Entanglement Entropy in Higher Spin Gravity

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University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

'Entanglement Entropy in Higher Spin Gravity'

by

Eva Llabres

10407715

July 2014

60 ECTS

September 2013 - July 2014

Supervisor:

Alejandra Castro, PhD

Prof. Jan de Boer, PhD

Examiner:

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1 Abstract

Two dierent candidates for a bulk observable which compute the entanglement entropy for higher spin gravity are proposed in [1] and [2]. These two dierent formulations give the same non-trivial results for the leading term in the UV cuto, when the symmetry group of SL(3, R) × SL(3, R). This lead us to think that they might be dierent versions of the same proposal. In this work we will review both formalisms. We will generalize the formalism in [1] to SL(N, R) × SL(N, R), and we will show that its leading term in the cuto is equivalent to the proposal in [2].

2 Introduction

Higher spin theories, as well known by Vasiliev gravity, are an extension of Einstein gravity. In this theory the graviton is coupled to an innite tower of massless spin elds in AdS background with highly non-linear dynamics [3]. Higher spin theories are particularly interesting for several reasons. Maybe the most important of them is that they might serve as tool to better understand stringy geometry. String Theory contains an innite tower of massive spin elds, and we would like to think that it is an spontaneously broken phase of Higher spin theory. In a Higgs-like mechanism, the broken symmetries will give mass to the higher spin elds, building with this the string theory. Actually, it is known that the spectrum of the tensionless limit of string theory matches the spectrum of higher spin gravity (For a review, see [4]). However, this tensionless limit of string theory it is particulary dicult to study in AdS background, which is required to construct consistent theories of higher spin. The recent work [5] uses the known AdS/CFT duality to shed more light on the relation between string theory and higher spin gravity in AdS background. They proved that a subsector of the CFT duals of the tensionless limit of string theory on AdS3× S3×T4, describes the CFT dual of higher spin gravity in AdS3.

Higher spin theories are as well interesting in the AdS/CFT framework. They posses and enlarged number of gauge symmetries. The free gauge elds can be translated into a large number of conserved currents in the boundary. This means that the CFT duals to higher spin theories are expected to be 'simple' and tractable, and makes HSAdS/CFT a good framework to better understand the holographic principle. Actually, the 4-dimensional case is conjectured to be dual to the free O(N) vector model, and the 3-dimensional is dual to WN minimal models.

We will be particulary interested in the (2+1)-dimensional case as toy model for the higher dimensional theories. Both, pure gravity and higher spin gravity become much simpler in three than in other dimensions. For example, in this case, gravity becomes a topological theory, i.e. with no propagating degrees of freedom. We can see this by considering the

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Einstein-Hilbert action with cosmological constant: SEH = 1 16πG3 Z M √ −g(R − 2Λ) , (2.1)

whose equations of motion are Einstein eld equations:

Rµν+ gµνR + gµνΛ = 0 . (2.2)

In three dimensions, the Riemann and the metric have 6 independent components, while there are only 6 independent eld equations. We can say that the theory is exactly solvable. For the case of 3-dimensional higher spin gravity, the massless elds are higher spin analogs of the graviton, and they do not propagate degrees of freedom either.

Another advantatge of working in 3 dimensions is that, a Chern-Simons theory whose symmetry group is two copies of the innite Lie group (hs[λ] × hs[λ]) is equivalent to higher spin gravity. While higher spin theories have highly non-linear equations of motion, Chern-Simons theories are easier to treat. Rewriting higher spin gravity in Chern-Simons formalism is a powerful tool to analyze the complicated higher spin theories that we cannot nd in other dimensions. Moreover, when λ is an integer number (λ = ±N), the symmetry group reduces to SL(N, R) × SL(N, R). In this case the innite tower of massless spin truncates to N − 1 elds with spin s 6 N. Truncating the tower of higher spin elds considerably facilitates the analysis. This situation is only found in the 3-dimensional case.

Now, we will turn to the issue of the entanglement entropy. In the usual AdS/CFT dictio-nary, to compute the entanglement entropy hollographically, we use the Ryu-Takayanagi proposal [6]:

SEE =

L(A) 4 G3

(2.3) where L(A) is the length of the minimal curve in the space dual to the CFT2 for which we

are computing the entanglement entropy, and G3 is the 3-dimensional Newton constant.

However, this formula is not valid when the concept of geometry is not well dened. Higher spin theories posses much more gauge symmetries than the dieomorsm and Lorentz. As a consequence, the metric is gauge dependent, namely, original features of the geometry, such a singularity, can be removed by gauge transformations. The lackness of the notion of geometry makes the Ruy-Takayanagi prescription not applicable to compute entanglement entropy in higher spin theories. Consequently, a new bulk observable needs to be proposed to compute the entanglement entropy in higher spin theories. The most fascinating aspect of nding such a quantity is the implications that it might have for the understanding of the emergence of space-time. Through the Ryu-Takayanagi prescription, the notion of classical geometry can be thought as emergent from the entanglement of degrees of freedom in the dual quantum eld theory [7]. We are tempted to think that a new concept of geometry can be dened taking into account the notion of the entanglement in the dual CFT as

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fundamental. This can be though as an actual realization of the emergence of space-time from the entanglement in the CFT.

In this work, we will review important aspect of entanglement entropy, 3-dimensional gravity, and higher spin theories. We will review as well the two current proposals in the literature for the entanglement entropy in higher spin theories ([1] and [2]). As original contributions, we will generalize the formalism in [1] to SL(N, R) × SL(N, R), and we will show that its leading term in the cuto is equivalent to the expression proposed in [2].

3 Entanglement entropy

We consider a quantum mechanical system at zero temperature, which is described by the pure ground state |Ψi in a Hilbert space H. The density matrix of this pure state is:

ρ = |ΨihΨ| , (3.1)

Now we separate the system in two, the subsystem A and the its complementary B. The Hilbert space is divided in two H = HA ⊗ HB. The subsystem A is described for the

reduced density matrix:

ρA=TrBρ , (3.2)

where the trace is taken in over the states that in the subspace HB. The entanglement

entropy for the subsystem A is dened as the the von Neumann entropy of the density matrix ρA:

SEE = −Tr(ρAlogρA) . (3.3)

The entanglement entropy is a measure of entanglement between the subsystem A and B. If the system A and B are not entangled, the reduced density matrix ρA represents a pure

state, and its entanglement entropy (3.3) will be zero.

We can consider a QFT on a manifold with d spatial dimensions, and 1 time direction. We dene a d-dimensional submanifold A at xed time, and its complementary B. The boundary which divides the manifold in A and B is called ∂A. We nd the entanglement entropy of the manifold A through formula (3.3). However, we will have to introduce an ultraviolet cuto , because the entanglement entropy in continuum theories is divergent. The leading term of the divergence is proportional to the area of the boundary ∂A [8]:

SEE = γ

Area(∂A)

d−1 +subleading terms , (3.4)

where γ is a constant which depends on the system. However, the formula (3.4) does not always holds. For example, for conformal eld theories in two dimensions, the entanglement

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entropy scales logarithmically with respect to the size of the subsystem A. In this section, we will compute the leading order of the entanglement entropy for a CFT2. We will do

the calculation for three dierent theories: a CFT living in an open interval, a CFT in an arc of a circumference, and a CFT at nite temperature. We will arrive to the same result via two dierent formalisms. In the rst subsection we will use the replica trick, which uses path-integral and conformal eld theory techniques [9]. In the second one, we use Ryu-Takayanagi prescription [6], which applies AdS/CFT correspondence to perform the computation.

