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

2. Motivation Black hole entropy calculations all rely on 2d CFT Can we use AdS/CFT to calculate entropy of D>3 AdS black holes? D=4: probably not, CFT not understood D=5: CFT is N=4 SYM… Need supersymmetric AdS 5 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 holes HSR 02, Gutowski 04 5D ungauged N=1 sugra + abelian vectors Introduce coordinates adapted to horizon Take near-horizon limit Impose supersymmetry: eqs on spatial cross- section of horizon Can determine general solution for compact horizon

5. SUSY 5D black holes HSR 02, Gutowski 04 All possible near-horizon geometries: Which arise from asymp flat black holes? Near-horizon BMPV from BMPV! AdS 3 xS 2 from BPS black rings Elvang et al 04 Flat T 3 horizon unlikely Galloway 06 Near-horizonHorizon geometry BMPVSquashed S 3 AdS 3 x S 2 S 1 x S 2 FlatT3T3

6. SUSY AdS Black Holes BPS limit of Reissner-Nordstrom-AdS is nakedly singular 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 holes Gutowski & HSR 04 Reduce IIB SUGRA on S 5 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 AdS 5 from Bergman manifold SU(2,1)/U(2)

8. 5D SUSY AdS black holes Gutowski & HSR 04 Seek SUSY black holes systematically by examining near-horizon geometry In near-horizon limit, conditions for SUSY are equations on 3-manifold General solution not known Particular homogeneous S 3 solution can be found (cf near-horizon BMPV)

9. 5D SUSY AdS black holes Gutowski & HSR 04 Near-horizon solution motivates cohomogeneity-1 Ansatz for full solution First examples of SUSY AdS 5 black holes! Base space singular, cohomogeneity-1, asymptotically Bergman space 1/16 BPS

10. Unequal Angular Momenta Chong, Cvetic, Lü & Pope 05 Guessed non-BPS charged rotating black hole solution of minimal gauged sugra (Einstein-Maxwell) Cohomogeneity-2, 4 parameters (M,J 1,J 2,Q) BPS limit: 2 parameter solution with J 1 ≠J 2

11. General solution Kunduri, Lucietti & HSR 06 Determine base space of BPS solution of minimal gauged sugra: singular, cohomogeneity-2, asymptotically Bergman Plug into BPS eqs of U(1) 3 gauged sugra, solve… BPS solution parametrized by J 1, J 2, Q 1, Q 2, Q 3 with one constraint Expect non-BPS generalization with independent M,J,Q (2 more parameters)

12. CFT description BPS AdS 5 black hole microstates are 1/16 BPS states of N=4 large N SYM on RxS 3 (equivalently BPS local operators on R 4 ) States classified by SO(4)xSO(6) quantum numbers J,Q Black hole quantum numbers O(N 2 ) Black hole entropy O(N 2 ) Entropy calculation: count all 1/16 BPS states with same quantum numbers as black hole

13. A Puzzle 1/16 BPS states have independent J,Q 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 corresponding non-SUSY solution would need more than just conserved charges to specify it BPS AdS black rings?

14. BPS AdS black rings? Kunduri, Lucietti & HSR 06 Most general BPS near-horizon geometry in 5D gauged sugra not known Assume existence of 2 rotational symmetries (true for all known 5d black holes): problem reduces to ODEs. 2 interesting solutions. One solution is near-horizon geometry of known S 3 black holes Another solution is a warped product AdS 3 xS 2 with horizon topology S 1 xS 2 …

15. BPS AdS black rings? Kunduri, Lucietti & HSR 06 …but with a conical singularity on S 2 Can’t eliminate singularity (without turning off cosmological constant) BPS AdS black rings with 2 rotational symmetries do not exist Oxidize to 10d: warped product AdS 3 xM 7 with M 7 =S 2 xS 5 (singular)

16. New 10d black holes? Solution is locally isometric to AdS 3 xM 7 solution of Gauntlett et al 06 They showed that solution can be made globally regular by choosing topology of M 7 appropriately (not S 2 xS 5 ) Resulting solution cannot be reduced to 5d Could this be near-horizon geometry of an asymptotically AdS 5 xS 5 black hole?

17. Resolutions of the puzzle? Are there BPS 10d black hole solutions that can’t be reduced to 5d? Are there BPS 5d black holes without 2 rotational symmetries? Non-abelian BPS black holes? Maybe we know most general black hole. 1/16 BPS states have 5 charges but perhaps only 4 charge subset has O(N 2 ) entropy Berkooz et al 06

18. CFT entropy calculation? Need to count 1/16 BPS states of N=4 SU(N) SYM on RxS 3 (or local operators on R 4 ) with same quantum numbers O(N 2 ) as black hole Black hole entropy O(N 2 ) States typically descendents but need large entropy O(N 2 ) in primaries

19. No 1/8 BPS black holes Roiban & HSR 04, Berenstein 05 1/8 BPS primaries built from N=1 superfields X i,W Commutators give descendents, so X i, W can be treated as commuting Diagonalize: O(N) degrees of freedom so entropy of primaries of length O(N 2 ) is O(N log N), too small for bulk horizon

20. Weakly coupled CFT Roiban & 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

21. Superconformal Index Kinney, 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

22. Summary There is a 4-parameter family of 1/16 BPS black holes in AdS 5 Why only 4 parameters? How do we calculate their entropy using N=4 SYM?