Transcript Anti de Sitter Black Holes

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
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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
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5D supergravity + vectors
Introduce coordinates adapted to horizon
Impose supersymmetry
Near-horizon geometry fully determined
Near-horizon
BMPV
AdS3 x S2
Flat
Horizon geometry
Squashed S3
S1 x S2
T3
• 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
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 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
• 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|
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)
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!
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
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(N2)
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 AdS5
• Why only 4 parameters?
• How do we calculate their entropy using
N=4 SYM?