Bellucci The Attractor Mechanism

Proceedings of the INFN-Laboratori Nazionali di Frascati School 2007
1. Auflage 2010
ISBN: 978-3-642-10736-8
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 346 Seiten

Autoren/Hrsg.

Bellucci, Stefano

Inhaltsverzeichnis

1;Preface
;6
2;Contents
;8
3;Contributors
;10
4;Chapter 1 SAM Lectures on Extremal Black Holes in d=4 ExtendedSupergravity;12
4.1;1.1 Introduction;12
4.2;1.2 N=2, d=4 Symmetric MESGT's;14
4.2.1;1.2.1 U-Duality ``Large'' Orbits;14
4.2.2;1.2.2 Classification of Attractors;21
4.2.2.1;1.2.2.1 12-BPS;21
4.2.2.2;1.2.2.2 Non-BPS;21
4.2.3;1.2.3 Critical Spectra and Massless Hessian Modes of VBH;27
4.2.3.1;1.2.3.1 12-BPS;28
4.2.3.2;1.2.3.2 Non-BPS, Z0;29
4.2.3.3;1.2.3.3 Non-BPS, Z=0;30
4.2.4;1.2.4 From Massless Hessian Modes of VBH to Moduli Spaces of Attractors;31
4.3;1.3 U-Duality ``Large'' Orbits and Moduli Spaces of Attractors in N3-Extended, d=4 Supergravities;34
4.4;1.4 Conclusions;35
4.5;References;36
5;Chapter 2 Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace;42
5.1;2.1 Introduction;42
5.2;2.2 U-Duality Symmetries of Maximal Supergravity in Various Dimensions;44
5.3;2.3 5D, N=2 Maxwell–Einstein Supergravity Theories;45
5.4;2.4 MESGT's with Symmetric Target Spaces and Euclidean Jordan Algebras of Degree 3;47
5.5;2.5 Rotation (Automorphism), Lorentz (Reduced Structure) and Conformal (Möbius) Groups of Jordan Algebras;51
5.6;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
5.7;2.7 4D, N=2 Maxwell–Einstein Supergravity Theories with Symmetric Target Spaces and Freudenthal Triple Systems;58
5.8;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
5.9;2.9 Quasiconformal Realizations of Lie Groups and Freudenthal Triple Systems;65
5.10;2.10 3D U-Duality Groups as Spectrum Generating Quasiconformal Groups of 4D Supergravity Theories and Quantum Attractor Flows;69
5.11;2.11 Harmonic Superspace, Minimal Unitary Representations and Quasiconformal Group Actions;75
5.11.1;2.11.1 4D, N=2 -Models Coupled to Supergravity in Harmonic Superspace;76
5.11.2;2.11.2 Minimal Unitary Representations of Non-compact Groups from Their Quasiconformal Realizations;79
5.11.3;2.11.3 Harmonic Superspace Formulation of N=2 Sigma Models and Minimal Unitary Representations of Their Isometry Groups;82
5.11.4;2.11.4 Implications;85
5.12;2.12 M/Superstring Theoretic Origins of N=2 MESGTs with Symmetric Scalar Manifolds;86
5.13;2.13 Recent Developments and Some Open Problems;88
5.14;References;92
6;Chapter 3 Attractors, Black Holes and Multiqubit Entanglement;96
6.1;3.1 Introduction;96
6.2;3.2 Bipartite Systems;98
6.2.1;3.2.1 Pure States;98
6.2.2;3.2.2 Mixed States;102
6.2.3;3.2.3 Real Bipartite Pure States;104
6.2.4;3.2.4 Real Bipartite Mixed States;107
6.3;3.3 Multipartite Systems;108
6.3.1;3.3.1 Pure States, Three Qubits;108
6.3.2;3.3.2 Pure States, Four Qubits;114
6.4;3.4 Error Correction, Hadamard Matrices and Graph States;120
6.4.1;3.4.1 Errors;120
