Canonical transformations generated by the boundary volume: unimodular and non-abelian teleparallel gravity
CCanonical transformations generated by the boundary volume:unimodular and non-abelian teleparallel gravity
Florian Girelli and Abdulmajid Osumanu
Department of Applied Mathematics, University of Waterloo,200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1
Wolfgang Wieland
Austrian Academy of Sciences, Institute for Quantum Optics andQuantum Information (IQOQI), Boltzmangasse 3, 1090 Vienna, Austria
Recently, a new choice of variables was identified to understand how the quantum group structureappeared in three-dimensional gravity [1]. These variables are introduced via a canonical transfor-mation generated by a boundary term. We show that this boundary term can actually be takento be the volume of the boundary and that the new variables can be defined in any dimensiongreater than three. In addition, we study the associated metric and teleparallel formalisms. Theformer is a variant of the Henneaux–Teitelboim model for unimodular gravity. The latter providesa non-abelian generalization of the usual abelian teleparallel formulation.
Contents
I. Introduction II. Gauge theory for a matched pair of Lie groups III. Canonical transformation induced by the boundary volume term
IV. Metric and deformed teleparallel formulation in the new variables
V. Outlook References I. INTRODUCTION
Quantum states are built from the representations of the symmetries, identified at the classical level. The structureof the symmetries depends on the choice of variables used to describe the phase space of the classical theory. Inthe gravity case, moving from the metric to the frame field variables (Palatini formalism) enlarges the set of gaugesymmetries since internal Lorentz rotations are absent in the usual metric formalism. On the other hand, treatinggravity as a constrained topological field theory [2, 3] has the opposite effect: the topological theory has a largersymmetry algebra. Gravity is recovered by imposing constraints that break the gauge symmetries down to thosethat appear in general relativity. Using the teleparallel formulation of gravity leads to yet another different type ofsymmetries, representing different representations at the quantum level (possibly related by dualities). Since both themetric and teleparallel formulations can be seen as a second-order formulation of gravity, it is interesting to study therelationship between the respective symmetries depending on the choice of variables. We refer to [4–12] for a recentdiscussion about these different issues in gravity.In field theory, canonical transformations are typically generated by adding topological invariants or boundaryterms to the action. Let us list some important examples in the context of gravity below (a more detailed discussioncan be found in e.g. [4–12] and references therein).• In any dimension, the Gibbons-Hawking-York (boundary) term generates the canonical transformation changingthe polarization from the superspace momentum representation, where the wave-functionals depend on the ADM a r X i v : . [ g r- q c ] J a n momentum ˜ π ab = πG √ det h ( K ab − h ab K ) , to the position representation, where the wave-functional Ψ[ h ab ] depends only on the intrinsic metric of the boundary, c.f. [13–15].• For gravity in four spacetime dimensions, the Holst term induces the Ashtekar–Barbero canonical transformationfrom the ADM-type of variables to the (complex) Ashtekar variables, which greatly simplifies the structure ofthe constraints, enabling the loop quantisation of gravity [15–18]• In three-dimensions, quantum groups play an important role in both the quantization of Chern–Simons theory[19] and the discrete path integral formulation [20] of gravity. Quantum groups appear as deformed gaugesymmetries, where the quantum deformation is parametrized by the cosmological constant. This is in strikingcontrast to the classical formulation of three-dimensional gravity in the first-order formalism, where the gaugesymmetries do not depend on the cosmological constant. It was recently found that a canonical transformation,induced by a boundary term, could be used to have a (bulk) theory defined in terms of new variables, such thatthe gauge symmetries do dependent now on the cosmological constant as well, see [1]. This provided a directmanner to retrieve the quantum group symmetries upon quantization.It has been conjectured that quantum groups could also be relevant for quantum gravity in four dimensions [21–23],so one might wonder whether the canonical transformation or the boundary term that was identified in [1] can alsobe introduced in four dimensions. We show here that this is indeed the case and that there is a straightforwardgeneralization of the boundary term in the first-order formalism. It is then natural to ask what happens to thesecond-order formalism and how the metric and teleparallel formulations of gravity could be expressed in terms of thenew variables. Here, we answer these questions. In the metric formalism, we have a new notion of Christoffel symbol.The volume term that is added to the action via the cosmological constant is replaced by the divergence of a vectorfield, whose norm depends on the cosmological constant. In fact, we recover a variant of the Henneaux–Teitelboimmodel for unimodular gravity [24]. On the other hand, in the teleparallel formalism [25], the new variables suggest adeformed notion of torsion, which defines a new and non-abelian version of the standard teleparallel formulation.The boundary term that generates the canonical transformation that was introduced in [1] depends on an internalvector r I , whose norm sets the value of the cosmological constant. In order to not increase the number of degreesof freedom, it was assumed that the vector is an auxiliary background structure, whose field-variations vanish on thecovariant phase space, i.e. δ [ r I ] = 0 . However, the boundary term [1] has a very strong resamblance with the volumeterm of the boundary. It seems natural, therefore, to relax the condition δ [ r I ] = 0 . In fact, we will show that if weactually interpret the vector r I as the internal normal n I to the boundary, the generating functional [1] is exactly thevolume of the boundary. In here, we show that given this interpretation, we can allow for δ [ n I ] (cid:54) = 0 and still performthe canonical transformation, without adding any new degrees of freedom in the bulk.The new formulation has some interesting relations with already existing frameworks.• The boundary term plays an important role in holographic renormalization [26, 27] so that our new variableswill also be relevant to this framework.• With the change of variables, the volume term that scales with the cosmological constant drops out of the actionand is replaced by the divergence of a vector field and a constraint which sets the cosmological constant as thenorm of this vector field. We will see below that the resulting action defines a variation of the Henneaux–Teitelboim model for unimodular gravity [24].One can often see the first-order formalism for gravity as a gauge theory, so that the canonical variables are Liealgebra-valued differential forms. This is obviously realized in three dimension in the Chern–Simons formulation. Infour dimensions, the McDowell–Mansouri formulation [28] provides a similar realization. Depending on the value ofthe cosmological constant and the signature of the metric, one usually deals with a (anti-)de Sitter or Poincaré (orEuclidian) gauge group, where the frame field has values in the boost or translation sector of the group. Accordingly,the (anti-)de Sitter or Poincaré curvature splits into the Lorentzian curvature sector, which depends on the cosmologicalconstant, and the translational or boost sector, which represents torsion and is independent of Λ . Since the (anti-)deSitter or Poincaré groups can be seen as the semi-cross product between the Lorentz transformations and the boostsor translations, there is a natural action of the Lorentz group on the frame field. On the other hand, there is no actionof translations back onto rotations (the commutator of a translation and a rotation has no translational part).The canonical transformation that we will study in here amounts to choosing a different decomposition of the(anti-)de Sitter group (the Poincaré case can also be treated in the same way). Indeed, if the standard formulationamounts to use the Cartan decomposition, the canonical transformation instead picks the Iwasawa decomposition.We have then a double cross product relation and a more symmetric treatment between the connection and the framefield: the connection acts on the frame field and the frame field acts back on the connection. As a consequence the(anti-)de Sitter or Poincaré curvature can be split differently than before. We get a generalized notion of torsion,which can be viewed as a (non-abelian) curvature in the frame field sector. This is especially relevant regarding theteleparallel formulation. Outline : The article is organized as follows. In section II, we review the gauge theory framework when dealingwith a matched pair (double cross product) of Lie algebras, which will conveniently set up the conventions for the restof the paper. In section III, we discuss how the canonical transformation can be implemented in any dimension by theboundary term given in terms of the volume of the boundary. In section IV A, we determine the metric formulationin the new variables, while in section IV B, we determine the dual teleparallel formulation.
