AdS/CFT, Black Holes, And Fuzzballs
AdS/CFT, Black Holes, And FuzzballsbyIda G. ZadehA thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of PhysicsUniversity of TorontoCopyright c 2013 by Ida G. Zadeh
AbstractAdS/CFT, Black Holes, And FuzzballsIda G. ZadehDoctor of PhilosophyGraduate Department of PhysicsUniversity of Toronto2013In this thesis we investigate two different aspects of the AdS/CFT correspondence. Wefirst investigate the holographic AdS/CMT correspondence. Gravitational backgroundsin d 2 dimensions have been proposed as holographic duals to Lifshitz-like theoriesdescribing critical phenomena in d 1 dimensions with critical exponent z 1. Wenumerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds assolutions. We show how to embed these solutions into AdS space for a range of valuesof z and d.We next investigate the AdS3 /CFT2 correspondence and focus on the microscopicCFT description of the D1-D5 system on T 4 S 1 . In the context of the fuzzball programme, we investigate deforming the CFT away from the orbifold point and studylifting of the low-lying string states. We start by considering general 2D orbifold CFTsof the form MN /SN , with M a target space manifold and SN the symmetric group. TheLunin-Mathur covering space technique provides a way to compute correlators in theseorbifold theories, and we generalize this technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, andshow how to account for these excitations when computing correlation functions in thecovering space. Second, we consider non-twist sector operators and show how to includethe effects of these insertions in the covering space.Using the generalization of the Lunin-Mathur symmetric orbifold technology and conii
formal perturbation theory, we initiate a program to compute the anomalous dimensionsof low-lying string states in the D1-D5 superconformal field theory. Our method entailsfinding four-point functions involving a string operator O of interest and the deformationoperator, taking coincidence limits to identify which other operators mix with O, subtracting conformal families of these operators, and computing their mixing coefficients.We find evidence of operator mixing at first order in the deformation parameter, whichmeans that the string state acquires an anomalous dimension. After diagonalization thiswill mean that anomalous dimensions of some string states in the D1-D5 SCFT mustdecrease away from the orbifold point while others increase.Finally, we summarize our results and discuss some future directions of research.iii
DedicationTo my parents Ali and Maryamiv
AcknowledgementsThe 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 productivediscussions, 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 hospitality 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. DavidBailey.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, HelenIyer, Carrie Meston, Julian Comanean, and Steven Butterworth.My partner Alijon. My mother Maryam, and my sister Anahita. Special thanks tomy father Ali for his constant encouragement, support, and sense of humour.v
Contents1 Introduction11.1String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Black holes in string theory . . . . . . . . . . . . . . . . . . . . . . . . .31.3AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.4AdS/CMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111.5Fuzzball proposal and black hole information paradox . . . . . . . . . . .131.5.1Mathur’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .151.5.2Fuzzball conjecture . . . . . . . . . . . . . . . . . . . . . . . . . .161.5.3Two-charge and three-charge fuzzballs . . . . . . . . . . . . . . .191.5.4AdS3 /CFT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211.5.5D1-D5 CFT at the orbifold point . . . . . . . . . . . . . . . . . .231.5.6Moving away from the orbifold point . . . . . . . . . . . . . . . .26Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281.62 Lifshitz black brane thermodynamics in higher dimensions2.130Analysis of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332.1.1Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332.1.2Perturbation theory at the horizon . . . . . . . . . . . . . . . . .372.1.3Perturbation theory at r : AdS asymptotics . . . . . . . . . .392.1.4Other considerations, and setup for numerics . . . . . . . . . . . .41vi
