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Microstates at the boundary of AdS
David Turton
Ohio State
SPOCK, Purdue, Jan 21 2012
Based on 1112.6413 with Samir Mathur
Outline
1. Black holes in string theory
2. D1-D5 system
3. AdS3 and asymptotic symmetries
4. Overview of the construction
5. Technical details
6. Discussion
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What is a Black Hole?
Physical: dark, heavy, compact bound state of matter
Quantum: bound state in
quantum gravity theory
Classical: geometry with horizon
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Why Black Holes?
• Strong evidence for existence of astrophysical Black holes
• Want a unified theory of laws of Nature
• Requires a theory of quantum gravity (QG)
• String theory: leading candidate
• Major test of any QG theory:
Information paradox.
Hawking ‘75
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The Information Paradox
BH Horizon:
normal lab physics
(small curvature)
Hawking radiation:
pair creation
 entangled pair
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The Information Paradox
• Each Hawking pair increases entanglement entropy by ln 2
When does process terminate? If no new physics
enters problem until BH becomes Planck-sized,
1.
BH evaporates completely ) unitarity
2.
Planck-sized remnant ) exotic objects
(arbitrarily high entanglement with surroundings)
Allowing arbitrary small corrections to this process, conclusions robust
Mathur ‘09
If a traditional horizon forms and persists,
the theory either violates unitarity or has exotic remnants.
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Black hole microstates
• A black hole has Bekenstein-Hawking entropy S proportional to its
horizon area
• In a quantum gravity theory, expect eS microstates
• What is the physics of individual microstates?
Do they modify the Hawking calculation?
• If not, Hawking’s argument shows the theory is sick.
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Hair for Black holes
• Hawking’s argument motivated many searches for Black hole ‘hair’:
deformations at the horizon.
• In classical gravity, many ‘no-hair’ theorems resulted
Israel ‘67, Carter ’71, Price ‘72, Robinson ’75,…
In String theory, we do find hair. Suggests that
• Quantum effects important at would-be-horizon (fuzz)
• Bound states have non-trivial size (ball)
“Fuzzball”
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Two-charge Black hole
• Make Black hole in String theory: must use fundamental quanta
• Simplest example: Multiwound fundamental string + momentum
• Entropy of small black hole reproduced by microscopic string states
Sen ‘94
• For classical profiles, string sources good supergravity background
• Transverse vibration modes only
 string occupies non-trivial size
Dabholkar, Gauntlett, Harvey, Waldram ‘95
Callan, Maldacena, Peet ‘95
Lunin, Mathur ‘01
• Classical profiles $ coherent states in usual way
• No horizons; source at location of string
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D1-D5 system: two-charge
• D1-D5 system is U-dual to F1-P
Lunin, Mathur ‘01
• Configurations are everywhere smooth in D1-D5 frame
Lunin, Maldacena, Maoz ‘02
• Geometric quantization yields entropy of D1-D5 system
Ryzchkov ‘05
• Generalizations to include fermionic condensates and T 4 profiles
Taylor ’05, Kanitscheider, Skenderis, Taylor ‘07
• Two-charge Black hole is string-scale sized.
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D1-D5-P: three charges
• Add momentum charge  macroscopic black hole with same charges
• Entropy reproduced from microscopic degrees of freedom
Strominger, Vafa ‘96
• Many three-charge states constructed
Giusto, Mathur, Saxena, Srivastava,…
Bena, Bobev, Wang, Warner,…
de Boer, Shigemori,…
• How large a subset of the degrees of freedom are well-described by
smooth horizonless supergravity solutions?
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Nonextremal: Hawking radiation
• Emission rate from D1-D5 system matches Hawking radiation rate
Das, Mathur ‘96
• Class of non-extremal microstate geometries known
Jejjala, Madden, Ross, Titchener (JMaRT) ‘05
• Ergoregion emission – classical instability
Cardoso, Dias, Hovdebo, Myers ‘05
• Matches (Hawking) emission rate from these states
Chowdhury, Mathur ’07, ‘08
• In fuzzball scenario, Hawking radiation is ordinary quantum emission
) unitary
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This talk
• Construct a D1-D5-P state as a perturbation of a D1-D5 background
• Motivated by old work on AdS3 and asymptotic symmetries
Brown, Henneaux ‘84
• Connect to ideas of states localized at boundary of AdS3
 U(1) in AdS5/CFT4 & Singletons
Witten ‘98
• Next: Review D1-D5 system
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D1-D5 system: setup
We work in type IIB string theory on
• Radius of S1 : Ry
• Wrap n1 D1 branes on S1
• Wrap n5 D5 branes on S1 £ T4
The bound state creates a geometry with D1 and D5 charges
Q1 =
1
V
(2¼)4g®03 n1
For simplicity, set Q1 = Q5 = Q.
