Entanglement Entropy and Black Holes in (2+1)-Dimensional Higher Spin Gravity Strong Sub-additivity of Higher Spin Entanglement Entropy

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University of Amsterdam

MSc Physics

Theoretical Phyiscs

Master Thesis

Entanglement Entropy and Black Holes in

(2+1)-Dimensional Higher Spin Gravity

Strong Sub-additivity of Higher Spin Entanglement Entropy

by

Jochem F. Knuttel

5975492

54 ECTS

September 2013-August 2014

Supervisor:

Dr. Alejandra Castro

Examiner:

Dr. Miranda Cheng

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Abstract

Higher spin theory in AdS3 can be constructed by the Chern-Simons formalism with a SL(3, R)

gauge group. As a generalization to the BTZ black hole, higher spin black holes have been con-structed, carrying a higher spin charge and a corresponding chemical potential. In this context, two black holes, that include the chemical potential to the connections in a different way, have been constructed in the literature. We will use higher spin entanglement entropy, which is defined by using a Wilson line, to check the validity of the discussed black holes. This is done by demanding that the entanglement entropy of the black holes behaves strongly sub-additive. We try to find an explanation for the fact that one of the black holes does not satisfy this requirement. In addition we aim to clarify the ambiguities that arise for the Wilson line entanglement entropy in higher spin theories.

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Acknowledgements

I am grateful to my supervisor Alejandra Castro, who guided me through this project and who was always available to answer my questions. I would also like to thank Miranda Cheng, who on very short terms agreed to be my second supervisor. Among my fellow students I owe special gratitude to Eva Llabr`es, for discussion on higher spin theories and to Alex Kieft, Gerben Oling and Manus Visser for discussion on more general subject in physics. Finally, I would like to thank them and my other fellow students for mental support and friendship during the process of this thesis.

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Introduction

The major question in theoretical physics is how to reconcile quantum theory with gravity. Since general relativity cannot be quantized, it is not possible to describe quantum gravity in terms of known theories and a complete new formalism is required. It is hoped that string theory can fill this hole in the landscape of physical theories. One of the situations where a quantum gravity comes into the picture is in the neighborhood of black holes. The spacetime curvature is expected to be strong enough to have effects on scales at which quantum theory becomes important. Moreover, there are still many unanswered questions about black holes, among which are the firewall paradox and the understanding of the inside of black holes. It is therefore necessary to study black holes in great detail.

A very interesting aspect of general relativity is that it admits black holes, objects to which thermodynamical laws can be assigned. Already some time ago it was shown that an entropy law can be written for black holes, now called the Bekenstein-Hawking formula [1,2],

SBH =

A 4GN

, (1)

where A is the area of the event horizon of a black hole and GNthe Newton constant. The

Bekenstein-Hawking formula was a precursor of the holographic principle. Holography states that the degrees of freedom in a (d+1) dimensional quantum gravity theory can be described by a many body quantum theory on a d-dimensional surface.

The most well-established holographic theory is the Anti de Sitter/Conformal Field Theory correspondence, AdS/CFT for short. AdS is particularly well fit for holography, since its boundary is at infinity and all the volume is ’concentrated’ on the boundary, such that area scales as volume. In the appropriate limit this correspondence is the example of a strong/weak duality: a strongly coupled field theory is related to a weakly coupled bulk theory, effectively resulting in classical gravity.

Among one of the most important developments in the AdS/CFT correspondence is the Ryu Takayanagi-formula of entanglement entropy [3],

SA=

Area(γA)

4GN

. (2)

In here γA is the length of the minimal surface area. This is a generalization to the

Bekenstein-Hawking entropy, since now any area in the bulk can be related to a quantum mechanical quantity. It is hoped that entanglement could explain us how classical spacetimes emerge from quantum theories. One way to understand this emergence was introduced in [4], where it was argued that geometry will change, when modifying the entanglement entropy. The fundamental reason for the emergence of geometry from quantum entanglement can be understood from quantum renormalization, where the renormalization flow of the microscopical quantum degrees of freedom result in the macroscopical quantity of geometry [5]. Then entanglement entropy could learn us how a possible quantum gravity theory would emerge to the classical gravity that we observe.

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CONTENTS 5

Gravity in three dimensions

A full string theory is expected to admit more than, the to us familiar, four dimensions. Since an increase in dimensions certainly does not simplify things, one could first try to conceptually understand theories with a lower amount of dimensions. Three dimensions is a good candidate because of its topological nature, which is a consequence of the lack of degrees of freedom. 3d gravity can be used as an interesting playground for investigating the AdS/CFT correspondence. Pioneers in the of AdS3/CFT2were Brown and Henneaux. They showed that the asymptotic symmetries of

AdS3are equivalent to the conformal group in two dimensions and related the central charge of the

CFT to the corresponding AdS radius in the following way [6],

c = 3` 2GN

, (3)

where c is the central charge of the CFT, ` is the radius of curvature in the bulk and GN is the

grav-itational constant. This Brown Henneaux-formula reveals that the already mentioned appropriate limit corresponds to small GN (or large `) and so large c, known as the semi-classical limit.1

There are two reasons that a topological theory as (2+1)-dimensional gravity is not trivial. The first reason is the existence of a boundary that gives rise to degrees of freedom that live on the boundary and the second is the existence of topologically different spacetimes, that arise from global identifications.

Higher spin theories

One of the useful consequences of the topological nature of 3d gravity is the possibility to write the AdS3 as a SL(2, R) × SL(2, R) Chern-Simons gauge theory [7]. An interesting extension of pure

AdS gravity is constructed by adding massless higher spin fields to the spectrum, which proves to be particularly easy in Chern-Simons formulation. It can be shown that instead of describing difficult nonlinear interactions of the higher spin fields, promoting the symmetry group of the Chern-Simons action to a SL(N, R) × SL(N, R), with N > 2, is sufficient to successfully describe a higher spin theory.

The other main advantage is that in three dimensions the tower of higher spin fields can be truncated by s 6 N [8].2 _{This truncation is a useful property that lacks in higher dimensional} analogs as the Vasiliev theory [9]. Similar to the Brown and Henneaux analysis, it can be shown that the boundary structure will obey a W-algebra for which the central charge is also given by (3). From this point of view, ordinary Einstein gravity in three dimensions is only a specific example of the wider group of gravity solutions whose asymptotic structure can by described by a conformal group.

Another reason for studying higher spin theories is that it is expected that massless higher spin fields will appear on tensionless strings.3 It is not very clear if there is a direct link between the three dimensional higher spin fields, but it is certainly a motivation to understand higher spin theories.

One of the major problems of higher spin theory in three dimensions is that gauge transformations of the higher spin field are able to change the metric. It is therefore not always clear if one can still use the concept of the metric, which normally fulfills a central role in a gravity theory. Consequently, the main challenge is to reconsider geometry and related notions in these theories. It is then hoped that the generalization to higher spin theories helps to improve our understanding of these concepts.

1_{’Classical’ because we are dealing with a small curvature or equivalently a very small G}

N∼ ~, so gravity can be

described by GR. ’Semi-’ because we are not putting ~ to zero, so quantum fluctuations are still allowed.

2_{Spin in here does not refer to the representation of SO(2), like in 3+1 dimensional gravity, but instead refers to}

the conformal symmetry at infinity. To be thorough, there is one more notion of spin used in the literature, which is the finite representation of the higher spin fields in SL(2, R). If we call the latter one s0, the relation between the two can be denoted as s = s0+ 1.

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CONTENTS 6

One of the natural steps would be to find a way to understand entanglement entropy in such theories. This reasoning led to two proposals in which the Ryu and Takayanagi formula was generalized with the help of Wilson lines [11,12].

