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Canonical structure of 3D gravity with torsion?

M. Blagojevi? c and B. Cvetkovi? c?

Institute of Physics, P. O. Box 57, 11001 Belgrade, Serbia

arXiv:gr-qc/0412134v1 30 Dec 2004

Abstract We study the canonical structure of the topological 3D gravity with torsion, assuming the anti-de Sitter asymptotic conditions. It is shown that the Poisson bracket algebra of the canonical generators has the form of two independent Virasoro algebras with classical central charges. In contrast to the case of general relativity with a cosmological constant, the values of the central charges are di?erent from each other.

1

Introduction

Faced with enormous di?culties to properly understand fundamental dynamical properties of gravity, such as the nature of classical singularities and the problem of quantization, one is naturally led to consider technically simpli?ed models with the same conceptual features. An important and useful model of this type is 3D gravity [1, 2]. In the last twenty years, 3D gravity has become an active research area, with a number of outstanding results. Here, we focus our attention on a particular line of development, characterized by the following achievements. In 1986, Brown and Henneaux introduced the so-called anti-de Sitter (AdS) asymptotic conditions in their study of 3D general relativity with a cosmological constant (GRΛ ) [3]. They showed that the related behavior of the gravitational ?eld allows for an extremely rich asymptotic structure—the conformal symmetry described by two independent canonical Virasoro algebras with classical central charges. Soon after that, Witten rediscovered and further explored the fact that GRΛ in 3D can be formulated as a Chern-Simons gauge theory [4]. The equivalence between gravity and an ordinary gauge theory was shown to be crucial for our understanding of quantum gravity. Then, in 1993, we had the discovery of the BTZ black hole [5], with a far-reaching impact on the development of 3D gravity. All these ideas have had a signi?cant in?uence on our understanding of the quantum nature of 3D black holes [2,6-13]. Following a widely spread belief that general relativity is the most reliable approach for studying the gravitational phenomena, the analysis of these issues has been carried out mostly in the realm of Riemannian geometry. However, there is a more general conception of gravity based on Riemann-Cartan geometry, in which both the curvature and the torsion characterize the structure of gravity (see, for instance, Refs. [14, 15]). Riemann-Cartan

Invited contribution to appear in Progress in General Relativity and Quantum Cosmology , vol. 2, ed. Frank Columbus (Nova Science Publishers, New York, 2005). ? Email addresses: mb@phy.bg.ac.yu, cbranislav@phy.bg.ac.yu

?

1

2

M. Blagojevi? c and B. Cvetkovi? c

geometry has been used in the context of 3D gravity since the early 1990s [16-18], with an idea to explore the in?uence of geometry on the dynamics of gravity. Recently, new advances in this direction have been achieved [19-24]. Asymptotic conditions are an intrinsic part of the canonical formalism, as they de?ne the phase space in which the canonical dynamics takes place. Their in?uence on the dynamics is particularly clear in topological theories, where the propagating degrees of freedom are absent, and the only non-trivial dynamics is bound to exist at the asymptotic boundary. General action for topological 3D gravity with torsion, based on Riemann-Cartan geometry of spacetime, has been proposed by Mielke and Baekler [16, 17]. The objective of the present paper is to investigate the canonical structure of the general topological 3D gravity with torsion, including its asymptotic behavior, in the AdS sector of the theory. This will generalize the results of Refs. [3, 4] and [20], where the speci?c choice of parameters corresponds to Riemannian and telaparallel vacuum geometry, respectively. Combining this approach with another interesting result, the existence of the Riemann-Cartan black hole [19, 22], we shall be able to explore the full in?uence of torsion on the canonical and asymptotic structure of 3D gravity. The paper is organized as follows. In Sect. 2 we review some basic features of Riemann– Cartan spacetime as the proper geometric arena for 3D gravity with torsion, and discuss the ?eld equations derived from the Mielke-Baekler action. In Sect. 3 we describe the Riemann-Cartan black hole solution, a generalization of the BTZ black hole. Then, in Sect. 4, we introduce the concept of asymptotically AdS con?guration, and derive the related asymptotic symmetry, which turns out to be the same as in general relativity—the conformal symmetry. In the next section, the asymptotic structure of the theory is incorporated into the Hamiltonian formalism by calculating the Poisson bracket (PB) algebra of the canonical generators. It has the form of two independent Virasoro algebras with classical central charges, the values of which di?er from each other, in contrast to what we have in Riemannian GRΛ and the teleparallel theory [3, 20]. Finally, Sect. 7 is devoted to concluding remarks, while Appendices contain some technical details. Our conventions are given by the following rules: the Latin indices refer to the local Lorentz frame, the Greek indices refer to the coordinate frame; the ?rst letters of both alphabets (a, b, c, ...; α, β, γ, ...) run over 1,2, the middle alphabet letters (i, j, k, ...; ?, ν, λ, ...) run over 0,1,2; the signature of spacetime is η = (+, ?, ?); totally antisymmetric tensor εijk and the related tensor density ε?νρ are both normalized so that ε012 = 1.

2

Topological 3D gravity with torsion

Theory of gravity with torsion can be formulated as Poincar? e gauge theory (PGT), with an underlying spacetime structure described by Riemann-Cartan geometry [14, 15]. PGT in brief. The basic gravitational variables in PGT are the triad ?eld bi and the Lorentz connection Aij = ?Aji (1-forms). The ?eld strengths corresponding to the gauge potentials bi and Aij are the torsion T i and the curvature Rij (2-forms): T i = dbi + Ai m ∧ bm , Rij = dAij + Ai m ∧ Amj . Gauge symmetries of the theory are local translations and local Lorentz rotations, parametrized by ξ ? and εij .