2.2Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Twist-nontwist correlators in M N /SN orbifold CFTs44513.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .513.2The Lunin-Mathur technique, and generalizations . . . . . . . . . . . . .543.2.1Lunin-Mathur . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543.2.2Generalization to the non twist sector. . . . . . . . . . . . . . . .60Example calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .653.3.1Excitations orthogonal to twist directions . . . . . . . . . . . . . .653.3.2Non-twist operator insertions . . . . . . . . . . . . . . . . . . . .71Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .813.33.44 String states mixing in the D1-D5 CFT near the orbifold point844.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .844.2Perturbing the D1-D5 SCFT . . . . . . . . . . . . . . . . . . . . . . . . .864.2.1The D1-D5 superconformal field theory . . . . . . . . . . . . . . .864.2.2Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .894.2.3Deformation operator . . . . . . . . . . . . . . . . . . . . . . . . .954.2.4Conformal perturbation theory . . . . . . . . . . . . . . . . . . .994.2.5Four-point functions and factorization channels . . . . . . . . . . 1054.34.4Dilaton warm-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.1Four-point function . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.2Mapping from the base to the cover . . . . . . . . . . . . . . . . . 1104.3.3Summing over images . . . . . . . . . . . . . . . . . . . . . . . . . 1144.3.4Lack of operator mixing . . . . . . . . . . . . . . . . . . . . . . . 115Lifting of a string state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.4.1Passing to the covering surface4.4.2Image sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120vii. . . . . . . . . . . . . . . . . . . 118
4.54.4.3Coincidence limit and operator mixing . . . . . . . . . . . . . . . 1214.4.4Conformal family subtraction and mixing coefficients . . . . . . . 121Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275 Conclusions130Bibliography134viii
List of Figures1.1Ensemble of fuzzball microstates . . . . . . . . . . . . . . . . . . . . . . .182.1The plots of ln (4 Gd 2 s) versus ln(LT ) for fixed µ̂ 1. . . . . . . . . . .452.2The plot of ln (c(z, d)) as a function of ln(z) for fixed value of µ̂ 1 . . .473.1Diagram of the three-fold cover of the base space . . . . . . . . . . . . .58ix
Chapter 1Introduction1.1String theoryString theory provides a consistent theory of quantum gravity which in the low-energylimit reduces to classical supergravity. Mathematical consistency requires string theoryto live in ten spacetime dimensions. Compactifying extra dimensions on small scalecompact manifolds provides a powerful tool for constructing lower dimensional effectivetheories. Existence of higher dimensions in string theory has made it possible to constructa wide variety of black holes and their extended counterparts which are localized in alower dimensional spacetime.Fundamental ingredients of the theory are one-dimensional open and closed stringswhose oscillations give rise to the elementary particles of the universe. The motion ofa string in the ten-dimensional spacetime is described by the worldsheet theory. Theworldsheet 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 stringspectrum contains gravitons, as well as a host of other particles. The interactions givenby the worldsheet theory provide a unified picture of all the fundamental forces of nature.The classical worldsheet theory is invariant under Weyl transformations. The quan1
2Chapter 1. Introductiontum theory, however, acquires Weyl anomalies. Let us consider the limit where theradius of curvature of the background, R, is much larger than the string length scale ls :ls R 1 1. We focus on massless states in this low-energy limit. We can then performperturbation theory in this limit and compute the beta functions. Requiring that thequantum worldsheet theory be Weyl-invariant boils down to vanishing of the beta functions. This yields Einstein’s equations in ten-dimensional spacetime and confirms thatclassical 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 andthe string coupling constant gs according to16π G10 (2π)7 ls8 gs2 .(1.1)In addition to the fundamental strings, string theory has a class of building blockswhich are extended objects called Dp-branes. Dp-branes have (p 1) spacetime dimensions. They are objects on which the endpoints of open strings are located. Equations ofmotion of the worldsheet theory allow for Dirichlet and Neumann boundary conditions forthe end points of open strings. Dirichlet boundary conditions violate Poincaré symmetryof the theory. In the presence of D-branes, Poincaré invariance is broken spontaneously.Open string endpoints satisfy Neumann boundary conditions in the (p 1) spacetimedirections 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 branesor D-branes. T-duality exchanges Neumann and Dirichlet boundary conditions. Thus,performing T-duality in a direction tangential (perpendicular) to the Dp-brane results ina Dp 1 (Dp 1 )-brane.The symmetry preserved by Dp-branes is SO(1, p) SO(9 p). The tension of D-
