String Theory and Black Holes 9 9 9 1 v o N A thesis submitted to the 2 University of Mumbai 1 for the v Ph.D. Degree in Physics 3 0 0 1 1 9 9 / h t - p e h : by v i X Justin Raj David r a Department of Theoretical Physics Tata Institute of Fundamental Research Mumbai 400 005, India. 1999 Abstract This thesis aims to make precise the microscopic understanding of Hawking radiation from the D1/D5 black hole. We present an explict construction of all the shortmultiplets of the = (4,4)SCFTonthesymmetricproductT˜4/S(Q Q ). Aninvestigationofthesymmerties 1 5 N of this SCFT enables us to make a one-to-one correspondence beween the supergravity moduli and the marginal opeerators of the SCFT. We analyse the gauge theory dynamics of the splitting of the D1/D5system into subsystems and show that it agrees with supergravity. WehaveshownthatthefixedscalarsoftheD1/D5systemcoupleonlyto(2,2)operatorsthus removing earlier discrepancies between D-brane calculations and semiclassical calculations. The absorption cross-section of the minimal scalars is determined from first principles upto a propotionality constant. We show that the absorption cross-section of the minimal scalars computed in supergravity and the SCFT is independent of the moduli. Acknowledgements Firstly, I would like to thank Spenta Wadia, my thesis advisor for inspiration, support and encouragement throughout my five years at TIFR. It is from him I learnt how to approach problems in physics. He has emphasized to me the importance of thinking from first princi- ples and attacking physically relevant problems. During my early years in TIFR, Spenta’s guidance helped me overcome the maze of superfluous literature and to concentrate on the basics. He emphasized to me the importance of independent thinking in physics. I thank him for sharing with me his wonderful ideas. His inspiring confidence helped me overcome moments of despair when all means of attacking a physics problem seemed to fail. During the course of my five years I worked very closely with Gautam Mandal and Avinash Dhar. I have learnt (not completely) from Gautam the art of reducing a physics problem to its very essentials. It is amazing to see how simple a seemingly complicated problem is, when reformulated by Gautam. Avinash helped me to realize the importance of being persistent and pursuing a physics problem to its entire logical conclusion. Avinash and I were closely working on a problem which would not have yielded results had it not been for the constant encouragement and persistence of Avinash. I owe most of my understanding of gauge theories to the wonderful course that Sumit Das took great pains to teach. The course stretched for the entire year of 1995. I thank the string community in TIFR for being active in fostering the nascent enthusiasm for the subject. There are other members of the Theory group working in different fields who provided in- spiration, help and guidance indirectly and unknowingly. They did it through their lectures, seminars and discussions. I have benefitted in this way from Deepak Dhar and Mustansir Barma. The students of the Theory group during the past five years contributed a lot to help me imbibe the spirit of search and discovery. I would like to mention Abhishek Dhar whose uncompromising attitude towards physics continues to be a source of inspiration. IwouldliketoacknowledgethehelpfromtheefficientandfriendlystaffoftheDepartment of Theoretical Physics. Finally I would like to thank my family andfriends for their support in helping me pursue physics as a career. Contents 1 Introduction 1 1.1 Black hole physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Black holes in string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The D1/D5 black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 The Microscopic Theory of the D1/D5 system 20 2.1 The gauge theory of the D1/D5 system . . . . . . . . . . . . . . . . . . . . . 20 2.2 The instanton moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 The SCFT on the orbifold . . . . . . . . . . . . . . . . . . . . . . . . . . 27 M 2.3.1 The = 4 superconformal algebra . . . . . . . . . . . . . . . . . . . 27 N 2.3.2 Free field realization of = (4,4) SCFT on the orbifold . . . . . 28 N M 2.3.3 The SO(4) algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.4 The supergroup SU(1,1 2) . . . . . . . . . . . . . . . . . . . . . . . . 31 | 2.4 Short multiplets of SU(1,1 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 | 2.5 The resolutions of the symmetric product . . . . . . . . . . . . . . . . . . . . 32 2.5.1 The untwisted sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5.2 Z twists. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 2.5.3 Higher twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 The chiral primaries of the = (4,4) SCFT on . . . . . . . . . . . . . . 37 N M 2.6.1 The k-cycle twist operator . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.2 The complete set of chiral primaries . . . . . . . . . . . . . . . . . . . 39 2.7 Shortmultiplets of = (4,4) SCFT on . . . . . . . . . . . . . . . . . . . 42 N M i 2.8 The location of the symmetric product . . . . . . . . . . . . . . . . . . . . . 43 2.9 The linear sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.9.1 The linear sigma model description of R4/Z . . . . . . . . . . . . . . 45 2 2.9.2 The gauge theory description of the moduli of the D1/D5 system . . 48 2.9.3 The case (Q ,Q ) (Q 1,Q )+(1,0): splitting of 1 D1-brane . . 52 1 5 1 5 → − 2.9.4 The dynamics of the splitting at the singularity of the Higgs branch . 53 3 Coupling with the bulk fields 56 3.1 Near horizon symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Classification of the supergravity modes . . . . . . . . . . . . . . . . . . . . 58 3.2.1 Comparison of supergravity shortmultiplets with SCFT . . . . . . . . 61 3.3 The supergravity moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 AdS /CFT correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3 2 3.5 Supergravity moduli and the marginal operators . . . . . . . . . . . . . . . . 63 3.6 Fixed scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.7 Intermediate Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.8 Supergravity from gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 The Hawking Process 73 4.1 Supergravity calculation of absorption/Hawking radiation in presence of moduli 73 4.2 Near horizon geometry of the D1/D5 black hole . . . . . . . . . . . . . . . . 