Canonical structure of 3D gravity with torsion

3

In 3D, we can simplify the notation by introducing the duals of Aij , Rij and εij : 1 1 1 ωi = ? εijk Ajk , Ri = ? εijk Rjk , θi = ? εijk εjk . 2 2 2 ? In local coordinates x , we can expand the triad and the connection 1-forms as bi = bi ? dx? , ω i = ω i ? dx? . Gauge transformation laws have the form δ0 bi ? = ?εi jk bj ? θk ? (?? ξ ρ )bi ρ ? ξ ρ?ρ bi ? ≡ δPGT bi ? , δ0 ω i ? = ?(?? θi + εi jk ω j ? θk ) ? (?? ξ ρ)ω i ρ ? ξ ρ ?ρ ω i ? ≡ δPGT ω i? , and the ?eld strengths are given as 1 T i = ?bi ≡ dbi + εi jk ω j ∧ bk = T i ?ν dx? ∧ dxν , 2 1 1 Ri = dω i + εi jk ω j ∧ ω k = Ri ?ν dx? ∧ dxν , (2.2) 2 2 where ? = dx? ?? is the covariant derivative. To clarify the geometric meaning of the above structure, we introduce the metric tensor as a speci?c, bilinear combination of the triad ?elds: g = ηij bi ? bj = g?ν dx? ? dxν , g?ν = ηij bi ? bj ν , ηij = (+, ?, ?) . Although metric and connection are in general independent dynamical/geometric variables, the antisymmetry of Aij in PGT is equivalent to the so-called metricity condition , ?g = 0. The geometry whose connection is restricted by the metricity condition (metric-compatible connection) is called Riemann-Cartan geometry . Thus, PGT has the geometric structure of Riemann-Cartan space. The connection Aij determines the parallel transport in the local Lorentz basis. Being a true geometric operation, parallel transport is independent of the basis. This property is incorporated into PGT via the so-called vielbein postulate , which implies the identity Aijk = ?ijk + Kijk , (2.3) where ? is Riemannian (Levi-Civita) connection, and Kijk = ? 1 (Tijk ? Tkij + Tjki ) is the 2 contortion. Topological action. In general, gravitational dynamics is de?ned by Lagrangians which are at most quadratic in ?eld strengths. Omitting the quadratic terms, Mielke and Baekler proposed a topological model for 3D gravity [16, 17], with an action of the form I = aI1 + ΛI2 + α3 I3 + α4 I4 + IM , where IM is a matter contribution, and I1 = 2 I2 = ? I3 = I4 = 1 3 bi ∧ Ri , εijk bi ∧ bj ∧ bk , 1 ω i ∧ dωi + εijk ω i ∧ ω j ∧ ω k , 3 (2.4b) (2.4a) (2.1)

bi ∧ Ti .

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M. Blagojevi? c and B. Cvetkovi? c

The ?rst term, with a = 1/16πG, is the usual Einstein-Cartan action, the second term is a cosmological term, I3 is the Chern-Simons action for the Lorentz connection, and I4 is an action of the translational Chern-Simons type. The Mielke-Baekler model can be thought of as a natural generalization of Riemannian GRΛ (with α3 = α4 = 0) to a topological gravity theory in Riemann-Cartan spacetime. Field equations. Variation of the action with respect to triad and connection yields the gravitational ?eld equations: ε?νρ aRiνρ + α4 Tiνρ ? Λεijk bj ν bk ρ = τ ? i , ε?νρ α3 Riνρ + aTiνρ + α4 εijk bj ν bk ρ = σ ? i , where τ ? i = ?δIM /δbi ? and σ ? i = ?δIM /δω i? are the matter energy-momentum and spin currents, respectively. For our purposes—to study the canonical structure of the theory in the asymptotic region—it is su?cient to consider the ?eld equations in vacuum, where τ = σ = 0. In the sector α3 α4 ? a2 = 0, these equations take the simple form Tijk = pεijk , Rijk = qεijk , where p= (2.5a) (2.5b)

α3 Λ + α4 a (α4 )2 + aΛ , q = ? . α3 α4 ? a2 α3 α4 ? a2 Thus, the vacuum con?guration is characterized by constant torsion and constant curvature. In Riemann-Cartan spacetime, one can use the identity (2.3) to express the curvature ? ij ?ν ≡ Rij ?ν (?) and the contortion: Rij ?ν (A) in terms of its Riemannian piece R ? ij ?ν + ?? K ij ν ? K i m? K mj ν ? (? ? ν ) . Rij ?ν (A) = R This relation, combined with the ?eld equations (2.5), leads to

1 Λ e? ≡ q ? p2 , (2.6) 4 where Λ e? is the e?ective cosmological constant. Equation (2.6) can be considered as an equivalent of the second ?eld equation (2.5b). Looking at (2.6) as an equation for the metric, one concludes that our spacetime has maximally symmetric metric [25]: ? ij ?ν = ?Λ e? (bi ? bj ν ? bi ν bj ? ), R for Λ e? < 0 (Λ e? > 0), the spacetime manifold is anti-de Sitter (de Sitter). There are two interesting special cases of the general Mielke-Baekler model, which have been studied in the past. For α3 = α4 = 0, the vacuum geometry becomes Riemannian , Tijk = 0. This choice corresponds to GRΛ [3, 4]; for (α4 )2 + aΛ = 0, the vacuum geometry is teleparallel , Rijk = 0. The vacuum ?eld equations are “geometrically dual” to those of GRΛ [20]. In the present paper, we shall investigate the general model (2.4) with α3 α4 ? a2 = 0, assuming that the e?ective cosmological constant is negative (anti-de Sitter sector): 1 (2.7) Λ e? ≡ ? 2 < 0 . ? The de Sitter sector with Λ e? > 0 is left for the future studies.

