Black hole bound states in AdS3

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Black hole bound states in AdS3
Jan de Boer, Amsterdam Jerusalem, April 10, 2008
Based on arXiv0802.2257 - JdB, Frederik
Denef, Sheer El-Showk, Ilies Messamah, Dieter
van den Bleeken to
appear - JdB, Sheer El-Showk, Ilies
Messamah, Dieter van den Bleeken
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Motivation

• What are their holographic duals? What about
  statement that black holes are dual to thermal
  states?
• Can find many smooth geometries applications to
  black hole microstates/fuzzballs.
• Clarify when objects/branes form bound states.
• Give alternative derivation of the wall-crossing
  formula.
• Clarify the nature of scaling solutions.
• Applications to OSV etc.

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Starting point 4d Black holes in type IIA string
theory on a CY

• Have charges
  corresponding to D6-D4-D2-D0 bound states
• Playground for OSV conjecture
• Feature so-called attractor flow near-horizon
  moduli indepent of moduli at infinity
• Entropy is related to number of D-brane bound
  states

Ooguri,Strominger,Vafa,
Kallosh,Ferrara,Strominger
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Actually, BPS states in d4 also carry angular
momentum. Define an index where p denotes the
magnetic charges and
q the electric charges then the
entropy is (believed to be) more precisely given
by
Unresolved puzzle why is the geometric entropy
equal to the index?
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Single centered black holes
Are described by a spherically symmetric
solution moduli follow attractor flowgradient
flow.
Here, are the Kähler
moduli and Z is the central charge.
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The charge is given by
basis for 4-cycles
basis for 2-cycles
Define
and
then
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The central charge at infinity determines the
unbroken N1 subalgebra of the N2 algebra that
is being preserved by the black hole.
Interestingly, there are also many multi-centered
configurations.
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Multi-center solutions in d4
DenefBates,Denef
where
(triple intersection product)
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The 4D solutions are described purely in terms of
harmonic functions (constant determines
asymptotics)
where labels the (D0,D2,D4,D6) charges of
each of the centers.
implies the necessary
integrability condition
These solutions are stationary and carry angular
momentum
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The existence of a given multi-center solution is
in general difficult to establish. Conjecture
they are in one-to-one correspondence with split
attractor flow trees
wall of marginal stability
cartoon of actual solution
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Walls of marginal stability can exist when
central charges align so that
At the same time the index jumps as given by the
wall-crossing formula
which counts the contributions from the
multi-centered configurations
Denef,Moore
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To improve understanding, take a decoupling
limit. Idea D4-branes
M5-branes N(0,4) SCFT
lift to
decoupling
M-theory
limit
Maldacena,Strominger,Witten
This decoupling limit involves sending
while keeping the size of the 11th
dimension, the masses of stretched membranes, and
the size of the CY in 11d Planck units fixed.
Advantage no states coming in from infinity,
well-defined theory of quantum gravity. The
decoupling limit can only be taken of the total
D6-brane charge vanishes.
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Not all solutions survive the decoupling limit
as we send we have to adjust the
locations of the centers. Can only be done if the
original solution exists at infinite Kähler
modulus
Crucially, one of the constants in the harmonic
functions survives the decoupling limit while the
other do not. Without this constant there would
not be any interesting bound states.
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One can show that the resulting space is
asymptotic to AdS3xS2. By studying the asympotics
we find that
These are not the actual values of .
Chern-Simons terms for the gauge fields give
additional subtle contributions but in e.g.
Cardys formula we should use the above values.
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D4-D2-D0
Example single center Corresponds to the usual
BTZ black hole
Example two centers
fixed
D6-D4-D2-D0
D6-D4-D2-D0
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Application 1
When the two centers correspond to pure fluxed
D6-branes, i.e. they correspond to D6-branes with
a non-trivial gauge field configuration there is
a coordinate change which maps the solution into
global AdS3. This coordinate transformation
correspond to spectral flow in the CFT.
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Example three centers
D4-D2-D0
D4-D2-D0
spectral
in global AdS3
flow
D6-D4-D2-D0
D6-D4-D2-D0
The three centers form a bound state, and so does
the single center in global AdS. This naturally
explains why there are no giant graviton states
in Poincare coordinates and why there are in
global coordinates.
Mandal,Raju,Smedbäck Raju Lunin, Mathur
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Application 2 The entropy enigma
Send . Recall that
Therefore
enigma!
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• The entropy enigma arises in a regime where Cardy
  is not reliable.
• In this regime, the single centered black hole is
  thermodynamically unstable against decaying in
  the two-centered one.
• The two-centered on that dominates the entropy is
  the one where one of the two-centers has zero
  entropy.
• The resulting configuration can be interpreted as
  a single black hole localized on the two-sphere
• Therefore, this is a supersymmetric version of
  the Gregory-Laflamme instability.
• OSV is weakly coupled exactly in this regime and
  yields the entropy of the single-centered
  solution.

