Master Thesis
Interacting Gravitational Wilson lines in
AdS
3
/ CFT
2
Dimitrios Toulikas
11743891
November 2019
Supervisor:
Dr. Alejandra Castro
Examiner:
Prof. Dr. Jan de Boer
The purpose of this thesis is to study interactions between gravitational Wilson lines, which are the objects that describe the propagation of a massive particle in the Chern-Simons formulation of three-dimensional gravity. We begin by motivating the study of three-dimensional gravity with negative cosmological constant and then we discuss its main features, from the lack of propagating degrees of freedom and the black hole solutions it admits to the way it is recast as a Chern-Simons theory with gauge-group SL(2, R) × SL(2, R). We review the path integral representation of a Wilson line [1], which only allows for a semi-classical analysis of the field that lives on it, and we move away from this limit by presenting a recent interpretation of the Wilson line [2], where the worldline degree of freedom is treated quantum mechanically. Then, following [3] we demonstrate how we can join Wilson lines using sl(2, R) Clebsch-Gordan coefficients. We apply this method to study boundary-anchored Wilson line networks in an AdS3
background using the language of [2] and we find that they compute global conformal partial waves in the dual CFT2. Finally, we consider an interaction between three
Wilson lines, where one of them has both its endpoints in the bulk, and we show that this object is the Chern-Simons analogue of a particular bulk geodesic Witten diagram.
I would like to thank Alejandra Castro for her constant guidance throughout this project and for the long meetings we had. I would also like to thank Jan de Boer for agreeing to be my second examiner and for taking the time to answer my questions. Moreover, I greatly appreciate the useful discussions I had with Eric Perlmutter, Nabil Iqbal and Mert Besken.
Among my fellow students, I would like to thank Gonzalo Contreras, with whom I extensively discussed Wilson lines, and Dimitris Patramanis and Savvas Malikis for conversations on more general physics topics.
I am grateful to the Alexander S. Onassis Foundation for awarding me a full schol-arship for the entire duration of my MSc studies.
1 Introduction 4
2 3d gravity 7
2.1 General Relativity in 2+1 dimensions . . . 7
2.2 Anti-de Sitter spacetime . . . 8
2.2.1 AdS3 as an embedded hyperboloid . . . 8
2.2.2 Global coordinates . . . 9
2.2.3 Poincar´e Coordinates . . . 10
2.2.4 AdS3 as the group SL(2, R) . . . 11
2.3 BTZ black hole . . . 11
2.3.1 Metric, properties and phase space . . . 11
2.3.2 BTZ from identifications . . . 13
2.4 Asymptotic symmetry group . . . 14
2.4.1 Asymptotically AdS3 spacetimes . . . 14
2.4.2 A stress tensor for AdS3 . . . 14
2.4.3 Asymptotic symmetry group . . . 16
3 3d gravity as a Chern-Simons gauge theory 21 3.1 First-order formalism of General Relativity . . . 21
3.2 Chern Simons action . . . 24
3.3 SO(2, 2) Chern-Simons Theory as Λ < 0 Gravity . . . 25
3.3.1 SL(2, R) × SL(2, R) Chern-Simons Theory as Λ < 0 Gravity . . 26
3.4 Asymptotic symmetry group in CS formulation . . . 27
3.4.1 Global charges . . . 28
3.4.2 Boundary conditions . . . 29
3.4.3 Asymptotically AdS3 spacetimes . . . 30
3.4.4 Asymptotic symmetry algebra . . . 31
4 Gravitational Wilson lines: Part I 33 4.1 Wilson line as a massive probe of the bulk . . . 33
4.2 Path integral representation of a Wilson line . . . 35
4.4 Computing the on-shell action . . . 38
4.4.1 Empty AdS3 . . . 40
5 Gravitational Wilson lines: Part II 43 5.1 Group and algebra prerequisites . . . 43
5.2 Rotated Ishibashi states . . . 44
5.3 Properties of the |U i states . . . 46
5.4 Wilson line as an inner product between |U i states . . . 47
5.4.1 An Example . . . 49
5.4.2 Comments . . . 50
5.5 Local Bulk Fields . . . 51
5.5.1 Application to global AdS3 . . . 52
6 Interacting Gravitational Wilson lines: Part I 55 6.1 Rotated Ishibashi states with non-zero spin . . . 55
6.2 Wilson line junctions . . . 58
6.3 Boundary anchored Wilson line networks . . . 61
6.3.1 Required matrix elements . . . 61
6.3.2 Prescription . . . 62
6.3.3 Two-point function . . . 63
6.3.4 Three-point function . . . 63
6.3.5 Four-point global conformal partial wave . . . 65
7 Interacting Gravitational Wilson lines: Part II 67 7.1 Geodesic Witten diagrams . . . 67
7.1.1 Proof by direct calculation . . . 68
7.1.2 Proof by the conformal Casimir equation . . . 70
7.2 Wilson line interactions as Geodesic Witten diagrams . . . 72
7.2.1 An explicit example . . . 76
8 Summary and Outlook 80 A AdS/CFT essentials 83 A.1 Brief introduction to CFT . . . 83
A.1.1 Conformal partial waves . . . 85
A.2 Scalar field in AdS . . . 87
A.3 CFT correlators from AdS . . . 88
B Clebsch-Gordan coefficients 90 B.1 Definition and derivation of sl(2, R) Clebsch-Gordan coefficients . . . 90
C Matrix Elements of section 6.3 95 C.1 phΣ, j|g(x)|h, ¯h; k, ¯ki . . . 95 C.2 hh, ¯h; k, ¯k|g−1(x)|Σ, jip . . . 97 C.3 Limit at ρ → ∞ . . . 99
Two of the most important achievements of theoretical physics in the 20th century were Einstein’s theory of general relativity and quantum theory. Both of these theories have been repeatedly shown to provide successful descriptions of a wide array of physical phenomena and continue to be the two main pillars of modern physics. However, they seem to be conceptually incompatible and finding a consistent theory of quantum gravity remains one of the biggest challenges of theoretical physics, with attempts for finding one dating back to the 1930s. The usual perturbative quantization approach of quantum field theory does not work with general relativity at high energies, since the theory is non-renormalizable, but more fundamentally, quantizing gravity would require to quantize spacetime itself, which is something we do not really know what it means.
Facing such issues leads us to consider simpler models, which nevertheless provide useful setups for exploring various aspects about quantum and classical gravity. General relativity in 2+1 dimensions is one such toy model, which even though has quite simpler dynamics than its 3+1 dimensional analogue shares several features with it. The dynam-ical simplicity has to do with the fact that three-dimensional gravity has no propagating degrees of freedom and all solutions to the Einstein equations are locally flat when the cosmological constant Λ vanishes and similarly locally de Sitter or anti de Sitter when Λ > 0 and Λ < 0 respectively. This does not mean, though, that three-dimensional gravity is trivial, since it is known to admit point particles that appear as conical de-fects [4,5], boundary dynamics, and in the presence of a negative cosmological constant black holes [6], which share common properties with four-dimensional black holes, such as an entropy obeying the Bekenstein-Hawking area law [7,8].
The Bekenstein-Hawking law states that black holes have an entropy proportional to their horizon’s area A:
SBH =
A 4GN
, (1.1)
where GN is Newton’s constant. This equation contrasts with the usual entropy of
ther-modynamic systems, which scales with the volume and is an example of a holographic behaviour, where the information inside a volume is contained on the surface that sur-rounds it. According to the holographic principle [9,10] this type of behaviour does not only concern black holes but it is a general property of quantum gravity. The prime example of a holographic duality is the AdS/CFT correspondence [11], which states that
a gravitational theory living in a d + 1-dimensional AdS spacetime is dual to a conformal field theory in d dimensions. The AdS/CFT correspondence is a strong-weak duality, since it relates the strongly coupled regime of the one side to the weakly coupled regime of the other side and hence it is an extremely useful framework to study both quantum gravity and strongly coupled quantum field theories.
In three dimensions we have a special situation, since the conformal algebra of the dual conformal field theory becomes infinite dimensional, which leads to a more con-trolled version of the duality and for this reason a very well studied one. Actually, in three dimensions AdS3/CFT2 was hinted over a decade before Maldacena’s paper, when
Brown and Henneaux showed in [12] that the symmetry algebra of asymptotically AdS3
spacetimes is generated by two copies of Viraroso algebra with a non-vanishing central charge, which is exactly the conformal algebra of local conformal transformations in two dimensions.
Another important feature of gravity in three dimensions is that, as it was shown by Achucarro and Townsend [13] and later by Witten [14] it can be equivalently (at least classically) recast as a Chern-Simons theory with gauge group the isometry group of the spacetime in question. This formulation has a lot of advantages and makes manifest the topological nature of three dimensional gravity, but at the same time geometrical aspects, such as proper distances, are not clear anymore.
