AdS/CFT, Black Holes, And Fuzzballs by Ida G. Zadeh A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright c 2013 by Ida G. Zadeh Abstract AdS/CFT, Black Holes, And Fuzzballs Ida G. Zadeh Doctor of Philosophy Graduate Department of Physics University of Toronto 2013 In this thesis we investigate two different aspects of the AdS/CFT correspondence. We first investigate the holographic AdS/CMT correspondence. Gravitational backgrounds in d + 2 dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in d + 1 dimensions with critical exponent z ≥ 1. We numerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds as solutions. We show how to embed these solutions into AdS space for a range of values of z and d. We next investigate the AdS3/CFT2 correspondence and focus on the microscopic CFT description of the D1-D5 system on T 4 × S1. In the context of the fuzzball pro- gramme, we investigate deforming the CFT away from the orbifold point and study lifting of the low-lying string states. We start by considering general 2D orbifold CFTs N of the form M =SN , with M a target space manifold and SN the symmetric group. The Lunin-Mathur covering space technique provides a way to compute correlators in these orbifold theories, and we generalize this technique in two ways. First, we consider exci- tations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. Using the generalization of the Lunin-Mathur symmetric orbifold technology and con- ii formal perturbation theory, we initiate a program to compute the anomalous dimensions of low-lying string states in the D1-D5 superconformal field theory. Our method entails finding four-point functions involving a string operator O of interest and the deformation operator, taking coincidence limits to identify which other operators mix with O, sub- tracting conformal families of these operators, and computing their mixing coefficients. We find evidence of operator mixing at first order in the deformation parameter, which means that the string state acquires an anomalous dimension. After diagonalization this will mean that anomalous dimensions of some string states in the D1-D5 SCFT must decrease away from the orbifold point while others increase. Finally, we summarize our results and discuss some future directions of research. iii Dedication To my parents Ali and Maryam iv Acknowledgements The following is a list of people to whom I would like to express my gratitude. My advisor, Prof. Amanda Peet, for help in sparking ideas, and for suggestions, guidance, encouragement, and support. Prof. Amanda Peet, Prof. Ben Burrington, and Dr. Gaetano Bertoldi, for productive discussions, rewarding and fun collaborations, and interesting courses. Prof. Ben Burrington, especially for inspiring discussions and valuable suggestions. Prof. Samir Mathur, for stimulating discussions, insightful comments, and for hospi- tality during my visits to The Ohio State University. Prof. Erich Poppitz, for helpful comments and advice. My committee members, Prof. Erich Poppitz, Prof. Peter Krieger, and Prof. David Bailey. Fellow student Daniel O’Keeffe, for helpful discussions and comments. Prof. Werner Israel at the University of Victoria, for his support. The staff of the Physics department, especially Teresa Baptista, Krystyna Biel, Helen Iyer, Carrie Meston, Julian Comanean, and Steven Butterworth. My partner Alijon. My mother Maryam, and my sister Anahita. Special thanks to my father Ali for his constant encouragement, support, and sense of humour. v Contents 1 Introduction 1 1.1 String theory . .1 1.2 Black holes in string theory . .3 1.3 AdS/CFT . .7 1.4 AdS/CMT . 11 1.5 Fuzzball proposal and black hole information paradox . 13 1.5.1 Mathur's theorem . 15 1.5.2 Fuzzball conjecture . 16 1.5.3 Two-charge and three-charge fuzzballs . 19 1.5.4 AdS3/CFT2 .............................. 21 1.5.5 D1-D5 CFT at the orbifold point . 23 1.5.6 Moving away from the orbifold point . 26 1.6 Outline of the thesis . 28 2 Lifshitz black brane thermodynamics in higher dimensions 30 2.1 Analysis of the model . 33 2.1.1 Reduction . 33 2.1.2 Perturbation theory at the horizon . 37 2.1.3 Perturbation theory at r = 1: AdS asymptotics . 39 2.1.4 Other considerations, and setup for numerics . 41 vi 2.2 Results and discussion . 44 N 3 Twist-nontwist correlators in M =SN orbifold CFTs 51 3.1 Introduction . 51 3.2 The Lunin-Mathur technique, and generalizations . 54 3.2.1 Lunin-Mathur . 54 3.2.2 Generalization to the non twist sector. 60 3.3 Example calculations . 