II. GAUGE THEORY FOR A MATCHED PAIR OF LIE GROUPS
In the following, we develop the notion of a gauge theory for a matched pair of Lie algebras. The algebraic backgroundbehind the matched pairs can be found in [29]. We call a matched pair G = g (cid:46)(cid:47) h of Lie algebras the pair of Lie algebras g and h , with respective Lie brackets [ · , · ] g and [ · , · ] h , equipped with two bilinear maps (cid:67) : h × g → h , ( x, α ) (cid:55)→ x (cid:67) α and (cid:66) : h × g → g , ( x, α ) (cid:55)→ x (cid:66) α , such that h is a right g -module and g is a left h -module. More intuitively, g actsfrom the right on h via (cid:67) and there is a back action (cid:66) of h on g . To guarantee the Jacobi identity on G , the left andright actions must satisfy the following compatibility conditions: ∀ α, β ∈ g , x, y ∈ h , x (cid:66) [ α, β ] g = [ x (cid:66) α, β ] g + [ α, x (cid:66) β ] g + ( x (cid:67) α ) (cid:66) β − ( x (cid:67) β ) (cid:66) α, (2.1a) [ x, y ] h (cid:67) α = [ x (cid:67) α, y ] h + [ x, y (cid:67) α ] h + x (cid:67) ( y (cid:66) α ) − y (cid:67) ( x (cid:66) α ) . (2.1b)As a vector space, g (cid:46)(cid:47) h is isomorphic to g ⊕ h . If we introduce a matched basis, such that ( α, x ) ∈ g ⊕ h ∼ g (cid:46)(cid:47) h ,we may express the Lie bracket of G in terms of [ · , · ] g , [ · , · ] h , (cid:67) and (cid:66) , (cid:2) ( α, x ) , ( β, y ) (cid:3) ≡ (cid:0) [ α, β ] g + x (cid:66) β − y (cid:66) α, [ x, y ] h + x (cid:67) β − y (cid:67) α (cid:1) , ∀ α, β ∈ g , x, y ∈ h . (2.2)In particular, there is the mixed bracket (cid:2) (0 , x ) , ( β, (cid:3) ≡ [ x, β ] = x (cid:67) β + x (cid:66) β ∈ g ⊕ h . (2.3)Notice that the left and right modules can be written in terms of the Lie bracket alone x (cid:67) β := [ x, β ] (cid:12)(cid:12) h , x (cid:66) β := [ x, β ] (cid:12)(cid:12) g , (2.4)where [ x, β ] | V is the projection of the Lie bracket [ · , · ] on the vector space V . With this notation, the compatibilityrelations (2.1) become x (cid:66) [ α, β ] g = (cid:2) [ x, α ] (cid:12)(cid:12) g , β (cid:3) g + (cid:2) α, [ x, β ] (cid:12)(cid:12) g (cid:3) g + (cid:2) [ x, α ] (cid:12)(cid:12) h , β (cid:3)(cid:12)(cid:12) g − (cid:2) [ x, β ] (cid:12)(cid:12) h , α (cid:3)(cid:12)(cid:12) g , (2.5a) [ x, y ] h (cid:67) α = (cid:2) [ x, α ] (cid:12)(cid:12) h , y (cid:3) h + (cid:2) x, [ y, α ] (cid:12)(cid:12) h (cid:3) h + (cid:2) x, [ y, α ] (cid:12)(cid:12) g (cid:3)(cid:12)(cid:12) h − (cid:2) y, [ x, α ] (cid:12)(cid:12) g (cid:3)(cid:12)(cid:12) h . (2.5b)Consider then a gauge connection A for such a Lie algebra, i.e. a G -valued one-form on a manifold M , and considerits different components with respect to the matched Lie algebra g (cid:46)(cid:47) h . This type of gauge theory was studied byMajid [29] at the discrete level of parallel propagators (holonomies). Below we will derive the infinitesimal picture.Let us denote by γ and h the components of such a gauge connection with respect to the Lie algebras g and h .The g (cid:46)(cid:47) h Lie algebra-valued gauge connection takes the form A = γ + h ∈ ( g ⊕ h ) ⊗ Λ ( M ) . (2.6)Due to the kick-back action between g and h , there is a non-trivial mixing between the components of the g (cid:46)(cid:47) h basis,see (2.4). Consider, for example, a G -valued p -form B = ( β, b ) . Its exterior covariant derivative is d A B = d β + d b + [ γ + h, β + b ] = d β + d b + [ γ, β ] g + [ γ, b ] + [ h, b ] h + [ h, β ]= (cid:16) d β + [ γ, β ] g + [ γ, b ] (cid:12)(cid:12) g + [ h, β ] (cid:12)(cid:12) g (cid:17) + (cid:16) d b + [ h, b ] h + [ γ, b ] (cid:12)(cid:12) h + [ h, β ] (cid:12)(cid:12) h (cid:17) . (2.7) We will relate ( γ, h ) to the spin connection ω and the frame field e . For the individual component fields ( β, and (0 , b ) , we have the covariant derivatives D | g β = d β + [ γ, β ] g + [ h, β ] (cid:12)(cid:12) g , (2.8) D | h b = d b + [ γ, b ] (cid:12)(cid:12) h + [ h, b ] h . (2.9)Due to the kick-back actions between the two Lie algebras, the covariant derivatives depend on both connectioncomponents γ and h . Consider then the components of the field strength F in the directions of g and h , F [ A ] = d A + 12 [ A ∧ A ] = d γ + d h + 12 (cid:2) ( γ + h ) ∧ ( γ + h ) (cid:3) = d γ + d h + 12 [ γ ∧ γ ] g + [ γ ∧ h ] (cid:12)(cid:12) g + [ γ ∧ h ] (cid:12)(cid:12) h + 12 [ h ∧ h ] h = (cid:18) d γ + 12 [ γ ∧ γ ] g + [ γ ∧ h ] (cid:12)(cid:12) g (cid:19) + (cid:18) d h + [ γ ∧ h ] (cid:12)(cid:12) h + 12 [ h ∧ h ] h (cid:19) ≡ F + T . (2.10)The physical significance of splitting F into F ∈ g ⊗ Λ ( M ) and T ∈ h ⊗ Λ ( M ) will be clear below. We will identify,in fact, F and T with a deformed version of h -valued torsion and g -valued curvature, F = d γ + 12 [ γ ∧ γ ] g + [ γ ∧ h ] (cid:12)(cid:12) g ≡ F + [ γ ∧ h ] (cid:12)(cid:12) g , (2.11) T = d h + [ γ ∧ h ] (cid:12)(cid:12) h + 12 [ h ∧ h ] h ≡ T + 12 [ h ∧ h ] h . (2.12)Again, the curvature components depend on both connections types. The Bianchi identity for each component is A F = d F + d T + (cid:2) ( γ + h ) ∧ ( F + T ) (cid:3) = (cid:16) d F + [ γ ∧ F ] g + [ h ∧ F ] (cid:12)(cid:12) g + [ γ ∧ T ] (cid:12)(cid:12) g (cid:17) + (cid:16) d T + [ γ ∧ T ] h + [ h ∧ F ] (cid:12)(cid:12) h + [ γ ∧ T ] (cid:12)(cid:12) h (cid:17) . (2.13)Let us consider then the gauge transformations of such a G -valued connection. Let λ = ( φ, t ) be a G -valued scalar,which generates an infinitesimal transformation of the connection A = ( γ, h ) , δ λ [ A ] = d λ + [ A , λ ] = (cid:16) d φ + [ γ, φ ] g + [ h, φ ] (cid:12)(cid:12) g + [ γ, t ] (cid:12)(cid:12) g (cid:17) + (cid:16) d t + [ h, t ] h + [ γ, t ] (cid:12)(cid:12) h + [ h, φ ] (cid:12)(cid:12) h (cid:17) . (2.14)For