Black Hole Entropy Function, Attractors and Precision Counting of Microstates

Abstract

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N=4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centered black holes as well.

LaTeX file, 196 pages, based on lectures given at various schools; v2: added appendix E containing analysis of the multiple D5-brane system, expanded discussion on duality orbits, other minor changes, references added; v3: equations (5.6.20) and (5.6.21) corrected; v4: minor corrections to equations (C.19), (C.20)

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• Non-supersymmetric Black Holes and Topological Strings
• Precision Entropy of Spinning Black Holes
• New Attractors, Entropy Function and Black Hole Partition Function
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• Attractors with Vanishing Central Charge
• Nonextremal black holes are BPS
• Black-Hole Attractors in N=1 Supergravity
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• On Attractor Mechanism and Entropy Function for Non-extremal Black Holes/Branes
• One entropy function to rule them all
• Spinning Strings as Small Black Rings
• Noncommutative $D_3$-brane, Black Holes and Attractor Mechanism
• Extremal non-BPS black holes and entropy extremization
• Fundamental Superstrings as Holograms
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• Fundamental Strings and Black Holes
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• On Entropy Function for Supersymmetric Black Rings
• From BPS to Non-BPS Black Holes Canonically
• How many black holes fit on the head of a pin?
• Entropy Function for Non-Extremal Black Holes in String Theory
• Extremal black holes in D=5: SUSY vs. Gauss-Bonnet corrections
• The alpha-prime stretched horizon in the Heterotic string
• The Attractor Mechanism in Gauss-Bonnet Gravity
• On the Entropy Function and the Attractor Mechanism for Spherically Symmetric Extremal Black Holes
• Non-Supersymmetric Attractors in BI black holes
• Extremal Black Holes in Supergravity
• Heterotic Coset Models of Microscopic Strings and Black Holes
• Entropy Function and Universal Entropy of Two-Dimensional Extremal Black Holes
• AdS_3, Black Holes and Higher Derivative Corrections
• Open String Attractors
• Partition function of non-supersymmetric black holes in the supergravity limit
• Non-supersymmetric Attractor with the Cosmological Constant
• Non-Abelian Magnetized Blackholes and Unstable Attractors
• Double Horizon Limit, AdS Geometry and Entropy Function
• Non-Supersymmetric Unattractors in Born-Infeld Black Holes
• Generalized entropy function for Schwarzschild and non-extremal black holes in string theory
• Entropy Function for Non-extremal D1D5 and D2D6NS5-branes
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• Holographic Entanglement Entropy: An Overview
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• Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
• Indices for Superconformal Field Theories in 3,5 and 6 Dimensions
• CFT Duals for Extreme Black Holes
• Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates
• Logarithmic Corrections to N=4 and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity
• Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function
• Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions
• A classification of near-horizon geometries of extremal vacuum black holes
• Lectures on Quantum Black Holes
• Wald's entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling
• Topology of black hole thermodynamics in Gauss-Bonnet gravity
• Near Extremal Kerr Entropy from AdS_2 Quantum Gravity
• Quantum black holes, localization and the topological string
• Greybody Factors and Charges in Kerr/CFT
• Supersymmetric Index from Black Hole Entropy
• Topology of critical points and Hawking-Page transition
• Arithmetic of Quantum Entropy Function
• Topology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet gravity
• The BPS limit of rotating AdS black hole thermodynamics
• String theory: a perspective over the last 25 years
• String Theory Effects on Five-Dimensional Black Hole Physics
• Extremal vacuum black holes in higher dimensions
• Holographic Complexity Equals Which Action?
