Erice Lectures on Black Holes and Attractors

Abstract

These lectures give an elementary introduction to the subject of four dimensional black holes (BHs) in supergravity and the Attractor Mechanism in the extremal case. Some thermodynamical properties are discussed and some relevant formulae for the critical points of the BH effective potential are given. The case of Maxwell-Einstein-axion-dilaton (super)gravity is discussed in detail. Analogies among BH entropy and multipartite entanglement of qubits in quantum information theory, as well moduli spaces of extremal BH attractors, are also discussed.

1+65 pages, 5 figures, 4 tables. Contribution to the Proceedings of the International School of Subnuclear Physics, 45th Course: Search for the "Totally Unexpected" in the LHC era, Erice, Italy, 29 August - 7 September 2007 (Directors: G. 't Hooft - A. Zichichi). v2: typos fixed in Eqs. (4.21), (4.24) and (4.25)

References in corpus (110)

• Microscopic Origin of the Bekenstein-Hawking Entropy
• Distributed Entanglement
• String Theory Dynamics In Various Dimensions
• Unity of Superstring Dualities
• Evidence for F-Theory
• N=2 Extremal Black Holes
• String Solitons
• D-brane Approach to Black Hole Quantum Mechanics
• Supersymmetry and Attractors
• Black Hole Entropy in M-Theory
• Black Hole Attractors and the Topological String
• Macroscopic Entropy of $N=2$ Extremal Black Holes
• Noncompact Symmetries in String Theory
• Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity
• Black Holes and Critical Points in Moduli Space
• Universality of Sypersymmetric Attractors
• The Extreme Kerr Throat Geometry: A Vacuum Analog of AdS_2 x S^2
• Black Hole Entropy Function, Attractors and Precision Counting of Microstates
• N=2 Supergravity and N=2 Super Yang-Mills Theory on General Scalar Manifolds: Symplectic Covariance, Gaugings and the Momentum Map
• Counting States of Near-Extremal Black Holes
• Vacuum interpolation in supergravity via super p-branes
• Supersymmetry as a Cosmic Censor
• Solitonic Strings and BPS Saturated Dyonic Black Holes
• Four Dimensional String/String/String Triality
• Duality Transformations in Supersymmetric Yang-Mills Theories coupled to Supergravity,
• Moduli, Scalar Charges, and the First Law of Black Hole Thermodynamics
• Non-Supersymmetric Attractors
• Black Hole Solutions in Heterotic String Theory on a Torus
• Rotating Attractors
• Duality Symmetries of 4D Heterotic Strings
• Flow Equations for Non-BPS Extremal Black Holes
• Lectures on Superstring and M Theory Dualities
• Thermodynamic curvature and phase transitions in Kerr-Newman black holes
• Non-Supersymmetric Attractors in String Theory
• Orbits of Exceptional Groups, Duality and BPS States in String Theory
• The Symplectic Structure of N=2 Supergravity and its Central Extension
• M-Theory (the Theory Formerly Known as Strings)
• STU Black Holes and String Triality
• Entropy Function for Heterotic Black Holes
• E(7) Symmetric Area of the Black Hole Horizon
• The Non-BPS Black Hole Attractor Equation
• New Attractors
• General Static Spherically Symmetric Black Holes of Heterotic String on a Six Torus
• Microscopic Entropy of N=2 Extremal Black Holes
• Hartle-Hawking Wave-Function for Flux Compactifications
• BPS Spectrum of the Five-Brane and Black Hole Entropy
• First-order flow equations for extremal black holes in very special geometry
• Branes, central charges and U-duality invariant BPS conditions
• Black Holes and U-Duality
• Charge Orbits of Symmetric Special Geometries and Attractors
• Non-Supersymmetric Attractor Flow in Symmetric Spaces
• String triality, black hole entropy and Cayley's hyperdeterminant
• General Matter Coupled N=2 Supergravity
• Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
• On N=8 attractors
• On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds
• New Attractors and Area Codes
• Strings, Black Holes, and Quantum Information
• First Order Description of Black Holes in Moduli Space
• Entropy and Action of Dilaton Black Holes
• Stationary Axion/Dilaton Solutions and Supersymmetry
• Entropy function and attractors for AdS black holes
• Charge Quantization of Axion-Dilaton Black Holes
• Exact Attractive Non-BPS STU Black Holes
• Unified geometric description of black hole thermodynamics
• Orbits and Attractors for N=2 Maxwell-Einstein Supergravity Theories in Five Dimensions
• Supersymmetric Black Holes
• Stationary black holes and attractor mechanism
• Stringy Black Holes and the Geometry of Entanglement
• On some properties of the Attractor Equations
• E_7 and the tripartite entanglement of seven qubits
• Non-Supersymmetric Attractors and Entropy Function
• N=8 non-BPS Attractors, Fixed Scalars and Magic Supergravities
• A C-Function For Non-Supersymmetric Attractors
• Dilaton Black Holes Near the Horizon
• New Attractors, Entropy Function and Black Hole Partition Function
• Non-supersymmetric Black Holes and Topological Strings
• Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane
• U-Invariants, Black-Hole Entropy and Fixed Scalars
• 4d/5d Correspondence for the Black Hole Potential and its Critical Points
• Non-Supersymmetric Attractors in $R^2$ Gravities
• Critical points of the Black-Hole potential for homogeneous special geometries
• Non-BPS Attractors in 5d and 6d Extended Supergravity
• Attractors with Vanishing Central Charge
• Wrapped branes as qubits
• Nonextremal black holes are BPS
• Black-Hole Attractors in N=1 Supergravity
• Black hole entropy functions and attractor equations
• Supersymmetric Black Holes in $N=8$ Supergravity
• BPS Microstates and the Open Topological String Wave Function
• N=2 Supersymmetric Black Attractors in Six and Seven Dimensions
• New attractor mechanism for spherically symmetric extremal black holes
• Moduli flow and non-supersymmetric AdS attractors
• Extremal non-BPS black holes and entropy extremization
• Black Hole Attractors and Pure Spinors
• On the Existence of Non-Supersymmetric Black Hole Attractors for Two-Parameter Calabi-Yau's and Attractor Equations
• Exact Solutions and the Attractor Mechanism in Non-BPS Black Holes
• Separation of Attractors in 1-modulus Quantum Corrected Special Geometry
• Attractor Horizon Geometries of Extremal Black Holes
• The Entropic Principle and Asymptotic Freedom
• On The Stability of Non-Supersymmetric Attractors in String Theory
• Microscopic entropy of the most general four-dimensional BPS black hole
• Non-supersymmetric Attractors in Born-Infeld Black Holes with a Cosmological Constant
• The Attractor Mechanism in Gauss-Bonnet Gravity
• N=2 Extremal Black Holes and Intersecting Branes
• Unification of String Dualities
• Quantum Lift of Non-BPS Flat Directions
• Entropy of Pairs of Dual Attractors in 6D/7D
• A Note on Exact Solutions and Attractor Mechanism for Non-BPS Black Holes
• Duality Invariant Non-Extreme Black Holes in Toroidally Compactified String Theory

