Quantum Gravity in Three Dimensions from Higher-Spin Holography by Hai Siong Tan A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Ori Ganor, Chair Professor Petr Horava Professor Vaughan Jones Spring 2013 1 Abstract Quantum Gravity in Three Dimensions from Higher-Spin Holography by Hai Siong Tan Doctor of Philosophy in Physics University of California, Berkeley Professor Ori Ganor, Chair In this thesis, I explore various aspects of quantum gravity in three dimensions from the per- spective of higher-spin holography in anti-de Sitter spacetime. The bulk theory is a higher- spin Vasiliev gravity theory of which topological sector can be described by a Chern-Simons theory equipped with some suitable Lie (super-)algebra, whereas the boundary conformal field theory is conjectured to be a coset minimal model which contains W symmetries. I present new black hole solutions and investigate the thermodynamics of these solutions, in particular, I establish a relationship among black hole thermodynamics, asymptotic sym- metries and W algebras. I also construct new conical defect solutions, supersymmmetric RG flow solutions in the bulk gravity theories, and present the bulk-boundary propagator for scalar fields interacting with a higher-spin black hole. The main examples used in this thesis are illustrated in the framework of SL(N) and SL(NjN − 1) Chern-Simons theories, and I point out how these new solutions can be used to yield some insights into the nature of quantum gravity. i Dedicated to my Mum, and in fond memory of my Dad. ii Acknowledgement It is a pleasure to hereby give thanks to a number of amazing individuals from whom I have received support over the past five years. Foremost, I would like to express my deepest gratitude to my thesis advisor Prof. Ori Ganor for his guidance. Ori is a tremendously creative and knowledgeable string theorist, and it has been an awe-inspiring experience for me to witness his genius at work. He seems to be able to derive anything in physics that is sensible from scratch. On top of his technical prowess, his good-natured and caring personality has always made it easy for me to seek his advice on various issues. From getting research funds to letters for summer schools and post-doc applications, he has always been there. I have learnt that being a graduate student in string theory is a very challenging position to be in, yet Ori has also made my journey much more joyful and memorable than it could have been. I have learnt so many things from him, about physics and beyond physics. I would like to thank Prof. Petr Hoˇrava for his generous support in many important matters. I am very priveleged to have the chance to learn many things from him when he taught the QFT and string theory courses, to have him on my qualifying exam, and to learn from him about his exciting proposal for a new model of quantum gravity back in 2009. His encouragement has played a crucial role in my learning process. I also thank Prof. Vaughan Jones for agreeing to be on my qualifying exam and thesis committee. It is a humbling experience to have him on my committee, and I thank him for his time and kindness. Prof. Sergey Cherkis, who was a former visitor taught me many things about physics, especially about instantons. I had many enjoyable discussions with him when he was at Berkeley. Prof. Neal Synderman, who was my instructor for a quantum mechanics course when I was an exchange undergraduate student at Berkeley many years ago, has inspired me much not only by his depth of physics knowledge, but also by his rich life experiences. I enjoy listening to his numerous interesting encounters with many big characters in physics. I am really glad to know Sergey and Neal in person. Apart from them, I had the chance to benefit from many conversations with a number of physicists, in particular, I thank Mina Aganagic, Jan de Boer, Steve Carlip, Sven Bjarke Gudnason, Yoon Pyo Hong, Bom Soo Kim, Per Kraus, Emanuele Latini, Juan Maldacena, Hitoshi Murayama, Eric Perlmutter and Meng Chwan Tan. Finally, I thank my good friend Phuntsok Tseten for many hours of badminton sparring sessions, and three angels Anne Takizawa, Donna Sakima and Kathy Lee for being so kind in helping me plough through various administrative issues along the way. iii Preface Some of the original materials in this thesis have been published in the following two papers 1. H. S. Tan, \Exploring Three-dimensional Higher-Spin Supergravity based on sl(N |N - 1) Chern-Simons theories," JHEP 1211, 063 (2012) [arXiv:1208.2277 [hep-th]]. 