• Einstein-Maxwell-scalar black holes: classes of solutions, dyons and extremality
• A Holographic Derivation of the Weak Gravity Conjecture
• On the Universality of Inner Black Hole Mechanics and Higher Curvature Gravity
• Black holes admitting a Freudenthal dual
• Exact asymptotically flat charged hairy black holes with a dilaton potential
• stu Black Holes Unveiled
• A Farey tale for N=4 dyons
• Black Hole Horizon Fluffs: Near Horizon Soft Hairs as Microstates of Three Dimensional Black Holes
• Jackiw-Teitelboim Gravity and Rotating Black Holes
• Thermodynamic Topology and Photon Spheres in the Hyperscaling violating black holes
• Black hole entropy in massive Type IIA
• Conformal Symmetry for Black Holes in Four Dimensions
• Modular Invariance, Tauberian Theorems, and Microcanonical Entropy
• All higher-curvature gravities as Generalized quasi-topological gravities
• Erice Lectures on Black Holes and Attractors
• A Comment on Kerr-CFT and Wald Entropy
• Ads(3)/CFT(2) to Ads(2)/CFT(1)
• CFT Duals for Attractor Horizons
• Black Hole Hair Removal
• Two Centered Black Holes and N=4 Dyon Spectrum
• On attractor mechanism of $Ad{S}_4$ black holes
• N=8 Dyon Partition Function and Walls of Marginal Stability
• d=4 Attractors, Effective Horizon Radius and Fake Supergravity
• Eisenstein series and automorphic representations
• Supersymmetry, Localization and Quantum Entropy Function
• BKM Lie superalgebras from dyon spectra in Z_N-CHL orbifolds for composite N
• Uniqueness of near-horizon geometries of rotating extremal AdS(4) black holes
• AdS${}_2$ Holography: Mind the Cap
• Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives
• d=4 Black Hole Attractors in N=2 Supergravity with Fayet-Iliopoulos Terms
• Black Hole Microstate Counting and its Macroscopic Counterpart
• Generalities of Quarter BPS Dyon Partition Function and Dyons of Torsion Two
• Trace Anomaly and Counterterms in Designer Gravity
• Asymptotic Expansion of the N=4 Dyon Degeneracy
• Corrections to the statistical entropy of five dimensional black holes
• Extremal Black Hole and Flux Vacua Attractors
• Non-BPS Attractors in 5d and 6d Extended Supergravity
• Holographic RG flow dual to attractor flow in extremal black holes
• Hidden conformal symmetry of rotating charged black holes
• Nearing Extremal Intersecting Giants and New Decoupled Sectors in N = 4 SYM
• Logarithmic Corrections to Entropy of Magnetically Charged AdS4 Black Holes
• Entropy-Product Rules for Charged Rotating Black Holes
• Counting all dyons in N =4 string theory
• Duality Orbits, Dyon Spectrum and Gauge Theory Limit of Heterotic String Theory on T^6
• Isometry-preserving boundary conditions in the Kerr/CFT correspondence
• Entropy increase during physical processes for black holes in Lanczos-Lovelock gravity
• EVH Black Holes, AdS3 Throats and EVH/CFT Proposal
• Thermodynamic geometry and interacting microstructures of BTZ black holes
• Attractors in Black
• Ruppeiner Geometry, Phase Transitions and Microstructures of Black Holes in Massive Gravity
• Emergent IR dual 2d CFTs in charged AdS5 black holes
• Discrete Information from CHL Black Holes
• Holographic screens in ultraviolet self-complete quantum gravity
• Gravitational Chern-Simons Lagrangians and black hole entropy
• Physical process first law and increase of horizon entropy for black holes in Einstein-Gauss-Bonnet gravity
• N=2 Supersymmetric Black Attractors in Six and Seven Dimensions
• BKM Lie superalgebras from counting twisted CHL dyons
• Conformal Symmetry of a Black Hole as a Scaling Limit: A Black Hole in an Asymptotically Conical Box
• NHEG Mechanics: Laws of Near Horizon Extremal Geometry (Thermo)Dynamics
• Thermodynamic Topology of Quantum Corrected AdS-Reissner-Nordstrom Black Holes in Kiselev Spacetime
• No entropy enigmas for N=4 dyons
• Precision Microstate Counting for the Entropy of Wrapped M5-branes
• A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two
• Black hole entropy from entanglement: A review
• Quantum energy teleportation in a quantum Hall system
• Extremal Black Holes and First Law of Thermodynamics
• Fricke S-duality in CHL models
• How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?