Canonical structure of 3D gravity with torsion

5

3

Exact vacuum solutions

Some aspects of the canonical analysis rely on the existence of suitable asymptotic conditions. A proper choice of these conditions is based, to some extent, on the knowledge of exact classical solutions in vacuum. For the Mielke-Baekler model (2.4), these solutions are well known [19, 22]. Their construction can be described by the following set of rules: For a given Λ e? , use Eq. (2.6) to ?nd a solution for the metric. This step is very simple, since the metric structure of maximally symmetric spaces is well known [25]. Given the metric, ?nd a solution for the triad ?eld, such that g = ηij bi ? bj . Finally, use Eq. (2.5a) to determine the connection ω i . For exact solutions with non-vanishing sources, the reader can consult Ref. [24]. Riemann-Cartan black hole. For Λ e? < 0, equation (2.6) has a well known solution for the metric — the BTZ black hole [5]. Using the static coordinates x? = (t, r, ?) (with 0 ≤ ? < 2π ), and units 4G = 1, it is given as ds2 = N 2 dt2 ? N ?2 dr 2 ? r 2 (d? + N? dt)2 , r2 J 2 J N 2 = ?2m + 2 + 2 , N? = 2 . ? r r

(3.1)

The parameters m and J are related to the conserved charges—energy and angular momentum. Since the triad ?eld corresponding to (3.1) is determined only up to a local Lorentz transformation, we can choose bi to have the simple, “diagonal” form: b0 = Ndt , b1 = N ?1 dr , b2 = r (d? + N? dt) . (3.2a)

Then, the connection is obtained by solving the ?rst ?eld equation (2.5a): p p J dt ? d? , ω 1 = N ?1 + 2 dr , 2 2 r p? dt J J p r d? . ? + r? ω2 = ? ? 2 r ? 2 r ω0 = N Equations (3.2) de?ne the Riemann-Cartan black hole. Riemann-Cartan AdS solution. In Riemannian geometry with negative Λ, the general solution with maximal number of Killing vectors is called the AdS solution [5, 25]. Although AdS solution and the black hole are locally isometric, they are globally distinct. The AdS solution can be obtained from (3.1) by the replacement J = 0, 2m = ?1. Similarly, there is a general solution with maximal symmetry in Riemann-Cartan geometry, the Riemann-Cartan AdS solution. It can be obtained from the black hole (3.2) by the same replacement (J = 0, 2m = ?1). Using the notation f 2 ≡ 1 + r 2 /?2 , we have: b0 = f dt , p ω0 = f dt ? d? , 2 b1 = f ?1 dr , p ω1 = dr , 2f b2 = rd? , r dt p? ω2 = ? ? d? . ? ? 2 (3.3a) (3.3b)

(3.2b)

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M. Blagojevi? c and B. Cvetkovi? c

In order to understand symmetry properties of (3.3), we note that the form-invariance of a given ?eld con?guration in Riemann-Cartan geometry is de?ned by the requirements δ0 bi ? = 0, δ0 ω i ? = 0, which di?er from the Killing equation in Riemannian geometry, δ0 g?ν = 0 (δ0 is the PGT analogue of the geometric Lie derivative). When applied to the Riemann-Cartan AdS solution (3.3), these requirements restrict (ξ ? , θi ) to the subspace ? i de?ned by the basis of six pairs (ξ( k ) , θ(k ) ) (k = 1, . . . , 6), given in Appendix A. The related symmetry group is the six-dimensional AdS group SO (2, 2).

4

Asymptotic conditions

Spacetime outside localized matter sources is described by the vacuum solutions of the ?eld equations (2.5). Thus, matter has no in?uence on the local properties of spacetime in the source-free regions, but it can change its global properties. On the other hand, global properties of spacetime a?ect symmetry properties of the asymptotic con?gurations, and consequently, they are closely related to the gravitational conservation laws. In 3D gravity with Λ e? < 0, maximally symmetric AdS solution (3.3) has the role analogous to the role of Minkowski space in the Λ e? = 0 case. Following this analogy, we could choose (3.3) to be the ?eld con?guration to which all the dynamical variables approach in such a way, that the asymptotic symmetry is SO (2, 2), the maximal symmetry of (3.3). However, such an assumption would exclude the important black hole geometries, which are not SO (2, 2) invariant. Having an idea to maximally relax the asymptotic conditions and enlarge the set of asymptotic states (and the relevant group of symmetries), we introduce the concept of the AdS asymptotic behavior , based on the following requirements [3, 26]: (a) asymptotic con?gurations should include the black hole geometries; (b) they should be invariant under the action of the AdS group SO (2, 2); (c) asymptotic symmetries should have well de?ned canonical generators. The conditions (a) and (b) together lead to an extended asymptotic structure, quite different from the standard, form-invariant vacuum con?guration, while (c) is just a technical assumption. AdS asymptotics. We begin our considerations with the point (a) above. The asymptotic behaviour of the black hole triad (3.2a) is given by r m? ? 0 0 ? ? r ? ? ? ? ? ? m?3 bi ? ? ? 0 + 3 0 ? ? ? , ? ? r r ? ? J 0 r r where the type of higher order terms on the right hand side is not written explicitly. Similarly, the asymptotic behaviour of the connection (3.2b) has the form ? p? r r m? ? m 0 ? + ? 2 ?2 ? r ? r ? ? ? 3 ? ? J? + pm? / 2 p? i ? ? . ω??? + 0 0 ? 3 ? ? 2 r r ? r pJ pr J ? ? 2+ 0 ? ? 2r 2 r

? ?

Canonical structure of 3D gravity with torsion

7

According to (a), asymptotic conditions should be chosen so as to include these black hole con?gurations. In order to realize the requirement (b), we start with the above black hole con?guration and act on it with all possible SO (2, 2) transformations, de?ned by the basis of six pairs (ξ(k) , θ(k) ), displayed in Appendix A. The result has the form δ(k) bi ? O1 O4 O1 ? ? ? ? O2 O3 O2 ? , O1 O4 O1

? ?

δ(k) ω i?

O1 O4 O1 ? ? ? ? O2 O3 O2 ? , O1 O4 O1

?

?

where On denotes a quantity that tends to zero as 1/r n or faster, when r → ∞. The family of the black hole triads obtained in this way is parametrized by six real parameters, say σi ; we denote it by B6 . In order to have a set of asymptotic states which is su?ciently large to include the whole B6 , we adopt the following asymptotic form for the triad ?eld: ? r ? r ? ? 0 0 + O O O 1 4 1 ? ? ? ? ? ? ? ? ? ? i ? ? ? ? ? ? + Bi? . ≡ b?=? O (4.1a) ? ? ? 0 0 + O O 2 3 2 ? ? ? ? r r O1 O4 r + O1 0 0 r The real meaning of this expression and its relation to B6 is clari?ed by noting that any c/r n term in B6 is transformed into the corresponding c(t, ?)/r n term in (4.1a), i.e. constants c = c(σi ) are promoted to functions c(t, ?). Thus, (4.1a) is a natural generalization of B6 . The triad family (4.1a) generates the Brown–Henneaux asymptotic form of the metric, r2 O3 O0 ? 2 + O0 ? ? ? ?2 =? ? + O4 O3 O ? 3 ? r2 2 O0 O3 ?r + O0

? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

g?ν

≡

r2 0 0 ?2 2 ? 0 0 ? 2 r 0 0 ?r 2

?

