The Attractor Mechanism (eBook)

Preface 
6 
Contents 
8 
Contributors 
10 
Chapter 1 SAM Lectures on Extremal Black Holes in d=4 ExtendedSupergravity 12
1.1 Introduction 12
1.2 N=2, d=4 Symmetric MESGT's 14
1.2.1 U-Duality ``Large'' Orbits 14
1.2.2 Classification of Attractors 21
1.2.2.1 12-BPS 21
1.2.2.2 Non-BPS 21
1.2.3 Critical Spectra and Massless Hessian Modes of VBH 27
1.2.3.1 12-BPS 28
1.2.3.2 Non-BPS, Z0 29
1.2.3.3 Non-BPS, Z=0 30
1.2.4 From Massless Hessian Modes of VBH to Moduli Spaces of Attractors 31
1.3 U-Duality ``Large'' Orbits and Moduli Spaces of Attractors in N3-Extended, d=4 Supergravities 34
1.4 Conclusions 35
References 36
Chapter 2 Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace 42
2.1 Introduction 42
2.2 U-Duality Symmetries of Maximal Supergravity in Various Dimensions 44
2.3 5D, N=2 Maxwell–Einstein Supergravity Theories 45
2.4 MESGT's with Symmetric Target Spaces and Euclidean Jordan Algebras of Degree 3 47
2.5 Rotation (Automorphism), Lorentz (Reduced Structure) and Conformal (Möbius) Groups of Jordan Algebras 51
2.6 U-Duality Orbits of Extremal Black Hole Solutions of 5D Supergravity Theories with Symmetric Target Manifolds and Their Spectrum Generating Conformal Extensions 54
2.7 4D, N=2 Maxwell–Einstein Supergravity Theories with Symmetric Target Spaces and Freudenthal Triple Systems 58
2.8 U-Duality Orbits of Extremal Black Holes of 4D, N=2 MESGTs with Symmetric Scalar Manifolds and of N=8 Supergravity and Their Spectrum Generating Quasiconformal Extensions 62
2.9 Quasiconformal Realizations of Lie Groups and Freudenthal Triple Systems 65
2.10 3D U-Duality Groups as Spectrum Generating Quasiconformal Groups of 4D Supergravity Theories and Quantum Attractor Flows 69
2.11 Harmonic Superspace, Minimal Unitary Representations and Quasiconformal Group Actions 75
2.11.1 4D, N=2 -Models Coupled to Supergravity in Harmonic Superspace 76
2.11.2 Minimal Unitary Representations of Non-compact Groups from Their Quasiconformal Realizations 79
2.11.3 Harmonic Superspace Formulation of N=2 Sigma Models and Minimal Unitary Representations of Their Isometry Groups 82
2.11.4 Implications 85
2.12 M/Superstring Theoretic Origins of N=2 MESGTs with Symmetric Scalar Manifolds 86
2.13 Recent Developments and Some Open Problems 88
References 92
Chapter 3 Attractors, Black Holes and Multiqubit Entanglement 96
3.1 Introduction 96
3.2 Bipartite Systems 98
3.2.1 Pure States 98
3.2.2 Mixed States 102
3.2.3 Real Bipartite Pure States 104
3.2.4 Real Bipartite Mixed States 107
3.3 Multipartite Systems 108
3.3.1 Pure States, Three Qubits 108
3.3.2 Pure States, Four Qubits 114
3.4 Error Correction, Hadamard Matrices and Graph States 120
3.4.1 Errors 120
3.4.2 The (7,4,3) Hamming Code, Designs and Steiner Triple Systems 121
3.4.3 Graph States 126
3.5 STU Black Holes 127
3.5.1 The Black Hole Potential as the Norm of a Three-Qubit State 131
3.5.2 BPS and Non-BPS Solutions 135
3.5.3 Entanglement and BPS Solutions 136
3.5.4 Entanglement and Non-BPS Solutions 140
3.5.5 N=8 Reinterpretation of the STU-Model: Density Matrices 144
3.5.6 A Unified Picture for the D2–D6 System 148
3.5.7 Real States in the STU Model 151
3.5.8 Summary 153
3.6 N=8 Supergravity and the Tripartite Entanglement of Seven Qubits 154
3.6.1 The Representation Space for the 56 of E7 154
3.6.2 The Generators of E7 157
3.6.3 The Generators of e7 as a Set of TripartiteTransformations 159
3.6.4 Cartan's Quartic Invariant as an Entanglement Measure 163
3.7 Conclusions 172
References 173
Chapter 4 From Special Geometry to Black Hole Partition Functions 176
4.1 Introduction 176
4.2 Lecture I: Special Geometry 178
4.2.1 Gauge Equivalence and the Stückelberg Mechanismfor Gravity 179
4.2.2 Gravity as a Constrained Gauge Theory of the Conformal Group 180
4.2.3 Rigid N=2 Vector Multiplets 182
4.2.4 Rigid Superconformal Vector Multiplets 185
4.2.5 N=2 Conformal Supergravity 186
4.2.6 N=2 Poincaré Supergravity 187
4.2.7 Problems 192
