Bellucci The Attractor Mechanism

Proceedings of the INFN-Laboratori Nazionali di Frascati School 2007
2010
ISBN: 978-3-642-10736-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 134, 346 Seiten

Reihe: Springer Proceedings in Physics

This book is based upon lectures presented in June 2007 at the INFN-Laboratori Nazionali di Frascati School on Attractor Mechanism, directed by Stefano Bellucci. The symposium included such prestigious lecturers as S. Ferrara, M. Günaydin, P. Levay, and T. Mohaupt. All lectures were given at a pedagogical, introductory level, which is reflected in the specific "flavor" of this volume. The book also benefits from extensive discussions about, and related reworking of, the various contributions.In addition, this volume contains contributions originating from short presentations of recent original results and an essay on the relation between complexity science and high-energy physics by A. Zichichi. It is the fourth volume in a series of books on the general topics of supersymmetry, supergravity, black holes and the attractor mechanism.

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