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Transcript
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
David Turton
2
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.
David Turton
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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.
David Turton
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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
David Turton
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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?
David Turton
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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
David Turton
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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
David Turton
Mathur, Saxena, Srivastava ‘03
27
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:
David Turton
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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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39
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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40
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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45
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.
David Turton
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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
David Turton
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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
David Turton
48
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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49
• 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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51
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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52