3.1 CFT

2

entanglement entropy

To nd the entanglement entropy of a CFT, we start noticing that equation (3.3) can be rewritten as: SEE = − lim n→1 ∂ ∂nlog(Trρ n A) . (3.5)

Therefore, to nd the entanglement entropy, we just need to nd the quantity Trρn A. We

start considering a quantum eld theory in the at Euclidean space (x, tE). The subsystem

A is situated in the spatial interval x ∈ (u, v). Following [6], we consider the density matrix of the total system ρφ0φ00 = Ψ(φ0) ¯Ψ(φ

0

0), where Ψ is the groundstate of the system

at tE = 0:

Ψ(φ0(x)) =

Z φ(tE=0,x)=φ0(x)

tE=−∞

Dφe−S(φ), (3.6)

where φ(tE, x)denotes the eld in the CFT2. We need to nd the reduced density matrix

of the subsystem A, so we will x φ0 = φ00, and integrate out φ0 when x ∈ B. It can be

proved that the resulting path integral is: [ρA]φ+φ− = (Z1) −1 Z tE=∞ tE=−∞ Dφe−S(φ)Y x∈A δ (φ(−0, x) − φ−(x)) δ (φ(+0, x) − φ+(x)) , (3.7)

which is represented in Figure 1a. The factor Z1 corresponds to the partition function of

the vacuum, i.e, the path integral over the surface in Figure 1a but without cuts. Z1 has

been added to get the right normalization TrρA= 1. We are ready now to nd TrρnA. If we

arrange n copies [ρA]φ1+φ1−[ρA]φ2+φ2−...[ρA]φn+φn−, taking the trace is equivalent to choose

φk,− = φk+1,−, with k = 1, 2, ...n, and integrate out φ(k)+. The resulting path integral

is: Trρn A = 1 Zn 1 Z (tE, x)∈Rn Dφe−S(φ)≡ Zn Zn 1 , (3.8)

where Rn is the Riemannian surface formed by n-copies of the eld theory which are glued

together through the boundaries of the interval A (see Figure 1b). We have dened the partition function over the Riemannian surface Rn as Zn. The procedure of calculating

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(a) (b)

Figure 1: (a) Graphic representation of the path integral [ρA]φ+φ− in (3.7). The elds φ+

and φ− belong to the interval A, and their time coordinate goes to zero from the right and

the left respectively. (b) Graphic representation of a Riemannian surface R3, where the

eld φ+ in the k-th sheet is connected to the eld φ− in the (k + 1)-th sheet.

the entanglement entropy using the path integral in n-sheeted Riemannian surface in (3.8) is called replica trick.

Now, we need to explicitly calculate this path integral. We introduce a replica eld φk in

every sheet (k = 1, 2, ..., n). To reproduce the Riemannian surface Rn , we can impose

boundary conditions:

φk(e2φi(ω − u)) = φk+1(w − u), φk(e2φi(ω − v)) = φk−1(w − v) , (3.9)

where w is a a coordinate in the complex plane w = x + itE. The conditions in (3.9) are

represented in Figure 2, and they are equivalent to insert twist operators in the complex plane C at the points w = u and w = v, with tE = 0 [9]. Denote a twist operator by

Tn(x, tE), and ¯Tn(x, tE)its holomorphic:

Trρn

A= hTn(u, 0), ¯Tn(v, 0)iC, (3.10)

It can be shown that the twist operators transform under a conformal transformations as primary operators with conformal dimension hn = ¯hn = (c/24)(n − 1/n). In general, the

two point function of a primary operator ψ with conformal dimension h = ¯h:

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Figure 2: This graphic represents twist conditions (3.9), to reproduce a Riemannian surface R3)

For the specic case of (3.10), the two point function is: hTn(u, 0), ¯Tn(v, 0)iC ∝ |u − v|

c 6(1−1/n

2₎

(3.12) Using (3.5), and dening ∆x = |u − v|, we nd the entanglement entropy of a CFT2 in an

open interval ∆x: SEE = c 6log ∆x , (3.13)

where we have introduced the UV cuto such that SEE(∆x = ) = 0.

In (3.10), we have seen that Trρn

A is a two point function of two primary operators. Since

we know how a two point function transforms under a conformal mapping, we can easily compute Trρn

Ain other geometries. Assuming the conformal transformation z → w = w(z),

where w is a coordinate in complex plane: Trρn A= hTn(z1, ¯z1), ¯Tn(z2, ¯z2)i = dw dz 2hn z=z1 d ¯w d¯z 2hn ¯ z=¯z2 hTn(w1, ¯w1), ¯Tn(w2, ¯w2)iC, (3.14)

where z1 and z2 must be substituted the endpoints of the subsystem A (z1 = ¯z1 = u,

z2 = ¯z1 = v). For example, we can compute the entanglement entropy of CFT in thermal

bath with inverse temperature β. The total density matrix of the system is:

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where H is the Hamiltonian. We consider the transformation that compacties the time direction tE → tE + β, and maps w-plane to the z-cylinder:

w = e2πβz. (3.16)

Using (3.14), we nd Trρn

A for in the cylinder tE → tE+ β:

Trρn A= β π sinh π β(u − v) −2dn . (3.17)

Using (3.5), we nd the entanglement entropy of a CFT with length ∆x = |u−v| at inverse temperature β: SEE = c 6log β πsinh π β∆x . (3.18)

Moreover we can consider the geometry that compacies the spatial direction instead of the time direction. If we substitute iβ → L, the formula (3.16) represents the transformation from the w-plane to the z-cylinder with x → x + L. Performing the same substitution in (3.18), we nd the entanglement entropy of a CFT2 living in circumference:

SEE = c 6log L πsin π L∆x , (3.19)

where ∆x is the length of the arc where the CFT2 lives. Following [10], we can compute as

well the entaglement entropy of a CFT with left and right movers at dierent temperature. The total density matrix is:

ρ = exp (−βH + iΩEP ) , (3.20)

where P is a momentum and Ω = iΩE is its associated potential. We regard β± =

(1 ± Ω), as the temperatures of the left and right movers. We now consider the conformal transformation: w = e 2π β−iΩEz_, _{w = e}_¯ 2π β+iΩEz¯_. (3.21)

which maps the plane into a geometry with periodicity z → z + (β − iΩE). If we apply

this transformation in (3.14), we nd: Trρn A= β+β− π2 sinh π∆x β− sinh π∆x β+ ₁₂c(n−1/n) (3.22) The entanglement entropy is:

SEE = c 6log β₊β− π2 sinh π∆x β− sinh π∆x β+ . (3.23)

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which is (3.18), when Ω = 0, and β+ = β− = β.

In summary, in this section we have described how to use the replica trick to compute the entanglement entropy, and we have found the entanglement entropy of three dierent spaces CFT2, in (3.31), (3.18), and (3.19).

3.2 Holographic entanglement entropy

To compute the entanglement entropy on a subsystem A of a CFT2, we can as well use

the Ryu-Takayanagi proposal [6]:

SEE =

L(A) 4 G3

, (3.24)

where L(A) is the length of a geodesic which connects the endpoints of the interval in the CFT2 through the bulk of its AdS3 dual. The geodesic is supposed to live in a constant

time slice. G3 is the 3-dimensional Newton constant. We will identify the Newton constant

with the central charge of the dual CFT using the Brown-Henneaux equation [11]: c

6 = ` 4G3

. (3.25)

In the UV regime, we send the boundary of AdS to innity, and the length of the geodesic L(A) is divergent. In the following we will compute the leading order of the length L(A) for three dierent backgrounds. These backgrounds are dual to states in the CFT's for which we have computed the entanglement entropy in the previous subsection. We will see that the results for the leading divergence of the geodesic exactly coincide with the leading term for the entanglement entropy found in the previous section.

3.2.1 Poincare patch AdS3 Entanglement Entropy

A vacuum stat for a CFT living on a line is dual to the the Poincare patch of AdS: ds2 = `

2

z2 dz

2_{− dt}2

+ dx2 , (3.26)

where −∞ < x, t < ∞, and 0 < z < ∞, and ` is the AdS radius.

To nd the entanglement entropy of a spatial interval ∆x at the boundary of AdS, we need to compute the length of the geodesic which connects the two endpoints. We start minimizing the length functional:

Length =Z ∆x/2

−∆x/2

dxp1 + (dz/dx)

2

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We know that the Hamilton-Jacobi equations minimize a functional of the type R dtL(x, ˙x). If we regard x as the time, and z as the position, we can use the Hamilton-Jacobi method for (3.27). We can as well construct the Hamiltonian which is invariant in the time x. We call z∗ to the energy associated to the system with lagrangian determined by (3.27), which

follows: dz dx = pz2 ∗− z2 z . (3.28)

The previous equation represents the minimal path between two points in x. Noticing that z∗ is the turning point we can nd its value from:

∆x 2 = Z x∗ 0 dx z pz2 ∗ − z2 = x∗ (3.29)

Using (3.28) and (3.29), we see that the minimal curve follows x2_{+ z}2 _{= (∆x/2)}2. We nd

the minimal length (3.27) to be:

Length(∆x) = −2 log(∆x

) . (3.30)

where we have considered the UV cuto z0 = ∆x/2to eliminate divergences in the integral,

and to normalize SEE(∆x = ) = 0. Using the prescription (3.24), the entanglement

entropy of a CFT living an interval ∆x is: SEE = Length(∆x) 4G3 = c 6log ∆x , (3.31)

which exactly coincides with the CFT computation (3.31). 3.2.2 Global AdS3 Entanglement Entropy

