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UCLA Electronic Theses and Dissertations UCLA UCLA Electronic Theses and Dissertations Title The Holographic Geometry of Conformal Blocks Permalink https://escholarship.org/uc/item/31s3h767 Author Snively, River Curry Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles The Holographic Geometry of Conformal Blocks A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by River Curry Snively 2018 c Copyright by River Curry Snively 2018 ABSTRACT OF THE DISSERTATION The Holographic Geometry of Conformal Blocks by River Curry Snively Doctor of Philosophy in Physics University of California, Los Angeles, 2018 Professor Per J. Kraus, Chair Conformal blocks are the building blocks of correlation functions in conformal field theory. They figure prominently in the study of quantum gravity in light of the AdS/CFT corre- spondence which identifies conformal field theory as a holographic representation of quantum gravity in Anti de-Sitter space. Thus while conformal blocks are fundamentally CFT ob- jects, it promises to be both conceptually and computationally useful to obtain for them a description on the AdS side. We lay out that description here, extending the holographic dictionary relating CFT objects to AdS objects by adding to it an entry for the conformal blocks themselves. The holographic dual to a global conformal block is a Feynman diagram- like object we call a geodesic Witten diagram, and a similar picture applies to semiclassical Virasoro blocks as well. We discuss ways in which this new picture for an old tool can help to further our understanding of how spacetime emerges from the quantum information structure of conformal field theory. ii The dissertation of River Curry Snively is approved. Eric D'Hoker Lincoln Chayes Per J. Kraus, Committee Chair University of California, Los Angeles 2018 iii This work is dedicated to my parents, who introduced me to blocks. iv TABLE OF CONTENTS 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Conformal field theory and conformal blocks . .2 1.1.1 Conformal symmetry . .3 1.1.2 The state/operator correspondence . .5 1.1.3 Primaries and the OPE . .6 1.1.4 Conformal blocks . .8 1.1.5 The conformal bootstrap . 10 1.2 Generalized Free CFT . 11 1.2.1 Large-N gauge theory . 12 1.2.2 The axioms of generalized free conformal field theory . 14 1.2.3 Multi-trace operators from the bootstrap . 15 1.3 AdS and Holography . 17 1.3.1 Introducing Anti-de Sitter space and its conformal boundary . 17 1.3.2 Quantum fields in AdS and holography . 18 1.3.3 Witten diagrams and conformal blocks . 20 1.4 Holographic conformal blocks and outline of this thesis . 22 2 Worldline approach to semi-classical conformal blocks ::::::::::: 23 2.1 Introduction . 23 2.2 Conformal blocks . 27 2.2.1 Definitions . 27 2.2.2 Heavy-light correlators, and the semi-classical limit . 28 2.2.3 Monodromy method . 30 v 2.3 Semi-classical conformal blocks from bulk geodesics . 33 2.3.1 Setup . 33 2.3.2 Result for general light operators . 36 2.4 Relation between geodesic and monodromy approaches . 40 2.5 Discussion and subsequent developments . 43 Appendices :::::::::::::::::::::::::::::::::::::::: 45 2.A Recursion relation . 45 2.B Extension to operators at different times . 46 3 Geodesic Witten diagrams ::::::::::::::::::::::::::::: 49 3.1 Introduction . 49 3.2 Conformal blocks, holographic CFTs and Witten diagrams . 54 3.2.1 CFT four-point functions and holography . 54 3.2.2 A Witten diagrams primer . 57 3.2.3 Mellin space . 62 3.2.4 Looking ahead . 64 3.3 The holographic dual of a scalar conformal block . 65 3.3.1 Proof by direct computation . 66 3.3.2 Proof by conformal Casimir equation . 70 3.3.3 Comments . 75 3.4 The conformal block decomposition of scalar Witten diagrams . 78 3.4.1 An AdS propagator identity . 78 3.4.2 Four-point contact diagram . 80 3.4.3 Four-point exchange diagram . 82 3.4.4 Further analysis . 85 vi 3.4.5 Taking stock . 87 3.5 Spinning exchanges and conformal blocks . 87 3.5.1 Known results . 88 3.5.2 Geodesic Witten diagrams with spin-` exchange: generalities . 89 3.5.3 Evaluation of geodesic Witten diagram: spin-1 . 91 3.5.4 Evaluation of geodesic Witten diagram: spin-2 . 92 3.5.5 General spin: proof via conformal Casimir equation . 93 3.5.6 Comparison to double integral expression of Ferrara et. al. 95 3.5.7 Decomposition of spin-1 Witten diagram into conformal blocks . 96 3.6 Discussion and future work . 103 4 Semiclassical Virasoro Blocks from AdS3 Gravity :::::::::::::: 110 4.1 Introduction . 110 4.2 Review of semiclassical Virasoro blocks . 113 4.2.1 The heavy-light semiclassical limit . 116 4.3 Semiclassical Virasoro blocks from AdS3 gravity . 117 4.3.1 Bulk prescription . 