Hamiltonian analysis of Einstein–Chern–Simons gravity

In this work we consider the construction of the Hamiltonian action for the transgressions ﬁeld theory. The subspace separation method for Chern–Simons Hamiltonian is built and used to ﬁnd the Hamiltonian for ﬁve-dimensional Einstein–Chern–Simons gravity. It is then shown that the Hamiltonian for Einstein gravity arises in the limit where the scale parameter l approaches zero.


Introduction
In the context of the general relativity 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 general relativity the geometry is dynamically determined. Therefore, the construction of a gauge theory of gravity requires an action that does not consider a fixed space-time background. An action for gravity fulfilling these conditions, albeit only in odd-dimensional spacetime, d = 2n + 1, was proposed long ago by Chamseddine [1,2], which is given by a Chern-Simons form for the anti-de Sitter (AdS) algebra. Chern-Simons gravities have been extensively studied; see, for instance, Refs. [3][4][5][6][7][8][9][10][11][12].
If Chern-Simons 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 general relativity.
Studies in this direction have been carried out in Refs. [13][14][15][16] (see also [17,18]). In these references it was found that standard, five-dimensional GR (without a cosmological constant) emerges as the → 0 limit of a CS theory for a certain Lie algebra B 5 . Here is a length scale, a coupling constant that characterizes different regimes within the theory. The B 5 algebra, on the other hand, is constructed from the AdS algebra and a particular semigroup by means of the S-expansion procedure introduced in Ref. [19].
Black hole type solutions and the cosmological nature of the corresponding fields equations satisfy the same property, namely, that standard black-holes solutions and standard cosmological solutions emerge as the → 0 limit of the black-holes and cosmological solutions of the Einstein-Chern-Simons field equations [14][15][16]. * Corresponding author.
The Einstein-Chern-Simons action was constructed using transgression forms and a method, known as subspace separation procedure [20]. This procedure is based on the iterative use of the Extended Cartan Homotopy Formula, and allows one to (i) systematically split the Lagrangian in order to appropriately reflect the subspaces structure of the gauge algebra, and (ii) separate the Lagrangian in bulk and boundary contributions.
However the Hamiltonian analysis of Einstein Chern-Simons gravity action as well as transgression forms is as far as we know an open problem.
In Ref. [21] was studied the Hamiltonian formulation of the Lanczos-Lovelock (LL) theory. The LL theory is the most general theory of gravity in d dimensions which leads to second-order field equations for the metric. The corresponding action, satisfying the criteria of general covariance and second-order field equations for d > 4 is a polynomial of degree [d/2] in the curvature, has [(d − 1)/2] free parameters, which are not fixed from first principles.
In Ref. [6] was shown, using the first order formalism, that requiring the theory to have the maximum possible number of degrees of freedom, fixes these parameters in terms of the gravitational and the cosmological constants. In odd dimensions, the Lagrangian is a Chern-Simons forms for the AdS group. The vielbein and the spin connection can be viewed as different components of an (A)dS or Poincare connection, so that its local symmetry is enlarged from Lorentz to (A)dS (or Poincare when = 0).
The principal motivation of this work is, using the first order formalism, find the Hamiltonian formalism for a Chern-Simons theory leading to general relativity in a certain limit.
In the first-order approach, the independent dynamical variables are the vielbein (e a ) and the spin connection (ω ab ), which obey first-order differential field equations. The standard secondorder form can be obtained if the torsion equations are solved for the connection and eliminated in favor of the vielbein-this step, however, cannot be taken, in general, because the equations for ω ab are not invertible for dimensions higher than four (for detail see Ref. [6]). This means it is not possible to find a Hamiltonian formulation of second order for a five-dimensional AdS as well as B 5 Chern-Simons theory.
The purpose of this work is to study (i) the Hamiltonian action for transgressions field theory, (ii) the subspace separation method in the context of Hamiltonian formalism, (iii) the Hamiltonian analysis of Einstein-Chern-Simons gravity, (iv) the relation between the Hamiltonian action of general relativity of Refs. [22,23] and the Hamiltonian action for Einstein-Chern-Simons gravity.
This paper is organized as follows: In Sec. 2 the Hamiltonian analysis of the five-dimensional Chern-Simons theory is briefly reviewed. The Hamiltonian analysis of the transgressions form Lagrangians is considered in Sec. 3 where the Extended Cartan Homotopy Formula is reviewed and used to find the triangle equation in its Hamiltonian form. In Sec. 4 the subspace separation method for Chern-Simons Hamiltonian is built and used to find the Hamiltonian for five-dimensional Einstein-Chern-Simons gravity. It is then shown that the Hamiltonian for Einstein gravity of Refs. [22,23] arises in the limit when the scale parameter l approaches zero.