In the present thesis we will mainly work in the Chern-Simons formulation of three dimensional gravity, since we will study gravitational Wilson lines, which represent the coupling of massive particles to gravity in this formulation. In particular, we will focus on networks of Wilson lines interacting in the bulk, which when boundary anchored have been shown to compute global conformal partial waves [3, 15]. However, we will try to move away from this case by also considering interacting Wilson lines with one of the external endpoints in the bulk.
This thesis is structured as follows. Chapter 2 provides an introduction to three-dimensional gravity with negative cosmological constant, discussing its lack of local degrees of freedoms, its solutions and its asymptotic symmetry group. In Chapter3we first show how the Einstein-Hilbert action, when written in the first order formalism, equals a Chern-Simons action for the gauge group SO(2, 2) ∼= SL(2, R) × SL(2, R) and then proceed to translate some of the 3d gravity aspects of Chapter 2 to the Chern-Simons language. In Chapter 4 we establish that a Wilson line is the Chern-Simons description of a massive particle propagating in the bulk, by using the path integral representation of a Wilson line, and in Chapter5we provide a better way to evaluate the Wilson line, which treats in a completely quantum mechanical way the auxiliary degree of freedom that lives on it. In Chapter 6 we turn our attention to interacting Wilson lines, firstly showing how one can treat junctions between them and then applying this prescription to the case of certain boundary anchored Wilson line networks in a Poincar´e AdS3background. In Chapter7we study an interaction between two boundary-anchored
geodesic Witten diagram in the metric formulation. We conclude the thesis by providing in Chapter 8a summary of our results and an outlook.
The setting of this thesis will be three-dimensional gravity with negative cosmological constant. Gravity in 2+1 dimensions has been proven a useful ground for testing various ideas about quantum gravity. The two main reasons behind that are that it is more simple compared to higher dimensional cases, since it has no local degrees of freedom1, and that in three dimensions we have the luxury of describing it in two equivalent ways, both as a usual gravitational theory and as a Chern-Simons gauge theory. In this chapter we will study the metric formulation of 3d gravity, beginning with discussing some general aspects of it and moving to the prime example of a space with negative cosmological constant, namely Anti-de-Sitter spacetime. Then, in section 2.3 we will present another solution to the Einstein equations that came as a surprise: the BTZ black hole. Finally, in section 2.4 we will discuss asymptotically AdS3 spacetimes, we
will derive a stress tensor for such spacetimes and we will describe their asymptotic symmetries.
2.1
General Relativity in 2+1 dimensions
A gravitational theory in three dimensions without matter content and with cosmological constant Λ is described by the Einstein-Hilbert action:
SEH = 1 16πG3 Z M d3x√−g(R − 2Λ) , (2.1)
where G3 is the gravitational constant in three dimensions and the Lagrangian is
inte-grated over a manifold M , which we take to have Lorentzian signature. The equations of motion, obtained by varying the action with respect to the metric, are the standard Einstein field equations:
Gµν = Rµν−
1
2gµνR + Λgµν = 0 . (2.2)
1
This does not hold for all gravitational theories in three dimensions. For example, theories with massive gravitons have propagating degrees of freedom.
The Riemann tensor can be written as Rµνρσ = Cµνρσ+ 2 d − 2(gµ[ρRσ]ν− gν[ρRσ]µ) − 2 (d − 1)(d − 2)Rgµ[ρgσ]ν, (2.3) where Cµνρσ is the Weyl tensor. Looking at the Einstein equations with matter Gµν =
8πGTµν, we see that outside the different energy-momentum-stress sources described
by Tµν, the Ricci tensor is zero. This does not mean, though, that the Weyl tensor is
zero as well, since this is the part of the curvature tensor (the traceless part) that caries the data of the sources to the observers. In contrast to gravity in four dimensions, in three dimensions the number of independent components of the Riemann and the Ricci tensors, d122(d2− 1) and 1
2d(d + 1) respectively, are both six and therefore the Weyl tensor
vanishes identically. This means that curvature is concentrated only at the location of matter, there are no propagating degrees of freedom and hence no gravitational waves. In the case of vacuum Einstein equations (2.1), which we will almost exclusively consider, the vanishing of the Weyl tensor implies that for Λ = 0 every solution is flat and for Λ 6= 0 every solution has constant curvature. Taking the Newtonian limit of the Einstein equations, we can also see that there is no force between static point masses. For a thorough review of gravity in 2 + 1 dimensions the reader is referred to [16].
2.2
Anti-de Sitter spacetime
2.2.1 AdS3 as an embedded hyperboloid
Anti-de Sitter spacetime is the maximally symmetric solution (i.e. it has the maximal number of linearly independent Killing vector fields) to the Einstein’s equations (2.1) with negative cosmological constant. AdSd+1 can be described by a hyperboloid
em-bedded in a d + 2 dimensional flat space with signature (−, +, +, ..., +, −). In the d = 2 case its equation is given by:
− X02+ X12+ X22− X32 = −`2, (2.4) where the parameter ` is called the AdS-radius and is related to the cosmological con-stant via Λ = −`12.
From (2.4) it is obvious that the isometry group of AdS3 is SO(2, 2). The six Killing
vectors that generate the isometries of the hyperboloid are given in terms of the above coordinates as [17] Jab= Xb ∂ ∂Xa − Xa ∂ ∂Xb , (2.5)
where Xa = (X0, X3, X1, X2) = (U, V, X, Y ). Out of (2.5), J
01= U ∂V − V ∂U generates
vectors obey the so(2, 2) algebra under Lie brackets
[Jab, Jcd] = ηbcJad+ ηadJbc− ηacJbd− ηbdJac (2.6)
with ηab = (−1, −1, +1, +1).
2.2.2 Global coordinates
Now, let us find intrinsic coordinates on the hyperboloid. The choice
X0 = ` cosh ρ cos t, X1= ` sinh ρ sin φ, X2 = ` sinh ρ cos φ, X3= ` cosh ρ sin t,
(2.7) with ρ ∈ R+, φ ∈ [0, 2π] and t ∈ [0, 2π], solves (2.4) and leads to the following metric:
ds2= `2(− cosh2ρdt2+ dρ2+ sinh2ρdφ2) . (2.8) These coordinates are called global coordinates of AdS3 because they cover the
hyper-boloid exactly once. Since ∂tis a timelike Killing vector on the whole manifold, we can
take it as the time, which being though an angle will lead to closed timelike curves. This is why we take t ∈ R, i.e. we do not identify t with t + 2π and the resulting space is the universal covering of anti-de Sitter, which is the space that will be implied when we refer to AdS.
In order to discuss the conformal boundary of AdS spacetime in this patch we make the following change of coordinates tan θ = sinh ρ to get the metric
ds2= `
2
cos2θ(−dt
2+ dθ2+ sin2θdφ2) , (2.9)
with the new radial coordinate θ ranging from 0 to π/2. The causal structure of this patch remains unchanged when we get rid of the overall factor 1/ cos2θ and if we also
add θ = π/2 corresponding to spatial infinity, we can say that global AdS3is conformally
equivalent to a solid cylinder.
The metric (2.9) has a second-order pole at θ = π/2. When we referred previously to the conformal boundary what we meant was that we do not have a single AdS3 boundary
but a conformal class of boundary metrics corresponding to functions b(xµ) which have the following properties:
b(x) =
positive smooth in the interior b has a simple zero at π/2
(2.10)
So, for b = cos θ, b cos θeais also a valid choice and therefore we have a class of boundary metrics, which are interrelated via conformal rescalings.
replacement of the radial coordinate r = ` sinh ρ: ds2 = − 1 +r 2 l2 dt2+ 1 +r 2 l2 −1 dr2+ r2dφ2, (2.11)
with the boundary located again at r → ∞.
2.2.3 Poincar´e Coordinates
Another useful parametrization of the hyperboloid is defined through:
X0= `2 2u+ u 2`2(` 2+ x2− t2), X 1 = ux ` , X2= `2 2u+ u 2`2(−` 2+ x2− t2), X 3 = ut ` (2.12) and results in the following metric:
ds2 = ` 2 u2du 2+u2 `2(−dt 2+ dx2) . (2.13)
Here t, x ∈ R and due to the range of r > 0, these coordinates cover only one-half of the hyperboloid.
We can think of Poincar´e AdS as a stack of flat spacetimes parametrized by t and x plus an extra warped dimension r. This means that for a fixed value of r we have a surface which is conformal to two dimensional Minkowski spacetime. The conformal boundary is now located at r → ∞, where the metric has again a second order pole.
