Asymptotic Symmetry Algebras in Non-Anti-de-Sitter Higher-Spin Gauge Theories
MMASTERARBEIT
Asymptotic Symmetry Algebras in
Non-Anti-de-Sitter Higher-Spin GaugeTheories
Ausgef¨uhrt am Institut f¨urTheoretische Physikder Technischen Universit¨at Wienunter der Anleitung von Daniel Grumillerdurch
Max Riegler
Schwarzspanierstraße 15/8/341090 Wien
Wien, 3.9.2012 Unterschrift a r X i v : . [ h e p - t h ] O c t ch erkl¨are hiermit, dass ich die eingereichte Masterarbeit selbstst¨andig verfassthabe und keine anderen als die angegebenen Quellen und Hilfsmittel benutztwurden. Weiters versichere ich, dass ich diese Masterarbeit bisher weder im In-noch im Ausland in irgendeiner Form als Pr¨ufungsarbeit vorgelegt habe.Wien, 2018 Max Riegler inleitung/Kurzfassung Von allen vier fundamentalen Wechselwirkungen ist die Gravitation die einzigeWechselwirkung, von der bis heute keine allgemein akzeptierte quantisierte Theorieexistiert. Ein m¨oglicher L¨osungsansatz zu diesem Problem basiert auf demholographischen Prinzip. Dieses Prinzip bezeichnet im Wesentlichen dieVermutung, dass eine Theorie der Quantengravitation in d + 1 Dimensionen durcheine ¨aquivalente Beschreibung einer Quantenfeldtheorie (ohne Gravitation) in d Dimensionen formuliert werden kann. In Analogie zu einem Hologramm, bei demman entweder das abgebildete Objekt in 3 Dimensionen, oder die gespeicherteIntensit¨at und Phase auf dem 2-dimensionalen Schirm als Beschreibungheranziehen kann, so hat man auch im Falle des Holographischen Prinzips zweigleichwertige, aber dennoch unterschiedliche Beschreibungen der Dynamik einesSystems zur Verf¨ugung.Ein Spezialfall dieses Prinzips, welcher im Kontext der Stringtheorie formuliertwurde, ist die so genannte Anti-de-Sitter/konforme Feldtheorie (AdS/CFT)Korrespondenz, bei der eine Theorie der Quantengraviation mit negativerkosmologischer Konstante durch eine Quantenfeldtheorie, welche unter konformenTransformationen invariant ist, beschrieben werden kann. Da diese KorrespondenzErgebnisse bei starker Kopplung mit solchen bei schwacher Kopplung verbindet,w¨are dies ein idealer Kandidat, um Quantengravitation auf Skalen, bei denenQuanteneffekte nicht mehr vernachl¨assigt werden k¨onnen, besser zu verstehen.Allerdings gibt es in mehr als 2+1 Dimensionen noch viele konzeptionelle undtechnische Schwierigkeiten. Daher wird beispielsweise versucht, mit Hilfe vontechnisch einfacheren Gravitationstheorien in 2+1 Dimensionen zuerst diekonzeptionellen Schwierigkeiten zu beseitigen, welche dem Verst¨andnis einervollst¨andigen Theorie der Quantengravitation im Wege stehen.Viel Aufmerksamkeit in diesem Kontext haben in den vergangenen Jahren auch sogenannte H¨ohere-Spin Gravitationstheorien in 2+1 Dimensionen erregt, durchderen Studium man sich ebenfalls ein besseres Verst¨andnis der AdS/CFTKorrespondenz erhofft.In dieser Masterarbeit befassen wir uns mit einer bestimmten H¨ohere-SpinGravitationstheorie in 2+1 Dimensionen. Wir f¨uhren eine kanonische Analysedurch und stellen konsistente Randbedingen vor, welche von den dynamischenFeldern der Theorie erf¨ullt werden m¨ussen. Weiters bestimmten wir die klassischeund quantisierte Symmetriealgebra der daraus resultierenden holographischenQuantenfeldtheorie am Rande der Raumzeit und versuchen sie physikalisch zuinterpretieren. bstract
We analyze asymptotic symmetry algebras in (2+1)-dimensional non-AdS higher-spin gravity with a focus on AdS × R and H × R . We find a consistent set ofboundary conditions for spin-3 gravity in the non-principal embedding and calcu-late the corresponding asymptotic symmetry algebra in the classical and quantummechanical case. In addition, we check for unitary representations of the resultingquantum W (2)3 algebra and give an interpretation of the corresponding CFT. cknowledgements At this point I want to thank all the people who supported me in the course of thisthesis.Special thanks go to my supervisor Daniel Grumiller whose support was invaluable.Thank you for your continued guidance since we first met three years ago.I also want to thank Michael Gary who provided me with a basic mathematica filethat I modified to suit my purposes and considerably sped up the process of findingconsistent boundary conditions for AdS × R and H × R .Furthermore I want to thank Hamid Reza Afshar, Radoslav Rashkov, Sabine Ertland Stefan Prohazka for enlightening discussions and valuable input.Very special thanks go to my family and friends and especially to my mother Ulrikeand aunt Karin. Without the continuous support of my mother and my aunt itwould have been impossible for me to finish my studies the way I did. Thank youfor everything! I consider myself very lucky to have such an awesome mother andaunt.Thanks a lot,Max ontents
1. Introduction 2
2. Basics of Chern-Simons Theories 11
3. AdS × R and H × R for sl (3) W (2)3 and u (1) Current Algebra 273.5 Unitarity of the Resulting CFT 303.5.1 ˆ k = − and ˆ c = 0 333.5.2 ˆ k = − c = 1 34
4. Conclusion 37A. Suitable Spin-3 Bases 40
A.1 Non-Principal Embedding 40
B. Gramian Matrices for ˆ k ∈ {− , − } C.1 2-2 Embedding 42C.2 2-1-1 Embedding 43– 1 – . Introduction
In this section we will give a short introduction to the basic concepts underlying thismaster thesis such as the AdS/CFT correspondence and the motivation to studyhigher-spin gauge theories.
One of the big open questions in physics of the last century is formulating a con-sistent theory of quantum gravity and in turn maybe also a theory of everythingthat explains all of the fundamental forces of nature. One possible solution forthis problem could be provided by string theory where the fundamental objects aredescribed by one-dimensional objects called strings rather than zero-dimensional ob-jects . One conjecture formulated in the framework of string theory and a possiblecandidate to understand quantum gravity better quantitatively is the so called Anti-de-Sitter/Conformal field theory (AdS/CFT) correspondence. This AdS/CFT cor-respondence, originally discovered by Maldacena in 1997, is one of the most strikingand unexpected discoveries of the last 20 years. Originally formulated as a correspon-dence between a N = 4 supersymmetric Yang-Mills theory in four dimensions anda type IIB superstring theory on AdS × S [4] the correspondence has been muchgeneralized since then and found many applications. The name of the special caseof a AdS/CFT correspondence originates from the canonical example according towhich the first space is the product of a p+1-dimensional Anti-de-Sitter spacetime and some closed manifold (a sphere for example) and the p-dimensional quantumfield theory defined on the boundary is a conformal field theory.The generalized conjecture is formulated as an equivalence of a gauge theory (stringtheory for example) defined on a specific background and a quantum field theorywithout gravity on the (conformal) boundary of this spacetime. This general princi-ple that the dynamics of a region of spacetime are encoded on the boundary of thisregion is also called the holographic principle . This terminology is indeed adequatebecause a hologram is completely analogous, i.e. a three-dimensional image that hasbeen saved on a two dimensional holographic screen but still retaining all informationpresent in three dimensions.There is also one prominent physical example that hints to the possibility that theholographic principle is actually present in nature: the entropy of a black hole.Initially black holes were thought of as objects that have zero entropy until Beken-stein [5] noted that this assumption would violate the second law of thermodynamics.One could for example throw a cup with hot gas and a certain amount of entropyinto a black hole and thus decrease the amount of entropy in the universe if the For introductional literature on string theory please refer to [1–3]. Anti-de-Sitter spacetimes are maximally symmetric solutions of the Einstein equations with anegative cosmological constant and have constant negative curvature. – 2 –ssumption of black holes with zero entropy would be true. Thus, black holes wouldhave to have entropy in order for the second law of thermodynamics to still hold.In fact black holes have more entropy per volume than any other object in the uni-verse. This can be understood by considering a sphere of fixed radius R containinga relativistic gas. The entropy of this gas increases as the energy increases and isonly limited by gravitational forces. When the energy exceeds a certain limit thegas collapses to a black hole and thus the resulting black hole has to contain atleast the same amount of entropy as the gas before the collapse. Bekenstein usedthis argument to conjecture an upper bound of the entropy of a black hole which isproportional to the area of the black hole. This conjecture was later confirmed byHawking [6]. Since in statistical physics entropy is proportional to the logarithm ofthe number of possible microstates, the Bekenstein-Hawking entropy suggests thatthe logarithm of the number of microstates of a black hole is proportional to its arearather than its volume. This is a statement that strongly hints at the validity of theholographic principle.Now back to the main reason why this AdS/CFT correspondence is a candidate dodeepen our understanding of quantum gravity. This duality is a strong/weak duality.This means that the coupling constants of the bulk and boundary theories are relatedin such a way that if one tuned the coupling of the bulk theory such that the theoryis strongly coupled, then the dual boundary theory would be weakly coupled andvice versa. This can be seen for example by considering the perturbative expansionof the partition function of a large N gauge theory and the loop expansion in stringtheory. The perturbative expansion of a large N gauge theory in N and g Y M N isgiven by [7] Z = (cid:88) g ≥ N − g f g ( λ ) , (1.1)where g Y M is the coupling constant of the (Yang-Mills) gauge theory, λ = g Y M N isthe so called ’t Hooft coupling and f g ( λ ) are arbitrary functions of λ . This expansionlooks very much like the loop expansion in string theory given by Z = (cid:88) g ≥ g g − s Z g , (1.2)where g s is the string coupling which is identified with N in large N dualities. Con-sidering the supergravity approximation of string theory on AdS × S this theoryhas three important parameters: • The string coupling constant g s which measures the string interaction strengthrelevant for splitting and joining of strings. • The string length l s measuring the size of the fluctuations of the string world-sheet. – 3 – The curvature radius (cid:96) of AdS and S .Four dimensional N = 4 super Yang-Mills (SYM) Theory with U ( N ) gauge grouphas the following two parameters: • The rank N of the gauge group. • The coupling constant g Y M determining the strength of interactions in SYMtheories.These parameters are related as follows g s = g Y M , (cid:18) (cid:96)l s (cid:19) = 4 πg Y M N = 4 πλ. (1.3)Looking at (1.1) we see that the SYM theory perturbative expansion is only validfor small g Y M and small λ . Since the supergravity approximation of string theory isonly valid for (cid:96)l s (cid:29)
1, which is equivalent to λ (cid:29)
1, we see that we have indeed astrong/weak duality. If • λ (cid:28) • λ (cid:29) λ (cid:29) . Thus CFTs in two dimensions are idealmodels to address fundamental questions about quantum gravity that are usuallyvery hard to answer in higher dimensional field theories. As a consequence we haveto consider gravity theories in three dimensions as appropriate dual theories to twodimensional field theories. N is the number of supersymmetry generators. As an introduction to CFTs please refer to [8–11]. – 4 – .2 Einstein-Hilbert Gravity as a Chern-Simons Theory
