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
or 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,R) and Chern-Simons level, k = l4G
, 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,R)
3D Higher spin gravity as Chern-Simons theory with gauge group SL(3,R)× SL(3,R)
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,R) embedding into sl(3,R).
sl(3,R) generators
sl(3,R) has eight generators which we split into:
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,R) embedding into sl(3,R).
sl(3,R) generators
sl(3,R) has eight generators which we split into:
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,R) into sl(3,R)
principle embedding:
take Ja = La as sl(2,R) generators
bulk degrees of freedom: metric gµν and Spin-3 field φµνρ given by
gµν =1
2trf (eµeν) , φµνρ =
1
6trf (e(µeνeρ)) e = eµdx
µ.
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,R) embedding into sl(3,R).
sl(3,R) generators
sl(3,R) has eight generators which we split into:
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,R) into sl(3,R)
diagonal embedding:
take J0 = L0/2, J±1 = ±W±2/4 as sl(2,R) generators
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− = −e
ρJ−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− = −e
ρJ−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 − dg g
−1
where g and g are functions of spacetime coordinates and are valued in SL(3,R).
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− = −e
ρJ−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 − dg g
−1
where g and g are functions of spacetime coordinates and are valued in SL(3,R).
Remarks
Some of the gauge transformations (namely g, g ∈ SL(2,R) ⊂ SL(3,R))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 + µWThe 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 + µWThe 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 = −TrAA logA .
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 = −TrAA logA .
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,R): field capturing the dynamics of the probe
P(s) ∈ sl(3,R): momentum conjugate to U(x)
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,R): field capturing the dynamics of the probe
P(s) ∈ sl(3,R): momentum conjugate to U(x)
S(U,P;A)C =
∫
ds(
Tr (PU−1
DsU) + λ2(Tr (P2)− c2) + λ3(Tr (P3)− c3))
where DsU = dds
U + AsU − UAs , As ≡ Aµdxµ
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,R): field capturing the dynamics of the probe
P(s) ∈ sl(3,R): momentum conjugate to U(x)
S(U,P;A)C =
∫
ds(
Tr (PU−1
DsU) + λ2(Tr (P2)− c2) + λ3(Tr (P3)− c3))
where DsU = dds
U + AsU − UAs , As ≡ Aµdxµ
ds,
Entanglement entropy: Take c3 = 0 and√
2c2 → c6
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,R) Cartan L0,W0:
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,R) Cartan L0,W0:
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 = 12L2
0 +38W 2
0 + · · · , C3 = 38W0
(
L20 − 1
4W 2
0
)
+ · · · .Acting with C2 and C3 on the highest weight state |h,w〉 we find
c2 = 12h2 + 3
8w2 , c3 = 3
8w(
h2 − 14w2)
.
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,R)× SL(2,R):
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 → c6
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,R)× SL(2,R)?
S(U,P;A)C =
∫
ds(
Tr(
PU−1
DsU)
+ λ(s)(
Tr(P2)− c2
))
,
The equations of motion reduce to
U−1
DsU + 2λP = 0 ,d
dsP + [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)µdxµ
ds
)
+ [Aµ,Auν ]
dxµ
ds
dxν
ds= 0, A
us ≡ U
−1 d
dsU + 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) = 1 is a solution
(but not for higher spin gravity in generic gauge field backgrounds)
Geodesic equation
d
ds
(
(A − A)µdxµ
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)νdxµ
ds
dxν
ds
)
=√
2c2
∫
C
ds
√
gµν(x)dxµ
ds
dxν
ds≡√
2c2LC
and thus
WR(C) = e−√
2c2LC
or, for the entanglement entropy we get
SEE = − log WR(C) =√
2c2LC =1
4GLC
using√
2c2 = c6= 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
3log
(
∆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
3log
(
β
πǫsinh
(
π∆φ
β
))
.
Results for the thermal entropy (BTZ black hole)
Sth = − log WR(C) = 2π√
2πkL+ 2π√
2πkL .
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 + e
2ρ
(
dr2 + 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 = limn→1
1
1 − nlog Trρn
A = limn→1
1
1 − nlog(WRn(C))
we get finally
SEE = − limn→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,R)× SL(N,R) and to Vasiliev theory
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,R)× SL(N,R) and to Vasiliev theory
Renyi entropies, more than one interval, ...
For more details
Wilson Lines & Entanglement Entropy in higher spin gravity
MA, Castro, Iqbal, arXiv: 1306.4338
or 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,R)× SL(N,R) and to Vasiliev theory
Renyi entropies, more than one interval, ...
For more details
Wilson Lines & Entanglement Entropy in higher spin gravity
MA, Castro, Iqbal, arXiv: 1306.4338
or just ask me!
Martin Ammon (FSU Jena) Entanglement entropy in higher spin gravity July 29, 2013 21 / 21
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