Another form of (2.13) is taken by defining z = `2/u:
ds2 = `
2
z2(dz
2− dt2+ dx2) , (2.14)
where the boundary is located at z = 0. Lastly, the transformation z = e−ρ leads to ds2= `2 dρ2+ e2ρ(−dt2+ dx2) . (2.15) The relation between Poincar´e (2.14) and global coordinates (2.9) is given by:
t0(t, z, x) = arctan 2z2t 1 + z2(1 + x2− t2) , θ(t, z, x) = arctan r z2x2+(1 − z2(1 − x2+ t2))2 4z2 ! , (2.16) φ(t, z, x) = arctan 1 − z 2(1 − x2+ t2) 2z2x ,
2.2.4 AdS3 as the group SL(2, R)
In this subsection we will show that AdS3 is isomorphic to the group SL(2, R). To do
that, we consider the matrices:
A = 1 ` " X0+ X1 X3+ X2 −X3+ X2 X0− X1 # . (2.17)
The group SL(2, R) is defined as the group of real 2 × 2 matrices with determinant one. In order, therefore, for the A matrices to belong to SL(2, R) they need to satisfy det(A) = 1. This condition leads to the hyperboloid equation that defines AdS3:
det(A) = 1 ↔ −X02+ X12+ X22− X32= −`2. (2.18) We see, thus, that each point in the hyperboloid corresponds to an SL(2, R) matrix and the isomorphism has been established.
The metric on the group manifold SL(2, R) will be given by the Killing-Cartan metric ds2 = 1
2Tr(A
−1
dA)2. (2.19)
This metric is invariant under transformations of the form A → LA and A → AR, where L, R ∈ SL(2, R). These left and right actions are independent and hence the isometry group of Lorentzian AdS3 is SL(2, R) × SL(2, R)/Z2 ∼= SO(2, 2), where we quotiented
by Z2 since (L, R) has the same action with (−L, −R).
2.3
BTZ black hole
Although gravity in 2 + 1 dimensions had become a popular testing ground for studying several ideas, it was believed that it was too simple to give important insight into real gravitational systems in 3 + 1 dimensions, since as we showed in the previous section it has no propagating degrees of freedom. Therefore, it was quite surprising when Ban˜ados, Teitelboim and Zanelli found a black hole solution in 2 + 1 dimensions [6]. This black hole2, however, has some important differences with the Schwarzschild and Kerr ones. It is asymptotically AdS rather than asymptotically flat and since all solutions to Einstein’s equations are locally AdS3, it cannot possess any curvature singularity. However, it
has an event horizon, which protects a causal singularity and it has thermodynamics properties similar to those of a black hole in 3 + 1 dimensions.
2.3.1 Metric, properties and phase space
A stationary and axisymmetric spacetime which approaches AdS3 when r → ∞ has
been shown [6] to be described in Boyer-Lindquist-like coordinates (t, r, φ), which are a
generalization of Schwarzschild coordinates, by:
ds2 = −N2(r)dt2+ dr
2
N2(r)+ r
2(dφ + Nφ(r)dt)2, (2.20)
with −∞ < t < ∞, 0 < r < ∞, φ ∈ [0, 2π] and the lapse and shift functions3 given by:
N2(r) = −8M G3+ r2 `2 + 16G23J2 r2 , N φ(r) = −4G3J r2 . (2.21)
One can explicitly check that this metric solves the vacuum Einstein field equations with negative cosmological constant. The two parameters M and J are the ADM mass and angular momentum, which are conserved charges computed by certain integrals over a circle at spacelike infinity and are associated with asymptotic invariance under time displacements and rotations respectively4.
The lapse function N (r) vanishes for two positive values of r
r± = ` p 4G3M v u u t1 ± s 1 − J M ` 2 (2.22)
where r+ is the event horizon of the black hole and r− is the inner horizon. In terms of
r±, (2.21) is written as: ds2 = −(r 2− r2 +)(r2− r2−) `2r2 dt 2+ `2r2 (r2− r2 +)(r2− r−2) dr2+ dφ −r+r− `r2 dt 2 (2.23)
and the ADM charges are:
M = r 2 ++ r2− 8G3`2 , J = r+r− 4G3` . (2.24)
The line element (2.20) describes black hole solutions for |J | ≤ M ` and M > 0, i.e. when the surfaces r = r± exist. In the extremal case |J | = M `, r+ coincides with r−
and when M → 0 we get the “empty space” solution. But (2.20) does not describe only black holes. For M > 0, |J | > M ` the singularity is exposed and for −1/8G3 < M < 0
we have solutions with a conical defect that can be thought of as particles in AdS3
sitting at the origin. When M = −1/8G3, J = 0 the conical singularity disappears and
we get global AdS3 and since solutions with naked singularities are usually discarded
from the spectrum, as they are considered unphysical, we observe that global AdS is a bound state [6] separated from the continuous black hole spectrum by a mass gap of −1/8G3.
3The names of these functions derive from the Hamiltonian formulation of general relativity, where
one defines them. For an excellent review of Hamiltonian general relativity see chapter 4 of [19].
2.3.2 BTZ from identifications
It is easy to see that the Ricci scalar R is everywhere equal to −6/`2and so the singularity at r = 0 cannot be a curvature singularity. In order to demonstrate that the singularity is of another kind, it is instructive to show how to get the BTZ black hole through identifications of AdS spacetime points [20].
The action of a Killing vector on points P ∈ AdS3 is: P → esξP , where s is in general
a continuous parameter, which we will take to be discrete with values 2πk, k ∈ Z. These points define the identification subgroup. Next, we consider a surface S0 and since all
points in S0 belong to a certain orbit of the Killing vector, we can take the surfaces
S(2πk) by applying the identification subgroup. Finally, in order to get the quotient space we identify these surfaces.
Since the transformations S(0) → S(2πk) are isometries, the resulting quotient space inherits from AdS3 a well-defined metric of constant negative curvature which remains
a solution to the Einstein equations. Since curves joining points that belong to the same orbit will be closed in the quotient space, we have to make sure that there will be no closed null or timelike curves. The necessary condition for that is
ξ · ξ > 0 , (2.25)
i.e. the Killing vector ξ should be spacelike.
In [17] it is shown that the non-extremal BTZ black hole solutions5 are obtained by making identifications using the Killing vector
ξ = r+ ` J12−
r−
` J03, (2.26)
where Jab are given by (2.5). Moreover, one can show that in the coordinate system,
where the above Killing vector is just ξ = ∂φ, AdS3 is represented by the metric (2.20)
but with φ ∈ R. We see, therefore, that in order to take the BTZ black hole we have to periodically identify points along ξ, which is an allowed process since the metric does not depend on φ and therefore gµν will not become a multivalued function of it.
The final thing to do is to check whether ξµis everywhere spacelike and if not remove the regions where ξ2 ≤ 0. It turns out that at r = 0, ∂φbecomes timelike and therefore
r = 0 is a causal singularity. We find, thus, another similarity with 3+1 dimensional black holes: the spacetime is geodesically incomplete and the only incomplete geodesics are those that reach the singularity.
As a final remark, even though the BTZ black hole is derived from identifications in a spacetime with six isometries, it has only 2 Killing vectors ∂t and ∂φ, since some
of the other vectors become multivalued and thus cannot be used as symmetries of the new space.
2.4
Asymptotic symmetry group
In the context of the AdS/CFT correspondence global properties of a spacetime are particularly relevant. In the case of AdS3they are even more important since the absence
of local degrees of freedom means that all physics is encoded at the boundary. In this section we will begin by defining what we mean by an asymptotically AdS spacetime (aAdS). We will continue by showing how to properly define a gravitational action for aAdS spacetimes, which will lead us to the derivation of a stress tensor for AdS3. We
will conclude with discussing the properties of this stress tensor and the appearance of an unexpected asymptotic symmetry group for aAdS3 spacetimes.
2.4.1 Asymptotically AdS3 spacetimes
The BTZ black hole is an example of an asymptotically AdS spacetime. In general, any asymptotically AdS3 spacetime can be written in the Fefferman-Graham form [21]:
ds2 = `2dρ2+ gij(xk, ρ)dxidxj, 0 < ρ < ∞, i, j, k = 1, 2 (2.27)
with gij having the following ρ expansion:
gij(xk, ρ) = e2ρ/`gij(0)(xk) + gij2(xk) + . . . . (2.28)
g(0)ij corresponds to the boundary metric and is only defined up to Weyl transformations, while the omitted terms are subleading terms in ρ, scaling at least as e−ρ/`.
In [22] it was shown that the most general solution to the three-dimensional vacuum Einstein equations with cosmological constant Λ = −1/`2 and up to trivial diffeomor-phisms is:
ds2 = `2dρ2+8πG3` L(dx+)2+ ¯L(dx−)2− `2e2ρ+ (8πG3)2LL¯e−2ρ dx+dx−, (2.29)
whereL(x+) and ¯L(x−) are arbitrary functions that parametrize the space of solutions. For example, global AdS3 is obtained for 2πL= 2π ¯L= −`/16G3 and a BTZ black hole
corresponds to 2πL= `(r++r−)2
16G3 and 2π ¯L= `
(r+−r−)2
16G3 .