2. H. S. Tan, \Aspects of Three-dimensional Spin-4 Gravity," JHEP 1202, 035 (2012) [arXiv:1111.2834 [hep-th]]. Other papers not directly related to the theme of this thesis and which were published during the course of my study are 3. O. J. Ganor, Y. P. Hong, R. Markov and H. S. Tan, \Static Charges in the Low-Energy Theory of the S-Duality Twist," JHEP 1204, 041 (2012) [arXiv:1201.2679 [hep-th]]. 4. O. J. Ganor, Y. P. Hong and H. S. Tan, \Ground States of S-duality Twisted N=4 Super Yang-Mills Theory," JHEP 1103, 099 (2011) [arXiv:1007.3749 [hep-th]]. 5. H. S. Tan, \First order Born-Infeld Hydrodynamics via Gauge/Gravity Duality," JHEP 0904, 131 (2009) [arXiv:0903.3424 [hep-th]]. My research journey has led me to a variety of project topics in string theory and quantum gravity. Projects 3 and 4 involve the study of duality-twisted field theories using string- theoretic techniques, and were done during my second to fourth year of study. They gave me many opportunities to learn from my advisor Prof. Ganor and his other graduate students. Purely to keep to a more uniform theme, I have chosen to write this thesis based on projects 1 and 2 which involve an application of the principle of holography to understand quantum gravity in three dimensions. Some of the materials discussed in this thesis were presented in a seminar at University of California at Davis, Department of Mathematics in April 2012. iv Contents Acknowledgement ii Preface iii 1 Introduction 1 2 On 3D Quantum gravity: sketches from different perspectives 6 2.1 On pure gravity in three dimensions . 6 2.2 Playing with Wilson loops . 9 2.2.1 Worm-holes and black holes from the Mobius transformations of the Poincar´edisk . 10 2.2.2 Where the Wilson loops are . 12 2.3 Notes on string theoretic realizations . 14 2.4 Higher-spin holography: quantum Vasiliev gravity from coset CFTs . 19 3 Adventures in Vasiliev Gravity I: probing spacetime with scalar fields 21 3.1 Some aspects of the master field equations . 21 3.2 On the vacuum/topological sector . 23 3.3 About the infinite-dimensional Lie algebra hs[λ] . 25 3.4 On coupling to the matter fields . 27 3.5 Generalized Klein-Gordan equations . 28 3.6 Propagators in the background of a higher-spin black hole . 29 CONTENTS v 3.6.1 Traces of master fields from an infinite-dimensional representation of sl(2) . 30 3.6.2 Chiral higher spin deformations of AdS3 . 32 3.6.3 Scalar propagator in the BTZ background and physical interpretations 34 3.6.4 A derivation of the scalar propagator on a higher-spin black hole . 35 4 Adventures in Vasiliev Gravity II: the topological sector 43 4.1 Chern-Simons theories as the topological sector of the bulk . 43 4.2 Non-principal Embeddings . 47 4.3 Supersymmetric generalizations and a class of higher-spin supergravity theories 50 4.3.1 General remarks on higher-spin AdS3 SUGRA as a Chern-Simons theory . 50 4.3.2 About sl(NjN − 1) in the Racah basis . 51 4.3.3 A class of gauge transformations . 53 4.3.4 On the supersymmetry of osp(1j2) and osp(2j2) Chern-Simons theories 55 4.3.5 N = 2 supersymmetry and Killing spinors . 57 4.3.6 Solving the Killing spinor equations . 58 4.3.7 On the u(1) gauge field . 60 5 Notes on the boundary CFT 62 5.1 A review of important points about the coset models . 62 5.2 Two different limits for the holography conjecture . 65 5.2.1 The 't Hooft limit . 65 5.2.2 The semiclassical limit . 68 6 Asymptotic Spacetime Symmetries and W algebras 71 6.1 On W algebras . 71 6.2 Asymptotic Spacetime Symmetries . 73 6.3 Ward Identities and the case of SL(4) Chern-Simons theory . 75 6.4 Super-W algebras and SL(NjN − 1) Chern-Simons theories . 79 CONTENTS vi 6.4.1 On super-W symmetries as asymptotic spacetime symmetry . 79 6.4.2 Recovering the N = 2 super-Virasoro algebra . 80 7 Higher-spin black holes 84 7.1 The role of Wilson loops . 84 7.2 Higher-spin black holes and spacetime geometry . 87 7.2.1 The case of SL(N) black holes . 87 7.3 On the first law of higher-spin black hole thermodynamics . 91 7.4 The case of SL(NjN − 1) black holes . 96 7.4.1 Holonomy conditions and higher-spin black holes . 96 7.5 Other classical solutions: RG flow solutions . 101 7.5.1 Supersymmetric solutions with higher-spin fields . 101 7.6 Extension of results to sl(NjN − 1) for a general finite N.