? ? ? ? + G?ν ? ?

,

but clearly, it is not uniquely determined by it (any Lorentz transform of the triad produces the same metric). Having found the triad asymptotics, we now use similar arguments to ?nd the needed asymptotic behavior for the connection: pr r + O1 O2 ? + O1 ? 2? ? ? p? ? =? O2 + O3 O2 ? 2r ? pr r O2 + O1 ? 2 + O1 ? 2

? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

ωi?

≡

pr 2? 0 r ? 2 ?

r ? 0 ? ? ? ? p? ? + ?i? . 0 ? 2r pr ? ? 0 2

(4.1b)

Note that the choice ω 0 1 , ω 2 1 = O2 , adopted in (4.1b), represents an acceptable generalization of the conditions ω 01 , ω 21 = O4 , suggested by the form of δ(k) ω i? (compare also with the conditions (C.1)). As we have seen, the requirements (a) and (b) are not su?cient for a unique determination of the asymptotic behavior. Our choice of the asymptotics was guided by the idea to obtain the most general asymptotic behavior compatible with (a) and (b), with arbitrary

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M. Blagojevi? c and B. Cvetkovi? c

higher-order terms B i ? and ? i ? . Although B i ? and ? i ? are arbitrary at this stage, certain relations among them will be established latter (Appendix C), using some additional requirements. One can verify that the asymptotic conditions (4.1) are indeed invariant under the action of the AdS group SO (2, 2). In the next step, we shall examine whether there is any higher symmetry structure in (4.1), which will be the real test of our choice. Asymptotic symmetries. Having chosen the asymptotic conditions in the form (4.1), we now wish to ?nd the subset of gauge transformations that respect these conditions. Acting on a speci?c ?eld satisfying (4.1), these transformations are allowed to change the form of the non-leading terms B i ? , ? i ? , as they are arbitrary by assumption. Thus, the parameters of the restricted gauge transformations are determined by the relations εijk θj bk? ? (?? ξ ρ )bi ρ ? ξ ρ ?ρ bi ? = δ0 B i ? , ??? θi + εijk θj ωk? ? (?? ξ ρ )ω iρ ? ξ ρ ?ρ ω i? = δ0 ? i ? . (4.2a) (4.2b)

The transformations de?ned in this way di?er from those that are associated to the forminvariant vacuum con?gurations (δ0 bi ? = 0, δ0 ω i? = 0). The restricted gauge parameters are determined as follows [20]. The symmetric part of (4.2a) multiplied by biν (six relations) yields the transformation rule of the metric: ?(?? ξ ρ )gνρ ? (?ν ξ ρ )g?ρ ? ξ ρ ?ρ g?ν = δ0 G?ν . Expanding ξ ? in powers of r ?1 , we ?nd the solution of these equations as 1 ? 2 T ?4 ξ =? T + + O4 , 2 ?t2 r 2 1 ? 2 S ?2 + O4 , ξ2 = S ? 2 ??2 r 2 ?T ξ 1 = ?? r + O1 , ?t

0

(4.3a) (4.3b) (4.3c)

where the functions T (t, ?) and S (t, ?) satisfy the conditions ?T ?S =? , ?? ?t ?S ?T =? . ?? ?t (4.4)

In GRΛ , these equations de?ne the two-dimensional conformal group at large distances [3]. The remaining three components of (4.2a) determine θi : θ0 = ? θ2 = ?2 ?0 ?2 T + O3 , r

?3 2 ? T + O3 , r 0 θ1 = ?2 T + O2 .

(4.3d)

The conditions (4.2b) produce no new limitations on the parameters.

Canonical structure of 3D gravity with torsion

9

Introducing the light-cone coordinates x± = x0 /? ±x2 , the conditions (4.4) can be written in the form ?± (T ? S ) = 0 , from which one easily ?nds the general solution for T and S : T + S = g (x+ ) , T ? S = h(x? ) , (4.5)

where g and h are two arbitrary, periodic functions. The commutator algebra of Poincar? e gauge transformations (2.1) is closed: we have ′ ′′ ′′′ ′ , where δ0 ≡ δ0 (ξ ′ , θ′ ) and so on, and the composition law reads: [δ0 , δ0 ] = δ0 ξ ′′′ ? = ξ ′ ρ ?ρ ξ ′′ ? ? ξ ′′ ρ ?ρ ξ ′ ? , θ′′′ i = εi mn θ′ m θ′′ n + ξ ′ · ?θ′′ i ? ξ ′′ · ?θ′ i . Substituting here the restricted form of the parameters (4.3) and comparing the lowest order terms, we ?nd the relations T ′′′ = T ′ ?2 S ′′ + S ′ ?2 T ′′ ? T ′′ ?2 S ′ ? S ′′ ?2 T ′ , S ′′′ = S ′ ?2 S ′′ + T ′ ?2 T ′′ ? S ′′ ?2 S ′ ? T ′′ ?2 T ′ ,

(4.6)

that are expected to be the composition law for (T, S ). To clarify the situation, consider the restricted form of the gauge parameters (4.3), and separate it into two pieces: the leading terms containing T and S , which de?ne the (T, S ) transformations, and the higher order terms that remain after imposing T = S = 0, which de?ne the residual (or pure) gauge transformations. If the relations (4.6) are to represent the composition law for the (T, S ) transformations, one has to check their consistency with higher order terms in the commutator algebra. As one can verify, the commutator of two (T, S ) transformations produces not only a (T, S ) transformation, with the composition law (4.6), but also an additional, pure gauge transformation. However, pure gauge transformations are irrelevant for our discussion of the conservation laws. Indeed, as we shall see in section 6, they do not contribute to the values of the conserved charges (their generators vanish weakly). Thus, we are naturally led to correct the non-closure of the (T, S ) commutator algebra by introducing an improved de?nition of the asymptotic symmetry [3, 26]: the asymptotic symmetry group is de?ned as the factor group of the gauge group determined by (4.3), with respect to the residual gauge group. In other words, two asymptotic transformations are identi?ed if they have the same (T, S ) pairs, and any di?erence stemming from the pure gauge terms is ignored. The asymptotic symmetry of our spacetime coincides with the conformal symmetry (see section 6). In conclusion, the set of asymptotic conditions (4.1) is shown to be invariant under the conformal symmetry group, which is much larger then the original AdS group SO (2, 2). The resulting con?guration space respects the requirements (a) and (b) formulated at the beginning of this section. The asymptotic structure of the whole phase space, as well as the status of the last requirement (c), will be examined in the next two sections.