Black holes in higher spin theories

The AdS/CFT correspondence does not only hold for pure AdS spacetimes. Demanding that a spacetime is only asymptotically AdS is sufficient to make sure that the correspondence is present. In three dimensions black holes are an example of such an asymptotic AdS space and are formed by global identification of global AdS, corresponding to CFTs with a finite temperature. The first black hole constructed in this way is the BTZ black hole [13]. Several proposals have been made to generalize the BTZ black holes in the context of higher spin theory [14,15,16]. However, it is not always clear how to define black holes that have an appropriate interpretation in the gravity side as well as in the CFT side. Ambiguities arise with the redefinition of thermodynamical properties, for example, there exist different ways to calculate the black hole entropy, since the lack of a gauge invariant horizon disqualifies the Bekenstein-Hawking formula as a definition for black hole entropy. Higher spin black holes are at the moment the best studied higher spin spacetimes and will be used to test the new entanglement entropy proposals. Not everything is yet clear about the higher spin entanglement entropy, so specific calculations could help us to get rid of the ambiguities. Recent results from the corresponding CFT found in [17] could help with this. The other way around, it is hoped that entanglement entropy can help us to judge the validity of the different higher spin black hole proposals, by examining if their entanglement entropy satisfies the right universal properties. Outline

This thesis has the following outline. In chapter 1 a brief introduction to AdS in three dimensions is given and translated into Chern-Simons language. By promoting the SL(2, R) symmetry group to SL(3, R) a higher spin theory is created. For the spacetimes that arise from both symmetry groups, a careful asymptotic analysis is given that strongly depends on the imposed boundary conditions, analogous to the derivation of Brown and Henneaux [6]. In chapter 2 black holes are described. First, as an introduction, the BTZ black hole is explained and its thermodynamical properties are analyzed. In the same way of reasoning, the different higher spin black holes are described. In chapter 3, a basic introduction to entanglement entropy is given. Then the derivation of entanglement entropies in CFTs is understood and it is showed how corresponding gravity solutions can be used to obtain the same answer. The holographic entanglement entropy is then generalized to higher spin theories and applied to some higher spin configurations.

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Chapter 1

Higher spin gravity in

(2+1)-dimensions

As mentioned in the introduction, we will focus on AdS gravity in 3d, which can be described in the Chern-Simons language. We have to be careful in the constructing of solutions of the Chern-Simons action, since we will have to impose boundary conditions on the connections, in order to construct a well-defined theory. The main goal of this chapter is to get familiar with the Chern-Simons formalism for 3d gravity and understand how (asymptotic) AdS solutions can be constructed in this context.

As a start, a brief description of ordinary 3d gravity and some relevant solutions of the Einstein-Hilbert will be given. This ordinary description of AdS3 will be translated into a Chern-Simons

language (the proof for this relation is given in Appendix A), which enables us to generalize to higher spin gravity. The boundary conditions on the connections are specified, such that well-defined solutions of the higher spin Chern-Simons action can be constructed. It will as well be discussed how the relation with the CFT can be understood. To conclude, the symmetries of two different vacua in higher spin spacetimes will be discussed.

1.1 Anti de Sitter gravity

The Einstein-Hilbert action for Anti-de Sitter space is given by SEH = 1 16πGN Z M d3x√−g(R − 2Λ) , (1.1)

where Λ is the cosmological constant. Here it will be taken to be Λ < 0. Physically such an action has no local degrees of freedom. This can easily be understood by counting the degrees of freedom. The Riemann tensor has in general ₁₂1n2(n − 1) independent components, and so in three dimensions there are six. Because the Einstein equations,

Rµν− gµν

1 2R − Λ

= 0 , (1.2)

are symmetric, there are exactly six independent equations, restricting al the local degrees of freedom of the Riemann tensor. However, the theory is not trivial for two reasons. The first reason is that there can be a difference in the topological structure of spacetimes, e.g. the difference of empty AdS and the BTZ black hole. The second reason comes from the fact that we are considering a spacetime with a boundary. More about this will be discussed later.

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1.1. ANTI DE SITTER GRAVITY 8

AdS can be constructed in the following way [18]. Write a four dimensional universal covering space R2,2 with symmetry group SO(2, 2),

ds2= dX₁2+ dX₂2− dT2 1 − dT

2

2 . (1.3)

Since AdS is characterized by a negative cosmological constant, Λ = −1

`2, the space should be

restricted to the submanifold

X₁2+ X₂2− T2 1 − T

2 2 = −`

2 _. _(1.4)

This constraint can be solved by considering the following coordinates, X1= l sinh ρ cos φ

X2= l sinh ρ sin φ

T1= l cosh ρ costl

T2= l cosh ρ sint_l ,

(1.5)

which lead to an induced metric, given by

ds2= −l2cosh2ρdt2+ l2sinh2ρdφ2+ l2dρ2 , (1.6) which is the global metric of AdS3. By making the transformation r = l sinh ρ, this metric can be

rewritten as ds2= (1 +r 2 `2)dt 2 + dr 2 1 + r_`22 + r2dφ2 . (1.7)

This metric is then a solution of the Einstein equations (1.2), which could be checked by explicitly constructing the Ricci tensor and scalar. The spacetime described by global AdS will be considered as the vacuum of the theory. Since AdS3 is a maximally symmetric space it has six symmetries:

three translation, one rotation and two boosts. One of the ways excitations can be introduced in the theory is by making global identifications, i.e. taking a quotient of the space. One then reduces the amount of symmetries of the solution. The vacuum is thus simply the theory with the most symmetries. The best known example of such a global identification is the BTZ black hole, which shall be discussed in the next chapter.

By writing the coordinates of the covering space in a two by two matrix, it becomes apparent that global AdS can also be described by the group SL(2, R) × SL(2, R)/Z2,

X = 1 ` X1+ T1 X2+ T2 X2− T2 −X1+ T1 . (1.8)

Equation (1.4_{) is then obtained by det(X) = 1, such that X ∈ SL(2, R). It can be shown that the} Killing metric

ds2= 1 2Tr(X

−1_dXX−1_{dX) ,} _(1.9)

will give back the coordinates of the covering space. It is invariant under left and right multiplication of SL(2, R) matrices: ΛlXΛr. The Z2, comes from the fact that (Λl, Λr) has the same effect as

(−Λl, −Λr). This is the most instructive explanation that the symmetry AdS3can be described by

two SL(2, R), i.e. a left and a right symmetry group. However, a more convenient description of AdS3 is however given in the Chern-Simons language. We will shortly denote some AdS3 metrics,

that will be useful in relating the Chern-Simons and metric formalism. Fefferman graham coordinates

The global metric (1.7) can be written in a Fefferman Graham form [19]. Take ` = 1, and consider the coordinate transformation

r = 1 z−

1

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1.2. CHERN-SIMONS FORMULATION OF 2+1 GRAVITY 9

The metric (1.7) is then written as

ds2= 1 z2 " dz2− 1 + 1 4z 2 2 dt2+ 1 −1 4z 2 2 dφ2 # . (1.11)

Subsequently the coordinate transformation z = e−ρ_{leads to}

ds2= dρ2− (eρ₊1 4e −ρ₎2_dt2_{+ (e}ρ₋1 4e −ρ₎2_dφ2 _. _(1.12) Poincare patch

The hyperboloid (1.4) can also be parameterized differently with a resulting metric, Xd = xz T1 = _zt Xd+ T2 = −1_z Xd− T2 = −t 2_+z2_+x2 z2 ,

where d = 1, 2, such that Xd describes the spacelike coordinates of the covering space. This choice

of coordinates results in the metric: ds2= `

2

z2 dz

2_{− dt}2_{+ dx}2

. (1.13)

It is clear that the boundary is planar. The cost is that these coordinates cover only a part of the manifold: the Poincaré metric describes the near boundary section of the spacetime. This can be seen by replacing z = `2/r, the Poincaré metric corresponds to 1.7 in the r >> 1 limit. For the purpose of Chern-Simons formulation, we again replace z = e−ρ and obtain the Poincaré metric

ds2= e2ρ(dx2− dt2_{) + dρ}2 _. _(1.14)

1.2 Chern-Simons formulation of 2+1 gravity

It was shown in [7,20] that the action of 2+1 gravity can be related to a Chern-Simons theory with SO(2, 2) ∼ SL(2, R) × SL(2, R), see [18] for an introduction. The Chern-Simons action is given by

SCS[A] = k 4π Z M Tr(A ∧ dA + 2 3A ∧ A ∧ A) , (1.15)

where the integral is taken over a 3-manifold, the trace denotes a metric on the Lie algebra, i.e. δab = Tr(TaTb) with Ta the generators of the gauge group G, k is the level of the Chern-Simons

theory and Aµ is a Lie-algebra valued one-form. The equations of motion of the action are given by

dA + A ∧ A = 0, (1.16)

which reflects the fact that the curvature has to be constant. In order to relate the Chern-Simons action to SO(2,2) gravity, Aµ is written in terms of the vielbein and the spin connection

Aµ= eµaPa+ ωµaJa , (1.17)

where Pa, Jaare the generators that respectively denote translations and rotations of SO(2,2).

