Holographic Entanglement Entropy for AdS3 in the Presence of Higher Spin Fields and a Gravitational Anomaly

University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

Holographic Entanglement Entropy for AdS

in the

Presence of Higher Spin Fields and a Gravitational

Anomaly

Entanglement Entropy Beyond Geometry

by

Vincent Min

10001751

August 2015

60 ECTS

September 2014 - August 2015

Supervisor:

dr. Alejandra Castro

Examiners:

dr. Alejandra Castro

prof. dr. Jan de Boer

Vincent Min: Holographic Entanglement Entropy for AdS3in the Presence of Higher Spin Fields

and a Gravitational Anomaly, Entanglement Entropy Beyond Geometry, c August 2015

s u p e rv i s o r s: dr. Alejandra Castro prof. dr. Jan de Boer u n i v e r s i t y:

University of Amsterdam i n s t i t u t e:

Institute for Theoretical Physics Amsterdam (ITFA) l o c at i o n:

Amsterdam t i m e f r a m e:

A B S T R A C T

Entanglement entropy in a CFT with an AdS dual is directly related to geodesic lengths through a proposal by Ryu and Takayanagi. We review the Wilson line proposal that allows for the gener-alization of Ryu-Takayanagi’s proposal in AdS3to include higher spin fields and a gravitational

anomaly. AdS3is topological and can be formulated as a Chern-Simons theory with gauge group

SL(2,R)× SL(2,R). The Chern-Simons formulation allows for the addition of a tower of higher spin fields up to spin N to the spectrum by enlarging the gauge group to SL(N ,R)× SL(N ,R). Furthermore, a gravitational anomaly can be added by including a gravitational Chern-Simons term. In these extended theories, the notion of geometry breaks down and Ryu-Takayanagi’s proposal is no longer valid. The Wilson line proposal remains valid in the extended theories and relates entanglement entropy to a bulk Wilson line. We discuss various issues that arise in the presence of higher spin fields and a gravitational anomaly. For example, constructing a black hole carrying higher spin charges is non-trivial and there exist two proposals in the literature. We explicitly evaluate the entanglement entropy for both proposals and observe that only one satisfies strong sub-additivity. Furthermore, we explore the fact that the Wilson line proposal allows for a natural definition of a spin three version of entanglement entropy. Additionally, we discuss what boundary conditions should be imposed on the Wilson line in the extended theories.

There is an art, it says, or rather, a knack to flying.

The knack lies in learning how to throw yourself at the ground and miss. Clearly, it is this second part, the missing, which presents the difficulties.

A C K N O W L E D G E M E N T S

There are many people that I would like to thank that helped me to write this thesis. First of all, I would like to thank Alejandra Castro for being my advisor on this wonderful topic and her guidance throughout the project. I am grateful to Jan de Boer for taking on the role as my second supervisor. Furthermore, I would like to thank Nabil Iqbal for the many insightful discussions, you have a gift for making even the most abstract problems seem intuitive.

I would like to thank everyone in the Master room for making my Master project such an enjoyable experience. Especially Jorrit, Lars and Jonas; you have been essential to my thesis. Thank you for all the serious and also the mindless conversations.

I would like to thank my parents. You have always been there for me and I would not have made it this far without your support.

Finally, I thank Sanne for looking after me. The past year has not been easy on us, but as long as we are together, the rest will fall in place.

And of course thank you to all of those that I forgot to mention, you all mean the world to me.

C O N T E N T S 1 i n t ro d u c t i o n 1 1.1 AdS/CFT 2 1.2 Higher Spin 3 1.3 Anomalies 3 1.4 Outlook 4

I

AdS

in the presence of higher spin fields and gravitational anomalies

5

2 c h e r n - s i m o n s 7

2.1 Einstein-Hilbert gravity and Chern-Simons theory 7 2.2 Chern-Simons action 10

2.3 Chern-Simons theory with higher spin fields 11

2.4 Chern-Simons theory with a gravitational anomaly 13

3 s o l u t i o n s i n 2 + 1 d i m e n s i o n a l g r av i t y a n d t h e i r p ro p e rt i e s 15 3.1 Lower spin solutions 15

3.2 Higher spin solutions 19 3.3 Thermodynamics 25

II

Entanglement Entropy

36

5.2 Entanglement entropy for N = 3 69

6 e n ta n g l e m e n t e n t ro p y f o r g r av i tat i o n a l a n o m a ly 79

6.1 Replica Trick 79

6.2 Holographic entanglement entropy in the presence of a gravitational anomaly 81 6.3 The factorized Wilson line prescription 82

7 d i s c u s s i o n a n d c o n c l u s i o n 88

a a p p e n d i x 91

a.1 Representation 91

a.2 Geodesics in Poincaré 92 a.3 Poisson Brackets 93

a.4 Alternative derivation of SE E for N = 2 95

a.5 Divergent behavior of the eigenvalues of M 96

a.6 Computing traces of M 98

a.7 Rewriting ∂( n∂∆φ1n2) 102

a.8 Reproducing geodesic lengths with the factorized Wilson line 102 b i b l i o g r a p h y 104

L I S T O F F I G U R E S

Figure 1 AdS/CFT in Poincaré coordinates. AdS3 lives inside the bulk for finite

radial coordinate ρ. The CFT lives at the boundary, i.e. ρ → ∞. 2

Figure 2 Plots for L, W, ST, F for the canonical black hole with the L0holonomy

conditions for all four branches. 30

Figure 3 Plot of W for the canonical black hole with the L0holonomy conditions

for all four branches with a longer range such that the uncharged point in branch III is visible. 31

Figure 4 Plots for L, W, ST, F for the canonical black hole with the W0holonomy

conditions for all three branches. 34

Figure 5 Illustration of the construction of a 3-sheeted Riemann surface. 40

Figure 6 AdS/CFT in Poincaré coordinates. AdS3 lives inside the bulk for finite

radial coordinate ρ. The CFT lives at the boundary, i.e. ρ → ∞. We are interested in the entanglement entropy at a constant time slice. 44

Figure 7 A geometrical proof of strong sub-additivity. We consider three regions,

A, B and C on the CFT. 45

Figure 8 SEE for the canonical black hole with the L0 holonomy conditions for

branch I and III. In all plots we have picked kcs=1 and =10−9. 72

Figure 9 S_EE(3) for the canonical black hole with the W0 holonomy conditions for

branch I and II. In all plots we have picked kcs=1. 74

Figure 10 SEE for the holomorphic black hole with the L0holonomy conditions for

branch I and III. In all plots we have picked kcs=1 and =10−9. 76

Figure 11 SEE for the holomorphic black hole as a function of the dimensionless

parameter∆φ/β?for fixed T =100 and varying µ. Furthermore we have picked kcs=1 and =10−9. 77

Figure 12 S_EE(3) for the holomorphic black hole with the W0 holonomy conditions

for branch I and II. In all plots we have picked kcs=1. 78

Figure 13 R is a spatial interval at a constant time slice and R0 is obtained by boosting R. RRis the spatial interval on the constant time slice for which

all signals send to the right still reach R0. RL is the spatial interval on

the constant time slice for which all signals send to the left still reach

R0. 81

1

I N T R O D U C T I O N

A major advancement in theoretical physics physics took place in 1997, when J. Maldacena published a now famous paper on the AdS/CFT correspondence [1]. In this paper J. Maldacena conjectured a duality between two at first sight completely unrelated theories, namely gravity in Anti-de Sitter space (AdS) in d+1 dimensions and a conformal field theory (CFT) in d dimensions. The fact that these theories are related at all is miraculous to say the least. The two theories are formulated in different variables and they don’t even share the same number of dimensions. One might say that the two theories are in fact describing the same physics, but in a different language.

