle . . . . . . . . . . . . . . . . . . . . . 1716.1.6 Coadjoint Representation of DiffðS1Þ . . . . . . . . . . . . . . 1726.1.7 Exponential Map and Vector Flows . . . . . . . . . . . . . . . 174

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6.2 Virasoro Cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.2.1 The Gelfand-Fuks Cocycle . . . . . . . . . . . . . . . . . . . . . . 1766.2.2 The Bott-Thurston Cocycle . . . . . . . . . . . . . . . . . . . . . . 1796.2.3 Primary Cohomology of VectðS1Þ . . . . . . . . . . . . . . . . . 1826.2.4 Primary Cohomology of DiffðS1Þ . . . . . . . . . . . . . . . . . 183

6.3 On the Schwarzian Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.3.1 The Schwarzian Derivative is a Cocycle . . . . . . . . . . . . 1846.3.2 Projective Invariance of the Schwarzian . . . . . . . . . . . . 187

6.4 The Virasoro Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.4.1 Centrally Extended Groups Revisited . . . . . . . . . . . . . . 1926.4.2 Virasoro Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.4.3 Adjoint Representation and Virasoro Algebra . . . . . . . . 1946.4.4 Coadjoint Representation. . . . . . . . . . . . . . . . . . . . . . . . 1966.4.5 Kirillov-Kostant Bracket . . . . . . . . . . . . . . . . . . . . . . . . 198

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7 Virasoro Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.1 Coadjoint Orbits of the Virasoro Group . . . . . . . . . . . . . . . . . . . 201

7.1.1 Centerless Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . 2027.1.2 Basic Properties of Centrally Extended Orbits . . . . . . . . 2037.1.3 Hill’s Equation and Monodromy . . . . . . . . . . . . . . . . . . 2067.1.4 Winding Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.2 Virasoro Orbit Representatives . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.2.1 Prelude: Conjugacy Classes of SLð2;RÞ . . . . . . . . . . . . 2147.2.2 Elliptic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2167.2.3 Degenerate Parabolic Orbits . . . . . . . . . . . . . . . . . . . . . 2197.2.4 Hyperbolic Orbits Without Winding . . . . . . . . . . . . . . . 2207.2.5 Hyperbolic Orbits with Winding . . . . . . . . . . . . . . . . . . 2227.2.6 Non-degenerate Parabolic Orbits . . . . . . . . . . . . . . . . . . 2257.2.7 Summary: A Map of Virasoro Orbits . . . . . . . . . . . . . . 228

7.3 Energy Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2307.3.1 Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2307.3.2 The Average Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.3.3 Orbits with Constant Representatives . . . . . . . . . . . . . . 2347.3.4 Orbits Without Constant Representatives. . . . . . . . . . . . 2377.3.5 Summary: A New Map of Virasoro Orbits . . . . . . . . . . 238

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8 Symmetries of Gravity in AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.1 Generalities on Three-Dimensional Gravity . . . . . . . . . . . . . . . . 242

8.1.1 Einstein Gravity in Three Dimensions . . . . . . . . . . . . . . 2428.1.2 Boundary Conditions and Boundary Terms . . . . . . . . . . 2448.1.3 Asymptotic Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 246

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8.2 Brown–Henneaux Metrics in AdS3 . . . . . . . . . . . . . . . . . . . . . . . 2508.2.1 Geometry of AdS3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.2.2 Brown–Henneaux Boundary Conditions . . . . . . . . . . . . 2538.2.3 Asymptotic Killing Vector Fields . . . . . . . . . . . . . . . . . 2568.2.4 On-Shell Brown–Henneaux Metrics . . . . . . . . . . . . . . . 2588.2.5 Surface Charges and Virasoro Algebra . . . . . . . . . . . . . 2598.2.6 Zero-Mode Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8.3 The Phase Space of AdS3 Gravity . . . . . . . . . . . . . . . . . . . . . . . 2638.3.1 AdS3 Metrics as CFT2 Stress Tensors . . . . . . . . . . . . . . 2648.3.2 Boundary Gravitons and Virasoro Orbits. . . . . . . . . . . . 2658.3.3 Positive Energy Theorems . . . . . . . . . . . . . . . . . . . . . . . 267

8.4 Quantization and Virasoro Representations. . . . . . . . . . . . . . . . . 2688.4.1 Highest-Weight Representations of slð2;RÞ. . . . . . . . . . 2688.4.2 Virasoro Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718.4.3 Virasoro Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2758.4.4 Dressed Particles and Quantization . . . . . . . . . . . . . . . . 278

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Part III BMS3 Symmetry and Gravity in Flat Space