pure rotations ( t = 0 ) and pure translations ( φ = 0 ), we find δ φ [ γ ] = d φ + [ γ, φ ] g + [ h, φ ] (cid:12)(cid:12) g = D | g φ, δ φ [ h ] = [ h, φ ] (cid:12)(cid:12) h ,δ t [ γ ] = [ γ, t ] (cid:12)(cid:12) g , δ t [ h ] = d t + [ h, t ] h + [ γ, t ] (cid:12)(cid:12) h = D | h t. (cid:41) (2.15)What is nice about these transformations is that the variation of γ or h in each of the different components is eithera commutator or a derivative. In other words, the two components of the curvature two-form behave completelyanalogous under gauge transformations. Since δ λ [ F ] = [ F , λ ] , we obtain [ F + T , φ + t ] = [ F , φ ] g + [ F , t ] (cid:12)(cid:12) g + [ T , φ ] (cid:12)(cid:12) g + [ T , t ] h + [ F , t ] (cid:12)(cid:12) h + [ T , φ ] (cid:12)(cid:12) h . (2.16)In other words, δ φ [ F ] = [ F , φ ] g + [ T , φ ] (cid:12)(cid:12) g , δ φ [ T ] = [ T , φ ] (cid:12)(cid:12) h δ t [ F ] = [ F , t ] (cid:12)(cid:12) g , δ t [ T ] = [ T , t ] h + [ F , t ] (cid:12)(cid:12) h . (cid:41) (2.17)A simple, but non trivial, example of such a matched pair of Lie algebras is the Poincaré Lie algebra. It is asemi-cross product G = g (cid:66) < h , where g is either of the Lie algebra so ( n ) or so ( n − , depending on the signature ofspacetime and h ∼ R n is the abelian Lie algebra of translations on R n . The respective generators are the generators of(Lorentz) rotations J MN = − J NM and translations P M , which are labelled by spacetime indices M, N, · · · = 1 , . . . , n .In the Poincaré context, there is no kick-back action of h on g (recall [ J, P ] ∝ P ). Given (2.4), we thus have ( φ, t ) ∈ g (cid:66) < h , t (cid:67) φ := [ t, φ ] (cid:12)(cid:12) h , t (cid:66) φ := [ t, φ ] (cid:12)(cid:12) g = 0 . (2.18)The translations commute, i.e. [ · , · ] h = 0 , and the covariant derivative for the component fields ( φ, and (0 , t ) withvalues in g or h is given by D | g φ = d φ + [ γ, φ ] g , D | h t = d t + [ γ, t ] (cid:12)(cid:12) h . (2.19)Consider then the components of the curvature two-form F in the rotational and translational directions g and h ofthe Poincaré Lie algebra, F [ A ] = (cid:18) d γ + 12 [ γ ∧ γ ] g (cid:19) + (cid:16) d h + [ γ ∧ h ] (cid:12)(cid:12) h (cid:17) = F + T. (2.20)We recognise the usual notion of curvature F and torsion T , provided we identify γ with the spin connection ω and h with the frame field e .The infinitesimal gauge transformations for pure rotations ( t = 0 ) and pure translations ( φ = 0 ) are δ φ [ γ ] = d φ + [ γ, φ ] g = D | g φ, δ φ [ h ] = [ h, φ ] (cid:12)(cid:12) h ,δ t [ γ ] = 0 , δ t [ h ] = d t + [ γ, t ] (cid:12)(cid:12) h = D | h t, (2.21)where δ φ [ γ ] is the so gauge transformation for the spin connection ω = γ , and δ t [ h ] is an infinitesimal translation ofthe frame field h = e . Finally, there are the Poincaré transformations of the curvature two-form δ φ [ F ] = [ F, φ ] g , δ φ [ T ] = [ T, φ ] (cid:12)(cid:12) h δ t [ F ] = 0 , δ t [ T ] = [ F, t ] (cid:12)(cid:12) h . (2.22)Although the translations t ∈ h are abelian, a general such translation will act non-trivially on the translationalcurvature (torsion). If we would choose, however, a connection γ such that F [ γ ] = 0 , both the torsion component T and the rotational (Lorentz) component F of the Poincaré connection would be translational invariant. This choiceunderlies the teleparallel equivalent of general relativity.In the following, we will consider the more complicated case of G = so ( n − , ∼ so ( n − (cid:46)(cid:47) an n − , which is anexample of a matched Lie algebra induced by the Iwasawa decomposition of the Lorentz Lie algebra so ( n, . III. CANONICAL TRANSFORMATION INDUCED BY THE BOUNDARY VOLUME TERMA. Warm up: three-dimensions
Consider the three-dimensional Einstein – Cartan action in first-order spin-connection variables with a cosmologicalconstant Λ together with the following boundary terms, whose significance will become clear below, S EC [ e, A | n ] = 116 πG (cid:20) (cid:90) M ε IJK e K ∧ (cid:16) R IJ [ A ] − Λ3 e I ∧ e J (cid:17) − (cid:90) ∂ M (cid:16) s ε IJK e K ∧ n I d A n J + (cid:112) | Λ | ε IJK n K e I ∧ e J (cid:17)(cid:21) . (3.1)The action in the bulk is a functional of the Lorentz curvature R IJ [ A ] = d A IJ + A IK ∧ A KJ and the triad e I .The boundary term consists of two parts. The first term is the usual Gibbons – Hawking – York boundary term forfirst-order spin connection variables, where d A [ · ] = d[ · ] + [ A, · ] is the covariant exterior derivative. The internal vectorfield n I is constrained to lie orthogonal to the boundary, i.e. n I ϕ ∗ ∂ M e I = 0 , where ϕ ∗ ∂ M : T ∗ M → T ∗ ∂ M is the pull-back. In addition, n I is normalised such that s = η IJ n I n J = n I n I = {± , } depending on whether the boundary isspacelike, timelike or null. Its orientation is such that n a := n I e aI is the outwardly oriented normal to the boundary.If the torsionless equation is satisfied, the first term is the integral of the trace of the extrinsic curvature. The secondterm is proportional to the induced volume of the boundary. The boundary conditions are h IJ ϕ ∗ ∂ M δ [ e J ] = 0 , ϕ ∗ ∂ M δ [ n I e I ] = 0 , (3.2) In the Lorentzian case, our metric signature is ( − ++ . . . ) . Internal spacetime indices I, J, K, . . . are raised and lowered with the internalmetric tensors η IJ and η IJ . Total antismmetrisation of indices I , I , . . . is obtained via ω [ I ...I n ] = n ! (cid:80) σ ∈ S n ( − σ ω I σ (1) ...I σ ( n ) . where δ [ · ] is a variation on the infinite-dimensional space of kinematical histories and h IJ = − sn I n J + η IJ is theinternal projector onto the boundary.It us also useful to evaluate the action (3.1) for