Defin-ing an infinitesimal gauge transformation Λ = ρa_P

a+ τaJa, A will transform covariantly as

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1.3. HIGHER SPIN GRAVITY 10

For the purpose of introducing higher spin theories, it will prove to be convenient to split the SO(2,2) group into two SL(2, R) groups. It is easy to show that when writing

J_a± =1

2(Ja± lPa) , (1.19)

both J+

a and Ja−will respect individually an sl(2, R) algebra and [Ja+, Ja−] = 0. The new connections

are then given by

A = (ω_µa+1 `e a µ)J + adx µ _, _{A = (ω}_¯ a µ − 1 `e a µ )Ja−dx µ _, _(1.20)

such that A = A + ¯A. The relation of the Chern-Simons action of these two gauge group with the Einstein Hilbert action is then given by (see Appendix A for a proof of this equality),

SEH = SCS[A] − SCS[ ¯A] . (1.21)

The equations of motion are then again given by

dA + A ∧ A = 0 , d ¯A + ¯A ∧ ¯A = 0. (1.22) This identification can be shown explicitly by inserting (1.20) together with the relation,

k = ` 4GN

, (1.23)

where GN is the 3d gravitational constant. The metric and the connections are related in the

following way gµν = 1 2tr(eµeν) = 1 2tr(A − A)µ(A − A)ν , (1.24) where the last equality can be derived using (1.20).

1.3 Higher spin gravity

It is shown in [21_{] that when promoting the SL(2, R) to a SL(N, R) gauge group, a theory arises}

that describes gravity coupled to a finite tower of higher spin fields. The simplest of these theories is given by promoting SL(2, R) to SL(3, R), whose explicit algebras can be found in Appendix B. The actual properties of the additional fields, are determined by how sl(2, R) associated to pure gravity is embedded into sl(3, R). In sl(3, R) there are two different possible embeddings, which cannot be related to each other by conjugation and represent two inequivalent extensions of pure gravity. A spin-3 field in a constant curvature background can be described using the higher spin equivalent of the dreibein and spin-connection: eµab, ωµab [21]. Similar to what is done in (1.20), we can write a

linear combinations of these tensors, t_µab= ω_µab+1 `e ab µ , ¯t ab µ = ω ab µ − 1 `e ab µ , (1.25)

such that the connections can be written as [14]

A = ω_µa+1 è a µ Ja+ ω_µab+1 è ab µ Tab dxµ ¯ A = ω_µa−1 è a µ Ja+ ω_µab−1 è ab µ Tab dxµ , (1.26)

where Ja represent the generators of the SL(2, R) embedding and Tab the other higher spin

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1.4. ASYMPTOTIC SYMMETRIES AND CHARGES 11

Einstein gravity that interacts nonlinearly with higher spin fields.1 In order to simplify this higher spin theory, another basis for the generators is chosen. A property of higher spin theories that makes them more complicated is that higher spin gauge transformations act nontrivially on the metric, resulting in the fact that the of geometry becomes gauge invariant.

The conventions on the SL(3, R) group and this explicit change of basis is written in Appendix B. We will now discuss the two different embeddings.

Principle embedding

To construct an embedding we simply have to pick three generators from the set of sl(3, R) generators that respect the sl(2, R) algebra, which in the case of the principal embedding will be L1, L0, L−1.

How the other part of the sl(3, R) algebra decomposes is easiest understood by looking at the adjoint representation:

adj_N ∼= 32⊕ 52 , (1.27)

which means that there are three generators in the adjoint representation that can be decomposed in a 3 × 3 and a 5 × 5 block structure, both satisfying the sl(2, R) algebra. From the perspective of the bulk, this corresponds to the metric field and a spin-3 field:

gµν =

1

2trf(eµeν), φµνρ= 1

3!trf(e(µeνeρ)) . (1.28) We will refer to these fields as the spectrum of the embedding.

Diagonal embedding

The diagonal embedding is found by choosing W−2, L0, W2 as a sub-algebra. The name of the

embedding comes from the fact that the adjoint representation decomposes into a block-diagonal structure:

adjN ∼= 32⊕ 2 · 22⊕ 12 , (1.29)

reflecting a spectrum consisting of the graviton, one spin-1 and two spin-3/2 fields. It turns out that this embedding includes negative norm states and will therefore not be possible to quantize [22].

1.4 Asymptotic symmetries and charges

In this section, we impose boundary conditions on the connections and explain how asymptotic symmetries and surface charges are defined, following [21,23,24]. Given a set of boundary conditions, the asymptotic symmetries are defined as the set of transformations that respect these boundary conditions. One can define surface charges, or simply charges, associated to these symmetries, that generate global transformations relating physically different states to each other. These charges will become a symmetry algebra under quantization, which will be discussed in section1.5.

Alternatively, one could understand this in the following way. The boundary conditions break some of the gauge symmetries of the theory. These would-be gauge degrees of freedom now appear as real degrees of freedom at the boundary, which can be translated into an asymptotic symmetry algebra. Gauge transformations that asymptotically lead to an element of this algebra will change the physical state of the system and are therefore called improper. The remaining gauge symmetries can now be interpreted as real gauge symmetries of the AdS3spacetime with a boundary. For that

reason they will be called proper and they do not change the physical state.

1_{This is actually only true for the principal embedding, since in the case of the diagonal embedding, AdS}

3Einstein

gravity couples to lower spin fields. Both extensions of pure gravity are nevertheless referred to as higher spin gravity. Conversely we will call solutions of the the Einstein-Hilbert action without these higher spin fields pure gravity.

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The procedure of specifying boundary conditions that lead to a certain asymptotic structure of the spacetime, was first carried out by Brown and Henneaux [6]. We will shortly explain how this can be done in a Chern-Simons formulation, focussing first on SL(2, R). The connections have to be chosen in such a way that at infinity we have,

At= Aφ , A¯t= − ¯Aφ , (1.30)

or in light-cone coordinates x± = t_l ± φ, A− = 0 and ¯A+ = 0. A Chern-Simons action with

these restrictions is already sufficient to find asymptotic symmetries that can be identified with two SL(2, R) copies. Now, in addition imposing that the connections approach AdS3at the boundary,

(A − AAdS)|boundary= O(1) , ( ¯A − ¯AAdS)|boundary= O(1) , (1.31)

leads to asymptotic boundary conditions that can be identified with two Virasoro algebras with central charge c = _2G3`

N. It should be emphasized that this choice of boundary conditions are not

unique and determine the physical properties of the theory. Asymptotic charges

Now we would like to understand how the asymptotic charges are defined, which are the quantities that encode the physical degrees of freedom of the spacetime. Consider a Chern-Simons theory on a three dimensional manifold with a boundary, namely M = R × Σ, where R represents the time component and Σ is a spacelike two dimensional manifold with a boundary ∂Σ. We can split the gauge fields into a time and a space component,

A = Atdt + Aidxi , (1.32)

where xiare the variables defined on Σ. This leads to an action, which can be written in a canonical form as2 SCS[A] = k 4π Z R×Σ Tr(AtFij− ijAiA˙j)dtdx2 , (1.33)

where denotes the anti-symmetric Levi-Civita symbol, but now only for the two dimensional space, with convention 12_{= 1. This form of the action reveals that A}

i, ˙Aj are canonically conjugate and

At the Lagrange multipliers. If the gauge group has dimension n, then there are 2n different fields

Ai (there are two spatial coordinates) and n Lagrange multipliers. With this canonical form of the

action, Poisson brackets for two functionals depending on the fields Ai can be written as

{F, H} = 2 π Z Σ dx2ijTr _δF δAi δG δAj . (1.34)

The Lagrange multipliers At constrain the generators of the internal gauge symmetries, which will

be called G:

G = k 4π

ij_F

ij . (1.35)

The smeared generator of a gauge transformation Λ is then defined as (’smeared’ because it is integrated over a timeslice),

G(Λ) = k 4π Z Σ eijFijΛdx2 , (1.36) such that δAi= {G(Λ), Ai} = ∂iΛ + [Ai, Λ] . (1.37)

2_{At least, up to a boundary term that needs to be introduced to cancel terms that arise after varying the action,}