The metric (2.29) is of the Fefferman-Graham form (2.28) and describes all possible asymptotically AdS3 spacetimes. The finite number of terms that this metric has stems
from the fact that in three dimensions the Fefferman-Graham expansion terminates at second order [23].
2.4.2 A stress tensor for AdS3
In general relativity it is not possible to define a local energy-momentum density for the gravitational field. We can see why this is the case by considering how we define a local energy density in Newtonian gravity or electromagnetism, where it is simply
proportional to the square of the field in question. However, in general relativity the gravitational field is zero in a locally flat coordinate system6. So, what Brown and York [25] did was to define a so-called quasilocal stress tensor at the boundary of a given spacetime region inspired by the Hamilton-Jacobi theory of classical mechanics, where the energy of a classical solution is given by E = −∂Scl/∂t with Scl the classical action.
In gravity, time intervals are measured by the metric and so the natural generalization of it is Tµν = √2 −γ ∂S ∂γµν , (2.30)
where the gravitational action S is considered a functional of the boundary metric γµν.
Let us consider the following gravitational action:
S = SEH+ SGHY+ Sct, (2.31)
where SEH is the Einstein-Hilbert action, SGHY is the Gibbons-Hawking-York term
and Sct is a counter-term, the role of which will be explained shortly. The
Gibbons-Hawking-York term is needed in the presence of a boundary in order for the action to be stationary around classical solutions, since the variation of the E-H term has a bulk piece proportional to the equations of motion and a boundary piece (coming from an integration by parts), which for boundary conditions that only fix the metric δg|∂M = 0
and not its derivative, does not vanish.
Now, in order to derive the stress-tensor let us consider spacetimes with the following metric:
ds2= dρ2+ gij(xk, ρ)dxidxj. (2.32)
The GHY term is given then by SGHY = 1 8πG Z ∂M d2x√gTrK, Kij = 1 2∂ρgij, (2.33)
where K is the extrinsic curvature7. The on-shell variation of the action with respect to gij (without the counterterm) will be
δS = − 1 16πG Z ∂M d2x√g(Kij− Kgij)δg ij. (2.34)
The quasilocal stress-energy tensor is defined then as: Tij = − 1
8πG(K
ij − TrKgij) , (2.35)
which is a result derived by Brown and York.
6
A thorough investigation of this issue is done in section 11.2 of [24].
Now, if we specialize on asymptotically AdS3 spacetimes, i.e. use
gij = e2ρ/`gij(0)+ gij(2)+ · · · (2.36)
we need to find a variational principle where the boundary metric gij(0) is held fixed and the subleading metric terms are allowed to vary. However, when ∂M goes to the AdS boundary, divergences arise and we have to add a counterterm to the action. This is allowed since it does not affect the equations of motion. The counterterm for AdS3 was
found in [26] to be8 Sct= − 1 8πG` Z ∂M d2x√g . (2.37)
This counterterm is uniquely determined by requiring it to be a local and covariant function of the intrinsic geometry of the boundary that cancels the arising divergences. The final result is, therefore:
Tij =
1 8πG`
g(2)ij − Tr(g(2))gij(0) . (2.38) One can show that this tensor has all the properties of a CFT2 one. For example, it has
a nonzero trace [27] Tr(T ) = − 1 8πG3` Tr(g(2)) = − ` 16πGR (0), (2.39)
which comparing to the CFT Weyl anomaly Tr(T ) = −24πc R gives the central charge c = 2G3`
3 that was firstly computed by Brown and Henneaux [12]. If we take g
(0)
ij dxidxj =
dzd¯z the only non-vanishing components are: Tzz = 1 8πG3` gzz(2)=L, Tz ¯¯z = 1 8πG3` g(2)¯z ¯z = ¯L (2.40)
and we observe that our stress tensor has a holomorphic/anti-holomorphic factorization. In the above expressionsLand ¯Lare the arbitrary functions we had in the metric (2.29). Finally, one can show [26] that the AdS3 stress tensor under conformal transformations
transforms as the CFT2 one.
2.4.3 Asymptotic symmetry group
Having this stress-tensor we can now define conserved charges in our gravitational theory [25]. For an asymptotically AdS spacetime in order to do so, we have to specify a spacelike surface Σ at the boundary described by the induced metric σab. Then, the
conserved charge associated with a Killing vector ξµ that generates some isometry at
8The authors compute the stress tensor for AdS
the boundary is given by:
Qξ=
Z
Σ
dd−1x√σ(uµTµνξν) , (2.41)
where uµ is the timelike unit normal vector to Σ. For example, the energy or mass of the spacetime is given by the conserved charge corresponding to time translations. Not all symmetries have, though, non-vanishing charges. In order to see which have and which don’t and how this relates to the asymptotic symmetry group, we have to make the distinction between global and local diffeomorphims.
General relativity is a diffeomorphism invariant theory, but only under local diffeo-morphisms. These are like gauge symmetries, since they only relate physically equivalent configurations. On the other hand, diffeomorphisms that reach the boundary are impor-tant, since they relate physically distinct solutions and are therefore actual symmetries. For example, a time reparametrization of the form t → t0(t, x) with t0 → t when r → ∞ is a local diffeomorphism, whereas a global shift t → t + a represents true time evolution. Another example is electrodynamics, where the action is invariant under infinitesimal transformations of the form δAµ= ∂µα(x). A local gauge symmetry, i.e. one for which
α(x → ∞) → 0, has zero charge and is a trivial gauge transformation, which is not surprising, since we know that electromagnetism has only one conserved charge, the electric charge which is associated to a global U (1) rotation. The asymptotic symmetry group is defined, therefore, as the group of symmetry transformations mod trivial gauge transformations.
In gravity the asymptotic symmetry group is generated by the conserved charges, which are associated with asymptotic Killing vector fields, i.e. fields that obey
∇(iξj)→ 0 at infinity . (2.42)
When we say that a charge generates a symmetry we mean that the Poisson bracket of the charge Q[ξ] with any function A of the canonical variables is
{Q[ξ], A} =LξA , (2.43)
where Lξ denotes a Lie derivative along ξ.
Now, following [28] we will provide a rough sketch of the original derivation for the asymptotic symmetry group of AdS3 that was done by Brown and Henneaux in 1986
[12] and is considered one of the precursors to the AdS/CFT correspondence. Let us begin with global AdS3 in the following form:
ds2 = − 1 +r 2 `2 dt2+ 1 +r 2 `2 −1 dr2+ r2dφ2. (2.44)
which approaches in a certain way the metric (2.44) in the limit r → ∞. This asymptotic behaviour can be expressed in terms of fall-off conditions at the boundary, which Brown and Henneaux [12] found to be
gµν = grr grφ grt gφr gφφ gφt gtr gtφ gtt = `2 r2 +O(r −4) O(r−3) O(r−3) O(r−3) r2+O(1) O(1) O(r−3) O(1) −r`22 +O(1) . (2.45)
In order to find these conditions, they acted on the above metric with all possible anti-de Sitter transformations, since we want the asymptotic symmetry group to at least contain the AdS3 one. We also want our boundary conditions to allow a wide class of
asymptotically AdS3 spacetimes, like the BTZ black hole, which even though was not
known until 1992, it is indeed included in the above conditions.
Having now (2.45), the asymptotic symmetries are described by vector fields ξµ, which transform metrics of the form (2.45) into themselves. These vector fields are, therefore, solutions to the following equations
Lξgtt =O(1), Lξgtr =O(r−3), Lξgrr =O(r−4) . . . . (2.46)
The most general solution ξ = ξµ∂µ can be written as
ξt= `(T + ¯T ) + ` 3 2r2(∂ 2T + ¯∂2T ) +¯ O(r−4) , ξφ= (T − ¯T ) − ` 2 2r2(∂ 2T − ¯∂2T ) +¯ O(r−4) , (2.47) ξr = −r(∂T + ¯∂ ¯T ) +O(r−1) , where T ≡ T (x+), ¯T ≡ ¯T (x−), x±= t/` ± φ, ∂ ≡ 12(`∂t+ ∂φ) and ¯∂ ≡ 12(`∂t− ∂φ).
In the above vector fields the subleading terms O(. . . ) are arbitrary and represent pure gauge transformations, since any vector whose components behave only like the
O(. . . ) terms connects configurations which are physically equivalent and has no asso-ciated charge.