Already in the year 1986 Brown and Henneaux found clues of the AdS/CFT correspondence by looking at the asymptotic symmetries of AdS3[2]. They concluded that “a nontrivial central

charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge”. Indeed, this is one entry in the AdS/CFT dictionary, specifically

c= 3`

2G, (1.1)

where c is the central charge, ` is the AdS radius, and G is Newton’s constant. Note that the central charge on the left-hand side is defined in the CFT, while the quantities at the right-hand side are defined in AdS.

While it was not obvious at the time, the appearance of the Virasoro algebra at the boundary of AdS turned out to be just the tip of the iceberg. Eleven years later, J. Maldacena published his paper [1] on the AdS/CFT correspondence, which is currently the most cited paper in high energy physics with more than ten thousand citations [3].

What makes the correspondence so interesting is that observables in one theory can be translated to observables in the other with the dictionary that comes with the AdS/CFT cor-respondence. One of these translations will play a central role in this thesis, which is the relation between entanglement entropy in the CFT and minimal surfaces in AdS. Entanglement entropy is an observable that is defined for a region of space, A, in the CFT. It is a measure of the entanglement between A and its complement A in the CFT. One can then ask the question what is the corresponding observable in AdS that corresponds to entanglement entropy, SEE,

and the answer was given by Ryu-Takayanagi [4]

SEE =

Area[∂A]

4G , (1.2)

where Area[∂A] is the area of a minimal surface attached to the boundary of A and G is

Newton’s constant. This is a remarkably simple relation between two very fundamental concepts, i.e. entanglement in the CFT and minimal surfaces in AdS. It should be noted that it was by no means necessary for the relation to be this simple and it seems to imply a fundamental connection between geometry and entanglement.

General relativity provides a geometrical interpretation of how gravity works, which has been very successful. However, efforts to combine general relativity with the other fundamental forces has been unfruitful so far. This is where the relation between entanglement entropy and geometry can guide us towards a unifying theory, i.e. a theory of quantum gravity. In fact, the advent ofEquation 1.2has led people to propose that gravity is an emergent, rather than a fundamental force [5, 6]. Taking Equation 1.2, one can even derive the linearized Einstein equations [7].

Still, it is unclear what a theory of quantum gravity should look like. We do have certain expectations. For example the notion of geometry is expected to break down at the quantum scale. While we are currently unable to treat gravity at a quantum level, there are other examples

1.1 ads/cft 2 Bulk AdS Boundary CFT t φ ρ

Figure 1: AdS/CFT in Poincaré coordinates. AdS3lives inside the bulk for finite radial coordinate ρ.

The CFT lives at the boundary, i.e. ρ → ∞.

where geometry break down which we can explore to further our intuition. The example that will be discussed in this thesis is the inclusion of higher spin fields, i.e. fields with spin s > 2. When we add such fields to our theory, we require the theory to be invariant under higher spin transformations. However, these higher spin transformations also act on the metric and therefore on the geometry. This means that geometry is no longer a well-defined concept, because it is not invariant under the symmetries of our theory. This provides us with a toy model that we can exploit to guide us in the search for a theory of quantum gravity. Interestingly, if geometry breaks down, then so does the relation between entanglement and geometry. However, as we will explain and utilize in this thesis, a generalization of Ryu-Takayanagi’s proposal still holds [8]. Investigating this generalized proposal will be one of the main subjects of interest in this thesis. Furthermore, we will consider the case of gravity with a gravitational anomaly. This anom-aly causes our theory to pick up a frame dependence, which acts as a small deviation from the conventional gravity that we have come to understand. Again, we will investigate what consequences this has for the proposal by Ryu-Takayanagi.

1.1 a d s / c f t

Let us provide some more background on the AdS/CFT correspondence. This correspondence states that the partition function of Conformal Field Theories in d dimensions (CFTd) is equal

to that of Anti-de Sitter gravity in d+1 dimensions (AdSd+1) [9].

Z_{CF T}(d) =Z_AdS(d+1) (1.3)

A CFT is a quantum field theory with conformal invariance which can be formulated in an arbitrary number of dimensions, d, usually denoted by CFTd. CFT’s arise in various areas in

physics, such as in condensed matter physics, where it is used to describe the collective behavior of particles, such as electrons, at their critical point. AdS describes gravity in the presence of a negative cosmological constant. Now the AdS/CFT correspondence is the statement that some CFTd are equivalent to an AdSd+1 theory. While the correspondence has not been rigorously

proven, it has passed a large number of non-trivial checks and we will verify some of these checks in the rest of this thesis. An insightful picture one can have in mind is to pick d= 2 where we can view the CFT2 as defined on a two dimensional cylinder which forms the boundary of

the three-dimensional bulk described by AdS3, see Figure 1. The fact that a quantum theory

in d dimensions is related to a d+1 dimensional theory of classical gravity is a very non-trivial statement, and also a very useful one. A calculation of an observable in the CFTd can be

1.2 higher spin 3 translated to a calculation in AdSd+1 and vice versa, using the CFT dictionary. Often times

a calculation is very hard in one theory, but tractable in the other. For example, this has been exploited to describe non-Fermi liquids [10]. Specifically, in this thesis we will use the correspondence to find the entanglement entropy of a region in a CFT2by doing a computation

in the dual AdS3.

1.2 h i g h e r s p i n

Higher spin fields, i.e. fields with spin s > 2, are a recurring theme in this thesis. As mentioned earlier, adding massless higher spin fields to AdS3, provides a tractable toy model to study

gravity in a setting where geometry is no longer well-defined. Additionally, higher spin fields are interesting in their own right. For example, higher spin fields play an important role in string theory, whose energy spectrum contains an infinite tower of massive higher spin fields. One could speculate that string theory is the spontaneously broken phase of a higher-spin gauge theory that features only massless higher spin fields [11]. Studying higher spin fields can therefore help us gain a better understanding of string theory. The AdS/CFT correspondence provides a powerful framework to study these massless higher spin fields. The correspondence relates massless higher spin fields in AdS to conserved currents in the CFT. This leads to a highly constrained and therefore often tractable CFT. For example, the free O(N)vector model is conjectured to be dual to AdS4 with an infinite tower of massless higher spin fields [12]. The

dual AdS theory with massless higher spin fields was studied by Vasiliev and is in general difficult to control due to the highly non-linear dynamics of the higher spin fields [13]. In this thesis, we restrict to the case of d+1 = 3, where the complexity of pure gravity, i.e. gravity without higher spin fields, as well as higher spin gravity decreases considerably. Pure gravity in 2+1 dimensions is topological, thus there are no propagating degrees of freedom. Furthermore, AdS3 can be recast as a Chern-Simons theory with gauge group SL(2,R)× SL(2,R), which

is a topological gauge theory [14]. The Chern-Simons framework allows for a relatively simple generalization to higher spin fields coupled to AdS3. One simply enlarges the gauge group

to SL(N ,R)× SL(N ,R) [15]. The resulting Chern-Simons theory describes the coupling of a tower of fields of spin 2 up to spin N to AdS3. This truncation to a finite number of spins is

special to AdS3. In d+1 > 3, one needs an infinite tower of fields in order for the theory to be

consistent [13]. We will describe the relation between AdS3 and Chern-Simons theory in more

detail inchapter 2.

1.3 a n o m a l i e s

The use of symmetries can be a very powerful tool. For example, imposing a theory to be invariant under a given symmetry restricts the allowed terms one can add to the Lagrangian of your theory. Therefore, symmetries play a very important role in theoretical physics and an important question to ask is whether a symmetry that is present at a classical level is still present at a quantum level. It turns out that some classical symmetries fail to survive quantization, resulting in an anomaly [16]. One can start of with a classical action that is invariant under a given symmetry and upon quantization, the resulting quantum action is still invariant. However, the measure in the path integral does not necessarily have to be invariant and thus the theory acquires an anomaly. In other words, restrictions from symmetries at a classical level are not always obeyed at a quantum level.

