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Einsteinian Gravity from a Spontaneously Broken Topological BF Theory

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Einsteinian Gravity from a Spontaneously Broken Topological BF Theory Physics Letters B 688 (2010) 273–277 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Einsteinian gravity from a spontaneously broken topological BF theory Eckehard W. Mielke Departamento de Física, Universidad Autónoma Metropolitana–Iztapalapa, Apartado Postal 55-534, C.P. 09340, México, D.F., Mexico article info abstract Article history: Recently, (anti-)de Sitter gauge theories of gravity have been reconsidered. We generalize this to a Received 26 February 2010 metric-free sl(5, R) gauge framework and apply spontaneous symmetry breaking to the corresponding Accepted 14 April 2010 topological BF scheme. Effectively, we end-up with Einstein spaces with a tiny cosmological constant Editor: S. Dodelson related to the scale of symmetry breaking. An induced ‘background’ metric emerges from a Higgs-like mechanism. It is indicated how the finiteness of such a deformed topological scheme may convert into Keywords: Topological gravity asymptotic safeness after quantization of the spontaneously broken model. BF scheme © 2010 Elsevier B.V. All rights reserved. Spontaneous symmetry breaking Cosmological constant 1. Introduction Simons (CS) term. The topological BF scheme and its modification by a potential in B is exemplified for the Abelian case in Sec- In general relativity (GR), the metric tensor g occupies a dou- tion 3. It is generalized to a metric-free SL(5, R) model of gravity μν ˆ ble rôle: First, it determines distances in spacetime, as the metrical in Section 4. A symmetry breaking via a potential in B induces ‘arena’ for all other quantum fields. Second, it is a field in its own a non-trivial dynamics which is effectively equivalent to the stan- {} dard Einstein equations with cosmological constant. In Section 5 right, and serves as a potential for the Christoffel connection Γαβ of Riemannian geometry. a spontaneously symmetry breaking of the SL(5, R) gauge group Rather early there were attempts to alleviate this dichotomy, via a Higgs-like mechanism is analyzed from which emerges, at considering the gauge connection as primordial and the metric the same time, the spacetime metric as an induced concept. The as a derived concept as, for instance, by Eddington [7].Lateron, smallness of the symmetry breaking scale is in Section 6 tenta- the metric was tentatively considered as the gravitational analogue tively related to the renormalization flow of the running coupling [14] of the Higgs field. A more recent proposal is that of ’t Hooft constants. An Outlook concludes the Letter. ∗ 2 α β [55] by constructing an ‘alternative’ metric ds = oαβ Dξ ⊗s Dξ from a ‘quartet’ ξ α of scalar fields. In the Cartan formalism [4,16], 2. Topological invariants in gravity with induced metric a related coset field naturally arise in the affine gauge theory [26,32,53] after imposing locally the gauge condition of vanishing Group contraction of the meta-linear group SL(5, R) yields the ∗ translational connection. According to Trautman [51],sucha“gen- graded affine group A (4, R), with two Abelian subgroups instead eralized Higgs field” “hides” the action of local translations. of one. The origin of the two sets of “translations” can be exhib- Here we depart from a metric-free SL(5, R) topological model ited by rewriting the sl(5, R) algebra generating this group as the ∗ ∗ in which only the dimensionality of the gauge group is increased graded algebra a (4, R) = R4 ⊕ gl(4, R) ⊕ R 4. Although this de- but not that of four-dimensional (4D) spacetime. Then, an explicit composition seemingly looks like a generalization of the conformal ∗ symmetry breaking mechanism is constructed. group,theR4 and R 4 pieces cannot be identify with translations Consequently, our approach generalizes, to some extent, de Sit- and special conformal transformations, respectively. ter gauge models [21,50,54,9,43,22,52] of gravity which, however, Let us consider a sl(5, R)-valued i.e. tracefree connection on a are from the outset metric-contaminated due to Cartan–Killing four-dimensional manifold: metric gˆ AB of orthogonal groups. ˆ B A β α β 4 4 α The Letter is organized as follows: In the next section, the Γ := ΓA L B = Γα L β + Γ4 L β + Γα L 4, (1) metric-free gauge framework of the SL(5, R) group is set up and then applied to the decomposition of the corresponding Chern– where A, B,... run from 0 to 4. Similarly as