Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Near-Extremal Vanishing Horizon AdS 5 Black Holes and Their CFT Duals Maria Johnstone 1 M.M. Sheikh-Jabbari 2 Joan Simón 3 Hossein Yavartanoo 4 1 University of Edinburgh, UK 2 Institute for Research in Fundamental Sciences, Iran 3 University of Edinburgh, UK 4 Kyung Hee University, Korea EMPG Seminar, Edinburgh 2013
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Black Holes solutions to general relativity 1 behave like thermodynamic systems: 2 satisfy thermodynamic laws have a thermodynamic entropy: S BH = A d 4 G d Question Why does the entropy scale like the horizon area? ⇒ Holography: “the fundamental degrees of freedom describing the system are described by a quantum field theory with one less dimension.”
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Question What are the the underlying states of this QFT giving rise to black hole entropy? Two commonly used tools: Near horizon geometry: Zoom in on region very close to 1 the event horizon r + . Extremality: T=0 black holes are more symmetric: AdS 2 2 factor in near horizon geometry
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Kerr/CFT (Extremal Black Hole/CFT) Correspondence Statement of Kerr/CFT: Near horizon quantum states ⇐ ⇒ quantum states of a chiral 2d CFT
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Chiral 2d CFT 2d CFT: 2d quantum field theory invariant under conformal transformations. Generators L n of conformal transformations obey Virasoro algebra: [ L m , L n ] = ( m − n ) L m + n + c 12 ( m 3 − m ) δ m + n , 0 . Central charge c: a number that characterises the CFT States in 2d CFT: split into left-moving and right-moving pieces in left and right moving sectors. Left-moving sector: L m , L n ; c L . Right-moving sector: ¯ L m , ¯ L n ; c R . Chiral 2d CFT: excited states exist in only the left-moving sector. One copy of Virasoro algebra with one c L .
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Kerr/CFT (Extremal Black Hole/CFT) Correspondence Statement of Kerr/CFT: Extremal black holes are holographically dual to chiral 2d conformal field theory. Near horizon geometry: ds 2 = ds 2 AdS 2 + ... Use near horizon data to compute c L 1 Frolov-Thorne temperature T L : “temperature of the dual 2 CFT“. Microscopic Cardy formula ⇒ macroscopic black hole entropy: S Cardy = π 2 3 c L T L = S BH
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Kerr/CFT (Extremal Black Hole/CFT) Correspondence Kerr/CFT: originally for 4d black holes. Generalised to higher dimensions. Vacuum degeneracy of chiral 2d CFT accounts for macroscopic black hole entropy. Little more information.
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction AdS/CFT Correspondence AdS/CFT Correspondence: Gravity in AdS d + 1 ⇐ ⇒ CFT d . 1:1 correspondence between local fields in the gravity theory and operators in the boundary QFT. AdS 3 /CFT 2 : non-chiral 2d CFT dual to gravity in AdS 3 . Question Can an extremal black hole have a near horizon AdS 3 throat that’s dual to the full non-chiral CFT 2 ?
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Introduction Q: Can an Extremal Black Hole have a Near Horizon AdS 3 ? Answer: Yes A H , T H → 0: Extremal Vanishing Horizon (EVH) black holes. EVH black holes: Near horizon geometry develops locally AdS 3 throat. Local AdS 3 near horizon ⇒ dual CFT 2 description: EVH/CFT Correspondence. A H , T H ∼ ǫ << 1: Near-EVH black holes: AdS 3 → BTZ black hole. Asymptotically AdS 5 × S 5 (near-)EVH black holes: 4d CFT dual: link with near horizon 2d CFT?