Besides the generalization of the regular Einstein-Hilbert gravity to include higher spins fields, we will consider the inclusion of a gravitational anomaly. This specific anomaly is present if the central charges of the left and right moving sectors of the boundary CFT are unequal. As a consequence, the stress tensor is no longer conserved and thus Lorentz symmetry is broken. Therefore, the theory will pick up a frame dependence and the resulting effects for entanglement entropy will be discussed inchapter 6.

1.4 outlook 4

1.4 o u t l o o k

The remaining part of this thesis will be structured as follows.

In Part I, we introduces the preliminaries that are necessary to understand entanglement entropy and its generalizations.

Inchapter 2, we will introduce the relation between gravity in three dimensions and Chern-Simons theory. The latter theory allows for a more straightforward generalization to higher spin.

Inchapter 3, we will discuss the construction and properties of the various geometries that will be considered in this thesis. We describe the construction of black holes that are charged under higher spin symmetries and study their thermodynamic properties. Furthermore, we explore smoothness conditions and their solutions for the higher spin black holes.

In Part II, with the knowledge from the previous chapters combined, we will discuss the relevant observable that is central to this thesis, namely entanglement entropy. The discussion inchapter 4will be restricted to pure AdS3gravity without higher spin fields or a gravitational

anomaly. We will discuss three approaches to deriving entanglement entropy, which are the replica trick, Ryu-Takayanagi’s proposal and the Wilson line proposal. Insection 4.4, we provide a detailed description of the Wilson line proposal. This includes a discussion on the boundary conditions that should be imposed on the Wilson line.

The content of chapter 5 is dedicated to the generalization of the Wilson line proposal to include higher spin fields. Besides the generalization of entanglement entropy to include higher spin fields, we will introduce a new observable, which is a spin three variant of entanglement entropy. Furthermore, we will investigate the behavior of the higher spin black holes that are introduced inchapter 3.

In chapter 6 we will discuss the generalization of the Wilson line proposal to include a gravitational anomaly.

Part I

AdS

₃

in the presence of higher spin

fields and gravitational anomalies

6

The main topic of this thesis is the extension of entanglement entropy to the case of higher spin fields and a gravitational anomaly. Before we can discuss this generalization, we need to provide a background for several concepts that will play a crucial role in this extension. We will do so in this part and leave the definition and extension of entanglement entropy forPart II. This part is structured as follows. Inchapter 2we describe how three dimensional gravity with a negative cosmological constant can be recast as a topological gauge theory called Chern-Simons theory. Moreover, we will discuss how the Chern-Chern-Simons theory can be generalized to include higher spin fields and a gravitational anomaly. Inchapter 3we will introduce solutions to AdS3(with higher spin fields) for which we will evaluate the entanglement entropy inPart II.

2

C H E R N - S I M O N S

In this thesis, we restrict ourselves strictly to gravity with a negative cosmological constant in three dimensions, i.e. AdS3. This leads to several simplifications, because gravity becomes

topological in this case. To see this, let us count the number of propagating degrees of freedom (d.o.f). In d dimensions, the metric, gµν has d(d+1)/2 d.o.f due to the symmetry gµν =gνµ.

With diffeomorphism invariance we can gauge away d of these degrees of freedom and the Bianchi identities remove another d d.o.f. This leaves us with

#d.o.f .=d(d+1)/2 − 2d=d(d − 3)/2. (2.1) Thus indeed, for d = 3, we see that there are no propagating degrees of freedom, thus the theory is topological. However, this does not mean that the theory is trivial. One can make global identifications which result in different topologies. Furthermore, there can be interacting degrees of freedom at the boundary. Exploiting the topological nature of gravity in d = 3, the theory can be recasted as a Chern-Simons theory, which is a three-dimensional topological gauge theory [17, 14]. The spacetime isometries of AdS3 are SO(2, 2) ∼ SL(2,R)× SL(2,R)

which will also be the gauge group of the Chern-Simons theory. As we will see in section 2.3

the generalization to include higher spin fields is especially simple in the framework of Chern-Simons theory. One simply needs to promote the gauge group to a larger gauge group which we will pick to be SL(N ,R)× SL(N ,R)in this thesis.

This chapter is structured as follows. In section 2.1, we will discuss the relation between Chern-Simons theory and AdS3. In section 2.2 we will provide a review of the properties of

Chern-Simons theory. Furthermore,section 2.3will focus on the generalization to include higher spin fields. Finally,section 2.4will discuss how to include a gravitational anomaly in the theory.

2.1 e i n s t e i n - h i l b e rt g r av i t y a n d c h e r n - s i m o n s t h e o ry

To describe the relation between Einstein-Hilbert gravity and Chern-Simons theory, it is more convenient to work with a non-coordinate basis in terms of the vielbein1, e, and spin connection,

ω, rather than with the metric gµν [18]. The metric and the vielbein are related in the following

way [8]

gµν =2eaµebνηab. (2.2)

Note that we now have two types of indices, Greek (µ, ν, . . . ) and Latin (a, b, c). The Greek indices belong to a coordinate basis, ˆe_(µ) = ∂µ,2 while the Latin indices belong to the

non-coordinate basis ˆe_(a). This non-coordinate basis is chosen to be orthonormal, i.e. the indices are raised by the flat metric ηab. The vielbein, eaµ, is the tensor that transforms these two bases

into each other

ˆeµ=eaµˆea. (2.3)

Besides the vielbein, we need to define we need the spin connection3, ω_{µ b}a , to take covariant derivatives in the non-coordinate basis.

∇µXab =∂µXab +ωµ ca Xcb− ωµ bc Xac. (2.4)

1 In three dimensions it is customary to refer to the vielbein as the dreibein, but we will simply use the term vielbein.

2 In this thesis we will use the notation ∂µ≡_∂x∂µ

3 The name spin connection was chosen historically because it can be used to take covariant derivatives of spinors.

2.1 einstein-hilbert gravity and chern-simons theory 8 Where Xa_b is a tensor in the non-coordinate basis. The spin connection is the non-coordinate analog of the Christoffel connection,Γσ_µν, and they are related by

Γσ

µν =eσaωbνa eµb +eσaeaµ,ν. (2.5)

For more details on the relation between the coordinate and non-coordinate bases we refer to appendix J of [18].

Let us now rewrite the Einstein-Hilbert action with a negative cosmological constant, i.e. corresponding to AdS3, in terms of the vielbein and spin connection [19].

IEH = 1 16πG Z d3x√g  R+ 2 `2  = 1 16πG Z d3x|e|  δρ<sub>[µ</sub>δσ<sub>ν]</sub>Rµν<sub>ρσ</sub>+ 2 `2  = 1 16πG Z d3x|e| 1 2λµν λρσ_Rµν ρσ+ 2 `2  = 1 16πG Z d3xabceaλebµecν  1 2 λρσ<sub>R</sub>µν ρσ+λµν 2 3!`2  = 1 16πG Z d3xabceaλ  1 2 λρσ_Rab ρσ+λµνebµecν 1 3`2  = 1 16πG Z d3xabcea∧  Rbc+ 1 3`2e b_{∧ e}c  , (2.6)

where |e|= _3!1µνρabceaµebνecρis the determinant of eaµwhich is related to the determinant of gµν

as4 √g = |e|. Furthermore we used the following identities between the Levi-Civita symbol5

and the Kronecker delta,

µνρσλγ=3!δ_[µσδλνδ γ ρ], (2.7) µνρµλγ=2δλ_[νδ_ργ_]. (2.8) Finally we defined Rab=dωab+ωa_c∧ ωcb, (2.9)

where we introduced the wedge notation that for a p-form, χ, we have

χ= 1 p!χ[µ1...µp]dx µ1_{∧ ... ∧ dx}µ1 = 1 p!χ[µ1...µp] µ1...µp_dx3_. (2.10)

For more details on the wedge notation, see [20].