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To get an AdS throat, we take √Q ¿ Ry and work with small ² =
The throat is then AdS3 £ S3 £ T 4 (Poincare patch)
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D1-D5 geometries: terminology
• The AdS throat ends in a `cap’ at its core
• The region where the AdS joins to flat space is called the `neck’ – this
forms a natural ‘boundary’ for the AdS region.
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AdS3
Global coordinates on AdS3:
Poincare patch: (large r)
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Asymptotic symmetries of AdS3
• Asymptotic symmetry group (ASG): symmetries preserving
asymptotics of the space
• ASG of AdS3 : VirasoroL £ VirasoroR .
Brown, Henneaux ‘84
• VirasoroL generator L-n : diffeomorphism generated by (Ry = 1)
• Can also obtain central charge from boundary terms:
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Entropy from Cardy formula
• Near-horizon AdS3 $ Boundary CFT2
• Can obtain entropy of BTZ Black Hole from the Cardy formula:
Strominger ’98
• Generalized to arbitrary black hole horizons:
Virasoro algebra of horizon deformations
Carlip ’98, ‘99
• We can count the states, but where are the states localized in the
gravity description?
 at the cap, the neck…?
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D1-D5 CFT
• Worldvolume gauge theory on D1-D5 bound state flows in IR to a
(4,4) SCFT.
• Orbifold point in moduli space: Free SCFT on (T 4)N/SN , N= n1n5.
• Ramond ground states can be visualized as follows:
….
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CFT2 chiral algebra
• Orbifold CFT: (4,4) SCFT on (T 4)N/SN
• Symmetry algebra:
(L)
VirasoroL £ VirasoroR
L-n
R symmetries SU(2)L £ SU(2)R
Ja-n
a = 1,2,3
U(1) currents on T 4
J®-n
® = 5,...,8
• L moving algebra:
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A puzzle
• In AdS3 the Virasoro generators are diffeomorphisms;
• In the CFT they raise the energy of a state.
So if we have
And if we write
Then is M2 just a diffeomorphism of M? Or does it have a higher energy?
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The resolution
• We will next construct the perturbation corresponding to
The perturbation is:
• Pure gauge in the throat & cap regions
• Normalizable at infinity
• Non-trivial in the neck region
Also do same for the generators Ja-n , J®-n .
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The perturbation can be thought of as arising as follows:
1.
Consider just the AdS throat & cap (a),(b)
2.
Make a diffeomorphism (c)
3.
Glue back to flat space – creating a physical perturbation (d)
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The background geometry
The background geometry we use is the simplest one:
• Cap geometry is global AdS
• Corresponds to particular Ramond ground state
obtained from spectral flow of NS vacuum
Maldacena, Maoz ’00,
Balasubramanian, de Boer, Keski-Vakkuri, Ross ‘00
This background corresponds to an
NS1-P profile which is a helix.
Lunin, Mathur ’01
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Full background solution:
where
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Matching process
Method for constructing the perturbation:
• Solve equations of motion in ‘outer’ and ‘inner’ regions separately
• Match the solutions in the region of overlap
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Mathur, Saxena, Srivastava ‘03
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Outer Region
• Outer region is ‘naive’ D1-D5 solution to leading order:
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Inner Region
• Take r ¿ √Q and define (spectral flow)
 cap background to leading order is AdS3 £ S3 £ T 4 (global coords):
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Overlap Region (Throat)
Throat background at leading order is AdS3 £ S3 £ T 4 (Poincare patch):
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Field equations
• For the L-n and Ja-n perturbations, work in 6D
• Background gauge field strength is self-dual in 6D, perturbation
retains this
• Perturbation:
• Equations of motion:
• For the J®-n perturbation, work in 7D:
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The perturbation expansion
• Solve the equations perturbatively in the outer and inner regions
& match the solutions in the throat
• Full outer region metric:
• Schematic eqn of motion for perturbation Á in full outer metric:
• Expand ¤ and Á as series in ² :
• At lowest order:
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The perturbation expansion
• Solve the equations perturbatively in the outer and inner regions
& match the solutions in the throat
• Full outer region metric:
• Schematic eqn of motion for perturbation Á in full outer metric:
• Expand ¤ and Á as series in ² :
• At lowest order:
Then do same in
inner region – get
Á0outer & Á0inner , etc
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Requirements of the perturbation
General requirements:
• Regular at r = 0 (Á0inner) and normalizable as r  infinity (Á0outer)
• Á0inner = Á0outer + O(²) in the overlap region
(same story at higher orders Á1 , Á2 ,…)
Specific requirements:
• Perturbation for L-n should reduce to the L-n diffeomorphism in the
throat + cap
• Perturbation should not be pure gauge everywhere!
