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General Relativity As a Biconformal Gauge Theory

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General Relativity As a Biconformal Gauge Theory Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 943 (2019) 114624 www.elsevier.com/locate/nuclphysb General relativity as a biconformal gauge theory James T. Wheeler Utah State University, Department of Physics, 4415 Old Main Hill, Logan, UT 84322-4415, United States of America Received 25 August 2018; received in revised form 16 April 2019; accepted 21 April 2019 Available online 26 April 2019 Editor: Clay Córdova Abstract We consider the conformal group of a space of dim n = p + q, with SO(p, q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to a curved 2n-dim geometry admits an action functional lin- ear in the curvatures. Because symmetry is maintained between the translations and the special conformal transformations in the construction, these spaces are called biconformal; this same symmetry gives bicon- formal spaces overlapping structures with double field theories, including manifest T-duality. We establish that biconformal geometry is a form of double field theory, showing how general relativity with integrable local scale invariance arises from its field equations. While we discuss the relationship between biconfor- mal geometries and the double field theories of T-dual string theories, our principal interest is the study of the gravity theory. We show that vanishing torsion and vanishing co-torsion solutions to the field equations overconstrain the system, implying a trivial biconformal space. With co-torsion unconstrained, we show that (1) the torsion-free solutions are foliated by copies of an n-dim Lie group, (2) torsion-free solutions generically describe locally scale-covariant general relativity with symmetric, divergence-free sources on either the co-tangent bundle of n-dim (p, q)-spacetime or the torus of double field theory, and (3) torsion- free solutions admit a subclass of spacetimes with n-dim non-abelian Lie symmetry. These latter cases include the possibility of a gravity-electroweak unification. It is notable that the field equations reduce all curvature components to dependence only on the solder form of an n-dim Lagrangian submanifold, despite the increased number of curvature components and doubled number of initial independent variables. © 2019 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. E-mail address: [email protected]. https://doi.org/10.1016/j.nuclphysb.2019.114624 0550-3213/© 2019 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 J.T. Wheeler / Nuclear Physics B 943 (2019) 114624 1. Introduction 1.1. Biconformal spaces and the biconformal action It was shown in the 1950s and 1960s that general relativity may be cast as a Lorentz [1]or Poincaré [2] gauge theory. Subsequent approaches [3–9] refined the methods and extended the initial symmetry to Weyl, deSitter and conformal. A systematic approach to the resulting gauge theories of gravity shows that it is possible to formulate general relativity in several ways [10]. 1.1.1. Conformally based theories of gravity Generally, the use of conformal symmetries for gravity theories (and in the MacDowell- Mansouri case, de Sitter) leads to action functionals which are quadratic in the curvature and apply only to 4-dimensional spacetimes. This is because conformal scaling by eφ changes the volume form by enφ in n-dimensions. In four dimensions this factor may be offset by two factors of the curvature, but in even dimension n = 2k, we require k factors of the curvature to make the action dimensionless. Various techniques allow these quadratic theories to nonetheless re- duce to general relativity [4,8,9,11]. The most studied case is that of Weyl (conformal) gravity, in which the difficulty of higher order field equations has been alternatively exploited and over- come. When only the metric is varied, the field equations are fourth order, and include solutions not found in general relativity [12–14]. Mannheim [15] attempts to use the additional scaling properties to explain galactic rotation curves (but also see [16]). Alternatively, it has been shown in [17] that by reformulating Weyl gravity as a gauge theory and varying all of the gauge fields, the additional field equations give the integrability conditions needed to reduce the order and exactly reproduce locally scale covariant general relativity. Thereo are tw exceptions to these higher-order curvature requirements, which have actions lin- ear in the curvature and can be formulated in any dimension. In the first of the two approaches, Dirac [3] builds on previous work with scalar-tensor theories [18–21], achieving a curvature- linear action by including a scalar field to help offset the scaling of the volume form. Dirac