Anti de Sitter Black Holes

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Anti de Sitter Black Holes

• Harvey Reall
• University of Nottingham

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Motivation

• Black hole entropy calculations all rely on 2d
  CFT
• Can we use AdS/CFT to calculate entropy of Dgt3
  AdS black holes?
• D4 probably not, CFT not understood
• D5 CFT is N4 SYM
• Need supersymmetric AdS5 black holes to evade
  strong coupling problem

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Plan

• SUSY asymptotically flat black holes
• SUSY AdS black holes in D3,4
• SUSY AdS black holes in D5
• CFT interpretation
• Collaborators J. Gutowski, R. Roiban, H.
  Kunduri, J. Lucietti

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SUSY 5D black holesHSR 02, Gutowski 04

• 5D ungauged N1 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

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SUSY 5D black holesHSR 02, Gutowski 04

• All possible near-horizon geometries
• Which arise from asymp flat black holes?
• Near-horizon BMPV from BMPV!
• AdS3xS2 from BPS black rings Elvang et al 04
• Flat T3 horizon unlikely Galloway 06

Near-horizon Horizon geometry
BMPV Squashed S3
AdS3 x S2 S1 x S2
Flat T3
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SUSY AdS Black Holes

• BPS limit of Reissner-Nordstrom-AdS is nakedly
  singular
• D3 BTZ is SUSY black hole iff MJgt0
• D4 Kerr-Newman-AdS (M,J,Q,P) saturates BPS
  bound if MM(Q), JJ(Q), P0 Kostalecky Perry
  95, Caldarelli Klemm 98
• SUSY AdS black holes must rotate

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5D SUSY AdS black holesGutowski HSR 04

• Reduce IIB SUGRA on S5 to N1 D5 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 SU(2,1)/U(2)

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5D SUSY AdS black holesGutowski 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 S3 solution can be found
  (cf near-horizon BMPV)

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5D SUSY AdS black holesGutowski HSR 04

• Near-horizon solution motivates cohomogeneity-1
  Ansatz for full solution
• First examples of SUSY AdS5 black holes!
• Base space singular, cohomogeneity-1,
  asymptotically Bergman space
• 1/16 BPS

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Unequal Angular MomentaChong, Cvetic, Lü Pope
05

• Guessed non-BPS charged rotating black hole
  solution of minimal gauged sugra
  (Einstein-Maxwell)
• Cohomogeneity-2, 4 parameters (M,J1,J2,Q)
• BPS limit 2 parameter solution with J1?J2

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General solutionKunduri, 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 J1, J2, Q1, Q2, Q3
  with one constraint
• Expect non-BPS generalization with independent
  M,J,Q (2 more parameters)

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CFT description

• BPS AdS5 black hole microstates are 1/16 BPS
  states of N4 large N SYM on RxS3 (equivalently
  BPS local operators on R4)
• States classified by SO(4)xSO(6) quantum numbers
  J,Q
• Black hole quantum numbers O(N2)
• Black hole entropy O(N2)
• Entropy calculation count all 1/16 BPS states
  with same quantum numbers as black hole

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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?

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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 S3
  black holes
• Another solution is a warped product AdS3xS2 with
  horizon topology S1xS2

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BPS AdS black rings?Kunduri, Lucietti HSR 06

• but with a conical singularity on S2
• Cant eliminate singularity (without turning off
  cosmological constant)
• BPS AdS black rings with 2 rotational symmetries
  do not exist
• Oxidize to 10d warped product AdS3xM7 with
  M7S2xS5 (singular)

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New 10d black holes?

• Solution is locally isometric to AdS3xM7 solution
  of Gauntlett et al 06
• They showed that solution can be made globally
  regular by choosing topology of M7 appropriately
  (not S2xS5)
• Resulting solution cannot be reduced to 5d
• Could this be near-horizon geometry of an
  asymptotically AdS5xS5 black hole?

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Resolutions of the puzzle?

• Are there BPS 10d black hole solutions that cant
  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(N2) entropy Berkooz et al 06

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CFT entropy calculation?

• Need to count 1/16 BPS states of N4 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

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No 1/8 BPS black holesRoiban HSR 04,
Berenstein 05

• 1/8 BPS primaries built from N1 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

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Weakly coupled CFTRoiban HSR 04, Kinney,
Maldacena, Minwalla Raju 05

• Goal at weak coupling, count operators in short
  1/16 BPS multiplets that cant become long at
  strong coupling
• Too hard! Count everything instead
• Find correct scaling of entropy with charge for
  large charge

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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 doesnt see black
  holes
• Cancellation between bosonic and fermionic BPS
  states dual to black hole

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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 N4 SYM?