• T duality and Wald entropy formula in the Heterotic Superstring effective action at first order in ${\alpha }^{\prime }$
• AdS${}_5$ Black Hole Entropy near the BPS Limit
• Restricted thermodynamic fluctuations and the Ruppeiner geometry of black holes
• Horizon Fluffs: Near Horizon Soft Hairs as Microstates of Generic AdS3 Black Holes
• Instantons, black holes and harmonic functions
• AdS${}_7$ Black-Hole Entropy and 5D $\mathcal{N}=2$ Yang-Mills
• Beyond the near-horizon limit: Stringy corrections to Heterotic Black Holes
• Thermodynamic Black Holes
• Beyond Logarithmic Corrections to Cardy Formula
• Are quantization rules for horizon areas universal?
• Separation of Attractors in 1-modulus Quantum Corrected Special Geometry
• The Largest Mathieu Group and (Mock) Automorphic Forms
• Extremal dyonic black holes in D=4 Gauss-Bonnet gravity
• A special road to AdS vacua
• Classification, geometry and applications of supersymmetric backgrounds
• Breaking away from the near horizon of extreme Kerr
• A Twist in the Dyon Partition Function
• Black Hole Hair Removal: Non-linear Analysis
• Small Orbits
• Ineffective Higher Derivative Black Hole Hair
• Charge Orbits of Extremal Black Holes in Five Dimensional Supergravity
• N=4 conformal supergravity: the complete actions
• Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity
• A New Spin on the Weak Gravity Conjecture
• Quantum black hole entropy and the holomorphic prepotential of N=2 supergravity
• Revisiting Leading Quantum Corrections to Near Extremal Black Hole Thermodynamics
• Thermodynamics of Born-Infeld Black Holes
• Weak Gravity Conjecture, Black Hole Entropy, and Modular Invariance
• Universal Formula for the Holographic Speed of Sound
• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• Non-supersymmetric Attractors in Born-Infeld Black Holes with a Cosmological Constant
• alpha'^2-corrections to extremal dyonic black holes in heterotic string theory
• Global solutions for higher-dimensional stretched small black holes
• Dynamically and thermodynamically stable black holes in Einstein-Maxwell-dilaton gravity
• On the dyon partition function in N=2 theories
• Freudenthal Dual Lagrangians
• Small Black Hole Explosions
• Extremal black holes, Holography & Coarse graining
• Entropy Function and Universality of Entropy-Area Relation for Small Black Holes
• Conformal Symmetry for Black Holes in Four Dimensions and Irrelevant Deformations
• Genus Two Surface and Quarter BPS Dyons: The Contour Prescription
• Boundary terms and conserved charges in higher-derivative gauged supergravity
• From black holes to baby universes in CGHS gravity
• Ultra-Spinning Gauged Supergravity Black Holes and their Kerr/CFT Correspondence
• Nearing 11d Extremal Intersecting Giants and New Decoupled Sectors in D = 3,6 SCFT's
• Protected couplings and BPS dyons in half-maximal supersymmetric string vacua
• State-space Correlations and Stabilities
• A Survey of Black Hole Thermodynamics
• The fuzzball nature of two-charge black hole microstates
• Horizon Strings as 3d Black Hole Microstates
• Exact charges from heterotic black holes
• Single-centered black hole microstate degeneracies from instantons in supergravity
• On walls of marginal stability in N=2 string theories
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• Negative discriminant states in N=4 supersymmetric string theories
• BKM Lie superalgebras from counting twisted CHL dyons -- II
• Kerr-CFT From Black-Hole Thermodynamics
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• Probing the EVH limit of supersymmetric AdS black holes
• Gravitational Anomalies in nAdS${}_2$/nCFT${}_1$
• Superselection Sectors of Gravitational Subregions
• Intersecting Attractors
• On Quantum Special Kaehler Geometry
• The Entropy Shift and Extremality Shift at Zero Temperature
• Graph Theory and Qubit Information Systems of Extremal Black Branes
• Siegel Modular Forms and Black Hole Entropy
• New higher derivative action for tensor multiplet in N=2 conformal supergravity in four dimensions
• Near horizon data and physical charges of extremal AdS black holes
• Computing 1/N^2 corrections in AdS/CFT
• The Scales of Black Holes with nAdS${}_2$ Geometry
• Universal near-horizon conformal structure and black hole entropy
• Equivalence of JT Gravity and Near-extremal Black Hole Dynamics in Higher Derivative Theory
• U-Duality and the Compactified Gauss-Bonnet Term
• Brewing moonshine for Mathieu
• Curvature squared action in four dimensional $N=2$ supergravity using the dilaton Weyl multiplet
• Entropy Functions with 5D Chern-Simons terms
• Four-derivative couplings and BPS dyons in heterotic CHL orbifolds
