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Dimensional Reduction and Unification Schemes in First Order Formalism

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Dimensional Reduction and Unification Schemes in First Order Formalism Dimensional reduction and unication schemes in rst order formalism D. Flores-Alfonso1, R.P. Martinez-y-Romero2, A. Much1, L. Patiño2 and H. Quevedo1;3;4 1Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, AP 70543, Ciudad de México 04510, Mexico 2Departamento de Física, Facutad de Ciencias, Universidad Nacional Autónoma de México, AP 70542, Ciudad de México 04510, Mexico 3Dipartimento di Fisica and ICRA, Università di Roma La Sapienza", I-00185 Roma, Italy 4Department of Theoretical and Nuclear Physics, Kazakh National University, Almaty 050040, Kazakhstan Abstract We present a formalism for dimensional reduction where both, the gravitational connection and the gauge potentials in our spacetime are part of a higher dimensional spin connection which, along with the vielbein, accounts for the fundamental degrees of freedom. In this work we focus on the particular case where we recover Einstein-Maxwell theory in a 4-dimensional spacetime from Einstein-Gauss-Bonnet theory of gravity in a 6-dimensional one. To achieve these results, and in agreement with current observations where no higher dimensions have been observed, we consider the 4D spacetime as a totally geodesic submanifold of the higher dimensional Riemannian manifold. In this rst approach, we also require that the higher dimensional metric depends only on the 4D spacetime coordinates, expecting that the removal of this restriction, within the context of our formalism, should lead to richer physics than those reported here. 1 I. INTRODUCTION The idea of using higher dimensional theories to unify the interactions we observe in four dimensions was rst proposed by Kaluza and Klein [1, 2]. Although the quantization of the Kaluza-Klein theory leads to results which are not compatible with observations, the classical counterpart is considered today as one of the precursors to string theory [3], one of the most serious candidates to unify the four interactions of Nature. Several unication theories have been proposed in the past decades, but none of them has reached the main goal, which consists in explaining both the classical and quantum aspects of the interactions we observe in Nature. Kaluza-Klein reduction relates elements of the metric in the higher dimensional theory directly with gauge potentials in the lower dimensional one. To work towards a quantum description of dimensional reduction, our perspective is that since a gauge boson is repre- sented by a connection from the lower dimensional perspective, it should be encoded in an object that is also a connection in the context of the higher dimensional theory. Our point of view ts perfectly in the context of the recent proposal presented in [4], where a dierent unication scheme for gravity and gauge theories was introduced. The starting point is a 4-dimensional spacetime which is endowed with a higher dimensional tan- gent space, whose symmetries are represented by the SO(1; 13) Lorentz group. A particular mechanism is then used to break this tangent group into SO(1; 3) and SO(10). The spin connection associated with the original group SO(1; 13) is shown to contain the full infor- mation about the spacetime connection and all the gauge potentials, under the assumption that the metricity condition is satised. The action of the general theory can be written in terms of the curvature invariants of the tangent group that contains as a particular case the Yang-Mills action for gauge elds. In this work, we propose an alternative approach in which the 4-dimensional spacetime is considered as a totally geodesic submanifold of a higher D−dimensional manifold. This assumption is based upon our current observations of the Universe in which only four di- mensions have been detected. Indeed, a totally geodesic manifold can be understood as containing all the geodesics that begin tangent to it, implying that no observer on this manifold could establish whether it is a submanifold of a higher dimensional manifold. The Kaluza-Klein approach assumes that D = 5 and proposes an ad hoc metric which contains 2 the electromagnetic potential explicitly. The result is an electromagnetic theory which is not compatible with observations. Here, we choose D = 6, and do not assume any par- ticular form for the metric. Instead, we assume the validity of the most general gravity theory which is described by second-order dierential equations, i.e., Lovelock gravity which in this particular case is described by the Einstein-Gauss-Bonnet action. We then study the properties of the corresponding 4-dimensional totally geodesic submanifold, and show that the reduced action describes pure Einstein-Maxwell theory when the higher dimensional metric[7] depends only