Anti de Sitter Black Holes Harvey Reall University

Anti de Sitter Black Holes Harvey Reall University of Nottingham

Motivation • Black hole entropy calculations all rely on 2 d CFT • Can we use Ad. S/CFT to calculate entropy of D>3 Ad. S black holes? • D=4: probably not, CFT not understood • D=5: CFT is N=4 SYM… • Need supersymmetric Ad. S 5 black holes to evade strong coupling problem

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

SUSY 5 D black holes HSR 02, Gutowski 04 • • 5 D ungauged N=1 sugra + abelian vectors Introduce coordinates adapted to horizon Take near-horizon limit Impose supersymmetry: eqs on spatial crosssection of horizon • Can determine general solution for compact horizon

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

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

5 D SUSY Ad. S 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 4 d Kähler “base space” Gauntlett & Gutowski 03, Gutowski & HSR 04 • Choice of base space not obvious e. g. get Ad. S 5 from Bergman manifold SU(2, 1)/U(2)

5 D SUSY Ad. S 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)

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

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

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)

CFT description • BPS Ad. S 5 black hole microstates are 1/16 BPS states of N=4 large N SYM on Rx. S 3 (equivalently BPS local operators on R 4) • States classified by SO(4)x. SO(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

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 Ad. S black rings?

BPS Ad. S black rings? Kunduri, Lucietti & HSR 06 • Most general BPS near-horizon geometry in 5 D gauged sugra not known • Assume existence of 2 rotational symmetries (true for all known 5 d 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 Ad. S 3 x. S 2 with horizon topology S 1 x. S 2…

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

New 10 d black holes? • Solution is locally isometric to Ad. S 3 x. M 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 x. S 5) • Resulting solution cannot be reduced to 5 d • Could this be near-horizon geometry of an asymptotically Ad. S 5 x. S 5 black hole?

Resolutions of the puzzle? • Are there BPS 10 d black hole solutions that can’t be reduced to 5 d? • Are there BPS 5 d 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

CFT entropy calculation? • Need to count 1/16 BPS states of N=4 SU(N) SYM on Rx. S 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 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(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 Ad. S 5 • Why only 4 parameters? • How do we calculate their entropy using N=4 SYM?