4.3 Lecture II: Attractor Mechanism, Variational Principle,and Black Hole Partition Functions 194
4.3.1 BPS States 194
4.3.2 BPS Solitons and BPS Black Holes 195
4.3.3 The Black Hole Variational Principle 201
4.3.4 Canonical, Microcanonical and Mixed Ensemble 203
4.3.5 R2-Corrections 207
4.3.6 Non-holomorphic Corrections 210
4.4 Lecture III: Black Holes in N=4 Supergravity 215
4.4.1 N=4 Compactifications 215
4.4.2 N=4 Supergravity in the N=2 Formalism 216
4.4.3 R2-Corrections for N=4 Black Holes 219
4.4.4 The Reduced Variational Principle for N=4 Theories 222
4.4.5 Small N=4 Black Holes 223
4.5 Lecture IV: N=4 State Counting and Black Hole Partition Functions 224
4.5.1 Counting 12-BPS States 225
4.5.2 State Counting for 14-BPS States 227
4.5.3 Partition Functions for Large Black Holes 231
4.5.4 Partition Functions for Small Black Holes 233
4.5.5 Problems 236
A Kähler Manifolds and Special Kähler Manifolds 238
A.1 Complex and Almost Complex Manifolds 238
A.2 Hermitian Manifolds 239
A.3 Kähler Manifolds 240
A.4 Affine Special Kähler Manifolds 241
A.4.1 Special Affine Coordinates and the Hesse Potential 244
A.5 Conical Affine Special Kähler Manifolds and Projective Special Kähler Manifolds 245
B Modular Forms 247
References 250
Chapter 5 Complexity at the Fundamental Level 253
5.1 The Basic Points on Complexity and Predictions 253
5.1.1 Complexity 254
5.1.2 Predictions 254
5.1.3 Complexity and Predictions 254
5.2 AFB Phenomena from Beethoven to the Superworld 255
5.2.1 Beethoven and the Laws of Acoustics 255
5.2.2 The Living Cell and QED 255
5.2.3 Nuclear Physics and the UEEC Events 256
5.2.4 Nuclear Physics and QCD 257
5.3 UEEC Events, from Galilei up to Present Days 257
5.4 Seven Definitions of Complexity 261
5.5 Complexity Exists at all Scales 263
5.6 Science, from Planck to Complexity 265
5.7 The Two Asymptotic Limits: History and Science 266
5.8 Conclusions 267
References 293
Chapter 6 Non-supersymmetric Attractors in Symmetric Coset Spaces 298
6.1 Introduction 298
6.2 Framework 300
6.2.1 3D Moduli Space M3D 300
6.2.2 Attractor Flow Equations 301
6.2.3 Models with M3D Being Symmetric Coset Spaces 302
6.2.3.1 Parametrization of M3D 303
6.2.4 Example: nV=1 304
6.3 Generators of Attractor Flows 306
6.3.1 Construction of Attractor Flow Generators 306
6.3.1.1 Construction of kBPS 306
6.3.1.2 Extremality Implies Nilpotency of Flow Generators 307
6.3.1.3 Construction of kNB 308
6.3.2 Properties of Attractor Flow Generators 308
6.3.2.1 Properties of kBPS 309
6.3.3 Properties of kNB 310
6.4 Single-Centered Attractor Flows 311
6.4.1 Single-Centered BPS Attractor Flows 313
6.4.2 Single-Centered Non-BPS Attractor Flows 316
6.5 Multi-Centered Attractor Flows 318
6.5.1 Multi-centered BPS Attractors 319
6.5.2 Multi-centered Non-BPS Attractors 321
6.6 Conclusion and Discussion 322
References 323
Chapter 7 Higher-Order String Effective Actions and Off-Shell d=4 Supergravity 325
7.1 Introduction and Plan 325
7.2 Superspace Geometry 326
7.2.1 Vielbein, Connection, Torsion and Curvature 326
7.2.2 Variational Equations 329
7.2.3 Choice of Constraints 330
7.3 N=1 Supergravity in Superspace 332
7.3.1 N=1 Superspace Geometry and Constraints 332
7.3.2 From Conformal to Poincaré Supergravity 333
7.3.2.1 Ungauged U(1) 333
7.3.2.2 Gauged U(1) 334
7.3.3 The Chiral Compensator and the Chiral Measure 335
7.3.4 Solution to the Bianchi Identities in ``old minimal'' N=1 Poincaré Supergravity 337
7.3.5 From Superspace to x-Space 337
7.4 N=2 Supergravity in Superspace 340
7.4.1 N=2 Conformal Supergravity 340
7.4.2 Degauging U(1) 341
7.4.3 Degauging SU(2) 343
7.4.4 From N=2 SU(2) Superspace to x-Space 343
7.4.5 The Chiral Density and the Chiral Projector 344
7.5 Superstring '3 Effective Actions and R4 Terms in d=4 345
7.5.1 N=1, 2 Supersymmetrization of W+2 W-2 346
7.5.1.1 N=1 346
7.5.1.2 N=2 346
7.5.2 N=1 Supersymmetrization of W+4 + W-4 347
7.5.2.1 W+4 + W-4 in Extended Supergravity 349
7.6 Applications to Black Holes in String Theory 350
7.7 Summary and Discussion 352
References 353