AdS/CFT: Implications for Higher Spin Gravity Martin Ammon Friedrich-Schiller Universität Jena Karl-Schwarzschild Meeting 2013 July 25 th, 2013

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

1 / 23

Outline

1

AdS/CFT and higher spin gravity

2

Higher Spin Gravity in 3 dimensions

3

Black hole solutions in 3d higher spin gravity

4

Entanglement entropy

5

Summary

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

2 / 23

Interesting questions ...

How does spacetime look like at short distances? For example in string theory or any other sensible quantum theory of gravity? How are the puzzles regarding black holes solved?

Is black hole creation and black hole evaporation unitary? What about the information loss paradox? Does anything special happen at the horizon (firewalls, black hole complementarity,...)?

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

3 / 23

Interesting questions ...

How does spacetime look like at short distances? For example in string theory or any other sensible quantum theory of gravity? How are the puzzles regarding black holes solved?

Is black hole creation and black hole evaporation unitary? What about the information loss paradox? Does anything special happen at the horizon (firewalls, black hole complementarity,...)? ... I won’t answer in this talk!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

3 / 23

Interesting questions ...

How does spacetime look like at short distances? For example in string theory or any other sensible quantum theory of gravity? How are the puzzles regarding black holes solved?

Is black hole creation and black hole evaporation unitary? What about the information loss paradox? Does anything special happen at the horizon (firewalls, black hole complementarity,...)? ... I won’t answer in this talk! Instead: I show how AdS/CFT may provide (partial) answers to these questions!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

3 / 23

AdS/CFT and higher spin gravity

Outline

1

AdS/CFT and higher spin gravity

2

Higher Spin Gravity in 3 dimensions

3

Black hole solutions in 3d higher spin gravity

4

Entanglement entropy

5

Summary

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

4 / 23

AdS/CFT and higher spin gravity

What is AdS/CFT? In general Quantum gravity theory in asymptotically d + 1 dim. AdS

Martin Ammon (FSU Jena)

⇐⇒

Conformal field theory in d spacetime dimensions

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

5 / 23

AdS/CFT and higher spin gravity

What is AdS/CFT? In general Quantum gravity theory in asymptotically d + 1 dim. AdS

⇐⇒

Conformal field theory in d spacetime dimensions

A specific example for AdS/CFT N = 4 Super Yang-Mills (SYM) theory with gauge group SU(N) and Yang-Mills coupling constant gYM is dynamically equivalent to √ type IIB superstring theory with string length ls = α′ and coupling constant gs on AdS5 × S 5 with radius of curvature L and N units of F(5) flux on S 5 .

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

5 / 23

AdS/CFT and higher spin gravity

What is AdS/CFT?

A specific example for AdS/CFT N = 4 Super Yang-Mills (SYM) theory with gauge group SU(N) and Yang-Mills coupling constant gYM is dynamically equivalent to √ type IIB superstring theory with string length ls = α′ and coupling constant gs on AdS5 × S 5 with radius of curvature L and N units of F(5) flux on S 5 . Mapping of parameters √ gYM and N are mapped to gs and L/ α′ by 2 gYM = 2πgs

Martin Ammon (FSU Jena)

and

√ 2 2gYM N = L/ α′ .

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

5 / 23

AdS/CFT and higher spin gravity

What is AdS/CFT? √ gYM and N are mapped to gs and L/ α′ by 2 gYM = 2πgs

and

√ 2 2gYM N = L/ α′ .

Interesting limits 2 Large N limit: Take N → ∞ but keep λ = NgYM fixed

⇒ gs ∼ gYM → 0, i.e. classical string theory on AdS5 × S 5 strong coupling limit: λ → ∞

L4 /α′ 2 → ∞, i.e. supergravity on AdS5 × S 5

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

6 / 23

AdS/CFT and higher spin gravity

What is AdS/CFT? √ gYM and N are mapped to gs and L/ α′ by 2 gYM = 2πgs

and

√ 2 2gYM N = L/ α′ .