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M. Blagojevi? c and B. Cvetkovi? c

5

Gauge generator

In gauge theories, the presence of unphysical variables is closely related to the existence of gauge symmetries. The best way to understand the dynamical content of these symmetries is to explore the related canonical generator, which acts on the basic dynamical variables via the PB operation. To begin the analysis, we rewrite the action (2.4) as I= 1 d3 xε?νρ abi ? Riνρ ? Λεijk bi ? bj ν bk ρ 3 1 1 + α3 ω i ? ?ν ωiρ + εijk ω i? ω j ν ω k ρ + α4 bi ? Tiνρ . 3 2

(5.1)

Hamiltonian and constraints. The basic Lagrangian variables (bi ? , ω i? ) and the corresponding canonical momenta (π i ? , Πi ? ) are related to each other through the set of primary constraints: φi0 ≡ πi 0 ≈ 0 , φiα ≡ πi α ? α4 ε0αβ biβ ≈ 0 , Φi 0 ≡ Πi 0 ≈ 0 , Φi α ≡ Πi α ? ε0αβ (2abiβ + α3 ωiβ ) ≈ 0 .

(5.2)

Explicit construction of the canonical Hamiltonian yields an expression which is linear in unphysical variables, as expected: Hc = bi 0 Hi + ω i0 Ki + ?α D α , Hi = ?ε0αβ aRiαβ + α4 Tiαβ ? Λεijk bj α bk β , Ki = ?ε0αβ aTiαβ + α3 Riαβ + α4 εijk bj α bk β , D α = ε0αβ ω i0 (2abiβ + α3 ωiβ ) + α4 bi 0 biβ . Going over to the total Hamiltonian, HT = bi 0 Hi + ω i0 Ki + ui ? φi ? + v i ? Φi ? + ?α D α , (5.3)

we ?nd that the consistency conditions of the sure primary constraints πi 0 and Πi 0 yield the secondary constraints: Hi ≈ 0, Ki ≈ 0 . (5.4a) These constraints can be equivalently written in the form: Tiαβ ≈ pεijk bj α bk β , Riαβ ≈ qεijk bj α bk β . (5.4b)

The consistency of the remaining primary constraints φi α and Φi α leads to the determination of the multipliers ui β and v i β (see Appendix B):

ijk ui β + εijk ωj 0 bkβ ? ?β bi 0 = pε bjo bkβ , v i β ? ?β ω i 0 = qεijk bj 0 bkβ .

(5.5a)

˙ i β = uiβ and ω Using the equations of motion b ˙ i β = v i β , these relations reduce to the ?eld equations T i 0β ≈ pεijk bj 0 bkβ , Ri 0β ≈ qεijk bj 0 bkβ . (5.5b)

Canonical structure of 3D gravity with torsion

11

The substitution of the determined multipliers (5.5a) into (5.3) yields the ?nal form of the total Hamiltonian: ? T + ?α D ?α , HT = H ? T = bi 0 H ? i + ωi0K ? i + ui 0 πi 0 + v i0 Πi 0 , H where ? i = Hi ? ?β φi β + εijk bj β pφkβ + q Φkβ , H ? i = Ki ? ?β Φi β ? εijk bj β φkβ , K ? α = D α + bi 0 φi α + ω i0 Φi α . D

(5.6a)

(5.6b)

Further investigation of the consistency procedure is facilitated by observing that the ?i, K ? i obey the PB relations (B.2). One concludes that the consissecondary constraints H tency conditions of the secondary constraints (5.4) are identically satis?ed, which completes the Hamiltonian consistency procedure. Complete classi?cation of the constraints is given in the following table. Table 1. Classi?cation of constraints First class Primary Secondary πi 0 , Πi 0 ? i, K ?i H Second class φi α , Φi α

Canonical gauge generator. The results of the previous analysis are su?cient for the construction of the gauge generator [27]. Starting from the primary ?rst class constraints πi 0 and Πi 0 , one obtaines: ? i ? εijk ω j 0 ? pbj 0 π k0 + qεijk bj 0 Πk0 , G[?] = ?˙i πi 0 + ?i H ? i ? εijk bj 0 π k0 + ω j 0 Πk0 G[τ ] = τ˙i Πi 0 + τ i K . (5.7)

The complete gauge generator is given by the expression G = G[?] + G[τ ], and its action on the ?elds, de?ned by δ0 φ = {φ, G}, has the form:

k j δ0 bi ? = ?? ?i ? pεi jk bj ? τ k + εi jk b ? τ , j k δ0 ω i? = ?? τ i ? qεi jk b ? ? .

