Entanglement entropy in higher spin gravity Martin Ammon - - PowerPoint PPT Presentation

Entanglement entropy in higher spin gravity

Martin Ammon Friedrich-Schiller Universität Jena based on work in collaboration with Alejandra Castro and Nabil Iqbal MA, Castro, Iqbal, 1306.4338 Gauge Gravity Duality 2013 July 29 th, 2013

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 1 / 21

Outline

1

Why Higher Spin gravity in context of AdS/CFT?

2

Higher Spin Gravity in 3 dimensions

3

Entanglement entropy The concrete proposal Checks for the entanglement proposal

4

Summary

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 2 / 21

Why Higher Spin gravity in context of AdS/CFT?

Why Higher Spin gravity in context of AdS/CFT?

From conceptional point of view: What is the gravity dual of non-interaction field theoies? Of minimal CFTs? For condensed matter applications: Higher Spin Gavity in 4D dual to O(N) models in the large N-limit. How do we compute entanglement entropy in higher spin gravity? For (quantum) gravity applications: Higher Spin Gravity as toy-model to study propeties of black holes in asymptotically AdS. Can we study black hole creation and evaporation explicitly since we have both sides under full control? What is geometry in higher spin gravity? We will see that both questions are connected. In particular I hope to convince you that we may learn something about geometry, causal structure, event horizons, etc. by studying entanglement entropy.

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 3 / 21

Why Higher Spin gravity in context of AdS/CFT?

Why Higher Spin gravity in context of AdS/CFT?

In this talk: I focus on higher spin gravity in three spacetime dimensions. advantage: We do not have to take into account the infinite tower of higher spins since we can truncate to a finite order. Here: I consider only ’minimal’ extensions of Einstein Gravity by adding a spin-3 degree of freedom.

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 4 / 21

Higher Spin Gravity in 3 dimensions

Review: 3D Gravity as Chern-Simons theory

Action S = 1 16πG

• M

d3x

• −g(R + 2

l2 ) −

• ∂M

ωa ∧ ea

• r equivalently

S = SCS[A] − SCS[A] A = ω + e/l , A = ω − e/l SCS[A] = k 4π

• Tr
• A ∧ dA + 2

3A ∧ A ∧ A

• with gauge fields A, A ∈ sl(2,

l 4G , l the radius of

curvature of AdS, which we set to one, l = 1. Equations of motion F = dA + A ∧ A = 0 , F = dA + A ∧ A = 0

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 5 / 21

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory

We only want to add a spin-3 field, so what do we have to modify 3D Gravity coupled to spin-3 field given by S = SCS[A] − SCS[A] A = ω + e/l , A = ω − e/l SCS[A] = k 4π

• Tr
• A ∧ dA + 2

3A ∧ A ∧ A

• with gauge fields A, A ∈ sl(3,

3D Higher spin gravity as Chern-Simons theory with gauge group SL(3,

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

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 6 / 21

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory II

Theory has two different AdS vacua depending of an sl(2,

sl(3,

sl(3,

L−1, L0, L1 generators with commutation relations [Li, Lj] = (i − j)Li+j Wj, (j = −2, −1, ..., 2) satisfying [Lj, Wm] = (2j − m)Wj+m

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 7 / 21

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory II

Theory has two different AdS vacua depending of an sl(2,

sl(3,

sl(3,

L−1, L0, L1 generators with commutation relations [Li, Lj] = (i − j)Li+j Wj, (j = −2, −1, ..., 2) satisfying [Lj, Wm] = (2j − m)Wj+m inequivalent embeddings of sl(2,

principle embedding: take Ja = La as sl(2,

bulk degrees of freedom: metric gµν and Spin-3 field φµνρ given by gµν = 1 2trf(eµeν) , φµνρ = 1 6trf(e(µeνeρ)) e = eµdxµ. Asymptotic symmetry algebra: W3 × W3

[Campeleoni, Fredenhagen, Penninger, Theisen, ’10] Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 7 / 21