Note that Rab = dωab+ωa_c ∧ ωcb_{, thus if we interpret e and ω as gauge fields, A, the}

Lag-rangian takes the form AdA+A ∧ A ∧ A. This should remind you of the Chern-Simons three

form. Also we observe that this would not have worked in d 6=3, because then |e| would have contained a different number of vielbeins. For example in four dimensions, this would have led to a Lagrangian of the form A ∧ A ∧(dA+A ∧ A), which does not resemble any gauge invariant action.

It is convenient to use the definition

ωa = 1 2 a bcω bc_{, (2.11)} such that abcRbc =2dωa+abcωb∧ ωc, (2.12)

4 Note that we could also have chosen√g=−|e|. Our convention,√g= +|e|, means that we work with e > 0. 5 We use the convention 012= +1.

2.1 einstein-hilbert gravity and chern-simons theory 9 which allows us to write

IEH = 1 16πG Z d3xabcea∧  Rbc+ 1 3`2e b<sub>∧ e</sub>c  = 1 16πG Z d3x  2eadωa+abcea∧  ωb∧ ωc+ 1 3`2e b_{∧ e}c  . (2.13)

Let us now make the relation between gravity in d= 3 and Chern-Simons theory explicit by rewriting (2.6) in terms of the variables

Aa=ωa+1 `e a<sub>, A</sub>a =ωa−1 `e a_{, (2.14)}

We will now separately write the two terms into the appropriate Chern-Simons form. The first term gives 2eadωa= ` 2(A a<sub>− A</sub>a<sub>)d(A</sub> a+Aa), (2.15) = ` 2 A a_dA a− A a dAa− d(AaAa)
, (2.16)

where the last term is a total derivative which we can drop. The second term becomes

abcea∧  ωb∧ ωc+ 1 3`2e b<sub>∧ e</sub>c  =`abc 23 (A a_{− A}a_)∧  (Ab+Ab)∧(Ac+Ac) + 1 3(A b_{− A}b_)∧(Ac_{− A}c₎  =4`abc 3 × 23(A a<sub>− A</sub>a )∧Ab∧ Ac+Ab∧ Ac+Ab∧ Ac =`abc 6  Aa∧ Ab∧ Ac− Aa∧ Ab∧ Ac, (2.17)

Thus, combining the results, we find

IEH = ` 32πG Z d3x  AadAa+ 1 3abcA a_{∧ A}b_{∧ A}c₋_A adA a +1 3abcA a ∧ Ab∧ Ac  . (2.18) Now the final step is to notice that we can write this as

IEH = ` 32πG Z Tr  A ∧ dA+2 3A ∧ A ∧ A −  A ∧ dA+2 3A ∧ A ∧ A  , (2.19) where A=AaLa, (2.20) A ∧ dA= 1 3!A a [µ∂νA b ρ]LaLbdxµ∧ dxν∧ dxρ, (2.21) A ∧ A ∧ A= 1 3!A a [µA b νAcρ]LaLbLcdxµ∧ dxν∧ dxρ, (2.22)

and Laare the generators of a Lie algebra. In the case we are considering, AdS3, this Lie algebra

is sl(2,R), which satisfies LaLb= 1 2δab+ 1 2abcJ c_{, (2.23)} such that [La, Lb] =abcJa, (2.24) Tr LaLb=δab, (2.25) Tr LaLbLc= 1 2abc. (2.26)

2.2 chern-simons action 10 See appendixA.1.1for the explicit form of the matrices in the fundamental representation. Thus we can conclude that we can write the Einstein-Hilbert action with a negative cosmological constant in three dimensions as a Chern-Simsons theory.

IEH =ICS[A]− ICS[A], (2.27) where ICS[A] = k 4π Z Tr  A ∧ dA+2 3A ∧ A ∧ A  (2.28) and A, A ∈ sl(2,R). Furthermore, we defined k, the Chern-Simons level, as

k= `

4G. (2.29)

Using Equation 1.1, we see that 6k = _2G3` = c, relating the gravitational, Chern-Simons and

CFT variables. Thus we have shown that the Einstein-Hilbert action in three dimensions can be rewritten as a Chern-Simons theory with gauge group SL(2,R)× SL(2,R).

2.2 c h e r n - s i m o n s ac t i o n

Let us investigate the Chern-Simons actionEquation 2.28 in some more detail. The equations of motion (e.o.m) for the Chern-Simons action can be found by varying the action

δICS[A] =

k

4π Z

A ∧ dδA+δA ∧ dA+2δA ∧ A ∧ A

= k 4π

Z

(−d(A ∧ δA) +2δA ∧(dA+A ∧ A)),

(2.30)

where the first term is a total derivative which vanishes for suitable boundary conditions [21]. We will discuss the proper choice of boundary conditions in detail insubsection 3.2.1. Thus we find the equation of motion

F ≡ dA+A ∧ A=0, (2.31)

where we introduced the field strength F . Thus varying the actionEquation 2.28with respect to A and A gives the equations of motion

F =0, F =0. (2.32)

The relation to the e.o.m. of the Einstein-Hilbert action is most clear if we take the following linear combinations of F and F and write them out in terms of e = eaLa, ω = ωaLa using

Equation 2.14 1 2 F+F
=dω+ω ∧ ω+ 1 `2e ∧ e=0, (2.33) ` 2 F − F
=de+ω ∧ e+e ∧ w=0, (2.34)

We can recognize the first constraint to be the Einstein equations without matter in the presence of a negative cosmological constant using R=dω+ω ∧ ω. The second constraint is the torsion

free condition, which implies that w is the Levi-Civita connection [14].

s i g n i n t e r m e z z o Note that one should be careful with the signs arising from the wedge notation when doing this calculation. For example, we integrated by parts to get the second line, but the result has a positive sign, rather than a negative sign. This is because we get a minus sign from partial integration and a minus sign from moving the exterior derivative d over Aµ.

Furthermore, we used δ(Tr A ∧ A ∧ A) =3 Tr δA ∧ A ∧ A. To see why this is true, it is easiest to write it out in components

2.3 chern-simons theory with higher spin fields 11 where Tr A ∧ δA ∧ A=Aa_µδAb_νAc_ρTr LaLbLcdxµ∧ dxν∧ dxρ = (−1)2δAb_νAc_ρA_µaTr LbLcLadxν∧ dxρ∧ dxµ =Tr δA ∧ A ∧ A, (2.36)

where we used the fact that each element Aa_µ is a scalar that commutes, the cyclicity of the trace and we pulled dxµthrough dxνdxρ_{in exchange for two minus signs. The same calculation}

can be repeated to find Tr A ∧ A ∧ δA=Tr δA ∧ A ∧ A

2.2.1 Gauge invariance

Now let us check the gauge invariance of the Chern-Simons actionEquation 2.28. Under a gauge transformation, g, the connection A transforms as