Global AdS is dual to a vacuum state living in a circumference of length L. The AdS metric in global coordinates is:

ds2 = `2 dρ2− cosh2ρdt2+ sinh2ρdϕ2 , (3.32) where −∞ < t < ∞, 0 < ρ < ∞, and ϕ → ϕ + 2π. The CFT is located at the boundary ρ = ρ0, and the minimal curve live in a time slice. The length of the geodesic connecting

two points ϕi = 0 and ϕf = 2π∆x/L can be explicitly computed:

cosh(Length(∆x)) = 1 + sinh2ρ0sin2

π∆x L

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where x is the coordinate that follows x = Lϕ/2π = `ϕ. In the UV regime (ρ0 → ∞), the

leading term entanglement entropy is: SEE = c 3log eρ0_sinπ∆x L = c 3log L πsin π∆x L , (3.34)

where we have dened the cuto eρ0 _{= L/π}. The previous result for the entanglement

entropy is equivalent to the CFT calculation (3.19). 3.2.3 BTZ black hole Entanglement Entropy The rotating BTZ black hole metric is [10]:

ds2 = (r 2_{− r}2 +)(r2− r−2) `2_r2 dt 2₊ `2r2 (r2_{− r}2 +)(r2− r−2) dr2+ r2dϕ + r+r− `2_r2 dt 2 , (3.35) where 0 < r < ∞, ϕ has periodicity ϕ → ϕ + 2π, and the time is as well periodic to assure an smooth geometry. The radius of AdS is represented by `. The mass M and angular momentum J of the black hole are:

M = r 2 ++ r−2 8G3` , J = r+r− 4G3 . (3.36)

Consider a CFT with dierent inverse temperatures β±, corresponding to the left and

right movers respectively. The rotating BTZ black hole is dual to the asymmetric CFT in the high temperature limit β± L, where L is the length of the total system. The

temperatures of the CFT are determined from the BTZ parameters as: β± = β(1 ± Ω) =

2π` r+± r−

, (3.37)

where Ω is the potential conjugated to the angular momentum.

In the same fashion, the symmetric CFT at nite temperature is dual to the non-rotating BTZ black hole. If we set r− = 0 in the metric (3.35), we nd a non-rotating black hole

with angular momentum J = 0, and mass M = r2 +/8G3: ds2 = (r 2_{− r}2 +) `2 dt 2₊ `2 (r2_{− r}2 +) dr2 + r2dϕ2, (3.38) which is dual to a symmetric CFT at nite temperature in the high temperature limit β L. The temperature of the CFT is:

β = 2π` r+

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As in the previous cases, the length of the geodesic of the non-rotating BTZ black hole can be computed explicitly. We set the boundary of AdS at r = r0 and consider a geodesic

living in a time-slice, with endpoints ϕi = 0 and ϕf = 2π∆x/L:

cosh(Length(∆x)) = 1 +2r 2 0 r2 + sinh2 π∆x β . (3.40)

Imposing r0 → ∞, the leading term in the entanglement entropy related to the non-rotating

BTZ black hole: SEE = c 3log 2r0 r+ sinhπ∆x β = c 3log β πsinh π∆x β (3.41) where we have dened the cuto 2r0/r+ = β/π. The previous result for the entanglement

entropy is equivalent to the CFT calculation (3.19).

For the rotating BTZ black hole, the Ryu-Takayanagi prescription (3.24) is not valid any-more. Since this background is not static, the geodesic which connects the two endpoint of the CFT cannot be assumed to be in a constant time slice. To compute the entangle-ment entropy, we need to use the covariant prescription in [10]. They proposed that the entanglement entropy associated with a region of the boundary is given by the area of a co-dimension two bulk surface with vanishing null geodesics. To nd the entanglement en-tropy of the rotating BTZ black hole, they compute the geodesic length no longer assuming that the curve live in constant time slice, and they nd:

SEE = c 6log β+β− π2 sinh π∆x β− sinh π∆x β+ , (3.42)

which exactly coincides with the CFT result (3.23)

Before moving on, we would like to comment on the thermal limit ∆x β. The entan-glement entropy of the rotating BTZ black hole in this regime is:

SEE −−−−→ ∆xβ c 6 π β+ + π β− ∆x . (3.43)

It is interesting to compare (3.43) with the thermal entropy of the BTZ. The thermal en-tropy of a black hole is computed using Bekenstein-Hawking area law Sth =Area(Σ)/4G3,

where Area(Σ) is the area of event horizon. For the case of the rotating BTZ black hole, this area reduces to the length of the outer horizon Area(r+) = 2πr+. Therefore, the

thermal entropy of the rotating BTZ black hole is: Sth = c 6 π β+ + π β− 2π` . (3.44)

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We now consider the entanglement entropy in the thermal limit (3.43), when the interval is equal to the whole CFT system (∆x = L = 2π`). In this case, the entanglement entropy is the thermal entropy of the black hole. This is expected from the fact that when the interval of the CFT is big enough, the minimal distance is the geodesic wrapping the horizon of the black hole, i.e the thermal entropy according to Bekenstein-Hawking (See Figure 3). Dene now the density entropy as sth = Sth/2π`. The direct relation between thermal and

entanglement entropy can be phrased as: SEE −−−−→

∆xβ sth∆x . (3.45)

Figure 3: The graphic represents a time slice of the BTZ geometry, where the black area represents the interior of a black hole. The blue line is the CFT region for which we compute the entanglement entropy, and the red line represent the minimal distance connecting the two endpoint of the CFT. Left: For a small region of the CFT, the minimal distance is given by a connected geodesic hanging from the boundary. Right: For a large interval, the minimal distance is disconnected; it has one component hanging form the boundary, and the other wrapping around the horizon. This last one, computes the thermal entropy of the black hole according to Bekenstein-Hawking.

4 (2+1)-dimensional Gravity as a Gauge Theory

4.1 Ordinary Gravity as Chern-Simons Theory

In this section, following reference [12], we will show explicitly that (2+1)-dimensional pure gravity can be interpreted as a gauge theory. We will develop the proof just in AdS-space, eventhough the calculations can be repeated analogously in de Sitter, and Minkowsky, the two other maximally symmetric spaces. This derivation will be carried out building a local gauge theory in SO(2, 2), and identifying its action with the one of AdS gravity, which has the same symmetry group. We will build as well the gauge theory with symmetry group

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SL(2,R)×SL(2, R), which is isomorphic to SO(2, 2), and we will prove that it is equivalent to AdS gravity too.

4.1.1 SO(2, 2) Chern-Simons Gauge Theory

We start with a theory invariant to global innitesimal transformations of the type: G(α) = eiαaLa _{' 1 + iα}a_L

a, (4.1)

with Lathe generators of the symmetry group of the theory and αaan innitesimal constant

parameter. If the symmetry group is SO(2, 2) (see Appendix B.1), α = αa_L

adecomposes in

two: one corresponding to Lorentz transformations (Jagenerators) and other corresponding

to translations (Pa generators):

α = αaLa= ρaPa+ τaJa. (4.2)

We see that ρa and τa are, respectively, the innitessimal parameters for translations and

Lorentz transformations. To promote the global parameters to local and keep the theory invariant, we have perfom the following minimal substitution in the lagrangian:

∂µ → Dµ≡ ∂µ− Aµ, (4.3)

where Aµ= AaµLa is gauge eld belong to SO(2, 2), which can be separated as:

Aµ= eaµPa+ ωaµJa. (4.4)

We can say that ea

µ is the gauge eld for translations and ωaµ for Lorentz transformations.

In order to keep the theory invariant under local transformations, Dµ must behave as

DµG(α)φ = G(α)Dµφ. This condition xes the transformation of the gauge eld as:

Aµ→ Aµ− (∂µα + [Aµ, α]) ≡ Aµ− Dµα , (4.5)

which is the innitesimal version of the gauge transformation:

Aµ→ GAµG−1+ G∂µG−1. (4.6)

We see the gauge elds transform in the adjoint representation. The transformation laws of the gauge elds ea

µ and ωaµ can be found substituting (4.2) and (4.4) in (4.5) and using

commutation relations of SO(2, 2) (B.10):

δea_µ= −∂µρa− abceµbτc− abcωµbρc, (4.7)

δω_µa = −∂µτa− abcωµbτc−

1 `2

abc

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where the greek indices are space-time indices, and the latin are Lie indices. abc _{is the}

Levi-Civita symbol.