Interesting limits 2 Large N limit: Take N → ∞ but keep λ = NgYM fixed

⇒ gs ∼ gYM → 0, i.e. classical string theory on AdS5 × S 5 strong coupling limit: λ → ∞

L4 /α′ 2 → ∞, i.e. supergravity on AdS5 × S 5 What is AdS/CFT good for? For strongly coupled field theories

[see talks by Gubser, Karch, ...]

Discovering integrability aspects of N = 4 SYM and type IIB string theory

[see talk by Forini]

Enlightening Quantum Gravity aspects [in this talk] Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

6 / 23

AdS/CFT and higher spin gravity

Why higher spin gravity? We want to study the dual gravity theory (string theory) at small distances, L/ls ≪ 1 Massive string states get massless in this limit: m2 ∼

L2 ≪1 ls2

String theory (is expected to) reduce to higher spin gravity new gauge symmetry present involving higher spin fields! higher spin gravity formulation as limit of string theory on AdS5 × S 5 not known! Therefore in this talk we restrict to higher spin gravity in 3 dimensions involving only spin-2 and spin-3 fields dual CFT has enhanced symmetry: Virasoro → W3 symmetry!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

7 / 23

Higher Spin Gravity in 3 dimensions

Outline

1

AdS/CFT and higher spin gravity

2

Higher Spin Gravity in 3 dimensions

3

Black hole solutions in 3d higher spin gravity

4

Entanglement entropy

5

Summary

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

8 / 23

Higher Spin Gravity in 3 dimensions

Review: 3D Gravity as Chern-Simons theory Action S=

1 16πG

or equivalently

Z

d 3x

M

p

−g(R +

2 )− l2

Z

∂M

ω a ∧ ea

A = ω +e, A = ω −e S = SCS [A] − SCS [A]   Z k 2 SCS [A] = Tr A ∧ dA + A ∧ A ∧ A 4π 3 gauge fields A, A ∈ sl(2,

R)

k is the Chern-Simons level, k =

l . 4G

Equations of motion F = dA + A ∧ A = 0 ,

F = dA + A ∧ A = 0

Metric can be computed by gµν = Martin Ammon (FSU Jena)

1 Tr(eµ eν ) , 2

e = eµ dx µ .

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

9 / 23

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory 3D Gravity coupled to spin-3 field given by A = ω +e, A = ω −e S = SCS [A] − SCS [A]   Z k 2 SCS [A] = Tr A ∧ dA + A ∧ A ∧ A 4π 3 gauge fields A, A ∈ sl(3,

R)

k is the Chern-Simons level, k =

l . 4G

Equations of motion F = dA + A ∧ A = 0 ,

F = dA + A ∧ A = 0

Metric and Spin-3 field can be computed by gµν =

1 Tr(eµ eν ) , 2

Martin Ammon (FSU Jena)

φµνρ =

1 Tr(e(µ eν eρ) ) 6

AdS/CFT: Implications for Higher Spin Gravity

e = eµ dx µ .

July 25, 2013

10 / 23

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory II Gauge connection for AdS in Poincare patch A = A+ dx + + A− dx − + L0 dρ, ρ

A+ = e L 1 ,

Martin Ammon (FSU Jena)

ρ

A = A+ dx + + A− dx − − L0 dρ

A− = −e L−1 ,

A− = A+ = 0

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

11 / 23

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory II Gauge connection for AdS in Poincare patch A = A+ dx + + A− dx − + L0 dρ, ρ

ρ

A = A+ dx + + A− dx − − L0 dρ

A− = −e L−1 ,

A+ = e L 1 ,

A− = A+ = 0

Gauge Transformation A A

→ →

g −1 A g + g −1 dg ˜Ag ˜ −1 − d g ˜g ˜ −1 g

˜ are functions of spacetime coordinates and are valued in SL(3, where g and g

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

R).

