Supertubes and the 4D black hole Per Kraus, UCLA with I. Bena: hep-th/0402144, hep-th/0408186, hep-th/0502xxx Supertubes and the 4D black hole – p.1/25
Introduction Much has been learned from relating the gravity and gauge theory descriptions of the D1-D5-P system. AdS 3 × S 3 × T 4 (or K 3 ) • NS-NS vacuum ↔ • low energy chiral primaries ↔ sugra perturbations BTZ × S 3 × T 4 • Thermal ensemble ↔ More recently, we have learned (Lunin, Mathur; Lunin, Maldacena, Maoz) • chiral primaries ↔ 2-charge supertubes: D1-D5 → kk More general 3-charge supertubes exist; where do they fit in the picture? What about related CFT with less susy, e.g. D1-D5-KK system? Relation to black hole entropy Mathur Supertubes and the 4D black hole – p.2/25
Review of 2-charge supertubes (Mateos, Townsend) Start with a flat Dp-brane in x 0 , 1 ,...p , and turn on worldvolume electric and magnetic fields 2 πF 02 = 1 , 2 πF 12 = B Induces F1-strings, D(p-2)-branes, and P 1 : F 1 N p − 2 ≈ BRL P 1 N F 1 RT p /B ≈ L P 1 RLT p ≈ X 2 X 1 2 R π • N p − 2 N F 1 − J = 0 , J ≡ P 1 R Born-Infeld action gives L BI = − ( − det[ η µν + 2 πF µν ]) 1 / 2 ≈ − B and so the energy is H = π E F 02 − L BI = Q F 1 + Q p − 2 • BPS, and no contribution from Dp-brane Supertubes and the 4D black hole – p.3/25
Open string quantization Fluxes described by open string metric: � X µ ( τ 1 ) X ν ( τ 2 ) � = − G µν ln | τ 1 − τ 2 | 2 + i 2 θ µν ǫ ( τ − τ ′ ) − 1 + B − 2 − B − 1 0 G µν = − B − 1 0 0 B − 2 0 0 • G 11 = 0 ! � X 1 ( z 1 ) X 1 ( z 2 ) � = 0 ⇒ So we can start with a zero momentum vertex operator ǫ µ ∂ n,t X µ and attach a factor e ip 1 X 1 to get a dimension 1 primary V = ǫ µ ∂ n,t X µ e ip 1 X 1 , G µν ǫ µ p ν = 0 • Adds momentum P 1 but no energy or other charge. • Multiple such operators can be added, and exponentiated Supertubes and the 4D black hole – p.4/25
The Dp-brane can change its shape and local flux density at no cost in energy J F1 F1 P In the tubular case J is angular momentum. For a circular tube J = N p − 2 N F 1 Adding open string excitations decreases J , and counting is same as for momentum of gas in 1 + 1 dim: S ∼ ( N p − 2 N F 1 − J ) 1 / 2 Supertubes and the 4D black hole – p.5/25 Counting also done by dualizing to FP or in Born-Infeld
Comments • Supertube radius is R 2 ∼ g s , so at weak coupling the tube structure is lost. Makes counting at weak coupling more subtle. • But since tubes become large at strong coupling, they are more directly related to finite size gravitational description. • Entropy of 2-charge tube too small to correspond to classical black hole horizon, but was given a stretched horizon type interpretation (Lunin, Mathur) . • Related work including higher derivative corrections (Dabholkar et. al) . Supertubes and the 4D black hole – p.6/25
