Black Hole Attractors and Superconformal Quantum Mechanics Davide Gaiotto dgaiotto@fas.harvard.edu Harvard University joint work with Aaron Simons, Andrew Strominger, Xi Yin hep-th/ 0412322 hep-th/ 0412179 Black Hole Attractors and Superconformal Quantum Mechanics – p.
Contents • Brief review: • BPS black holes in N = 2 supergravity • Near horizon geometry: attractor mechanism Black Hole Attractors and Superconformal Quantum Mechanics – p.
Contents • Brief review: • BPS black holes in N = 2 supergravity • Near horizon geometry: attractor mechanism • D 0 -brane probes in near horizon geometry • Superconformal Quantum Mechanics • Short multiplets Black Hole Attractors and Superconformal Quantum Mechanics – p.
Contents • Brief review: • BPS black holes in N = 2 supergravity • Near horizon geometry: attractor mechanism • D 0 -brane probes in near horizon geometry • Superconformal Quantum Mechanics • Short multiplets • BH entropy from D 2 -brane wrapping horizon. Black Hole Attractors and Superconformal Quantum Mechanics – p.
N = 2 SUGRA • IIA (IIB) on Calabi Yau threefold Black Hole Attractors and Superconformal Quantum Mechanics – p.
N = 2 SUGRA • IIA (IIB) on Calabi Yau threefold • gravitational multiplet g µν , ψ µ , A µ • n = h 1 , 1 vector multiplets A i µ , λ i , z i • h 2 , 1 + 1 hypermultiplets Black Hole Attractors and Superconformal Quantum Mechanics – p.
N = 2 SUGRA • IIA (IIB) on Calabi Yau threefold • gravitational multiplet g µν , ψ µ , A µ • n = h 1 , 1 vector multiplets A i µ , λ i , z i • h 2 , 1 + 1 hypermultiplets Black Hole Attractors and Superconformal Quantum Mechanics – p.
N = 2 SUGRA • IIA (IIB) on Calabi Yau threefold • gravitational multiplet g µν , ψ µ , A µ • n = h 1 , 1 vector multiplets A i µ , λ i , z i • h 2 , 1 + 1 hypermultiplets � F A � � X A • Sp (2 n + 2) symmetry: � G A F A Black Hole Attractors and Superconformal Quantum Mechanics – p.
N = 2 SUGRA • IIA (IIB) on Calabi Yau threefold • gravitational multiplet g µν , ψ µ , A µ • n = h 1 , 1 vector multiplets A i µ , λ i , z i • h 2 , 1 + 1 hypermultiplets � F A � � X A • Sp (2 n + 2) symmetry: � G A F A • z i = X i X 0 Black Hole Attractors and Superconformal Quantum Mechanics – p.
N = 2 SUGRA • IIA (IIB) on Calabi Yau threefold • gravitational multiplet g µν , ψ µ , A µ • n = h 1 , 1 vector multiplets A i µ , λ i , z i • h 2 , 1 + 1 hypermultiplets � F A � � X A • Sp (2 n + 2) symmetry: � G A F A • z i = X i X 0 � � D abc X a X b X c • F A = ∂ A F = ∂ A + · · · X 0 Black Hole Attractors and Superconformal Quantum Mechanics – p.
BPS Black Holes • Graviphoton: X A G A − F A F A p A F A − q A X A � � � • Mass: Z ∞ = � ∞ Black Hole Attractors and Superconformal Quantum Mechanics – p.
BPS Black Holes • Graviphoton: X A G A − F A F A p A F A − q A X A � � � • Mass: Z ∞ = � ∞ • Depends on moduli at infinity. Black Hole Attractors and Superconformal Quantum Mechanics – p.
BPS Black Holes • Graviphoton: X A G A − F A F A p A F A − q A X A � � � • Mass: Z ∞ = � ∞ • Depends on moduli at infinity. • Dependence is forgotten near the horizon • X A ( r ) → X A fixed ( p A , q A ) r → 0 as • Scalars flow to an attractor point Black Hole Attractors and Superconformal Quantum Mechanics – p.