In this section we try to give a short introduction as to how it is possible to refor-mulate Einstein-Hilbert gravity in 2+1 dimensions as a Chern-Simons gauge theory.2+1 dimensional pure gravity without matter fields is described by the Einstein-Hilbert action I EH = 116 πG N (cid:90) M d x √− g (cid:18) R + 2 (cid:96) (cid:19) , (1.4)with G N being Newton’s constant, R the Ricci scalar and (cid:96) = − Λ is the cosmo-logical constant where (cid:96) denotes the AdS radius. One of the main problems oneencounters when trying to quantize gravity in dimensions d > D = 2 + 1 is non-renormalizable.However, the theory in 2+1 dimensions is trivial in the bulk on the classical level inthe sense that there are no propagating local degrees of freedom, i.e. gravitons. Sincequantized theories that are trivial on the classical level are usually renormalizablethere was a high probability that gravity in 2+1 dimensions is actually renormal-izable. And indeed it was then shown by Witten in [13] that in 2+1 dimensionsrenormalization is indeed possible. The main reason for this is that in 2+1 dimen-sions the Riemann tensor R abcd can be expressed in terms of the Ricci tensor R ab ,the Ricci scalar R and the metric g ab as R abcd = g ac R bd + g bd R ac − g ad R bc − g bc R ad − R ( g ac g bd − g ad g bc ) . (1.5)Since the equations of motion of pure Einstein-Hilbert gravity given by R ab − g ab (cid:18) R + 2 (cid:96) (cid:19) = 0 (1.6)tell us that the Ricci tensor is equal to some scalar times the metric one can ex-press all curvature invariants R abcs , R ab and R in terms of the metric g ab . This inturn implies that all possible counterterms , which should cancel the infinite dia-grams can be written as a multiple of (cid:82) d x (cid:112) | g | , which is equivalent to an on shellrenormalization of the cosmological constant. If there are counterterms present thatvanish on shell then these terms can be absorbed by a redefinition of the metric [14] g ab → g + αR ab + . . . , where α is some constant and the ellipsis denotes higher orderterms of curvature invariants. Thus in 2+1 dimensions all divergencies in pertur-bation theory can be removed by a redefinition of the metric and the cosmological The lowest order pure curvature corrections to the Einstein-Hilbert action look for example like R (cid:112) | g | , R ab R ab (cid:112) | g | , R abcd R abcd (cid:112) | g | . – 5 –onstant [15].In order to see that pure Einstein gravity and a Chern-Simons theory in three di-mensions are equivalent up to boundary terms, it is convenient to formulate 2+1 di-mensional general relativity in terms of a local orthonormal basis for the (co)tangentspace called dreibein e , which can be interpreted as a local inertial frame and a spinconnection ω . The dreibein and spin connection in terms of a cotangent basis aregiven by e a = e aµ dx µ , ω ab = ω abµ dx µ . (1.7)Latin letters a, b, . . . denote local Lorentz indices, greek letters µ, ν, . . . denote space-time indices and ω abµ = − ω baµ . The spacetime metric and the dreibein are relatedas g µν = e aµ e bν η ab , (1.8)where η ab denotes the Minkowski metric with signature ( − , + , +). A very conve-nient feature of 2+1 dimensions is that one can ”dualize” the spin connection in thefollowing way ω a = 12 (cid:15) abc ω bc , (1.9)where (cid:15) abc is the Levi-Civita symbol and we omitted the spacetime index µ for thesake of brevity. The curvature 2-form in terms of the dualized spin connection isthen given by R a = dω a + (cid:15) abc ω b ∧ ω c . (1.10)One can now regard the dreibein e and the spin connection ω as the new dynamicalvariables of the theory and reformulate the Einstein-Hilbert action in terms of thesenew variables. I EHP = 116 πG N (cid:90) M R a ∧ e a + 23 (cid:96) (cid:15) abc e a ∧ e b ∧ e c , (1.11)The action (1.11) already looks very familiar in comparison to the Chern-Simonsaction I CS [ A ] = k π (cid:90) M Tr( A ∧ d A + 23 A ∧ A ∧ A ) , (1.12)where A is a Lie algebra valued one form, which in the case of pure Einstein-Hilbertgravity will be sl (2). Combining the dreibein and the dualized spin connection intothe following connection one forms A a = ω a + 1 (cid:96) e a (1.13)¯ A a = ω a − (cid:96) e a , (1.14)Witten showed in [13] that the combination I = I CS [ A ] − I CS [ ¯ A ] (1.15)– 6 –s equivalent to the Einstein-Hilbert action up to boundary terms. Let L a , a = − , , sl (2) generators then the normalization [16]Tr( L a L b ) = 12 η ab (1.16)leads to the following identification k = (cid:96) G N . (1.17)In a similar fashion one can construct a first order formulation of the dynamics offree massless bosonic symmetric fields with spin s ≥ sl (2) construction. Dualizing the spinconnection as in (1.9) only works for gauge groups that have dimension 3, like sl (2).Thus in higher-spin gravity one starts with A a and ¯ A a and defines the correspondingspin connection and zuvielbein as e a = (cid:96) A a − ¯ A a ) (1.18) ω a = 12 ( A a − ¯ A a ) . (1.19)The spacetime metric is then defined as g µν := ( (cid:0) A µ − ¯ A µ (cid:1) (cid:0) A ν − ¯ A ν (cid:1) , (1.20)where ( A and ¯ A and the spacetime metric g µν . This isa viable choice since the metric defined in this way is manifestly gauge invariant andin the case of sl (2) coincides with (1.8). It is, however, also possible that one couldadd further terms to (1.20) and thus a unique definition of the spacetime metric interms of the gauge potentials A and ¯ A is still an open topic as of yet. Thus takingthese subleties into account one can construct bosonic massless higher-spin gaugetheories via G × G Chern-Simons theories [19], where G denotes the gauge group ofthe theory.As already mentioned, gravity theories as well as Chern-Simons theories are trivialin the sense that they have no propagating local degrees of freedom. This does,however, not mean that these theories are trivial. In fact they are quite nontrivial assoon as one introduces a boundary. In general there is even a symmetry enhancementof the bulk symmetries occurring at the boundary. The by now famous example ofa SL (2) × SL (2) bulk isometry algebra that is enhanced to two copies of a Vira-soro algebra with a central charge c = 6 k at the boundary has first been studied Since in general for higher-spin theories the Lie algebras considered have more than threegenerators and thus do not have to match the number of spacetime indices, e aµ is called zuvielbeinrather than dreibein, as it was called in the sl (2) case. – 7 –y Brown and Henneaux [20]. By adding further massless higher-spin excitationsthe asymptotic symmetries are even further enhanced to non linear algebras called W -algebras . In this section we will motivate why it is interesting to study higher-spin gravity in2+1 dimensions.Higher-spin excitations appear quite naturally in (super)string theories. In additionto massless modes of lower spin s ≤ T andspin s . Since the string tension is inverse proportional to the string length squared,these higher-spin modes are very heavy and thus unobservable at low energies. Nev-ertheless these higher-spin excitations are necessary for the consistency of a stringtheory describing all the fundamental interactions. In general, quantum field theo-ries containing massive particles with spin s ≥ T →
0. Thus, inthis limit one should observe a symmetry enhancement of string theory by higher-spin symmetry, and one can regard string tension generation as a mechanism of thespontaneous symmetry breaking of the higher-spin symmetry. If this conjecture wastrue, then this could be very useful in understanding string theory and in particularthe AdS/CFT correspondence. However, it is not easy to build such theories withhigher-spin gauge symmetries in flat space. There is in particular one theorem byColeman and Mandula which has been generalized to arbitrary dimensions by Pelcand Horwitz [23], which states that symmetries of the S-matrix in a non-trivial (i.e.interacting) field theory in a flat space can only have sufficiently low spins. Thistheorem can, however, be circumvented if one is considering AdS spacetimes [24].And indeed for AdS there exist interacting higher-spin theories of massless parti-cles [25–28]. There exists also a general statement that the cosmological constant Λin dimensions
D > (cid:54) = 0.After spontaneous symmetry breaking via some mechanism (dimensional compactifi-cation for example) one could indeed end up in principle with a theory where m (cid:54) = 0for higher-spin fields and Λ = 0 (or Λ very small) [30]. The modification of the For an introduction to W -algebras please refer to [21]. – 8 –osmological constant could then be due to some fields that acquire a non-zero vac-uum expectation value via the spontaneous symmetry breaking and thus modify thevacuum energy . Thus, it would be possible in principle to start with a masslesshigher-spin theory and Λ (cid:54) = 0, and after spontaneous symmetry breaking one couldend up with a string theory containing massive higher-spin fields and a very smallcosmological constant.In general, higher-spin gauge theories contain an infinite set of spins 0 ≤ s ≤ ∞ .Thus it is generically not possible to just consider particles up to spin n in a higher-spin gauge theory. The only known exception to this is the case of 2 + 1-dimensions,where it is possible to truncate this otherwise infinite tower of higher-spin fields atarbitrary spin n so that all fields have spin s ≤ n [31], which is another reason whyit is interesting to work in 2+1 dimensions. There are many applications that require a generalization of the AdS/CFT corre-spondence to a gauge/gravity duality that does not involve spacetimes asymptotingto AdS, or asymptoting to AdS in a weaker way as compared to Brown-Henneauxboundary conditions [20, 32]. Some examples are given by • null warped AdS spacetimes, which arise in proposed holographic duals of non-relativistic CFTs describing cold atoms [33, 34] • Schr¨odinger spacetimes, which generalize null warped AdS by introducing anarbitrary scaling exponent [35] • Lifshitz spacetimes, which arise in gravity duals of Lifshitz-like fixed points [36]and also have a scaling exponent parametrizing spacetime anisotropy • AdS/log CFT correspondence [37,38], which requires a relaxation of the Brown-Henneaux boundary conditions [39–41] • Flat space holography , which requires the spacetime in the bulk to be asymp-totically flat [42–44].A priori it is not clear that higher-spin gravity can accommodate such non-AdSbackgrounds. There is, however, one example of a theory that is very similar tohigher-spin gravity in 2+1 dimensions, namely conformal Chern-Simons gravity [45–47]. Conformal Chern-Simons gravity has • no local physical degrees of freedom, • a Chern-Simons formulations with a gauge group bigger than SL (2) × SL (2), This is due to the fact that a cosmological constant has the same effect as an intrinsic energydensity of the vacuum. – 9 – gauge symmetries that relate non-diffeomorphic metrics to each other, • and the asymptotic symmetry group can be larger than two copies of the Vi-rasoro algebra [48, 49].The axisymmetric stationary solutions of conformal Chern-Simons gravity includeAdS as well as AdS × R , which means that at least for conformal Chern-Simonsgravity non-AdS backgrounds exist. And indeed it has been shown in [50] that higher-spin gravity with an appropriate variational principle is indeed capable of generatingspacetimes that asymptote to AdS (with weaker boundary conditions than Brown-Henneaux), AdS × R , Schr¨odinger, Lifshitz and warped AdS spacetimes.– 10 – . Basics of Chern-Simons Theories In this section we will give a short introduction to the basics of Chern-Simons theo-ries and explain the variational principle we will be using in order to accommodateasymptotic backgrounds beyond AdS. Since there already exist excellent books ex-plaining the basics of constrained hamiltonian systems and canonical analysis, wewill not go much into detail regarding these topics. We refer the interested readerto [51, 52] for example.
