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Holographic Conformal Blocks University of California Los Angeles Holographic Conformal Blocks A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Eliot Hijano Cubelos 2016 c Copyright by Eliot Hijano Cubelos 2016 Abstract of the Dissertation Holographic Conformal Blocks by Eliot Hijano Cubelos Doctor of Philosophy in Physics University of California, Los Angeles, 2016 Professor Michael Gutperle, Chair The conformal block decomposition of field theory correlation functions is a powerful way of disentangling dynamical from universal properties of a field theory. The work on this thesis focuses on the holographic description of conformal blocks in the framework of the AdS/CFT correspondence. We begin by stablishing the correspondence between global conformal blocks in any number of space-time dimensions and \geodesic Witten diagrams" which are essentially ordinary Witten diagrams, except that vertices are not integrated over all of AdS, but only over bulk geodesics connecting boundary operators. We then turn to the study of Virasoro blocks in two dimensions. These are much richer objects that are known only in certain limits for the central charge and the conformal di- mensions of the operators in question. We provide a holographic description that correctly reproduces all known Virasoro blocks. We finish with the bulk description of blocks involving more complicated CFT symme- tries. The holographic picture involves Wilson lines merging at a bulk vertex. ii The dissertation of Eliot Hijano Cubelos is approved. Terence Chi-Shen Tao Per J. Kraus Michael Gutperle, Committee Chair University of California, Los Angeles 2016 iii Para mis padres . por haberlo hecho todo en la vida para que sus hijos pudieran lograr sus sue~nos iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Bulk theory . .3 1.2 Boundary theory . .4 1.3 The holographic dictionary . .5 1.4 Correlators in the field theory - Conformal partial waves . .7 1.5 Correlators in the bulk - Witten diagrams . .8 1.6 Partial waves VS Witten diagrams . 10 2 Witten diagrams revisited: The AdS geometry of conformal blocks ::: 12 2.1 Introduction . 12 2.2 Conformal blocks, holographic CFTs and Witten diagrams . 17 2.2.1 CFT four-point functions and holography . 17 2.2.2 A Witten diagrams primer . 21 2.2.3 Mellin space . 26 2.2.4 Looking ahead . 28 2.3 The holographic dual of a scalar conformal block . 28 2.3.1 Proof by direct computation . 30 2.3.2 Proof by conformal Casimir equation . 34 2.3.3 Comments . 40 2.4 The conformal block decomposition of scalar Witten diagrams . 43 2.4.1 An AdS propagator identity . 43 2.4.2 Four-point contact diagram . 45 2.4.3 Four-point exchange diagram . 48 v 2.4.4 Further analysis . 50 2.4.5 Taking stock . 53 2.5 Spinning exchanges and conformal blocks . 53 2.5.1 Known results . 54 2.5.2 Geodesic Witten diagrams with spin-` exchange: generalities . 55 2.5.3 Evaluation of geodesic Witten diagram: spin-1 . 57 2.5.4 Evaluation of geodesic Witten diagram: spin-2 . 59 2.5.5 General `: proof via conformal Casimir equation . 60 2.5.6 Comparison to double integral expression of Ferrara et. al. 62 2.5.7 Decomposition of spin-1 Witten diagram into conformal blocks . 63 2.6 Discussion and future work . 71 3 Semiclassical Virasoro Blocks from AdS3 Gravity :::::::::::::: 76 3.1 Introduction . 76 3.2 Review of semiclassical Virasoro blocks . 80 3.2.1 The heavy-light semiclassical limit . 83 3.3 Semiclassical Virasoro blocks from AdS3 gravity . 84 3.3.1 Bulk prescription . 84 3.3.2 Evaluating the geodesic Witten diagram . 87 3.4 Semiclassical Virasoro blocks from AdS3 higher spin gravity . 91 3.4.1 Higher spin fields in AdS3 ........................ 92 3.4.2 Setup . 96 3.4.3 Detailed calculation . 98 3.5 Final comments . 104 3.A Evaluation of the geodesic integrals . 105 vi 3.B Recovering the worldline approach . 108 3.B.1 Equivalence of the minimization prescriptions . 109 3.C Details of some higher spin gravity calculations . 111 3.C.1 sl(N; R) algebra . 111 3.C.2 Deriving (3.4.39) . 112 4 Holographic conformal blocks from interacting Wilson lines ::::::: 114 4.1 Introduction . 114 4.2 Correlation functions: general formulation . 118 4.2.1 Preliminaries . 118 4.2.2 Correlators . 119 4.3 Four-point functions . 122 4.3.1 Conformal block decomposition . 123 4.3.2 Conformal blocks from singlets . 124 4.3.3 Crossing symmetry . 125 4.4 General SL(2) result . 127 4.5 SL(3) Result . 131 4.6 An SL(N) example . 137 4.6.1 Exchange of 1 ............................... 138 4.6.2 Exchange of Adj ............................. 139 4.6.3 The four-point function G ................... 140 φ+φ+φ+φ+ 4.7 Heavy-light Virasoro blocks . 142 4.8 Discussion . 145 4.A Conformal invariance of correlators . 