BPS Black Holes • X A fixed extremizes | Z | = | p A F A − q A X A | • Re CX A fixed = p A Re CF A fixed = q A Black Hole Attractors and Superconformal Quantum Mechanics – p.
BPS Black Holes • X A fixed extremizes | Z | = | p A F A − q A X A | • Re CX A fixed = p A Re CF A fixed = q A • S BH ( p, q ) = π | Z | 2 fixed • exp S BH ( p, q ) ≃ Number of microstates of black hole Black Hole Attractors and Superconformal Quantum Mechanics – p.
Entropy from microstates � p A � • Charges from D-branes in Calabi-Yau q A ( p 0 , p a , q a , q 0 ) ( D 6 , D 4 , D 2 , D 0) Black Hole Attractors and Superconformal Quantum Mechanics – p.
Entropy from microstates � p A � • Charges from D-branes in Calabi-Yau q A ( p 0 , p a , q a , q 0 ) ( D 6 , D 4 , D 2 , D 0) • Counting of bound states of D 4 , D 2 , D 0 • S BH ( p, q ) ≃ 2 π √ D ˜ q 0 q 0 = q 0 + 1 • D = D abc p a p b p c 12 D ab q a q b ˜ Black Hole Attractors and Superconformal Quantum Mechanics – p.
Entropy from microstates � p A � • Charges from D-branes in Calabi-Yau q A ( p 0 , p a , q a , q 0 ) ( D 6 , D 4 , D 2 , D 0) • Counting of bound states of D 4 , D 2 , D 0 • S BH ( p, q ) ≃ 2 π √ D ˜ q 0 q 0 = q 0 + 1 • D = D abc p a p b p c 12 D ab q a q b ˜ • Different computations: M-theory, others. 1 • � (1 − x n ) 6 D Black Hole Attractors and Superconformal Quantum Mechanics – p.
Probing near the horizon • Bertotti-Robinson AdS 2 × S 2 × CY | fixed • Maximal supersymmetry: 8 supercharges Black Hole Attractors and Superconformal Quantum Mechanics – p.
Probing near the horizon • Bertotti-Robinson AdS 2 × S 2 × CY | fixed • Maximal supersymmetry: 8 supercharges • D-branes can be BPS for generic charges • Same SUSY preserved by charges ( u, v ) u A q A − v A p A > 0 • Many ways to probe near horizon geometry. Black Hole Attractors and Superconformal Quantum Mechanics – p.
Probing near the horizon • Bertotti-Robinson AdS 2 × S 2 × CY | fixed • Maximal supersymmetry: 8 supercharges • D-branes can be BPS for generic charges • Same SUSY preserved by charges ( u, v ) u A q A − v A p A > 0 • Many ways to probe near horizon geometry. Black Hole Attractors and Superconformal Quantum Mechanics – p.
Plan Superconformal QM of probe D-branes Black Hole Attractors and Superconformal Quantum Mechanics – p.
Plan Superconformal QM of probe D-branes • Black Hole Entropy: • microstates as bound states in near-horizon geometry Black Hole Attractors and Superconformal Quantum Mechanics – p.
Plan Superconformal QM of probe D-branes • Black Hole Entropy: • microstates as bound states in near-horizon geometry • AdS 2 /CFT 1 correspondence: • Superconformal Matrix Quantum Mechanics? Black Hole Attractors and Superconformal Quantum Mechanics – p.
D 0 branes in AdS 2 × S 2 × CY • Action in Poincare coordinates Black Hole Attractors and Superconformal Quantum Mechanics – p.
D 0 branes in AdS 2 × S 2 × CY • Action in Poincare coordinates � � • − m R 2 θ 2 +sin 2 θ ˙ d t y µ ˙ σ 2 ) − R 2 ( ˙ y ν φ 2 ) − 2 g µν ˙ σ 2 (1 − ˙ Black Hole Attractors and Superconformal Quantum Mechanics – p.