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Hence G generates the gauge transformations on the connections. However, when Σ has a boundary, an additional term has to be added to (1.36): G0(Λ) = G(Λ) + Q(Λ). The extra term ensures that the variation of G is well-defined, i.e does not give any surface terms, leading to

Q(Λ) = − k 2π

Z

δΣ

dxiTr(ΛAi) . (1.38)

We shall refer to Q(Λ) as the asymptotic charge. Note that G(Λ) will vanish on-shell by (1.35), but Q(Λ) will not. The transformations that are generated by G(Λ) such that Q(Λ) vanishes, are now the actual gauge transformation of the theory. The others for which Q(Λ) does not vanish are not true gauge transformation anymore but become global degrees of freedom that transform physical different states (i.e. states defined on the boundary) into each other. These gauge transformations are generated in the following way,

δAi= {Q(Λ), Ai} = ∂iΛ + [Ai, Λ] . (1.39)

In this formulation, it is very clear that the boundary causes some of the gauge freedoms to become actual global degrees of freedom (located at the boundary, but since there are no degrees of freedom in the bulk, they are the only ones). It needs to be emphasized how important the boundary is for the theory: if no boundary would be present, all the different configurations of the spacetime would be connected to each other with a proper gauge transformation, modulo the different holonomies,3_{and all the solutions would be equivalent to A = 0, by a gauge transformation} defined in (1.37). In the presence of a boundary, one could still relate any connection algebraically to A = 0, however, they would not represent the same physical configuration. Both configurations will be solutions to the equations of motion, but there will be one or more Q(Λ)s that will take a different value in each state.

Constructing a connection

It was shown in [21] that using the gauge freedom, the radial dependence can be fixed and the connections can always be written as,

A(φ, t, ρ) = b(ρ)−1a(φ, t)b(ρ) + b(ρ)−1db(ρ) , ¯

A(φ, t, ρ) = b(ρ)−1a(φ, t)b(ρ) + b(ρ)¯ −1db(ρ) ,

b(ρ) = eρL0 _, _(1.40)

where ρ is the radial component of the torus. We will refer to this gauge as the radial gauge and in the majority of this thesis we will use a, ¯a instead of A, ¯A to specify the connections.

The most general solution of the Einstein equation that is asymptotically AdS is given by [25]

ds2= `2 dρ2+L(x +₎ k (dx +₎2₊L(x¯ −) k (dx −₎2₋ e2ρ+L(x +_)L(x−₎ k e −2ρ _(1.41)

It is not hard to find connections that satisfy (1.40), and yield this metric through (1.24), a(x+) = L1− L(x+₎ k L−1 dx+ (1.42) ¯ a(x−) = L−1− ¯ L(x−₎ k L−1 dx− . (1.43)

3_{We will be using the notion of holonomy frequently throughout this thesis, so let us define here what holonomy is.}

The holonomy is defined as HolC(A) = P exp(

H

CA), and similar for the anti-holomorphic part. It is a path ordered

element of the gauge group. If the holonomy is non-trivial, one cannot find a single valued, globally defined g such that A = g−1dg. Intuitively the holonomy indicates whether the gauge field along the path C is wrapped around something non-trivial as a black hole.

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Figure 1.1: Empty AdS, a loop in the φ direction can be shrunken to a point.

These connections of course satisfy our chosen boundary conditions (1.30). Global AdS can be con-structed by demanding trivial holonomy around the φ-cycle (see figure1.1for an intuitive picture). This holonomy condition results in L = ¯L = −1

4, such that (1.42) corresponds to the metric (1.12).

Generalizing to SL(3, R)

We would like to generalize the boundary conditions to higher spin spacetimes. It was proposed in [23_{] that the boundary conditions have to be defined in a similar way as for the SL(2, R):}

A−= 0 , (A − AAdS)|boundary= O(1) ,

¯

A+= 0 , ( ¯A − ¯AAdS)|boundary= O(1) . (1.44)

and that the connections should be able to be written as (1.40). Instead of the two Virasoro algebras, for the principle embedding, one can identify two W3algebras at the boundary, again with a central

charge c = _2G3`

N, as will be shown in the next section. In the case of the diagonal embedding, two

W₃(2)_{algebras will arise. Consider the following form of a SL(3, R) connection respecting (}1.40) and A−= 0 , a(x+) =   1 X i=−1 li(x+)Li+ 2 X j=−2 wj(x+)Wj  dx +_. _(1.45)

In order for the condition (1.44) to be satisfied, the terms that interfere with the asymptotic behavior are demanded to be zero:

l1= w1= w2= 0 . (1.46)

Interfering terms consist of generators that through (1.40) give higher order fall off terms in ρ than L1. Some of the terms can be chosen to be zero by gauge freedom [21]:

l0= w0= w−1 = 0. (1.47)

by which we are not loosing any information, it is merely a choice of gauge, which we will refer to as the highest weight gauge (L−1, W−2 are the highest weight generators). This leaves us with the

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1.5. RELATION WITH CFT 15

following connections, (a similar analysis leads to the anti-holomorphic part), a(x+) = L1− L k(x +_)L −1− W 4k(x +_)W −2 dx+ a(x−) = − L−1− ¯ L k(x −_)L 1+ ¯ W(x+₎ 4k W2 dx− . (1.48)

1.5 Relation with CFT

In this section we will show how the chosen boundary conditions lead to a CFT. The AdS/CFT correspondence can be stated in the most simple way by the equivalence of the partition functions on both sides,

ZAdS= ZCF T . (1.49)

For example, the black hole entropy can be derived when this identity is evaluated in the high energy regime [26]. One of the advantages of three dimensional AdS is that the topological nature of the theory ensures that the only degrees of freedom live at the boundary. It is therefore very clear how to establish the holographic correspondence: the asymptotic symmetry algebra is defined by the gauge transformations that preserve these boundary conditions. This analysis is simplified by exploiting the possibility of writing AdS3 in Chern-Simons formalism. By now it is clear how to pick a gauge

and boundary conditions in order to obtain well-defined AdS spacetimes.

As mentioned before, in pure gravity, the asymptotic structure leads to two copies of the Virasoro algebra [6], [Lm, Ln] = (m − n)Lm+n+ c 12(m 3_{− m)δ} m+n,0 , (1.50)

where c is the central charge, that can be related to the level of the Chern-Simons theory c = 6k =

3l

2GN. We will carry out such an analysis for a connection of the principal embedding including a

higher spin charge, finding that the asymptotic symmetries can be identified with a W3-algebra [21].

Consider the most general gauge transformation

λ = 1 X i=−1 iLi+ 2 X j=−2 χjWj , (1.51)

where we have already taken out the ρ dependence, i.e. Λ = e−ρL0_λeρL0_{, such that gauge}

trans-forming A with Λ is the same as transtrans-forming a with λ. We can then consider the action of such a gauge transformation on the connection

δa = ∂λ + [a, λ] , (1.52)

which will give us 8 differential equations for the gauge parameters, i.e. one for each generator. For a we will choose (1.48) and we will demand that after the gauge transformation the connection is still in the highest weight gauge. This means that the pre-factor of L1is equal to one and the pre-factors of

L0, W2, W1, W0, W−1will remain zero. This gives us 6 equations where {0, −1, χ1, χ0, χ−1, χ−2}

depend on 1_{, χ}2_{, L, W and their derivatives:}

0 _{= −}0 −1 = 1₂00− kL + 2χ k W χ1 _{= −χ}0 χ0 ₌ 1 2χ 00₋2χ k L χ−1 = −1₆χ000+_3k5χ0L + 2 3kχL 0 χ−2 = ₂₄1χ0000− 2 3kχ 00_{L −} 7 12kχ 0_L0₋ 1 6kχL 00₊χL2 k2 − 1 4kW , (1.53)

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1.6. SYMMETRIES OF THE VACUUM 16

which we will refer to as the 6 auxiliary equations. To understand how L, W transform under the gauge transformations, we will use the six auxiliary equations to write δL, δW in terms of 1, χ2, L, W and their derivatives. We will denote 1 _{= , χ}2 _{= χ and the derivative with respect to x}

+ as a

prime.4 _{Now the transformations of L and W are given by}

δL = L0+ 20L + k 4π 000_{+ 2χW}0_{+ 3χ}0_W δW = W0+ 30W −1 3 2χL000+ 9χ0L00+ 15χ00L0+ 10χ000L + k 4πχ (5)₊64π k (χLL 0_{+ χ}0_L2₎ . (1.54) The charges that generate these transformations, c.f. (1.38) are then given by