9 Classical BMS3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.1 BMS Metrics in Three Dimensions. . . . . . . . . . . . . . . . . . . . . . . 287

9.1.1 Three-Dimensional Minkowski Space . . . . . . . . . . . . . . 2889.1.2 Poincaré Symmetry at Null Infinity . . . . . . . . . . . . . . . . 2909.1.3 BMS3 Fall-Offs and Asymptotic Symmetries . . . . . . . . 2919.1.4 On-Shell BMS3 Metrics. . . . . . . . . . . . . . . . . . . . . . . . . 2959.1.5 Surface Charges and BMS3 Algebra . . . . . . . . . . . . . . . 2969.1.6 Zero-Mode Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

9.2 The BMS3 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3009.2.1 Exceptional Semi-direct Products . . . . . . . . . . . . . . . . . 3019.2.2 Defining BMS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3069.2.3 Adjoint Representation and bms3 Algebra. . . . . . . . . . . 3099.2.4 Coadjoint Representation. . . . . . . . . . . . . . . . . . . . . . . . 3119.2.5 Some Cohomology* . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

9.3 The BMS3 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3159.3.1 Phase Space as a Coadjoint Representation . . . . . . . . . . 3159.3.2 Boundary Gravitons and BMS3 Orbits . . . . . . . . . . . . . 3179.3.3 Positive Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . 317

9.4 Flat Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3189.4.1 From DiffðS1Þ to BMS3 . . . . . . . . . . . . . . . . . . . . . . . . 3189.4.2 From Witt to bms3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3209.4.3 Stress Tensors and Central Charges. . . . . . . . . . . . . . . . 3229.4.4 The Galilean Conformal Algebra. . . . . . . . . . . . . . . . . . 323

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

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10 Quantum BMS3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.1 BMS3 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

10.1.1 Orbits and Little Groups . . . . . . . . . . . . . . . . . . . . . . . . 33010.1.2 Mass, Supermomentum, Central Charge . . . . . . . . . . . . 33310.1.3 Measures on Superrotation Orbits . . . . . . . . . . . . . . . . . 33610.1.4 States of BMS3 Particles . . . . . . . . . . . . . . . . . . . . . . . . 33810.1.5 Dressed Particles and Quantization . . . . . . . . . . . . . . . . 34110.1.6 The BMS3 Vacuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34410.1.7 Spinning BMS3 Particles . . . . . . . . . . . . . . . . . . . . . . . . 34610.1.8 BMS Particles in Four Dimensions? . . . . . . . . . . . . . . . 347

10.2 BMS Modules and Flat Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 35010.2.1 Poincaré Modules in Three Dimensions . . . . . . . . . . . . 35110.2.2 Induced Modules for bms3 . . . . . . . . . . . . . . . . . . . . . . 35510.2.3 Representations of the Galilean Conformal Algebra . . . 359

10.3 Characters of the BMS3 Group. . . . . . . . . . . . . . . . . . . . . . . . . . 36210.3.1 Massive Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36210.3.2 Comparison to Poincaré and Virasoro . . . . . . . . . . . . . . 36510.3.3 Vacuum Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

11 Partition Functions and Characters . . . . . . . . . . . . . . . . . . . . . . . . . . 37511.1 Rotating Canonical Partition Functions . . . . . . . . . . . . . . . . . . . . 376

11.1.1 Heat Kernels and Method of Images . . . . . . . . . . . . . . . 37611.1.2 Bosonic Higher Spins . . . . . . . . . . . . . . . . . . . . . . . . . . 37811.1.3 Partition Functions and BMS3 Characters . . . . . . . . . . . 38511.1.4 Relation to Poincaré Characters . . . . . . . . . . . . . . . . . . . 386

11.2 Representations and Characters of Flat WN . . . . . . . . . . . . . . . . 38811.2.1 Higher Spins in AdS3 and WN Algebras. . . . . . . . . . . . 38811.2.2 Flat W3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39211.2.3 Flat W3 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39511.2.4 Flat WN Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

11.3 Flat W3 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39911.3.1 Ultrarelativistic and Non-relativistic Limits of W3 . . . . 40011.3.2 Induced Modules for the Flat W3 Algebra . . . . . . . . . . 402

11.4 Super-BMS3 and Flat Supergravity. . . . . . . . . . . . . . . . . . . . . . . 40411.4.1 Fermionic Higher Spin Partition Functions . . . . . . . . . . 40511.4.2 Supersymmetric BMS3 Groups . . . . . . . . . . . . . . . . . . . 40911.4.3 Supersymmetric BMS3 Particles . . . . . . . . . . . . . . . . . . 413

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

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