As reviewed in section 1.2 Einstein gravity with a negative cosmological constant inthree dimensions can be reformulated as the difference of two Chern-Simons actionsgiven by I = I CS [ A ] − I CS [ ¯ A ] (2.1)with I CS [ A ] = k π (cid:90) M Tr( A ∧ d A + 23 A ∧ A ∧ A ) + B [ A ] . (2.2) I CS [ ¯ A ] is given by just replacing A → ¯ A in (2.2). Hence we will focus on the canonicalanalysis of the I CS [ A ] term. The canonical analysis for I CS [ ¯ A ] can then be obtainedin complete analogy to the one performed with I CS [ A ] just by replacing k → − k and A → ¯ A in all relevant formulas. In addition we will also set the AdS radius (cid:96) to 1.The action given by (2.2) is defined on a Manifold M with the topology M = R × Σand coordinates x µ = ( t, ρ, ϕ ), µ = 0 , ,
2. In addition, we assume that Σ has thetopology of a disk and is parameterized by ϕ and ρ , where ρ = const. correspondsto the boundary. B [ A ] is a boundary term defined on ∂ M = ∂ Σ × R to ensure awell defined variational principle and gauge invariance of the action if one wants toconsider spacetimes that do not asymptote to AdS. Without this boundary term theresulting restrictions on the connection A that would ensure a well defined variationalprinciple and gauge invariance of the action would only allow the resulting spacetimesto asymptote to AdS . The A ’s are Lie algebra valued 1-forms that can be writtenas A = A aµ d x µ T a , (2.3)with T a being a basis of the Lie algebra g one is considering . If one chooses such abasis then Tr( T a T b ) can be interpreted as a non-degenerate bilinear form on the Liealgebra. In components one can write (2.2) as I CS [ A ] = k π (cid:90) M d x(cid:15) µνλ g ab (cid:18) A aµ ∂ ν A bλ + 13 f acd A cµ A dν A bλ (cid:19) + B [ A ] , (2.4)where g ab = Tr( T a T b ), (cid:15) tρϕ = 1 and f abc are the structure constants of the Lie algebragiven by [ T a , T b ] = f cab T c . (2.5)– 11 –ie algebra indices ( a, b, . . . ) are raised and lowered with g ab and spacetime indices( µ, ν, . . . ) with the background metric g µν of the spacetime considered. In order to obtain the equations of motion one has to vary (2.2). This yields δI CS [ A ] = k π (cid:90) M Tr( δA ∧ F ) + k π (cid:90) ∂ M Tr( δA ∧ A ) + δB [ A ] , (2.6)with F = dA + A ∧ A . One could consider for example the following boundary term B [ A ] = k π (cid:90) ∂ M d x Tr( A ϕ A t ) . (2.7)In order to have a well defined variational principle we have to require δI CS [ A ] = 0which specifies the equations of motion as F = 0 . (2.8)In addition, we have to restrict the boundary conditions such that the total boundaryterm vanishes, i.e. δI CS [ A ] (cid:12)(cid:12)(cid:12) on-shell = k π (cid:90) ∂ M d x g ab A aϕ δA bt = 0 . (2.9)This can be achieved by demanding either A ϕ (cid:12)(cid:12)(cid:12) ∂ M = 0 or δA t (cid:12)(cid:12)(cid:12) ∂ M = 0 . (2.10)Since A ϕ (cid:12)(cid:12)(cid:12) ∂ M = 0 is a slightly stronger boundary condition on the connection than δA t (cid:12)(cid:12)(cid:12) ∂ M = 0, we will use the latter one because we do not want to put too manyrestrictions on the connection. Another important consistency condition is gauge invariance of the action. Since theconnection has to satisfy δA t (cid:12)(cid:12)(cid:12) ∂ M = 0, the form of the allowed gauge transformationswill also necessarily be restricted. Writing finite gauge transformations as A → g − ( ˜ A + d ) g, (2.11)with g ∈ G where G is the gauge group one is considering, we can calculate thechange of the action (2.2) under (2.11). This leads to I CS [ A ] = I CS [ ˜ A ] + δI CS [ ˜ A ] + δB [ ˜ A ] , (2.12)– 12 –here δI CS [ ˜ A ] = − k π (cid:90) M Tr( g − dg ∧ g − dg ∧ g − dg ) − k π (cid:90) ∂ M Tr( dgg − ∧ ˜ A ) (2.13)and δB [ ˜ A ] = − k π (cid:90) ∂ M d x Tr( ∂ ϕ g∂ t g − − ˜ A ϕ ∂ t gg − − ˜ A t ∂ ϕ gg − ) . (2.14)Hence the Chern-Simons action is gauge invariant if either • g → ∂ M ; or • the gauge transformations are certain infinitesimal gauge transformations asspecified below.Infinitesimal gauge transformations connected to the identity are given by g (cid:39) λ a T a . (2.15)This leads to δI CS [ ˜ A ] + δB [ ˜ A ] = − k π (cid:90) ∂ M d x g ab (cid:16) (cid:15) ρij ∂ i λ a ˜ A bj − ˜ A aϕ ∂ t λ b − ˜ A at ∂ ϕ λ b (cid:17) = k π (cid:90) ∂ M d x g ab ˜ A bϕ ∂ t λ a = 0 . (2.16)Since we do not want to impose additional constraints on the connection, one canconclude that (2.2) is gauge invariant for infinitesimal gauge transformations satis-fying at the boundary ∂ t λ a (cid:12)(cid:12)(cid:12) ∂ M = 0 . (2.17) In order to proceed with the canonical analysis it is convenient to use a 2 + 1 decom-position of the action (2.4) [53]. The 2 + 1 decomposition of (2.4) is given by I CS [ A ] = k π (cid:90) R d t (cid:90) Σ d x(cid:15) ij g ab (cid:16) ˙ A ai A bj + A a F bij + ∂ j (cid:0) A ai A b (cid:1)(cid:17) + B [ A ] , (2.18)with F aij = ∂ i A aj − ∂ j A ai + f abc A bi A cj and (cid:15) ij = (cid:15) tij . Since the EOM require F aij = 0, the form of (2.18) already specifies A a as a Lagrange multiplier and A ai asthe dynamical fields. The Lagrangian density L is then given by L = k π (cid:15) ij g ab (cid:16) ˙ A ai A bj + A a F bij + ∂ j (cid:0) A ai A b (cid:1)(cid:17) . (2.19)Calculating the canonical momenta π aµ ≡ ∂ L ∂ ˙ A aµ corresponding to the canonical vari-ables A aµ one finds the following primary constraints φ a := π a ≈ φ ai := π ai − k π (cid:15) ij g ab A bj ≈ . (2.20)– 13 –he Poisson brackets of the canonical variables are given by { A aµ ( x ) , π bν ( y ) } = δ ab δ µν δ ( x − y ) . (2.21)The next step is to calculate the canonical Hamiltonian density via the followingLegendre transformation H = π aµ ˙ A aµ − L = − k π (cid:15) ij g ab (cid:0) A a F bij + ∂ j (cid:0) A ai A b (cid:1)(cid:1) . (2.22)Since we are dealing with a constrained Hamiltonian system, we have to work withthe total Hamiltonian given by H T = H + u aµ φ aµ , (2.23)where u aµ are some arbitrary multipliers. Since the primary constraints should beconserved after a time evolution, we require˙ φ aµ = { φ aµ , H T } ≈ , (2.24)which leads to the following secondary constraints K a ≡ − k π (cid:15) ij g ab F bij ≈ D i A a − u ai ≈ , (2.26)where D i X a = ∂ i X a + f abc A bi X c is the covariant derivative. One can now use theHamilton equations of motion, which are given by˙ A ai = ∂ H T ∂π ai = u ai (2.27)to determine the Lagrange multipliers u ai and rewrite (2.26). This yields the followingweak equality D i A a − u ai = D i A a − ∂ A ai = F ai ≈ . (2.28)The total Hamiltonian can now be written in the following form H T = A a ¯ K a + u a φ a + ∂ i ( A a π ai ) , (2.29)with ¯ K a = K a − D i φ ai . (2.30)One can use the canonical commutation relations (2.21) to determine the followingPoisson brackets which will be necessary to determine the Poisson algebra of the– 14 –onstraints { φ a ( x ) , A b ( y ) } = − δ ab δ ( x − y ) , (2.31a) { φ ai ( x ) , A bj ( y ) } = − δ ab δ ij δ ( x − y ) , (2.31b) { φ ai ( x ) , π bj ( y ) } = − k π (cid:15) ij g ab δ ( x − y ) , (2.31c) { φ ai ( x ) , π bj ( y ) } = − k π (cid:15) ij g ab δ ( x − y ) , (2.31d) { A ai ( x ) , D j φ bj ( y ) } = [ δ ab ∂ i + f abc A ci ( y )] δ ( x − y ) , (2.31e) { π ai ( x ) , D j φ bj ( y ) } = − k π (cid:15) ij [ g ab ∂ j + f abc A cj ( y )] δ ( x − y ) + f abc φ ci ( y ) δ ( x − y ) , (2.31f) { φ ai ( x ) , D j φ bj ( y ) } = − k π (cid:15) ij [ g ab ∂ j + f abc A cj ( y )] δ ( x − y ) + f abc φ ci ( y ) δ ( x − y ) , (2.31g) { π ai ( x ) , K b ( y ) } = − k π (cid:15) ij [ g ab ∂ j + f abc A cj ( y )] δ ( x − y ) , (2.31h) { φ ai ( x ) , K b ( y ) } = − k π (cid:15) ij [ g ab ∂ j + f abc A cj ( y )] δ ( x − y ) , (2.31i) { D i φ ai ( x ) , K b ( y ) } = − k π (cid:15) ij f abc D i A cj δ ( x − y ) , (2.31j) { φ ai ( x ) , ¯ K b ( y ) } = − f abc φ ci δ ( x − y ) , (2.31k) { D i φ ai ( x ) , D j φ bj ( y ) } = − k π (cid:15) ij f abc D i A cj δ ( x − y ) − f abc D i φ ci δ ( x − y ) , (2.31l)where ∂ i denotes ∂∂y i . Using these relations one finds the following algebra of con-straints { φ ai ( x ) , φ bj ( y ) } = − k π (cid:15) ij g ab δ ( x − y ) , (2.32a) { φ ai ( x ) , ¯ K b ( y ) } = − f abc φ ci δ ( x − y ) , (2.32b) { ¯ K a ( x ) , ¯ K b ( y ) } = − f abc ¯ K c δ ( x − y ) , (2.32c)which are the only non-vanishing Poisson brackets of the constraints φ aµ and ¯ K a .Hence φ a and ¯ K a are first class constraints and φ ai are second class constraints.Thus we can use the second class constraints φ ai to restrict our phase space anddefine the corresponding Dirac bracket of the remaining canonical variables. In thiscase the only non-vanishing Dirac bracket of the dynamical fields is given by thefollowing relation { A ai ( x ) , A bj ( y ) } D.B = 2 πk g ab (cid:15) ij δ ( x − y ) . (2.33) As a next step we are interested in the generators that correspond to the gaugetransformations induced by the first class constraints φ a and ¯ K a . A useful way to– 15 –onstruct the generators is given by Castellani’s algorithm [52]. In the general casethe gauge generator is given by G = λ ( t ) G + ˙ λ ( t ) G , (2.34)with ˙ λ ( t ) ≡ dλ ( t ) dt . The constraints G and G then have to fulfill the followingrelations G = C P F C , (2.35a) G + { G , H T } = C P F C , (2.35b) { G , H T } = C P F C , (2.35c)where C P F C denotes a primary first class constraint. These relations are fulfilled for G = ¯ K a and G = φ a = π a . The smeared generator of gauge transformations hasthe following form G [ λ ] = (cid:90) Σ d x (cid:0) D λ a π a + λ a ¯ K a (cid:1) . (2.36)Using (2.31) one can show by a straightforward calculation that this generator gen-erates the following gauge transformations via δ λ • = {• , G [ λ ] } δ λ A a = D λ a , (2.37a) δ λ A ai = D i λ a , (2.37b) δ λ π a = − f abc λ b π c , (2.37c) δ λ π ai = k π (cid:15) ij g ab ∂ j λ b − f abc λ b π ci , (2.37d) δ λ φ ai = − f abc λ b φ ci . (2.37e)The generator G that we have constructed via this method is only a preliminaryresult, since the presence of a boundary in our theory prevents that the generator G is properly functionally differentiable. We will fix this by first computing the fullvariation of the generator for a field independent gauge parameter λ a δG [ λ ] = (cid:90) Σ d x ( δ ( D λ a π a ) + λ a δ ¯ K a ) = (cid:90) Σ d x (cid:18) ˙ λ a δπ a − λ a f abc ( δA b π c + A b δπ c ) − k π (cid:15) ij g ab ∂ j λ a δA bi + ∂ i λ a δπ ai − λ a f abc ( δA bi π ci + A bi δπ ci ) − ∂ i (cid:18) k π (cid:15) ij g ab λ a δA bj + λ a δπ ai (cid:19)(cid:19) = (cid:90) Σ d x (cid:18) f abc λ c π aµ δA bµ + D µ λ a δπ aµ + k π (cid:15) ij g ab ∂ i λ a δA bj − ∂ i (cid:18) k π (cid:15) ij g ab λ a δA bj + λ a δπ ai (cid:19)(cid:19) . (2.38)– 16 –he first three terms are regular bulk terms and thus do not spoil functional differen-tiability. The last term on the other hand is a boundary term that spoils functionaldifferentiability. Thus in order to fix this one has to add a suitable boundary term tothe gauge generator such that the variation of this additional boundary term cancelsexactly the boundary term in (2.38) i.e. δ ¯ G [ λ ] = δG [ λ ] + δQ [ λ ] , (2.39)with δQ [ λ ] = (cid:90) Σ d x ∂ i (cid:18) k π (cid:15) ij g ab λ a δA bj + λ a δπ ai (cid:19) . (2.40)Setting the second class constraints φ ai ≈ δQ [ λ ] = k π (cid:90) d ϕg ab λ a δA bϕ . (2.41)If we assume that the gauge parameter is field independent, then the boundary charge Q [ λ ] is trivially integrable. This yields the following canonical boundary charge Q [ λ ] = k π (cid:90) d ϕg ab λ a A bϕ . (2.42) After performing the canonical analysis and having identified all the constraints wecan turn our attention to an appropriate choice of gauge. Since we have foundtwo first class constraints we are free to impose two sets of gauge conditions. Oneappropriate partial gauge fixing choice is given by [52] A ρ = b − ( ρ ) ∂ ρ b ( ρ ) , (2.43a) A ϕ = b − ( ρ ) a ϕ ( ϕ, t ) b ( ρ ) , (2.43b) A t = b − ( ρ ) a t ( ϕ, t ) b ( ρ ) , (2.43c)with the group element b ( ρ ) = e ρL . (2.44)This choice of gauge automatically solves the flatness conditions F tρ = 0 and F ϕρ = 0.– 17 – . AdS × R and H × R for sl (3) In this section we present appropriate boundary conditions on the connection A withan AdS × R or H × R background with H being the Lobachevsky plane. In orderto construct such a background, an embedding that contains at least one singlet withTr( S ) (cid:54) = 0 is necessary. This leads to an embedding with three sl (2) generators L n ( n = − , , ψ ± n ( n = − , ) of spin and one singlet S ofspin 1. These generators fulfill the following commutation relations [ L n , L m ] = ( n − m ) L n + m , (3.1a)[ L n , S ] = 0 , (3.1b)[ L n , ψ ± m ] = ( n − m ) ψ ± n + m , (3.1c)[ S, ψ ± m ] = ± ψ ± m , (3.1d)[ ψ + n , ψ − m ] = L m + n + 32 ( m − n ) S. (3.1e)The existence of the two doublets in our representation allows us to consider linearcombinations of the corresponding generators without spoiling (3.1c), and we usedthis freedom to define the ψ ± n in such a way that they are eigenstates of the adjointaction of the singlet S .Consider the connections A ρ = L ¯ A ρ = − L (3.2a) A ϕ = σe ρ L ¯ A ϕ = − e ρ L − (3.2b) A t = 0 ¯ A t = √ S (3.2c)with some constant σ = ±
1. Using the following definition g µν := 12 Tr (cid:0) A µ − ¯ A µ (cid:1) (cid:0) A ν − ¯ A ν (cid:1) , (3.3)one obtains the following asymptotic background metricd s = d t + d ρ − σe ρ d ϕ . (3.4)Depending on the sign of σ this metric is asymptotically AdS × R ( σ = 1) or H × R ( σ = −
1) with an Euclidean signature.
