Standard General Relativity from Chern-Simons Gravity

Chern-Simons models for gravity are interesting because they provide with a truly gauge-invariant action principle in the fiber-bundle sense. So far, their main drawback has largely been the perceived remoteness from standard General Relativity, based on the presence of higher powers of the curvature in the Lagrangian (except, remarkably, for three-dimensional spacetime). Here we report on a simple model that suggests a mechanism by which standard General Relativity in five-dimensional spacetime may indeed emerge at a special critical point in the space of couplings, where additional degrees of freedom and corresponding"anomalous"Gauss-Bonnet constraints drop out from the Chern-Simons action. To achieve this result, both the Lie algebra g and the symmetric g-invariant tensor that define the Chern-Simons Lagrangian are constructed by means of the Lie algebra S-expansion method with a suitable finite abelian semigroup S. The results are generalized to arbitrary odd dimensions, and the possible extension to the case of eleven-dimensional supergravity is briefly discussed.


Introduction
Three of the four fundamental forces of nature are consistently described by Yang-Mills (YM) quantum theories. Gravity, the fourth fundamental interaction, resists quantization in spite of General Relativity (GR) and YM theories having a similar geometrical foundation. There exists, however, a very important difference between YM theory and GR (for a thorough discussion, see, e.g., Ref. [1]).
YM theories rely heavily on the existence of the "stage" -the fixed, non-dynamical, background metric structure with which the spacetime manifold M is assumed to be endowed.
In GR the spacetime is a dynamical object which has independent degrees of freedom, and is governed by dynamical equations, namely the Einstein field equations. This means that in GR the geometry is dynamically determined. Therefore, the construction of a gauge theory of gravity requires an action that does not consider a fixed spacetime background.
If CS theories are to provide the appropriate gaugetheory framework for the gravitational interaction, then these theories must satisfy the correspondence principle, namely they must be related to GR.
An interesting research in this direction has been recently carried out [18,19]. In these references it was found that the modification of the CS theory for AdS gravity following the expansion method of Ref. [20] is not sufficient to produce a direct link with GR. In fact, it was shown that, although the action reduces to Einstein-Hilbert (EH) when the matter fields are switched off, the field equations do not. Indeed, the corresponding field equations impose severe restrictions on the geometry, which are so strong as to rule out, for instance, the five-dimensional Schwarzschild solution.
It is the purpose of this paper to show that standard, five-dimensional GR (without a cosmological constant) can be embedded in a CS theory for a certain Lie algebra B. The CS Lagrangian is built from a B-valued, one-form gauge connection A [cf. eq. (26)] which depends on a scale parameter ℓ-a coupling constant that characterizes different regimes within the theory. The B algebra, on the other hand, is constructed from the AdS algebra and a particular semigroup S by means of the S-expansion procedure introduced in Refs. [21,22]. The field content induced by B includes the vielbein e a , the spin connection ω ab and two extra bosonic fields h a and k ab . The full CS field equations impose severe restrictions on the geometry [18,19], which at a special critical point in the space of couplings (ℓ = 0) disappear to yield pure GR.
The paper is organized as follows. In Sec. 2 we briefly review CS AdS gravity. An explicit action for five-dimensional gravity is considered in Sec. 3, where the Lie algebra S-expansion procedure is used to obtain a B-invariant CS action that includes the coupling constant ℓ. It is then shown that the usual EH theory arises in the strict limit where the scale parameter ℓ equals zero. Sec. 4 concludes the work with a comment about possible developments.

Chern-Simons anti-de Sitter Gravity
The CS AdS Lagrangian for gravity in d = 2n + 1 dimensions is given by [2,3] where the c k constants are defined as e a corresponds to the one-form vielbein, and R ab = dω ab + ω a c ω cb to the Riemann curvature in the first-order formalism.
The Lagrangian (5) is off-shell invariant under the AdS Lie algebra so (2n, 2), whose generatorsJ ab of Lorentz transformations andP a of AdS boosts satisfy the commutation relations The Levi-Civita symbol ε a1···a2n+1 in (5) is to be regarded as the only non-vanishing component of the symmetric, so (2n, 2)-invariant tensor of rank r = n+1, namely In order to interpret the gauge field associated with a translational generatorP a as the vielbein, one is forced to introduce a length scale ℓ in the theory. To see why this happens, consider the following argument. Given that (i) the exterior derivative operator d = dx µ ∂ µ is dimensionless, and (ii) one can always choose Lie algebra generators T A to be dimensionless as well, the one-form connection fields A A = A A µ dx µ must also be dimensionless. However, the vielbein e a = e a µ dx µ must have dimensions of length if it is to be related to the spacetime metric g µν through the usual equation g µν = e a µ e b µ η ab . This means that the "true" gauge field must be of the form e a /ℓ, where ℓ is a length.
Therefore, following Refs. [2,3], the one-form gauge field A of the CS theory is given in this case by It is important to notice that once the length scale ℓ is brought in to the CS theory, the Lagrangian splits into several sectors, each one of them proportional to a different power of ℓ, as we can see directly in eq. (5).
CS gravity is a well-defined gauge theory, but the presence of higher powers of the curvature makes its dynamics very remote from that for standard EH gravity. In fact, it seems very difficult to recover EH dynamics from a pure gauge, off-shell invariant theory in odd 3 dimensions (see, e.g., Refs. [18,19]).