6.4.2;3.4.2 The (7,4,3) Hamming Code, Designs and Steiner Triple Systems;121
6.4.3;3.4.3 Graph States;126
6.5;3.5 STU Black Holes;127
6.5.1;3.5.1 The Black Hole Potential as the Norm of a Three-Qubit State;131
6.5.2;3.5.2 BPS and Non-BPS Solutions;135
6.5.3;3.5.3 Entanglement and BPS Solutions;136
6.5.4;3.5.4 Entanglement and Non-BPS Solutions;140
6.5.5;3.5.5 N=8 Reinterpretation of the STU-Model: Density Matrices;144
6.5.6;3.5.6 A Unified Picture for the D2–D6 System;148
6.5.7;3.5.7 Real States in the STU Model;151
6.5.8;3.5.8 Summary;153
6.6;3.6 N=8 Supergravity and the Tripartite Entanglement of Seven Qubits;154
6.6.1;3.6.1 The Representation Space for the 56 of E7;154
6.6.2;3.6.2 The Generators of E7;157
6.6.3;3.6.3 The Generators of e7 as a Set of TripartiteTransformations;159
6.6.4;3.6.4 Cartan's Quartic Invariant as an Entanglement Measure;163
6.7;3.7 Conclusions;172
6.8;References;173
7;Chapter 4 From Special Geometry to Black Hole Partition Functions;176
7.1;4.1 Introduction;176
7.2;4.2 Lecture I: Special Geometry;178
7.2.1;4.2.1 Gauge Equivalence and the Stückelberg Mechanismfor Gravity;179
7.2.2;4.2.2 Gravity as a Constrained Gauge Theory of the Conformal Group;180
7.2.3;4.2.3 Rigid N=2 Vector Multiplets;182
7.2.4;4.2.4 Rigid Superconformal Vector Multiplets;185
7.2.5;4.2.5 N=2 Conformal Supergravity;186
7.2.6;4.2.6 N=2 Poincaré Supergravity;187
7.2.7;4.2.7 Problems;192
7.3;4.3 Lecture II: Attractor Mechanism, Variational Principle,and Black Hole Partition Functions;194
7.3.1;4.3.1 BPS States;194
7.3.2;4.3.2 BPS Solitons and BPS Black Holes;195
7.3.3;4.3.3 The Black Hole Variational Principle;201
7.3.4;4.3.4 Canonical, Microcanonical and Mixed Ensemble;203
7.3.5;4.3.5 R2-Corrections;207
7.3.6;4.3.6 Non-holomorphic Corrections;210
7.4;4.4 Lecture III: Black Holes in N=4 Supergravity;215
7.4.1;4.4.1 N=4 Compactifications;215
7.4.2;4.4.2 N=4 Supergravity in the N=2 Formalism;216
7.4.3;4.4.3 R2-Corrections for N=4 Black Holes;219
7.4.4;4.4.4 The Reduced Variational Principle for N=4 Theories;222
7.4.5;4.4.5 Small N=4 Black Holes;223
7.5;4.5 Lecture IV: N=4 State Counting and Black Hole Partition Functions;224
7.5.1;4.5.1 Counting 12-BPS States;225
7.5.2;4.5.2 State Counting for 14-BPS States;227
7.5.3;4.5.3 Partition Functions for Large Black Holes;231
7.5.4;4.5.4 Partition Functions for Small Black Holes;233
7.5.5;4.5.5 Problems;236
7.6;A Kähler Manifolds and Special Kähler Manifolds;238
7.6.1;A.1 Complex and Almost Complex Manifolds;238
7.6.2;A.2 Hermitian Manifolds;239
7.6.3;A.3 Kähler Manifolds;240
7.6.4;A.4 Affine Special Kähler Manifolds;241
7.6.4.1;A.4.1 Special Affine Coordinates and the Hesse Potential;244
7.6.5;A.5 Conical Affine Special Kähler Manifolds and Projective Special Kähler Manifolds;245
7.7;B Modular Forms;247
7.8;References;250
8;Chapter 5 Complexity at the Fundamental Level;253
8.1;5.1 The Basic Points on Complexity and Predictions;253
8.1.1;5.1.1 Complexity;254
8.1.2;5.1.2 Predictions;254
8.1.3;5.1.3 Complexity and Predictions;254