Regarding the fluctuations of the background on the boundary which we assumeto be located at ρ → ∞ , we consider as a starting point the following boundary For the matrix representations of these generators and the corresponding Killing form used forthe computations in this section, please refer to Appendix (A.1). – 18 –onditions g µν = O ( e − ρ ) O ( e − ρ ) O (1) · O ( e − ρ ) O (1) · · − σe ρ + O ( e ρ ) µν , (3.5)where the coordinates are ordered as t, ρ, ϕ . These fluctuations are chosen in sucha way that they agree with the asymptotic behavior of the first descendant of theAdS × R [ H × R ] vacuum, which can be found in [54]. In fact it is even possible tohave a bit stricter boundary conditions that still agree with the first descendant of thevacuum, which we will present in the following subsection. Since the structure of (3.2)suggests that A and ¯ A are treated differently we will also state boundary conditionsand the corresponding boundary charges for A and ¯ A differently. A convenientnotation for A ( ¯ A ) is given by the following splitting A µ = A (0) µ + A (1) µ , (3.6)where A (1) µ will contain all the subleading parts that do not appear in the canonicalboundary charge. Using the gauge choice given by (2.43) and (2.44) one can write(3.6) as A µ = b − a µ b = b − (cid:0) a (0) µ + a (1) µ (cid:1) b (3.7)and ¯ A µ as ¯ A µ = b ¯ a µ b − = b (cid:0) ¯ a (0) µ + ¯ a (1) µ (cid:1) b − . (3.8)the connection A has to obey the following boundary conditions for ρ → ∞ in orderto fulfill (3.5) a (0) ρ = L , (3.9a) a (0) ϕ = σL + 2 πk (cid:18) −L ( ϕ ) L − + W + ( ϕ ) ψ + − − W − ( ϕ ) ψ −− + 32 W ( ϕ ) S (cid:19) , (3.9b) a (0) t = 0 , (3.9c) a (1) µ = O ( e − ρ ) . (3.9d)And for ¯ A the boundary conditions are given by¯ a (0) ρ = ( − B ( ϕ ) e − ρ ) L + O ( e − ρ ) S, (3.10a)¯ a (0) ϕ = O (1) L + ( − B ( ϕ ) e − ρ ) L − + O (1) ψ + + O (1) ψ − − πk ¯ W ( ϕ ) S, (3.10b)¯ a (0) t = (cid:16) √ O ( e − ρ ) (cid:17) S, (3.10c)¯ a (1) µ = O ( e − ρ ) . (3.10d)We have chosen a normalization for the fields L , W ± , W and ¯ W such that thecorresponding canonical charge is conveniently normalized. The specific form and– 19 –hus the appearance of the function B ( ϕ ) as the subleading terms in (3.10a) and(3.10b) is a result of the requirement to fulfill the EOM asymptotically, i.e. F | ∂M →
0. If the subleading part of these two terms was not the same function, then theresulting connection would not be an asymptotically flat one. This is thus the onlycaveat if one tries to fulfill the boundary conditions (3.5). The fluctuations appearingin g ρρ and g φφ are not independent and are in fact identical. The requirement ofasymptotical flatness of the connection also restricts the subleading terms of A and¯ A to be functions of only ϕ and ρ .As already mentioned, it is also possible to consider fluctuations of the metric thatare a bit more restricted than (3.5). Considering the fluctuations given by g µν = O ( e − ρ ) O ( e − ρ ) O (1) · O ( e − ρ ) O (1) · · − σe ρ + O (1) µν , (3.11)then the corresponding connection A has to obey the following boundary conditions a (0) ρ = L , (3.12a) a (0) ϕ = σL + 2 πk (cid:18) −L ( ϕ ) L − + W + ( ϕ ) ψ + − − W − ( ϕ ) ψ −− + 32 W ( ϕ ) S (cid:19) , (3.12b) a (0) t = 0 , (3.12c) a (1) µ = O ( e − ρ ) . (3.12d)For the ¯ A -sector the connection has to obey asymptotically¯ a (0) ρ = − L , (3.13a)¯ a (0) ϕ = O (1) L − L − + O (1) ψ + + O (1) ψ − − πk ¯ W ( ϕ ) S, (3.13b)¯ a (0) t = √ S, (3.13c)¯ a (1) µ = O ( e − ρ ) . (3.13d)The fluctuations resulting from (3.12) and (3.13) obey (3.11) and are completelyarbitrary in contrast to the boundary conditions (3.10) and (3.13), which yieldedfluctuations of the metric that had to be of a specific form. The boundary condition preserving gauge transformations and the resulting canoni-cal boundary charges are the same for both boundary conditions presented, thus thefollowing discussion applies to both cases. Since A and ¯ A obey different boundaryconditions, the corresponding boundary condition preserving gauge transformations– 20 –nd canonical boundary charges will be treated separately as well.A gauge transformation with gauge parameter (cid:15) preserves a given set of boundaryconditions if δ (cid:15) A aµ = D µ (cid:15) a = ∂ µ (cid:15) a + f abc A bµ (cid:15) c = O (cid:0) A aµ (cid:12)(cid:12) ∂ M (cid:1) . (3.14)To be a little more specific on the notation: A aµ (cid:12)(cid:12) ∂ M denotes the subleading terms ofthe connection as ρ → ∞ . For the gauge choice (2.43) and the boundary conditions(3.12b) this would mean for example that a boundary preserving gauge transforma-tion has to satisfy δ (cid:15) A L ϕ = O ( e − ρ ) , δ (cid:15) A L ϕ = O ( e − ρ ) , δ (cid:15) A L − ϕ = O ( e − ρ ) ,δ (cid:15) A ψ +12 ϕ = O ( e − ρ ) , δ (cid:15) A ψ + − ϕ = O ( e − ρ ) ,δ (cid:15) A ψ − ϕ = O ( e − ρ ) , δ (cid:15) A ψ −− ϕ = O ( e − ρ ) ,δ (cid:15) A Sϕ = O (1) . (3.15)Using (3.14) one finds that the gauge transformations that preserve (3.9) [and (3.12)]are given by ˆ (cid:15) = b − (cid:0) (cid:15) (0) + (cid:15) (1) (cid:1) b. (3.16)The first part is given by (cid:15) (0) = ( (cid:15) L + (cid:15) L + (cid:15) L − + (cid:15) ψ + + (cid:15) ψ + − + (cid:15) ψ − + (cid:15) ψ −− + (cid:15) S ) (3.17)with (cid:15) = (cid:15) ( ϕ ) , (cid:15) = − σ (cid:15) (cid:48) ( ϕ ) , (3.18a) (cid:15) = 12 σ (cid:15) (cid:48)(cid:48) ( ϕ ) − πσk (cid:18) L ( ϕ ) (cid:15) ( ϕ ) + 12 (cid:16) W − ( ϕ ) (cid:15) + ( ϕ ) + W + ( ϕ ) (cid:15) − ( ϕ ) (cid:17)(cid:19) , (3.18b) (cid:15) = (cid:15) + ( ϕ ) , (cid:15) = − σ (cid:18) (cid:15) + (cid:48) ( ϕ ) − πk (cid:18) W + ( ϕ ) (cid:15) ( ϕ ) − W ( ϕ ) (cid:15) + ( ϕ ) (cid:19)(cid:19) , (3.18c) (cid:15) = (cid:15) − ( ϕ ) , (cid:15) = − σ (cid:18) (cid:15) − (cid:48) ( ϕ ) + 2 πk (cid:18) W − ( ϕ ) (cid:15) ( ϕ ) − W ( ϕ ) (cid:15) − ( ϕ ) (cid:19)(cid:19) , (3.18d) (cid:15) = (cid:15) ( ϕ ) . (3.18e)and the subleading parts are given by (cid:15) (1) = O ( e − ρ ) . (3.19)Having found the boundary condition preserving gauge transformations we are nowinterested how the fields L , W ± and W transform under these gauge transforma-tions. Since A L − ϕ = − πk L e − ρ , A ψ ±− ϕ = ± πk W ± e − ρ , A Sϕ = 3 πk W , (3.20)– 21 –t is easy to see that δA L − ϕ = − πk δ L e − ρ , δA ψ ±− ϕ = ± πk δ W ± e − ρ , δA Sϕ = 3 πk δ W , (3.21)with δ = δ ˆ (cid:15) = δ (cid:15) + δ (cid:15) + δ (cid:15) +12 + δ (cid:15) − . In order to find the correct boundary preservinggauge transformations we already calculated δA L − ϕ , δA ψ ±− ϕ and δA Sϕ . Thus, one onlyhas to look at the leading order contributions of these expressions and read off thetransformation properties of the fields L , W ± and W . This leads to the followingtransformations δ (cid:15) W = k π (cid:15) (cid:48) , δ (cid:15) W ± = ∓ (cid:15) W ± , δ (cid:15) L = 0 , (3.22a) δ (cid:15) W = 0 , δ (cid:15) L = − k π (cid:15) (cid:48)(cid:48)(cid:48) + σ (2 (cid:15) (cid:48) L + (cid:15) L (cid:48) ) ,δ (cid:15) W ± = σ (cid:18) (cid:15) (cid:48) W ± + (cid:15) W ± (cid:48) ± πk (cid:15) W ± W (cid:19) , (3.22b) δ (cid:15) ± W = ∓W ∓ (cid:15) ± , δ (cid:15) ± W ∓ = 0 ,δ (cid:15) ± W ± = ± (cid:15) ± L − σ (cid:18) (cid:15) ± (cid:48) W + 32 (cid:15) ± W (cid:48) ± π k (cid:15) ± W W ± k π (cid:15) ± (cid:48)(cid:48) (cid:19) δ (cid:15) ± L = σ (cid:18) ± πk W W ∓ (cid:15) ± + W ∓ (cid:48) (cid:15) ∓ + 3 W ∓ (cid:15) ± (cid:48) (cid:19) . (3.22c)Please note that in order to obtain (3.22) we used σ = 1 in order to simplify someof the expressions. If σ (cid:54) = ± σ → σ and − k π (cid:15) (cid:48)(cid:48)(cid:48) → − k πσ (cid:15) (cid:48)(cid:48)(cid:48) in (3.22) and in all formulas appearing in section 3.3 to get the correct prefactors.The corresponding variation of the boundary charge is then given by δ Q (ˆ (cid:15) ) = (cid:90) d ϕ (cid:16) δ L (cid:15) + δ W (cid:15) + δ W + (cid:15) − + δ W − (cid:15) + (cid:17) . (3.23)Since the gauge parameters (cid:15) , (cid:15) ± , (cid:15) are field independent one can write the corre-sponding canonical charge as Q (ˆ (cid:15) ) = (cid:90) d ϕ (cid:16) L (cid:15) + W (cid:15) + W + (cid:15) − + W − (cid:15) + (cid:17) . (3.24)Finding the boundary condition preserving gauge transformations and boundarycharge for the ¯ A -sector works exactly like for the A -sector. The only difference is thatwe have to preserve a different set of boundary conditions. The gauge transformationspreserving (3.10) [and (3.13)] are given by¯ (cid:15) ( ϕ ) = b (cid:0) ¯ (cid:15) (0) + ¯ (cid:15) (1) (cid:1) b − , (3.25)– 22 –ith ¯ (cid:15) (0) = ¯ (cid:15) ( ϕ ) S (3.26)and ¯ (cid:15) (1) = O ( e − ρ ) . (3.27)Comparing these gauge transformations with (3.18a), we see that the gauge transfor-mations in the ¯ A -sector are a lot more restricted. The reason for this is the presenceof the singlet term at leading order in ¯ A St . Without this singlet term the boundarycondition preserving gauge transformations for the ¯ A -sector would look like (3.18a).This singlet term, however, is crucial in our discussion of AdS × R [ H × R ] sincethe singlet term generates the d t part of the background. Thus, the same analysisas for the A -sector yields the following transformation δ ¯ (cid:15) ¯ W = − k π ¯ (cid:15) (cid:48) . (3.28)The corresponding variation of the boundary charge is given by δ ¯ Q (¯ (cid:15) ) = (cid:90) d ϕδ ¯ W ¯ (cid:15) . (3.29)Again, the gauge parameters are field independent and thus the canonical charge forthe ¯ A -sector is given by ¯ Q (¯ (cid:15) ) = (cid:90) d ϕ ¯ W ¯ (cid:15) . (3.30) After computing (3.9) and (3.10) one can calculate the Dirac brackets correspondingto the symmetry present at the boundary [16]. The latter then yields the asymptoticsymmetry algebra. Actually, there is a convenient short-cut that avoids the tediouscalculation of Dirac brackets. Given two fields V , W and a canonical boundarycharge ˆ Q ( λ ) = (cid:82) d ϕ λ ( ϕ ) V ( ϕ ) one can use δ λ W ( ¯ ϕ ) = −{ ˆ Q ( λ ) , W ( ¯ ϕ ) } = − (cid:90) d ϕ λ ( ϕ ) {V ( ϕ ) , W ( ¯ ϕ ) } , (3.31)to determine {V ( ϕ ) , W ( ¯ ϕ ) } , given that δ λ W ( ¯ ϕ ) has been calculated beforehand. TheDirac bracket {L ( ϕ ) , L ( ¯ ϕ ) } for example can be calculated via δ (cid:15) L ( ¯ ϕ ) = −{Q ( (cid:15) ) , L ( ¯ ϕ ) } = − (cid:90) d ϕ (cid:15) ( ϕ ) {L ( ϕ ) , L ( ¯ ϕ ) } . (3.32)Equation (3.32) can be satisfied for {L ( ϕ ) , L ( ¯ ϕ ) } = − k π δ (cid:48)(cid:48)(cid:48) ( ϕ − ¯ ϕ ) + σ (2 L ( ¯ ϕ ) δ (cid:48) ( ϕ − ¯ ϕ ) − L (cid:48) ( ¯ ϕ ) δ ( ϕ − ¯ ϕ )) , (3.33)– 23 –ith δ (cid:48) ( ϕ − ¯ ϕ ) = ∂ ϕ ( ϕ − ¯ ϕ ). This can also be written in terms of δ (cid:15) L as {L ( ϕ ) , L ( ¯ ϕ ) } = − δ (cid:15) L ( ¯ ϕ ) (cid:12)(cid:12)(cid:12) ∂ n ¯ ϕ (cid:15) ( ¯ ϕ )=( − n ∂ nϕ δ ( ϕ − ¯ ϕ ) . (3.34)Using (3.31) this procedure can be repeated for all the other remaining fields andgauge parameters appearing in (3.22). This yields the following Dirac brackets withthe convention that all fields appearing on the right hand side depend on ¯ ϕ and δ (cid:48) ( ϕ − ¯ ϕ ) ≡ ∂ ϕ δ ( ϕ − ¯ ϕ ). {W ( ϕ ) , W ( ¯ ϕ ) } = k π δ (cid:48) ( ϕ − ¯ ϕ ) , (3.35a) {W ( ϕ ) , L ( ¯ ϕ ) } = 0 , (3.35b) {W ( ϕ ) , W ± ( ¯ ϕ ) } = ±W ± δ ( ϕ − ¯ ϕ ) , (3.35c) {L ( ϕ ) , L ( ¯ ϕ ) } = − k π δ (cid:48)(cid:48)(cid:48) ( ϕ − ¯ ϕ ) + σ (2 L δ (cid:48) ( ϕ − ¯ ϕ ) − L (cid:48) δ ( ϕ − ¯ ϕ )) , (3.35d) {L ( ϕ ) , W ± ( ¯ ϕ ) } = σ (cid:18) W ± δ (cid:48) ( ϕ − ¯ ϕ ) − W ± (cid:48) δ ( ϕ − ¯ ϕ ) ∓ πk W ± W δ ( ϕ − ¯ ϕ ) (cid:19) , (3.35e) {W + ( ϕ ) , W − ( ¯ ϕ ) } = L δ ( ϕ − ¯ ϕ ) + σ (cid:18) − W δ (cid:48) ( ϕ − ¯ ϕ ) + 32 W (cid:48) δ ( ϕ − ¯ ϕ ) − π k W W δ ( ϕ − ¯ ϕ ) − k π δ (cid:48)(cid:48) ( ϕ − ¯ ϕ ) (cid:19) (3.35f) {W + ( ϕ ) , W + ( ¯ ϕ ) } = {W − ( ϕ ) , W − ( ¯ ϕ ) } = 0 . (3.35g)This algebra is written in a non-primary basis, as one can see by looking at (3.35b)and (3.35e). This can be fixed by a shift of L given by L → L + 3 πσ k W W ≡ ˆ L . (3.36)After applying this shift, the non-vanishing Dirac brackets for the A -sector are givenby {W ( ϕ ) , W ( ¯ ϕ ) } = k π δ (cid:48) ( ϕ − ¯ ϕ ) , (3.37a) {W ( ϕ ) , ˆ L ( ¯ ϕ ) } = σ W δ (cid:48) ( ϕ − ¯ ϕ ) , (3.37b) {W ( ϕ ) , W ± ( ¯ ϕ ) } = ±W ± δ ( ϕ − ¯ ϕ ) , (3.37c) { ˆ L ( ϕ ) , ˆ L ( ¯ ϕ ) } = − k π δ (cid:48)(cid:48)(cid:48) ( ϕ − ¯ ϕ ) + σ (cid:16) L δ (cid:48) ( ϕ − ¯ ϕ ) − ˆ L (cid:48) δ ( ϕ − ¯ ϕ ) (cid:17) , (3.37d) { ˆ L ( ϕ ) , W ± ( ¯ ϕ ) } = σ (cid:18) W ± δ (cid:48) ( ϕ − ¯ ϕ ) − W ± (cid:48) δ ( ϕ − ¯ ϕ ) (cid:19) , (3.37e) {W + ( ϕ ) , W − ( ¯ ϕ ) } = ˆ L δ ( ϕ − ¯ ϕ ) + σ (cid:18) − W δ (cid:48) ( ϕ − ¯ ϕ ) + 32 W (cid:48) δ ( ϕ − ¯ ϕ ) − πk W W δ ( ϕ − ¯ ϕ ) − k π δ (cid:48)(cid:48) ( ϕ − ¯ ϕ ) (cid:19) . (3.37f)– 24 –he Dirac brackets for the ¯ A -sector are given by { ¯ W ( ϕ ) , ¯ W ( ¯ ϕ ) } = − k π δ (cid:48) ( ϕ − ¯ ϕ ) . (3.38)Thus, the boundary conditions (3.9) and (3.10) give rise to one copy of a classical W (2)3 algebra, which is also called Polyakov-Bershadsky algebra for the A -sector and a u (1) current algebra with a central extension for the ¯ A -sector. Therefore, we obtain W (2)3 ⊕ u (1) as the asymptotic symmetry algebra. Since we are interested in thecentral charges of the corresponding boundary theory, we will first express (3.35)in terms of its Fourier modes but without taking normal ordering into account andthus obtain the ”classical central charges” of the boundary theory. After obtainingthis algebra and replacing the Dirac brackets with commutators, we will focus onnormal ordering issues and determine the effective central charge of the boundarytheory. Since the central terms in (3.35) can be rescaled arbitrarily and thus thecentral charges are not unique, one has to find a way to fix this. In the case we areconsidering, at least in the A -sector one has the additional information of the algebrawithout a central extension given by (3.1). Thus if one rescales the relations (3.35)in such a way that the corresponding non-centrally extended part of the commutatoralgebra agrees with (3.1), then the correct central charges can be read off directly.Using the following mode expansion L ( ϕ ) = σ π (cid:88) n ∈ Z L n e − inϕ and δ ( ϕ − ¯ ϕ ) = 12 π (cid:88) n ∈ Z e − in ( ϕ − ¯ ϕ ) , (3.39)where we have shifted the zero mode as L → L − k δ n, , (3.40)and plugging this expansion into (3.35d) one obtains (cid:88) n,m ∈ Z e − i ( nϕ + m ¯ ϕ ) { L n , L m } = (cid:88) n,p ∈ Z e − ip ¯ ϕ − in ( ϕ − ¯ ϕ ) (cid:18) L p − k δ p, (cid:19) ( − in + ip ) − k (cid:88) n ∈ Z in e − in ( ϕ − ¯ ϕ ) . (3.41)Making an appropriate shift p − n = m and using the orthogonality property of thecomplex exponential function (cid:82) d ϕe i ( n − m ) ϕ = 2 πδ n,m one obtains { L n , L m } = i ( m − n ) L m + n − i k n ( n − δ m + n, . (3.42)Replacing the Dirac bracket with a commutator using i {· , ·} → [ · , · ] one obtains[ L n , L m ] = ( n − m ) L m + n + c n ( n − δ m + n, . (3.43)– 25 –ith the central charge c = 6 k . Using the same procedure with all remaining Diracbrackets (3.35) and the following mode expansions W ( ϕ ) = i π (cid:88) n ∈ Z J n e − inϕ and W ± ( ϕ ) = ( iσ ) ∓ π (cid:88) n ∈ Z + G ± n e − inϕ , (3.44)one obtains the following (classical) commutation relations[ J n , J m ] = − k nδ n + m, , (3.45a)[ J n , L m ] = 0 , (3.45b)[ J n , G ± m ] = ± G ± m + n , (3.45c)[ L n , L m ] = ( n − m ) L m + n + c n ( n − δ n + m, , (3.45d)[ L n , G ± m ] = (cid:16) n − m (cid:17) G ± n + m ± k (cid:88) p ∈ Z ( G ± m + n − p J p + J p G ± m + n − p ) , (3.45e)[ G + n , G − m ] = L m + n + 32 ( m − n ) J m + n + 94 k (cid:88) p ∈ Z J m + n − p J p + k ( n −
14 ) δ m + n, , (3.45f)[ G + n , G + m ] = [ G − n , G − m ] = 0 . (3.45g)Please note that in order to calculate (3.45f) the following definition for δ ( ϕ − ¯ ϕ ) hasbeen used δ ( ϕ − ¯ ϕ ) = 12 π (cid:88) n ∈ Z + e − in ( ϕ − ¯ ϕ ) . (3.46)This definition is necessary in order to satisfy (cid:90) d ϕ(cid:15) ± ( ϕ ) δ ( ϕ − ¯ ϕ ) = (cid:15) ± ( ¯ ϕ ) , (3.47)with (cid:15) ± ( ϕ ) = 12 π (cid:88) n ∈ Z + (cid:15) ± n e − inϕ . (3.48)Thus if one tries to write a Dirac bracket in terms of Fourier modes that has beenobtained by satisfying δ (cid:15) ± W ± ( ¯ ϕ ) = − (cid:90) d ϕ(cid:15) ± ( ϕ ) {W ∓ ( ϕ ) , W ± ( ¯ ϕ ) } , (3.49)one has to use (3.46) rather than (3.39). In addition, we used( W ± W )( ¯ ϕ ) = 12 ( W ± W + W W ± )( ¯ ϕ ) . (3.50)– 26 –he algebraic relations (3.45) can again be brought into a form where all fieldsappearing are proper Virasoro primaries. This is a similar shift to the one done inthe case of the Dirac bracket algebra and is given by L n → ˆ L n ≡ L n − k (cid:88) p ∈ Z J n − p J p . (3.51)This yields the following algebra[ J n , J m ] = − k nδ n + m, , (3.52a)[ J n , ˆ L m ] = nJ n + m , (3.52b)[ J n , G ± m ] = ± G ± m + n , (3.52c)[ ˆ L n , ˆ L m ] = ( n − m ) ˆ L m + n + c n ( n − δ n + m, , (3.52d)[ ˆ L n , G ± m ] = (cid:16) n − m (cid:17) G ± n + m , (3.52e)[ G + n , G − m ] = ˆ L m + n + 32 ( m − n ) J m + n + 3 k (cid:88) p ∈ Z J m + n − p J p + k ( n −
14 ) δ m + n, , (3.52f)[ G + n , G + m ] = [ G − n , G − m ] = 0 . (3.52g) W (2)3 and u (1) Current Algebra
Since we are interested in the quantum mechanical version of (3.45), we also have totake normal ordering into account whenever products of Fourier modes appear. Thesymbol : : denotes normal ordering which we defined as follows (cid:88) p ∈ Z : J n − p J p := (cid:88) p ≥ J n − p J p + (cid:88) p< J p J n − p . (3.53)However, since the algebraic relations (3.52) are given in terms of large c or equiv-alently large k , it is possible that all coefficients that contain factors of k obtainquantum corrections of O (1). Thus when introducing normal ordering these correc-tions have to be determined. Possibly the easiest way to do this is to consider the– 27 –ollowing algebra[ J n , J m ] = C nδ n + m, , (3.54a)[ J n , ˆ L m ] = nJ n + m , (3.54b)[ J n , G ± m ] = ± G ± m + n , (3.54c)[ ˆ L n , ˆ L m ] = ( n − m ) ˆ L m + n + ˆ c n ( n − δ n + m, , (3.54d)[ ˆ L n , G ± m ] = (cid:16) n − m (cid:17) G ± n + m , (3.54e)[ G + n , G − m ] = C ˆ L m + n + 32 C ( m − n ) J m + n + C (cid:88) p ∈ Z : J m + n − p J p : + C ( n −
14 ) δ m + n, , (3.54f)[ G + n , G + m ] = [ G − n , G − m ] = 0 , (3.54g)and calculate the Jacobi identities which yield relations between these coefficientsthat allow us to fix them such that we have a consistent algebra (at least consistentwith respect to the Jacobi identities). Even though the coefficients C and C areequal to 1 in (3.52) and do not contain factors of k , a rescaling of G ± n by a factor of √ k could easily produce such a k dependence and thus one has to consider these twocoefficients not to be fixed to 1. In addition, we fixed the shift of the normal orderedVirasoro modes to be L n → ˆ L n ≡ L n + 12 C (cid:88) p ∈ Z : J n − p J p : . (3.55)This normalization ensures that normal ordering of the Virasoro modes results in theexpected shift of the central charge c = 6 k by +1, thus yielding a preliminary shift ofthe central charge c → c + 1. This shift, however, will also be further modified oncethe Jacobi identities have to be satisfied. Calculating the Jacobi identities yields thefollowing relations between the coefficients C + 3 C + 2 C C = 0 , C + 32 C C = 0 , (3.56a) C ˆ c − C + 2 C C = 0 , C − C − C = 0 . (3.56b)Since we have four equations but six free parameters, we have the freedom to fixtwo of them and the remaining four coefficients are then determined by the relations(3.56). Since we already know what the algebra looks like in the classical case forlarge k , one viable choice of coefficients would be C = − k , C = 1 . (3.57)– 28 –his yields the following coefficients and shifted central charge C = 1 + 84 k − , C = 124 k − , C = k (cid:18) k − (cid:19) , (3.58a)ˆ c = 32 k k − k. (3.58b)This is essentially the quantum W (2)3 algebra found by Polyakov and Bershadskyin [55, 56] but with a different k and different normalization of the spin- modes. Inorder to bring this algebra in a more familiar form, we apply the following shift of k and renormalization of G ± n k → − (cid:18) ˆ k + 32 (cid:19) , G ± n (cid:113) − (ˆ k + 3) → ˆ G ± n . (3.59)This results in the following algebra[ J n , J m ] = 2ˆ k + 33 nδ n + m, , (3.60a)[ J n , ˆ L m ] = nJ n + m , (3.60b)[ J n , ˆ G ± m ] = ± ˆ G ± m + n , (3.60c)[ ˆ L n , ˆ L m ] = ( n − m ) ˆ L m + n + ˆ c n ( n − δ n + m, , (3.60d)[ ˆ L n , ˆ G ± m ] = (cid:16) n − m (cid:17) ˆ G ± n + m , (3.60e)[ ˆ G + n , ˆ G − m ] = − (ˆ k + 3) ˆ L m + n + 32 (ˆ k + 1)( n − m ) J m + n + 3 (cid:88) p ∈ Z : J m + n − p J p : +(ˆ k + 1)(2ˆ k + 3)2 ( n −
14 ) δ m + n, , (3.60f)[ ˆ G + n , ˆ G + m ] = [ ˆ G − n , ˆ G − m ] = 0 , (3.60g)with ˆ c = 25 − k + 3 − k + 3) = − (2ˆ k + 3)(3ˆ k + 1)ˆ k + 3 . (3.61)It is easy to see that the central charge ˆ c is only non-negative for a small rangeof ˆ k , which is given by the interval − ≤ ˆ k ≤ − . The maximum value of thecentral charge is ˆ c = 1, which is obtained for ˆ k = −