a spin connection A IJ , which is torsionless (by going half-shell ).We obtain S EC [ A, e | n ] (cid:12)(cid:12)(cid:12) d A e =0 = S EH [ g | n ] = 116 πG (cid:20)(cid:90) M d v (cid:0) R [ g ] − (cid:1) + 2 (cid:90) ∂ M d v (cid:0) sK − (cid:112) | Λ | (cid:1)(cid:21) , (3.3)where R [ g ] is the Ricci scalar for the metric g ab = e Ia e Ib and K = ∇ a n a is the trace of the extrinsic curvature. Inaddition, d v and d v are the canonical volume elements on M and ∂ M . The volume term appears in the definitionof the bulk plus boundary action (3.3) to cancel infrared divergencies that would otherwise appear when the boundary ∂ M is sent to infinity [26].Consider then a region Σ ⊂ ∂ M within the boundary. The first variation of the action for given boundary conditions(3.2) determines the pre-symplectic potential Θ EC Σ on the space of physical histories, i.e. the space of solutions to thefield equations. A straightforward calculation gives, Θ EC Σ ( δ ) = 18 πG (cid:90) Σ (cid:16) s ε IJ δ [ e I ] ∧ K J − (cid:112) | Λ | ε IJ e I ∧ δ [ e J ] (cid:17) − πG (cid:73) ∂ Σ s ε IJ e I δ [ n J ] , (3.4)where K I = ϕ ∗ ∂M d A n I is the extrinsic curvature (a one-form along the boundary) and ε IJ = n K ε KIJ is the internalarea element at the boundary. Notice that the variation δ [ n I ] of the internal normal only affects a corner term tothe pre-symplectic potential (3.4). This is a consequence of the torsionless condition d A e I = 0 and the boundaryconstraints n I n I = s and ϕ ∗ ∂ M ( n I e I ) = 0 , hence n I δ [ n I ] = 0 and ϕ ∗ ∂ M [ (cid:15) IJK δ [ n I ] e J ∧ K K ] = 0 .Equation (3.4) is a manifestation of the well-known fact that the pull-back of the triad and the extrinsic curvature areconjugate variables. From the perspective of the Chern–Simons, and Ponzano–Regge quantisation of three-dimensionalgravity, a connection representation is more appropriate. Following [15], we consider the canonical transformation,which is generated by the Gibbons–Hawking–York boundary term Θ Σ ( δ ) := Θ EC Σ ( δ ) − πG δ (cid:20) (cid:90) Σ s ε IJ e I ∧ K J (cid:21) . (3.5)Going back to (3.1), we now immediately have Θ Σ ( δ ) = − πG (cid:90) Σ ε IJK e I ∧ δ (cid:104) A JK − (cid:112) | Λ | n [ J e K ] (cid:105) ≡ − πG (cid:90) Σ ε IJK e I ∧ δ [Ω JK ] , (3.6)where we introduced a new connection Ω IJ . Notice that the variation δ [Ω IJ ] contains a variation δ [ n I ] (cid:54) = 0 . Since,however, the boundary conditions (3.2) must be satisfied, we obtain δ [ n I ] ⊥ n I . In addition, the vector n a = n I e aI liesorthogonal to the boundary, hence δ [ n I ] e aI lies tangential to the boundary. Therefore ϕ ∗ ∂ M ( (cid:15) IJK δ [ n I ] e J ∧ e K ) = 0 ,such that πG (cid:90) Σ ε IJK e I ∧ δ [ n J e K ] = 18 πG (cid:90) Σ ε IJK e I ∧ n J δ [ e K ] = 116 πG δ (cid:20)(cid:90) Σ ε IJK e I ∧ n J e K (cid:21) . (3.7)To solve the equations of motion in terms of the new variables ( e Ia , Ω IJa ) , it is then necessary to smoothly extendthe internal vector n I into the bulk. We thus write Ω IJ = A IJ + e [ I p J ] ≡ A IJ − I IJ , (3.8) I IJ ≡ p [ I e J ] = 12 C IJ K e K , C IJ K = ( p I δ JK − p J δ IK ) , (3.9)where p I is an internal Lorentz vector that satisfies the constraints p I p I = − , p I (cid:12)(cid:12) ∂ M = 2 (cid:112) | Λ | n I . (3.10) The volume elements are p -forms d v = ε IJK e I ∧ e J ∧ e K and d v = n I ε IJK e J ∧ e K . The next step ahead is to write the action in terms of the new connection Ω IJ . Consider first the curvature, R IJ [ A ] = R IJ [Ω] + d Ω I IJ + I IK ∧ I KJ = R IJ [Ω] + d Ω I IJ + 14 (cid:0) p ∧ ( p I e J − p J e I ) − p K p K e I ∧ e J (cid:1) , (3.11)where p = p I e I and d Ω I IJ = d I IJ + Ω IK ∧ I KJ + I IK ∧ Ω KJ = d I IJ + [Ω ∧ I ] IJ and we used the fact that e I ∧ e I = 0 .With the above expression, the first term of the action (3.1) is ε JKI e I ∧ R JK [ A ] = ε JKI e I ∧ (cid:18) R JK [Ω] + d Ω I JK + 12 p [ J p ∧ e K ] − p M p M e J ∧ e K (cid:19) = ε JKI e I ∧ (cid:18) R [Ω] JK + d Ω I JK − p M p M e J ∧ e K (cid:19) . (3.12)Going from the first line to the second line, we used the following identity n M n M ε IJK e I ∧ e J ∧ e K = 12 n I ε IJK n ∧ e J ∧ e K , (3.13)for all n I : n I n I = ∈ { , ± } . Returning to the definition of the action (3.1), we obtain S EC [ e, A | n ] − πG (cid:90) ∂ M s ε JK e J ∧ K K = 116 πG (cid:20)(cid:90) M ε IJK e I ∧ (cid:18) R JK [Ω] + d Ω I JK (cid:19) − (cid:90) ∂ M ε IJK e I ∧ e J p K (cid:21) = 116 πG (cid:90) M ε IJK e I ∧ (cid:18) R [Ω] JK + d Ω I JK + (d Ω e [ J ) p K ] − e [ J ∧ d Ω p K ] (cid:19) = 116 πG (cid:90) M ε IJK e I ∧ (cid:18) R JK [Ω] + 12 e [ J ∧ d Ω p K ] (cid:19) . (3.14)Equation (3.14) suggests to introduce a new action S [ e, Ω , p ] = 116 πG (cid:90) M ε IJK e I ∧ (cid:18) R [Ω] JK + 12 e [ J ∧ d Ω p K ] (cid:19) , (3.15)where p I satisfies the mass shell condition p I p I = − . To obtain the equations of motion, we vary the action forfixed boundary conditions. Given the action (3.15), the appropriate boundary conditions are p I (cid:12)(cid:12) ∂M = 2 (cid:112) | Λ | n I , h [ IK h J ] L ϕ ∗ ∂ M δ [Ω KL ] = 0 , (3.16)where h IJ is the projector h IJ = − sn I n J + δ IJ and ϕ ∗ ∂ M : T ∗ M → T ∗ ( ∂ M ) is the pull-back. The resulting equationsof motion are F IJ = R IJ [Ω] + e [ I ∧ d Ω p J ] = 0 , (3.17) T I = d Ω e I + 14 C JKI e J ∧ e K = 0 . (3.18)The variation with respect to p I is redundant. Taking into account that δ [ p I p I ] = 2 p I δ [ p I ] = 0 , i.e. p I ⊥ δ [ p I ] , weobtain that the variation of the action with respect to p I vanishes provided ε IJK d Ω ( e J ∧ e K ) ∝ p I d v. (3.19)Given (3.18), this condition is always satisfied, since ε IJK d Ω ( e J ∧ e K ) = − ε IJK p ∧ e J ∧ e K = − p I d v .The field equations (3.17, 3.18) have a simple geometric meaning. They can be rearranged into a single flatnessconstraint for a so (1 , (cid:46)(cid:47) an matched connection. Consider the so (1 , (cid:46)(cid:47) an Lie algebra-valued one-form A a = 12 Ω IJa ⊗ J IJ + e Ia ⊗ P I , (3.20)where J IJ are the (Lorentz) generators of so (1 , and P I are the generators of an . The commutation relations are (cid:2) J IJ , J I (cid:48) J (cid:48) (cid:3) = 4 δ [ RI δ S ] J η SS (cid:48) δ [ S (cid:48) I (cid:48) δ R (cid:48) ] J (cid:48) J RR (cid:48) , (3.21a) (cid:2) P K , J IJ (cid:3) = 2 δ K [ I P J ] + J K [ I p J ] , (3.21b) (cid:2) P I , P J (cid:3) = p [ I P J ] . (3.21c)Going back to (2.3), we identify the action of the right (left) module, P K (cid:67) J IJ = 2 δ K [ I P J ] ∈ an , P K (cid:66) J IJ = J K [ I p J ] ∈ so (1 , . (3.22)To introduce the curvature of this connection, consider first the following covariant derivative, defined by its actionon the basis elements of the so (1 , (cid:46)(cid:47) an algebra d J IJ := 0 , d P I := 12 d p K ⊗ J KI . (3.23)The definition (3.23) extends naturally to all so (1 , (cid:46)(cid:47) an Lie algebra-valued p -forms ω ∈ Ω p ( M : so (1 , (cid:46)(cid:47) an ) via d ( ω + f ω ) = d ω + f d ω + d f ∧ ω for all f : M → R and ω , ω ∈ Ω p ( M : so (1 , (cid:46)(cid:47) an ) . Notice alsothat the derivative is flat, i.e. d = 0 . Consider then a second such covariant derivative D , which is defined by thedeformation D = d + [ A , · ] . Its curvature D = [ F , · ] is given by F [ A ] = d A + 12 [ A ∧ A ] = 12 F IJ ⊗ J IJ + T I ⊗ P I = 0 , (3.24)where the deformed so (1 , -valued curvature F and an -valued torsion T are defined as (3.17, 3.18). Let us alsostress that there always exists a gauge such that d p = 0 , such that the notion of curvature and torsion F , T wouldalso coincide with (2.11, 2.12). If the field equations (3.17, 3.18) are satisfied, this derivative is flat, i.e. F [ A ] = 0 .The torsion two-form T I = d Ω e I in now sourced by the cosmological constant, see (3.18). On the other hand, thefield equations for the so (1 , Lorentz part of the curvature two-form admit solutions where the spin connection Ω IJ is flat and p I is constant, i.e. d Ω p I = 0 . Hence there is some Lorentz gauge element Λ IJ : M → SO (1 , such that Ω IJ = Λ IK dΛ KJ .Given some internal vector p I on M , such a gauge element Λ IJ can always be found (unless there are topologicalobstructions). In other words, the homogenous curvature of the underlying spacetime metric g ab = η IJ e Ia e Jb hasbeen encoded into a flat so (1 , connection with non-vanishing an -valued torsion T I = d Ω e I . This was a key featurethat simplified the theory at the discrete level [1] and this simplification will also be relevant for us to introduce ateleparallel equivalent of gravity in the case of homogeneously curved geometries that we will discuss below.Before going to the general case, let us compare what we have just done with respect to [1]. There, the startingaction is S [ e, A ] ≡ πG (cid:20) (cid:90) M ε IJK e K ∧ (cid:16) R IJ [ A ] − Λ3 e I ∧ e J (cid:17) − (cid:90) ∂ M (cid:16) ε IJK r K e I ∧ e J (cid:17)(cid:21) . = 116 πG (cid:90) M ε IJK e I ∧ (cid:18) R JK [Ω] + 12 e [ J ∧ d Ω r K ] (cid:19) , (3.25)but the internal Lorentz vector r I is not interpreted as the normal of the boundary. In order to not increase thenumber of degrees of freedom, it is assumed that δ [ r I ] = 0 . This ensures that the so (1 , connection A IJ is shifted to ˜Ω IJ = A IJ − r [ I e J ] just as in (3.6). The vector r I is also normalized as in (3.10) in order to cancel the volume term. r I r I = − . (3.26)Without loss of generality, the vector r is assumed to have only one non-zero component, which is proportional tothe square root of the (absolute value of) the cosmological constant. As a consequence, d r I = 0 and the equations ofmotion then coincide with the generalized curvature and torsion being zero. We note that there is always a choice ofgauge such that the normal p I can be the constant vector r I , so that the two approaches are the same. B. Beyond three dimensions
In the last section, we considered gravity in three dimensions. Our next step is to generalise the construction toarbitrary spacetime dimensions d ≥ . Given the so (1 , d − spin connection A IJ , its conjugate momentum is thegravitational B -field, which is a bivector-valued ( d − -form, B IJ [ e ] = 1( d − ε IJK ··· K d − e K ∧ · · · ∧ e K d − , (3.27)where e I is the co-frame that defines the metric g ab = η IJ e Ia e Jb . Consider then the usual Einstein – Cartan actionwith a boundary term proportional to the volume of the boundary S EC [ e, A, p ] = 116 πG (cid:20)(cid:90) M (cid:18) B IJ [ e ] ∧ R IJ [ A ] − d ! ε I ...I d e I ∧ · · · ∧ e I d (cid:19) − d − (cid:90) ∂ M p J ε JI ...I d − e I ∧ · · · ∧ e I d − (cid:21) , (3.28)where R IJ [ A ] is the so (1 , d − -valued curvature two-form R IJ [ A ] = d A IJ + A IK ∧ A KJ and p I is an internal vector(a vector-valued -form). The critical points of the action are found by imposing the following boundary conditions, h IK h JL ϕ ∗ ∂ M δ [ A KL ] = 0 , δ [ p I p I ] = 0 , p I ϕ ∗ ∂ M e I = 0 , (3.29)where ϕ ∗ ∂ M : T ∗ M → T ∗ ( ∂ M ) is the pull-back and h IJ = sn I n J + h IJ is the projector onto the boundary such that n I n I = s ∈ {± } , and h IJ ϕ ∗ ∂ M e J = ϕ ∗ ∂ M e I . The resulting field equations are the d -dimensional Einstein equationsfor the metric g ab = e Ia e Ib with a cosmological constant Λ .Consider then a region Σ within the boundary, i.e. Σ ⊂ ∂ M . The pre-symplectic potential is obtained from thefirst variation of the action (3.28). Taking into account the boundary conditions (3.29), we obtain Θ Σ ( δ ) = ( − ( d − πG (cid:90) Σ B IJ ∧ δ (cid:104) A IJ − p [ I e J ] (cid:105) ≡ ( − ( d − πG (cid:90) Σ B IJ ∧ δ [Ω IJ ] . (3.30)Next, we need to express the field equations in terms of the new and shifted connection Ω . This requires to smoothlyextend the internal boundary vector p I into the bulk such that p I ϕ ∗ ∂ M e I = 0 is still satisfied. Given such an extensionof p I from the boundary into the bulk, we introduce the shifted connection Ω IJ = A IJ + e [ I p J ] ≡ A IJ − I IJ , (3.31) I IJ ≡ p [ I e J ] = 12 C IJ K e K , C IJ K = ( p I δ JK − p J δ IK ) . (3.32)Just as in three-dimensions, we reformulate the action in terms of the new connection. Consider first the curvaturescalar, B IJ [ e ] ∧ R IJ [ A ] = B IJ [ e ] ∧ (cid:0) R IJ [Ω] + d Ω I IJ + I IK ∧ I KJ (cid:1) = B IJ [ e ] ∧ (cid:18) R IJ [Ω] + d Ω I IJ + 12 p [ I p ∧ e J ] − p K p K e I ∧ e J (cid:19) , (3.33)where p denotes the one-form p = p I e I . We then also have the identity p M p M d ! ε I ...I d e I ∧ · · · ∧ e I d = 1( d − p J ε JI ...I d − p ∧ e I . . . e I d − . (3.34)We thus have, B IJ ∧ (cid:18) p I p ∧ e J − p K p K e I ∧ e J (cid:19) = − p M p M d − d − d ! ε I ...I d e I ∧ · · · ∧ e I d . (3.35)If we impose the mass shell condition p I p I = − d − d − , (3.36)the action (3.28) simplifies S EC [ e, Ω , p ] = 116 πG (cid:20)(cid:90) M B IJ ∧ (cid:16) R IJ [Ω] + d Ω I IJ (cid:17) − ( − ( d − d − (cid:90) ∂ M B IJ ∧ p I e J (cid:21) = 116 πG (cid:90) M (cid:20) B IJ ∧ R IJ [Ω] + B IJ ∧ (cid:18) d Ω I IJ − p [ I d Ω e J ] − d − Ω p [ I ) ∧ e J ] (cid:19)(cid:21) . (3.37)0Taking into account that I IJ = p [ I e J ] , one finally arrives at the expression S EC [ e, Ω , p ] = 116 πG (cid:90) M (cid:16) B IJ ∧ R IJ [Ω] − ( d − − d − B I ∧ d Ω p I (cid:17) , (3.38)where we introduced the vector-valued ( d − -form, B I = 1( d − ε IK ...K d − e K ∧ . . . e K d − . (3.39)The Einstein equations are the saddle points of the action (3.38), in the space of all fields that satisfy the massshell constraint (3.36) and boundary conditions h IK h JL ϕ ∗ ∂ M δ [Ω KL ] = 0 , ϕ ∗ ∂ M ( p I e I ) = 0 . (3.40)The variation of the frame field for boundary conditions (3.40) yields the curvature constraint d − ε IJKL ...L d − e L ∧ · · · ∧ e L d − ∧ R JK [Ω] + e [ J ∧ d Ω p K ] (cid:124) (cid:123)(cid:122) (cid:125) F JK = 0 . (3.41)By varying the action (3.38) with respect to Ω and taking into account the boundary conditions (3.40), we find thatthe deformed torsion vanishes T I = d Ω e I + 14 C KIJ e I ∧ e J = 0 , (3.42)where the structure constants C KIJ are given in (3.32). Finally, we should also consider the variations of p I . Takinginto account the mass shell condition (3.36), we obtain d Ω B I ∝ p I . (3.43)This condition is satisfied once we solve for (3.42). Indeed, (3.42) implies: d Ω B I = 12 1( d − ε IK ...K d − p ∧ e K ∧ . . . e K d − = d − p I d v. (3.44)As in three dimensions, the new variables deform the torsion two-form T I = d Ω e I , which is now sourced by thecosmological constant. In addition, we also see that a flat so (1 , d ) connection Ω IJ solves the curvature equation (3.41)provided d Ω p I = 0 . This observation will be relevant once we consider a lattice approach, where the field equationsare solved by imposing that the connection is piecewise flat.The nature of the boundary (spacelike, timelike or null) depends via the mass shell condition (3.36) on the sign of Λ . The following table summarises the situation for both Euclidean and Lorentzian signature in arbitrary dimensions. Euclidian LorentzianFlat:
Λ = 0 p I = 0 or p I is Grassmanian p I = 0 or p I is light-like AdS: Λ < p I is real p I is space-like or imaginary time-like dS: Λ > p I is imaginary p I is time-like or imaginary space-like IV. METRIC AND DEFORMED TELEPARALLEL FORMULATION IN THE NEW VARIABLESA. Recovering unimodular gravity
Generalized covariant derivative.
The Levi-Civita connection defines the unique torsionless covariant deriva-tive that is metric compatible. In the last section, we deformed the notion of torsion. Let us now introduce thecorresponding covariant derivative on the tangent bundle, i.e. the corresponding metric formulation. Provided e I ∧ · · · ∧ e I d (cid:54) = 0 . d -bein e Ia and given Lorentz vector p I , the field equation d Ω e I + 14 C JKI e J ∧ e K = T I + 14 C JKI e J ∧ e K = 0 (4.1)has a unique solution ◦ Ω IJa [ e, p ] for the connection Ω IJa in terms of e Ia and p I . If, in fact, ∇ a denotes the usualLevi-Civita metric compatible and torsionless covariant derivative on the tensor bundle, we find ◦ Ω IJa [ e, p ] = e Ib ∇ a e bJ + C IJK e Ka = e Ib ∇ a e bJ + C IJa . (4.2)Equation (4.2) suggest to introduce the difference tensor C abc = C IJK e Ia e Jb e Kc . (4.3)The corresponding metric compatible covariant derivative is defined for any smooth vector field V a ∈ T M via ◦ ∇ a V b = ∇ a V b + C bac V c . (4.4)This definition generalises naturally to arbitrary tensor fields. For any two such vector fields, we may then definecurvature and torsion of this new connection ◦ R cdab V d = ◦ ∇ a ◦ ∇ b V c − ◦ ∇ b ◦ ∇ a V c − ◦ T dab ◦ ∇ d V c , (4.5) ◦ T abc U b V c = U b ◦ ∇ b V a − V b ◦ ∇ b U a − [ U, V ] a . (4.6)Inserting (4.4) back into (4.6), we obtain the components of the torsion two-form, ◦ T abc = − C a [ bc ] , with C abc = 2 p [ a g b ] c , (4.7)which is, of course, the same as (4.1), but now written in the standard tensor language. Note that we can eitherinterpret the Christoffel symbol ◦ Γ abc = Γ abc + C abc as having torsion in the usual sense, or it has vanishing generalizedtorsion T Iab = 0 . Recovering unimodular gravity.