Due to the periodicity of the φ coordinate, we can expand T and ¯T in Fourier modes: T ∼ eimx+ and ¯T ∼ eimx−. Doing so, we find
ξm = eimx + ∂ −` 2m2 2r2 ∂ −¯ imr 2 ∂r , ¯ ξm= eimx − ¯ ∂ −` 2m2 2r2 ∂ − imr 2 ∂r . (2.48)
These modes obey two copies of the Witt algebra [ξm, ξn] = (m − n)ξm+n,
[ ¯ξm, ¯ξn] = (m − n) ¯ξm+n,
[ξm, ¯ξn] = 0 .
(2.49)
The asymptotic symmetry algebra will be generated by the conserved charges Lm =
Q[ξm] and ¯Lm= Q[ ¯ξm] associated with ξm and ¯ξm respectively. The algebra that these
charges obey is isomorphic to the algebra that the asymptotic Killing vectors obey up to a possible central extension
{Q[ξ], Q[η]} = Q[[ξ, η]] + K[ξ, η] (2.50)
where { } denote Poisson brackets9.
The phase space function that generates a flow along a vector ξµ is given by H[ξ] =
Z
dnxξµHµ(x) + Q[ξ] , (2.51)
whereHµcontains the usual Hamiltonian and momentum constraints that we encounter
in the Hamiltonian formulation of general relativity10.The charge Q[ξ] is a surface in-tegral that is determined in such a way that its variation cancels the unwanted surface terms in the variation of H[ξ]. This fixes Q[ξ] up to an arbitrary constant, which is chosen such that Q[ξ] vanishes for global AdS3. Now, noting that the following Dirac
bracket
{Q[ξ], Q[η]} = δηQ[ξ] (2.52)
equals the change of Q[ξ] under the surface deformation generated by Q[η] and evaluating (2.50) for global AdS, the computation of the central charge reduces to
K[ξ, η] = δηQ[ξ] . (2.53)
For the particular form of the charges and the explicit computation, we refer the reader to the original paper of Brown and Henneaux [12], who found that the conserved charges obey {Lm, Ln} = (m − n)Lm+n+ c 12m(m 2− 1)δ m+n,0, { ¯Lm, ¯Ln} = (m − n) ¯Lm+n+ c 12m(m 2− 1)δ m+n,0, {Lm, ¯Ln} = 0 , (2.54) 9
Actually in the most general case, { } denote Dirac brackets, which are a generalization of the Poisson brackets in the case of physical systems with secondary class constraints. For more details see e.g. [29] .
with central charge
c = 3`
2G. (2.55)
We see, therefore, that studying the asymptotic symmetry group of spacetimes that approach AdS3, which has the six generator SL(2, R) × SL(2, R) as its symmetry group,
we end up with two copies of the infinite dimensional Virasoro algebra, which is the algebra of the conserved charges of a CFT2.
gauge theory
In this chapter we will present the second way in which we can study gravity in 2 + 1 dimensions, which is that of a Chern-Simons gauge theory. This was discovered by Achucarro and Townsend [13] and later by Witten [14]. We will begin by reviewing the first-order, Palatini or vielbein formalism of general relativity, which we will use in order to explicitly show following [14] how 3d gravity can be reformulated as a Chern-Simons theory. Using this new formalism, we will then express some aspects of 3d gravity mentioned in the previous chapter in the Chern-Simons language.
3.1
First-order formalism of General Relativity
In order to demonstrate the relation between General relativity and Chern-Simons the-ory we will have to use a non-coordinate basis for our tangent spaces. Usually, we take the tangent space at a point of our manifold to be spanned by the vector fields {∂/∂xµ} and the cotangent space to be spanned by their dual 1-forms {dxµ}. Instead, we are now going to consider the following vector fields as basis vectors1:
ˆ
e(a)= eµaeˆµ= eµa∂µ (3.1)
and similarly their dual 1-forms eaµdxµ. The auxiliary quantity eaµ is an invertible n × n matrix and is called frame field or vielbein (from the German for “many legs”)2. The inverse frame field eµa satisfies eµaeaν = δ
µ
ν and eaµe µ
b = δba. Notice that we use now
two kinds of indices; Greek indices correspond to the coordinate system, while Latin indices represent the non-coordinate basis. These non-coordinate vectors are chosen to be orthonormal
g ˆe(a)eˆ(b) = gµνeµaeνb = ηab, (3.2)
where in the case of a Euclidean signature the above condition would have δab in the
right-hand side.
1For a detailed presentation of the vielbein formalism see Appendix J of [30]. 2
In three dimensions we call it dreibein and in four vierbein or tetrad (from the Greek word tetras for four).
Using the vielbein, we can transform between Greek and Latin indices:
Vµν = eµaVaν = ebνVµb= ebνeµaVab, (3.3) where the Greek indices are lowered and raised using gµν and the Latin using ηab.
For a given frame field eµa(x) the metric gµν is uniquely determined. However, for
a given metric, the vielbein is not unique. Indeed, from gµν = eaµebνηab we can see that
vielbeins related by a Lorentz transformation e0aµ = Λabebµ at each point give the same metric gµν. Therefore, using this basis we have the freedom of performing local Lorentz
transformations along with our usual freedom of performing coordinate transformations. In fact, we can do both at the same time:
Ta0µ0b0ν0 = Λa 0 a ∂xµ0 ∂xµΛ b b0 ∂xν ∂xν0T aµ bν. (3.4)
In order to take covariant derivatives of tensors in the non-coordinate basis, we have to define the spin connection3 ωa
µ b. Then, we have:
∇µTab = ∂µTab+ ωµ ac Tcb− ωµ cb Tac. (3.5)
The condition of metric compatibility ∇µηab implies that the spin connection should be
antisymmetric in its frame indices: ωabµ = −ωbaµ. The spin connection is related to the Christoffel one via
Γνµλ= eνa∂µeλa+ eνaeλbωµ ba , ωµ ba = eνaeλbΓµλν − eλb∂µeλa. (3.6)
Having introduced the frame indices, we can know think of various objects as tensor-valued differential forms. In particular, any tensor that possesses some antisymmetric lower Greek indices and some Latin indices can be considered a differential form taking values in some tensor bundle. Having this viewpoint allows us to use all useful opera-tions such as exterior derivatives and wedge products to manipulate these forms4. For example, we can write the vielbein and the spin connection as:
ea= eµadxµ, ωab= ωµ ba dxµ. (3.7) Using these we can express the torsion Tµνa and the Riemann tensor Rabµν as
Ta= dea+ ωab∧ eb, (3.8)
Rab = 1 2R
ab
µνdxµ∧ dxν = dωab+ ωac∧ ωcb, (3.9)
which are known as the first and second Cartan structure equations respectively.
3The name derives from the fact that the spin connection is used to take covariant derivatives of
spinors.
Now, we have all the required ingredients to express the Einstein-Hilbert action (2.1) in the first-order formalism. Let us firstly express √−g in terms of the vielbein5:
√
−g =q− det(ea
µebνηab) =
√
e2 = e . (3.10)
The Ricci scalar can be written as:
R = Rµνµν = Rαβγδgαµgβνgγµgδν = Rαβγδδαγδβδ = 1 2R αβ γδδ γ [αδ δ β] (3.11)
and we will also need the following relations: e = 1 3!abc µνρea µebνecρ ⇒ µνρe = abceaµebνecρ, δ[αγδβ]δ = καβκγδ, d3x√−g = edx0dx1dx2 = 1 3!eµνρdx µ∧ dxν∧ dxρ= 1 6abce a∧ eb∧ ec. (3.12)
So, we can write now: SEH = 1 16πG Z M d3x√−g(R − 2Λ) = 1 16πG Z M d3xe 1 2δ ρ [µδ σ ν]R µν ρσ− 2Λ = 1 16πG Z M d3xe 1 2λµν λρσRµν ρσ− 2Λ = 1 16πG Z M d3xabc 1 2e a λebµecνλρσRµνρσ− Λ 3e a∧ eb∧ ec (3.13) = 1 16πG Z M d3xabc 1 2 λρσea λRbcρσ− Λ 3e a∧ eb∧ ec = 1 16πG Z M abc ea∧ Rbc− Λ 3e a∧ eb∧ ec = 1 16πG Z M 2ea∧ Ra− Λ 3abce a∧ eb∧ ec ,
where we defined Ra≡ 12abcRbc= dωa+ 21abcωb∧ ωc, with ωµa= 12 abcω
bcµ.
Using Rabµν = ∂µωνab− ∂νωabµ + ωµacωνcb − ωνacωbµc we can write the above action in
the following form as well: SEH = 1 16πG Z M d3xµνρ eµa(∂νωρa− ∂ρωνa) + abceaµωνbωρc− Λ 3abce a µebνecρ . (3.14)
5We will assume that det(ea µ) > 0.