3Chapter 1. Introductionbranes is related to gs and ls through the relationτDp 1.(2π)p lsp 1 gs(1.2)The tension of the D-branes is proportional to the inverse of the coupling constant.Therefore, they are non-perturbative objects. However, one can perform perturbationtheory in the presence of D-brane backgrounds by considering low-energy fluctuations ofopen strings attached to the D-branes.A great breakthrough was made by Polchinski in 1995 [3] by showing that Dp-branescouple to the p-form fields of the R-R sector of superstring theory and therefore carryR-R charges. This discovery set the stage for the construction of black holes that carryconserved R-R charges in string theory and investigation of microscopic structure of blackholes. We will discuss these constructions in the next section.1.2Black holes in string theoryAstronomical observations have provided ample evidence of the existence of macroscopicphysical black holes in our universe. The evidence includes observations of SagittariusA at the centre of our Milky Way galaxy, X-ray binary systems, and supermassive blackholes at the centre of other galaxies and clusters of galaxies [4, 5, 6]. Black holes havebeen studied in theoretical physics for more than four decades. The laws of black holemechanics were formulated by Bardeen, Carter, and Hawking [7]. The analogy betweenthese laws and the laws of thermodynamics led Bekenstein to conjecture that the entropyof a black hole is proportional to the surface area of its event horizon [8]. Semiclassicalanalysis of Hawking [9, 10] showed that black holes are thermodynamic systems thatemit black body radiation. Hawking’s discovery resulted in the precise formulation of
4Chapter 1. Introductionblack hole entropySBH Ad,4 Gd(1.3)where Ad is the area of the event horizon, Gd is the d-dimensional Newton’s constant,and c kB 1.The breakthrough in understanding thermodynamic properties of black holes raisedtwo fundamental issues. First, equation (1.3) indicates that there are eSBH microscopicdegrees of freedom that contribute to the exponentially large value of the black holeentropy. Second, Hawking’s results showed that black holes emit radiation through pairproduction out of vacuum at the horizon. The emitted radiation does not reveal anyinformation about the matter that has made the black hole. These statements raisedchallenging conundrums in black hole physics: what are the microscopic degrees of freedom of black holes? Where are these degrees of freedom located? Can we retrieve theinformation about the matter that has fallen inside a black hole? We discuss the firstquestion in the remainder of this subsection. The second and the third questions will beaddressed in section 1.5.String theory has provided a powerful framework for constructing black holes andstudying the statistical mechanical description of their entropy. The key poin
analysis of Hawking [9, 10] showed that black holes are thermodynamic systems that emit black body radiation. Hawking’s discovery resulted in the precise formulation of. Chapter 1. Introduction 4 black hole entropy S BH A d 4G d; (1.3) where A d is the area of the event horizon, G d is the d-dimensional Newton’s constant, and c k B 1. The breakthrough in understanding .
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Black holes have entropy S. Black holes have Hawking temperature T H, consistent with thermodynamic relation between energy, entropy and temperature. Thermodynamics S A 4 where Ais the area of the event horizon. T H 2 ˇ where in the surface gravity of the black hole. Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor: Ruth GregoryBlack Holes, Vortices and Thermodynamics. Path .
mass black holes, no credible formation process is known, and indeed no indications have been found that black holes much lighter than this \Chandrasekhar limit" exist anywhere in the Universe. Does this mean that much lighter black holes cannot exist? It is here that one could wonder about all those fundamental assumptions that underly the theory of quantum mechanics, which is the basic .
However, in addition to black holes formed by stellar collapse, there might also be much smaller black holes which were formed by density fluctua-202 S. W. Hawking tions in the early universe [9, 10]. These small black holes, being at a higher temperature, would radiate more than they absorbed. They would therefore pre- sumably decrease in mass. As they got smaller, they would get hotter and .
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