74 4.3 The coupling with the bulk fields in the Ramond sector . . . . . . . . . . . . 77 4.4 Determination of the strength of the coupling µ . . . . . . . . . . . . . . . . 77 4.4.1 Evaluation of the tree-level vertices in supergravity . . . . . . . . . . 80 4.4.2 Two-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 The black hole state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Absorption cross-section as thermal Green’s function . . . . . . . . . . . . . 86 4.7 Absorption cross-section of minimal scalars from the D1/D5 SCFT . . . . . 88 4.8 Independence of Hawking radiation on D1/D5 moduli . . . . . . . . . . . . . 89 4.9 Entropy and area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ii 5 Concluding remarks and discussions 97 1 Chapter 1 Introduction A unified theory of fundamental interactions should include within its framework a quan- tum theory of gravity. The quantization of Einstein’s theory of gravity using conventional methods poses a host of problems. Einstein’s theory of gravity is a non-renormalizable field theory. Therefore one can not extract meaningful answers from quantum perturbation theory. The existence of black holes as solutions in Einstein’s theory of gravity is another stumbling block. Arguments involving physics of black holes within the framework of quan- tum mechanics seem to lead to the conclusion that quantum mechanical evolution of black hole is not unitary. This contradicts the basic rules of quantum mechanics. At present string theory is the leading candidate for a unified theory of fundamental interactions. The massless spectrum of string theory includes the graviton. The low energy effective action of string theory includes the Einstein action. String theory is also consistent with the rules of quantum mechanics. Thus the natural generalization of Einstein’s theory within the framework of quantum mechanics is string theory. Furthermore string theory is perturbatively finite. This cures the non-renormalizability of gravity. The understanding of non-perturbative spectra of string theory like D-branes has paved the way for addressing the problems of black hole physics. In this thesis we will attempt to make precise the description of black hole physics within the framework of string theory. 1.1 Black hole physics Let us briefly review some general properties of black holes. [1] Black holes are objects which result as end points of gravitational collapse of matter. For masses greater than 3.6 solar mass, the gravitational force overcomes all other forces and the matter generically collapses 1 into a black hole (in some exceptional cases a naked singularity might result). This would suggest that to specify a black hole it is necessary to give in detail the initial conditions of the collapse. As we will see below a black hole is completely specified by a few parameters only. To introduce various concepts related to black hole we will discuss two examples of black holes. First, let us consider the Schwarzschild black hole in 3+ 1 dimensions. It is a time independent, spherically symmetric solution of pure Einstein gravity. Its metric is given by 2G M 2G M 1 ds2 = 1 N dt2 + 1 N − dr2 +r2dΩ2 (1.1) − − r − r (cid:18) (cid:19) (cid:18) (cid:19) where t refers to time, r the radial distance, Ω the solid angle in 3 dimensions and G the N Newton’s constant. We have chosen units so that the velocity of light, c = 1. The surface r = 2G M is called the event horizon. It is a co-ordinate singularity but not a curvature N singularity. Light-like geodesics and time-like geodesics starting at r < 2G M end up at N r = 0 (the curvature singularity) in finite proper time. This means that classically the black hole is truly black, it cannot emit anything. Note that the solution is completely specified by only one parameter M, the mass of the black hole. Next we consider theReissner-N¨ordstrom blackhole. Itisatimeindependent, spherically symmetric solution of Einstein gravity coupled to the electromagnetic field. The solution is given by the following backgrounds. ds2 = 1 2GNM + GNQ2 dt2 + 1 2GNM + GNQ2 −1dr2+r2dΩ2, (1.2) − − r r2 ! − r r2 ! Q A = . 0 r where A is the time component of the vector potential. This solution carries a charge Q. 0 There are two co-ordinate singularities at r = G M + G2 M2 G Q2 (1.3) + N N − N q and r = G M G2 M2 G Q2 (1.4) − N − N − N q The event horizon is at r = r . When M = Q /√G , the outer horizon at r = r and the + N + | | inner horizon at r = r coincide. A black hole with coincident inner and outer horizons is − 2 called an extremal black hole. Note that in this case the black hole is completely specified by its mass M and the charge Q. In general, collapsing matter results in black holes which are completely specified by the mass M, the U(1) charges Q and the angular momentum J. This is called the no hair i theorem. Whatever other information (for example, multipole moments) present decays ex- ponentially fast during the collapse. Thus, all detailed information carried by the collapsing matter is completely lost. So far we have discussed the black holes only classically. In the seventies, the works of Bekenstein, Hawking and others furthered the understanding of black holes within the framework of quantum mechanics. It was found by Hawking that the Schwarzschild black holeisnottrulyblack. Asemi-classicalcalculationbyHawkingshowedthatitemitsradiation with the spectrum of a black body at a temperature T given by h¯ T = (1.5) 8πG M N The quantum nature of this effect is clearly evident from the fact that the temperature is proportional to h¯. For the Reissner-N¨ordstrom black hole the temperature of Hawking radiation is (r r )h¯ + T = − − (1.6) 4πr2 + One notes that the extremal Reissner-N¨ordstrom black hole does not Hawking radiate. In general the Hawking temperature turns out to be function of mass, the charges and the angularmomentumalone. Thusevensemi-classicaleffectsdonotprovidefurtherinformation of the black hole. The works in the seventies culminated in the following laws of black holes which are analogous to the laws of thermodynamics. i. First Law: Two neighboring black hole equilibrium states are related by A dM = Td +Φ dQ +ΩdJ (1.7) i i 4G h¯ (cid:20) N (cid:21) where A is the area of the event horizon, Φ the electric surface potential and Ω the angular velocity. For the special case of the Reissner-N¨ordstrom black hole the first law reduces to A dM = Td +ΦdQ. (1.8) 4G h¯ (cid:20) N (cid:21) 3