As emphasized for example in [30], the gravitational force can be encodedinto various geometric entities, namely an affine connection ˜Γ abc , non-metricity ˜ Q abc or torsion ˜ T abc . Our choiceof variables provides a connection ◦ Γ abc where ◦ Q abc = 0 but ◦ T abc (cid:54) = 0 . The corresponding second-order action isobtained by going half-shell , i.e. by reinserting the solution for the connection ◦ Ω I Ja in terms of p I and e Ia back intothe first-order action (3.37). A short calculation gives S EH [ g, p ] := S EC (cid:104) e, ◦ Ω[ e, p ] , p (cid:105) = 116 πG (cid:90) M d d v g (cid:16) g cb ◦ R acab [ g, p ] + ( d − ◦ ∇ a p a (cid:17) , (4.8)where d d v g is the usual metrical volume element. This action resembles the Henneaux-Teitelboim model for unimod-ular gravity [24]. In fact, to obtain the Einstein equations, this action is to be extremized in the space of all fields ( g ab , p c ) that satisfy the mass shell condition (3.36) that now simply reads g ab p a p b = − d − d − . (4.9)It seems now natural to relax this condition, namely by adding a Lagrange multiplier that will impose the mass-shell constraint at the dynamical level. At the same time, this also allows us to unfreeze Λ , obtaining a version ofunimodular gravity. Consider, in fact, a ( d − -form τ . Its exterior derivative defines a volume density ˜ N = d τ .Varying the action S unimod [ g, p, τ ] = 116 πG (cid:90) M (cid:20) d d v g (cid:16) g cb ◦ R acab [ g, p ] + ( d − ◦ ∇ a p a (cid:17) − g ab p a p b d τ (cid:21) (4.10)with respect to τ will tell us then that g ab p a p b is constant for some Λ , while all other variations return the usualEinstein equations with respect to the deformed connection ◦ ∇ a . In this way, τ provides a notion of time, p a plays therole of a space-time momentum and (cid:112) | Λ | represent the rest mass of p a . The volume element d d v g is the d -form, whose components are the Levi-Civita tensor, i.e. ( d d v g ) abc... = ε abc... . B. Deformed Teleparallel Gravity
The new variables deform curvature and torsion, see (3.17) and (3.18). The gravity action can be encoded in thecurvature when there is no torsion or vice-versa in the torsion if there is no curvature. The latter viewpoint underliesthe teleparallel framework. We want to see now how the new variables provide a non-abelian version of teleparallelism,where the translational connection takes values in an d − , whereas the Lorentzian part of the so (1 , d − (cid:46)(cid:47) an d − connection is flat.Consider a generic so (1 , d − (cid:46)(cid:47) an d − connection A . The space of such so (1 , d ) (cid:46)(cid:47) an d − connections is an affinespace. Since it is an affine space, any such connection can be written as a sum of some arbitrary reference connection • A and a difference tensor ∆ , A = • A − ∆ . (4.11)With this decomposition, we have F [ A ] = • F − • D ∆ + 12 [ ∆ ∧ ∆ ] , (4.12)where • F is the curvature of • A and • D ( · ) = d ( · ) + [ • A , · ] is the covariant derivative as in (3.23), (3.20) above.The deformed an d − Lie-algebra valued equivalent of teleparallel gravity corresponds to the following choice ofreference connection, • A = 12 • Ω IJ ⊗ J IJ + e I ⊗ P I , (4.13)where J IJ and P I are the generators of so (1 , d − (cid:46)(cid:47) an d − with commutation relations (3.21a, 3.21b, 3.21c). Noticethat A and • A have the same projection onto an d . Therefore ∆ is a pure rotation, ∆ = 12 ∆ IJ ⊗ J IJ . (4.14)We are now left to specify the reference connection • Ω IJ , and we choose it to satisfy the following two conditions R IJ [ • Ω] = d • Ω IJ + • Ω IK ∧ • Ω KJ = 0 , (4.15) d Ω p I = d p I − • Ω JI p J = 0 . (4.16)Notice that there always exists a Lorentz transformation Λ I J such that p I = Λ J I r J , where r J is a constant vector,i.e. d r J = 0 . We can then choose • Ω IJ = (Λ − dΛ) I J . (4.17)which will then solve both (4.15) and (4.16). Therefore, the curvature of the so (1 , d − (cid:46)(cid:47) an d − connection hasonly a an d − part, • F = 12 • F IJ ⊗ J IJ + • T I ⊗ P I = • T I ⊗ P I , (4.18)where we defined the an d Lie algebra-valued torsion two-form • T I = d • Ω e I + 14 C IJK e J ∧ e K ≡ • ∇ e I + 14 C IJK e J ∧ e K . (4.19)If the field equation (3.42) is satisfied, i.e. by going half-shell , we can express the components of the difference tensor ∆ IJ = ∆ IJK e K in terms of the components of the deformed an d Lie algebra-valued torsion, ∆ IJK = − (cid:16) • T IJK + • T JKI − • T KIJ (cid:17) . (4.20)3Inserting this solution back into the first-order action (3.38), we obtain S EC (cid:104) e, • Ω + ∆ , p (cid:105) = 116 πG (cid:90) M (cid:20) B IJ ∧ R IJ [ • Ω] − ( − d − B IJK ∧ (d • Ω e K ) ∧ ∆ IJ − B IJ ∧ ∆ IL ∧ ∆ LJ − ( d − − d − B I ∧ d • Ω p I + ( d − − d − B I p J ∧ ∆ IJ (cid:21) − πG (cid:90) ∂ M ( − d − B IJ ∧ ∆ IJ = 116 πG (cid:90) M (cid:20) B IJ ∧ F IJ [ • Ω] − ( − d − B IJK ∧ • T K ∧ ∆ IJ − B IJ ∧ ∆ IL ∧ ∆ LJ − B IJ ∧ e I ∧ d • Ω p J + 12 ( − d − B IJK ∧ p ∧ e K ∧ ∆ IJ (4.21) − ( d − − d − B I ∧ d • Ω p I + ( d − − d − B I p J ∧ ∆ IJ (cid:21) − πG (cid:90) ∂ M ( − d − B IJ ∧ ∆ IJ , (4.22)where we performed a partial integration and used the definition of • F = • F IJ ⊗ J IJ + • T I ⊗ P I for a generic so (1 , d − (cid:46)(cid:47) an d − connection • A , see also (3.17) and (3.18). In addition, B I ...I n := 1( d − n )! ε I ...I n K ...K d − n e K ∧ · · · ∧ e K d − n . (4.23)If we then also use the algebraic identity n ! B I ...I n ∧ e J ∧ . . . e J n = ( − n ( d − n ) d d v δ [ J I . . . δ J n ] I n , (4.24)we can write the action (4.22) in terms of contractions of • T IKJ alone. A straight-forward calculation gives S EC (cid:104) e, • Ω + ∆ , p (cid:105) = 116 πG (cid:90) M d d v (cid:20) • F IJIJ − e aI • ∇ a p I + 12 • T KIJ • T IJK − • T KIJ • T IJK + • T JIJ • T IKK (cid:21) (4.25) − πG (cid:90) ∂ M ( − d − B IJ ∧ ∆ IJ , (4.26)where • F IJKL are the components of the curvature two-form • F IJ = • F IJKL e K ∧ e L .Since we can always find a reference connection • Ω IJ such that • ∇ a p I = 0 and R IJ [ • Ω] = 0 , we can now introducethe following teleparallel action, which is quadratic in the an d − Lie algebra-valued field strength • T I , S tele [ e, • Ω , p ] = 116 πG (cid:90) M d d v (cid:20) • T KIJ • T IJK − • T KIJ • T IJK + • T JIJ • T IKK (cid:21) . (4.27) V. OUTLOOK
