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 represented 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 tangent 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 information 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 dimensions 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 particular 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 coordinates. 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 ∇¯ , and an isometric embedding M → M¯ such that the restriction of ∇¯ on M coincides with the Levi-Civita connection ∇ 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 {ea¯}, or as X ∂µ¯ in local coordinates. For two vector elds X¯ and Y¯ on M¯ , we have that

¯ ¯ ¯ µ¯ ¯ ν¯ ν¯ ¯ λ¯ (1) ∇X¯ 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 ∈ TM

¯ µ ν ν λ µ ν Λ ∇X Y = X ∂µY + Γ µλY ∂ν + X Y Γ µν∂Λ

= ∇X 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

¯ ∇X Y = ∇X Y ∈ TM. (3)

Some equivalent conditions for M to be a totally geodesic submanifold of M¯ are:

1. For arbitrary vectors X,Y ∈ TM, the shape tensor Π vanishes identically

¯ Π(X,Y ) = ∇X Y − ∇X Y = 0 . (4)

2. Every geodesic of M is also a geodesic of M¯ .

¯ 3. For v ∈ Tp(M) tangent to M, then the geodesic γv lies initially in M.

4. If α is a curve in M and v ∈ 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 M|M = 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) ω b = ω bµdx ω B = ω 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- ω ¯b M SO(1, 5)− ferential form. However, when restricted to , it possesses only two split parts. The a M ω b part is an valued connection which describes the curvature of , while A is an SO(1, 3)− M ω 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 = dω ¯b + ω c¯ ∧ ω ¯b , which for the totally geodesic submanifold M yields

a a a c¯ Ω b = dω b + ω c¯ ∧ ω b a a c (12) = dω b + ω c ∧ ω b whereas for the normal directions, we obtain

A A A c¯ Ω B = dω B + ω c¯ ∧ ω B A (13) = dω 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.

IV. EFFECTIVE EINSTEIN-MAXWELL THEORY

We now assume that M¯ is the spacetime of a 6-dimensional gravitational eld described by the Lovelock action. Lovelock gravity [6] is the most general ghost-free metric theory of gravity yielding conserved second order equations of motion for any given dimension D. In dimensions three and four, it coincides with Einstein's gravity, whereas for higher dimensions it represents a generalization of Einstein's theory which can lead to dierent physical eects. For D = 5 and D = 6, it is dynamically equivalent to the Gauss-Bonnet gravity. We thus consider a 6-dimensional manifold M¯ in which a gravity theory is described by means of the Lagrangian density ¯ ¯ ¯ L = c0 + c1R + c2LGB , (15) where c0, c1 and c2 are coupling constants and bars make reference to 6-dimensional objects. Moreover, the Gauss-Bonnet Lagrangian density is dened as

2 ab abcd LGB = R − 4R Rab + R Rabcd . (16)

To compute the eective Lagrangian density dened on the totally geodesic submanifold M, whose geometric properties are induced from M¯ , we use the fact that, according to our starting assumption, the dependence of all the elds lies entirely within M. Then, the Lagrangian on M can be written as

ABcd L = c0 + c1R + c2LGB + c2R RABcd . (17)

Since the only non-zero components of the curvature tensor are a and A , the Ricci R bcd R Bcd scalars of M¯ and M are related by

¯ a¯¯b ab (18) R = R a¯¯b = R ab = R.

6 As for the second-order curvature terms, similar reductions can be performed. For instance, for the contraction of the Ricci tensor with itself, the last two indices must be lowercase, i.e., a¯¯b a¯ d¯¯b c¯ a d¯b c (19) Ra¯¯bR = R ¯ba¯c¯R d¯ = R ¯bacR d . Moreover, the rst two indices must be of the same type, i.e., both lowercase or both uppercase. Then, a¯¯b a db c ab (20) Ra¯¯bR = R bacR d = RabR . Finally, we analyze the Kretschmann scalar, and obtain

¯ a¯¯bc¯d¯ K = R Ra¯¯bc¯d¯

a¯¯bcd = R Ra¯¯bcd

abcd ABcd = R Rabcd + R RABcd = K + 2F 2 , (21) where we have dened 4 and 2 ab (22) Fab = R 5ab F = F Fab To nish, we notice that the Lagrangian density (17) contains a redundant term. The second order curvature terms, the Gauss-Bonnet contribution, correspond to a topological term, i.e., a boundary term that does not aect the dynamics of the elds. Consequently, using the results given in Eq.(22), the Lagrangian (17) can be written as

cd L = c0 + c1R + 2c2F Fcd . (23)

By choosing a negative Gauss-Bonnet coupling constant, we conclude that the resulting Lagrangian density determines an eective Einstein-Maxwell theory with cosmological term on M.