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Plan of the Talk Describe asymptotically AdS 5 × S 5 black hole solutions to 1 10d IIB supergravity Criteria: EVH and near-EVH black holes 2 Near horizon limit: AdS 3 3 IR dual CFT 2 and compare with UV CFT 4 4 1st Law of Thermodynamics in near-EVH limit 5 Compare results with Kerr/CFT 6 Summarise and Discuss 7
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT 5d Supergravity Solution Black hole solution to U(1) 3 5d gauged supergravity: � ρ 2 ( dt − a sin 2 θ d φ − b cos 2 θ d ψ − X H − 4 ds 2 ) 2 = 3 Ξ a Ξ b + C ρ 2 ( ab dt − b sin 2 θ d φ − a cos 2 θ d ψ ) 2 Ξ a Ξ b f 3 f 2 f 1 Z sin 2 θ ) 2 + W cos 2 θ ) 2 � ( a dt − 1 d φ ( b dt − 1 d ψ + ρ 2 ρ 2 f 3 f 2 Ξ a f 3 f 1 Ξ b 3 ( ρ 2 X dr 2 + ρ 2 2 d θ 2 ) , + H ∆ θ Gauge fields: A 2 = P 1 ( dt − a sin 2 θ d φ − b cos 2 θ d ψ A 1 = ) Ξ a Ξ b P 3 ( b sin 2 θ d φ + a cos 2 θ d ψ A 3 = ) Ξ a Ξ b
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Scalar fields: X 1 = X 2 = H − 1 2 3 , X 3 = H 3 H , ρ, ˜ ρ, f i , ∆ θ , C , Z , W , Ξ a , Ξ b , P i : functions of ( r ; a , b , q , m ) . Horizon function: X ( r + ) = X ( r − ) = 0 r 2 ( a 2 + r 2 )( b 2 + r 2 ) − 2 m + ( a 2 + r 2 + q )( b 2 + r 2 + q ) X ( r ) = 1 ℓ 2
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Thermodynamic Quantities Hawking Temperature : + ( ℓ 2 + a 2 + b 2 + 2 q ) − a 2 b 2 ℓ 2 T H = 2 r 6 + + r 4 2 π r + ℓ 2 [( r 2 + + a 2 )( r 2 + + b 2 ) + qr 2 + ] Beckenstein-Hawking Entropy: S BH = π 2 [( r 2 + + a 2 )( r 2 + + b 2 ) + qr 2 + ] 2 G 5 Ξ a Ξ b r +
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Thermodynamic Quantities Rotation in φ , ψ : Angular velocities: + b 2 + r 2 + q + ℓ 2 b 2 + ℓ 2 r 2 Ω a = a ( r 4 + + r 2 + ) , ℓ 2 ( a 2 + r 2 + )( b 2 + r 2 + ) + ℓ 2 qr 2 + + a 2 + r 2 + q + ℓ 2 a 2 + ℓ 2 r 2 Ω b = b ( r 4 + + r 2 + ) . ℓ 2 ( a 2 + r 2 + )( b 2 + r 2 + ) + ℓ 2 qr 2 + Angular momenta: π a ( 2 m + q Ξ b ) J b = π b ( 2 m + q Ξ a ) J a = , . 4 G 5 Ξ b Ξ 2 4 G 5 Ξ a Ξ 2 a b parametrised by a,b.
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Thermodynamic Quantities Gauge Fields A i : Chemical Potentials: � q 2 + 2 mq r 2 + Φ 1 = Φ 2 = , ( a 2 + r 2 + )( b 2 + r 2 + ) + qr 2 + qab Φ 3 = . ( a 2 + r 2 + )( b 2 + r 2 + ) + qr 2 + Electric Charges: � q 2 + 2 mq Q 2 = π π abq Q 3 = − Q 1 = , . 4 G 5 ℓ 2 Ξ a Ξ b 4 G 5 Ξ a Ξ b parametrised by q. Note: Q 3 ∼ ab not independent.
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Thermodynamic Quantities First Law of Thermodynamics: 3 � T H dS BH = dE − Ω a dJ a − Ω b dJ b − Φ i dQ i i = 1 Integrate ⇒ Black hole mass: E = π [ 2 m ( 2 Ξ a + 2 Ξ b − Ξ a Ξ b ) + q ( 2 Ξ 2 a + 2 Ξ 2 b + 2 Ξ a Ξ b − Ξ 2 a Ξ b − Ξ 2 b Ξ 8 G 5 Ξ 2 a Ξ 2 b
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT 10d Embedding Solution to 10d IIB supergravity: � 2 � ℓ 2 � ds 2 ∆ ds 2 10 = 5 + � d � ∆ 5 ds 2 5 : 5d black hole metric deformed S 5 : 2 3 � � X − 1 ( d µ 2 i + µ 2 i ( d ψ i + A i /ℓ ) 2 ) . d = i 5 i = 1 also: F 5 = ⋆ F 5 with flux N Newton’s constants: ℓ 3 π 3 ℓ 5 = π 1 G 5 = G 10 N 2 . 2
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT 10d Embedding 10d Embedding 5d electrostatic potential Φ i = 10d angular velocity Ω i on S 5 . 5d electric charge Q i = 10d angular momentum J i on S 5 .
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT Dual 4d Description AdS/CFT: Black Hole in AdS 5 × S 5 ↔ mixed state in dual N = 4 SYM. States carry conserved charges given by gravity conserved charges: � q 2 + 2 mq N 2 , ∆ = ℓ E , J 1 = J 2 = Q 1 = 2 ℓ 2 Ξ a Ξ b S a = J a = a ( 2 m + q Ξ b ) S b = J b = b ( 2 m + q Ξ a ) N 2 , N 2 . 2 ℓ 3 Ξ 2 2 ℓ 3 Ξ 2 a Ξ b b Ξ a
Introduction Charged rotating AdS 5 black holes The Set of EVH Black Holes Near-EVH Black Holes Relation between EVH/CFT The Set of EVH Black Holes EVH Black Holes Horizon equation: X ( r + ) = 0 ⇒ m = m ( r + ) 4-dimensional black hole parameter space: ( a , b , q , m ) ↔ ( a , b , q , r + ) EVH black holes: A BH = T H = 0 ⇒ r + = 0 and ab = 0 . Two types of EVH configurations for these black holes: Rotating: b = r + = 0 , a � = 0 ( J b = 0, J a � = 0) 1 Static: a = b = r + = 0 ( J a = J b = 0) 2 Note: EVH limit ⇒ angular momentum ∼ ab:J 3 = 0
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