considers a Weyl geometry in which the curvature is coupled to the Weyl vector and a scalar field. Generalizing his action functional to n-dimensions, it takes the form − αβ β 2( n 4 ) αμ βν S = κ2g R − κ n−2 g g αβ 2 αβ μν − √ 4(n 1) αβ 2n n − g D κD κ − λκ n−2 −gd x n − 2 α β =−n−2 = − where κ is a scalar field of conformal weight wκ 2 and αβ Wα,β Wβ,α is the di- latational curvature. The only occurrence of the Weyl vector is in the dilatational curvature, αβ , which must vanish or be unmeasurably small to avoid unphysical size changes. Dirac in- terpreted αβ as the electromagnetic field and the time dependence of the scalar field κ as a time-dependent gravitational constant. The electromagnetic interpretation is untenable, but solu- tions with αβ = 0or with other interpretations remain to be explored. The second curvature-linear action arises when the volume element is like that of a phase space, since the “momentum” directions have opposite conformal weight from the “space” di- mensions. Such a space arises naturally from the quotient of the conformal group by its Weyl subgroup (SO(p, q) and dilatations), with the most general curvature-linear action built from the SO(p, q) and dilatational curvatures [11] being J.T. Wheeler / Nuclear Physics B 943 (2019) 114624 3 = be···f a + a + a ∧ ∧ c ∧···∧ d ∧ ∧···∧ S eac···d α b βδ b γ e fb e e fe ff (1) a Here, b is the curvature of the SO(p, q) gauge field, is the dilatational curvature, and a α, β and γ are dimensionless constants. The differential forms e and fa are the gauge fields of translations and special conformal transformations, respectively. Together the latter give an orthonormal frame field on a 2n-dimensional manifold. Because of the symmetry maintained between the translations of the conformal group and the special conformal transformations, these spaces are called biconformal. Here we explore the large class of torsion-free biconformal spaces, showing that they reduce to general relativity on Lagrangian submanifolds. It is the consequences of the action, Eq. (1) that will occupy our present inquiry. 1.1.2. The biconformal gauging The construction of a biconformal space begins with a flat, n-dimensional space with SO(p, q) invariant metric, p + q = n. Compactifying this space by adding an appropriate point or null cones at infinity (See Appendix A) allows us to define its conformal group, SO(p + 1, q + 1). The biconformal quotient [8,9]is then SO(p+ 1, q + 1)/SO(p, q) × SO(1, 1), where SO(1, 1) transformations represent dilatations and the full subgroup W ≡ SO(p, q) × SO(1, 1) is the homogeneous Weyl group. The quotient gives rise to a principal fiber bundle with 2n di- mensional base manifold and homogeneous Weyl fibers. The connection of this flat biconformal space is then generalized, giving rise to conformal Lie algebra-valued 2-form curvatures. These are required to be horizontal and the resulting Cartan equations integrable. The 2n-dimensional curved base manifolds are biconformal spaces while local SO(p, q) and dilatational invariance remain, together comprising the biconformal bundle. Biconformal gravity is the gravity theory following from variation of the action, Eq. (1), with respect to each of the conformal gauge fields, together with the Cartan structure equations to define the curvatures in terms of the connection, and the generalized Bianchi identities arising as integrability conditions. The construction of these models is described in full detail in [22]. See also [8,9,11,23]. 1.2. Relationship to double field theory Biconformal spaces share many features in common with double field theories. Double field theory is a means of making the O(d, d) symmetry of T -duality manifest. By introducing scalars to produce an additional d dimension, Duff [27] doubled the X(σ, τ) string variables to make this O(d, d) symmetry manifest. Siegel brought the idea to full fruition by deriving results from superstring theory [24–26]. Allowing fields to depend on all 2d coordinates, Siegel introduced generalized Lie brackets, gauge transformations, covariant derivatives, and a section condition on the full doubled space, thereby introducing torsions and curvatures in addition to the manifest T-duality. There has been substantial subsequent development. Much of this development is reviewed in [28]; the introduction to [29]gives a concise summary. Briefly, double field theory arises by making T -duality manifest in string theory. When we compactify n dimensions on a torus, the windings of string about the torus can be interpreted as momenta. T -duality is a mapping between the original spatial directions and these momenta. Double field theory arises when these two n-spaces are kept present simultaneously, making T -duality manifest and leading to an overall O(n, n) symmetry.
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