This result looks more like a standard gauge transformation, with no trace of the expected local Poincar? e transformations. However, after introducing the new parameters ?i = ?ξ ? bi ? , one easily obtains δ0 bi ? = δPGT bi ? ? ξ ρ T i ?ρ ? pεijk bj? bkρ , δ0 ω i? = δPGT ω i? ? ξ ρ Ri ?ρ ? qεijk bj? bkρ . τ i = ?(θi + ξ ? ω i? ) ,

12

M. Blagojevi? c and B. Cvetkovi? c

Thus, on-shell , we have the transformation laws that are in complete agreement with (2.1). Expressed in terms of the new parameters, the gauge generator takes the form G = ?G1 ? G2 , ˙ρ bi ρ πi 0 + ω iρ Πi 0 + ξ ρ bi ρ H ? i + ω iρ K ? i + (?ρ bi )πi 0 + (?ρ ω i 0 )Πi 0 , G1 ≡ ξ 0 ˙i Πi 0 + θi K ? i ? εijk bj 0 π k0 + ω j 0 Πk0 G2 ≡ θ , (5.8)

˙ i ? and ω where the time derivatives b ˙ i? are shorts for ui? and v i ? , respectively. Note, in particular, that the time translation generator is determined by the total Hamiltonian: ˙0 bi 0 πi 0 + ω i 0 Πi 0 ? ξ 0 H ?T . G ξ 0 = ?ξ In the above expressions, the integration symbol when necessary, it will be restored. d3 x is omitted for simplicity; later,

Asymptotics of the phase space. In order to extend the asymptotic conditions (4.1) to the canonical level, one should determine an appropriate asymptotic behavior of the whole phase space, including the momentum variables. This step is based on the following general principle: the expressions than vanish on shell should have an arbitrary fast asymptotic decrease, as no solutions of the ?eld equations are thereby lost. By applying this principle to the primary constraints (5.2), one ?nds the following asymptotic behavior of the momentum variables: ?, πi 0 = O ?, Πi 0 = O ? πi α = α4 ε0αβ biβ + O ?. Πi α = 2aε0αβ biβ + α3 ε0αβ ωiβ + O (5.9)

We shall use this principle again in connection to the consistency requiremets (5.4b) and (5.5b), in order to re?ne the general asymptotic canditions (4.1) and (5.9) (Appendix C).

6

Canonical structure of the asymptotic symmetry

In this section, we study the in?uence of the adopted asymptotics on the canonical structure of the theory: we construct the improved gauge generators, examine their canonical algebra and prove the conservation laws. Improving the generators. The canonical generator acts on dynamical variables via the PB operation, which is de?ned in terms of functional derivatives. A phase-space functional F = d2 xf (φ, ?φ, π, ?π ) has well de?ned functional derivatives if its variation can be written in the form δF = d2 x [A(x)δφ(x) + B (x)δπ (x)] , where terms δφ,? and δπ,ν are absent. In order to ensure this property for our generator (5.8), we have to improve its form by adding certain surface terms [28]. Let us start the procedure by examining the variations of G2 : ? i + R = θ i δ Ki + ? O ?+R δG2 = θi δ K ?+R = ?2ε0αβ θi (a?α δbiβ + α3 ?α δωiβ ) + ? O ? + R = ? O2 + R , = ?2ε0αβ ?α aθi δbiβ + α3 θi δωiβ + ? O

Canonical structure of 3D gravity with torsion

13

where the last equality follows from the asymptotic relations θi δbiβ , θi δωiβ = O2 . The total divergence term ? O2 gives a vanishing contribution after integration, as follows from the Stokes theorem:

M2

d2 x?α v α =

? M2

v α dfα =

2π 0

v 1 d?

(dfα = εαβ dxβ ) ,

where the boundary of the spatial section M2 of spacetime is taken to be the circle at in?nity, parametrized by the angular coordinate ?. Thus, the boundary term for G2 vanishes, and G2 is regular as it stands, without any correction. Going over to G1 , we have: ? i + ωiρδK ? i + ?O ?+R δG1 = ξ ρ bi ρ δ H ? +R. = ?2ε0αβ ?α ξ ρbi ρ δ (aωiβ + α4 biβ ) + ξ ρ ω iρ δ (abiβ + α3 ωiβ ) + ? O Using the adopted asymptotic conditions, the preceding result leads to δG1 = ??α ξ 0 δ E α + ξ 2 δ Mα + ? O2 + R = ?δ?α ξ 0 E α + ξ 2 Mα + ? O2 + R , where E α ≡ 2ε0αβ a+ ap 0 a α3 α3 p ω 0 β + α4 + b β + b2 β + ω 2 β b0 0 , 2 2 ? ? α3 p ap 2 a 0 α3 2 a+ ω β + α4 + b β + b β + ω 0β b2 2 . 2 2 ? ?

Mα ≡ ?2ε0αβ

(6.1)

Thus, the improved form of the complete gauge generator (5.8) reads: ? =G+Γ , G Γ =? dfα ξ 0 E α + ξ 2 Mα = ?

2π 0

d? ?T E 1 + S M1 .

(6.2)

? is ?nite and di?erentiable functional. The adopted asymptotic conditions guarantee that G The boundary term Γ depends only on T and S , not on any pure gauge term in (4.3). The improved time translation generator has the form ? [ξ 0] = G[ξ 0 ] ? E [ξ 0 ] , G E [ξ 0 ] ≡

2π 0

d? ξ 0 E 1 .

(6.3a)

? T , and the corresponding boundary term has the For ξ 0 = 1, the generator G reduces to ?H meaning of energy: 2π ?T = H ?T + E , H E= d? E 1 . (6.3b)

0

The improved spatial rotation generator is given by ? [ξ 2 ] = G[ξ 2 ] ? M [ξ 2 ] , G M [ξ 2 ] ≡

2π 0

d? ξ 2M1 ,

(6.4a)

14

M. Blagojevi? c and B. Cvetkovi? c

where M is a ?nite integral. The boundary term for ξ 2 = 1, M= is the angular momentum of the system. Canonical algebra. The PB algebra of the improved generators could be found by a direct calculation, but we shall rather use another, more instructive method, based on the results of Refs. [29] and [20]. Let us ?rst recall that our improved generator (6.2) is a di?erentiable phase space functional that preserves the asymptotic conditions (4.1) and (5.9), hence, it satis?es the conditions of the main theorem in Ref. [29]. Introducing a ?′ ≡ G ? [T ′ , S ′ ], G ? ′′ ≡ G ? [T ′′ , S ′′ ], the main theorem states that the convenient notation, G ? ′′ , G ? ′ } of two di?erentiable generators is itself a di?erentiable generator. Taking into PB {G ? is de?ned only up to an additive, constant account that any di?erentiable generator G ? on the phase space), the phase-space functional C (which does not change the action of G main theorem leads directly to ? ′′′ + C ′′′ , ? ′′ , G ?′ = G G (6.5)

2π 0

d? M1 ,

(6.4b)

? ′′′ are de?ned by the composition law (4.6), and C ′′′ is an unknown, where the parameters of G ?eld-independent functional, C ′′′ ≡ C ′′′ [T ′ , S ′; T ′′ , S ′′ ]. The term C ′′′ is known as the central charge of the PB algebra. ? ′′ , G ? ′} = δ′ G ? ′′ ≈ δ ′ Γ ′′ , where the weak equality In order to calculate C ′′′ , we note that {G 0 0 ′ is a consequence of the fact that δ0 is a symmetry operation that maps constraints into ? ′′′ ≈ Γ ′′′ , Eq. (6.5) implies constraints. Combining this result with G

′ ′′ δ0 Γ ≈ Γ ′′′ + C ′′′ .