A → g−1(A+d)g, (2.37)

where g ∈ SL(2,R). Thus the action transforms as

ICS → k 4π Z Tr  g−1(A+d)g ∧ d g−1(A+d)g
+2 3g −1_{(A+d)g ∧ g}−1_{(A+d)g ∧ g}−1_(A+d)g = k 4π Z Tr  A ∧ dA+2 3A ∧ A ∧ A − g −1_Adgg−1_dg+dgg−1_{dA −}1 3g −1_dgg−1_dgg−1_dg =ICS− k 4π Z Tr  d g−1Adg
+1 3g −1_dgg−1_dgg−1_dg_. (2.38) Note that the action is invariant up to two terms. The first term is a total derivative which van-ishes for suitable boundary conditions on A [22]. The second term seems to be more troublesome and is in fact a topological term involving the winding number w(g)defined as

w(g) = 1

24π2Tr g

−1_dgg−1_dgg−1_{dg
. (2.39)}

For so called “proper” gauge transformations that leave the boundary invariant, the integral of

w(g)is zero and thus the action is gauge invariant. For so called “improper”6gauge transform-ations that do alter the boundary, one can show that the integral of the winding number is an integer [22]. Thus under large gauge transformations the Chern-Simons action transforms as

ICS → ICS− 2πkN , (2.40)

where we dropped the total derivative and N ∈Z. We see that the action is not strictly invariant, but the path integral involving eiICS _{will be if we impose that k is also an integer. Given the}

identification k = `/4G, this implies that `/G is also quantized, see [25] for a discussion on this remarkable feature. In conclusion, the Chern-Simons action is invariant under gauge transformations if we impose that the Chern-Simons level k is discrete.

2.3 c h e r n - s i m o n s t h e o ry w i t h h i g h e r s p i n f i e l d s

The main motivation for recasting AdS3 gravity as a Simons theory is that the

Chern-Simons formalism admits an elegant generalization to higher spin fields as shown in [15,26]. One can simply enlarge the gauge group of the Chern-Simons action to obtain a more interesting field content. In this thesis we will consider the higher spin theory obtained by promoting

SL(2,R)× SL(2,R)→ SL(N ,R)× SL(N ,R). (2.41)

6 We use the terminology introduced in [23]. Another popular convention is to call these gauge transformations “global” gauge transformations as in [24].

2.3 chern-simons theory with higher spin fields 12

Embedding the gravitational subgroup

The field content of the theory depends on how we embed the gravitational subgroup SL(2,R) in SL(N ,R). For N = 3, there are two non-trival embeddings. The one we are interested in is called the principal embedding, which corresponds to picking the set of generators {Ja} for

sl(2,R) from the set of generators {Ta} for sl(3,R)as

J1=L1, J0=L0, J−1=L−1, (2.42)

where La and their algebra are specified insubsection A.1.2. The adjoint representation then

decomposes as [27]

Adj₃=32⊕ 52, (2.43)

where(2j+1)2denotes a (2j+1)-dimensional irreducible representation of SL(2,R). Thus, the

algebra decomposes into two representations of j= 1 and j=2. From the perspective of the bulk, a(2j+1)-dimensional representation corresponds to a field of spin j+1 [28]. Thus the in the principal embedding, the spectrum contains a spin two and spin three field. There exists another embedding, called the diagonal embedding, corresponding to the choice

J1= W2 4 , J1= L0 2 , J−1=− W−2 4 . (2.44)

With this choice, the generators Jastill satisfy the sl(2,R)algebra. However, the decomposition

of the adjoint representation is now [27]

Adj₃=32⊕ 2 · 22⊕ 12. (2.45)

The spectrum now contains a spin two field, spin 3₂ field and a spin one field. Thus the diagonal embedding for N =3 does not contain a spin three field in the spectrum.

For arbitrary N , the principal embedding leads to [27]

Adj₃=32⊕ 52⊕ · · · ⊕(2N − 1)2, (2.46)

which leaves us with a spectrum including a tower of massless spin fields of spin two up to spin

N .

Chern-Simons theory with gauge group SL(3,R)× SL(3,R)

Let us now focus on Chern-Simons theory with gauge group SL(3,R)× SL(3,R)in the principal embedding, i.e. AdS3in the presence of a spin three field. The bulk connections A,A are valued

in the enlarged gauge group and we can decompose them as

A=A(2)a La+A(3)a Wa, A=A(2)a La+A (3) a Wa, (2.47) A(2)a ≡ ωa(2)+ 1 `e (2) a , A (2) a ≡ ω (2) a − 1 `e (2) a , (2.48) A(3)a ≡ ωa(3)+ 1 `e (3) a , A(3)a ≡ ω (3) a − 1 `e (3) a , (2.49)

where La are the three generators of the subalgebra sl(2,R)inside sl(3,R)and Wa to be the

remaining five generators of sl(3,R)(see subsection A.1.1for our conventions). Now e(2)a , ωa(2)

are the frame field and connection associated to the gravitational spin two gauge symmetries (diffeomorphisms and Lorentz transformations) generated by Laand e(3)a , ωa(3)are the frame field

and connection associated to the spin three gauge symmetries generated by Wa. The metric

and spin three fields are then defined as [29]

gµν = 1 2Trf  e_(µe_ν), ϕµνρ= 1 3Trf  e_(µe_νe_ρ). (2.50)

2.4 chern-simons theory with a gravitational anomaly 13 With this identification, the equations of motion of the Chern-Simons theory (F =F =0) can be interpreted as describing the coupling of the metric to the spin three field.

Note that if we had attempted to write down Trf



e_(µeνeρ)



in SL(2,R), this would have vanished. The reason being that eµ would only be valued in SL(2,R), thus this would reduce

to Trf  e_(µeνeρ)  =Trf  e(2)a_(µe(2)b_νe(2)c_ρ)LaLbLc  (2.51) = 1 2e (2)a (µe(2)bνe(2)cρ)abc (2.52) =0, (2.53)

where we used Equation 2.24. Thus the spin three field is not present without enlarging the gauge group.

Under general gauge transformations the metric gµν and the spin three field ϕµνρ mix in a

complicated way [26]. The concept of geometry depends on the metric gµν and is thus no longer

gauge invariant in the presence of higher spin fields. Thus geometry is no longer well-defined in the presence of higher spin fields. Therefore it is convenient to work with the Chern-Simons formulation of AdS3 in terms of the gauge fields A, A which transform nicely under gauge

transformations.

To summarize, the action

I=ICS[A]− ICS[A], ICS[A] = kcs 4π Z Tr  A ∧ dA+2 3A ∧ A ∧ A  , (2.54) kcs= 1 2 Tr(L0L0) ` 4G, (2.55)

with gauge group SL(N ,R)× SL(N ,R)describes the coupling of a tower of fields of increasing spin 2, 3, . . . , N to gravity with a negative cosmological constant in three dimensions. Note that we defined kcs with a factor 2 Tr(L0L0), where the trace is taken in the fundamental

representation. This is due to the fact that the trace on sl(3,R) reduces to the metric on SL(2,R) only up to this normalization factor. For SL(3,R) we have Tr(L0L0) = 2, thus we

identify kcs = _16G` = ₂₄c, where we usedEquation 1.1 to relate the Chern-Simons level to the

central charge.

2.4 c h e r n - s i m o n s t h e o ry w i t h a g r av i tat i o n a l a n o m a ly

The relation between gravity in three dimensions and Chern-Simons theory was first noted in [17] and later expanded upon in [14]. In the latter paper, it was noticed that there is another action, besides ICS[A]− ICS[A], that is gauge invariant, namely ICS[A] +ICS[A]. Adding this

piece does not change the equations of motion at a classical level, i.e. the field strengths F , F still vanish. However, at a quantum mechanical level, adding this piece will result in a different theory. Adding the extra piece with constant coefficient −1/ν to the action results in7

IP OG=ICS[A]− ICS[A]− 1 ν ICS[A] +ICS[A]
= kL 4π Z Tr  A ∧ dA+2 3A ∧ A ∧ A  +kR 4π Z Tr  A ∧ dA+2 3A ∧ A ∧ A  , (2.56) where kL≡ k  1 − 1 ν  , kL≡ k  1+ 1 ν  , (2.57)

7 Note that we use ν rather than µ in contrast to [30] since we will use µ to denote a higher spin deformation in

chapter 5. Furthermore, in this section and in the rest of the thesis we set `=1 for notational convenience. The factors of ` can be reinstated with dimensional analysis at any point.

