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Lovelock Gravity with Spontaneous Dimensional Breaking

We consider D-dimensional Lovelock gravity with only one term of higher-order Lovelock Lagrangian densities, and show that a product of Minkowski space-time and n-spheres is its vacuum solution. The most interesting feature of our model is that the spontaneous compactification of the extra dimensions results in reproduction of the Einstein gravity with no cosmological constant.


§1. Introduction
Lovelock gravity 1) is a natural extension of Einstein gravity to higher dimensions, and it is of great interest in theoretical physics as it describes a wide class of models. The equations of motion of Lovelock gravity do not contain more than second order derivatives with respect to metric, as in the case of general relativity.
Recently, Maeda and Dadhich 2) presented vacuum solutions of the form M m × H n in Gauss Bonnet gravity theory. Cai, Cao, and Ohta 3) generalized the solutions in Ddimensional Lovelock gravity. Here, M m is the m-dimensional manifold and H n is a negative constant curvature space.
In this paper, we consider D-dimensional Lovelock gravity with only one term of higherorder Lovelock Lagrangian densities, and show that a product of Minkowski space-time and n-spheres (positive constant curvature space) is its vacuum solution. This vacuum solution indicates the possibility that compactification of the extra dimensions takes place spontaneously. We show that the usual Einstein gravity with no cosmological constant is reproduced as a result of the spontaneous compactification.
To illustrate our idea, let us consider the simplest example. We take the following GB is the six-dimensional Gauss-Bonnet term (second order Lovelock Lagrangian density) L (6) GB ≡ R µνρσ R µνρσ − 4R µν R µν + R 2 . Let us compactify the six-dimensional spacetime to a product of a four-dimensional manifold and a two-sphere, that is, M D → M 4 ×S 2 .
By substituting R abcd = r 2 (η ac η bd −η ad η bc ) in the two-sphere part, eqn. (1 . 1) can be rewritten as where V (2) represents the volume of S 2 and we have used d 4 x √ −gL GB = 0. Note that we obtained the Einstein-Hilbert action and (1 . 2) does not include the cosmological constant.
This example could not reproduce the Standard Model, because it could only yield the gauge symmetry of SU(2), which comes from S 2 , when compactified down to four dimensions. In the next section, we generalize the above treatment to higher dimensions. §2. The D-dimensional model We consider the following Lagrangian The δ symbol denotes a totally anti-symmetric product of Kronecker deltas, normalized to take values 0 and ±1.
Varying the action (2 . 1), we obtain the equations of motion as where T µν is the energy-momentum tensor of the matter field derived from S matter and Let us compactify the D-dimensional space-time to a product of an m-dimensional manifold and an n-sphere, that is, M D → M m × S n . For S n , the Riemann curvature can be calculated as where r (n) is the constant radius of the n-sphere. We indicate the vectors in the mdimensional manifold by subscripted Roman letters i, j, k, etc. and the vectors in the n-sphere by subscripted Roman letters a, b, c, etc. Greek letters such as µ and ν are used where p − t ≥ 1, and we have used the identities for the Lovelock tensors: 2) and the identity where r denotes the range of the index (r = m for M m and r = n for S n ) and δ µ 1 ···µ p−2 ν 1 ···ν p−2 ≡ 1 for p = 2. The other components (such as G where p − t ≥ 1. Let us consider the most important case M 4 × S n . By substituting m = 4 in (2 . 6), (2 . 8), and (2 . 12), we obtain (2 . 14) (2 . 15) Note that the usual Einstein gravity with no cosmological constant can be reproduced when For example, when M 13 → M 4 × S 9 (n = 9) and p = 5, by using (2 . 13) and (2 . 14), the equations of motion (2 . 3) can be decomposed as for the four-dimensional parts, and − δ a b R (4) + r 2 (9) L (4) for the nine-dimensional parts. By using (2 . 15), (2 . 1) can be rewritten as where V (n) is the volume of S n and κ 2 (4) ≡ r 8 Incidentally, the space-time M 4 × S 9 has a gauge symmetry of SO(10) which comes from S 9 . In the case of vacuum (T µν = 0), Minkowski space-time is a trivial solution of (2 . 17) and (2 . 18), namely, the product of a four-dimensional Minkowski and an n-sphere is a solution of (2 . 3). This vacuum solution indicates the possibility that compactification of the extra dimensions takes place spontaneously.
Because SO(3) = SU(2) and SO(6) = SU(4), the space-time M 4 × S 2 × S ′ 2 × S 5 has the gauge symmetry of SU(4) ⊗ SU(2) ⊗ SU(2) as the internal symmetry. First, we compactify M 13 to M 8 × S 5 . By using (2 . 6), the eight-dimensional parts of G (2)xy = 0, G (2) R (4) = 10κ 2 (4) T a b , In this paper, we considered D-dimensional Lovelock gravity with only one term of the higher-order Lovelock Lagrangian densities. We show that a product space of Minkowski space-time and n-spheres is its vacuum solution. Furthermore, we showed that the Einstein gravity with no cosmological constant can be obtained as a result of spontaneous compactification of the extra dimensions. It is remarkable that without introducing negative constant curvature spaces and adjusted coefficients, we were able to obtain the compactified vacuum solutions and the zero cosmological constant.