65 3.3.1 Excitations orthogonal to twist directions . 65 3.3.2 Non-twist operator insertions . 71 3.4 Discussion . 81 4 String states mixing in the D1-D5 CFT near the orbifold point 84 4.1 Introduction . 84 4.2 Perturbing the D1-D5 SCFT . 86 4.2.1 The D1-D5 superconformal field theory . 86 4.2.2 Cocycles . 89 4.2.3 Deformation operator . 95 4.2.4 Conformal perturbation theory . 99 4.2.5 Four-point functions and factorization channels . 105 4.3 Dilaton warm-up . 108 4.3.1 Four-point function . 108 4.3.2 Mapping from the base to the cover . 110 4.3.3 Summing over images . 114 4.3.4 Lack of operator mixing . 115 4.4 Lifting of a string state . 116 4.4.1 Passing to the covering surface . 118 4.4.2 Image sums . 120 vii 4.4.3 Coincidence limit and operator mixing . 121 4.4.4 Conformal family subtraction and mixing coefficients . 121 4.5 Summary and outlook . 127 5 Conclusions 130 Bibliography 134 viii List of Figures 1.1 Ensemble of fuzzball microstates . 18 2.1 The plots of ln (4 Gd+2 s) versus ln(LT ) for fixedµ ^ = 1. 45 2.2 The plot of ln (c(z; d)) as a function of ln(z) for fixed value ofµ ^ = 1 . 47 3.1 Diagram of the three-fold cover of the base space . 58 ix Chapter 1 Introduction 1.1 String theory String theory provides a consistent theory of quantum gravity which in the low-energy limit reduces to classical supergravity. Mathematical consistency requires string theory to live in ten spacetime dimensions. Compactifying extra dimensions on small scale compact manifolds provides a powerful tool for constructing lower dimensional effective theories. Existence of higher dimensions in string theory has made it possible to construct a wide variety of black holes and their extended counterparts which are localized in a lower dimensional spacetime. Fundamental ingredients of the theory are one-dimensional open and closed strings whose oscillations give rise to the elementary particles of the universe. The motion of a string in the ten-dimensional spacetime is described by the worldsheet theory. The worldsheet is the two-dimensional surface that the string sweeps out during its motion. Upon quantization of the classical worldsheet action one finds that the closed string spectrum contains gravitons, as well as a host of other particles. The interactions given by the worldsheet theory provide a unified picture of all the fundamental forces of nature. The classical worldsheet theory is invariant under Weyl transformations. The quan- 1 Chapter 1. Introduction 2 tum theory, however, acquires Weyl anomalies. Let us consider the limit where the radius of curvature of the background, R, is much larger than the string length scale ls: 1 ls R− 1. We focus on massless states in this low-energy limit. We can then perform perturbation theory in this limit and compute the beta functions. Requiring that the quantum worldsheet theory be Weyl-invariant boils down to vanishing of the beta func- tions. This yields Einstein's equations in ten-dimensional spacetime and confirms that classical gravity emerges as the low-energy effective description of string theory [1, 2]. The Newton's constant in ten dimensions, G10, is given in terms of the string length and the string coupling constant gs according to 7 8 2 16π G10 = (2π) ls gs : (1.1) In addition to the fundamental strings, string theory has a class of building blocks which are extended objects called Dp-branes. Dp-branes have (p + 1) spacetime dimen- sions. They are objects on which the endpoints of open strings are located. Equations of motion of the worldsheet theory allow for Dirichlet and Neumann boundary conditions for the end points of open strings. Dirichlet boundary conditions violate Poincar´esymmetry of the theory. In the presence of D-branes, Poincar´einvariance is broken spontaneously. Open string endpoints satisfy Neumann boundary conditions in the (p + 1) spacetime directions tangent to the brane and satisfy Dirichlet boundary conditions in the (9 − p) directions transverse to the branes. For this reason the branes are called Dirichlet branes or D-branes. T-duality exchanges Neumann and Dirichlet boundary conditions. Thus, performing T-duality in a direction tangential (perpendicular) to the Dp-brane results in a Dp 1 (Dp+1)-brane. − The symmetry preserved by Dp-branes is SO(1; p) × SO(9 − p). The tension of D- Chapter 1. Introduction 3 branes is related to gs and ls through the relation 1 τD = : (1.2) p p p+1 (2π) ls gs The tension of the D-branes is proportional to the inverse of the coupling constant. Therefore, they are non-perturbative objects.