2.4 chern-simons theory with a gravitational anomaly 14 which we will refer to as Parity Odd Gravity (POG), since the extra term is parity odd. Written in terms of the vielbein and spin connection, the additional term reads

ICS[A] +ICS[A] = kcs 2π Z Tr  ω ∧ dω+2 3ω ∧ ω ∧ ω  +e ∧ F − F
 , (2.58)

where we recognize the Chern-Simons functional for the spin connection in the first term and the torsion tensor in the second. Indeed, varyingEquation 2.56with respect to A, A still leads to the e.o.m. F =F =0, so classically the theory remains the same. A more interesting scenario is described by Topologically Massive Gravity (TMG), for which the action reads [30]

IT M G=  1 −1 ν  ICS[A]−  1 −1 ν  ICS[A]− 1 ν kcs 4π Z Tr β ∧ F −F

, (2.59)

where we added a degree of freedom characterized by β. In terms of the metric formulation, Topologically Massive Gravity can be obtained from Einstein-Hilbert gravity by adding the Chern-Simons functional for the Christoffel connectionΓ [30,31]

IT M G=IEH− 1 ν kcs 2π Z Tr  Γ ∧ dΓ+2 3Γ ∧ Γ ∧ Γ  . (2.60)

Adding this term does change the equations of motion, which are now [30]

Rµν−

1

2gµνR − gµν =− 1

νCµν, (2.61)

where Cµν is the Cotton tensor defined as

Cµν =αβµ ∇α  R_βν−1 4gβνR  . (2.62)

Furthermore, the Cotton tensor is symmetric, transverse and traceless. The tracelessness of Cµν

means that the solutions of TMG still have a constant Richi scalar R=−6, thus we can write the e.o.m. as

Rµν+2gµν =−

1

νCµν (2.63)

Note that for conformally flat spacetimes, the Cotton tensor vanishes, thus solutions to pure AdS3such as the BTZ black hole are still valid solutions in TMG.

These theories are interesting to consider, because the left and right moving sectors of the dual CFT have different central charges

c_L= kL 6 =  1 −1 ν  k 6, cR= kR 6 =  1+ 1 ν  k 6. (2.64)

This leads to an anomaly at the quantum level. Furthermore, the addition of a gravitational Chern-Simons term breaks diffeomorphism invariance, leading to a non-conserved stress tensor

∇iTij =

cL− cR

96π 

kl_∂

k∂mΓmjl. (2.65)

Thus TMG constitutes a theory where our notion of geometry is again altered, in the sense that Lorentz symmetry is violated. We will consider TMG again in chapter 6, where we will consider the effects of the gravitational anomaly on the entanglement entropy.

3

S O L U T I O N S I N 2 + 1 D I M E N S I O N A L G R AV I T Y A N D T H E I R P R O P E RT I E S

In this chapter we will discuss solutions to gravity with a negative cosmological constant in three dimensions. The solutions that will be reviewed are empty AdS3, thermal AdS3, the

BTZ black hole and two black holes carrying higher spin charges. Furthermore, properties such as their charges, entropy and free energy will be examined. In chapter 4, chapter 5and

chapter 6we will evaluate the entanglement entropy for the solutions that are introduced in this chapter. Therefore, this chapter will function as a reference for coming chapters. Furthermore, this chapter will provide some background on how to construct quantities such as the thermal entropy and free energy. In section 3.1 we introduce solutions without higher spin charges. In

section 3.2we will discuss how to construct a black hole with a spin 3 charge. Furthermore, we will examine what smoothness conditions we should impose on the solutions in the presence of higher spin fields.

3.1 l ow e r s p i n s o l u t i o n s 3.1.1 Empty AdS3

The most symmetric solution, i.e. having the maximum number of killing vectors, to the Einstein equations of motion with a negative cosmological constant is empty AdS3. There is a particularly

elegant way to view AdS3, which is as an embedded hypersurface inR2,2 [32]. Let Xi be the

coordinates ofR2,2, then AdS3 is the hypersurface for which

X₀2+X₁2− X₂2− X₃2=1. (3.1) This constraint is invariant under rotations in SO(2, 2) which is the group of spacetime iso-metries for AdS3. Furthermore, these embedding coordinates can be packaged into a matrix X

from which the metric can be extracted as

X = X0− X2 X3− X1 X3+X1 X0+X2 ! , det X=1, (3.2) ds2= 1 2Tr X −1_dX
2 . (3.3)

Written in this way, the SL(2,R)× SL(2,R) symmetry in AdS3 is particularly evident, since ds2is invariant under

X → g_LX, X → Xg_R, g_L,R∈ SL(2,R), (3.4)

where gL, gR are constant matrices. One can obtain empty AdS3 with the following global

coordinates in Lorentzian signature

X0=cosh ρ cos t, X1=cosh ρ sin t,

X2=sinh ρ cos φ, X3=sinh ρ sin φ, (3.5)

which leads to the metric

ds2=dρ2− cosh2ρdt2+sinh2ρdφ2

= (1+ρ02)−1dρ02−(1+ρ02)dt2+ρ02dφ2, (3.6) where we defined the coordinate ρ0=sinh ρ.

3.1 lower spin solutions 16 Another coordinate system is Poincaré coordinates for which

X0=rt, X1= r 2(1+r −2_+x2_{− t}2₎ X2=rx, X3= −r 2 (−1+r −2_+x2_{− t}2_{). (3.7)}

For these coordinates the metric takes the form

ds2 =r−2dr2+r2(dx2− dt2) = 1

z2 dz

2_{− dt}2_+dx2
_, (3.8)

where we defined the coordinate z = r−1. Written in this way, it is especially clear that the boundary metric, z → 0, is conformal to flat Minkowski space,R1,1.

c h e r n - s i m o n s As described inchapter 2, the geometry can also be described with connec-tions, A,A ∈ sl(2,R)which are related to the metric by

gµν =

1 2Tr



(A − A)µ(A − A)ν , (3.9)

where we usedEquation 2.2andEquation 2.14 to write the metric in terms of the connection. It was shown in [24] that we can use the gauge freedom of the Chern-Simons theory to gauge out the radial dependence of the connections such that

A=b−1(a+d)b, A=b(a+d)b−1, b=eρL0_{. (3.10)}

where both A and a are flat, i.e. F =F =0, and a only depends on t, φ but not on the radial coordinate ρ.