1. Thus, it is not possible toobtain a unitary field theory dual of AdS × R or H × R in the semi-classical limit | ˆ k | → ∞ [57].In the ¯ A -sector we have only one Poisson bracket corresponding to a u (1) currentalgebra with a central extension. The value of the central charge correspondingto this central extension is not unique, since we can always rescale the fields orcorresponding Fourier modes. Thus, with the following mode expansion¯ W ( ϕ ) = 12 π (cid:88) n ∈ Z ¯ J n e − inϕ , (3.62)– 29 –nd the same shift of the Chern-Simons level k as in (3.59), one obtains the followingcommutator algebra [ ¯ J n , ¯ J m ] = 2ˆ k + 33 nδ n + m, . (3.63) Having found the asymptotic symmetry algebra we are now interested if it is possibleto obtain a unitary CFT for certain values of ˆ k or not. Thus, we have to check if thereare any unphysical states i.e. states with negative norm present. Since the ¯ A -sectoronly consists of a u (1) algebra and it is not hard to find unitary representations forthis algebra, we will focus with our analysis on the A -sector containing the W (2)3 algebra where the existence of unitary representations is not obvious at first glance.Let | a ; N (cid:105) , with a = 1 , . . . , N denote a basis of states for a given level N . Then anystate at level N can be written as the following linear combination | ψ ; N (cid:105) = N (cid:88) a =1 λ a | a ; N (cid:105) , (3.64)where λ a ∈ C are some arbitrary constants. The norm of such a state is then givenby (cid:104) ψ ; N | ψ ; N (cid:105) = N (cid:88) a,b λ † a (cid:104) a ; N | b ; N (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) K ( N ) ab λ b , (3.65)where K ( N ) ab denotes the Gramian matrix at level N . Thus, in order to have a unitarytheory the Gramian matrix has to be positive semidefinite.Since all modes appearing in the W (2)3 algebra are proper Virasoro primaries, theiraction on the vacuum state is given by L n | (cid:105) = 0 , J n | (cid:105) = 0 , ˆ G ± n | (cid:105) = 0 for n > − h i (3.66)where h i (with i = L, J, ˆ G ) denotes the conformal weight of the primary fields thatcorrespond to the given modes L n , J n and ˆ G ± n respectively. The hermitian conjugateof the modes is defined as( L n ) † ≡ L − n , ( J n ) † ≡ J − n , (cid:16) ˆ G ± n (cid:17) † ≡ ˆ G ∓− n . (3.67)The hermitian conjugate of ˆ G ± n may look strange, but is in fact a direct consequenceof the quantum W (2)3 algebra. After defining ( L n ) † and ( J n ) † one can look at (cid:16)(cid:104) ˆ G + n , ˆ G − m (cid:105)(cid:17) † = (cid:20)(cid:16) ˆ G − m (cid:17) † , (cid:16) ˆ G + n (cid:17) † (cid:21) == − (ˆ k + 3) ˆ L † m + n + 32 (ˆ k + 1)( n − m ) J † m + n + 3 (cid:88) p ∈ Z (: J m + n − p J p :) † +(ˆ k + 1)(2ˆ k + 3)2 ( n −
14 ) δ m + n, . (3.68)– 30 –fter using ( L n ) † ≡ L − n and ( J n ) † ≡ J − n one obtains the following (cid:20)(cid:16) ˆ G − m (cid:17) † , (cid:16) ˆ G + n (cid:17) † (cid:21) = − (ˆ k + 3) ˆ L − ( m + n ) + 32 (ˆ k + 1)( n − m ) J − ( m + n ) + 3 (cid:88) p ∈ Z : J − ( m + n ) − p J p : +(ˆ k + 1)(2ˆ k + 3)2 ( n −
14 ) δ m + n, = (cid:104) ˆ G + − m , ˆ G −− n (cid:105) . (3.69)Thus, in general the hermitean conjugate of ˆ G ± n is given by (cid:16) ˆ G ± n (cid:17) † ≡ ( α ) ± ˆ G ∓− n , (3.70)were α could in principle be any complex number. Since in any quantum field theorythe n-point correlation functions have to be real valued functions, we get an additionalrestriction on α . Consider for example the norm of the state ˆ G + − | (cid:105) . Reality of thenorm requires (cid:104) | (cid:16) ˆ G + − (cid:17) † ˆ G + − | (cid:105) = (cid:18) (cid:104) | (cid:16) ˆ G + − (cid:17) † ˆ G + − | (cid:105) (cid:19) † . (3.71)Using (3.70) one gets α (cid:104) | ˆ G − ˆ G + − | (cid:105) = α ∗ (cid:104) | ˆ G − ˆ G + − | (cid:105) (3.72)and thus α = α ∗ . (3.73)Hence without loss of generality we can set α = 1 and arrive at the relations givenby (3.67). Having properly defined the hermitean conjugate of the modes present inthe W (2)3 algebra, one can look for possible negative norm states on the first levels ofthe resulting CFT.On level 1 there is only the state J − | (cid:105) present. In this case the Gramian matrix issimply the norm of the state and is given by (cid:104) | J J − | (cid:105) = 2ˆ k + 33 , (3.74)which is non-negative for ˆ k ≥ − . The level contains two states ˆ G + − | (cid:105) andˆ G −− | (cid:105) . Thus, the Gramian matrix at level is given by K ( ) = (ˆ k + 1)(2ˆ k + 3) (cid:18) − (cid:19) , (3.75)with the basis vectors arranged as ˆ G + − | (cid:105) , ˆ G −− | (cid:105) . At this level we encounter acrucial difference to the similar and maybe more familiar N = 2 superconformalalgebra [58]. In the case of the W (2)3 algebra where the modes ˆ G + n and ˆ G − n obey– 31 –ommutation rather than anticommutation relations the norm of the states ˆ G + − n | (cid:105) and ˆ G −− n | (cid:105) will always differ by a sign. However in case of the N = 2 superconformalalgebra the norm of these states would be the same. Thus while in the superconformalcase it is possible to have states corresponding to the modes ˆ G ±− n which have positivenorm this is not possible for the W (2)3 algebra. Hence we arrive at the followingconclusion: • Unless the states ˆ G + − n | (cid:105) and ˆ G −− n | (cid:105) are null states there are no unitary rep-resentations of the W (2)3 algebra for our choice of the vacuum given by (3.66) . Looking at (3.75) we see that the only values where ˆ G ±− | (cid:105) are null are ˆ k = − k = − . Choosing one of these two values of ˆ k does not automatically ensure thatthe resulting CFT is unitary. One still has to check whether the remaining states inthe theory that are not null spoil unitarity or not.In order to simplify the following discussion, it is beneficial to check if the field contentof the two CFTs we are looking at is maybe more restricted than one initially thinks.The following discussion applies only to the two values of ˆ k for which ˆ G ±− n | (cid:105) are nullstates and can thus set ˆ G ±− n | (cid:105) = 0 ∀ n ∈ Z . (3.76)This in turn also implies that [ ˆ G + − n , ˆ G −− m ] | (cid:105) = 0. Thus, also the right hand side of(3.60) has to be zero. This leads to the following relationˆ L − ( m + n ) | (cid:105) = 1ˆ k + 3 (cid:32)
32 (ˆ k + 1)( m − n ) J − ( m + n ) + 3 m + n − (cid:88) p> J − p J − ( m + n )+ p (cid:33) | (cid:105) . (3.77)Thus, we see that for ˆ k ∈ {− , − } the states ˆ L − ( m + n ) | (cid:105) are a linear combinationof other states which simplifies the theory considerably.In order to have a well defined basis of states at level N , we employ the followingordering of operators J m − n . . . J m p − n p L m p +1 − n p +1 . . . L m N − n N | (cid:105) , (3.78)with the following restrictions on the indices m i and n i m i ∈ N , (3.79) n , . . . , n p ∈ N \{ } , (3.80) n p +1 , . . . , n N ∈ N \{ , } , (3.81) n > . . . > n p , (3.82) n p +1 > . . . > n N , (3.83) N (cid:88) i =1 m i n i = N. (3.84)– 32 –ince L m p +1 − n p +1 . . . L m N − n N | (cid:105) can be rewritten as linear combinations of states of the form J m p +1 − n p +1 . . . J m N − n N | (cid:105) we can express every state at a given level N as J m − n . . . J m N − n N | (cid:105) , (3.85)where now n i ∈ N \{ } ∀ i = 1 , . . . , N and n > . . . > n N .It is also possible to write all states at a given level N as L − and J − descendants oflevel N − J − | (cid:105) present. Acting with either L − or J − on that state one gets the following two states L − J − | (cid:105) =[ L − , J − ] | (cid:105) = J − | (cid:105) , (3.86) J − J − | (cid:105) = J − | (cid:105) , (3.87)which are all possible states at level 2. One can repeat this process indefinitely andobtain in such a way all possible states at a given level N . Please note that withthis way of generating states, starting with M states at level N would generate 2 M states for level N + 1. Since the number of possible states at level N is given bythe number of possible partitions of N , this procedure will in general produce ”toomany” states. This usually happens when a state | a ; N + 1 (cid:105) at level N + 1 can begenerated by the action of either L − or J − on two different states | b ; N (cid:105) , | c ; N (cid:105) atlevel N i.e. | a ; N + 1 (cid:105) ∝ L − | b ; N (cid:105) , (3.88) | a ; N + 1 (cid:105) ∝ J − | c ; N (cid:105) . (3.89)Since this will only manifest as additional zero eigenvalues of the Gramian matrix,this would not spoil any unitarity analysis. Thanks to this procedure of generatingstates it is possible to write all states as descendants of the lowest level states, whichis very convenient. If for example all states at a given level N are null states, thenone can show that all states at levels M > N are also null states. This just followsfrom the fact that one can write all states at level N + 1 as descendants of level N states, which are again null states, since descendants of null states are also nullstates.The same arguments regarding descendant states also apply to the half integer valuedlevels. ˆ k = − and ˆ c = 0This case is trivial: The only state present in our Hilbert space is the vacuum stateitself. This can easily be seen by looking at (3.74) and (3.75), which are null for thisvalue of ˆ k . Thus, all states at integer level n > m > arenull, since all states at a given level m, n can be written as descendants of the level 1– 33 –nd level states. This leads to the conclusion stated at the beginning that the onlystate present in the theory is the vacuum state. This statement is true in general forthe quantum W (2)3 algebra, if one only has (3.60) as a starting point where the onlyrestriction on ˆ k is that ˆ k (cid:54) = −
3. However, we obtained this algebra starting from aChern-Simons action with original level k . This gives us another quick argument asto why the resulting CFT is trivial for this value of ˆ k . Looking at (3.59) we see that k = 0 for ˆ k = − . We see that already at the level of the action (2.2) the theory istrivial because the action itself is zero for this value of ˆ k . ˆ k = − and ˆ c = 1For this value all half integer valued levels contain again only null states. This canagain easily be seen by the same argument used for the case ˆ k = . Thus, the onlystates remaining are the states at integer valued levels since the norm of the level 1state is positive for this value of ˆ k . In order to check if negative norm states appearat higher levels, we need to calculate the Gramian matrix for any level N . Since ageneral state at a given level N is given by (3.85), the entries of the level N Gramianmatrix will be given by (cid:104) | J ¯ m N ¯ n N . . . J ¯ m ¯ n J m − n . . . J m N − n N | (cid:105) . (3.90)Using the algebraic relations (3.60a) and the property (3.66) one can immediatelysee that these entries will be zero unless ¯ n i = n i ∀ i = 1 , . . . , N and ¯ m i = m i ∀ i =1 , . . . , N . Hence we see that the Gramian matrix at level N will always be diagonaland thus the eigenvalues will be equal to the norm of the states present at level N .Calculating these norms yield (cid:104) | J m N n N . . . J m n J m − n . . . J m N − n N | (cid:105) = N (cid:89) i =1 m i ! n i (cid:32) k + 33 (cid:33) m i = N (cid:89) i =1 m i ! n i (cid:18) (cid:19) m i . (3.91)Since the eigenvalues of the Gramian matrix are all positive, we see that all statesin our theory have positive norm and thus we have a unitary theory for ˆ k = − c = 1.Having found a unitary theory for ˆ k = − k = − L n via the u (1) currents J n . i.e.ˆ L − n | (cid:105) = 32 n − (cid:88) p> J − p J − n + p | (cid:105) = 32 (cid:88) p ∈ Z : J − n − p J p : | (cid:105) . (3.92) The Gramian matrices of the levels 1, , 2, 3, 4, 5 as well as a general expression for arbitraryinteger valued level N can be found in Appendix B. – 34 –ince we obtained ˆ L n via a shift given by (3.55), this in turn also means that theunshifted Virasoro modes L n annihilate the vacuum ∀ n ∈ Z . Thus, the resultingtheory looks very similar to a theory based on a u (1) current algebra given by[ J n , J m ] = nκδ m + n, (3.93)and the following Sugawara construction for the Virasoro modesˆ L n = 12 κ (cid:88) p ∈ Z : J − n − p J p : . (3.94)This would yield the following algebra[ J n , J m ] = nκδ n + m, , (3.95a)[ J n , ˆ L m ] = nJ n + m , (3.95b)[ ˆ L n , ˆ L m ] = ( n − m ) ˆ L m + n + ˆ c κ n ( n − δ n + m, , (3.95c)with ˆ c = 1. For κ = one would obtain the same field content as for the W (2)3 algebrafor ˆ k = −