Hamiltonian analysis of the five-dimensional Chern-Simons theory
In this Section we briefly review of the Hamiltonian analysis of Chern-Simons theory studied in Refs. [24][25][26].

(4 + 1)-dimensional case
The Chern-Simons action in 4 + 1 dimensions is given by μ T a is the gauge connection field, T a are de generators of the corresponding gauge group, · · · denote the symmetrized trace and M is an orientable 5-dimensional manifold on which the connection A is defined. If M has the topology M = R × where R can be considered as the temporal line and as a spatial section, then we can split the gauge field in time and space components A μ dx μ = Adx 0 + A i dx i (i = 1, 2, 3, 4). The action then takes the form where g abc = T a T b T c . Defining for convenience we arrive to Since the canonical momenta are given by we find that the primary constraints are and therefore the canonical Hamiltonian and the total Hamiltonian are then given by

Hamiltonian analysis of transgression field theory
In this section we consider the Hamiltonian analysis of transgression field theory introduced and studied in Refs. [27][28][29][30]. This results allow us to find the Hamiltonian triangular equation. This equation together with a method, known as subspace separation method, will be used to obtain the Chern-Simons Hamiltonian for the algebras so(4, 2) and B 5 .

(2 + 1)-dimensional case
The action for a 3-dimensional transgression gauge field theory (TGFT) is given by · · · denote the symmetrized trace and M is an orientable 3-dimensional manifold on which the connection A is defined. If M has the topology M = R × where R can be considered as the temporal line and as a spatial section, then we can split the gauge field in time and space components where the 3-dimensional TGFT Lagrangian is given by boundary term and D t i is the covariant derivatives for the A t connection.
From (7) we can see that when Ā = 0 we obtain the 3-dimensional Chern-Simons Lagrangian.
Using the definition of Hamiltonian we find From (9) we can see that when Ā = 0 we obtain the 3-dimensional Chern-Simons Hamiltonian of Refs. [25,26].

The 3-dimensional triangle equation
In the Lagrangian formalism the so called triangle equation is given by [27][28][29][30] This equation can be read off as saying that a transgression form "interpolating" between Ā and A may be written as the sum of two transgressions which introduce an intermediate, ancillary oneform Ã plus a total derivative. It is important to note here that Ã is completely arbitrary, and may be chosen according to convenience. From the triangle equation, and fixing the middle connection to zero (Ã = 0), we can see Using a method, known as subspace separation method, the equation (12) allows to construct the Chern-Simons Lagrangian for the B algebra, from which emerges as l → 0 the standard (2n + 1)-dimensional Lagrangian for general relativity.
In the case of the Hamiltonian formalism, triangular equation plays a similar role. The Hamiltonian triangular equation allows obtaining the Chern-Simons Hamiltonian for algebra B. This Hamiltonian empties into the Hamiltonian for five-dimensional relativity in the limit l → 0. This is not achievable in the case of the Hamiltonian for AdS algebra. Writing is the volume form) and integrating for n = 1 we have (2) [A μ ,Ā ν ]. (13) From (7) and (9) we can write where so that (13) takes the form which can be written as and (2) From here we can see that Eq. (17) is similar to Lagrangian triangular equation except for the term P (3) defined in equation (18).
Since this term depends on the speed and the Hamiltonian can not dependent on it, it is convenient to analyze the aforementioned term. In fact, from we can see that P (3) is a boundary term so that the triangular equation (17) can be written as where B contains (21). This result can be extended to the case where the intermediate connection Ã is nonzero. In this case (22) takes the form where the main difference with (22) resides in the fact that the term P (3) , which appears in B (3) [A μ , Ā μ , Ã μ ], contains two terms more than his counterpart (21).
The above results lead to show that the Hamiltonian transgression is the difference between two Chern-Simons Hamiltonians term plus a boundary term. The boundary term appears due to contributions of the Legendre transformation on the triangular Lagrangian equation. This result together to a method, known as subspace separation method, allows to construct the Chern-Simons Hamiltonian for an arbitrary Lie algebra.