The metric in Poincaré coordinates,Equation 3.8, is reproduced by the connections

a=L1dx+, ⇔ A=eρL1dx++L0dρ, (3.11)

a=−L−1dx−, ⇔ A=−eρL−1dx−− L0dρ, (3.12)

where we introduced the light-cone coordinates x± =t ± φ and ρ is a radial coordinate1related to r in Equation 3.8 by ρ = − log r. Furthermore, La are the generators of sl(2,R), see

sec-tion A.1. We can also reproduce the metricEquation 3.6for empty AdS3 in global coordinates

with the following gauge fields

a=  L1+ 1 4L−1  dx+, a=−  L−1+ 1 4L1  dx−. (3.13) 3.1.2 BTZ black hole

The most symmetric solution is empty AdS3which we discussed above and applies to the zero

temperature and non-rotating case. If we extend the discussion to non-zero temperature and non-zero rotation, there is a new solution called the BTZ black hole [33,34]. The metric for the BTZ black hole in Euclidean coordinates is given by

ds2= r 2 (r2_{− r}2 +)(r2− r2−) dr2+(r 2_{− r}2 +)(r2− r2−) r2 dt 2 E+r2  dφ+r+|r−| r2 dtE 2 , (3.14)

where tE is the euclidean time coordinate obtained by Wick-rotating t → −itE and for

nota-tional convenience we drop the subscript E from now on. This solution has two horizons denoted by r± where the inner horizon r− = −i|r−| is imaginary in Euclidean signature. In the

non-rotating case we have r− =0 and in the extremal case we have r+=r−. To find the mass and 1 Note that this ρ is different from the ρ in global coordinates. We hope this will not cause confusion, but there

3.1 lower spin solutions 17 angular momentum for the BTZ black hole it is convenient to work with a metric that takes the form

ds2 =dρ2+γijdxidxj, (3.15)

where ρ denotes the radial coordinate and γij is an arbitrary function of ρ and the remaining

coordinates. This can be accomplished by redefining the coordinates as [24]

r2=r2₊cosh2(ρ − ρ0)− r−2 sinh2(ρ − ρ0), e2ρ0 ≡

r₊2 − r2 −

4 , (3.16)

such that the metric reads

ds2=dρ2+L k(dx +₎2₊L k(dx −₎2₋  e2ρ+L k L ke −2ρ_dx+_dx−_{, (3.17)} where2 L k = (r++r−)2 2 , L k = (r+− r−)2 2 . (3.18)

The parameters L,L are related to the zero modes of the boundary stress tensor. To see why this is true we have to take a step back and construct the boundary stress tensor. The paper [36] and the lecture notes by Per Kraus [9] provide a nice reference on how to construct the stress tensor in AdS. We consider AdS3 on a manifold M with boundary ∂M with a gravitational

action Igrav. The gravitational action includes the Einstein-Hilbert action, IEH, but we have to

supplement it with boundary terms such that the variational principle is well defined and the boundary stress tensor is finite. The boundary stress tensor Tij is defined from the variation of the gravitational action Igrav as

δIgrav= 1 2 Z ∂M √ γTijδγij. (3.19)

The variation of the Einstein-Hilbert action contains a boundary term proportional to δ∂ργij

which spoils the variational principle. We can resolve this by adding the Gibbons-Hawking term

IGH = 1 16πG Z ∂M d2x√γγij∂ργij, (3.20)

which cancels this contribution. We can capture the divergence in the metric when ρ → ∞, by taking the Fefferman-Graham expansion of γij

γij =e2ργ_ij(0)+γ_ij(2)+. . . . (3.21)

The variation of the Einstein-Hilbert action supplemented with the Gibbons-Hawking term takes the form

δ(I_EH+I_GH) =− 1 32πG Z ∂M d2x√γ∂ργij− γkl∂ργklγij  δγij, (3.22)

such that the boundary stress tensor would be

Tij =− 1 16πG  ∂ργij− γkl∂ργklγij  . (3.23)

However, plugging in the Fefferman-Graham expansion shows that this boundary stress tensor diverges in the limit ρ → ∞. We can obtain a finite boundary stress tensor by adding yet another boundary term which is

Ict= −1 8πG Z ∂M dx2√g. (3.24)

2 A different but popular convention is to include a factor of 2π in the charges, such as in [8]. We will not do so and stick to the conventions of [35].

3.1 lower spin solutions 18 Taking the variation of Igrav=IEH+IGH+Ictand using Fefferman-Graham expansion results

in δIgrav= 1 2 Z ∂M q γ(0)Tijδγ_ij(0), (3.25) where Tij = 1 8πG  γ_ij(2)− g_kl(2)gkl₍₀₎g_ij(0). (3.26) Evaluating this expression for the BTZ metric, Equation 3.17, with k = c/6 = 1/(4G) we obtain T++= L 2π, T−−= L 2π, (3.27)

thus indeed L, L are the zero modes3of T++, T−−respectively, rescaled by a factor 2π. We can

also define a mass and angular momentum for the black hole using

M = Z 2π 0 dφTtt=L+L, J = Z 2π 0 dφTtφ=L − L. (3.28)

t e m p e r at u r e a n d avo i d i n g t h e c o n i c a l s i n g u l a r i t y We require the solution to be smooth everywhere, which will lead to identifications on the coordinates t and φ [37]. Specifically, there should be no conical singularity at the horizon. We will see shortly what is exactly meant by this statement. We can consider the BTZ metric very close to the outer horizon, i.e. r → r+. To do this in a controlled manner, we define r=r++u2/r+and take the

limit u → 0 inEquation 3.14 lim u→0ds 2_≈ 2 r2 +− r−2 du2+2(r 2 +− r−2)u2 r2 + dt2+r2₊  dφ+|r−| r+ dt 2 , (3.29)

If we use the coordinates x=q 2

r2 +−r−2 u, θ = r 2 +−r−2 r+ t we can recognize R 2_{× S} squashed, where

Ssquash denotes a squashed sphere. With these coordinates we have

ds2≈ dx2+x2dθ2 | {z } R2_{in polar coordinates} + r₊2  dφ+ |r−| r2₊− r2 − dθ 2 | {z } squashed sphere . (3.30)

However, for the first part to beR2 in polar coordinates(x, θ), we need θ ∼ θ+2π. Otherwise we obtain a cone, rather than flat space. Furthermore, the squashed sphere should also be invariant under this identification. This forces us to make the identifications

(φ, t)∼(φ+Ω, t+β), Ω ≡ − 2π|r−| r₊2 − r2 − , β ≡ 2πr+ r2₊− r2 − , (3.31)

In terms of the Euclidean version of the light-cone coordinates (z = φ+it, z = φ − it), the

identifications are

(z, z)∼(z+2πτ , z+2πτ), 2πτ ≡Ω+iβ, 2πτ ≡Ω − iβ. (3.32) Thus, the boundary topology is that of a torus with modular parameter τ . Furthermore, the

t-cycle is contractible, while the φ-cycle is non-contractible, because it wraps around the black

hole singularity.

3 For the BTZ black hole the stress tensor is a constant, but in general T±± can depend on the boundary

coordinates x±. One can then decompose T±± in Fourier modes T±±(x±) = _2π1 P_ne−inxL±n. For the BTZ black hole we then find the zero mode L₀+=L/2π, L−₀ =L/2π.

3.2 higher spin solutions 19 Let us state the relations between L, L, τ , τ , β,Ω, r+, r− once more below for completeness.

L k = M +J 2k = (r++r−)2 2 =− 1 2τ2 =− 2π2 (Ω+iβ)2, (3.33) L k = M − J 2k = (r+− r−)2 2 =− 1 2τ2 =− 2π2 (Ω − iβ)2. (3.34) In conclusion, there is a conical singularity present at the horizon unless the Euclidean co-ordinates t, φ are periodic in β,Ω respectively. With these identifications only the coordinate singularity remains at the horizon.

c h e r n - s i m o n s Again, there is an equivalent description of the BTZ black hole in terms of the Chern-Simons connections A, A. The extension of empty AdS3to the BTZ case is especially

nice in this setup and takes the form

a=  L1− L kL−1  dx+, a=−  L−1− L kL1  dx−. (3.35) Or in Euclidean coordinates a=  L1− L kL−1  dz, a=  L−1− L kL1  dz. (3.36)

Furthermore, we provide a Chern-Simons interpretation of requiring the solution to be smooth by avoiding the conical singularity. The equivalent statement from the Chern-Simons point of view is that the holonomy around the contractible direction is equal to the center of SL(2,R), i.e. [29]

Holτ ,τ(a) =e2π(τ az+τ az)=e2πiL0 =−1, (3.37)

Holτ ,τ(a) =e2π(τ az+τ az)=e2πiL0 =−1, (3.38)

where we used the fundamental representation of SL(2,R), seesubsection A.1.1. One can check that these constraints are indeed solved by the relations between L, L and τ , τ that are stated in

Equation 3.33. Note that we needed the holonomies to be equal to −1 in order for consistency withEquation 3.33. This poses no problem, because ±1 both belong to the center of SL(2,R). For more information on the origin of the minus sign, we refer to [29].