1. There is, however, a crucial difference between these two theories. Whilein the case of (3.95) the parameter κ can take arbitrary values , the central chargeˆ c κ remains unaffected by a change of κ . Thus, for this theory there is no preferredvalue of κ . In the case of the W (2)3 algebra on the other hand there is a preferredvalue of ˆ k and the central charge ˆ c is also not independent of ˆ k .One way to understand this is to think back to the canonical analysis and the con-straints associated with the states ˆ G ±− n | (cid:105) . Looking at (3.60f) one can see that forˆ k = − k = − ) the central term vanishes. This in turn means that theconstraints associated with ˆ G ±− n | (cid:105) remain first class even at the boundary and thusthere is an additional gauge symmetry present at the boundary that constrains ourtheory . This additional gauge symmetry then restricts the normalization of the J n modes and forces κ to take the value . Thus demanding unitarity of the quantumtheory leads to an additional symmetry enhancement of the CFT which in turn fur-ther constrains the theory.In order to give this fixed value of κ a physical interpretation one can consider forexample a free boson defined on the complex plane described by the action [9] S = 14 πκ (cid:90) d z d¯ z ∂X · ¯ ∂X, (3.96) The only restriction is that κ > In the case of ˆ k = − all central extensions in the W (2)3 algebra vanish which is another reasonwhy this theory is trivial. All first class constraints remain first class at the boundary and thus thetheory has no degrees of freedom left after fixing the gauge. – 35 –here ∂X · ¯ ∂X denotes (cid:112) | g | g ab ∂ a X ¯ ∂ b X , with g ab d x a d x b = d z d¯ zz ¯ z . One can thendefine the following chiral [anti-chiral] fields with conformal weight 1 j ( z ) = i∂X ( z, ¯ z ) , ¯ j (¯ z ) = i ¯ ∂X ( z, ¯ z ) , (3.97)whose Laurent modes, which are given by j ( z ) = 12 π (cid:88) n ∈ Z J n z − n − , ¯ j (¯ z ) = 12 π (cid:88) n ∈ Z ¯ J n ¯ z − n − , (3.98)obey the commutation relations given by (3.93). Since κ appearing in (3.93) isthe same as in (3.96) this parameter is essentially the coupling constant of thistheory. Thus, demanding unitarity and nontriviality of the CFT based on a W (2)3 algebra would fix this coupling constant to a certain value and one could interpretthe resulting theory as a free boson with a coupling constant fixed by an additionalgauge symmetry. This follows from the equations of motion ∂ ¯ ∂X ( z, ¯ z ) = 0. – 36 – . Conclusion The goal of this thesis was to analyze asymptotical symmetry algebras of (2+1)-dimensional non-AdS higher-spin gravity with a focus on AdS × R and H × R . Wefound a consistent set of boundary conditions that yield finite, integrable, conservedand nontrivial boundary charges. Then we determined the classical symmetry alge-bra of these boundary charges and found a classical W (2)3 ⊕ u (1) symmetry algebraat the boundary.Since we were also curious to find out what kind of CFT this symmetry algebrawould yield, we tried to obtain the quantized version of the W (2)3 algebra by satisfy-ing the Jacobi identities. Analyzing the field content of the quantized version of the W (2)3 algebra and looking for possible unitary representations of the W (2)3 algebra wefound some quite interesting features. For our definition of the vacuum we did findtwo unitary representations of the W (2)3 algebra. The reason that we only found twopossible unitary representations is that the modes ˆ G ± n appearing in the W (2)3 algebraare bosonic and hence obey commutator relations. Because of this the norms of themodes ˆ G + − n and ˆ G −− n differ by a sign and hence these states have to be null in orderto have a chance of obtaining a unitary theory. Thus, by demanding unitarity, ghoststates are automatically projected out of the theory.Taking a closer look at the two unitary representations that we obtained we foundthat one of these representations is trivial and the other one is very similar to atheory described via u (1) currents. However, in the case of the W (2)3 algebra theresulting theory is more restricted than for the pure u (1) case. The reason for this isan additional gauge symmetry enhancement that occurs at the two special values of ˆ k where the states corresponding to ˆ G ±− n are null. This gauge symmetry enhancementcould also explain the value of certain coupling constants such as for example thefree boson as we suggested.Having this example of a theory where the coupling constant is fixed by an enhancedgauge symmetry originating from a higher-spin algebra it is tempting to think thatthis might also work for other theories. One could for example try to do the sameanalysis for AdS × R and H × R as we did in this thesis, but for spin-4 gravity andthe 2-2 embedding of sl (4). For the interested reader we provided a suitable basis inappendix C.1.Another interesting question is related to the two values of ˆ k , for which the resultingCFT is unitary. It could be possible that there is another choice of vacuum for whichwe have more values of ˆ k that allow unitary representations. This could for examplebe realized by different embeddings of the non principal sl (3) embeddings in sl ( N ),with N >
3. One could for example consider the 2-1-1 embedding of sl (4) andagain perform the same analysis as in this thesis. At first glance it seems like thisanalysis is straightforward and should be analog to the spin-3 case that we analyzed A suitable basis is given in appendix C.2. – 37 –n this thesis. After having determined the quantum asymptotic symmetry algebra,one would then have to check whether or not the sl (4) invariant vacuum allows morevalues of ˆ k , for which the W [2]3 algebra that should be contained in the resultingasymptotic symmetry algebra is unitary. In principle it could also be possible thatthere are even less possible values of ˆ k since there are more Jacobi identities to befulfilled for the quantum version of the asymptotic symmetry algebra.Since the W [2]3 algebra is very similar to the N = 2 superconformal algebra, it isalso interesting to check what happens if we start the canonical analysis with a su-persymmetric theory rather than a sl (3) invariant theory. One would again expecta correlation between the unshifted Chern-Simons level k and the central charge c .Since in the supersymmetric case the modes ˆ G ± n obey anticommutation relations andthus the norms of the states corresponding to ˆ G + − n and ˆ G −− n have the same sign, thereshould be a wider range of k that allow for unitary representations.The specific example of AdS × R [ H × R ] and the non-principal embedding of sl (3)provided in this thesis can also be used to employ a general procedure in analyzinghigher-spin gravity theories formulated via a SL ( N ) × SL ( N ) Chern-Simons formu-lation [59].One important starting point of this analysis is the correct choice of embedding of sl (2) in sl ( N ). If the embedding cannot reproduce the chosen background, then it willbe impossible to find boundary conditions that are consistent with the backgroundand the fluctuations. Thus, if a chosen set of boundary conditions is not consistentwith the background and the fluctuations, then the reason for this is not necessarilya bad choice of boundary conditions. The inconsistencies could also be the result ofa bad choice of embedding or spin- N theory. We will list in the following the basicsteps one has to do in order to analyze higher-spin gravity theories. A more detailedversion of this procedure is given by figure 1 in terms of a flowchart.1. Identify the bulk theory and propose a variational principle.2. Choose boundary conditions of the connections A and ¯ A that lead to the desiredbackground (BG) solution and are compatible with a given set of fluctuationsof the BG and the variational principle employed.3. Determine the boundary condition preserving gauge transformations (BCPGT).4. Calculate the canonical boundary charge.5. Determine the classical asymptotic symmetry algebra.6. Quantize the classical asymptotic symmetry algebra if necessary. This canbe done for example by introducing normal ordering and imposing the Jacobiidentities on the quantum level.7. Analyze the field content of the resulting CFT.– 38 – roposevariationalprincipleidentifybulktheoryidentify gravitytheory chooseboundaryconditionsconsistentwith BG andfluctuationsconsistentwithvariationalprincipledetermine BCPGTcalculate canonicalboundary chargecharge isnon-trivival,finite,conservedandintegrable determine classicalasympt. symm. algebrasymmetryalgebra is ofdesired formconsistent atquantumlevelquantize algebradetermine unitary rep-resentations of algebraIdentifydual fieldtheoryyesno noyesno yesno yesno yescorrespondenceholographic Figure 1:
Flowchart depicting the procedure of analyzing higher-spin gravity theories – 39 – . Suitable Spin-3 Bases