An action which is invariant to the gauge transformation (4.6), is the the Chern-Simons action: SCS[A] = k 4π Z Tr A ∧ dA + 2 3A ∧ A ∧ A . (4.9)

The symbol M represents a (2+1)-dimensional manifold, and the overall factor k/4π is added following conventions. The constant k is called Chern-Simons level. The notation Tr(...) is a short-cut for the contraction using the Lie algebra metric ηab related to the

basis of generators of SO(2, 2) that we choose. The eld equations of the action (4.20) are explicitly:

0 = δSCS δA =

k

2πF , F = dA + A ∧ A , (4.10) where F is the curvature tensor. Consequently, the curvature of the gauge theory van-ishes:

F = dA + A ∧ A = 0 , (4.11) and Chern-Simons is regarded as a theory of at connections. Under the gauge transfor-mation in (4.6), the curvature tensor transforms:

F → G−1F G , (4.12)

and we see that the equations of motion are invariant to local gauge transformations. Substituting (4.4) in the Chern-Simons action (4.9), and using the commutation relations and the Lie algebra of SO(2, 2), we arrive to :

SCS[A] = k 4πl Z M µνσ eµa(∂νωaσ− ∂σωaν) + abceaµωνbωcσ− 1 3`2abce a µebνecσ , (4.13)

The previous action is the Einstein-Hilbert action in vielbein formalism in (C.12), if we do the identication:

k = ` 4G3

. (4.14)

Then, identifying the components ea

µ and ωaµ of the gauge eld (4.4) with the vielbein and

the spin connection, we recover the a pure gravity action with a gauge theory. It can be proven that the gauge symmetries are (4.7), are actually symmetries of the action (4.13), and they correspond to dieomorsms in vielbein formalism. The equations of motion (4.11) are Einstein-Cartan equations.

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In summary, we have proved that (2+1)-dimensional AdS gravity is equivalent to a SO(2, 2) gauge theory, where the vielbein is the gauge eld for translations, and the spin connection for Lorentz transformations. This proof can be analogously carried out for de Sitter back-grounds changing ` → i`. Doing this substitution in (B.10), the commutation relations correspond to SO(3, 1), exactly the symmetry of de Sitter space. For at space we require ` → ∞, and the symmetry group is ISO(2, 1).

4.1.2 SL(2, R) × SL(2, R) Chern-Simons Gauge Theory

Since SO(2, 2) is isomorphic to two non-interacting copies of SL(2, R) (Appendix B.1), we can expect that from a SL(2, R) × SL(2, R) gauge theory, we can reconstruct as well pure gravity. In order to formulate a gauge theory SL(2, R) × SL(2, R), we will proceed analogously as in the previous section. Denote R as the right handed (-) group element, and L is the left handed (+) elds:

L = eiα+aJa+ _{' 1 + iα}+a_J+

a , R

−1

= eiα−aJa− _{' 1 + iα}−a_J−

a (4.15)

where J+

a and J −

a are the generators of two non-interacting copies of sl(2, R) algebras,

related to so(2, 2) generators following (B.11). In order to keep the theory invariant to local gauge transformations, we need to add two independent gauge elds A+ _{and A}−_:

A±_µ → A±_µ − ∂µα±−A±µ, α

±_{= (A}a±

µ − ∂µαa±− abcA±_µbα±c)J ±

a , (4.16)

which are innitessimal version of the nite transformations: A+_µ → L(A+_µ + ∂µ)L−1, A−µ → R

−1

(A−_µ + ∂µ)R , (4.17)

A+ and A− are related to the gauge eld A ∈ SO(2, 2) by: Aµ= Aa+µ J + a + A a− µ J − a . (4.18)

Using the relation between the generators of the algebra (B.11), we can relate the gauge elds of SL(2, R) × SL(2, R) with the ones of SO(2, 2):

A+a= ωa+1 `e a _A−a = ωa− 1 `e a_. _(4.19)

The Chern-Simons action SCS[A+] is invariant under the left gauge transformation (4.17),

and the analogous holds for SCS[A−]. Consequently, any linear combination SCS[A+] and

SCS[A−] will be invariant under the group G = SL(2, R) × SL(2, R). In particular, the

following combination can be proven to reproduce Einstein-Hilbert action (C.12) if k = `/4G3:

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SEH[e, ω] = S[A+]CS− S[A−]CS , (4.20)

The proof is carried out substituting (4.19) in (4.20), and using the commutation relations and Lie metric of SL(2, R). The equations of motion of the action (4.20) are:

F+= dA++ A+∧ A+ _{= 0 ,} _F−

= dA−+ A−∧ A−= 0 . (4.21) Equations (4.21) are naturally equivalent to the equations of motion of pure gravity with a negative cosmological constant. Specically, plugging (4.19) in F+_{− F}− _{= 0}_{, we nd}

Ta

µν = 0, where Tµνa is the torsion in the rst Cartan equation (C.8). With equation

F++ F− = 0, we nd the Einstein eld equations for empty space with negative negative cosmological constant.

In summary, in this section we have proven that AdS gravity can be reformulated in terms of SL(2, R) × SL(2, R), identifying actions, equations of motion and symmetries.

4.1.3 AdS3 geometry as a gauge connection

In this section we will nd how to represent usual AdS metric in terms of gauge connections, and we will explicitly study the most common backgrounds. In the rest of this work we will change the notation respect the previous sections: A+_{, A}− _{→ A, ¯}_A_{. Using (4.19), we}

can nd an expression that relates gµν with the gauge elds A, ¯A :

Tr Aµ− ¯Aµ Aν − ¯Aν = 4 `2ηab e a µe b ν = 2 `2gµν, (4.22)

Therefore, when gravity is treated in Chern-Simons formalism, it is common to indier-ently exchange the terms `background' and `gauge connection'.

We will now nd the connections related to common AdS backgrounds. In [13], it is shown that the following metric is solution 3-dimensional gravity with negative cosmolog-ical constant: ds2 = `2 dρ2+ 2πL(x +₎ k (dx + )2+2π ¯L(x −₎ k (dx − )2− e2ρ+ e−2ρ(2π) 2_L(x+_{) ¯}_L(x−₎ k2 dx+dx− (4.23) where x±_{= t ± x}_{, and k is again the Chern-Simons level. The previous metric is solution}

for any L(x+₎ _{and ¯}_L(x−₎_{. There are three interesting cases for constant values of the}

charges L(x+₎ _{and ¯}_L(x−₎_{. First, we consider:}

2πL = 1 2(M + J ) = k π2 β2 + , 2π ¯L = 1 2(M − J ) = k π2 β2 − , (4.24)

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with β+ and β− following (3.37). If we perform the change of variables ϕ = 2πx/L,

r2 _{= r}2 +cosh

2_{(ρ − ρ}

∗) + r−2 sinh2(ρ − ρ∗), with e2ρ∗ = (r2+− r2−)/4, we see that the metric

(4.23) corresponds to the rotating BTZ (3.35). Moreover, we can recover the global AdS solution (3.32) if we set the charges to:

2πL = 2π ¯L = −k

4, (4.25)

with J = 0 and M = −k/2, and consider x = ϕ. Finally, xing L = ¯L = 0 (M = J = 0), and taking r = e−ρ_{, we recover the AdS metric in Poincare coordinates (3.26).}

Now we will write the above solutions in Chern-Simons formalism. We use the gauge freedom of the theory to dierently parametrize the gauge connections. Using (4.22), we can prove that (4.23) can be rewritten as:

A = b−1(ρ)db(ρ) + b−1(ρ)a(x+_{, x}−_{)b(ρ) ,} _(4.26)

¯

A = b(ρ)db−1(ρ) + b(ρ)¯a(x+_{, x}−_)b−1_{(ρ) ,}

with b(ρ) = exp(ρL0), and:

a = L+− 2πL(x+₎ k L− dx+, (4.27) ¯ a = −L−− 2π ¯L(x −₎ k L+ dx−,

We can as well parametrize the solutions following the empty gauge, which stands for choosing A = ¯A = 0in the transformation (4.17), leading to:

A = LdL−1, A = R¯ −1dR . (4.28) where L and R are:

R(x±, ρ) =exp Z x 0 dxia¯i b−1(ρ) (4.29) L(x±, ρ) = b−1(ρ)exp − Z x 0 dxiai .

Specically, for the metric (4.23) when L(x+₎ _{and ¯}_L(x−₎ _{are constants, L and R will take}

the form:

L = b−1(ρ) exp L+− 2πL_k L− x+ , (4.30)

R = exph−L−−2π ¯_kLL+

x−ib−1(ρ) .