3-charge supertubes (Bena, P.K.) To compare with black hole physics would like a tube carrying D1-D5-P charges. But more convenient to dualize and take D0-D4-F1 since F1 appears in supertube construction. Starting from D 0 + F 1 d 2 → and dualizing, we have D 4 + F 1 → d 6 D 0 + D 4 ns 5 → • So we expect a tube with 3 independent dipole charges: d2, d6, and ns5. • For now set ns5 dipole to zero, since we can’t describe it via flux in Born-Infeld. Can include by T-dualizing ns5 → kk ≈ A N singularity. Or, work in M-theory (Elvang et. al.) Supertubes and the 4D black hole – p.7/25
On a D6-brane turn on fluxes F 02 , F 12 , F 34 , F 56 to induce charges F 02 ∼ F1 − strings , F 12 ∼ D4 − branes , F 12 F 34 F 56 ∼ D0 − branes But also have D2-branes from F 12 F 34 , F 12 F 56 , F 34 F 56 First two are unwanted; last will give wanted d2 dipole. • Cancel unwanted D2-branes by introducing second D6-brane with flipped signs of F 34 and F 56 . • Generalizing to N 6 such D6-branes, we get a BPS configuration with energy H = Q F 1 + Q D 0 + Q D 4 and momentum J = P 1 R = N F 1 N D 4 N D 6 Supertubes and the 4D black hole – p.8/25
Quantizing the neutral open strings proceeds just as before. Again find BPS fluctuations of shape and flux profiles, and can form circular tube. Spectrum of charged strings more involved (e.g. Callan et. al.) . Need to work with superstring. Zero mode problem in x 3 , 4 , 5 , 6 like charged particle in magnetic field [ P 3 , P 4 ] ≈ iF 34 , [ P 5 , P 6 ] ≈ iF 56 s Get a Landau level degeneracy � N F 1 N D 4 � p V 3456 F 34 F 56 • Combine these with massless states from R or NS sector. • Including X 0 , 1 , 2 part, we can again attach e ip 1 X 1 factors at no cost in energy. • With N 6 D6-branes, have number of species N 2 6 V 3456 F 34 F 56 ≈ N 6 n 2 • Entropy is therefore S ∼ N 6 n 2 − J = n 2 N F 1 N D 4 − N 6 n 2 J N D 6 Supertubes and the 4D black hole – p.9/25
Comments • Still too small to correspond to black hole area. Need the ns5 dipole! • Enhancement of entropy compared to 2-charge case came from Landau degeneracy. Corresponds to changes in non-abelian part of flux. • Since states are described by Landau levels, wavefunctions are inhomogeneous in x 3 , 4 , 5 , 6 . • So sugra solutions for microstates need to capture non-abelian degrees of freedom, and inhomogeneity on T 4 . Supertubes and the 4D black hole – p.10/25
Including the ns5 dipole charge • Including NS5 in the flat case yields a brane carrying charges D2-D6-NS5-P . These are the standard ingredients of the 4d black hole, after compactification on T 6 . • Entropy given by quartic E 7(7) invariant: √ S = 2 π J 4 x ij y jk x kl y li − x ij y ij x kl y kl / 4 − J 4 = ǫ ijklmnop ( x ij x kl x mn x op + y ij y kl y mn y op ) + with the charges identified as x 12 = N D 0 , x 34 = N D 4 , x 56 = N F 1 , x 78 = 0 y 34 = n d 2 , y 56 = n ns 5 , y 78 = J y 12 = n d 6 , • System now has finite size S 2 × T 6 horizon. As before, we can instead curl up one direction into a circle and compactify on T 5 . Result should be a horizon of topology S 1 × S 2 in D = 5 — a black ring. Entropy should agree with above. Related approach (Cyrier, Guica, Mateos, Strominger) . Supertubes and the 4D black hole – p.11/25