(4 + 1)-dimensional case
In the (4 + 1)-dimensional case the action is given by Following the procedure of the above section we find so that Splitting the gauge field in time and space components where g abc = T a T b T c and B (5) (27) we can see that when Ā = 0 we obtain the 5-dimensional Chern-Simons Lagrangian.
Using the definition of Hamiltonian we find From (29) we can see that when Ā = 0 we obtain the

Five-dimensional triangular equation
From (11) we have that the 5-dimensional triangular equation is given by (4) [A μ ,Ā μ ]. (31) From (27) and (29) we can see that where P (5) so that (31) takes the form wherẽ P (5) and B (5) with Q (5) [A μ , Ā μ ] = κdQ (4) From here we can see that the Eq. (34) is similar to Lagrangian triangular equation except for the term, P (5) , defined in equation (35). Since this term depends on the velocity and the Hamiltonian can not dependent on it, it is convenient to analyze the aforementioned term. In fact, from (37) we have that the triangular equation is given by where B contains (37).
This result together with a procedure, known as subspace separation method, allows to construct the Chern-Simons Hamiltonian for the so(4, 2) and B 5 Lie algebra.
From (45) we can see that when Ā = 0 we find the (2n I.e., 3.6. The (2n + 1)-dimensional triangle equation Following the procedure used in the above sections we have From (40) and (45) we have where so that (48) takes the form where, This term is a boundary term, so that the triangular equation is finally given by

Hamiltonian for five-dimensional general relativity
The action for five-dimensional Einstein gravity is given by where M is an orientable 5-dimensional manifold. If M has the topology I × , then M can be foliated in four-dimensional Cauchy surfaces along I ∈ R, i.e., to carry out a separation (4 + 1). Since e a = e a μ dx μ and R ab = 1 2 R ab μν dx μ dx ν we have This means that (62) is the Hamiltonian for the five-dimensional general relativity in the first order formalism.

Hamiltonian for five-dimensional AdS Chern-Simons gravity
To construct the AdS Chern-Simons Hamiltonian we use the so called subspace separation method [20]. This procedure is based on the triangle equation (38), and embodies the following steps: i) Identify the relevant subspaces present in the so(4, 2) algebra, i.e., write g ii) Write the connections as a sum of pieces valued on every subspace, i.e., iii) The triangle equation (38) leads to To calculate the terms present in (67) we use the following information: • The non-vanishing components of the invariant tensor for fivedimensional AdS algebra are J ab J cd P e = 4 3 ε abcde .
• The connection A t μ and the tensors μ and F t μν are given by From this Hamiltonian it is apparent that neither the l → ∞ nor the l → 0 limit yields the Hamiltonian for general relativity.

Lagrangian Einstein-Chern-Simons gravity
In Ref. [13] was shown that the standard, five-dimensional general relativity can be obtained from Chern-Simons gravity theory for a certain Lie algebra B [13], whose generators { J ab , P a , Z ab , Z a } satisfy the commutation relations given in Eq. (7) of Ref. [14]. This algebra was obtained from the anti de Sitter (AdS) algebra and a particular semigroup S by means of the S-expansion procedure introduced in Ref. [19].
Following the dual procedure of S-expansion developed in Ref. [19] we find the five-dimensional Chern-Simons Lagrangian for the B algebra is given by L (5) EChS = α 1 l 2 ε abcde e a R bc R de + 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 + dB (4) EChS , where the surface term B and where α 1 , α 3 are parameters of the theory, l is a coupling constant, R ab = dω ab + ω a c ω cb corresponds to the curvature 2-form in the first-order formalism related to the 1-form spin connection, and e a , h a and k ab are others gauge fields presents in the theory.
The Lagrangian (73) shows that standard, five-dimensional General Relativity emerges as the l → 0 limit of a Chern-Simons theory for the generalized Poincaré algebra B. Here l is a length scale, a coupling constant that characterizes different regimes within the theory.

Hamiltonian for Einstein-Chern-Simons gravity
Now we will find the Hamiltonian for Einstein-Chern-Simons gravity, which is a Chern-Simons form for the so called B algebra.
Following the procedure of the above subsection, we have: i) The relevant subspace of the B algebra can be identify in the following form: ii) The connections A μ , Ā μ , Ā 1 μ , Ā 2 μ and Ã μ are: