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Defects in Conformal Field Theories

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Defects in Conformal Field Theories Scuola Normale Superiore di Pisa CLASSE DI SCIENZE Ph.D. Thesis Defects in Conformal Field Theories. Candidato: Relatore: Marco Meineri Prof. Davide Gaiotto Supervisore interno: Prof. Enrico Trincherini Anno Accademico 2015/2016 i My supervisor, Davide Gaiotto, deserves a special acknowledgment, not only for his support during the years of my Ph.D. Meeting him catapulted me into an entirely different world, in many senses. A new continent, where squirrels stroll around the cities instead of cats, the same shops and restaurants appear replicated on the main street, and winter teaches people to be courteous. A new physics institute, whose almost scary perfection is thankfully mitigated by the considerable amount of entropy that a large group of physicists generates. And new fields of research, more abstract, scattered with fundamental questions and filled with a myriad of technical details. As for the questions, I think I was never able to answer even one of those that Davide posed during these years. However, those questions helped me finding my own, less ambitious, problems to solve, and even the sketch of a research program. The interaction with Davide has something magic, or perhaps esoteric. When explaining the terms of a problem to him, I would typically first understand from a comment that I had been thinking about it all wrong, and then witness the development of a completely different approach, in real time, which would culminate in some final observation, much past the point where I stopped following the details. Those comments were often not just the keys to the solution of a specific issue, but the germs of new – and undoubtedly deeper – projects than those I was pursuing. Some of his remarks became suddenly crucial months after being made, when my knowledge had matured enough to understand them. Some others I try to carry in mind, hoping that sooner or later those seeds will sprout. I would like to thank Enrico Trincherini, my internal supervisor at the Scuola Normale, for his support and his constant advice throughout my Ph.D. His sharp sense of humor and unprejudiced mindset always made our conversations interesting and enjoyable. It is a pleasure to thank my collaborators, Lorenzo Bianchi, Marco Billò, Michele Caselle, Shira Chapman, Xi Dong, Damian Galante, Ferdinando Gliozzi, Vasco Gonçalves, Edoardo Lauria, Pedro Liendo, Rob Myers, Roberto Pellegrini, Antonio Rago, Misha Smolkin. I have learned something from each of them, but I would especially like to reiterate my gratitude to Rob Myers, whose guidance literally illuminated the path in the last year of my Ph.D., and to Vasco Gon¸alves, whose humor as a friend and whose skills as a physicist meant for me much more than he probably knows. The years of my Ph.D. have been split between two great institutions. On one hand, the Scuola Normale, with its rigor and its solid traditions, embodied in the incomparable beauty of Palazzo della Carovana. On the other hand, the Perimeter Institute, whose dynamism is also reflected in the almost audacious lines of its architecture. In between, the Physics Department of the University of Pisa, in which I spent almost a year surrounded by a friendly and stimulating atmosphere. I am indebted to the Scuola Normale for allowing me to spend so much time away from my home institution, and Perimeter Institute for the challenging, breathtaking experience of working and studying in its environment. I would also like to thank Ettore Vicari, who welcomed me in his research group at the University of Pisa, and helped me move the first steps into my Ph.D. Of the many people I met in the last four years, a few quickly started playing a role in my life, and I am proud to call them my friends. I am grateful to Damian for his inexhaustible good mood, to Shira for uncountable passionate conversations and for a lot of music, to Dalimil for his kindness, and for what I can only describe as an immense intellectual power. My friendship with Lorenzo is old and deep: knowing that someone like him is around always made me feel better, and always will. My gratitude towards my mother grows daily, along with my debt, for nearly twenty- eight years of care, which is just plain impossible to properly acknowledge. My sister used to complete the family, but this year decided to accomplish so much more, and made it ii bigger, with the arrival of our brand new Viola. Marta, troppo a lungo, ti sei svegliata con qualche parola inviata da lontano, invece di una carezza. In compenso, ho tenuto per te «la luce di quando fa giorno» e anche, e soprattutto, «l’attesa, che diventa ritorno». iii A Stefano, sebbene non sia abbastanza. A Viola, chissà che un giorno non ne sia incuriosita. iv v We know our friends by their defects rather than by their merits. W. Somerset Maugham vi Contents Introduction ix 1 Defect Conformal field theories 1 1.1 Conformal symmetry, the Operator Product Expansion and the CFT data .1 1.1.1 CFT data in the presence of a defect . .6 1.2 Tensors as polynomials and CFTs on the light-cone . .7 1.2.1 Defect CFTs on the light-cone . 