Note that we can also reproduce empty AdS3 in global coordinates by picking L/k=L/k=

1/4. For this choice of parameters one can check that the holonomy around the φ-cycle is trivial

Holφ(a) =e2πaφ =e2πiL0. (3.39)

This is consistent, since there is no singularity in global AdS that prevents us from shrinking a circle around the φ-cycle to zero.

3.2 h i g h e r s p i n s o l u t i o n s

In this section we will discuss the construction of solutions to AdS3 in the presence of a spin

3 field. As discussed in section 2.3, AdS3 coupled to a spin 3 field is described by a

Chern-Simons theory with gauge group SL(3,R)× SL(3,R). This theory has an enlarged set of gauge transformations, compared to pure gravity. Under a general gauge transformation, the spin 3 field mixes with the metric. Therefore geometry is no longer gauge invariant and we will not provide a metric description of the higher spin solutions. Rather, we will work in the Chern-Simons formulation described in chapter 2. As we will see shortly, the construction of black holes that carry higher spin charges is not without ambiguities. In the literature there exist two proposals for the construction of a black holes with spin 3 charges. The first higher spin black hole was constructed by Gutperle and Kraus in 2011 [38], which we will refer to from now on

3.2 higher spin solutions 20 as the holomorphic black hole. The second higher spin black hole was constructed by Bunster

et al. in 2014 [39], which we will refer to from now on as the canonical black hole. The two

proposals differ in the implementation of the higher spin chemical potential that sources the higher spin charges.

This section is structured as follows. Insubsection 3.2.1we follow the arguments presented in [26] to construct a generic solution to the field equations that has an extension of the Virasoro algebra as an asymptotic symmetry group. InEquation 3.2.2we review how to include the higher spin chemical potential in the connections from a Lagrangian point of view, which produces the holomorphic black hole. InEquation 3.2.2 we review how to include the higher spin chemical potential in the connections from a Hamiltonian point of view, which produces the canonical black hole. In section 3.3we compute and discuss the thermodynamic properties of both the canonical and holomorphic black hole.

3.2.1 Spin three charges and boundary conditions

The aim of this section is to provide a well-defined procedure for constructing a solution with higher spin charges. We are constrained in the number of ways that we can modify the connec-tions to incorporate these charges. One requirement is that the asymptotic symmetry group of the solution is the Virasoro algebra or at most an extension of this algebra. Another require-ment is that the on-shell variation of the Chern-Simons action vanishes. Both constraints can be satisfied if we impose the correct boundary conditions on the gauge fields A,A [26].

We now restrict to the Chern-Simons gauge group SL(3,R)× SL(3,R). Let us first see which boundary conditions are allowed such that the variation of the Chern-Simons action vanishes. In section 2.2, we saw that the variation of the Chern-Simons action on a manifold M with respect to A is invariant up to a boundary term

δICS[A] = kcs 4π Z M Tr(−d(A ∧ δA) +2δA ∧ F). (3.40)

Working in the light-cone coordinates x± = t ± φ, we can write the on-shell variation of the

Chern-Simons action as δI_CS[A] =−kcs 4π Z ∂M Tr(A ∧ δA), =−kcs 4π Z ∂M dx+∧ dx−(A₊δA₋− A−δA+), (3.41)

where the integral now runs over the boundary of M, denoted by ∂M. For the variation of the Chern-Simons action to vanish, we need to impose suitable boundary conditions. We consider the boundary condition

A−|boundary =0, δA−|boundary=0, (3.42)

for which Equation 3.41 indeed vanishes. These boundary conditions constrain the ways in which we can add charges to the background gauge fields. Let us again pick the gauge such that

Aρ =b−1(ρ)∂ρb(ρ), b(ρ) =eρL0. (3.43)

The equations of motion then constrain the other components of A

Fρ−=∂ρA−+ [Aρ, A−] =0, (3.44)

which is solved by

A−=b−1(ρ)a−(x+, x−)b(ρ), (3.45)

for some arbitrary function ˜a(x₊, x−). However, the boundary conditionsEquation 3.42imply a−(x+, x−) =0, thus A− does not only vanish at the boundary but also in the bulk. Similarly Fρ+=0 implies

3.2 higher spin solutions 21 where a+(x+, x−)is an arbitrary function which is not constrained by the boundary conditions.

The e.o.m. F+−=0 then implies

∂−a+(x+, x−) =0. (3.47)

Thus in the gauge Equation 3.43and with boundary conditionsEquation 3.42, the connection

A takes the form

Aρ=b−1(ρ)∂ρb(ρ), A− =0, A+=b−1(ρ)a+(x+)b(ρ). (3.48)

However, as discussed in [26], the asymptotic symmetry group corresponding to these con-nections is in general not the Virasoro algebra with central charge c=3`/2G. We are trying to construct an extension of Einstein gravity, so we should impose additional boundary conditions toEquation 3.42such that the asymptotic symmetry group is at most an extension of the Viras-oro algebra. As was shown in [2], global AdS does have the correct asymptotic symmetry group. Therefore we should impose that we can only modify the connections such that asymptotically they do not deviate from global AdS by more than a finite amount. Indeed it was shown in [26] that solutions that are asymptotically AdS have an asymptotic extended conformal symmetry. Where we call a solutions asymptotically AdS when their connections satisfy

(A − AAdS)|boundary=O(1), (3.49)

where AAdS is the connection for empty AdS3 in global coordinates given inEquation 3.13.

Let us restrict to N =3 for now and write a generic AdS solution of the field equations that satisfiesEquation 3.42andEquation 3.49. First we decompose a₊(x₊)in terms of the sl(3,R) generators as a₊(x₊) = i=1 X i=−1 li(x₊)Li+ i=2 X i=−2 wi(x₊)Wi, (3.50)

where li(x₊), wi(x₊) are arbitrary functions of only x+. The constraint Equation 3.49 then

implies that l1 =1 and w1 =w2 =0. Furthermore, we can use the residual gauge symmetry left after choosing Equation 3.43 to choose l0 = w0 = w−1 = 0 [26]. Thus a generic AdS solution of the field equations takes the form

Aρ=b−1(ρ)∂ρb(ρ), A− =0, A+=b−1(ρ)a+(x+)b(ρ), (3.51) where, a+(x+) =L1− L(x+) k L−1− W(x+) 4k W−2, (3.52) with k ≡ 2 Tr(L0L0)kcs= 1 4G, (3.53)

where we chose a convenient rescaling of the parameters. A similar analysis for the barred sector will result in Aρ=b(ρ)∂ρb(−ρ), A+=0, A− =b(ρ)a−(x−)b−1(ρ), (3.54) a−(x−) =−  L−1+ L(x−) k L1+ W(x−) 4k W2  . (3.55)

Now let us see if these connections indeed have an extended Virasoro algebra as an asymp-totic symmetry group. We will again follow [26]. Let us consider the residual gauge symmetry that remains after gauging out the radial dependence by picking Equation 3.43. The gauge transformations that preserve this gauge choice are of the form