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L-n perturbation
• Recall L-n as diffeomorphism (Ry restored):
• Make perturbation: work to linear order in ² .
• In the AdS throat this diffeo creates the pure gauge perturbation
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L-n : outer region
• Extend to a solution in the full outer region
 ansatz: multiply by f(r) and solve eqns of motion.
• Solution:
• Reduces to pure gauge in strict throat limit r ¿ √Q
• Non-trivial in the neck region
• Normalizable at infinity
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L-n : inner region
• Outer region solution in throat = pure gauge + remainder
• Pure gauge part trivially matches to inner region  match r2 piece.
• Ansatz in AdS3 £ S3 :
• Equations of motion:
Deger, Kaya, Sezgin, Sundell ‘98
¤ : AdS3
laplacian
• Solution:
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L-n : matching in throat
• Throat: a ¿ r ¿ √Q
• Need to compare inner and outer regions solutions in same gauge
• Take r ¿ √Q in outer region solution
and apply diffeo to
1.
‘undo’ the pure gauge part - L-n diffeo
2.
Go to same gauge as inner region solution
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L-n : matching in throat
• Diffeo on outer region solution:
• Resulting fields (outer):
• Take r À a in inner region solution:
• Match solutions: choose const A
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L-n perturbation: summary
• Found solutions in inner and outer regions and matched in throat to
leading order
• Upon quantization, only n > 0 should be physical excitations
• Energy = momentum =
n
as expected
Ry
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Sphere diffeomorphism Ja
• SU(2)L on the sphere:
• Work just with V(3) – to the order we work, we have spherical
symmetry (broken by spectral flow at higher orders)
• Add same v dependence as before: consider diffeo generated by
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• Two-form is invariant under gauge transformation:
• Combine with diffeo to get solution in nice gauge:
• Pure gauge perturbation generated:
• Extend to full outer region solution: £ f(r) & solve eqns
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Ja perturbation: inner region
• AdS3£S3 cap: make ansatz
Mathur ‘01
• Choose spherical harmonic:
• Find solution
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Ja : matching in throat
• Take r ¿ √Q in outer region solution and make
‘undoing’ diffeo & gauge trans
• Resulting fields:
• Take r À a in inner region solution:
• Match solutions: choose const C
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Torus diffeomorphism J®
• Let z be a direction on torus
• Torus diffeomorphism: generated by
• Add gauge transformation with parameter
• Pure gauge perturbation generated:
• Extend to full outer region solution: £ f(r) & solve eqns
Solution:
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J® perturbation: inner region
• Reduce onto AdS3£S3 cap: ansatz
• Find solution
• Same soln for Z as for sphere perturbation but there had K = -2Z
 basic difference is that S3 has curvature, but torus doesn’t.
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J® : matching in throat
• Take r ¿ √Q in outer region solution and make
‘undoing’ diffeo & gauge trans
• Resulting fields:
• Take r À a in inner region solution:
• Match solutions: choose const D
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Summary of results
• Constructed perturbations generated by chiral algebra generators &
U(1) currents
• Saw that each perturbation was pure gauge in throat limit,
non-trivial in neck
• Corrections from perturbation theory in geometry
matched to leading order
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Comments
• Background geometry used in this talk:
• Saw that global L-n perturbation gave perturbation at neck
• Suggests a way to understand the states which live at the cap
(majority of states)
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• Background with small sub-throat:
• Neck of sub-throat » flat space (on scale of sub-throat)
• L-n perturbation of the sub-throat should be the state
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• Background with two large sub-throats:
• Antisymmetric combination of L-n perturbations of sub-throats:
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Conclusions
• Constructed a three-charge perturbation
• Suggests a way to think of all three-charge states
Open problems:
• Extend perturbation to non-linear order in ²
(in progress)
• Non-linear order in ² (size of perturbation)
• Make perturbation on different backgrounds
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