Einstein-Hilbert Action from five-dimensional Chern-Simons gravity
In this section we show how to recover five-dimensional GR from CS Gravity. The generalization to an arbitrary odd dimension is given in Appendix A.

S-Expansion Procedure
The Lagrangian for five-dimensional CS AdS gravity can be written as From this Lagrangian it is apparent that neither the ℓ → ∞ nor the ℓ → 0 limits yield the EH term ε abcde R ab e c e d e e alone. Rescaling κ properly, those limits will lead to either the Gauss-Bonnet term (Poincaré CS gravity) or the cosmological constant term by itself, respectively. The Lagrangian (12) is arrived at as the CS form for the AdS algebra in five dimensions. This algebra choice is crucial, since it permits the interpretation of the gauge fields e a and ω ab as the fünfbein and the spin connection, respectively. It is, however, not the only possible choice: as we explicitly show below, there exist other Lie algebras that also allow for a similar identification and lead to a CS Lagrangian that touches upon EH in a certain limit.
Following the definitions of Ref. [21], let us consider the S-expansion of the Lie algebra so (4, 2) using S E as the relevant finite abelian semigroup. After extracting a resonant subalgebra and performing its 0 S -reduction, one finds a new Lie algebra, call it B, with the desired properties. In simpler terms, consider the Lie algebra generated by {J ab , P a , Z ab , Z a }, where these new generators can be written as HereJ ab andP a correspond to the original generators of so (4, 2), and the λ α belong to a finite abelian semigroup. The semigroup elements {λ 0 , λ 1 , λ 2 , λ 3 , λ 4 } are not real numbers and they are dimensionless. In this particular case, they obey the multiplication law An explicit matrix representation for the λ α is given in Table 1.
Using Theorem VII.2 from Ref. [21], it is possible to show that the only non-vanishing components of a symmetric invariant tensor for the B algebra are given by E .
where α 1 and α 3 are arbitrary independent constants of dimension [length] −3 . In order to write down a CS Lagrangian for the B algebra, we start from the B-valued, one-form gauge connection and the associated two-form curvature Consistency with the dual procedure of S-expansion in terms of the Maurer-Cartan (MC) forms [22] demands that h a inherits units of length from the fünfbein; this is why it is necessary to introduce the ℓ parameter again, this time associated to h a .
It is interesting to observe that J ab are still Lorentz generators, but P a are no longer AdS boosts; in fact, we have [P a , P b ] = Z ab . However, e a still transforms as a vector under Lorentz transformations, as it must be in order to recover gravity in this scheme.