117 4.3.2 Evaluating the geodesic Witten diagram . 120 4.3.3 Evaluating the geodesic integrals . 124 4.4 Recovering the worldline approach . 125 4.4.1 Equivalence of the minimization prescriptions . 127 4.5 Final comments . 128 References ::::::::::::::::::::::::::::::::::::::::: 130 vii LIST OF FIGURES 1.1 Conformal map from cylinder to plane . .3 1.2 Generic insertion of four operators . .8 1.3 Large-N counting in gauge theory . 13 1.4 Holographic dual of a conformal block . 21 2.1 Worldline picture for semiclassical blocks . 26 3.1 Geodesic Witten diagram for the exchange of a symmetric traceless tensor . 52 3.2 Tree-level four-point Witten diagrams for external scalar operators . 59 3.3 A scalar propagator identity . 79 3.4 Conformal partial wave decomposition of a four-point scalar contact diagram . 81 3.5 Conformal partial wave decomposition of a four-point scalar exchange diagram . 83 3.6 Decomposition of a vector exchange diagram . 97 3.7 Some examples of loop diagrams that can be decomposed using our methods . 104 3.8 Five-point conformal partial wave? . 105 4.1 The spectrum of gravity duals of large c Virasoro blocks . 112 4.2 Geodesic Witten diagram for a global conformal block . 118 4.3 Holographic semiclassical Virasoro conformal blocks . 120 viii ACKNOWLEDGMENTS I am deeply indebted to my advisor Per Kraus for sharing his knowledge and time, for always being willing to spend an hour listening to mediocre ideas from me, and for his readiness to meet seemingly any question with a perfectly well-formed pedagogical answer from first principles. It is also my pleasure to thank my coauthors on the work included in this thesis, Eliot Hijano and Eric Perlmutter, both for their contributions to the research and for being exem- plars of graduate school success. Though it is not included in this thesis, I am very proud of the non-block research I have undertaken with Allic Sivaramakrishnan and would like to thank him for his contributions and his uncanny enthusiasm for physics, which kept me going through the many phases of that endeavor. Also, to all the people I have shared an office with, I am grateful for the thought-provoking conversations about grammar, comparative linguistics, and occasionally conformal field theory. I would also like to thank David Bauer for all the inspiration he has provided over the years in the form of absurd excerpts from the arXiv. The introduction to this thesis is informed by all my attempts at explaining the context of my research to my parents and parents-in-law. I want to thank all of them for always showing an interest and for their understanding and support throughout my graduate career. Finally, I wish to thank the most important physicist in my life, my wife Emma. Bringing us together is the greatest thing physics has done for me, and that includes holding me to the planet's surface. Attached to each small quantum of knowledge in the work described below there are a thousand happy memories of our life together while the work was in progress. Co-author contributions. Chapter 2 of this thesis was published as [1] and written in collaboration with Eliot Hijano and Per Kraus. Chapter 3 was published as [3] and chapter 4 as [2], both in collaboration with Eliot Hijano, Per Kraus, and Eric Perlmutter. Each has undergone minor adjustments in the interest of cohesion, and final sections have been expanded to mention interesting follow-up work. ix Figure permission. Figure 1.4 (a) originally appeared in [5] and has been reproduced with permission from Kyriakos Papadodimas. The question answered in this thesis { what is the bulk dual of a conformal block? { I first saw posed in this figure, and I could not resist including it. x VITA 2012 B.A. Mathematics and B.A. Physics, University of California, Berkeley. 2014 M.Sc. Physics, University of California, Los Angeles. PUBLICATIONS [1] Eliot Hijano, Per Kraus, and River Snively. Worldline approach to semi-classical con- formal blocks. Journal of High Energy Physics, 07:131, 2015. [2] Eliot Hijano, Per Kraus, Eric Perlmutter, and River Snively. Semiclassical Virasoro blocks from AdS3 gravity. Journal of High Energy Physics, 12:077, 2015. [3] Eliot Hijano, Per Kraus, Eric Perlmutter, and River Snively. Witten Diagrams Re- visited: The AdS Geometry of Conformal Blocks. Journal of High Energy Physics, 01:146, 2016. [4] Per Kraus, Allic Sivaramakrishnan, and River Snively. Black holes from CFT: Univer- sality of correlators at large c. Journal of High Energy Physics, 08:084, 2017. xi CHAPTER 1 Introduction Black hole formation introduces an upper bound on the amount of information a region of space can contain, a bound proportional to the area of the region's boundary [6].
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