Anti de Sitter Black Holes

1. Anti de Sitter Black Holes Harvey Reall University of Nottingham

2. Motivation • Black hole entropy calculations all rely on 2d CFT • Can we calculate BH entropy using higher dimensional CFT? • Need supersymmetric AdS black holes to evade strong coupling problem

3. Plan • SUSY asymptotically flat black holes • SUSY AdS black holes in D=3,4 • SUSY AdS black holes in D=5 • CFT interpretation • Collaborators: J. Gutowski, R. Roiban, H. Kunduri, J. Lucietti

4. SUSY 5D black holesHSR 02, Gutowski 03 • 5D supergravity + vectors • Introduce coordinates adapted to horizon • Impose supersymmetry • Near-horizon geometry fully determined

5. SUSY horizons: S3, S1 x S2, T3 • Only asymptotically flat SUSY BH with S3 horizon is BMPV • SUSY black rings discovered recently Elvang et al 04, Bena & Warner 04, Gauntlett & Gutowski 04 • T3 seems unlikely

6. SUSY AdS Black Holes • D=3: BTZ is SUSY black hole iff M=|J|>0 • D=4: Kerr-Newman-AdS (M,J,Q,P) saturates BPS bound if M=M(Q), J=J(Q), P=0 Kostalecky & Perry 95, Caldarelli & Klemm 98 • SUSY AdS black holes must rotate

7. 5D SUSY AdS Black holesGutowski & HSR 04 • Reduce IIB SUGRA on S5 to N=1 D=5 U(1)3 gauged SUGRA Cvetic et al 99 • Canonical form for SUSY solutions involves specifying 4d Kähler “base space” Gauntlett & Gutowski 03, Gutowski & HSR 04 • Choice of base space not obvious e.g. get AdS5 from Bergman manifold • Try to find SUSY black holes systematically by examining near-horizon geometry

8. In near horizon limit, conditions for SUSY reduce to equations on 3-manifold • General solution unknown but particular S3 solution can be found • Use this to guess form of base space of corresponding black hole solution • First examples of SUSY AdS5 black holes • Preserve 1/16 SUSY • Cohomogeneity 1, 3 parameters • SO(6) R-charge (Q1,Q2,Q3), J1=J2=J(Q), BPS relation M=|J1|+|J2|+|Q1|+|Q2|+|Q3|

9. Unequal Angular Momenta • Chong, Cvetic, Lü & Pope 05: SUSY black holes with unequal angular momenta • Cohomogeneity 2, 2 parameters • Kunduri, Lucietti & HSR 06: most general known SUSY solution • parameterized by J1, J2, Q1, Q2, Q3 with one constraint • Expect non-BPS generalization with independent M,J,Q (2 more parameters)

10. A Puzzle • Why do BPS black holes have a constraint relating J,Q? • Is there a more general family of SUSY black holes with independent J,Q? • But then must have a more general family of non-BPS black holes: specifying M,Q,J not enough to determine topologically S3 black holes!

11. CFT entropy calculation? • Need to count 1/16 BPS states of N=4 SU(N) SYM on RxS3 (or local operators on R4) with same quantum numbers O(N2) as black hole • Black hole entropy O(N2) • States typically descendents but need large entropy O(N2) in primaries

12. No 1/8 BPS black holesRoiban & HSR 04, Berenstein 05 • 1/8 BPS primaries built from N=1 superfields Xi,W • Commutators give descendents, so Xi, W can be treated as commuting • Diagonalize: O(N) degrees of freedom so entropy of primaries of length O(N2) is O(N log N), too small for bulk horizon

13. Weakly coupled CFTRoiban & HSR 04, Kinney, Maldacena, Minwalla & Raju 05 • Goal: at weak coupling, count operators in short 1/16 BPS multiplets that can’t become long at strong coupling • Too hard! Count everything instead… • Find correct scaling of entropy with charge for large charge

14. Superconformal IndexKinney, Maldacena, Minwalla & Raju 05 • Vanishing contribution from states in short multiplets that can combine into long ones • Independent of N at large N: doesn’t “see” black holes • Cancellation between bosonic and fermionic BPS states dual to black hole

15. Summary • There is a 4-parameter family of 1/16 BPS black holes in AdS5 • Why only 4 parameters? • How do we calculate their entropy using N=4 SYM?