10 1.2.2 Projection to physical space: flat defect . 12 1.2.3 Projection to physical space: spherical defect . 14 1.3 Correlation functions in a Defect CFT . 17 1.3.1 One-point function . 17 1.3.2 Bulk-to-defect two-point function . 18 1.3.3 Two-point function of bulk primaries . 19 1.3.4 Parity odd correlators . 21 1.4 Scalar two-point function and the conformal blocks . 21 1.4.1 Defect channel Casimir equation . 22 1.4.2 Bulk channel Casimir equation . 23 1.5 Ward identities and the displacement operator . 27 1.5.1 The Ward identities for diffeomorphisms and Weyl transformations . 28 1.5.2 Constraints on CFT data . 32 1.5.3 Displacement and reflection in two dimensional CFTs . 34 1.5.4 Examples . 36 1.6 Summary and outlook . 41 Appendix . 42 1.A Notations and conventions . 42 1.B OPE channels of scalar primaries and the conformal blocks . 43 1.B.1 Bulk and defect OPEs . 43 1.B.2 The Casimir equations for the two-point function . 44 1.B.3 The bulk-channel block for an exchanged scalar . 46 1.C Differential geometry of sub-manifolds . 48 2 Boundaries and interfaces in spin systems 51 2.1 Crossing symmetry and the conformal bootstrap . 51 2.2 Defect CFTs and the method of determinants . 54 2.3 The boundary bootstrap and the 3d Ising and O(N) models . 58 2.3.1 The ordinary transition . 60 2.3.2 The extraordinary transition . 60 2.3.3 The special transition . 64 2.4 Renormalization group domain wall for the O(N) model . 66 vii viii CONTENTS 2.4.1 The -expansion and the role of the displacement operator . 67 2.4.2 Leading order mixing of primary operators . 72 2.4.3 The interface bootstrap . 73 2.5 Conclusions and outlook . 77 Appendix . 78 2.A RG domain wall: details on the -expansion . 78 2.A.1 One loop computations . 78 2.A.2 Two-point functions across the interface . 81 3 A twist line in the 3d Ising model 83 3.1 A monodromy defect on the lattice . 83 3.2 The spectrum of defect operators . 85 3.3 Simulations and results . 89 4 Rényi entropies as conformal defects 95 4.1 Entanglement in quantum systems . 96 4.1.1 Entanglement in quantum field theory and holography . 99 4.2 Shape dependence of Rényi entropies in a CFT . 103 4.2.1 The replica trick and the twist operator . 104 4.3 Rényi and entanglement entropy across a deformed sphere . 109 4.4 The cone conjecture . 111 4.4.1 Conical deformation from the displacement operator . 112 4.4.2 Wilson lines in supersymmetric theories and entanglement in d = 3 . 115 4.5 Rényi entropy and anomalies in 4d Defect CFTs . 116 4.5.1 fc and the expectation value of the stress tensor . 117 4.5.2 fb and the two-point function of the displacement operator . 119 4.6 Twist operators and the defect CFT data . 121 4.7 Discussion . 124 Appendix . 125 4.A Displacement operator for the free scalar . 125 5 Shape dependence of holographic Rényi entropies 131 5.1 The CFT Story . 133 5.2 The AdS story . 136 5.2.1 Holographic Setup . 137 5.2.2 Einstein Gravity . 137 5.3 Discussion . 142 Appendix . 143 5.A Expanding Two Point Functions as Distributions . 143 5.A.1 Derivation of the Kernel Formula . 144 5.A.2 List of Formulas for Kernels . 145 5.B Details of Holographic Renormalization . 146 Final remarks and outlook 149 Introduction This thesis explores some features of extended probes in conformal field theories (CFTs). Therefore, it seems appropriate to spend a few words justifying why it is interesting to study conformal field theories in the first place. In theoretical physics, symmetries are often used as a tool to simplify hard problems. They always reduce the room for dynamics, and therefore make life easier. This quest for symmetries, even in cases in which the physical system under inquiry does not possess them, is especially necessary in studying quantum field theories (QFTs). Indeed, in a generic situation, our analytic control over the dynamics of a QFT is extremely limited, it is essentially reduced to perturbation theory around some point in parameter space in which the theory is exactly solvable.
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