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Smooth solutions

• There are many smooth solutions this happens
  when all the centers have zero entropy.
• One way to phrase the fuzzball proposal is that
  the space of smooth supergravity solutions with
  given charges at infinity is a phase space.
  Quantizing the phase space yields a Hilbert space
  which should have enough states to account for
  the entropy the black hole with the same charges.
  
• The black hole is the result of coarse graining
  over all those microstates.
• It is therefore interesting to quantize the phase
  space of multicentered black hole geometries.

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Mandal
Grant,
Maoz, Marsano, Papadodimas, Rychkov
Takayama, Tsuchiya Donos, Jevicki Rychkov
Moduli space quantization
Idea restrict the symplectic form of
supergravity to a space of BPS solutions. For
families of static solutions the result will
vanish, but for stationary solutions it can give
rise to a non-degerate symplectic form ? on the
BPS moduli space.
Given such a symplectic form one quantizes the
moduli space by finding a line bundle such
that . The Hilbert space of BPS
states is then given by
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Here it is very difficult to evaluate the
symplectic form directly in supergravity.
Instead we will use the D-brane probe
approximation and invoke a non-renormalization
theorem. This yields the following result for the
symplectic form
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Two-center case
Here, the distance is fixed and .
Result
This is in complete agreement with the
wall-crossing formula.
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Three-center case
The moduli space is much more complicated. It can
be parameterized by the angles on S3 and the size
of the angular momentum. The symplectic form
becomes
This is invariant under two U(1)s and the moduli
space is in fact a toric Kähler manifold can use
known technology to write explicitly wave
functions etc. (generalized Landau levels) Count
number of states use toric polytope
Guillemin
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?
Toric polytope
j
agrees with wall-crossing
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Some further issues

• If either a bound state does not survive the
  decoupling limit, or if the symplectic form is
  degenerate, there are no corresponding BPS states
  in the dual CFT.
• Sometime centers can move off to infinity. Using
  form of wavefunctions we can check that this does
  not spoil asymptotic AdS boundary conditions.
• Sometime centers can approach each other
  arbitrarily closely (scaling solutions, abysses).
  An infinite deep throat develops. The throat is
  capped off by the quantized wavefunctions.

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To which black holes does a given smooth geometry
belong? Conjecture partition the centers of the
smooth solution into groups that each can
approach each other arbitrarily closely (i.e. are
of scaling type). Then
contributes to
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The single black hole for large charges is dual
to a thermal density matrix. Recall that the
supergravity limit is very much like a classical
limit in quantum mechanics. Not all states have a
good classical limit, in fact most dont. We
have seen that for large charges, the thermal
density matrix is a classical state and dual to a
large black hole. For small L0 this is no longer
true the thermal density matrix is dual to a sum
over geometries, one for each orientation of the
two-centered black hole solution.
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Conclusions/open problems

• Extend quantization to more than three centers.
• Understand the decomposition of the dual CFT
  Hilbert space dual to the various black hole
  configurations. Renormalization group flow may
  shed light on this?
• Count the number of states obtained from smooth
  geometries. Are there enough? Are some of them
  typical?
• Understand Coulomb/Higgs mismatch for the number
  of states obtained from scaling solutions.
• Many things to do!