July 25, 2013

11 / 23

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory II Gauge connection for AdS in Poincare patch A = A+ dx + + A− dx − + L0 dρ, ρ

ρ

A = A+ dx + + A− dx − − L0 dρ

A− = −e L−1 ,

A+ = e L 1 ,

A− = A+ = 0

Gauge Transformation A A

→ →

g −1 A g + g −1 dg ˜Ag ˜ −1 − d g ˜g ˜ −1 g

˜ are functions of spacetime coordinates and are valued in SL(3, where g and g

R).

Remarks ˜ ∈ SL(2, Some of the gauge transformations (namely g, g correspond to diffeomorphisms.

R) ⊂ SL(3, R))

Higher spin gauge transformations may change the causal structure of the spacetime. What is the notion of geometry in higher spin gravity? Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

11 / 23

Black hole solutions in 3d higher spin gravity

Outline

1

AdS/CFT and higher spin gravity

2

Higher Spin Gravity in 3 dimensions

3

Black hole solutions in 3d higher spin gravity

4

Entanglement entropy

5

Summary

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

12 / 23

Black hole solutions in 3d higher spin gravity

Black holes in 3D Higher Spin Gravity I Can we find black holes in 3D Higher spin gravity? Yes, ... BTZ black hole is also a solution of 3D higher spin gravity.

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

13 / 23

Black hole solutions in 3d higher spin gravity

Black holes in 3D Higher Spin Gravity I Can we find black holes in 3D Higher spin gravity? Yes, ... BTZ black hole is also a solution of 3D higher spin gravity. There exist also black holes with higher spin charge [Gutperle, Kraus, ’11, MA, Gutperle, Kraus, Perlmutter, ’11]

SCFT → SCFT + µW

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

13 / 23

Black hole solutions in 3d higher spin gravity

Black holes in 3D Higher Spin Gravity I Can we find black holes in 3D Higher spin gravity? Yes, ... BTZ black hole is also a solution of 3D higher spin gravity. There exist also black holes with higher spin charge [Gutperle, Kraus, ’11, MA, Gutperle, Kraus, Perlmutter, ’11]

SCFT → SCFT + µW

The gauge connection is known explicitly.

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

13 / 23

Black hole solutions in 3d higher spin gravity

Black holes in 3D Higher Spin Gravity I Can we find black holes in 3D Higher spin gravity? Yes, ... BTZ black hole is also a solution of 3D higher spin gravity. There exist also black holes with higher spin charge [Gutperle, Kraus, ’11, MA, Gutperle, Kraus, Perlmutter, ’11]

SCFT → SCFT + µW

The gauge connection is known explicitly.

The causal structure is not invariant under higher spin transformations.[ MA, Gutperle, Kraus, Perlmutter, ’11]

For example, a higher spin black hole in one gauge can look like a traversable wormhole in another gauge, even though they describe the same physics. Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

13 / 23

Black hole solutions in 3d higher spin gravity

Black holes in 3D Higher Spin Gravity II

Thermodynamics of charged higher spin black holes are only consistent if Holonomy condition is satisfied. The Holonomy condition The holonomies associated with the Euclidean time circle ω = 2π(τ A+ − τ A− )

ω = 2π(τ A+ − τ A− )

have eigenvalues (0, 2πi, −2πi) as in the case of the BTZ black hole. Gauge invariant characterization of higher spin black holes!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

14 / 23

Entanglement entropy

Outline

1

AdS/CFT and higher spin gravity

2

Higher Spin Gravity in 3 dimensions

3

Black hole solutions in 3d higher spin gravity

4

Entanglement entropy

5

Summary

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

15 / 23

Entanglement entropy

Review: Entanglement entropy in CFT & AdS/CFT Entanglement entropy in CFT Consider quantum system described by a density matrix ̺, and divide it into two subsystems A and B = Ac . Reduced density matrix ̺A of subsystem A: ̺A = TrAc ̺ Entanglement entropy SEE = von Neumann entropy associated with ̺A : SEE = −TrA ̺A log ̺A .