The Lagrangian
Using the extended Cartan homotopy formula as in Ref. [24], and integrating by parts, it is possible to write down the CS Lagrangian in five dimensions for the B algebra as L (5) CS = α 1 ℓ 2 ε abcde R ab R cd e e + α 3 ε abcde 2 3 R ab e c e d e e + +2ℓ 2 k ab R cd T e + ℓ 2 R ab R cd h e .
Two important points can now be made: 1. The Lagrangian (28) is split in two independent pieces, one proportional to α 1 and the other proportional to α 3 . The piece proportional to α 1 corresponds to thė Inönü-Wigner contraction of the Lagrangian (12), and therefore it is the CS Lagrangian for the Poincaré Lie group ISO (4, 1). The piece proportional to α 3 contains the EH term ε abcde R ab e c e d e e plus non-linear couplings between the curvature and the bosonic "matter" fields k ab and h a . These couplings are all proportional to ℓ 2 . 2. When the constant α 1 vanishes, the Lagrangian (28) almost exactly matches the one given in Ref. [18], the only difference being that in our case the coupling constant ℓ 2 appears explicitly in the last two terms. This difference has its origin in the fact that, in Ref. [18], both the symmetry and the Lagrangian arise through the process of Lie algebra expansion (see Ref. [20]), using 1/ℓ as an expansion parameter. In contrast, no parameter has been used here to create the new B-symmetry and the Lagrangian. Instead, they were constructed through the S-expansion procedure, using the dimensionless elements of a finite abelian semigroup (which in general cannot be represented by real numbers, but rather by matrices).
The presence or absence of the coupling constant ℓ in the Lagrangian (28) may seem like a minor or trivial matter, but it is not. As the authors of Ref. [18] clearly state, the presence of the EH term in this kind of action does not guarantee that the dynamics will be that of GR. In general, extra constraints on the geometry appear, even around a "vacuum" solution with k ab = 0, h a = 0. In fact, the variation of the Lagrangian, modulo boundary terms, can be written as Therefore, when α 1 vanishes, the torsionless condition is imposed, and a solution without matter (k ab = 0, h a = 0) is singled out, we are left with δL (5) CS = 2α 3 ε abcde R ab e c e d δe e + α 3 ℓ 2 ε abcde R ab R cd δh e .
In this way, besides the GR equations of motion [first term in (29)], the equations of motion of pure Gauss-Bonnet theory [second term in (29)] also in general appear as an anomalous constraint on the geometry. It is at this point where the presence of the ℓ parameter makes the difference. In the present approach, it plays the rôle of a coupling constant between geometry and "matter." Remarkably, in the strict limit where the coupling constant ℓ equals zero, we obtain solely the EH term in the Lagrangian In the same way, in the limit where ℓ = 0 the extra constraints just vanish, and δL It is interesting to observe that the argument given here is not just a five-dimensional accident. In every odd dimension, it is possible to perform the S-expansion in the way sketched here, take the vanishing coupling constant limit ℓ = 0 and recover EH gravity (see Appendix A).

Comments and Possible Developments
The present work shows the difference between the possibilities of the S-expansion procedure [21,22] (using semigroups) and the MC forms expansion (using a parameter).
The S-expansion procedure allows us to study in a deeper way the rôle of the ℓ parameter. In fact, it makes possible to recover odd-dimensional EH gravity from a CS theory in the strict limit where the coupling constant ℓ equals zero while keeping the effective Newton's constant fixed. It is only at this point (ℓ = 0) in the space of couplings that the "anomalous" Gauss-Bonnet constraints disappear from the on-shell system. This is in strong contrast with the standard CS AdS gravity [2,3] or the result of expansion using a real parameter [18,19].
The system of extra constraints on the geometry arises for any finite value of the scale parameter (coupling constant ℓ = 0). In other words, for ℓ = 0 the system has to obey Einstein's equations plus a set of on-shell Gauss-Bonnet constraints. In this way, GR corresponds to a special critical point, ℓ = 0, in the space of couplings of the CS gravitational theory.
The simple model and procedure considered here could play an important rôle in the context of supergravity in higher dimensions. In fact, it seems likely that it might be possible to recover the standard eleven-dimensional Cremmer-Julia-Scherk Supergravity from a CS/transgression form principle, in a way reminiscent to the one shown here. In this way, the procedure sketched here could provide us with valuable information on what the underlying geometric structure of Supergravity and M theory in d = 11 could be. at the MPI für Gravitationsphysik in Golm, where part of this work was done. He is also grateful to the Deutscher Akademischer Austauschdienst (DAAD), Germany, and the Comisión Nacional de Investigación Científica y Tecnológica (CONICYT), Chile, for financial support. P. S. was supported by Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT, Chile) Grants 1080530 and 1070306 and by the Universidad de Concepción, Chile, through DIUC Grants 208.011.048 -1.0. F. I. was supported by FONDECYT Grant 11080200, the Vicerrectoría de Asuntos Internacionales y Cooperació of the Universitat de València, Spain, and the Dirección de Perfeccionamiento y Postgrado of the Universidad Católica de la Santísima Concepción, Chile. E. R. was supported by FONDECYT Grant 11080156.

A. Extension to Higher Odd Dimensions
The CS AdS Lagrangian for gravity in d = 2n + 1 dimensions is given by [cf. eq. (5)] where the c k constants are defined as e a corresponds to the one-form vielbein, and R ab = dω ab + ω a c ω cb to the Riemann curvature in the first-order formalism.
Simple inspection of (31) shows that neither the ℓ → ∞ nor the ℓ → 0 limits produce EH gravity.