Higher Spin Gravity in 3 dimensions

3D Higher Spin Gravity as Chern-Simons theory II

Theory has two different AdS vacua depending of an sl(2,

sl(3,

sl(3,

L−1, L0, L1 generators with commutation relations [Li, Lj] = (i − j)Li+j Wj, (j = −2, −1, ..., 2) satisfying [Lj, Wm] = (2j − m)Wj+m inequivalent embeddings of sl(2,

diagonal embedding: take J0 = L0/2, J±1 = ±W±2/4 as sl(2,

bulk degrees of freedom: spin-2 field, a pair of spin-1 U(1) gauge fields and of spin 3/2 bosonic fields Asymptotic symmetry algebra: W(2)

3

× W(2)

3 Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 7 / 21

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− + J0 dρ, A = A+ dx+ + A− dx− − J0 dρ A+ = eρJ1, A− = −eρJ−1, A− = A+ = 0 where x± = t ± φ and ρ is the radial direction

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 8 / 21

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− + J0 dρ, A = A+ dx+ + A− dx− − J0 dρ A+ = eρJ1, A− = −eρJ−1, A− = A+ = 0 where x± = t ± φ and ρ is the radial direction Gauge Transformation A → g−1 A g + g−1 dg A → ˜ g A ˜ g−1 − d ˜ g ˜ g−1 where g and ˜ g are functions of spacetime coordinates and are valued in SL(3,

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 8 / 21

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− + J0 dρ, A = A+ dx+ + A− dx− − J0 dρ A+ = eρJ1, A− = −eρJ−1, A− = A+ = 0 where x± = t ± φ and ρ is the radial direction Gauge Transformation A → g−1 A g + g−1 dg A → ˜ g A ˜ g−1 − d ˜ g ˜ g−1 where g and ˜ g are functions of spacetime coordinates and are valued in SL(3,

Remarks Some of the gauge transformations (namely g, ˜ g ∈ SL(2,

correspond to diffeomorphisms. 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) Entanglement entropy in higher spin gravity July 29, 2013 8 / 21

Higher Spin Gravity in 3 dimensions

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) Entanglement entropy in higher spin gravity July 29, 2013 9 / 21

Higher Spin Gravity in 3 dimensions

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) Entanglement entropy in higher spin gravity July 29, 2013 9 / 21

Higher Spin Gravity in 3 dimensions

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) Entanglement entropy in higher spin gravity July 29, 2013 9 / 21

Higher Spin Gravity in 3 dimensions

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) Entanglement entropy in higher spin gravity July 29, 2013 9 / 21

Higher Spin Gravity in 3 dimensions

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(τA+ − τA−) ω = 2π(τA+ − τA−) have eigenvalues (0, 2πi, −2πi) as in the case of the BTZ black hole. Gauge invariant characterization of higher spin black holes! There is another interesting black hole in diagonal embedding

[Castro, Hijano, Lepage-Jutier, Maloney, ’11] Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 10 / 21

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 = −TrA̺A log ̺A .

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 11 / 21

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 = −TrA̺A 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 = m(A) 4GN .

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 11 / 21

Entanglement entropy The concrete proposal

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) Entanglement entropy in higher spin gravity July 29, 2013 12 / 21

Entanglement entropy The concrete proposal

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. WR(C) = trR(P exp

• C

A) =

• DU exp(−S(U, P; A)C)

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 12 / 21

Entanglement entropy The concrete proposal

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. WR(C) = trR(P exp

• C

A) =

• DU exp(−S(U, P; A)C)

R contains information about quantum numbers of probe U(s) ∈ SL(3,

P(s) ∈ sl(3,

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 12 / 21

Entanglement entropy The concrete proposal

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. WR(C) = trR(P exp

• C

A) =

• DU exp(−S(U, P; A)C)

R contains information about quantum numbers of probe U(s) ∈ SL(3,

P(s) ∈ sl(3,

S(U, P; A)C =

• ds
• Tr (PU−1DsU) + λ2(Tr (P2) − c2) + λ3(Tr (P3) − c3)
• where DsU =

d ds U + AsU − UAs ,

As ≡ Aµ dxµ

ds , Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 12 / 21

Entanglement entropy The concrete proposal

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. WR(C) = trR(P exp

• C

A) =

• DU exp(−S(U, P; A)C)