Since the entanglement entropy for higher spin theory will be calculated in Chern-Simons formalism, we will use the previous results in the following sections.

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4.2 Higher spin gravity as a Chern-Simons Theory

We now consider again the action (4.20), and promote the symmetry group to SL(N, R) × SL(N,R). In this section we will analyze to which physical theory does this action cor-respond to. In order to do this, rst, we will notice that we can decompose the algebra sl(N,R) in direct sums of dierent-dimensional representations of sl(2, R). Each of these decompositions will be called an embedding. Second, we will calculate the decompositions of the adjoint representation in order to nd the spectrum of the theory. We will see that the SL(N, R) × SL(N, R) action corresponds to 3-dimensional higher spin gravity, with AdS background. Finally, we will study some of the gauge connections that are commonly used in higher spin gravity.

4.2.1 sl(2, R) embedded into sl(N, R)

In this section we will analyze the embeddings of sl(2, R) in sl(N, R) following reference [14]. Denote L(Λhw_{; N )}_{a nite dimensional representation of the algebra sl(N, R), with highest}

weight Λhw1_{. In general, we can decompose it in a direct sum of irreducible representations}

of sl(2, R):

L(Λhw; N ) ∼=M

j∈N₂

nj(Λhw; i)(2j + 1)2, (4.31)

where (2j + 1)2 is the (2j + 1)-dimensional irreducible representation of sl(2, R), and

nj(Λhw; i) is the multiplicity of (2j + 1)2 representation in a specic embedding, denoted

by i. This collection of numbers nj(Λhw; i)is called the branching rule for the embedding

i of the representation Λhw_{. Formula (4.31) means that the three matrices belonging to}

the sl(2, R) subgroup in a representation of sl(N, R) can be decomposed in diagonal blocks which satisfy separedtly the sl(2, R) algebra. The dimensions such structure are deter-mined by the branching rules.

In order to nd the branching rules for a general L(Λhw_{; N )}_{, the following can be proved}

[15]:

• The branching rules in the fundamental representation of sl(N, R) , denoted as NN,

will be simply determined by the partitions of N. There will be P (N) (number of partitions of N) independent embeddings. For example, for sl(3, R), the number of embeddings will be P (3) = 3 and the partitions of 3 give us the branching rule of the fundamental representation as 33 ∼= 32, 22 ⊕12, 3 · 12. The rst decomposition

corresponds to what it is called the principal embedding and the second one is the diagonal embedding (The last one is a trivial embedding where only singlets appear in the decomposition). In the branching rules for the general sl(N, R) the decomposition

1_{Actually, in this section, we will be only interested in analyzing branching rules for the N-dimensional}

fundamental and the (N2_{− 1}_{)-dimensional adjoint representation that will be denoted as N}

N and adjN

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of the form NN ∼=N2 will always denote the principal embedding, and NN ∼=22 ⊕

(N − 2)12 the diagonal embedding.

• The dierent embeddings of sl(2, R) into sl(N, R) in any representation are deter-mined by the branching rule of the fundamental representation. This means that if two embeddings are equivalent in the fundamental representation, they will be equivalent in any other representation.

Consequently, for any representation of sl(N, R), we have P (N) dierent embeddings, and once we know the branching rules in the fundamental representation we can determine them in any other. It turns out that the analysis of the branching rules in the adjoint rep-resentation is especially interesting, because this is the reprep-resentation in where the gauge elds live. Moreover, the adjoint representation has dimension N2_{− 1}_{, which agrees with}

the number of generators in sl(N, R). This means the matrices in these representation de-termine the structure of the algebra. From the fundamental representation, the branching rules for the adjoint representation of sl(N, R), denoted by adjN:

adjN ⊕12 ∼= M j∈N n2_j · (12⊕32⊕ ... ⊕ (4j + 1)2) M j6=j0 njnj0· ((2|j − j'| + 1)₂⊕ ... ⊕ (2|j + j'| + 1)₂) , (4.32)

where nj is the set of numbers that correspond to the branching rule in the fundamental

representation of the embedding that we want to translate (NN ∼=L_j∈N

2 nj(2j + 1)2). In

order to obtain the branching rules in the adjoint representation we still have to subtract a singlet in the right side. For example, just applying this formula, we can easily calculate the decomposition for two example embeddings, the principal and diagonal:

adjN ∼=32⊕52⊕ ... ⊕ (2N − 1)2 principal embedding

adjN ∼=32⊕ 2(N − 2) ·22⊕ (N − 2)2·12 diagonal embedding (4.33)

In the following, we will use the knowledge gained in this subsection to nd to which physical theories will correspond the Chern-Simons theory with symmetry SL(N, R) × SL(N,R).

4.2.2 Spectrum of the gauge theory

We consider now two non-interacting Chern-Simons theories with gauge elds, A, ¯A ∈ SL(N,R). We have shown that there are dierent possibilities to pick sl(2, R) embedded into sl(2, R). These choices will determine inequivalent SL(2, R) vacuums that correspond to dierent physical theories. In this subsection we explain a way to nd their spectrum.

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We start denoting the gauge eld as: Aµ= (A(2)µ ) a La+ jµaJa, (4.34) ¯ Aµ= (A (2) µ ) a_¯ La+ j a µJ¯a,

where the generators La and ¯La correspond to the generators in the sl(2, R) subgroup in

sl(N,R), and Ja and ¯Ja are the rest of generators. As proved before, there will be P (N)

ways to choose this SL(2, R) subgroup in SL(N, R). From the branching rules in the ad-joint representation, we can deduce the structure of the sl(N, R) algebra. In the following, we will identify to which theory the remaining generators in (4.34) correspond for dierent embeddings.

We start with the principal embedding. From the branching rules in (4.33) we infer that there are 3 generators representing the vacuum, 5 generators in the spin-2 representation of sl(2, R), 7 generators in the spin-3 representation, ..., and the 2N − 1 generators in the spin-(N − 1) representation of sl(2, R). In this case, we can rewrite the gauge eld:

Aµ = X s=2, ...N ta1a2...as−1 µ Ta1a2...as−1, A¯µ= X s=1, ...N −1 ¯ ta1a2...as µ T¯a1a2...as. (4.35)

where Ta1a2...as−1 are the generators in the spin-(s − 1) representation of sl(2, R), and

ta1a2...as−1 are components of the gauge eld that transform under this representation.

When s = 2, ta1 _{= (A}(2)

µ )a1, and Ta1 are the generators in the sl(2, R) subgroup.

Following [16], we now dene: ta1a2...as−1 µ ≡ ωµ+ 1 `eµ a1a2...as−1 , ¯ta1a2...as−1 µ ≡ ωµ− 1 `eµ a1a2...as−1 , (4.36) and the uctuations hµ and vµ around the background eµ and ωµ:

ea1a2...as−1

µ → (eµ+ hµ)a1a2...as−1, ωµa1a2...as−1 → (ωµ+ vµ)a1a2...as−1 (4.37)

We substitute the connections (4.35) in Chern-Simons equations of motion (4.21), and we consider only linear uctuations around the vacuum. We identify the linearized equations with a the Fronsdal equations in three dimensions with a tower of spin elds s = 2, ...N [17]. The equations are in Cartan formalism, and the spin-s gauge eld ϕµ1...µs is recovered

from the vielbein with:

ϕµ1...µs = 1 se a1 (µ1... e as−1 µs−1eµs)a1...as−1. (4.38)

The spin-2 gauge eld correspond to the metric or graviton gµ1µ2 ≡ ϕµ1µ2 = (1/2)(e

a

(µ1eµ2)a)

As an another example, we will nd now the spectrum of particles for the diagonal embed-ding. From (4.33) we will have 2(N − 2) couples of generators interacting as the spin-1/2

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representation of sl(2, R), (N − 2)2 _{generators that represent spin-0 singlets, and, three}

generators that follow spin-1 sl(2, R) algebra , i.e. AdS vacuum. We will use the basis of generators {La, Ub, Sc}, where La is the sl(2, R) subgroup, Ub are the generators

trans-forming in the spin-0 representation, and Sc correspond to the spin-1 representation. We

can split the gauge connection:

A = (A(2))aLa+ χbUb+ ΨcSc, A = ( ¯¯ A(2))aL¯a+ ¯χbU¯b+ ¯ΨcS¯c. (4.39)

where A(2) _{is be related with the vielbein (A}±(2) _{= ω ± e/`}_{). The component χ is a}

one-form, and Ψ is a two component spinor:

ΨcSc= Ψ+cSc++ Ψ −c

S_c−, (4.40)

Considering the linearized uctuations around the background, we nd that each of the spin-0 representation give an spin-1 gauge eld, and the spin-1/2 representations result into spin-3/2 gauge eld. The metric-eld is recovered with gµν = (1/2)(ea_(µe_ν)a). We would

like to mention that there exists a particular interesting truncation of the diagonal embbe-ding, which follows from considering two copies of a Chern-Simons theory with symmetry SL(N,R) × U(1)N −2. The spectrum of this theory is pure gravity coupled to 2(N − 2)

spin-1 gauge eld with symmetry U(1).