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

16 / 23

Entanglement entropy

Review: Entanglement entropy in CFT & AdS/CFT Entanglement entropy in CFT Consider quantum system described by a density matrix ̺, and divide it into two subsystems A and B = Ac . Reduced density matrix ̺A of subsystem A: ̺A = TrAc ̺ Entanglement entropy SEE = von Neumann entropy associated with ̺A : SEE = −TrA ̺A log ̺A . Gravity dual of entanglement entropy (supergravity limit)

Construct minimal spacelike surface m(A) which is anchored at the boundary ∂A of the region A and extends into the bulk spacetime. SEE =

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

m(A) . 4GN July 25, 2013

16 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity I

Geodesics will not work: What is spacetime geometry in higher spin gravity? Can we find a bulk object that correctly calculates the entanglement entropy?

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

17 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity I

Geodesics will not work: What is spacetime geometry in higher spin gravity? Can we find a bulk object that correctly calculates the entanglement entropy? Proposal for Entanglement Entropy in Higher Spin Gravity

[MA, Castro, Iqbal, ’13; see also de Boer,

Jottar,’13 for a similar proposal]

Entanglement Entropy may be calculated from a Wilson line in infinite dim. rep. Z Z WR (C) = trR (P exp A) = DU exp(−S(U, P; A)C ) C

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

17 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity I Geodesics will not work: What is spacetime geometry in higher spin gravity? Can we find a bulk object that correctly calculates the entanglement entropy? Proposal for Entanglement Entropy in Higher Spin Gravity

[MA, Castro, Iqbal, ’13; see also de Boer,

Jottar,’13 for a similar proposal]

Entanglement Entropy may be calculated from a Wilson line in infinite dim. rep. Z Z WR (C) = trR (P exp A) = DU exp(−S(U, P; A)C ) C

R contains information about quantum numbers of probe

R): field capturing the dynamics of the probe P(s) ∈ sl(3, R): momentum conjugate to U(x) U(s) ∈ SL(3,

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

17 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity I Geodesics will not work: What is spacetime geometry in higher spin gravity? Can we find a bulk object that correctly calculates the entanglement entropy? Proposal for Entanglement Entropy in Higher Spin Gravity

[MA, Castro, Iqbal, ’13; see also de Boer,

Jottar,’13 for a similar proposal]

Entanglement Entropy may be calculated from a Wilson line in infinite dim. rep. Z Z WR (C) = trR (P exp A) = DU exp(−S(U, P; A)C ) C

R contains information about quantum numbers of probe

R): field capturing the dynamics of the probe P(s) ∈ sl(3, R): momentum conjugate to U(x) Z U(s) ∈ SL(3,

S(U, P; A)C =

where Ds U =

d U ds

  ds Tr (PU −1 Ds U) + λ2 (Tr (P 2 ) − c2 ) + λ3 (Tr (P 3 ) − c3 )

+ As U − UAs ,

Martin Ammon (FSU Jena)

µ

As ≡ Aµ dxds ,

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

17 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity I Geodesics will not work: What is spacetime geometry in higher spin gravity? Can we find a bulk object that correctly calculates the entanglement entropy? Proposal for Entanglement Entropy in Higher Spin Gravity

[MA, Castro, Iqbal, ’13; see also de Boer,

Jottar,’13 for a similar proposal]

Entanglement Entropy may be calculated from a Wilson line in infinite dim. rep. Z Z WR (C) = trR (P exp A) = DU exp(−S(U, P; A)C ) C

R contains information about quantum numbers of probe

R): field capturing the dynamics of the probe R):Zmomentum conjugate to U(x)

U(s) ∈ SL(3, P(s) ∈ sl(3,

S(U, P; A)C = where Ds U =

d U ds

  ds Tr (PU −1 Ds U) + λ2 (Tr (P 2 ) − c2 ) + λ3 (Tr (P 3 ) − c3 )

+ As U − UAs ,

µ

As ≡ Aµ dxds ,

Entanglement entropy: SEE = − log(WR (C)) Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

17 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity II Two possible choices for C:

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

18 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity II Two possible choices for C:

Wilson Line does not depend on path. Geodesic equation is irrelevant to reproduce proper distance.