Q(λ) = Z

dx+ (x+)L(x+) + χ(x+)W(x+)

. (1.55)

The poisson brackets can be written down by using (1.39) and the two following steps: translate L, W into Fourier modes,

L(x+_{) = −} 1 2π X p Lpe−ipx + W(x+_{) = −} 1 2π X p Wpe−ipx + (1.56)

and shift the vacuum energy,

Lp→ Lp−

k

4δp,0 . (1.57)

A W3 algebra can then be identified (see Appendix B, where the classical W3 algebra is denoted,

i.e. without the terms including the central charge) [21],

i{Lp, Lq} = (p − q)Lp+q+ c 12(p 3_{− p)δ} p+q,0 i{Lp, Wq} = (2p − q)Wp+q i{Wp, Wq} = 1 3 h (p − q)(2p2+ 2q2− pq − 8)Lp+q+ 96 c (p − q)Λp+q + c 12p(p 2 − 1)(p2− 4)δp+q,0 i , (1.58)

where Λp ≡P_q∈ZLp+qL−q. The central charge is again related to the level of the Chern-Simons

theory as

c = 6k = 3l

2G , (1.59)

where in the last step (1.23) is used and the Brown-Henneaux formula is recovered [21]. W can be identified as a weight (3,0) operator and L as the stress energy tensor with weight (2,0).5 _{Note that} the first line of (1.58) represents a Virasoro algebra. Hence, after turning off the W’s, one obtains the Virasoro algebra, as expected. For the similar procedure of the diagonal embedding resulting in a asymptotic symmetry algebra W₃(2), we refer to [28].

1.6 Symmetries of the vacuum

Before we will consider the more interesting black hole spacetimes, the vacuum of the theory will be studied, since it is supposed to be the most basic spacetime to work with. The vacuum will therefore

4_{Compared to [}₂₁_{], we will take L}

here= −2πLthere, Where= 2πWthere, which is in line with [27]. 5_{Which can be seen by performing an operator product expansion for both W, L.}

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serve as a pedagogical introduction to higher spin theories. We will as well try to find the proper gauge transformations that are the ‘generalized’ Lorentz transformations, or global symmetries. With generalized we mean the following: by extending to higher spin gravity and promoting the symmetry group to SL(3, R) we increase the amount of symmetries, which are now a combination of the pure gravity Lorentz transformations and the higher spin symmetries. For the vacuum, L and W are constant, so (1.54) reduces to

δL = 20L + k 4π 000_{+ 3χ}0_W δW = 30W −1 3 10χ000L + k 4πχ (5)₊64π k χ 0_L2 . (1.60)

We can set these variations of L, W to zero, in order to find the global symmetries. It turns out that when using the connections of the black hole by [15], we find a second vacuum in addition to the one described in the previous section. We will refer to those two vacua as vacuum A and B.

1.6.1 Vacuum A

We have seen that the first vacuum can be constructed with the following connections a = (L1+ 1 4L−1)dx + ¯ a = −(L−1+ 1 4L1)dx − _(1.61)

We will call the space that is described by these connections vacuum A. For these connections the equations in (1.60) reduce to6

000+ 0= 0

χ(5)+ 5χ000+ 4χ0= 0 . (1.62)

Gauge transformations that represent the generalized Lorentz transformation have to satisfy both equations at the same time. The first equation gives three linearly independent solutions, = c1einx, n = (−1, 0, 1), χ = 0 corresponding to the SL(2, R) symmetries. If we allow the constants to

be complex, there will be 5 independent solutions for the second equation, = 0, χ = c2eimx, m =

(−2, −1, 0, 1, 2). We could then use these values to find 8 linearly independent λ’s, explicitly, λ1= e−ix+L1+ ie−ix+L0− 1/4e−ix+L−1

λ2= L1+ 1/4L−1

λ3= eix+L1− ieix+L0− 1/4eix+L−1

λ4= e−2ix+W2+ 2ie−2ix+W1−

3 2e −2ix+_W 0− 1 2ie −2ix+_W −1+ 1 16e −2ix+_W −2 λ5= e−ix+W2+ ie−ix+W1+ 1 4ie −ix+_W −1− 1 16e −ix+_W −2 λ6= W2+ 1 2W0+ 1 16W−2 λ7= eix+W2− ieix+W1− 1 4ie ix+_W −1− 1 16e ix+_W −2

λ8= e2ix+W2− 2ie2ix+W1−

3 2e 2ix+_W 0+ 1 2ie 2ix+_W −1+ 1 16e 2ix+_W −2 . (1.63)

6_{Since the formulas will be quite complicated, we need to simplify where possible. For the purpose of this section}

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In conclusion, there are 8 linearly independent gauge transformation that leave the connections invariant, which is expected for a vacuum of AdS3 with SL(3, R) symmetry. There should be three

λs that satisfy a SL(2, R) symmetry. In the case of Vacuum A, it is easy to see that {λ1, λ2, λ3}

satisfy SL(2, R) and thus describe the Lorentz transformations of pure gravity, because they only consist of generators of the SL(2, R) embedding.

1.6.2 Vacuum B

Vacuum A is not the most interesting solution of SL(3, R) gravity, since no higher spin charges can be involved.7 Another vacuum can be defined, by using the connections of a black hole with a higher spin charge found in [15]. The most general form for such a connection would be,

a =(L1− LL−1− WW−2)dx+

µ(W2+ w1W1+ w0W0+ w−1W−1+ w−2W−2+ l−1L−1)dx− , (1.64)

where w1, w0, w−1, w−2, l−1 are constants that can be specified by the equations of motion Fx+_x−:

a =(L1− 1 kLL−1− 1 4kWW−2)dx + + µ(W2− LW0+ 2 3∂+LW−1+ L2₋1 6∂+L W−2+ 2WL−1)dx− . (1.65)

We can then follow the same procedure as with Vacuum A to check if we indeed find eight gauge transformations again. Now there is an additional equation that the gauge transformation should satisfy (there were two before but the one with x− was trivial.) Hence, δAx+ = D_x+Λ = ∂_x+λ +

[Ax+, Λ] will give the same set of differential equations, resulting in (1.60). In contrast, δA_x− =

Dx−Λ = ∂_x−λ + [A_x−, Λ] will give us the following extra restrictions [27,29]

∂−χ = 2µ∂+ ∂− = − 2µ 3 ∂ 3 +χ + 32µL∂+χ . (1.66)

Using (1.65) with W, L, µ constant, we can find the trivial holonomy restrictions for W, L, µ, 32 27L 2_{µ 9 − 16Lµ}2_{+ 16µ}3 W2+ 16Lµ2W + W = 0 L + 3Wµ +16L 2_µ2 3 = −1/4 . (1.67)

To solve these equations, it is convenient to introduce the constant C and write (analogous to what is done in [15] for a black hole),

L = − (2C − 3) 2 16 1 −_4C3 (C − 3)2 µ = 3 √ C 4(2C − 3)√L W = −64L 2_µ2_{− 12L − 3} 36µ . (1.68)

7_{By demanding trivial holonomy, W will always disappear. This was the initial reason for the authors of [}₁₅_{] to}

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We can use (1.60), (1.66) and (1.68) to find solutions for the gauge parameters that leave the connections of vacuum B invariant, similar to what is done in [27]. We will start solving (1.66), leading to a solution,

(x+, x−) = αeiαx−+iβx+ χ(x+, x−) = 2µβeiαx−+iβx+ . (1.69) By imposing periodicity around the φ circle, we find that α = β − m, with m integer. The second equation of (1.66) gives us,

(β − m)2= 4 3µ

2_β2_(β2_{+ 16L) .} _(1.70)

Then (1.60) restricts these solutions to

12βWµ = (β − m) 4L + β2

9W(β − m) = β(β2+ 4L) β2+ 16L µ . (1.71) These last three equations can be solved, resulting in a solution where β is a function of c. It turns out, in agreement with [27], that there only exist solutions for m = ±1, ±2, which are given by,

m = 1 → β = (2C − 3) 4C ± 3 √ 4C − 3 + 3 8C2_{− 30C + 18} m = −1 → β = −(2C − 3) 4C ± 3 √ 4C − 3 − 3 8C2_{− 30C + 18} m = 2 → β = 2C − 3 C − 3 m = −2 → β = −2C − 3 C − 3 . (1.72)