V. DISCUSSION

The central idea behind our approach is that not only the gravitational connection, but also the gauge bosons in our spacetime are part of a spin connection in a higher dimensional manifold of which we can only perceive four dimensions. In this approach the fundamental degrees of gravitational freedom are the vielbein and the connection.

7 With this in mind, we work out a particular and most simple example, where we assume that the 4-dimensional spacetime M is a totally geodesic submanifold of a 6-dimensional pseudo-Riemannian manifold M¯ . This implies that for an observer on M¯ , the spacetime submanifold is at, because its extrinsic curvature vanishes. On the other hand, the 4- dimensional spacetime contains all its geodesics, consequently no observer can detect the presence of the additional dimensions. This assumption is in agreement with current observations in which no higher dimensions have been reported so far. In addition, we assume that the 6-dimensional metric depends only on the spacetime coordinates. This simplify- ing assumption allows us to show that only a reduced number of curvature components is non-vanishing, but as we will mention below, it is not essential for the formalism. We then consider Lovelock's gravity theory on M¯ , and compute the eective theory that arises on the 4-dimensional spacetime. The Ricci scalar and the quadratic Ricci tensor scalar turn out to coincide on M¯ and M, due to the assumptions that M is a totally geodesic submanifold and that the metric on M¯ depends on the coordinates of M alone. As a result, we obtain that the eective theory on M is equivalent to the Einstein-Maxwell theory with cosmological term. Our results corroborate the importance of Lovelock gravity as the only higher dimensional theory which leads to conserved second order dierential equations, i.e., it is the most natural generalization of Einstein's gravity to higher dimensions. Moreover, our dimensional reduction approach serves as an alternative unication scheme in which a higher dimensional gravity theory contains Einstein's gravity and additional gauge elds in four dimensions, as long as spacetime is a totally geodesic submanifold. This motivates a straightforward extension of the simple case presented here, that is, to consider our spacetime as a totally geodesic submanifold of a 14-dimensional Riemannian one. In an identical way to the one shown in this work, the local SO(13 + 1) symmetry group will split as SO(3 + 1) ⊕ SO(10), where the rst component will encode the gravitational content, while the second will be a gauge group, opening the possibility of a more general unication scheme. A comment concerning the extra dimensions is that, for the conservative taste, they can be thought of merely as a tool for representing the gauge elds, depleted of any other meaning. From our presentation the only claim we can solidly make is that no trajectory can deviate from our spacetime to the extra dimensions nor the other way around. Up to this point, all we have said has to be understood in the context that we worked

8 on, where no matter elds are present. To go pass this limitation, we are currently working to determine if it is possible to accommodate in the right representations the additional eective elds that appear in 4D when relaxing the condition of the metric of the higher dimensional manifold to depend only on the spacetime coordinates. The vielbein plays a major roll in this extension.

VI. ACKNOWLEDGMENTS

This work has been partially supported by the UNAM-DGAPA-PAPIIT, Grant No. IN111617 and Grant No. IN113115.

[1] T. Kaluza, Zum Unitätsproblem in der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) 966 (1921). [2] O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie, Zeitschrift für Physik A 37, 895 (1926). [3] J. Polchinsky, String theory (Cambridge University Press, Cambridge, UK, 2005). [4] A. H. Chamseddine and V. Mukhanov, On unication of gravity and gauge interactions JHEP 020, 03 (2016). [5] B. O'Neill, Semi-Riemannian geometry with applications to relativity (Academic Press, New York, 1983). [6] D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12, 498 (1971). [7] Given our point of view, in which the degrees of gravitational freedom are in the vielbein and the connection, any time we mention the metric it should be understood as abusing the language to compactly refer to the information carried by both these objects.