3.2
Chern Simons action
The Chern-Simons action for a gauge group G is given by:
SCS[A] = k 4π Z M Tr A ∧ dA +2 3A ∧ A ∧ A . (3.15)
In the above expression M represents a (2+1)-dimensional manifold and the constant k is called the Chern-Simons level. The gauge field A is a Lie algebra-valued one-form A = Aµdxµ = AaµTadxµ, where Ta are the generators of the Lie algebra that generates
the gauge group G. Tr denotes a non-degenerate invariant bilinear form on the Lie algebra6, which can be used to raise and lower the indices of the Lie algebra generators and so can be thought of as a metric of the algebra that contracts the generators in the above expression Tr(A ∧ dA) = Tr(TaTb)Aa∧ dAb.
The action (3.15) is invariant under a gauge transformation7
Aµ→ gAµg−1+ g∂µg−1, (3.16)
where g = eiαa(x)Ta is a member of the gauge group G with αa denoting a spacetime
dependent infinitesimal parameter. The infinitesimal version of this transformation is Aµ→ Aµ− Dµα ≡ Aµ− (∂µα + [Aµ, α]) , (3.17)
where Dµ is the covariant derivative.
In order to find the equations of motion we vary the action with respect to A δSCS[A] =
k 4π
Z
M
Tr (δA ∧ dA + A ∧ dδA + 2δA ∧ A ∧ A)
= k
4π Z
M
Tr ((−d(A ∧ δA) + 2δA ∧ (dA + A ∧ A))
= k
4π Z
M
Tr (2δA ∧ (dA + A ∧ A)) − k 4π
Z
∂M
Tr (A ∧ δA) .
(3.18)
For suitable boundary conditions [33] the second term vanishes and the equation of motion is:
F ≡ dA + A ∧ A = 0 , (3.19)
6
A non-degenerate invariant bilinear form, or an invariant metric, is a bilinear form h· , ·i : g × g → K on a Lie algebra g with field K which has the following three properties:
1. Symmetry: hX, Y i = hY, Xi ∀ X, Y ∈ g.
2. Invariance: h[Z, X], Y i + hX, [Z, Y ]i = 0 ∀ X, Y, Z ∈ g. 3. Non-degeneracy: If hX , Y i = 0 ∀ Y ∈ g then X = 0.
7The action (3.15) is invariant under (3.16) for suitable boundary conditions and for gauge
transfor-mations that are continuously connected to the identity transformation (small gauge transfortransfor-mations). For large gauge transformations, the action transforms as SCS → SCS− 2πkN , with N ∈ Z. If the
Chern-Simons level is taken to be an integer, the path integral, where the action appears as eiSCS, will
where F is the field strength two-form. From these equations we can see that locally A is a gauge transformation of the trivial field configuration
A = g−1dg , (3.20)
or in other words A is pure gauge, which shows that a Chern-Simons theory is purely topological.
3.3
SO(2, 2) Chern-Simons Theory as Λ < 0 Gravity
The isometry group of AdS3 is SO(2, 2) and since we want to show that in (2+1)
dimensions pure gravity is equivalent to a Chern-Simons theory we will take SO(2, 2) to be the gauge group G. This is what Witten in [14] showed in the case of a negative cosmological constant, whereas for Λ = 0 the gauge group is ISO(2, 1) and for Λ > 0 SO(3, 1).
The so(2, 2) algebra has six generators, three translations Pa and three rotations
Ja ≡ 12abcJbc, where Jab are the usual Lorentz generators. Using these generators
so(2, 2) is written as:
[Ja, Jb] = abcJc, [Ja, Pb] = abcPc, [Pa, Pb] = abcJc. (3.21)
The invariant metric of the above algebra is [34]
hJa, Jbi = 0 = hPa, Pbi , hJa, Pbi = ηab. (3.22)
In order to make the connection with gravity, we should take the gauge field to be Aµ=
1 `e
a
µPa+ ωaµJa, (3.23)
where ` is the AdS radius and we identify the flat indices of the vielbein and spin connection with the Lie algebra ones. Using this gauge field the first term of (3.15) becomes: Tr(A ∧ dA) = 1 `e aP a+ ωaJa, 1 `de bP b+ dωbJb = 1 `(e a∧ dωb+ ωa∧ deb)η ab = 1 `(e a∧ dωb− dωa∧ eb+ d(ωa∧ eb))η ab= 2 `e a∧ dω a, (3.24)
where in order to get to the final expression we threw away the total derivative term and used the antisymmetry of the wedge product.
The second term of (3.15) can be written as 2 3Tr(A ∧ A ∧ A) = 1 3Tr([Aµ, Aν]Aρ)dx µ∧ dxν∧ dxρ = 1 3Tr([ 1 `e a µPa+ ωaµJa, 1 `e b νPb+ ωνbJb]Aρ)dxµ∧ dxν∧ dxρ = 1 3Tr[( 1 `2e a µebνabdJd+ ωµaωbνabdJd+ 2 `e a µωνbabdPd)× (3.25) × (1 `e c ρPc+ ωρcJc)]dxµ∧ dxν ∧ dxρ= 1 3`3abce a∧ eb∧ ec+1 `abce a∧ ωb∧ ωc.
Adding (3.24) and (3.25) we finally have: SCS = k 4π` Z M 2ea∧ dωa+ abcea∧ ωb∧ ωc + 1 3`2abce a∧ eb∧ ec. (3.26)
Comparing with (3.13) for ΛAdS = −1/`2, we see that the SO(2, 2) Chern-Simons
action is equal to the Einstein-Hilbert one if we set the Chern-Simons level to k = `
4G. (3.27)
Showing that the Chern-Simons and Einstein-Hilbert actions are the same is, though, not enough in order to claim that the respective theories are equivalent. We should also make sure that the transformation laws of the vielbein and spin connection under an infinitesimal gauge transformation with parameter u = ρaPa+ τaTa
δeaµ= −∂µρa− abceµbτc− abcωµbρc,
δωµa= −∂µτa− abcωµbτc+ 1 l2 abce µbρc (3.28)
coincide with the transformations of (2+1) dimensional gravity. In [14] it is shown that this is the case on shell and up to local Lorentz transformations.
3.3.1 SL(2, R) × SL(2, R) Chern-Simons Theory as Λ < 0 Gravity
Since there is an isomorphism between SO(2, 2) and SL(2, R) × SL(2, R) we expect that a Chern-Simons theory with gauge group SL(2, R) × SL(2, R) will also be equivalent to pure AdS gravity. This splitting of the SO(2, 2) group translates in terms of the Lie algebras to so(2, 2) ' sl(2, R) ⊕ sl(2, R). We can express the generators of the two copies of sl(2, R) in terms of those of so(2, 2) as Ja±≡ 12(Ja± Pa) with the plus and minus signs
denoting the right and left algebras respectively. The algebra (3.21) reads now: [Ja+, Jb+] = abcJ+c, [Ja−, J
−
b ] = abcJ −c
The two gauge fields are taken to be: A = (1 `e a+ ωa)J+ a, A = (¯ 1 `e a− ωa)J− a . (3.30)
One can then easily show following the lines of our previous proof that the following linear combination gives the Einstein-Hilbert action:
SCS[A] − SCS[ ¯A] = SEH[e, ω] . (3.31)
The equations of motion are again the conditions that the gauge connections are flat: F ≡ dA + A ∧ A = 0, F ≡ d ¯¯ A + ¯A ∧ ¯A = 0 . (3.32) Combining them we get
1 2(F + ¯F ) = R + 1 `2e ∧ e = 0 , (3.33) 1 2(F − ¯F ) = de + ω ∧ e = 0 . (3.34)
The first of these is the vacuum Einstein equation with negative cosmological constant and the second is the torsion free equation.
Some comments
Before concluding this section, we would like to make a few comments about Chern-Simons theories.
• There exist Chern-Simons actions only in odd dimensions [34].
• For a given gauge group G, the existence or not of a Chern-Simons action depends on whether there is a non-degenerate invariant bilinear on the corresponding Lie algebra. For some cases, the SO(2, 2) being one of these, there can be more than one.
• The equivalence between Chern-Simons theory and gravity was proven only at the classical level. It is not certain if this correspondence holds at the quantum level as well.
3.4
Asymptotic symmetry group in CS formulation
When we computed the variation of the Chern-Simons action in (3.18), we said that for suitable boundary conditions the boundary term vanishes. Now, we are going to spell out these conditions, but before doing that we will show how one defines global charges in a Chern-Simons theory with boundary, which will be useful later on when we will derive the asymptotic symmetry algebra in the Chern-Simons formulation.