R contains information about quantum numbers of probe U(s) ∈ SL(3,

P(s) ∈ sl(3,

S(U, P; A)C =

• ds
• Tr (PU−1DsU) + λ2(Tr (P2) − c2) + λ3(Tr (P3) − c3)
• where DsU =

d ds U + AsU − UAs ,

As ≡ Aµ dxµ

ds ,

Entanglement entropy: Take c3 = 0 and √ 2c2 → c

6 and compute SEE = − log(WR(C)) Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 12 / 21

Entanglement entropy The concrete proposal

Entanglement entropy in higher spin gravity II

Infinite dimensional highest-weight state |h, w with definite eigenvalues under the elements of the SL(3,

L0|h, w = h|h, w , W0|h, w = w|h, w , and which is annihilated by the positive modes of the algebra: L1|h, w = 0 , W1,2|h, w = 0 . We may now generate other excited states by acting with L−1, W−1,−2 on this ground state, filling out an infinite dimensional unitary and irreducible representation.

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 13 / 21

Entanglement entropy The concrete proposal

Entanglement entropy in higher spin gravity II

Infinite dimensional highest-weight state |h, w with definite eigenvalues under the elements of the SL(3,

L0|h, w = h|h, w , W0|h, w = w|h, w , and which is annihilated by the positive modes of the algebra: L1|h, w = 0 , W1,2|h, w = 0 . We may now generate other excited states by acting with L−1, W−1,−2 on this ground state, filling out an infinite dimensional unitary and irreducible representation. Relationship between Casimirs c2, c3 and h, w C2 = 1

2L2 0 + 3 8W 2 0 + · · · ,

C3 = 3

8W0

• L2

0 − 1 4W 2

• + · · · .

Acting with C2 and C3 on the highest weight state |h, w we find c2 = 1

2h2 + 3 8w2 ,

c3 = 3

8w

• h2 − 1

4w2

. We consider heighest weight representation with w = 0, h = c/6 implying c3 = 0.

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 13 / 21

Entanglement entropy The concrete proposal

Entanglement entropy in higher spin gravity III

Two possible choices for C: Open Wilson Line determines the entanglement entropy of interval X Wilson Loop computes the thermal entropy of a black hole

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 14 / 21

Entanglement entropy The concrete proposal

Entanglement entropy in higher spin gravity III

Two possible choices for C: Open Wilson Line determines the entanglement entropy of interval X Wilson Loop computes the thermal entropy of a black hole Wilson Line does not depend on path. Geodesic equation is irrelevant to reproduce proper distance.

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 14 / 21

Entanglement entropy Checks for the entanglement proposal

Overview: Checks for our proposal

Why do we think our proposal is correct? In the case of CS theory with SL(2,

Entanglement entropy determined by geodesics. Perfect agreement with CFT results (where available) Wilson line induces conical defect if backreaction is included

[see also Lewkowycz, Maldacena,’13]

Result of this argument: √2c2 → c

6 Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 15 / 21

Entanglement entropy Checks for the entanglement proposal

Checks for our proposal I

How to get the geodesics for 3D spin-2 gravity descibed by CS theory with SL(2,

S(U, P; A)C =

• ds
• Tr
• PU−1DsU
• + λ(s)
• Tr(P2) − c2
• ,

The equations of motion reduce to U−1DsU + 2λP = 0 , d ds P + [¯ As, P] = 0 . Tr(P2) = c2 Plugging the eom back into the action we obtain S(U; A)C = √c2

• C

ds

• Tr (U−1DsU)2

The eom with respect to U(s) are d ds

• (Au − ¯

A)µ dxµ ds

• + [¯

Aµ, Au

ν]dxµ

ds dxν ds = 0, Au

s ≡ U−1 d

ds U + U−1AsU provided s is the proper distance (reparameterization invariance!)