We can can repeat the previous analysis for other embeddings. Generally, we will see that a spin-(s − 1) representation of sl(2, R) in the branching rule of sl(N, R) in the ad-joint, will result into a spin-s gauge eld in the spectrum of the theory. The metric will always be computed from the vielbein using:

gµν =

1 2(e

a

(µeν)a) (4.41)

In order to identify the actions of the SL(N, R)×SL(N, R) Chern-Simons with higher spin gravity, we need to relate the constants of both theories in the same way as we did in (4.14). To do so, we will use the fact that the SL(2, R) subgroup in (4.34) reproduces the action of AdS pure gravity. In section 4.1.2, to do the identication of the Chern-Simons action with the Einstein-Hilbert action, we used the sl(2, R) algebra in (B.13). However, the generators of the sl(2, R) subgroup in sl(N, R) have a dierent Lie metric in every embedding. This adds an overall normalization to the Chern-Simons action, that generalizes relation (4.14) to: 2ktrf(L (P ) 0 L (P ) 0 ) = ` 4G3 , (4.42) where L(P )

0 is the Cartan generator of the sl(2, R) subagebra in sl(N, R) with partition P .

Naturally, if L(P )

0 is the Cartan generator of the sl(2, R) algebra (B.13), we recover

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SL(N,R) is an alternative way to describe 3-dimensional higher spin theories with AdS background. The spectrum of the theory depends on the embedding of sl(2, R) in sl(N, R) we choose.

Before moving on, we would like to remark that it is often useful convenient to set a basis for generators of sl(N, R) which accommodates dierent embeddings. Specically for sl(3,R), a convenient basis is found in Appendix B.2.

4.2.3 Higher spin connections. Higher spin black holes.

In this subsection we will study valid solutions for gµν when the theory has SL(N, R) ×

SL(N,R) symmetry. For higher spin theories the graviton is related to the vielbein through formula (4.41). Analogously as we did to derive formula (4.22), we can relate the back-ground metric to the gauge connections. However, the generators of sl(2, R) subgroup not necessarily follow the algebra metric (B.14). Redoing the derivation of (4.22) considering a general Lie algebra metric, the resulting relation is:

trf h A(2)_µ − A(2)_µ A(2)_ν − A_ν(2)i= 4trf(eµeν) = 4trf(L (P ) 0 L (P ) 0 )gµν (4.43)

Evidently, formula (4.43) reproduces (4.22) if we consider L(P )

0 the generator of the original

sl(2,R) algebra. To simplify notation, form now on we will write L(P )₀ ≡ L0. So, L0 will

be the Cartan generator of the sl(2, R) subagebra in sl(N, R) with partition P .

In the following we will write explicit solutions of the SL(3, R) × SL(3, R) theory in terms of gauge connections. First, we start with solutions that do not carry higher spin charges, i.e., in a SL(3, R)×SL(3, R), the connections are A, ¯A ∈ SL(2,R). A solution of the theory is the AdS metric (4.23), and its related gauge connection will have as well the form (4.26) with (4.27). However, we need to substitute the old sl(2, R) generators for the sl(N, R) generators that are in the sl(2, R) subgroup. For the principal embedding in sl(3, R):

a =L+− 2πL(x +₎ k L− dx+_{, ¯}_{a = −}_L −− 2π ¯ L(x−₎ k L+ dx−, b(ρ) = eL0ρ_, (4.44)

where the generators belong to sl(3, R), and an appropriate basis in the fundamental rep-resentation is found in Appendix B.2. Analogously, for the diagonal embedding of sl(3, R) the connections are:

a =W+2 4 − 2πL(x+₎ k W−2 4 dx+, ¯a = −W−2 4 − 2π ¯L(x−₎ k W+2 4 dx−, b(ρ) = eL02 ρ, (4.45)

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In the following, we will analyze solutions carrying mass, angular momentum, and higher spin charges, which will be interpreted as black holes. The gauge dependence of the metric in higher spin theories complicates the denition of black hole respect to the pure gravity case. In [18], they consider 3 gauge independent conditions that are higher spin black holes:

1. The higher spin black hole solution tends smoothly to the BTZ black hole when the higher spin charges are sent to zero.

2. The Lorentzian horizon must be smooth, and have an Euclidean continuation. This implies that the holonomies around the thermal cycle should be trivial.

3. The thermodynamics quantities assigned to the black hole obey the laws of thermo-dynamics.

We will now elaborate in the previous conditions. The rst condition is a natural re-quirement. Since the higher spin parameters are independent, we should be able to con-tinuously send them to zero, and smoothly recover the black hole solution in SL(2, R) × SL(2,R).

The second condition is the generalization of the denition of a smooth black hole horizon for higher spin theories. A black hole is dened as a metric solution whose time component vanishes for some value of the radial coordinate. In Euclidean signature, the angular and time coordinate must be periodic, for smoothness of the solution. Transforming to euclidean time (t → itE), we introduce the following periodicities :

x+_{→ z = x + it}

E, z ' z + 2πτ , (4.46)

x− → −¯z = −x + itE, z ' ¯¯ z + 2π¯τ .

where τ = i(β − Ω)/2π and ¯τ = −i(β − Ω)/2π, with β representing the temperature and Ω the angular potential. Since the metric is gauge dependent in higher spin theories, we need to impose a gauge invariant condition to assure smoothness on the horizon. We will require that any black hole solution has a trivial holonomy around the thermal cycles z ' z + 2πτ and z ' z + 2πτ. This is equivalent to impose that the holonomy:

Holτ,¯τ(A) = P exp

I τ Azdz + I ¯ τ A¯zd¯z , (4.47)

is equal to the identity of the gauge group, which we will denote by ei2πL0. The same must

hold for the barred version of (4.47). Using (4.26), the previous conditions are equivalent to:

τ λz+ ¯τ λ¯z = iL0, τ ¯λz+ ¯τ ¯λz¯= iL0, (4.48)

where λz and λz¯ are the eigenvalues of az and a¯z, and we assume L0 in its diagonal basis.

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For the non-rotating case (Ω = 0), the thermal cycle is just detemined for the period-icity around the time coordinate, tE → tE + β/2π. In this case, the holonomy condition

implies:

τ λt− ¯τ ¯λt = iL0, τ ¯λt+ ¯τ ¯λt= iL0. (4.49)

with periodicities reduced to τ = −¯τ = iβ/(2π).

The third requirement imposses that the solution is arising from a partition function, which allow us to give thermodynamical interpretation to the parameters of the connection. For simplicity, we will ellaborate this condition for solutions with SL(3, R)×SL(3, R) symmetry group. In the SL(3, R) × SL(3, R) case, we expect 4 global conserved charges, each one associated to a dierent Cartan generator:

QL0 = hL0i , QW0 = hW0i , QL¯0 = h ¯L0i , QW¯0 = h ¯W0i . (4.50)

We can write the partition function as:

Z(τ, ¯τ , α, ¯α) =Trei2πτ L0_e−i2π¯τ ¯L0_ei2παW0_e−i2π ¯α ¯W0_, (4.51)

where α, τ, ¯α, and ¯τ are the conjugated potentials to the charges generated by L0, W0,

¯

L0, and ¯W0, respectively. The conserved charges can be computed from the partition

function: QL0 = 1 2iπ ∂ log Z ∂τ , QW0 = 1 2iπ ∂ log Z ∂α , (4.52)

The previous formulas, and its barred versions, yield to: ∂QL0 ∂α = ∂QW0 ∂τ , ∂Q_L¯₀ ∂ ¯α = ∂Q_W¯₀ ∂τ , (4.53)

which is called the integrability condition. The integrability condition is required in a black hole solution, because we must be able to associate a partition function like (4.51) to it. Applying (4.53), we do not need go through the dicult task to construct the partition function, but we assure that the associated thermodynamical quantities will follow the rst law of thermodynamics.