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

18 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity III Why do we think our proposal is correct? Perfect agreement with CFT results (where available)

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

19 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity III Why do we think our proposal is correct? Perfect agreement with CFT results (where available) Wilson line induces conical defect if backreaction is included

p Martin Ammon (FSU Jena)

2c2 →

c 6

AdS/CFT: Implications for Higher Spin Gravity

[see also Lewkowycz, Maldacena,’13]

July 25, 2013

19 / 23

Entanglement entropy

Entanglement entropy in higher spin gravity IV How to get the geodesics for 3D spin-2 gravity? Just set U(s) = 1 (not possible for higher spin gravity) Geodesic equation d ds



(A − A)µ

dx µ ds



+ [Aµ , Aν ]

dx µ dx ν =0 ds ds

Proper distance appears in on-shell action s   Z √ dx µ dx ν SC = c2 ds Tr (A − A)µ (A − A)ν ds ds C r Z ν p µ dx dx = 2c2 ds gµν (x) ds ds C and thus

SEE = e −SC

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

20 / 23

Summary

Outline

1

AdS/CFT and higher spin gravity

2

Higher Spin Gravity in 3 dimensions

3

Black hole solutions in 3d higher spin gravity

4

Entanglement entropy

5

Summary

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

21 / 23

Summary

Summary I In this talk We focused on gravity + spin-3 field in AdS3 Black hole solution in higher spin gravity with non-trivial higher spin charge

Causal structure/curvature singularities not invariant under higher spin gauge transformations Entanglement Entropy dual to a Wilson line in an infinite dimensional rep. Generalizations & Outlook Generalization to Vasiliev theory (gravity + infinite tower of higher spin fields + matter) dual to minimal two-dimensional CFTs (which can be solved)

[Gaberdiel, Gopakumar]

Higher spin black hole solutions known Have to construct Wilson Line Next: Study Black hole creation & evaporation in this theory!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

22 / 23

Summary

Summary II Possible Caveats Is higher spin gravity in 3D non-trivial enough to create black holes from scalar fields? Can we “higgs” higher spin gravity to say something useful for ordinary gravity in AdS? Can we learn something for four-dimensional asymptotically flat black holes from AdS black holes?

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

23 / 23

Summary

Summary II Possible Caveats Is higher spin gravity in 3D non-trivial enough to create black holes from scalar fields? Can we “higgs” higher spin gravity to say something useful for ordinary gravity in AdS? Can we learn something for four-dimensional asymptotically flat black holes from AdS black holes? For more details Wilson Lines & Entanglement Entropy in higher spin gravity MA, Castro, Iqbal, arXiv: 1306.4338 Review on Higher Spin black holes MA, Gutperle, Kraus, Perlmutter, arXiv: 1208.5182 or just ask me!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

23 / 23

Summary

Summary II Possible Caveats Is higher spin gravity in 3D non-trivial enough to create black holes from scalar fields? Can we “higgs” higher spin gravity to say something useful for ordinary gravity in AdS? Can we learn something for four-dimensional asymptotically flat black holes from AdS black holes? For more details Wilson Lines & Entanglement Entropy in higher spin gravity MA, Castro, Iqbal, arXiv: 1306.4338 Review on Higher Spin black holes MA, Gutperle, Kraus, Perlmutter, arXiv: 1208.5182 or just ask me!

Martin Ammon (FSU Jena)

AdS/CFT: Implications for Higher Spin Gravity

July 25, 2013

23 / 23