• Near Horizon Analysis of Extremal AdS5 Black Holes
• BPS State Counting in N=8 Supersymmetric String Theory for Pure D-brane Configurations
• Logarithmic Corrections to Twisted Indices from the Quantum Entropy Function
• Subleading Correction to Statistical Entropy for BMPV Black Hole
• Conductivities from attractors
• ${\alpha }^{\prime }$-corrections and their double formulation
• On Thermodynamics and Phase Space of Near Horizon Extremal Geometries
• Covariant Noether Charge for Higher Dimensional Chern-Simons Terms
• Quantum Corrected Polymer Black Hole Thermodynamics: Mass Relations and Logarithmic Entropy Correction
• Near Horizon Extremal Geometry Perturbations: Dynamical Field Perturbations vs. Parametric Variations
• Duality Symmetry and the Cardy Limit
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• Hot Attractors
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• The Hodge-elliptic genus, spinning BPS states, and black holes
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• Stringy Effects and the Role of the Singularity in Holographic Complexity
• Entropy function from the gravitational surface action for an extremal near horizon black hole
• The full space of BPS multicenter states with pure D-brane charges
• Dyon degeneracies from Mathieu moonshine
• Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-like Formula for Near-extremal Black Holes
• Subleading Microstate Counting in the Dual to Massive Type IIA
• Thermodynamic Geometry: Evolution, Correlation and Phase Transition
• A Toy Model for Topology Change Transitions: Role of Curvature Corrections
• On small black holes, KK monopoles and solitonic 5-branes
• Constraints on "rare" dyon decays
• Near-Extremal Fluid Mechanics
• Killing Spinors for Finite Temperature Euclidean Solutions at the BPS Bound
• Flux Attractors and Generating Functions
• Stationary rotating black holes in theories with gravitational Chern-Simons Lagrangian term
• Five-dimensional EVH heterotic black holes and its dual IR-CFT
• Five-dimensional heterotic black holes and its dual IR-CFT
• BKM superalgebras from counting dyons in N=4 supersymmetric type II compactifications
• On the internal structure of dyons in $\mathcal{N}=4$ super Yang-Mills theories
• N=4 Topological Amplitudes and Black Hole Entropy
• Properties of dyons in $\mathcal{N}=4$ theories at small charges
• Generalized Attractors in Five-Dimensional Gauged Supergravity
• Quantum entropy of BMPV black holes and the topological M-theory conjecture
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• Attraction, with Boundaries
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• On Black Objects in Type IIA Superstring Theory on Calabi-Yau Manifolds
• Extreme Kerr black hole microstates with horizon fluff
• Spontaneous scalarization in (A)dS gravity at zero temperature
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• Near-extremal dynamics away from the horizon
• Schwarzschild Mass Uncertainty
• Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics
• A Generalised Garfinkle-Vachaspati Transform
• Expanding on the Cardy-like limit of the superconformal index of 4d $\mathcal{N}=1$ ABCD SCFTs
• Quantum Black Hole Entropy, Localization and the Stringy Exclusion Principle
• Entanglement signatures of phase transition in higher-derivative quantum field theories
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• Horizon states and the sign of their index in $\mathcal{N}=4$ dyons
• Non-perturbative black holes in Type-IIA String Theory vs. the No-Hair conjecture
• A Kaluza-Klein Subttractor
• The fate of flat directions in higher derivative gravity
• Mock modularity at work, or black holes in a forest
• Flows of Extremal Attractor Black Holes
• Higher-derivative corrections to small black rings
• Attractor horizons in six-dimensional type IIB supergravity
• Gravitational entropy and thermodynamics away from the horizon
• An approach to BPS black hole microstate counting in an N=2 STU model
• Quantum entropy of supersymmetric black holes
• Attractor invariants, brane tilings and crystals
• Entropy increases at linear order in scalar-hairy Lovelock gravity
• Higher curvature self-interaction corrections to Hawking Radiation
• Generalized Covariant Entropy Bound in Lanczos-Lovelock Gravity
• Exact Half-BPS Black Hole Entropies in CHL Models from Rademacher Series
• BPS state counting on singular varieties
• Mock Modularity In CHL Models
• Absorption of scalars by extremal black holes in string theory
• Generalised Garfinkle-Vachaspati Transform With Dilaton
• Ramanujan and Quantum Black Holes
• Corrections to Bekenstein-Hawking entropy --- Quantum or not-so quantum?