146 4.B Computation of three-point function . 148 vii 4.C SL(N) Conventions and Facts . 150 4.C.1 Conventions . 150 4.C.2 Irreducible Tensors . 151 4.D Removing Traces of Symmetric Tensors . 152 4.E Details of SL(3) Calculation . 154 References ::::::::::::::::::::::::::::::::::::::::: 159 viii Acknowledgments Firstly, I would like to express my sincere gratitude to my advisors Profs. Per Kraus and Michael Gutperle for their continuous guidance during my PhD studies. Their patience, enthusiasm and immense knowledge have motivated me to keep doing research in physics, and I hope one day I can emulate the way they work and interact with graduate students. I would also like to thank Prof. Eric D'Hoker for insightful comments and encouragement that have proven useful to understand and widen the contents of this thesis. A great amount of people have contributed enormously to the research performed in this work, and I owe my gratitude to them. This includes Dr. Eric Perlmutter, River Snively, Ashwin Hegde, Mert Besken, Prof. Alejandra Castro, Prof. Jan de Boer, Dr. Juan Jottar, and Dr. Joshua Samani. I would like to show my ingratitude to Francis Duplessis and Grant Salton. Without them this thesis would have been much more complete. In the same line of thought I would like to acknowledge my brother Oliver and my friend Pau. Their continuous ruining of my gaming experiences has led me to spend much more time doing physics. I would also like to thank my fellow officemate Allic Sivaramakrishnan for stimulating, infuriating conversations. Last but not least I would like to thank my family; My parents, my brother, and the Hu for supporting me psychologically during the last four years. Without them, this adventure would have been much less pleasant. Contribution of authors Chapter 2 and 3 of this thesis has been published as [1] and [2] in August 2015 with the collaboration of Per Kraus, Eric Perlmutter and River Snively. Chapter 4 has been published as [3] in March 2016 with the collaboration of Mert Besken, Ashwin Hedge, and Per Kraus. The papers have been modified only to agree with UCLA's thesis format. ix Vita 2010 B.S. (Physics) , UNED (Madrid, Spain). 2012 M.Sc. (Physics), McGill, Canada. 2014-2015 Teaching Assistant, Physics Department, UCLA. Publications \Holographic conformal blocks from interacting Wilson lines," M. Besken, A. Hegde, E. Hijano and P. Kraus, arXiv:1603.07317 [hep-th]. \Semiclassical Virasoro blocks from AdS3 gravity," E. Hijano, P. Kraus, E. Perlmutter and R. Snively, JHEP 1512, 077 (2015). doi:10.1007/JHEP12(2015)077. [arXiv:1508.04987 [hep-th]]. \Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks," E. Hijano, P. Kraus, E. Perlmutter and R. Snively, JHEP 1601, 146 (2016). doi:10.1007/JHEP01(2016)146. [arXiv:1508.00501 [hep-th]]. \Worldline approach to semi-classical conformal blocks," E. Hijano, P. Kraus and R. Snively, JHEP 1507, 131 (2015). doi:10.1007/JHEP07(2015)131. [arXiv:1501.02260 [hep-th]]. \Higher spin entanglement and N conformal blocks," W x J. de Boer, A. Castro, E. Hijano, J. I. Jottar and P. Kraus, JHEP 1507, 168 (2015). doi:10.1007/JHEP07(2015)168. [arXiv:1412.7520 [hep-th]]. \A new spin on entanglement entropy," E. Hijano and P. Kraus, JHEP 1412, 041 (2014). doi:10.1007/JHEP12(2014)041. [arXiv:1406.1804 [hep-th]]. \Lifshitz black holes in higher spin gravity," M. Gutperle, E. Hijano and J. Samani, JHEP 1404, 020 (2014). doi:10.1007/JHEP04(2014)020. [arXiv:1310.0837 [hep-th]]. \Matching four-point functions in higher spin AdS3/CFT2," E. Hijano, P. Kraus and E. Perlmutter, JHEP 1305, 163 (2013). doi:10.1007/JHEP05(2013)163. [arXiv:1302.6113 [hep-th]]. \Smoothed Transitions in Higher Spin AdS Gravity," S. Banerjee, A. Castro, S. Hellerman, E. Hijano, A. Lepage-Jutier, A. Maloney and S. Shenker, Class. Quant. Grav. 30, 104001 (2013). doi:10.1088/0264-9381/30/10/104001 [arXiv:1209.5396 [hep-th]]. \Unitarity Bounds in AdS3 Higher Spin Gravity," A. Castro, E. Hijano and A. Lepage-Jutier, JHEP 1206, 001 (2012). doi:10.1007/JHEP06(2012)001. [arXiv:1202.4467 [hep-th]]. \Black Holes and Singularity Resolution in Higher Spin Gravity," A. Castro, E. Hijano, A. Lepage-Jutier and A. Maloney, JHEP 1201, 031 (2012). doi:10.1007/JHEP01(2012)031. [arXiv:1110.4117 [hep-th]]. xi CHAPTER 1 Introduction The AdS/CFT correspondence relates two completely different theories of physics to one another. On one hand there is a theory of dynamical space-time like string theory or general relativity (AdS), and on the other hand there is an interacting quantum field theory with conformal symmetry (CFT). The two sides of the duality are interesting independently of each other. String theory addresses many deep fundamental questions about the nature of our universe.
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