p Supertubes in sugra • Consider D1-D5 system in NS-NS vacuum. Corresponds to CFT on cylinder with antiperiodic fermions. Vacuum preserves full conformal SL (2 , R ) L × SL (2 , R ) R symmetry, and SO (4) ≈ SU (2) L × SU (2) R R-symmetry. Unique choice of bulk geometry is AdS 3 × S 3 ( × T 4 ) r 2 r 2 ds 2 = − (1 + ˜ d ˜ t 2 + r 2 dχ 2 + ℓ 2 ( d ˜ θ 2 + sin 2 ˜ ψ 2 + cos 2 ˜ θd ˜ θd ˜ φ 2 ) ℓ 2 ) d ˜ + ˜ r 2 1 + ˜ ℓ 2 ℓ 2 = Q 1 Q 5 • Want to extend to asymptotically flat region R (1 , 4) × S 1 × T 4 . • By susy, fermions are periodic on S 1 , so CFT in RR sector. • R sector related to NS sector by spectral flow L 0 → L 0 + ηJ + c J → J + c 6 η 2 , 3 η This is redefinition of generators, so is just a diffeomorphism in AdS 3 × S 3 : χ = x 5 t φ = φ + x 5 ˜ ˜ , ψ = ψ − , R 5 R 5 R 5 Also rescale r and t . Supertubes and the 4D black hole – p.12/25
q p p q • Also write r cos ˜ θ r 2 + R 2 sin 2 ˜ ρ = θ, cos θ = r 2 + R 2 sin 2 ˜ θ with R = ℓ 2 R 5 . • Then metric takes form similar to standard form 1 ds 2 = [ − ( dt + k ) 2 + ( dx 5 − k − s ) 2 ] + Z 1 Z 2 ( dρ 2 + ρ 2 d Ω 2 √ Z 1 Z 5 3 ) with Z 1 , 5 = 1 + Q 1 , 5 ( ρ 2 − R 2 ) 2 + 4 R 2 ρ 2 cos 2 θ , Σ = Σ • Z 1 , 5 are harmonic functions sourced on ring ρ = r, cos θ = 0 . Asymptotically flat solution obtained by including 1 as usual. • 1-forms k and s are essentially vector potentials sourced by currents on ring. • Solution is BPS with M = Q 1 + Q 5 and angular momenta J L = J R = N 1 N 5 Solution is completely smooth due to expansion D 1 − D 5 → kk . Easy to generalize: just replace ring by arbitrary curve in R 4 . Supertubes and the 4D black hole – p.13/25
Microscopic description of black rings (I. Bena, P.K.) Supergravity solution for 3-charge supertube was found by (Elvang, Emparan, Mateos, Reall) and generalized further by (Bena, Warner; EEMR; Gauntlett, Gutowski) In IIB frame solutions carries charges N 1 D 1(5) , N 2 D 5(56789) , N 3 P (5) and dipole charges n 1 d 5( x 6789) , n 2 d 1( x ) , n 3 kk ( x 56789) p • N i are conserved charges measured at infinity. These differ from charges N i at ring: N 1 = N 1 − n 2 n 3 , and permutations • Similarly, “harmonic" functions Z i are no longer harmonic Σ + q 2 q 3 ρ 2 Z 1 = 1 + Q 1 Σ 2 ( ρ 2 − R 2 ) 2 + 4 R 2 ρ 2 cos 2 θ . with Σ = • 1 / Σ is a harmonic function sourced on the ring: ρ = R, cos θ = 0 . R = 0 gives BMPV. Supertubes and the 4D black hole – p.14/25
• Solution carries angular momenta J φ = J BMP V = − 1 2Σ n i N i − n 1 n 2 n 3 J ψ = − J BMP V + J tube with R KK V 4 (2 π ) 4 ( α ′ ) 4 g 2 ( q 1 + q 2 + q 3 ) R 2 J tube = • Entropy is 2 π [ − 1 2 2 2 4( n 2 1 + n 2 2 + n 2 S = 1 N 2 N 3 N 3 ) 1 2( n 1 n 2 N 1 N 2 + n 1 n 3 N 1 N 3 + n 2 n 3 N 2 N 3 ) − n 1 n 2 n 3 ( J ψ + J φ )] 1 / 2 + √ = 2 π J 4 • Solutions have 7 free parameters, but only 5 conserved charges. So these black objects have “hair". Makes it especially interesting to understand them on gauge theory side. Supertubes and the 4D black hole – p.15/25
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