In addition there are two solutions given by (, χ) = (1, 0), (0, 1). For the first six solutions, the general gauge transformation can then be written as,

λi=(β − m)eγL1− iβ(β − m)eγL0+

4βµW −1 2β 2_{(β − m) − L(β − m)} eγL−1

+ 2βµeγW2− 2iβ2µeγW1− β3µ + 4βLµ eγW0+

1 3iβ 4_{µ +}10 3 iβ 2_Lµ eγW−1 + 1 12β 5_{µ +}4 3β 3_{Lµ + 2βL}2_{µ −}1 4W(β − m) eγW−2 , (1.73)

where i = {1, 6}, γ = i(x−(β − m) + βx+) and L, W, µ satisfy (1.68). The other two with constant

pre-factors are given by:

λ7= L1− LL−1−

W 4 W−2

λ8= 2WL−1+ W2− 2LW0+ L2W−2 . (1.74)

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Chapter 2

Black holes in (2+1) dimensional

higher spin gravity

One of the reasons that 3d higher spin gravity is interesting is because of the existence of black holes. Because of the lack of degrees of freedom of 3d gravity, it was a surprise that a black hole in three dimensions was found [13], now known as the BTZ black hole. The authors of a more recent paper [15] proposed to generalize the BTZ black hole in the context of higher spin gravity (see also [30]). A similar proposal was given in [16], with the difference that the chemical potential assigned to the higher spin charge was introduced in a different way. The chief aim of this chapter is to construct and understand these black holes and their thermodynamics.

The outline of the chapter is the following. First the construction of the BTZ black hole is discussed in metric as well as Chern-Simons formulation. Subsequently the thermodynamics that can be assigned to the BTZ black hole will be explained. These same steps are then generalized to the two different higher spin black holes, in which we are primarily interested. The different ways of calculating the thermal entropy of higher spin black holes will be discussed in detail. Finally, we will discuss black holes that can be constructed in the diagonal embedding.

2.1 BTZ black hole

Here we will show how the BTZ black hole is constructed. For reviews on the BTZ black hole, see the work of Carlip [18, 23, 31, 32] and Ba˜nados [24, 25]. There are some differences with e.g. the (3+1) dimensional Schwarzschild black hole: from the equations of motion it is not allowed to have a curvature singularity at its origin. Moreover the BTZ black hole has an asymptotic anti-de Sitter spacetime instead of flat spacetime, which makes it an interesting configuration for studying the AdS/CFT correspondence. On the other hand there are sufficient similarities with higher dimensional black holes to call it a black hole. It has a horizon (2 in the rotating case) and there are thermodynamical quantities as temperature and entropy that can be assigned to it. The BTZ solution to the three dimensional Einstein Hilbert action (1.1) with a negative cosmological constant was given by [13]

ds2= −N2dt2+ N−2dr2+ r2(Nφdt + dφ)2 , (2.1) where, N2(r) = −8GM +r 2 l2 + 16G2_J2 r2 , N φ_{(r) = −}4GJ r2 . (2.2) 20

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2.1. BTZ BLACK HOLE 21

Here J can be interpreted as the angular momentum and M as the mass. The BTZ black hole displays two horizons, determined by finding where the time component equals zero, i.e. N (r) = 0,

r±= 4GM l2  1 ± " 1 − J M l 2# 1 2  . (2.3)

From this expression it is clear that M > 0, |J | 6 M l in order for the horizon to exist. The mass and angular momentum can be related to the horizons,

M =r 2 ++ r−2 8Gl2 , J = r+r− 4Gl . (2.4)

Anticipating on the Chern-Simons formulation, it is useful to write light-cone coordinates with the transformation x± = tl ± φ and ρ is chosen such that l

2_dρ2 _{= N}−2_dr2 _{(for details see [}₂₅_{]). The}

metric can then be written as (1.41), where L, ¯L are constant and related to the mass and the angular momentum: L = (r+− r−) 2 16Gl = M l − J 2 , ¯ L = (r++ r−) 2 16Gl = M l + J 2 . (2.5)

L, ¯Lcould be removed by a gauge transformation that would not satisfy the boundary conditions at infinity, and are therefore an example of the would-be degrees of freedom that arise from the existence of the boundary. The BTZ black hole in Chern-Simons formulation, written in the radial gauge, is given by the connections

a = (J1− 1 kLJ−1)dx + _(2.6) ¯ a = −(J−1− 1 k ¯ LJ1)dx− . (2.7)

These connections will exactly lead to (1.41), with constant L, ¯L constant, when using (1.24). Since there are still no fluctuations allowed in the curvature, the BTZ black hole is locally AdS. On the other hand, globally, the BTZ black hole is a quotient space of AdS3. This quotient space can

explicitly be constructed by making the identification φ = φ + 2πn, which in terms of the universal covering space corresponds to a boost. Correspondingly, the matrix X as defined in (1.8) transforms as X → Λn LXΛnR,1 where ΛL= eπ(r+−r−) ₀ 0 eπ(r+−r−) , ΛR= eπ(r++r−) ₀ 0 eπ(r++r−) . (2.8)

2.1.1 Thermodynamics of the BTZ black hole

The natural next step is to investigate the thermodynamical properties of black holes. For the purpose of deriving the thermodynamics of the black hole, we will write the BTZ black hole in the Euclidean signature by doing a Wick rotation. Algebraically the metric (2.1) becomes Euclidean when t = iτ, J = iJE. The metric can then be written as

ds2_E= −N_E2dτ2+ N_E−2dr2+ r2(N_Eφdτ + dφ)2 , (2.9) with NE= −8GM +r 2 l2 − 16G2J_E2 4r2 1/2 , N_Eφ = −iNφ , (2.10)

1_{The coordinates of the global covering space are then given by three coordinate patches, for the three regions}

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where the inner horizon will become imaginary. Under a certain coordinate transformation (for details see [32]), it can be shown that (2.9) is locally AdS, provided that the following identification of the coordinates is made,

(φ, it) ∼ (φ + Φ, i(t + β)) , (2.11) where Φ = 2πl|r−| r2 +− r2− , iβ = 2πl 2_r + r2 +− r−2 . (2.12)

The resulting spacetime has the topology of a solid torus, with τ the modular parameter. On the torus the time cycle is contractible, whereas the φ-cycle is not. Without these specific values for Φ and β, a conical singularity would appear at the horizon r+. The Hawking temperature that can

thus be assigned to the Black hole is given by T = 1 β = r2 +− r2− 2πl2_r + . (2.13)

The Bekenstein-Hawking entropy gives the entropy of the black hole S = A

4G = 2πr+

4G , (2.14)

where A is the area of the black hole. The entropy could be derived in several ways: e.g. from the Euclidean path integral or with the Wald formula (see [31] for an overview of these derivations).

These physical properties of the BTZ black hole are related to each other by what is known as the first law of black hole thermodynamics:

dM = T dS + ΩdJ (2.15)

where M ,S and J are mass, the entropy and angular momentum of the black hole and T and Ω are the temperature and angular velocity of the horizon. These black hole dynamics are formally analogous to ordinary thermodynamics.2 _{We will use (2.15) in the following to calculate the entropy} of the black hole, whenever it is not clear anymore how to define the area such as in the case of higher spin black holes.

Temperature in Chern-Simons formulation

The Euclidean metric can also be written in Chern-Simons formulation. Using the coordinates, z = φ + it_l and its complex conjugate, in addition to the φ periodicity of the coordinates,3_{we have} (z, ¯z) ∼= (z + 2πτ, ¯z + 2π ¯τ ) , (2.16) where τ = il r+− r− = i 4 r l GL = ik 2 1 √ kL (2.17) ¯ τ = −il r++ r− = −i 4 r l G ¯L = −ik 2 1 √ k ¯L . (2.18)

2_{The other thermodynamical laws can also be stated, Zeroth law: surface gravity is constant on the horizon (where}

the surface gravity κ is proportional to the temperature: κ ∼ T ) ; second law: an always increasing area corresponds to an always increasing entropy; third law: it is not possible to construct a black hole with no surface gravity.