8.2;5.2 AFB Phenomena from Beethoven to the Superworld;255
8.2.1;5.2.1 Beethoven and the Laws of Acoustics;255
8.2.2;5.2.2 The Living Cell and QED;255
8.2.3;5.2.3 Nuclear Physics and the UEEC Events;256
8.2.4;5.2.4 Nuclear Physics and QCD;257
8.3;5.3 UEEC Events, from Galilei up to Present Days;257
8.4;5.4 Seven Definitions of Complexity;261
8.5;5.5 Complexity Exists at all Scales;263
8.6;5.6 Science, from Planck to Complexity;265
8.7;5.7 The Two Asymptotic Limits: History and Science;266
8.8;5.8 Conclusions;267
8.9;References;293
9;Chapter 6 Non-supersymmetric Attractors in Symmetric Coset Spaces;298
9.1;6.1 Introduction;298
9.2;6.2 Framework;300
9.2.1;6.2.1 3D Moduli Space M3D;300
9.2.2;6.2.2 Attractor Flow Equations;301
9.2.3;6.2.3 Models with M3D Being Symmetric Coset Spaces;302
9.2.3.1;6.2.3.1 Parametrization of M3D;303
9.2.4;6.2.4 Example: nV=1;304
9.3;6.3 Generators of Attractor Flows;306
9.3.1;6.3.1 Construction of Attractor Flow Generators;306
9.3.1.1;6.3.1.1 Construction of kBPS;306
9.3.1.2;6.3.1.2 Extremality Implies Nilpotency of Flow Generators;307
9.3.1.3;6.3.1.3 Construction of kNB;308
9.3.2;6.3.2 Properties of Attractor Flow Generators;308
9.3.2.1;6.3.2.1 Properties of kBPS;309
9.3.3;6.3.3 Properties of kNB;310
9.4;6.4 Single-Centered Attractor Flows;311
9.4.1;6.4.1 Single-Centered BPS Attractor Flows;313
9.4.2;6.4.2 Single-Centered Non-BPS Attractor Flows;316
9.5;6.5 Multi-Centered Attractor Flows;318
9.5.1;6.5.1 Multi-centered BPS Attractors;319
9.5.2;6.5.2 Multi-centered Non-BPS Attractors;321
9.6;6.6 Conclusion and Discussion;322
9.7;References;323
10;Chapter 7 Higher-Order String Effective Actions and Off-Shell d=4 Supergravity;325
10.1;7.1 Introduction and Plan;325
10.2;7.2 Superspace Geometry;326
10.2.1;7.2.1 Vielbein, Connection, Torsion and Curvature;326
10.2.2;7.2.2 Variational Equations;329
10.2.3;7.2.3 Choice of Constraints;330
10.3;7.3 N=1 Supergravity in Superspace;332
10.3.1;7.3.1 N=1 Superspace Geometry and Constraints;332
10.3.2;7.3.2 From Conformal to Poincaré Supergravity;333
10.3.2.1;7.3.2.1 Ungauged U(1);333
10.3.2.2;7.3.2.2 Gauged U(1);334
10.3.3;7.3.3 The Chiral Compensator and the Chiral Measure;335
10.3.4;7.3.4 Solution to the Bianchi Identities in ``old minimal'' N=1 Poincaré Supergravity;337
10.3.5;7.3.5 From Superspace to x-Space;337
10.4;7.4 N=2 Supergravity in Superspace;340
10.4.1;7.4.1 N=2 Conformal Supergravity;340
10.4.2;7.4.2 Degauging U(1);341
10.4.3;7.4.3 Degauging SU(2);343
10.4.4;7.4.4 From N=2 SU(2) Superspace to x-Space;343
10.4.5;7.4.5 The Chiral Density and the Chiral Projector;344
10.5;7.5 Superstring '3 Effective Actions and R4 Terms in d=4;345
10.5.1;7.5.1 N=1, 2 Supersymmetrization of W+2 W-2;346
10.5.1.1;7.5.1.1 N=1;346
10.5.1.2;7.5.1.2 N=2;346
10.5.2;7.5.2 N=1 Supersymmetrization of W+4 + W-4;347
10.5.2.1;7.5.2.1 W+4 + W-4 in Extended Supergravity;349
10.6;7.6 Applications to Black Holes in String Theory;350
10.7;7.7 Summary and Discussion;352
10.8;References;353