D 0 branes in AdS 2 × S 2 × CY • Action in Poincare coordinates � � • − m R 2 θ 2 +sin 2 θ ˙ d t y µ ˙ σ 2 ) − R 2 ( ˙ y ν φ 2 ) − 2 g µν ˙ σ 2 (1 − ˙ � � d t • q e R σ + q m R d φ cos θ Black Hole Attractors and Superconformal Quantum Mechanics – p.
D 0 branes in AdS 2 × S 2 × CY • Action in Poincare coordinates � � • − m R 2 θ 2 +sin 2 θ ˙ d t y µ ˙ σ 2 ) − R 2 ( ˙ y ν φ 2 ) − 2 g µν ˙ σ 2 (1 − ˙ � � d t • q e R σ + q m R d φ cos θ � u A � � u A F A − v A X A � � • Charges m = � v A • q m = p A v A − q A u A q 2 e + q 2 m = m 2 Black Hole Attractors and Superconformal Quantum Mechanics – p.
Hamiltonian • Hamiltonian to quadratic order ( σ = ξ 2 ) 8 mR ˆ ξ 2 ˆ ξ 2 ˆ 1 1 1 P 2 L 2 ξ + S 2 + ∆ CY • 2 mR ˆ 2 mR ˆ Black Hole Attractors and Superconformal Quantum Mechanics – p. 10
Hamiltonian • Hamiltonian to quadratic order ( σ = ξ 2 ) 8 mR ˆ ξ 2 ˆ ξ 2 ˆ 1 1 1 P 2 L 2 ξ + S 2 + ∆ CY • 2 mR ˆ 2 mR ˆ • N = 4 superconformal symmetry SU (1 , 1 | 2) • ˆ 2 { ˆ ξ, ˆ K = 2 mR ˆ ˆ D = 1 ξ 2 P ξ } Black Hole Attractors and Superconformal Quantum Mechanics – p. 10
Hamiltonian • Hamiltonian to quadratic order ( σ = ξ 2 ) 8 mR ˆ ξ 2 ˆ ξ 2 ˆ 1 1 1 P 2 L 2 ξ + S 2 + ∆ CY • 2 mR ˆ 2 mR ˆ • N = 4 superconformal symmetry SU (1 , 1 | 2) • ˆ 2 { ˆ ξ, ˆ K = 2 mR ˆ ˆ D = 1 ξ 2 P ξ } • SU (2) R-symmetry ˆ = ˆ i + ˆ L spin L S 2 L tot i i • ˆ αβ = ˆ L tot L tot i σ i αβ Black Hole Attractors and Superconformal Quantum Mechanics – p. 10
Fermions • Fermionic partners: D 0 brane has 16 in 10 d spinor Θ u + ¯ • Θ = λ α u + η a η ¯ a α Γ a u + ¯ α Γ ¯ a ¯ λ α ¯ u Black Hole Attractors and Superconformal Quantum Mechanics – p. 11
Fermions • Fermionic partners: D 0 brane has 16 in 10 d spinor Θ u + ¯ • Θ = λ α u + η a η ¯ a α Γ a u + ¯ α Γ ¯ a ¯ λ α ¯ u partners of ξ, ˆ � λ α • 4 goldstinos λ A � α = L S 2 (3 , 4 , 1) ¯ λ α Black Hole Attractors and Superconformal Quantum Mechanics – p. 11
Fermions • Fermionic partners: D 0 brane has 16 in 10 d spinor Θ u + ¯ • Θ = λ α u + η a η ¯ a α Γ a u + ¯ α Γ ¯ a ¯ λ α ¯ u partners of ξ, ˆ � λ α • 4 goldstinos λ A � α = L S 2 (3 , 4 , 1) ¯ λ α • { λ A α , λ B β } = 2 ǫ AB ǫ αβ Black Hole Attractors and Superconformal Quantum Mechanics – p. 11
Fermions on CY • 12 η a η ¯ y ¯ α partners of y a , ¯ a a α , ¯ 3(2 , 4 , 2) Black Hole Attractors and Superconformal Quantum Mechanics – p. 12
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