3_{The relation of the identification of τ with β, Φ is given by τ = i(Φ +} i

lβ). For the anti-holomorphic part:

¯

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For the last step, (2.5) is used. The connections can be written in the same way as in the Lorentzian signature, a = (J1− 1 kLJ−1)dz (2.19) ¯ a = (J−1− 1 kLJ1)d¯z . (2.20)

Demanding the absence of a conical singularity at the horizon has a Chern-Simons language equiv-alent: the holonomy around the contractible cycle has to be equal to the center of the group:

Holτ,¯τ(a) = eω , Holτ,¯τ(a) = eω , (2.21)

where ω, ¯ω are defined as,

ω = 2π(τ az+ ¯τ az¯) , ω = 2π(τ ¯¯ az+ ¯τ ¯a¯z). (2.22)

In the case of the non-rotating black hole, (2.21) reduces to demanding that the holonomy around the time circle has to be trivial:

HolC(at) = e2πτ at . (2.23)

The eigenvalues of ω can be calculated to be ±2πτ

√ Λ √

k . Inserting (2.17), shows that the holonomy is

given by {−1, −1}, which is in the center of the group, and therefore indicates a trivial holonomy.4 In the next section the aforementioned properties of the BTZ black hole will be generalized for the higher spin black hole.

Entropy and the integrability condition

A useful way to calculate the entropy for a black hole is to write the partition function and calculate the entropy by assuming that the first law of thermodynamics (2.15) holds. The black hole partition function can be defined as

Z(τ, ¯τ ) = Trei2πτ L−i2π ¯τ ¯L= eS+i2πτ L−i2π ¯τ ¯L , (2.24) where in the last equality the thermodynamical limit is taken.5 _{From this expression of the partition} function the charges can be written in the following way,

L = − i 2π ∂ ln Z ∂τ , ¯ L = i 2π ∂ ln Z ∂ ¯τ . (2.26)

Moreover, the entropy of the black hole can be written in terms of the partition function and the charges and chemical potential (equivalently instead of ln(Z), we could have written the free energy F = −T ln(Z))

S = ln(Z) − i2πτ L + i2π ¯τ ¯L . (2.27)

4_{There is a subtlety with these holonomy conditions, coming from the fact that identifying the Chern-Simons}

theory with gravity, the solutions are manifolds with a spin structure, i.e. odd spins will pick up a minus sign when going around the thermal cycle. The result is that minus the identity also has to be interpreted as a trivial holonomy. In other words, the holonomy has to be equal to the center of the group. For more details see [14].

5_{Note that the relation between the partition functions of AdS and CFT is particularly evident, (see e.g. [}₂₆_]).

The CFT partition function can be written as ZCF T = Tr

h

e2πiτ (L0−24c)−2πi¯τ ( ¯L0−24c)

i

. (2.25)

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2.2. GK BLACK HOLE 24

Now since the dependence of L in terms of τ is known, ln Z can be found by integrating (2.26): ln Z = i2π Z Ldτ − Z ¯ Ld¯τ = i2πk 4( 1 τ − 1 ¯ τ) = π√kL + πpk ¯L, (2.28) resulting in an entropy, Sth= 2π √ kL + 2πpk ¯L . (2.29)

When using the relations (2.5) and (1.23), this entropy is equal to the Bekenstein-Hawking entropy. By trading k, L back for the CFT variables c, L0, it is clear that we have have found the Cardy

formula, which is the universal formula for the entropy in the high temperature limit of the CFT [33].

We could in general add a charge to the black hole, which gives an extra component to the first law.6 _{We then can write a new partition function:}

Z(τ, ¯τ , α, ¯α) = eS+i2π(τ L−¯τ ¯L+αQ− ¯α ¯Q) , (2.30) where L, Q, ¯L, ¯Q are the charges and τ, α, ¯τ , ¯α are the conjugate potentials associated to these charges. Until now this is a straightforward generalization of the BTZ black hole. However, in order for the partition function to exist, an extra restriction on the charges is given [15]. Consider the analog of (2.26), L = − i 2π ∂ ln Z ∂τ , Q = − i 2π ∂ ln Z ∂α , (2.31)

and similar for the anti-holomorphic part. Relating these two expression implies a condition on the charges:7 ∂L ∂α _τ = ∂Q ∂τ _α . (2.32)

We will refer to this relation as the integrability condition, and it will show to be important in defining black holes in higher spin theories.

2.2 GK black hole

Because SL(2, R) can be embedded in SL(3, R), a pure gravity solution can always be found as a solution of higher spin gravity. One can simply choose the connections in such a way that they only include generators of the SL(2, R) subgroup. Hence, the BTZ black hole is also a solution of higher spin gravity. These are not the most interesting black holes though. We would prefer to study black holes with a higher spin charge. Connections that allow such a higher spin charge were already constructed in section1.6. The BTZ black hole was constructed by demanding the absence of a conical defect in the t-direction, resulting in a consistent thermodynamical picture. Defining a black hole with a higher spin charge as the generalization of the BTZ black hole asks for a similar procedure. Some difficulties that arise, will be discussed here.

6_{Here we are anticipating on the generalization to higher spin theories, where black holes can be assigned with}

higher spin charges. For now we will not talk about higher spin and try to understand what happens with the above derivation for the entropy whenever a general charge Q is added to the thermodynamical picture.

7_{Equivalently, we could have simply used the free energy, from which it is clear that L = −}∂F ∂τ αand Q = − ∂F ∂α τ,

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Let us restate the connections of vacuum B. We will call the black hole that is constructed with these connections, as was first done by Gutperle and Kraus, the GK black hole [15]. Compared to (1.65), L, W are taken to be constant and the anti-holomorphic part is denoted as well,

a = L1− 1 kLL−1− 1 4kWW−2 dx+ + µ W2− 2L k W0+ L k 2 W−2+ 2W k L−1 ! dx− ¯ a = − L−1− 1 k ¯ LL1− 1 4k ¯ WW2 dx− − ¯µ W−2− 2 ¯L k W0+ _¯ L k 2 + ¯ 2W k L1 ! dx+ , (2.33)

which can also be written in Euclidean coordinates by x+ _{→ z, x}− _{→ −¯}_z. _{Note that these}

connections violate the boundary condition (1.44). Instead of carefully performing an asymptotic analysis where the construction of the connection follows from the chosen boundary conditions, the authors of [15] justified this form of black hole by showing that these connections can be identified with the Ward identities of a W3algebra. µ is then identified as the source of a spin 3 current, only

when it is introduced in the way of (2.33).

The black hole that is defined by (2.33) can be interpreted as a generalization of the BTZ black hole. Taking µ = ¯µ = W = ¯W = 0 returns the connections of a rotating BTZ black hole (and non-rotating in the case of L = ¯L). We then interpret W as a higher spin charge (Q in the previous section) and µ as its corresponding chemical potential. A higher spin black hole (i.e. a black hole that caries a higher spin charge) is then defined by the following three conditions [15]:

1. The geometry in euclidean signature is smooth, as is also demanded for the BTZ black hole. The holonomy of the connections around the contractible cycle has to be trivial.

2. Since the spin-3 black hole is to be interpreted as a generalization of the BTZ black hole, it should give back the BTZ black hole in the µ → 0 limit (implying that in this limit W → 0 as well). In fact, we want any higher spin solution to have a well-defined pure gravity limit. 3. As discussed in the previous section, the integrability condition

∂L ∂α =

∂W

∂τ (2.34)

has to be satisfied. Here α = τ µ, such that there is consistency with the holonomy condition and µ can be interpreted as the chemical potential of W.

We will focus on the non-rotating case, where ¯L = L, ¯W = −W, ¯µ = −µ. In that case the connections depend on four parameters: L, W, µ and the inverse temperature β that is related to modular parameter of the torus in Euclidean signature, τ = _2πiβ. The trivial holonomy condition around the time cycle is then again given by (2.23).