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 16 / 21

Entanglement entropy Checks for the entanglement proposal

Checks for our proposal II

For spin-two gravity, U(s) =

(but not for higher spin gravity in generic gauge field backgrounds) Geodesic equation d ds

• (A − A)µ dxµ

ds

• + [Aµ, Aν]dxµ

ds dxν ds = 0 Proper distance appears in on-shell action SC = √c2

• C

ds

• Tr
• (A − A)µ(A − A)ν dxµ

ds dxν ds

• =
• 2c2
• C

ds

• gµν(x)dxµ

ds dxν ds ≡

• 2c2LC

and thus WR(C) = e−√

2c2LC

• r, for the entanglement entropy we get

SEE = − log WR(C) =

• 2c2LC =

1 4G LC using √2c2 = c

6 = k = 1 4G Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 17 / 21

Entanglement entropy Checks for the entanglement proposal

Checks for our proposal III

Result for the open interval at zero temperature (i.e. in AdS) SEE = c 3 log ∆φ ǫ

• ,

where ǫ ≡ e−ρ0 and ∆φ = φ(s = sf) − φ(s = 0) Results for the open interval at finite temperature (i.e. in BTZ) SEE = c 3 log β πǫ sinh π∆φ β

• .

Results for the thermal entropy (BTZ black hole) Sth = − log WR(C) = 2π √ 2πkL + 2π

• 2πk ¯

L .

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 18 / 21

Entanglement entropy Checks for the entanglement proposal

Checks for our proposal IV

Consider interval A = [x0, x1] and determine backreaction of Wilson line near x0 [see also

Lewkowycz, Maldacena,’13]

metric near x0 ds2 = dρ2 + e2ρ

• dr 2 + r 2

√2c2 k − 1 2 dθ2

• Martin Ammon (FSU Jena)

Entanglement entropy in higher spin gravity July 29, 2013 19 / 21

Entanglement entropy Checks for the entanglement proposal

Checks for our proposal IV

Deficit angle has to be 2π/n and thus for n → 1

• 2c2 = k(n − 1) = c

6(n − 1) . Determining the entanglement entropy from Renyi entropies, SEE = lim

n→1

1 1 − n log Trρn

A = lim n→1

1 1 − n log(WRn(C)) we get finally SEE = − lim

n→1

1 1 − n

• 2c2LC = c

6LC , where we assumed WRn(C) = e−√

2c2LC Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 20 / 21

Summary

Summary

In this talk we focused on gravity + spin-3 field in AdS3 and put forward the proposal: Entanglement entropy in CFT = Wilson Line in CS theory in inf. dim representation Proposal passes non-trivial checks: collapses to geodesics for spin-2 gravity, reproduces CFT results Wilson line creates correct conical deficit Results which I have not shown: Thermal Entropy for higher spin black holes

[agrees with De Boer, Jottar,’13]

possible Generalizations Generalizations to TMG, to SL(N,

Renyi entropies, more than one interval, ...

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 21 / 21

Summary

Summary

In this talk we focused on gravity + spin-3 field in AdS3 and put forward the proposal: Entanglement entropy in CFT = Wilson Line in CS theory in inf. dim representation Proposal passes non-trivial checks: collapses to geodesics for spin-2 gravity, reproduces CFT results Wilson line creates correct conical deficit Results which I have not shown: Thermal Entropy for higher spin black holes

[agrees with De Boer, Jottar,’13]

possible Generalizations Generalizations to TMG, to SL(N,

Renyi entropies, more than one interval, ... For more details Wilson Lines & Entanglement Entropy in higher spin gravity MA, Castro, Iqbal, arXiv: 1306.4338

• r just ask me!

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 21 / 21

Summary

Summary

In this talk we focused on gravity + spin-3 field in AdS3 and put forward the proposal: Entanglement entropy in CFT = Wilson Line in CS theory in inf. dim representation Proposal passes non-trivial checks: collapses to geodesics for spin-2 gravity, reproduces CFT results Wilson line creates correct conical deficit Results which I have not shown: Thermal Entropy for higher spin black holes

[agrees with De Boer, Jottar,’13]

possible Generalizations Generalizations to TMG, to SL(N,

Renyi entropies, more than one interval, ... For more details Wilson Lines & Entanglement Entropy in higher spin gravity MA, Castro, Iqbal, arXiv: 1306.4338

• r just ask me!

Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 21 / 21