Cited by in corpus (46)

• Universal BPS structure of stationary supergravity solutions
• First-order flow equations for extremal and non-extremal black holes
• First Order Description of D=4 static Black Holes and the Hamilton-Jacobi equation
• E7(7) formulation of N=2 backgrounds
• Freudenthal triple classification of three-qubit entanglement
• Constituent Model of Extremal non-BPS Black Holes
• Holographic RG flow dual to attractor flow in extremal black holes
• Black holes in supergravity and integrability
• The full integration of black hole solutions to symmetric supergravity theories
• Instantons, black holes and harmonic functions
• N=8 Supergravity on the Light Cone
• Split Attractor Flow in N=2 Minimally Coupled Supergravity
• Generalized Attractor Points in Gauged Supergravity
• Extremal black holes, Holography & Coarse graining
• Moduli and electromagnetic black brane holography
• Type II compactifications on manifolds with SU(2) x SU(2) structure
• On Quantum Special Kaehler Geometry
• Near Horizon Structure of Extremal Vanishing Horizon Black Holes
• Topics in Cubic Special Geometry
• Three Theorems on Near Horizon Extremal Vanishing Horizon Geometries
• Black Strings, Black Rings and State-space Manifold
• Special Vinberg Cones and the Entropy of BPS Extremal Black Holes
• Thermodynamic Geometry: Evolution, Correlation and Phase Transition
• On the Uniqueness of Supersymmetric Attractors
• Multiple Single-Centered Attractors
• Black holes and general Freudenthal transformations
• Black Holes and Groups of Type E7
• Spontaneous scalarization in (A)dS gravity at zero temperature
• New Branches of Non-supersymmetric Attractors in N=2 Supergravity
• Extremal Black Attractors in 8D Maximal Supergravity
• A comment on no-force conditions for black holes and branes
• Primordial black holes, astrophysical systems and the Eddington-Weinberg relation
• Fermions, Wigs, and Attractors
• Multi-Centered Invariants, Plethysm and Grassmannians
• On Black Attractors in 8D and Heterotic/Type IIA Duality
• Intersecting Black Attractors in 8D N=1 Supergravity
• Stability of Bianchi attractors in Gauged Supergravity
• Quantization of Black Holes entropy and its cosmological consequences
• On the Stability of Non-Supersymmetric Quantum Attractors in String Theory
• Peccei-Quinn Transformations and Black Holes : Orbit Transmutations and Entanglement Generation
• Topological solutions in ungauged Supergravity
• BPS Black Hole Entropy and Attractors in Very Special Geometry. Cubic Forms, Gradient Maps and their Inversion
• Stable Bianchi III attractor in $U(1{)}_R$ gauged supergravity
• On Symmetries of Extremal Black Holes with One and Two Centers
• Freudenthal Duality in Conformal Field Theory
• On the Supersymmetry of Bianchi attractors in Gauged supergravity