The easiest way to solve the holonomy conditions is by finding the eigenvalues of the exponent. We will be focussing on the branch in which the limit of the higher spin charge to zero will give back the BTZ black hole thermodynamics, in line with the the second condition of the higher spin black hole. The easiest way to solve the holonomy condition is by solving the characteristic polynomial where λ = 0, ±2iπ: λ3−1 2tr(ω 2_{)λ − det(ω) = 0 .} _(2.35) It is solved by det(ω) = 0 , tr(ω2) + 8π2= 0 . (2.36)

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These two equations can explicitly be written in terms of the four parameters,

− 512µ3_L3_{+ 288kµL}2_{− 432kµ}2_{WL + 432kµ}3_W2_{− 27k}2_{W = 0} _(2.37)

64µ2L2_{+ 12kL − 36kµW =} 12k2π2

β2 . (2.38)

There are different solutions allowed, corresponding to different thermodynamical branches, studied in [34]. We choose the one that gives back the BTZ black hole in the appropriate limit, which is called the BTZ branch in [34]. It will proof to be very helpful to solve these two equation in terms of one parameter C (not to be confused with the central charge c), similar to what is done in the vacuum. W, L, µ can be written as

W = 4(C − 1) C3/2 L r L k, µ = 3√C 4(2C − 3) r k L, L = kπ2 β2 C(3 − 2C)2 (C − 3)2_{(4C − 3)} . (2.39)

It can be proven that the equations in (2.37) satisfy the integrability condition [15]. It is easy to see that C → ∞ implies W = µ = 0, corresponding to the BTZ black hole.

Back to a metric

By using (1.24) it is possible to write a metric for the connections (2.33). It will be of the form ds2= −F (ρ)dt2+ G(ρ)dφ2+ dρ2, (2.40) where F (ρ) = 2µeρ+W 2kWe −2ρ₋ 2 k2µL 2_e−2ρ 2 + eρ−L ke −ρ₊2 kµWe −ρ 2 . (2.41) In line with the BTZ black hole, one expects that there is a point at which F (ρ) vanishes. This is, however, not the case. One solution of the condition F (ρ) = 0 is given by W = µ = 0, which is the BTZ black hole and is therefore not the solution we are looking for. Another solution is found by

k + 32µ2(µW − L)

2 = 0 . (2.42)

With this extra restriction, the temperature of the black hole that is found by demanding the absence of a conical singularity does not agree with defining conditions 2 and 3. In other words, there is no physical solution that admits a horizon, at least not in the way we are used to define a horizon, i.e. by the vanishing metric component. In a metric formulation the discussed connections describe a wormhole connecting two asymptotic AdS3 spacetimes [30]. In [28], a gauge transformation is

found conserving all the properties discussed above, but obtaining a metric with a horizon that does not contradict the other conditions. On the one hand, this gauge transformation was an interesting verification for the definition of the black hole. On the other hand, it indicates a subtle point of higher spin gravity: geometry is not well-defined anymore. By a gauge transformation, the geometry changed while leaving the physics unchanged. The black hole in higher spin gravity should therefore only be defined in terms of holonomy conditions (2.36), which are invariant under such gauge transformations.

2.2.1 Thermodynamics of the GK black hole

Because of the lack of a notion of geometry, it is not possible to write an expression for the area of the horizon. The Bekenstein-Hawking entropy can therefore not be used. The original derivation that was introduced together with the definition of the black holes shall be discussed here. Similar to the discussion of the BTZ black hole, it is also possible to find an expression for the entropy of

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the black hole through the partition function. In addition, the integrability condition relates the two charges as a generalization of (2.27). Following the derivation of [15], we write the entropy as (for the non-rotating black hole),

S = ln Z − i2πτ L − i2παW . (2.43)

From this expression it is clear that the standard expression for entropy with respect to charges and canonical chemical potential, is given by

τ = i 2π ∂S ∂L _α , α = i 2π ∂S ∂W _τ . (2.44)

In the appropriate limit it is required that the BTZ entropy is obtained. So for now lets say that the entropy should be written as

S = 4π√Lf (x(C)) , (2.45)

where

x(C) = 27(C − 1)

2

2C3 . (2.46)

Since we want to obtain the entropy of the BTZ black hole in the C → ∞ limit, it is required that f (0) = 1. To find the explicit form of f , (2.38) is used, now explicitly in terms of L, W, τ, α,

64µ2L2_{+ 12L − 36W}α

τ = − 3µ2

α2 . (2.47)

Then inserting (2.47) in (2.44) gives a differential equation in terms of f (x),

36x(2 − x)(f0(x))2+ f (x)2− 1 = 0 , (2.48) which can be solved by

f (x) = cos φ , φ =1 6arctan p x(2 − x) 1 − x ! . (2.49)

When writing x explicitly in terms of C, the function f (x(C)) simply reduces to

f (C) = r

1 − 3

4C . (2.50)

Hence, we have obtained the following entropy for the non-rotating black hole,

S = 4π√L r

1 − 3

4C . (2.51)

Indeed the limit C → ∞ gives the BTZ black hole entropy, which is what we expect from the second point of the definition of the higher spin black hole.

We will first discuss the entropy that was found through a different procedure, contradicting (2.51). Afterwards the validity of both equations shall be discussed.

Entropy through the Euclidean action

Another way to define the entropy of a black hole is proposed in [35]. In short, the entropy is derived in the following way. First a boundary term introduced, such that the variation of the action in

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Euclidean signature is well-defined. Then the partition function is defined through the variation of the Euclidean action. The partition function can be obtained from the Euclidean on-shell action:

ln Z = −S_osE . (2.52)

The free energy is defined as usual as F = −T ln Z. The entropy can be determined by, F = U − T S → S = ln Z + βU = ln Z − β∂ ln Z ∂β = β ∂SE os ∂β − S E os , (2.53) where SE os= SCS os + SbeyE

_os, with S_bdyE the term that is introduced to ensure a well-defined action variation principle. It is argued in [35] that in the Lorentzian signature this boundary term has to be chosen as, Sbdy= − k 2π Z ∂M dx2Tr [(a+− 2L1)a−] − k 2π Z ∂M dx2Tr [(¯a−+ 2L−1)¯a+] , (2.54)

because then together with the boundary term that arises from the Chern-Simons action, δSCS _os= − k 2π Z ∂M dx2Tr (a+δa−− a−δa+− ¯a+δ¯a−+ ¯a−δ¯a+) (2.55)

we will find a variation of the full on-shell action, δ (SCS+ Sbdy) _os= − k π Z ∂M dx2 Ldτ + ¯Ld¯τ + Wdα + ¯Wd ¯α , (2.56) which vanishes when we choose the τ, α to be constant. Now we would like to write this expression in Euclidean signature and plug it into (2.53). Because the potential of L, τ is incorporated through identification of the coordinates, we have to be careful with writing the Euclidean variation, i.e. we want to write Euclidean action in such a way that the coordinates have a fixed periodicity. This can be done with the following choice of coordinates [26],

z = 1 − iτ 2 w +

1 + iτ

2 w ,¯ (2.57)

such that the identification of the coordinates comes down to

w → w + 2π → w + 2πi , (2.58)

where w, ¯w are now the new Euclidean coordinates. Then the equivalent equation in the Euclidean signature of (2.54), (2.55) are given by (for the Jacobian and the explicit change of the connections under these coordinate transformations see [35]),

Sbdy= − k 2π Z ∂M dz2Tr [(az− 2L1)az¯] − k 2π Z ∂M dz2Tr [(¯az¯+ 2L−1)¯az] , (2.59) and δSCSE = − iπk Z ∂M dz2

4π2_{Im(τ )}Tr [(¯τ − τ )(azδa¯z− a¯zδaz) + (az+ az¯)(δτ az− δ¯τ a¯z)]

+ iπk Z

∂M

dz2

Entanglement Entropy and Black Holes in (2+1)-Dimensional Higher Spin Gravity Strong Sub-additivity of Higher Spin Entanglement Entropy

University of Amsterdam

MSc Physics

Theoretical Phyiscs

Master Thesis

Entanglement Entropy and Black Holes in

(2+1)-Dimensional Higher Spin Gravity

Strong Sub-additivity of Higher Spin Entanglement Entropy

by

Jochem F. Knuttel

5975492

54 ECTS

September 2013-August 2014

Supervisor:

Dr. Alejandra Castro

Examiner:

Dr. Miranda Cheng

Acknowledgements

Contents

Introduction

Chapter 1

Higher spin gravity in

(2+1)-dimensions

1.1

Anti de Sitter gravity

1.2

Chern-Simons formulation of 2+1 gravity

1.3

Higher spin gravity

1.4

Asymptotic symmetries and charges

1.5

Relation with CFT

1.6

Symmetries of the vacuum

1.6.1

Vacuum A

1.6.2

Vacuum B

Chapter 2

Black holes in (2+1) dimensional

higher spin gravity

2.1

BTZ black hole

2.1.1

Thermodynamics of the BTZ black hole

2.2

GK black hole

2.2.1

Thermodynamics of the GK black hole