for the affine connec- tion [32,11], the additional connection pieces are related via α = α − α 4 = − ∗ E-mail address: [email protected]. Γ4 ϑ Dξ ,Γβ θβ Dξβ , (2) 0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2010.04.030 274 E.W. Mielke / Physics Letters B 688 (2010) 273–277 α respectively, to two sets of soldering one-forms ϑ and θβ of identically. Consequently, the sl(5, R) CS term decomposes [11] fi- the spacetime manifold. The physical dimension of is [length], nally into later on dynamically related to the cosmological constant (31). ˆ α ∗ C = CRR − 2CTT, (10) The ‘would-be Goldstone bosons’ (octet of scalars) ξ and ξβ are affine vectors resembling the ‘radius vectors’ of Cartan and are ∗ where living on to the coset space SL(5, R)/GL(4, R) ≈ R4 ⊕ R 4 of the − sig graded affine group. This construction generalizes the anti-de Sit- 1 α β ( 1) ∗ CTT := gαβ ϑ ∧ T =− A (11) ter gauge model of Stelle and West [50,19], where the coframe 22 22 α ˚ αβ =− ˚ βα ϑ and the Lorentz connection Γ Γ can be derived from denotes the translational CS term of Nieh and Yan (NY) [39,38] and 1 the original SO(2, 3)-connection via a nonlinear realization in- ∗ α i A := (ϑα ∧ T ) = Ai dx the axial torsion one-form. volving Goldstone type ξ –fields, there parameterizing the coset Accordingly, the (anholonomic) spacetime metric gαβ is induced SO(2, 3)/SO(1, 3). by the reduction (10) of the SL(5, R) Chern–Simons term, indicat- α =∗ ∗ =∗ After imposing the covariant constraint Dξ 0 and Dξβ 0, ing a possible topological origin. and in view of the dimensionality 1/[length] of the generators Pα β and P∗ of pseudo-translations, the connection decomposes into 3. Modified BF scheme ˆ = β α + α + β Γ Γα L β ϑ Pα θβ P∗ , (3) The BF formalism for topological field theory is a rather new i.e., it becomes a Cartan connection [16,57]. development, although Plebanski [45] anticipated a related con- The SL(5, R) curvature takes the form straint formalism for gravity. In the Abelian case, the connection one-form A = A dxi and an auxiliary2 two-form B = 1 B dxi ∧ dx j i 2 ij ˆ B A B C B A are varied independently. In its primordial form it starts from the R := R A L B := dΓA − ΓA ∧ ΓC L B metric-independent Lagrangian four-form β 1 β α β α = Rα − θα ∧ ϑ L β + T Pβ + Dθα P∗ . (4) =− ∧ =− ∧ 2 LBF B F B dA. (12) = The corresponding SL(5, R) Chern–Simons (CS) term reads Independent variations with respect to A and B lead to dB 0 and the constraint of vanishing field strength F := dA = 0, implying ˆ 1 B A 2 B C A that such a topological model has no local degrees of freedom. C =− ΓA ∧ dΓB − ΓA ∧ ΓB ∧ ΓC . (5) 2 3 This pure BF system can be modified [13,3] viaatermquadratic in B: Not surprisingly, it contains the CS term ˜ 1 LBF := −B ∧ dA + B ∧ B 1 β α 1 β γ α 2 CRR := − Γα ∧ Rβ + Γα ∧ Γβ ∧ Γγ 2 3 =∼ − ∧ + B dA dC. (13) 1 β α 2 β γ α Now, independent variations provide the definition of the field =− Γα ∧ dΓβ − Γα ∧ Γβ ∧ Γγ , (6) 2 3 strength F together with the corresponding Bianchi identity ∼ ∼ built from the linear GL(4, R) gauge fields in 4D. In order to isolate B = dA := F , dB = dF ≡ 0, (14) the remaining terms, we perform an expansion and find respectively, in compliancy with the Poincaré lemma dd ≡ 0. It still ˆ 1 β α defines a topological theory since, ‘on shell’, Eq. (13) differs from C = CRR − ϑ ∧ dθβ + θα ∧ dϑ 22 (12) only by a boundary term dC derived from a CS three-form C. For an Abelian connection A, the CS term and the Pontrjagin in- 2 β α γ α β γ − Γα ∧ θβ ∧ ϑ + θα ∧ ϑ ∧ Γγ + ϑ ∧ Γβ ∧ θγ variant are simply given by 3 = 1 ∧ = 1 ∧ 1 β α α β C A F , dC F F (15) = CRR − ϑ ∧ dθβ + θα ∧ dϑ + 2θα ∧ Γβ ∧ ϑ 2 2 22 − 1 such that we end up, ‘on shell’, in (13) with dC. In general, it = C − ϑβ ∧ Dθ + θ ∧ T α . (7) is well known [10] that Bianchi type identities can be recovered RR 2 β α 2 via the variation of the associate Pontrjagin term, e.g. δ dC/δ A = Since the one–form ϑβ corresponds to the R4 part of the graded dF ≡ 0 in the Abelian case. affine algebra, its exterior derivative Since Bianchi identities do not allow for non-trivial couplings, α := α = α + α ∧ β in an “embedding” into realistic physical models such as Maxwell’s T Dϑ dϑ Γβ ϑ (8) theory or QCD, the corresponding BF Lagrangian denotes the torsion two-form. =− ∧ + 1 ∧ ∗ + The CS decomposition (7) would acquire the usual form, if we LMax B dA B B Lmatter (16) 2 identify θα, after some symmetry reduction specified later, with ∗ the soldering coframe ϑβ by means of necessarily involves the Hodge dual depending on the metric, though.
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