(6.6a)

This relation determines the value of C ′′′ only weakly, but since C ′′′ is a ?eld-independent ′ ′′ quantity, the weak equality is easily converted into the strong one. The calculation of δ0 Γ is based on the relations δ0 ?E 1 = ?M1 ?2 T ? ?E 1 ?2 S ? ?2 M1 T + ?E 1 S

3 3 +(2a + α3 p)??2 S ? 2α3 ?2 T + O2 ,

δ0 M1 = ??E 1 ?2 T ? M1 ?2 S ? ?2 ?E 1 T + M1 S

3 3 S + O2 , T ? 2α3 ?2 +(2a + α3 p)??2

which follow from the re?ned asymptotic conditions derived in Appendix C, and the transformation rules de?ned by the parameters (4.3). Substituting the calculated expression for ′ ′′ δ0 Γ into (6.6a) yields the following value for the central charge C ′′′ : C ′′′ = (2a + α3 p)? ?2α3

2π 0 2π 0 2 ′ 2 ′′ d? ?2 S ′′ ?2 T ? ?2 S ′ ?2 T

2 ′ 2 ′ d? ?2 T ′′ ?2 T + ?2 S ′′ ?2 S .

(6.6b)

Canonical structure of 3D gravity with torsion

15

Conservation laws. As we noted in section 5, the improved total Hamiltonian is one ? [1, 0] = ??H ? T . A direct calculation based on the PB algebra (6.5) shows of the generators, G ? [T, S ] is conserved [20]: that the asymptotic generator G d ? ? ? ? H ? T ≈ ? Γ [T, S ] ? 1 Γ [?2 S, ?2 T ] = 0 . G= G + G, dt ?t ?t ? (6.7)

This also implies the conservation of the boundary term Γ . Now, we wish to clarify the meaning of the conserved charges by calculating their values for the black hole solution (3.2). First, note that the black hole solution depends on the radial coordinate only, and consequently, the terms E 1 and M1 in Γ behave as constants. Second, the parameters (T, S ) are periodic functions, equation (4.5), so that only zero modes in the Fourier expansion of (T, S ) survive the integration in Γ . Thus, there are only two independent non-vanishing charges for the black hole solution, given by two inequivalent choices of the constants T and S . If we take, for instance, (T = 1, S = 0) as the ?rst choice, and (T = 0, S = 1) as the second one, all the other non-zero charges will be given as linear combinations of these two. For (T = 1, S = 0) we have Γ [1, 0] = ??E , and the corresponding conserved charge is the energy E . Its value for the black hole solution is found to be E (black hole) = 4π m a + α3 p α3 J ? 2 . 2 ? (6.8a)

The second choice (T = 0, S = 1) leads to Γ [0, 1] = ?M . The corresponding conserved charge is the angular momentum M , and its black hole value reads M (black hole) = 4π J a + α3 p ? α3 m . 2 (6.8b)

Our expressions for the conserved charges (6.8) coincide with the results obtained in Ref. [19]. In the sector α3 = 0 (GRΛ and the teleparallel theory), we have E = m and M = J (in units 4G = 1), while for α3 = 0, the constants m and J do not have directly the meaning of energy and angular momentum, respectively. Geometrically, the two independent charges (6.8) parametrize the family of globally inequivalent, asymptotically AdS spaces. Central charge. Using the Fourier expansion, one can rewrite the canonical algebra (6.5) in a more familiar form. The parameters (T, S ) can be Fourier decomposed as follows:

+∞

T =

?∞

an e

inx+

+∞

+a ?n e

inx?

,

S=

?∞

an einx ? a ?n einx

+

?

.

The asymptotic generator is a linear, homogeneous function of the parameters, so that: ? [T, S ] = ?2 G The previous relations imply: ? [T = S = einx+ ] , 2Ln = ?G ? n = ?G ? [T = ?S = einx? ]. 2L (6.9a)

+∞ ?∞

?n . an Ln + a ?n L

16

M. Blagojevi? c and B. Cvetkovi? c

? n , the canonical algebra takes the Expressed in terms of the Fourier coe?cients Ln and L form of two independent Virasoro algebras with classical central charges: c 3 in δn,?m , 12 ? 3 ?n, L ? m = ?i(n ? m)L ? m+n ? c in δn,?m , L 12 ? m} = 0 . {Ln , L {Ln , Lm } = ?i(n ? m)Ln+m ?

(6.9b)

The central charges, given in the standard string theory normalization, have the form: c = 12 · 2π a? + α3 c ? = 12 · 2π a? + α3 p? ?1 2 p? +1 2 , . (6.10)

Thus, the gravitational sector with α3 = 0 has the conformal asymptotic symmetry with two di?erent central charges, while α3 = 0 implies c = c ? = 3?/2G. The general classical central charges c and c ? di?er from each other, in contrast to the results obtained in GRΛ and the teleparallel theory [3, 20]. ?0 → L ?0 + c By rede?ning the zero modes, L0 → L0 + c/24, L ?/24, the Virasoro algebra takes its standard form. One should note that the central term for the SO (2, 2) subgroup, ? ?1 , L ? 0, L ? 1 ), vanishes. This is a consequence of the fact generated by (L?1 , L0 , L1 ) and (L that SO (2, 2) is an exact symmetry of the AdS vacuum [3].