• Dyonic AdS_4 black hole entropy and attractors via entropy function
• Black hole thermodynamics at 4 derivatives, natural variables and BPS limits
• Entanglement entropy due to near horizon degrees of freedom
• Automorphic Black Hole Entropy
• Killing Horizons and Spinors
• Microscopic State of BHs and an Exact One Body Method for Binary Dynamics in General Relativity
• Comments on Sen's Classical Entropy Function for Static and Rotating AdS${}_4$ Black Holes
• Two-dimensional Black Holes in a Higher Derivative Gravity and Matrix Model
• Siegel Paramodular Forms from Exponential Lifts: Slow versus Fast Growth
• On Exponentially Suppressed Corrections to BMPV Black Hole Entropy
• On the Stability of Non-Supersymmetric Quantum Attractors in String Theory
• Attractor Flows in st^2 Black Holes
• Generalized Hot Attractors
• From Special Geometry to Black Hole Partition Functions
• Elliptic genera from multi-centers
• Quantum Black Hole Entropy and Localization in Supergravity
• Attractors with Large Complex Structure for One-Parameter Families of Calabi-Yau Manifolds
• Rholography, Black Holes and Scherk-Schwarz
• The Hot Attractor Mechanism: Decoupling Without Deep Throats
• M theory and the Coulomb phase of higher rank DT invariants
• Charge Orbits and Moduli Spaces of Black Hole Attractors
• Entropy of Black Holes in N=2 Supergravity
• On Symmetries of Extremal Black Holes with One and Two Centers
• M5-brane prongs, string soliton bound states and wall-crossing
• Fluctuating Black Hole Horizons
• Higher derivative corrections to BPS black hole attractors in 4d gauged supergravity
• Exact holography and black hole entropy in N=8 and N=4 string theory
• Near horizon symmetry of extremal spacelike-stretched black holes
• Scale invariance with fundamental matters and anomaly: A holographic description
• Scalar Moduli, Wall Crossing and Phenomenological Predictions
• Stable Bianchi III attractor in $U(1{)}_R$ gauged supergravity
• Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the ${\mathbb{Z}}_2$ Model
• Contributions of Riemann invariants to the Entropy of Extremal Black Holes
• Thermodynamic Split Conjecture and an Observational Test for Cosmological Entropy
• Off-shell phase diagram of BPS black holes in AdS${}_5$
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• Higher-order string effective actions and off-shell d=4 supergravity
• Higher genus Siegel forms and multi-center black holes in N=4 supersymmetric string theory
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• Aspects of Higher Spin Theories, Conformal Field Theories and Holography
• Agujeros Negros con Pelo y Dualidad AdS/CFT
• Bubbling saddles of the gravitational index
• On the Supersymmetry of Bianchi attractors in Gauged supergravity
• First Law of Black Hole in the Gravitational Electromagnetic System
• Small black holes in the large D limit
• Mock Modularity Of Twisted Index In CHL Models
• Non-supersymmetric black holes with ${\alpha }^{\prime }$ corrections
• Gauss-Bonnet Contributions to the Entropy of Extremal Black Holes in the Gauge-Gravity sector
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