on the 4D coordinates. This work is organized as follows. In Sec. II, we review the main properties of totally geodesic Riemannian submanifolds. In Sec. III, we study the particular case in which the 4- dimensional spacetime can be considered as a totally geodesic submanifold of a 6-dimensional pseudo-Riemannian manifold whose metric depends explicitly only on the spacetime coordi- nates. The properties of the corresponding connections and curvature tensors are analyzed in detail. In Sect. IV, we start from the 6-dimensional Lovelock gravity theory, and show that its projection on a 4-dimensional spacetime, which is a totally geodesic submanifold, leads to an eective Einstein-Maxwell theory with cosmological constant. In Sec. V, we discuss our results and comment on future works. II. TOTALLY GEODESIC SUBMANIFOLDS Consider a Riemannian manifold M¯ with Levi-Civita connection r¯ , and an isometric embedding M ! M¯ such that the restriction of r¯ on M coincides with the Levi-Civita connection r of M. We will use bared lowercase letters for indices on M¯ , lowercase letters for M, and uppercase letters for the complement of M in M¯ . So, for instance, a vector eld ¯ ¯ ¯ a¯ ¯ µ¯ X on M may be written as X ea¯ in a local vielbein fea¯g, or as X @µ¯ in local coordinates. For two vector elds X¯ and Y¯ on M¯ , we have that ¯ ¯ ¯ µ¯ ¯ ν¯ ν¯ ¯ λ¯ (1) rX¯ Y = X @µ¯Y + Γ µ¯λ¯Y @ν¯ ; where ν¯ are the Christoel symbols. Then, for arbitrary vectors , we obtain Γ µ¯ρ¯ X; Y 2 TM ¯ µ ν ν λ µ ν Λ rX Y = X @µY + Γ µλY @ν + X Y Γ µν@Λ = rX Y + BXY ; (2) 3 ¯ where BXY is the second fundamental form. If M is a totally geodesic submanifold of M, then for any and , i.e., Λ . Accordingly, this is equivalent to BXY = 0 X Y Γ µν = 0 ¯ rX Y = rX Y 2 TM: (3) Some equivalent conditions for M to be a totally geodesic submanifold of M¯ are: 1. For arbitrary vectors X; Y 2 TM, the shape tensor Π vanishes identically ¯ Π(X; Y ) = rX Y − rX Y = 0 : (4) 2. Every geodesic of M is also a geodesic of M¯ . ¯ 3. For v 2 Tp(M) tangent to M, then the geodesic γv lies initially in M. 4. If α is a curve in M and v 2 Tα(0)(M), then the parallel transport of v along α is the same for M and M¯ . The tangent space over M splits orthogonally as ¯ T MjM = TM ⊕ NM: (5) Observers in M¯ do not detect any curvature of M, i.e., M is extrinsically at. The manifolds M¯ and M share the same intrinsic curvature. Totally geodesic submanifolds are the simplest Riemannian submanifolds. They are special classes of minimal submanifolds, and generalize the notion of geodesics to higher dimensions. Clearly, totally geodesic submanifolds of dimension one are just geodesics. Some of the components of the Riemann tensor µ¯ are trivial, if a totally geodesic R ν¯λ¯τ¯ submanifold exists. For instance, the components Λ Λ Λ Λ λ¯ Λ λ¯ (6) R ρµν = @µΓ νρ − @νΓ µρ + Γ µλ¯Γ νρ − Γ νλ¯Γ µρ = 0; due to the vanishing of the Christoel symbols Λ Γ µν = 0: III. A TOTALLY GEODESIC SPACETIME Let M be a 4-dimensional totally geodesic submanifold of a 6-dimensional pseudo- Riemannian manifold M¯ . The metric on M¯ may be written as µ¯ ν¯ a¯ ¯b (7) g = gµ¯ν¯dx ⊗ dx = ηa¯¯bθ ⊗ θ ; 4 where a¯ a¯ µ¯ (8) θ = e µ¯dx ; is a vielbein in M¯ . We adopt a notation in which a¯, ¯b range from 0 to 5. The tetrad indices in M are denoted by a, b = 0; 1; 2; 3 and the normal directions are denoted by A; B = 4; 5. Coordinate indices are denoted as before with lowercase and uppercase Greek letters. The main assumption is that the geometry of M does not depend on points outside M, i.e., a¯ . This implies, in particular, that the only non-zero components of the spin @Ae µ¯ = 0 connection are a a µ A A µ (9) ! b = ! bµdx ! B = ! Bµdx where we have used the fact that M is extrinsically at. In general, a¯ conform the connection one-form on ¯ , which is a valued dif- ! ¯b M SO(1; 5)− ferential form. However, when restricted to , it possesses only two split parts. The a M ! b part is an valued connection which describes the curvature of , while A is an SO(1; 3)− M ! B SO(2)−valued one-form on M. There is, therefore, an eective symmetry breaking of the form SO(1; 5) ! SO(1; 3) × SO(2) : (10) The SO(2) part of the connection can be associated with an electromagnetic potential, since SO(2) and U(1) are isomorphic groups. The curvature two-form over M¯ is dened by means of Cartan's second structure equation a¯ a¯ a¯ c¯ (11) Ω ¯b = d! ¯b + ! c¯ ^ ! ¯b ; which for the totally geodesic submanifold M yields a a a c¯ Ω b = d! b + ! c¯ ^ ! b a a c (12) = d! b + ! c ^ ! b whereas for the normal directions, we obtain A A A c¯ Ω B = d! B + ! c¯ ^ ! B A (13) = d! B : 5 Equations (12) and (13) imply that the only non-vanishing components of the curvature tensor are a and A (14) R bcd R Bcd : The rst term determines all the components of the curvature of M, and the second term is the curvature of the normal directions which has components along M only.
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