7

Concluding remarks

In this paper, we investigated the canonical structure of 3D gravity with torsion. (1) The geometric arena for the topological 3D gravity with torsion, de?ned by the Mielke-Baekler action (2.4), has the form of Riemann-Cartan spacetime. (2) There exists an exact vacuum solution of the theory, the Riemann-Cartan black hole (3.2), which generalizes the standard BTZ black hole in GRΛ . (3) Assuming the AdS asymptotic conditions, we constructed the canonical conserved charges. Energy and angular momentum of the Riemann-Cartan black hole are di?erent from the corresponding BTZ values. (4) The PB algebra of the canonical generators has the form of two independent Virasoro algebras with classical central charges. The values of the central charges are di?erent from each other, in contrast to the situation in GRΛ and the teleparallel theory. The implications of this result for the quantum structure of black hole are to be explored.

Acknowledgements

This work was supported by the Serbian Science foundation, Serbia. One of us (MB) would like to thank Milovan Vasili? c, Friedrich Hehl and Yuri Obukhov for a critical reading of the manuscript and many useful suggestions.

Canonical structure of 3D gravity with torsion

17

A

Symmetries of the AdS vacuum

The invariance conditions δ0 bi ? = 0 for the AdS triad (3.3a) yield the set of requirements on the parameters (ξ ? , θi ), the general solution of which has the form [20] ξ 0 = ?σ1 ?

0

r ?2 Q , f

ξ 1 = ?2 f ?0 ?2 Q , θ1 = Q ,

ξ 2 = σ2 ? θ2 =

?2 f ?0 Q , r (A.1)

?2 θ = ? ?0 Q , r where

1 ?2 Q , f

Q ≡ σ3 cos x+ + σ4 sin x+ + σ5 cos x? + σ6 sin x? ,

(A.2)

and σi are six arbitrary dimensionless parameters. The invariance conditions δ0 ω i ? = 0 for the AdS connection (3.3b) do not produce any new restrictions on (ξ ? , θi ). For each k = 1, 2, . . . , 6, we can choose σk = 1 as the only non-vanishing constant, and ?nd the ? corresponding basis of six independent Killing vectors ξ( k) : ξ(1) = (?, 0, 0) , ξ(2) = (0, 0, 1) , r ?f ξ(3) = sin x+ , ??f cos x+ , sin x+ , f r ?f r cos x+ , ?f sin x+ , cos x+ , ξ(4) = f r r ?f ξ(5) = ? sin x? , ?f cos x? , sin x? , f r ?f r cos x? , ?f sin x? , ? cos x? , ξ(6) = f r

(A.3)

? i i and similarly for θ( k ) . As one can explicitly verify, the six pairs (ξ(k ) , θ(k ) ) fall into the class of asymptotic parameters (4.3), and de?ne the algebra of the AdS group SO (2, 2).

B

The algebra of constraints

The structure of the PB algebra of constraints is an important ingredient in the analysis of the Hamiltonian consistency conditions. For the nontrivial part of the PB algebra involving (φi α , Φi α , Hi , Ki ), we have the following result: {φiα , φj β } = ?2α4 ε0αβ ηij δ , {Φi α , Φj β } = ?2α3 ε0αβ ηij δ , {φi α , Φj β } = ?2aε0αβ ηij δ ,

{φiα , Hj } = 2ε0αβ α4 ηij ?β δ ? εijk α4 ω k β ? Λbk β δ , {φiα , Kj } = 2ε0αβ aηij ?β δ ? εijk aω k β + α4 bk β δ , {Φi α , Hj } = 2ε0αβ aηij ?β δ ? εijk aω k β + α4 bk β δ , {Φi α , Kj } = 2ε0αβ α3 ηij ?β δ ? εijk α3 ω k β + abk β δ . (B.1)

18

M. Blagojevi? c and B. Cvetkovi? c

? i, K ? i ) is given by The essential part of the PB algebra involving the ?rst class constraints (H the following relations: ? j } = εijk pφkα + q Φkα δ , {φi α , H ? j } = ?εijk φkα δ , {Φi α , H ? i, H ? j } = εijk pH ? k + qK ?k δ , {H ? i, K ? j } = ?εijk K ? kδ . {K ? j } = ?εijk φkα δ , {φi α , K ? j } = ?εijk Φkα δ , {Φi α , K ? i, K ? j } = ?εijk H ?kδ , {H (B.2)

C

Asymptotic form of the constraints

Here, we analyze the in?uence of the secondary constraints (5.4) and relations (5.5) for the determined multipliers, on the basic asymptotic conditions (4.1) and (5.9), using the principle formulated at the end of section 5. Let us start with the secondary constraints (5.4b). Using (4.1) and (5.9), these constraints imply the following asymptotic relations: ω 0 1 = O4 , ?1 (re2 ) = O3 , ω 2 1 = O4 , ?1 (rm2 ) = O3 ,

?1 r (B 2 2 ? B 0 2 ) = ? ? 2 2 ? ? 0 2 + r 2 ? 1 1 p? r2 B 2 2 ? B 0 2 + B 1 1 + O3 , 2 ? 2 r p? = B 0 2 + B 2 2 + B 1 1 ? ?? 0 2 + O3 , 2 ? + 1?

?1 rB 2 2 where:

(C.1)

α3 p ap 0 a α3 ω 0 ? + α4 + b ? + b2 ? + ω 2 ? , 2 2 ? ? ap a α3 α3 p ω 2 ? + α4 + b2 ? + b0 ? + ω 0 ? . m? = a + 2 2 ? ? e? = a + From the expressions (C.1), one easily concludes that the terms E α and Mα , included in the surface integral (6.2) for Γ , satisfy the following asymptotic conditions: ?1 E 1 = O3 , E 2 = O3 , ?1 M1 = O3 , M2 = O3 . (C.2)

In a similar manner, equations (5.5b) lead to: p? 2 r2 B 0 + 2 B 1 1 ? ?? 2 0 + O3 , 2 ? ?1 (re0 ) = O3 , ?1 (rm0 ) = O3 , ?2 e0 ? ?0 e2 = O3 , ?2 m0 ? ?0 m2 = O3 , ?e0 + m2 = O3 , ?m0 + e2 = O3 . ?1 rB 0 0 = B 0 0 +

(C.3)

Canonical structure of 3D gravity with torsion

19

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