3.4.1 Global charges
We consider again a Chern-Simons theory with gauge group G generated by the Lie algebra g defined on M = R × Σ, where Σ is a two dimensional manifold with boundary ∂Σ ∼= S1. By decomposing the gauge field in the following way
A = Atdt + Aidxi, (3.35)
with the time coordinate parametrizing R, the action becomes SCS = k 4π Z M dt ∧ dxi∧ dxjTr(AtFij − AiA˙j) + k 4π Z R×S1 dt ∧ dxiTr(AtAi) . (3.36)
The Poisson bracket of two differentiable phase-space functionals F [Ai] and H[Ai] is
given by {F, H} = 2π k Z Σ dxi∧ dxjTr δF δAi(x) δH δAj(x) . (3.37)
From (3.36) it is evident that Atis a Lagrange multiplier and thus its equation of motion
leads to the first class constraint
G = k
4π
ijF
ij = 0 , (3.38)
which generates gauge transformations. By this we mean that the smeared generator G(Λ) = k
4π Z
Σ
dxi∧ dxjTr(ΛFij) + Q(Λ) , (3.39)
has the following Poisson bracket
{G(Λ), Ak} = DkΛ = δΛAk, (3.40)
where Dk is the gauge-covariant derivative.
In (3.39) Q(Λ) is a boundary term, which is added in order for the smeared generator to be differentiable8. If the gauge parameter Λ is independent of the gauge fields, Q(Λ) is given by Q(Λ) = − k 2π Z ∂Σ dxiTr(ΛAi) (3.41)
and the smeared generators satisfy the following Poisson algebra: {G(Λ), G(Γ)} = G([Λ, Γ]) + k
2π Z
∂Σ
dxiTr(Λ∂iΓ) . (3.42)
Gauge transformations with gauge parameters Λ such that Q(Λ) = 0 are proper gauge transformations, whereas gauge transformations for which Q(Λ) is non-zero are
8
By differentiable we mean that the functional derivative of the smeared generator has no ill-defined surface term.
global symmetries, which transform physically inequivalent configurations into each other.
As expected, the Q(Λ) are the charges of the Chern-Simons theory and generate global symmetries
δΛF = {Q(Λ), F } , (3.43)
where F is a general phase-space functional. Finally, these charges obey the same algebra with the smeared generators but now under Dirac brackets
{Q(Λ), Q(Γ)} = Q([Λ, Γ]) + k 2π Z ∂Σ dxiTr(Λ∂iΓ) . (3.44) 3.4.2 Boundary conditions
In order to find the boundary conditions for which the boundary term in (3.18) vanishes, it is best to work in light-cone coordinates x±= t/` ± θ, where θ parametrizes the S1 at the boundary. The boundary term in the variation of the Chern-Simons action becomes in these coordinates δSCS boundary= − k 4π Z R×S1 Tr(A ∧ δA) = − k 4π Z R×S1
dx+dx−Tr(A+δA−− A−δA+) .
(3.45) If we impose
A−= 0 and δA0 = 0 at the boundary , (3.46)
the boundary term will always vanish. Let us now pick a gauge in which
Aρ= b−1(ρ)∂ρb(ρ) , (3.47)
where ρ is the radial coordinate of the disc Σ. To show that this choice is always possible, we start with a gauge field A0 and we perform a gauge transformation U in order to cast it in the form of (3.47)
U−1A0ρU + U−1∂ρU = b−1∂ρb . (3.48)
Expressing U as U0b, (3.48) becomes
∂ρU0= −A0ρU0, (3.49)
which is always solved by the following path-ordered exponential: U =Pe−RρA0ρdρ0U
0b . (3.50)
where U0 is chosen such that U = 1 at the boundary, in order for the boundary condition
Let us now employ the equations of motion. Fρθ = 0 gives
∂ρAθ+ [Aρ, Aθ] = 0 , (3.51)
which has as solution
Aθ(t, ρ, θ) = b−1(ρ)a(t, θ)b(ρ) , (3.52)
with a(t, θ) being an arbitrary g valued function. Fρ0 fixes the ρ-dependence of A0 and
the boundary condition A− = 12(A0− Aθ) = 0 implies then that A0 = Aθ not only at
the boundary but everywhere, since the radial dependence is completely fixed. Finally, Ftθ= 0 gives
∂tAθ− ∂θAt+ [At, Aθ] = (∂t− ∂θ)Aθ= 0 → ∂−a(t, θ) = 0 . (3.53)
We see, thus, that the gauge fields A are parametrized by a(x+). Therefore, in the gauge (3.47) and with boundary conditions (3.46), the connection is given by
Aρ= b−1(ρ)∂ρb(ρ), A−= 0, A+ = b−1(ρ)a(x+)b(ρ) . (3.54)
Similarly, for the right sector we will have ¯
Aρ= b(ρ)∂ρb−1dρ, A¯+= 0, A¯−= b−1(ρ)¯a(x+)b(ρ) . (3.55)
One can show [33] that the gauge choice (3.47) is preserved by gauge transformations with parameters
Λ(t, ρ, θ) = b−1(ρ)λ(t, θ)b(ρ), ∂−λ = 0 . (3.56)
3.4.3 Asymptotically AdS3 spacetimes
As discussed in [33], the boundary conditions (3.54) are not enough to fully describe asymptotically AdS3 spacetimes and they result in an affine Lie algebra for the
asymp-totic symmetry group and not the Virasoro with central charge c = 3`/2G that we have wanted. We have, therefore, to supplement (3.54) with [33]
(A − AAdS)
boundary=O(1) (3.57)
and similarly for the right sector. Connections that satisfy this asymptotic behaviour correspond to asymptotically AdS spacetimes. In (3.57) AAdS and ¯AAdSare given by
AAdS= b−1 L1+ 1 4L−1 bdx++ b−1∂ρbdρ , ¯ AAdS= −b 1 4L1+ L−1 b−1dx−+ b∂ρb−1dρ , (3.58)
with b(ρ) = eρL0. These connections are special cases of A = b−1 L1+ 2π k L(x +)L −1 bdx++ b−1∂ρbdρ , ¯ A = −b 2π k ¯ L(x−)L1+ L−1 b−1dx−+ b∂ρb−1dρ , (3.59)
which are the analogues of (2.29) in the Chern-Simons formulation and satisfy (3.54) and (3.55). Connections that have different values for L and ¯L are physically distinct, since they cannot be related by true gauge transformations.
3.4.4 Asymptotic symmetry algebra
Let us now derive the asymptotic symmetry algebra in the Chern-Simons formulation. We saw previously that the aAdS3 connections are parametrized by a(x+). Let us
expand a(x+) in the {Li} basis:
a(x+) = `−1L−1+ `0L0+ `1L1. (3.60)
Evaluating the coefficient of dx+in (3.58) we get (eρL1+ e−ρL−1) dx+, which shows that
at the boundary ρ → ∞ the first term is dominant and we should, therefore, impose `1= 1. Moreover, using the residual gauge symmetry we can set `0 = 0 and a(x+) takes the form a(x+) = L1+ 2π k L(x +)L −1, (3.61)
where we chose a convenient form for the factor of L−1. We expand now λ(x+), which
describes the global gauge transformations (3.56) of the connections parametrized by a(x+), in the {Li} basis
λ(x+) = −1L−1+ 0L0+ 1L1 (3.62)
and identify those transformations that leave the structure of (3.60) invariant. Using ≡ 1 and denoting x+ derivatives by primes, we get
0 = −0, −1 = 1 2 00+2π k L. (3.63)
Under this transformationL transforms as
δL= L0+ 20L+
k 4π
which shows that L plays the role of the energy momentum tensor in the boundary theory. The charges that generate (3.64) are given by (3.41)
Q(λ) = Z
∂Σ
L. (3.65)
Using (3.43) we now get
{L(θ),L(θ0)} = −(δ(θ − θ0)L0(θ) + 2δ0(θ − θ0)L(θ)) − k 4πδ
000
(θ − θ0) . (3.66) If we expandL in Fourier modes
L(θ) = − 1 2π
X
n∈Z
Lne−inθ, (3.67)
shift the zero mode in the following way
L0 →L0−
k
4 (3.68)
and substitute {. . . } → −i[. . . ] we finally get [Ln,Lm] = (n − m)Ln+m+
c 12(n
3− n)δ
n+m,0, (3.69)
Part I
In this chapter we introduce Wilson lines, which are the natural observables in the Chern-Simons theory and the main objects of interest of this thesis. We will follow [1], the authors of which resorted to Wilson lines in order to derive a generalization of the Ryu-Takayanagi prescription [35] for the holographic computation of entanglement entropy in CFT’s with higher spin fields, where there is no meaningful notion of proper distance and hence the conventional Ryu-Takayanagi prescription can not be used. Their proposal was
SEE = − log(WR(C)) , (4.1)
which implies that in the spin 2 setting the Wilson line must be related to geodesic length. We will indeed show that this is the case. In section4.1we define the Wilson line and in section4.2we present its path integral representation, treating in a semiclassical way the auxiliary field that lives on it. In section 4.3 we construct the action for this probe field, showing that computing it on-shell the geodesic distance arises, and finally, in section 4.4we demonstrate for an open interval a different way to evaluate a Wilson line.