We have shown in this paper how the canonical transformation induced by the boundary volume term was similar tothe canonical transformation considered in [1] to recover the notion of quantum group for gravity in three-dimensions.The canonical transformation results essentially in a shift of the connection by the frame field. A IJ → Ω IJ = A IJ + e [ I p J ] , (5.1)where p is either interpreted as the normal to the boundary or as a constant vector. In both cases, the norm of p isconstrained to be proportional to the cosmological constant. This is true in any dimension higher or equal than three.In two dimensions, the frame field does not appear in the B -field, so the shift cannot be done by such boundary term.This can also be seen from the normalization condition p = − d − d − .4 S EC [ e, A, p ] = 116 πG !" M B IJ [ e ] ∧ R IJ [ A ] − d ! ε I ...I d e I ∧· · ·∧ e I d $ − d − " ∂ M p J ε JI ...I d − e I ∧· · ·∧ e I d − % S tele [ e, • Ω , p ] = 116 πG " M d d v ! • T KIJ • T IJK − • T KIJ • T KIJ + • T JIJ • T KIK & S EC [ e, Ω , p ] = 116 πG " M ’ B IJ ∧ R IJ [Ω] − ( d − − d − B I ∧ d Ω p I ( S EH [ g, p ] = 116 πG " M d d v g ’ g cb ◦ R acab [ g, p ] + ( d − ◦ ∇ a p a ( integrating out localLorentz transformations introducing a flatreference connection • Ω IJ shift: Ω IJ = A IJ + p [ I e J ] Figure 1: Bief summary of the paper: First of all, we performed a change of variables going from the spin connection toa shifted connection Ω . The new connection Ω depends on a boundary field p , whose norm determines the cosmologicalconstant. Integrating out the internal Lorentz transformations, we obtain a new second-order metric formulation, whichresembles unimodular gravity (the norm of p a is determined by the cosmological constant). A non-abelian version of tele-parallel gravity is found by introducing a specific flat reference connection • Ω with d • Ω p I = 0 . We would like to emphasize that the change of variables can also be performed when
Λ = 0 . In this case, p will be anull vector in the Lorentzian case. In the Euclidean case, we can use a Grassmanian number θ to parametrize p suchthat θ = 0 .We identified the two associated second-order theories, namely the metric and the teleparallel formulations, see thefigure above. The metric formalism associated to the new variables can be seen as a new version of the Henneaux–Teitelboim model for unimodular gravity. We note that the first-order action (3.38) can also be interpreted as afirst-order formulation for unimodular gravity, since it can be supplemented with the normalizing constraint of thevector p . S EC [ e, Ω , p ] = 116 πG (cid:90) M (cid:16)(cid:16) d − ε IJK ··· K d − e K ∧ · · · ∧ e K d − (cid:17) ∧ R IJ [Ω] − ( d − − d − ( d − (cid:16) ε IK ...K d − e K ∧ . . . e K d − (cid:17) ∧ d Ω p I − g ab p a p b d τ (cid:17) (5.2)As in the metric formulation, variation with respect to τ leads to the norm of p being constant, hence recovering thecosmological constant as a constant of integration. The teleparallel formulation, on the other hand, is now expressedin terms of a non-abelian field strength (4.19) that enters the action quadratically (4.27). With respect to the usualabelian teleparallel formulation, the abelian Lie algebra R d is deformed into the non-abelian Lie algebra an d − .The results of this paper open a number of interesting new directions to explore. The boundary volume term plays animportant role in the holographic renormalization context [26]. It would be interesting to see whether the formulationusing the new variables can also be useful, in particular its connection with unimodular gravity.The boundary term is the key to understand why quantum groups appear in three dimensions [1]. In the four-dimensional context it has been conjectured that they should also be relevant [21–23]. It would then be important tostudy how the charge algebra is affected by adding this term to the theory. In particular, in the list of works [4–6],it was always assumed that Λ = 0 . It would be worthwhile to study how these results are affected if Λ (cid:54) = 0 and theboundary volume term is also considered.The non-abelian formulation of teleparallel gravity is likely to be the classical continuum counterpart of the (de-formed) dual BF vacuum [31]. It would be interesting to construct the discrete picture which leads upon quantizationto such deformed dual BF vacuum, generalizing [32, 33] This is work in progress.The origin of this work came from studying three-dimensional gravity. Recently a most general bulk action forthree dimensional gravity was introduced [34]. It would be interesting to see how the change of variables could be ofuse there, either for discretizing or in recovering the different second order formalisms.5Finally, the canonical transformation is implemented for a constant homogenous curvature, parameterized by thecosmological constant. One might wonder whether we could generalize the construction in the case of a varyingcurvature. We leave this intriguing question for later investigations. [1] M. Dupuis, L. Freidel, F. Girelli, A. Osumanu, and J. Rennert, “On the origin of the quantum group symmetry in 3dquantum gravity,” arXiv:2006.10105 .[2] L. Freidel, K. Krasnov, and R. Puzio, “BF description of higher dimensional gravity theories,” Adv. Theor. Math. Phys. (1999) 1289–1324, arXiv:hep-th/9901069 .[3] L. Freidel and A. Perez, “Quantum gravity at the corner,” Universe (2018), no. 10, 107, arXiv:1507.02573 .[4] L. Freidel, M. Geiller, and D. Pranzetti, “Edge modes of gravity. Part I. Corner potentials and charges,” JHEP (2020)026, arXiv:2006.12527 .[5] L. Freidel, M. Geiller, and D. Pranzetti, “Edge modes of gravity. Part II. Corner metric and Lorentz charges,” JHEP (2020) 027, arXiv:2007.03563 .[6] L. Freidel, M. Geiller, and D. Pranzetti, “Edge modes of gravity - III: Corner simplicity constraints,” arXiv:2007.12635 .[7] E. De Paoli and S. Speziale, “A gauge-invariant symplectic potential for tetrad general relativity,” JHEP (2018) 040, arXiv:1804.09685 .[8] R. Oliveri and S. Speziale, “Boundary effects in General Relativity with tetrad variables,” arXiv:1912.01016 .[9] W. Wieland, “Null infinity as an open Hamiltonian system,” arXiv:2012.01889 .[10] W. Wieland, “New boundary variables for classical and quantum gravity on a null surface,” Class. Quantum Grav. (2017) 215008, arXiv:1704.07391 .[11] W. Wieland, “Fock representation of gravitational boundary modes and the discreteness of the area spectrum,” Ann.Henri Poincaré (2017) 3695–3717, arXiv:1706.00479 .[12] W. Wieland, “Conformal boundary conditions, loop gravity and the continuum,” JHEP (2018) 089, arXiv:1804.08643 .[13] R. Arnowitt, S. Deser, and C. Misner, The dynamics of general relativity , ch. 7, pp. 227–264. Wiley, New York, 1962. arXiv:gr-qc/0405109v1 .[14] C. Kiefer,
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