For the sl (2) generators we use the following conventions[ L n , L m ] = ( n − m ) L n + m (A.1)where L ± := L ± . The commutation relations of the remaining generators of the W -Algebras are given by [ L n , W l [ a ] m ] = ( nl − m ) W l [ a ] n + m . (A.2)The index l appearing in W l [ a ] m is an sl (2) quantum number, while [ a ] is a color index.The traces of these generators are given bytr( W k [ a ] m W l [ b ] n ) = ( − l − m ( l + m )!( l − m )!2 l ! δ k,l δ m + n, N a,bl (A.3)with the normalization N a,bl := tr( W l [ a ] l W l [ b ] − l ) . (A.4)Whenever singlets fall into an sl (2) on their own their generators will be defined suchthat they obey [ S [ n ] , S [ m ] ] = ( n − m ) S [ n + m ] . (A.5)In addition, we also use the notation S [ n ] := W n ]0 . If there is only one singlet presentin our representation we just denote it by S . Doublets are denoted by ψ [ a ] n := W [ a ] n . A.1 Non-Principal Embedding
For the non-principal embedding of sl (2) in sl (3) in section (3) we used the followingset of generators obeying the commutation relations given by 3.1 . L = 12 − L + = L − = −
10 0 00 0 0 (A.6)Doublets: ψ + = − ψ + − = (A.7) ψ − = ψ −− = (A.8)Singlet: S = 13 − − (A.9)– 40 –illing form: g ab = − − − − , (A.10)with the generators are ordered as L , L , L − , ψ + , ψ + − , ψ − , ψ −− , S . B. Gramian Matrices for ˆ k ∈ {− , − } A general expression for calculating all coefficients for a Gramian matrix at integervalued level N of the W (2)3 algebra is given by (cid:104) | J ¯ m N ¯ n N . . . J ¯ m ¯ n J m − n . . . J m N − n N | (cid:105) = N (cid:89) i =1 m i ! n i (cid:32) k + 33 (cid:33) m i δ m i , ¯ m i δ n i , ¯ n i . (B.1)Level 2: K (2) = k +33
00 2 (cid:16) k +33 (cid:17) , (B.2)with the basis vectors arranged as J − | (cid:105) , J − | (cid:105) .Level : K ( ) = (ˆ k + 1)(2ˆ k + 3) − − (ˆ k + 3) 00 2 0 (ˆ k + 3) , (B.3)with the basis vectors arranged as G + − | (cid:105) , G −− | (cid:105) , G + − J − | (cid:105) and G −− J − | (cid:105) .Level 3: K (3) = k + 3 0 00 2 (cid:16) k +33 (cid:17)
00 0 6 (cid:16) k +33 (cid:17) , (B.4)with the basis vectors arranged as J − | (cid:105) , J − J − | (cid:105) , J − | (cid:105) .– 41 –evel 4: K (4) = k +33 (cid:16) k +33 (cid:17) (cid:16) k +33 (cid:17) (cid:16) k +33 (cid:17)
00 0 0 0 24 (cid:16) k +33 (cid:17) , (B.5)with the basis vectors arranged as J − | (cid:105) , J − J − | (cid:105) , J − | (cid:105) , J − J − | (cid:105) , J − | (cid:105) .Level 5: K (5) = k +33 (cid:16) k +33 (cid:17) (cid:16) k +33 (cid:17) (cid:16) k +33 (cid:17) (cid:16) k +33 (cid:17) (cid:16) k +33 (cid:17)
00 0 0 0 0 0 120 (cid:16) k +33 (cid:17) , (B.6)with the basis vectors arranged as J − | (cid:105) , J − J − | (cid:105) , J − J − | (cid:105) , J − J − | (cid:105) , J − J − | (cid:105) , J − J − | (cid:105) , J − | (cid:105) . C. Suitable Spin-4 Bases
In this section we present two different embeddings of sl (2) in sl (4). We use the samenotation as in [50] where one my also find further embeddings for sl (4). C.1 2-2 Embedding sl (2) generators: L = 12 − − L + = L − = − −
10 0 0 00 0 0 0 (C.1)– 42 –ther triplets: T [1]0 = 12 − T [1]+ = T [1] − = − (C.2a) T [2]0 = 12 − −
10 0 0 0 T [2]+ = T [2] − = − −
10 0 0 00 0 0 00 0 0 0 (C.2b) T [3]0 = 12 − − T [3]+ = T [3] − = − −
10 0 0 00 0 0 0 (C.2c)Singlets: S [0] = 12 − − S [+] = S [ − ] = − −
10 0 0 0 (C.3)
C.2 2-1-1 Embedding sl (2) generators: L = 12 − L + = L − = −
10 0 0 00 0 0 00 0 0 0 (C.4)Doublets: G [1]+ = G [2]+ = G [3]+ = G [4]+ = − G [1] − = −
10 0 0 0 G [2] − = G [3] − = −
10 0 0 00 0 0 0 G [4] − = − (C.5)– 43 –inglets: S [0] = 12 − S [+] = S [ − ] = − (C.6a) S = − − (C.6b)– 44 – eferences [1] B. Zwiebach, A First Course in String Theory . Cambridge University Press, 2004.[2] J. Polchinski,
String theory . Cambridge University Press, 1998. Vol. 1: AnIntroduction to the Bosonic String.[3] J. Polchinski, “String theory. Vol. 2: Superstring theory and beyond,”. Cambridge,UK: Univ. Pr. (1998) 531 p.[4] J. M. Maldacena, “The large N limit of superconformal field theories andsupergravity,”
Adv. Theor. Math. Phys. (1998) 231–252, hep-th/9711200 .[5] J. D. Bekenstein, “Universal upper bound on the entropy-to-energy ratio forbounded systems,” Phys. Rev.
D23 (1981) 287–298.[6] S. W. Hawking, “Particle creation by black holes,”
Commun. Math. Phys. (1975)199–220.[7] J. de Boer, Introduction to the AdS/CFT correspondence . University of AmsterdamInstitute for Theoretical Physics (ITFA) Technical Report 03-02, 2003.[8] P. Di Francesco, P. Mathieu, and D. Senechal,
Conformal Field Theory . Springer,1997.[9] R. Blumenhagen,
Introduction to Conformal Field Theory . Springer, 2009.[10] M.Schottenloher,
A Mathematical Introduction to Conformal Field Theory . Springer,2008.[11] P. H. Ginsparg, “Applied conformal field theory,” hep-th/9108028 .[12] J. M. Maldacena, “Gravity, particle physics and their unification,”
Int.J.Mod.Phys.
A15S1 (2000) 840–852, hep-ph/0002092 .[13] E. Witten, “(2+1)-dimensional gravity as an exactly soluble system,”
Nucl. Phys.
B311 (1988) 46.[14] G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theory ofgravitation,”
Annales Poincare Phys. Theor.
A20 (1974) 69–94.[15] E. Witten, “Three-Dimensional Gravity Revisited,” .[16] A. Campoleoni, S. Fredenhagen, and S. Pfenninger, “Asymptotic W-symmetries inthree-dimensional higher-spin gauge theories,”
JHEP (2011) 113, .[17] M. A. Vasiliev, “’Gauge Form of Description of Massless Fields with Arbitrary Spin.(in Russian),”
Yad.Fiz. (1980) 855–861.[18] V. Lopatin and M. A. Vasiliev, “Free Massless Bosonic Fields of Arbitrary Spin ind-Dimensional De Sitter Space,” Mod.Phys.Lett. A3 (1988) 257. – 45 –
19] M. Blencowe, “A Consistent Interacting Massless Higher Spin Field Theory in D =(2+1),”
Class.Quant.Grav. (1989) 443.[20] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization ofAsymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun.Math. Phys. (1986) 207–226.[21] P. Bouwknegt and K. Schoutens, “W symmetry in conformal field theory,”
Phys.Rept. (1993) 183–276, hep-th/9210010 .[22] D. Sorokin, “Introduction to the Classical Theory of Higher Spins,”
AIP Conf.Proc. (2005) 172–202, hep-th/0405069v3 .[23] O. Pelc and L. P. Horwitz, “Generalization of the Coleman-Mandula Theorem toHigher Dimension,”
J.Math.Phys. (1997) 139–172, hep-th/9605147 .[24] M. Vasiliev, “Higher Spin Gauge Theories: Star-Product and AdS Space,” hep-th/9910096v1 .[25] E. S. Fradkin and M. A. Vasiliev, “Cubic interaction in extended theories of masslesshigher spin fields,” Nucl. Phys.
B291 (1987) 141.[26] E. S. Fradkin and M. A. Vasiliev, “On the gravitational interaction of masslesshigher spin fields,”
Phys. Lett.
B189 (1987) 89–95.[27] M. Vasiliev, “Consistent Equation for Interacting Gauge Fields of all Spins in(3+1)-Dimensions,”
Phys. Lett.
B243 (1990) 378–382.[28] M. Vasiliev, “Nonlinear Equations for Symmetric Massless Higher Spin Fields in( A ) dS ( d ) ,” Phys. Lett.
B567 (2003) 139–151, hep-th/0304049 .[29] M. A. Vasiliev, “Consistent equation for interacting gauge fields of all spins in(3+1)-dimensions,”
Phys.Lett.
B243 (1990) 378–382.[30] M. A. Vasiliev, “Higher spin gauge theories: Star product and AdS space,” hep-th/9910096 .[31] C. Aragone and S. Deser, “Hypersymmetry in d = 3 of Coupled Gravity MasslessSpin 5 / Class. Quant. Grav. (1984) L9.[32] M. Henneaux and S.-J. Rey, “Nonlinear W ∞ Algebra as Asymptotic Symmetry ofThree-Dimensional Higher Spin Anti-de Sitter Gravity,”
JHEP (2010) 007, hep-th/1008.4579 .[33] D. T. Son, “Toward an AdS/cold atoms correspondence: a geometric realization ofthe Schroedinger symmetry,”
Phys. Rev.
D78 (2008) 046003, .[34] K. Balasubramanian and J. McGreevy, “Gravity duals for non-relativistic CFTs,”
Phys. Rev. Lett. (2008) 061601, . – 46 –
35] A. Adams, K. Balasubramanian, and J. McGreevy, “Hot Spacetimes for ColdAtoms,”
JHEP (2008) 059, .[36] S. Kachru, X. Liu, and M. Mulligan, “Gravity Duals of Lifshitz-like Fixed Points,” Phys. Rev.
D78 (2008) 106005, .[37] D. Grumiller and N. Johansson, “Instability in cosmological topologically massivegravity at the chiral point,”
JHEP (2008) 134, .[38] S. Ertl, D. Grumiller, and N. Johansson, “Erratum to ‘Instability in cosmologicaltopologically massive gravity at the chiral point’, arXiv:0805.2610,” .[39] D. Grumiller and N. Johansson, “Consistent boundary conditions for cosmologicaltopologically massive gravity at the chiral point,” Int. J. Mod. Phys.
D17 (2009)2367–2372, .[40] M. Henneaux, C. Martinez, and R. Troncoso, “Asymptotically anti-de Sitterspacetimes in topologically massive gravity,”
Phys.Rev.
D79 (2009) 081502R, hep-th/0901.2874 .[41] A. Maloney, W. Song, and A. Strominger, “Chiral Gravity, Log Gravity andExtremal CFT,”
Phys.Rev.
D81 (2010) 064007, hep-th/0903.4573 .[42] L. Susskind, “Holography in the flat space limit,” hep-th/9901079 .[43] G. Arcioni and C. Dappiaggi, “Exploring the holographic principle in asymptoticallyflat space-times via the BMS group,”
Nucl.Phys.
B674 (2003) 553–592, hep-th/0306142 .[44] G. Arcioni and C. Dappiaggi, “Holography in asymptotically flat space-times andthe BMS group,”
Class.Quant.Grav. (2004) 5655, hep-th/0312186 .[45] S. Deser, R. Jackiw, and S. Templeton, “Three-dimensional massive gauge theories,” Phys. Rev. Lett. (1982) 975–978.[46] S. Deser, R. Jackiw, and S. Templeton, “Topologically massive gauge theories,” Ann.Phys. (1982) 372–411.[47] J. H. Horne and E. Witten, “Conformal gravity in three-dimensions as a gaugetheory,”
Phys. Rev. Lett. (1989) 501–504.[48] H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller, and N. Johansson, “Holograms ofConformal Chern-Simons Gravity,” Phys.Rev.
D84 (2011) 041502, .[49] H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller, and N. Johansson, “ConformalChern-Simons holography - lock, stock and barrel,”
Phys.Rev.
D85 (2012) 064033, .[50] M. Gary, D. Grumiller, and R. Rashkov, “Towards non-AdS holography in3-dimensional higher spin gravity,” hep-th/1201.0013v3 . – 47 –
51] M. Henneaux and C. Teitelboim,
Quantization of Gauge Systems . PrincetonUniversity Press, Princeton, New Jersey, 1992.[52] M. Blagojevic,
Gravitation and Gauge Symmetries . Institute of Physics Publishing,Bristol and Philadelphia, 2002.[53] M. Ba˜nados, “Global charges in Chern-Simons field theory and the (2+1) blackhole,”
Phys. Rev.
D52 (1996) 5816, hep-th/9405171 .[54] M. Bertin, S. Ertl, P. Ghorbani, D. Grumiller, and D. Vassilevic, “Chiral warpedAdS .” in preparation.[55] M. Bershadsky, “Conformal field theories via Hamiltonian reduction,” Commun.Math.Phys. (1991) 71–82.[56] A. M. Polyakov, “Gauge Transformations and Diffeomorphisms,”
Int.J.Mod.Phys. A5 (1990) 833.[57] A. Castro, E. Hijano, and A. Lepage-Jutier, “Unitarity Bounds in AdS Higher SpinGravity,”
JHEP (2012) 001, .[58] M. Ademollo, L. Brink, A. D’Adda, R. D’Auria, E. Napolitano, et al. ,“Supersymmetric Strings and Color Confinement,”
Phys.Lett.
B62 (1976) 105.[59] H. Afshar, M. Gary, D. Grumiller, R. Rashkov, and M. Riegler, “Non-AdSholography in 3-dimensional higher spin gravity - General recipe and example,” ..