4.1
Wilson line as a massive probe of the bulk
The Wilson loop is defined as
WR(C) = TrR Pexp − I C A . (4.2)
Here R is a representation of the gauge group G, which we choose to be SL(2, R) × SL(2, R). In the right-hand side, we trace the path ordered exponential of the connection over the specific representation we pick. The path ordering is needed when the gauge group is non-Abelian and is a shorthand notation for
exp (−Aµ(x0)δx0) exp (−Aµ(x1)δx1) . . . , (4.3)
If we want to compute an open Wilson line, the corresponding expression is a tran-sition amplitude between an initial state |ii and a final state |ji
WR(C) ≡ hj|Pexp − Z C A |ii . (4.4)
If the curve C is closed, the resulting Wilson loop is invariant under a global gauge transformation
A → A0= Λ−1AΛ + Λ−1dΛ , (4.5)
whereas for an open interval, the Wilson line is gauge covariant in the following sense
WR(C) → Λ−1(xf)WR(C)Λ(xi) , (4.6)
As we said in the introduction of this chapter, we want the Wilson line to give information on the geodesic distance between its two endpoints. Since a geodesic curve can be understood as the path that a massive particle follows in the bulk, our goal is to describe in the Chern-Simons formulation the coupling of this massive particle to 3d gravity.
The Wilson line depends on the curve C and the representation R. Due to the flatness of the connections, all paths with the same endpoints that can be continuously deformed into one another are equivalent. As for the representation, in principle we can choose any representation we want, but our choice is dictated by the interpretation we want to give to our Wilson line. A massive particle is characterized by its mass and spin. At least in the classical limit, these numbers should have a continuous spectrum. This suggests that we should choose an infinite dimensional representation, in order to be able to store the continuously tunable information about the mass and the spin. Besides this, SL(2, R) is a non compact group and therefore there are no finite dimensional unitary representations, which is something that in a fully quantum theory we do not want.
We will take an infinite dimensional highest-weight representation for sl(2, R)1. This is defined with the highest weight state |hi, which is an eigenstate of L0and is annihilated
by L1:
L0|hi = h |hi , L1|hi = 0 , (4.7)
where h is the parameter that characterizes the representation. The sl(2, R) algebra is given by
[Li, Lj] = (i − j)Li+j, (4.8)
with i, j = −1, 0, 1. By acting with L−1 on the highest weight state we obtain an infinite
number of descendants states
(L−1)n|hi ∼ |h, ni . (4.9)
1
Other choices are of course possible and their physical implications might be interesting to investi-gate.
These states form the irreducible, unitary and infinite dimensional representation of sl(2, R). Each representation is labeled by the value of the quadratic Casimir operator
C2= 2L20− L1L−1− L−1L1, (4.10)
which is an operator quadratic in Li that commutes with all Lie algebra elements and
thus its action on any state |h, ni gives the constant 2h(h − 1). For each copy we will have a Casimir and it might be already clear that in the context of the AdS/CFT cor-respondence the two highest weights (h, ¯h) will correspond to the conformal dimensions of the dual to the Wilson line CFT operator. In particular, h + ¯h will be related to the bulk mass and h − ¯h to the bulk spin.
4.2
Path integral representation of a Wilson line
Looking at (4.2) we see that we have to compute a trace in an infinite dimensional repre-sentation, which we will do using a technique developed in [36]. The idea is to generate the states of the representation with an auxiliary quantum mechanical system that lives on the Wilson line. This should not be very surprising, since infinite dimensional highest weight representations appear in quantum mechanics as the Hilbert spaces of various systems. Then, the trace is replaced by a path integral over a field U ∈ SL(2, R) with global symmetry SL(2, R) × SL(2, R), the dynamics of which are chosen in such a way that upon quantization the Hilbert space of this system will be the representation R. The Wilson line is written, therefore, as
WR(C) = Z
DU exp[−S(U ; A, ¯A)C] . (4.11)
The field U is coupled to the gauge connections A and ¯A and the action in (4.11) can be decomposed as
S(U ; A, ¯A)C = S(U )C,free+ S(U ; A, ¯A)C,int, (4.12)
where the free action has SL(2, R) × SL(2, R) as global symmetry and in the interacting action this symmetry is promoted to a local one.
We choose the probe field U to be described by the action S(U, P )free= Z C ds Tr P U−1dU dS + λ(s) Tr(P2) − c2 , (4.13)
where s ∈ [0, sf] parametrizes the curve C, P is the conjugate momentum to U living
in the Lie algebra sl(2, R) and λ is a Lagrange multiplier that constraints the value of P to that of the quadratic Casimir of the representation we pick. Tr denotes again the
contraction using the Killing form of the Lie algebra
Tr(P2) = PaPbδab= 2P02− (P−1P1+ P1P−1) . (4.14)
In our conventions the Lie algebra metric is given by δ00= −
1
2, δ+−= δ−+= −1 . (4.15)
The action (4.13) has SL(2, R)×SL(2, R) as a global symmetry with the following action on U and P
U (s) → LU (s)R, P → R−1P (s)R , (4.16)
where L and R belong in the left and right SL(2, R) respectively.
When we quantize the system the resulting Hilbert space is the desired infinite dimensional highest-weight representation of SL(2, R) × SL(2, R) with Casimir given by Tr(P2) = c2 = 2h(h − 1) for ¯h = h.
We now promote the global SL(2, R) × SL(2, R) symmetry to a local one
U (s) → L(xµ(s))U R(xµ(s)), P (s) → R−1(xµ(s))P R(xµ(s)) , (4.17) by coupling the system to the gauge connections A and ¯A, which describe the bulk spacetime through which the Wilson line traces a path x(s). The bulk is invariant under local gauge transformations of the form
Aµ→ L(x)(Aµ+ ∂µ)L−1(x), A¯µ→ R−1(x)( ¯Aµ+ ∂µ)R(x) . (4.18)
These transformations affect the worldline field U (s), since it depends on the bulk path x(s). If we do the minimal substitution
d dsU → DsU = d dsU + AsU − U ¯As, As≡ Aµ dxµ ds , (4.19)
where the covariant derivative DsU transforms under (4.17) in the same manner
DsU → L(xµ(s))(DsU )R(xµ(s)) , (4.20)
the following action is invariant under the combined local gauge transformations (4.17) and (4.20)
S(U, P, A, ¯A)C =
Z
C
ds Tr(P U−1DsU ) + λ(s)(Tr(P2) − c2) . (4.21)
In order to compute an open ended Wilson line we also have to specify boundary conditions for the field U , which we will discuss in the next sections. Before moving, we have to note that (4.12) is not the only choice we could make, since other choices of
probes and actions can give the same trace.
4.3
Appearance of the geodesic equation
We said in the beginning of this chapter that Wilson lines represent the coupling of a massive particle to gravity in the Chern-Simons formulation. In this chapter, we will establish this equivalence by looking at the equations of motion of (4.21). In order to facilitate our computation, we will eliminate P and λ from (4.21):
S(U ; A, ¯A)C = √ c2 Z C dspTr(U−1D sU )2. (4.22)
Using the freedom to do reparametrizations of the worldline parameter, we will pick s to be the proper distance of the probe which implies that λ and equivalentlypTr(U−1D
sU )2
are constant. Varying now (4.22) with respect to U we get d ds (Au− ¯A)µ dxµ ds + [ ¯Aµ, Auν] dxµ ds dxν ds = 0 , (4.23)
where U enters in the above expression through the definition of Au
Auµ≡ U−1 d dsU + U
−1A
µU . (4.24)
Since our goal is to compute geodesic lengths using a Wilson line, we need the open Wilson line to be Lorentz invariant. From the path integral expression of (4.11) we see that the Wilson lines depends on the boundary values of the probe field U . As we know by now, under a simultaneous left and right gauge transformation, U transforms as
U (s) → LU (s)R, L = eiαaJa, R−1= ei ¯αaJ¯a, (4.25)
with the generators written as Ja=
1
2(Ma+ `Pa), J¯a = 1
2(Ma− `Pa) . (4.26)
From (4.25) and (4.26) we see that Lorentz transformations correspond to gauge trans-formations with L = R−1 (since then the infinitesimal parameters corresponding to translations Pashould be zero) and that in order for U to be invariant under all Lorentz
transformations, it has to commute with all L, i.e. belong to the center of SL(2, R) which is composed of ±1. We will choose, therefore, the following boundary conditions U (s = 0) ≡ Ui = 1, U (s = sf) ≡ Uf = 1 . (4.27)
