Anti de Sitter Black Holes Harvey Reall University of Nottingham

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

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

SUSY 5D black holes HSR 02, Gutowski 03 5D supergravity + vectors Introduce coordinates adapted to horizon Impose supersymmetry Near-horizon geometry fully determined Near-horizonHorizon geometry BMPVSquashed S 3 AdS 3 x S 2 S 1 x S 2 FlatT3T3

SUSY horizons: S 3, S 1 x S 2, T 3 Only asymptotically flat SUSY BH with S 3 horizon is BMPV SUSY black rings discovered recently Elvang et al 04, Bena & Warner 04, Gauntlett & Gutowski 04 T 3 seems unlikely

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

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 Try to find SUSY black holes systematically by examining near-horizon geometry

In near horizon limit, conditions for SUSY reduce to equations on 3-manifold General solution unknown but particular S 3 solution can be found Use this to guess form of base space of corresponding black hole solution First examples of SUSY AdS 5 black holes Preserve 1/16 SUSY Cohomogeneity 1, 3 parameters SO(6) R-charge (Q 1,Q 2,Q 3 ), J 1 =J 2 =J(Q), BPS relation M=|J 1 |+|J 2 |+|Q 1 |+|Q 2 |+|Q 3 |

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 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)

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 S 3 black holes!

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

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

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

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

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?