Precision holography and its applications to black holes - Contents

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Precision holography and its applications to black holes

Kanitscheider, I.

Publication date

2009

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Citation for published version (APA):

Kanitscheider, I. (2009). Precision holography and its applications to black holes.

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C

ONTENTS

1 Holography and the AdS/CFT correspondence 1

1.1 Introduction . . . 1

1.2 The holographic dictionary . . . 3

1.3 A preview of holographic renormalization . . . 5

1.4 Chiral primaries and the Kaluza-Klein spectrum . . . 8

1.5 Kaluza-Klein holography . . . 9

2 The fuzzball proposal for black holes 13 2.1 Black hole puzzles . . . 13

2.2 Black hole entropy counting by string theory . . . 15

2.3 The D1-D5 toy model . . . 17

2.4 AdS/CFT supports the fuzzball proposal . . . 18

2.5 Are astrophysical black holes fuzzballs? . . . 19

3 Holographic anatomy of fuzzballs 21 3.1 Introduction, summary of results and conclusions . . . 21

3.2 FP system and perturbative states . . . 30

3.2.1 String quantization . . . 30

3.2.2 Relation to classical curves . . . 32

3.2.3 Examples . . . 33

3.3 The fuzzball solutions . . . 34

3.3.1 Compactification to six dimensions . . . 35

3.3.2 Asymptotically AdS limit . . . 37

3.4 Harmonic expansion of fluctuations . . . 37

3.4.1 Asymptotic expansion of the fuzzball solutions . . . 38

3.4.2 Gauge invariant fluctuations . . . 42

3.5 Extracting the vevs systematically . . . 43

3.5.1 Linearized field equations . . . 43

3.5.2 Field equations to quadratic order . . . 45

3.5.3 Reduction to three dimensions . . . 46

3.5.4 Holographic renormalization and extremal couplings . . . 48

viii CONTENTS

3.6 Vevs for the fuzzball solutions . . . 52

3.6.1 Higher dimension operators . . . 53

3.7 Examples . . . 55

3.7.1 Circular curves . . . 55

3.7.2 Ellipsoidal curves . . . 58

3.8 Dual field theory . . . 59

3.8.1 R ground states and vevs . . . 61

3.9 Correspondence between fuzzballs and chiral primaries . . . 63

3.9.1 Correspondence with circular curves . . . 63

3.9.2 Non-circular curves . . . 65

3.9.3 Testing the new proposal . . . 65

3.10 Symmetric supergravity solutions . . . 68

3.10.1 Averaged geometries . . . 69

3.10.2 Disconnected curves . . . 72

3.10.3 Discussion . . . 73

3.11 Dynamical tests for symmetric geometries . . . 73

3.12 Including the asymptotically flat region . . . 76

3.A Appendix . . . 77

3.A.1 Properties of spherical harmonics . . . 77

3.A.2 Proof of addition theorem for harmonic functions on R4 _{. . . . 81}

3.A.3 Six dimensional field equations to quadratic order . . . 82

3.A.4 3-point functions . . . 85

3.A.5 Holographic 1-point functions . . . 90

3.A.6 Three point functions from the orbifold CFT . . . 91

4 Fuzzballs with internal excitations 93 4.1 Introduction . . . 93

4.2 Fuzzball solutions on T4 _{. . . . 96}

4.2.1 Chiral null models . . . 96

4.2.2 The IIA F1-NS5 system . . . 97

4.2.3 Dualizing further to the D1-D5 system . . . 98

4.3 Fuzzball solutions on K3 . . . 100

4.3.1 Heterotic chiral model in 10 dimensions . . . 100

4.3.2 Compactification on T4 _{. . . 101}

4.3.3 String-string duality to P-NS5 (IIA) on K3 . . . 102

4.3.4 T-duality to F1-NS5 (IIB) on K3 . . . 103

4.3.5 S-duality to D1-D5 on K3 . . . 105

4.4 D1-D5 fuzzball solutions . . . 107

4.5 Vevs for the fuzzball solutions . . . 111

4.5.1 Holographic relations for vevs . . . 111

4.5.2 Application to the fuzzball solutions . . . 115

CONTENTS ix

4.6.1 Dual field theory . . . 117

4.6.2 Correspondence between geometries and ground states . . . 118

4.6.3 Matching with the holographic vevs . . . 120

4.6.4 A simple example . . . 121

4.6.5 Selection rules for curve frequencies . . . 124

4.6.6 Fuzzballs with no transverse excitations . . . 125

4.7 Implications for the fuzzball program . . . 127

4.A Appendix . . . 129

4.A.1 Conventions . . . 129

4.A.2 Reduction of type IIB solutions on K3 . . . 132

4.A.3 Properties of spherical harmonics . . . 136

4.A.4 Interpretation of winding modes . . . 137

4.A.5 Density of ground states with fixed R charges . . . 141

5 Precision holography of non-conformal branes 145 5.1 Introduction . . . 145

5.2 Non-conformal branes and the dual frame . . . 147

5.3 Lower dimensional field equations . . . 153

5.4 Generalized conformal structure . . . 156

5.5 Holographic renormalization . . . 160

5.5.1 Asymptotic expansion . . . 161

5.5.2 Explicit expressions for expansion coefficients . . . 165

5.5.3 Reduction of M-branes . . . 168

5.5.4 Renormalization of the action . . . 170

5.5.5 Relation to M2 theory . . . 172

5.5.6 Formulae for other Dp-branes . . . 173

5.6 Hamiltonian formulation . . . 175

5.6.1 Hamiltonian method for non-conformal branes . . . 176

5.6.2 Holographic renormalization . . . 179

5.6.3 Ward identities . . . 181

5.6.4 Evaluation of terms in the dilatation expansion . . . 181

5.7 Two-point functions . . . 188

5.7.1 Generalities . . . 188

5.7.2 Holographic 2-point functions for the brane backgrounds . . . 190

5.7.3 General case . . . 195

5.8 Applications . . . 200

5.8.1 Non-extremal D1 branes . . . 200

5.8.2 The Witten model of holographic Y M4theory . . . 202

5.9 Discussion . . . 204

5.A Appendix . . . 206

5.A.1 Useful formulae . . . 206

x CONTENTS

5.A.3 Reduction of M5 to D4 . . . 209

5.A.4 Explicit expressions for momentum coefficients . . . 212

6 Hydrodynamics of non-conformal branes 213 6.1 Introduction . . . 213

6.2 Lower dimensional field equations . . . 216

6.3 Universal Hydrodynamics . . . 221

6.4 Generalized black branes . . . 224

6.5 Generalized black branes in Fefferman-Graham coordinates . . . 225

6.6 Transformation to Eddington-Finkelstein coordinates . . . 229

6.7 Discussion . . . 232

6.A Appendix . . . 232 6.A.1 The asymptotic expansion of metric and scalar beyond the non-local mode 232

Bibliography 235

Summary 247

Zusammenfassung 251

Samenvatting 255