In order to check that a solution follows (4.53), we need nd the global charges and the conjugated potentials in terms of the parameters of the connection2_{. The global charges}

can be found from the global symmetries of the solution using standard Chern-Simons techniques. The potentials τ and ¯τ are dened in terms of the parameters from the holon-omy condition (4.48). We will able to relate α and ¯α to the parameters of the solution that source the spin-3 particles, which we will call µ and ¯µ:

α = τ µ , α = ¯¯ τ ¯µ . (4.54)

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Consequently, we nd as well α and ¯α in function of the parameters of the solution. Following this procedure we will be able to check if the integrability condition (4.53) is accomplished.

Black hole solutions

Connections that accomplish the previous three conditions have been proposed in the literature. In the following we will briey review them. We will not prove that they accomplish all three previous conditions (for this analysis we refer the reader to [19]). The rst solution is the GK spin-3 black hole, proposed in [18]:

a = L+− 2πL_k L−− πW_2k W−2 dx+ +µ W2− 4πL_k W0+ 4π 2_L2 k2 W−2+ 4πW k L− dx−, ¯ a = −L−− 2π ¯_kLL+− π ¯_2kWW2 dx− +¯µW−2− 4π ¯_kLW0+4π 2_L¯2 k2 W−2+ 4π ¯W k L−1 dx+_, b = eL0ρ_.

The previous connections have to fullll the holonomy condition in (4.48). For simplicity, we will consider the non-rotating black hole:

L = ¯L , W = − ¯W µ = −¯µ , (4.55) Then, we impose the holonomy condition only around the time coordinate (4.49), which constraints the parameters µ, W, L, and τ. Solving the holonomy condition explicetely, we nd four branches of solutions. The only one compatible with the BTZ black hole limit gives the following relations:

W = 4(C − 1) C3/2 L r 2πL k , µ = 3√C 4(2C − 3) r k 2πL, τ = i(2C − 3) 4(C − 3) q 1 −_4C3 r k 2πL, (4.56) where C is a dimensionless constant which C > 3, and C → ∞ is the BTZ limit.

In [20], they propose an alternative to the connections which represent spin-3 black hole, which we will call generalized spin-3 black hole:

a = L+− 2πL_k L−−πW_2k W−2 dϕ + ξ L+− 2πL_k L−− πW_2k W−2 + µW2− 4πL_k W0+ 4π 2_L2 k2 W−2+ 4πW k L− i dt , ¯ a = −L−− 2π ¯_kLL+− π ¯_2kWW+2 dϕ +h ¯ξL−− 2π ¯_kLL+−π ¯_2kWW+2 + ¯ µ W−2− 4π ¯_kLW0+4π 2_L_¯2 k2 W+2+ 4π ¯W k L+ i dt , b = eL0ρ_,

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We will consider as well the non-rotating solution (4.55), with ξ = −¯ξ = 1. The parameters are as well restricted for the holonomy condition (4.49). Noticing that the time compo-nent of the connections (4.57) is equal to (4.55), we observe that the constraints for the parameter are as well (4.56).

In cite [19], another black hole solution is proposed: a =W2 4 + ω W−2 4 − qW0 dx++η₂W0dx−, ¯ a =W−2 4 + ¯ω W2 4 − ¯qW0 dx−+η₂¯W0dx+, b = eL02 ρ, (4.57)

where ω, q and η, are constants representing the mass, the charge and the chemical po-tential of the black hole respectively. It is often called higher spin black hole in diagonal embedding. This solution does not carry higher spin charges. It is an standart black hole coupled to a pair of U(1) elds (χ = −qdx+_{+ (η/2)dx}−_{, ¯χ = −qdx}−_{+ (η/2)dx}+₎

Using (4.43) it can be found the associated metrics of (4.55) and (4.57). Even though both solutions accomplish the three conditions to be higher spin black holes, their metric do not posses an horizon. Actually, both solutions are found to be wormholes. However, it can be shown that a complicated gauge transformation converts these solutions into black holes ([21], [19]).

We will use the connections presented in this section to compute results for the entangle-ment entropy in higher spin theories in next sections.

5 Entanglement Entropy and Composite Wilson lines

In [2], they propose following formula to compute the leading divergence of the entangle-ment entropy for CFT with higher spin gravity dual:

SEE = k σ1/2 log lim ρ0→∞ W_Rcomp(sf, si) ρ0=ρf=ρi , (5.1)

where (sf, si) are the endpoints of the CFT on the boundary. They are connected in the

bulk through a composite Wilson line Wcomp

R (sf, si)dened as: W_Rcomp(sf, si) = TrR " Pexp Z sf si ¯ A Pexp Z si sf A !# , (5.2)

with of A and ¯A representing higher spin backgrounds in SL(N, R) and P denoting the path ordering. The trace is taken in a nite-dimensional representation R which is dierent

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for every N. How to nd this representation will be specied later in this section. When there are semi-integer spins in the spectrum, we x σ1/2 = 2, and σ1/2 = 1 otherwise. The

Chern-Simons level is related to the central charge c of the dual CFT using the Brown and Henneaux formula c/6 = l/4G3, which considering (4.42) becomes:

k = c 12trf(L (P ) 0 L (P ) 0 ) (5.3)

In this section we will review the motivation and construction of this proposal. We will as well show some results for particular embeddings.

5.1 Composite Wilson lines and the geodesic

In this subsection we study the formula (5.1) for the case of pure gravity (A, ¯A ∈ SL(2,R)). First, we will motivate the proposal by analyzing that the symmetries of the composite Wilson line, and, then, we will show that it recovers Ryu-Takayanagui prescription. We start by performing the gauge transformation (4.17) in the connections A, and ¯A. A single Wilson line functional transforms as:

Pexp Z si sf A ! → L(si) Pexp Z si sf A ! L−1(sf). (5.4)

With (5.4), we observe that the composite Wilson line (5.2) is invariant to the subgroup of gauge transformations with R = L−1_{. For pure gravity, the entanglement entropy is}

computed holographically via a geodesic connecting the two endpoints of the CFT (see section 3.1). Consequently, if we want a functional to reproduce the entanglement entropy, it must be invariant to Lorentz transformations, which is a symmetry of the geodesic in AdS. From (B.11), we see the gauge transformations with R = L−1 _{correspond to pure}

Lorentz. Then, the composite Wilson line (5.2) has the same symmetries of a geodesic. On the contrary, we see that it is not invariant under a general gauge transformations of the connection. This seems to violate the required gauge invariance for the entanglement entropy of a CFT. However, only the gauge symmetries that leave invariant the asymptotic behavior of the gauge elds are true symmetries of the quantum eld theory. The leading term when ρ0 → ∞ of the composite Wilson line is invariant to these true symmetries by

denition. As a consequence, the proposal for entanglement entropy (5.2) is invariant to the gauge symmetries of the quantum eld theory. The above reasoning shows that the composite Wilson line shares symmetries whit the entanglement entropy, and, consequently is a reasonable proposal.

Now, we will show that composite Wilson line gives the length of the geodesic for SL(2, R), and recovers the Ryu-Takayangi prescription. Since the connections are locally at, we

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can express them as in equation (6.60) and the composite Wilson line (5.2) can be rewrit-ten:

W_Rcomp(sf, si) =TrRR−1(sf)R(si)L(si)L−1(sf) . (5.5)

If we plug the SL(2, R) connection (4.27) in the fundamental representation, and consider ρ0 = ρf = ρi and ∆x = xf− xi, we get the following expression for the "composite" Wilson

line: W_fcomp(sf, si) =2 cosh r 2πL k ∆x ! cosh   s 2πL k ∆x   (5.6) + k 2π √ LLe 2ρ0 ₊2π √ LL k e −2ρ0 ! sinh r 2πL k ∆x ! sinh   s 2πL k ∆x  . Considering the results in Section 4.1.3, we observe that the previous formula computes the length of a geodesic L(sf, si)in AdS for Poincare coordinates, global AdS and non-rotating

BTZ black hole. Explicitly:

W_fcomp(sf, si) = 2 cosh L(sf, si) . (5.7)

We want to nd the entanglement entropy from the composite Wilson line. From Ryu-Takayanagi, we interpret the radial slice where the CFT lives (ρi = ρf = ρ0) as a UV

cuto. We need to consider ρ0 → ∞, and the length of the geodesic L(sf, si)diverges. We

perform the following approximation: lim

ρ0→∞

W_fcomp(sf, si) → exp L(sf, si) . (5.8)

With the previous equation and Brown-Henneaux formula (3.25) we see that the proposal (5.1) when R reproduces Ryu-Takayanagi prescription for pure gravity in terms of Chern-Simons formalism. With this, we can compute the entanglement entropy for the important cases of the metric 4.23:

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SEE = ₆clog ∆x Poincare patch SEE = ₆clog _πL sin π∆x_L global AdS SEE = ₆clog k 2π2 √ LLsinh q 2πL k ∆x sinh q 2πL k ∆x

rotating BTZ black hole (5.9) where we have xed the cuto dierently in every case in order to normalize the entan-glement entropy as we did in the Section 3.2. We observe that the two rst formulas are standard CFT results recovered for the holographic prescription of Ryu-Takayanagi (see Section 3). The third result reproduces as well via Ryu-Takayanagui the entanglement entropy for a CFT at inverse temperature β dual to a non rotating BTZ black hole, if L = L = kπ/2β2_{. However, for a rotating BTZ (L 6= L), the result for the entanglement}

entropy cannot be calculated using Ruy-Takayanagi prescription because it is not an static background (the geodesic in the bulk is not necessary in a constant time slice). To compute this kind of cases, a covariant prescription is proposed [10]. Their result for the rotating BTZ is equal to (5.9). Since the functional (5.1) has a direct generalization for higher spin gravity, it is plausible that it will compute the entanglement entropy for higher spin gravity. We would like as well to compute the thermal entropy of a black hole using composite Wilson lines. According to Bekenstein-Hawking, the entropy of a (2+1)-dimensional black hole in pure gravity is proportional the geodesic around its horizon. Therefore, from (5.7) we see that the thermal entropy can be found in terms of Wilson lines. Specically, we have to compute a composite Wilson loop around the spatial coordinate (∆ϕ = 2π, ∆x = L), evaluated at the ρ which minimizes the functional, i.e. the outer-horizon of the BTZ eρ∗ _{= (π)}2_`/β

+β−. The result is the known thermal entropy of the BTZ black hole:

Sth= k cosh−1 1 2W comp f (∆x = 2π`) ρ0=ρ∗ = c 6 π β+ + π β− 2π` . (5.10) As opposite to the case for the entanglement entropy, it is not clear how to generalize the previous formula for the higher spin case. The horizon is not well dened for higher spin theories, and the minimization procedure above loses its meaning. Consequently, it is not possible to nd the thermal entropy for a higher spin black hole from a composite Wilson loop. We could have expected that the composite Wilson loop was not computing thermal entropy, because it is not invariant to gauge transformations of the type (5.4).

From an object which is the generalization of a geodesic in higher spin gravity, we would expect that it computes as well the thermal entropy for closed curves. Consequently, we consider that the composite Wilson (5.2) is not a good denition of geodesic in higher spin theories. However, we will see in the following subsections that it properly computes the

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leading divergence of the entanglement entropy. For example, we will see that for a xed representation of SL(N, R), the entanglement entropy form will recover the thermal result as expected when ∆x β. Moreover, we compute the entanglement entropy for specic backgrounds of SL(3, R), and we will nd non-trivial and satisfactory results.

5.2 Choosing a representation

In this section we will comment on the role of the representation R in (5.1). We have seen that for SL(2, R) an appropriate representation to compute the entanglement entropy is the fundamental. However this will not be the case for SL(N, R). In this section we will x the representation imposing physical requirements.

In subsection 3.2.3, we have seen that the entanglement entropy of a CFT2 in an interval

∆xat temperature β−1, should reproduce the thermal limit when ∆x β. If we want the formula (5.1) to compute the entanglement entropy holographically, it should recover the thermal limit as well. In [22], it is shown that the following formula computes the thermal entropy for a higher spin black hole:

sth = ktrf(λx− λx)L0 , (5.11)

where λx and λx are the eigenvalues of ax and ax, and L0 is the Cartan generator of

the sl(2, R) subgroup in the sl(N, R) algebra, which changes for every embedding. As a consequence, we require that formula (5.1) recovers the thermal limit:

SEE −−−−→

∆xβ ktrf(λx− λx)L0 ∆x . (5.12)

It turns out that the previous is accomplished for an specic choice of representation R of the composite Wilson line. In the following we will analyze in which representations the thermal limit is recovered. If we want to analize (5.5) in terms of the representation, we need to know how to generally characterize the connections. We will do this using representation theory (Appendix A). First we need to bring ax and ¯ax into the Cartan

subalgebra h. We call λx and ¯λx the Cartan conjugated of ax and ¯ax. The corresponding

elements in the dual space h∗ _are −→_λ x and

− →

λx. We can nd their diagonal elements (λx) (j) R

in a specic representation R with: (λx) (j) R = h − → λx, − → Λ(j)_Ri , (5.13)

where j = 1, ..., dim(R). The inner product in the weight space h∗ _{is h... , ...i, dened in}

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We could analogously characterize the eigenvalues of λx. With the denition (5.13), we

can rewrite the composite Wilson line (5.2) as: W_Rcomp(sf, si) = X j χj(ρ) eh −→ λx− −→ λx,Λ(j)_Ri∆x_, (5.14)

where the coecients χj(ρ) depend on the eigenvectors of λx, and on b(ρ). For the limit

∆x β, the term with biggest exponent in the sum (5.14) will be the dominant. In this limit, the entanglement entropy according to (5.1) is:

SEE −−−−→ ∆xβ k σ1/2 h−→λx− − → λx, − → Λmax_R i∆x , (5.15) where −→Λmax

R is the weight of the representation that maximizes the exponent in (5.14).

To do the identication in (3.45) , it will be useful to re-express (5.11) in terms of inner products. By denition, we can write the thermal entropy of a higher spin black hole (5.11) as: sth = k h − → λx− − → λx, − → l0i , (5.16)

where −→l0 is the dual element of L0. In the next subsections, we will choose

− →

Λmax

R to

properly reproduce (5.16). This will characterize the representation R in which we have to compute the composite Wilson line. We will perform the analysis when the higher spins charges in the background are disconnected, and the connections represent the BTZ black hole. In this case, the Cartan conjugated of ax and ¯ax of are proportional to L0, and the

dual elements are represented by:

−→ λx− −→ λx∼ − → l0 . (5.17)

If we choose the correct representation in the BTZ point, we can say that the same rep-resentation will still be appropriate when the higher spin charges are continuously turned on. In the following we will do the analysis rst for the principal embedding, and then for the rest of embeddings.

5.2.1 Principal embedding

First, we will nd the weight that maximizes the inner product in the principal embedding. In this case, −→l0 is a dominant weight. By denition of dominant weight,

− →

l0 has positive

and integer Dynkyn labels and follows:

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where −→αi, i = 1, ..., N − 1 are the simple roots of the algebra. We know as well that,

among all the weights of a representation, it exists a unique highest weight which has the biggest (positive) coecients in the basis of the simple roots. In other words, all the other weights are calculated substracting simple roots to the highest weight:

− → Λ(j)_R =−→ΛhwR − X i n(j)_i −→αi, (5.19) where n(j)

i are positive integers that can be found for every weight

− →

Λ(j)_R following a known algorithm [23]. From (7.42) and (7.43), we can write the following inequality:

h−→l0, − → Λhw_R i − h−→l0, − → Λ(j)_R i =X i n(j)_i h−→l0, −→αii ≥ 0 . (5.20)

From the previous equation, we see that the maximum value of the inner product h−→l0,

− →

Λ(j)_R i arises when −→Λ(j)_R = −→Λhw_R . Consequently, equation (5.15) in the principal embedding be-comes, close to the BTZ point:

SEE −−−−→ ∆xβ kh − → l0, − → Λhw_R i∆x , (5.21) where we have used that σ1/2 = 1, because in the principal embedding there are not

half-integer spin particles in the spectrum. For comparison with (5.16), we clearly recover the thermal limit when the chosen representation has highest weight −→Λhw

R =

− →

l0. We can

calculate which will be the dimension of this representation via the Weyl formula: dim(R) = Y

− →_{α >0}

h−→Λhw_R + −→ρ , −→α i

h−→ρ , −→α i , (5.22) where −→ρ is the Weyl vector and −→α are the positive roots. In Appendix B, we show that −

→

l0 for the principal embedding is equal to the Weyl vector. Knowing that the number of

positive roots of sl(N, R) is N(N −1)/2. Consequently, the dimension of the representation that properly calculates the entanglement entropy in (5.1):

dim(R) = 2N (N −1)

2 . (5.23)

For example, for N = 3 the suitable representation is eight dimensional, and corresponds to the adjoint. In Section 5.3, we will compute the entanglement entropy for dierent SL(3,R) connections using the adjoint representation in (5.1). We will see that we nd consistent and non-trivial results.