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Conformal and Nearly Conformal Theories at Large N Conformal and Nearly Conformal Theories at Large N Grigory M. Tarnopolskiy A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Advisers: Professors Igor R. Klebanov and Alexander M. Polyakov September 2017 c Copyright by Grigory M. Tarnopolskiy, 2017. All rights reserved. Abstract In this thesis we present new results in conformal and nearly conformal field theories in various dimensions. In chapter two, we study different properties of the conformal Quantum Electrodynamics (QED) in continuous dimension d. At first we study conformal QED using large Nf methods, where Nf is the number of massless fermions. We compute its sphere free energy as a function of d, ignoring the terms of order 1=Nf and higher. For finite Nf we use the expansion. Next we use a large Nf diagrammatic approach to calculate the leading corrections to CT , the coefficient of the two-point function of the stress-energy tensor, and CJ , the coefficient of the two- point function of the global symmetry current. We present explicit formulae as a function of d and check them versus the expectations in 2 and 4 dimensions. − In chapter three, we discuss vacuum stability in 1 + 1 dimensional conformal field theories with external background fields. We show that the vacuum decay rate is given by a non-local two-form. This two-form is a boundary term that must be added to the effective in/out Lagrangian. The two-form is expressed in terms of a Riemann-Hilbert decomposition for background gauge fields, and is given by its novel “functional” version in the gravitational case. In chapter four, we explore Tensor models. Such models possess the large N limit dominated by the melon diagrams. The quantum mechanics of a real anti-commuting rank-3 tensor has a large N limit similar to the Sachdev-Ye-Kitaev (SYK) model. We also discuss the quantum mechanics of a complex 3-index anti-commuting tensor and argue that it is equivalent in the large N limit to a version of SYK model with complex fermions. Finally, we discuss models of a commuting tensor in dimension d. We study the spectrum of the large N quantum field theory of bosonic rank-3 tensors using the Schwinger-Dyson equations. We compare some of these results with the 4 expansion, finding perfect agreement. We also study the spectra of bosonic − theories of rank q 1 tensors with φq interactions. − iii Acknowledgements I am deeply grateful to my advisers, Igor Klebanov and Alexander Polyakov. For the past five years, I have had the privilege of collaborating with Igor Kle- banov on various amazing projects in theoretical physics. I first met Igor in the Labyrinth Books store five years ago, when I was buying a book in preparation for taking his class on String Theory. In this class, I was impressed by Igor’s outstanding ability to explain complicated concepts in a simple manner. His lectures always had a very clear structure and logic. His remarkable intuition in physics has always made our discussions both interesting and inspiring. Additionally, his deep knowledge of physics, history and literature, combined with his wonderful sense of humor, has made for many enjoyable hours of conversation. Igor has invested an extraordinary amount of time and care in my education. Without the steady guidance, constant encour- agement and generous support he has provided throughout my time at Princeton, I would not be where I am today. I am also very grateful to Alexander Markovich Polyakov for sharing his time and amazing ideas with me. In my second year I had the good fortune of taking his class on Quantum Field Theory. I was so impressed with this learning experience that I have since taken the same class every year; if I had more time at Princeton, I would have continued to do so. Each year this class is full of new striking ideas and topics in quantum field theory. Working with Alexander Markovich is an exciting experience; he can be relied upon to divine intuitively solutions to problems that normally require several pages of mathematical calculations to resolve. His physical perception and physical approach to theoretical problems in quantum field theory is nothing short of incredible. I would also like to express my sincerest gratitude to Simone Giombi, with whom I have collaborated on many projects during my time at Princeton. I have extensively enjoyed doing physics with Simone, whose insight and deep knowledge of theoretical iv physics have always motivated me. I am additionally very grateful to Simone for agreeing to be a reader of this thesis. It also has been a wonderful experience collaborating with Ksenia Bulycheva, Shai Chester, Kenan Diab, Lin Fei, Luca Iliesiu, Mark Mezei, Alexey Milekhin, Silviu Pufu, Guilherme Pimentel and Benjamin Safdi. I would like to thank Daniel Marlow and Herman Verlinde for serving on my thesis committee. I am extremely grateful to my previous adviser, Alexander Belavin, for educat- ing me in quantum field theory and theoretical physics at the Landau Institute for Theoretical Physics, Moscow. I would also like to thank the Graduate School and Physics Department for awarding me with the Myhrvold-Havranek scholarship and Kusaka Memorial Prize in Physics. I would like to thank my friends and colleagues in the Physics Department: Vasily Alba, Ilya Belopolski, Farzan Beroz, Ksenia Bulycheva, Shai Chester, Will Coul- ton, Kolya Dedushenko, Kenan Diab, Yale Fan, Lin Fei, Joshua Hardenbrook, Huan He, Po-Shen Hsin, Luca Iliesiu, Jiaqi Jiang, Vladimir Kirilin, Dima Krotov, Aitor Lewkowycz, Aaron Levy, Jeongseog Lee, Mark Mezei, Lauren McGough, Debayan Mitra, Alexey Milekhin, Victor Mikhaylov, Sarthak Parikh, Fedor Popov, Justin Rip- ley, Zach Sethna, Yu Shen, Siddharth Mishra Sharma, Joaquin Turiaci, Bin Xu, Zhenbin Yang and Sasha Zhiboedov. Finally, I would like to express gratitude to my family for their love and support. I would especially like to express my heartfelt thanks to my cousin, Misha Gorokhovich, who supported me during all my time at Princeton. These years would have been much harder without his constant encouragement and help. v To my parents vi Contents Abstract..................................... iii Acknowledgements............................... iv 1 Introduction1 1.1 Quantum Field Theory and Critical Phenomena............1 1.2 Landau theory...............................3 1.3 Wilson’s approach to the Renormalization Group...........5 1.4 Wilson-Fisher critical point and expansion..............9 1.5 Large N approximation.......................... 11 1.6 Conformal Quantum Electrodynamics................. 13 1.7 Tensor models............................... 14 1.8 Overview of the thesis.......................... 17 2 Conformal QED 20 2.1 Introduction and Summary........................ 20 2.1.1 Conformal Quantum Electrodynamics............. 20 2.1.2 Sphere free energy and the F -theorem in QED........ 22 2.1.3 Two point function of the stree-energy tensor in QED..... 24 2.2 Sphere free energy of Maxwell theory on Sd .............. 28 2.3 Conformal QED at large N ....................... 34 2.4 Sphere free energy of the QED at large N ............... 36 vii 2.4.1 Comments on d > 4 ........................ 41 2.5 Sphere free energy of the QED in the expansion........... 43 2.6 Padé approximation and the F -theorem................ 48 2.7 Calculation of CJ1 and CT 1 ....................... 53 top 2.8 CJ for the Topological Current in d = 3 ................ 59 2.9 4 Expansion of CJ and CT ..................... 60 − 2.10 Another Estimate for Symmetry Breaking in QED3 .......... 63 2.11 CT for Large Nf QCDd .......................... 64 2.12 Appendix A. Eigenvalues of the kernel Kµν ............... 67 2.13 Appendix B. Zonal spherical harmonics in continuous dimesion.... 70 2.13.1 Rank 0 zonal harmonics..................... 71 2.13.2 Rank 1 zonal harmonics..................... 72 2.13.3 Computations in stereographical coordinates.......... 73 2.13.4 Rank 1 kernel decomposition................... 77 2.14 Appendix C. Calculation of G2 and G4 ................. 78 2.15 Appendix D. Calculation of ZT ..................... 82 2.16 Appendix E. Results for JJ and TT diagrams........... 87 h i h i 3 Particle Production in 1 + 1 CFT 91 3.1 Introduction................................ 91 3.2 Vacuum decay in an Abelian background................ 93 3.3 Vacuum decay in a non-Abelian background.............. 97 3.4 Vacuum decay in the gravitational field................. 102 3.5 Appendix A. Boundary conditions on induced currents and alternative derivation of the boundary actions.................... 106 3.6 Appendix B. Gauge-gravity duality in two dimensions......... 110 3.7 Appendix C. Non-Abelian and gravitational corrections to Caldeira- Leggett formula.............................. 112 viii 4 Tensor Models 115 4.1 Introduction................................ 115 4.2 Melonic Dominance in the O(N)3 Symmetric Theories........ 121 4.3 O(N)3 Quantum Mechanics and the SYK Model............ 125 4.3.1 Models with a Complex Fermion................ 132 4.4 O(N)3 bosonic tensors.......................... 134 4.4.1 Spectrum of two-particle operators............... 136 4.4.2 Spectrum of higher-spin operators................ 139 4.4.3 Complex Large N Fixed Point in d = 4 ........... 142 − 4.4.4 Generalization to Higher q .................... 145 4.4.5 Higher spin operators....................... 148 4.4.6 A Melonic φ6 Theory in 2:99 Dimensions............ 149 4.5 Appendix A. Matrix model in d = 4 dimension........... 151 − Bibliography 154 ix Chapter 1 Introduction 1.1 Quantum Field Theory and Critical Phenomena Various physical systems containing large numbers of particles can be well described by Statistical Mechanics. In many cases it can be described by weakly interacting constituents which are called quasiparticles. Amazingly, this approximation often gives very good and precise results. Nevertheless, there are phenomena in Nature, which cannot be approximated by non-interacting or weakly-interacting particles. One well-known example of this kind is a phase transition.
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