Even-dimensional topological gravity from Chern-Simons gravity

It is shown that the topological action for gravity in 2n-dimensions can be obtained from the 2n+1-dimensional Chern-Simons gravity genuinely invariant under the Poincare group. The 2n-dimensional topological gravity is described by the dynamics of the boundary of a 2n+1-dimensional Chern-Simons gravity theory with suitable boundary conditions. The field $\phi^{a}$, which is necessary to construct this type of topological gravity in even dimensions, is identified with the coset field associated with the non-linear realizations of the Poincare group ISO(d-1,1).


I. INTRODUCTION
Twenty years ago A.H Chamseddine [1], [2] constructed topological actions for gravity in all dimensions. Chamseddine showed that the odd-dimensional theories are based on Chern-Simons forms with the gauge groups taken to be ISO(2n, 1) or SO(2n + 1, 1) or SO(2n, 2) depending on the sign of the cosmological constant. The use of the Chern-Simons form was essential so as to have a gauge invariant action without constraints.
The even-dimensional theories use, in addition to the gauge fields, a scalar multiplet in the fundamental representation of the gauge group. For even-dimensional spaces there is no natural geometric candidate such as the Chern-Simons form. The wedge product of n of the field strengths can make the required 2n-form in a 2n-dimensional space-time. The natural gauge group is ISO(2n − 1, 1) or SO(2n, 1) or SO(2n − 1, 2). To form a group invariant 2n-form, the n-product of the field strength is not enough, but will require in addition a scalar field φ a in the fundamental representation.
It is the purpose of this paper to show that the topological action for gravity in 2ndimensions can be obtained from the Chern-Simons gravity in (2n+ 1)-dimensions genuinely invariant under the Poincaré group. This Letter is organized as follows: In section 2 we shall review some aspects of (a) topological gravity [1], [2], (b) Lanczos-Lovelock gravity theory [6], [7], (c) the Stelle-West formalism [12] and of the Lanczos-Lovelock gravity theory genuinely invariant under the AdS group [9], [10], [11]. In section 3 it is shown that the topological action for gravity in 2n-dimensions, introduced in ref. [2] can be obtained from Chern-Simons gravity in (2n + 1)-dimensions genuinely invariant under the Poincaré group. Section 4 contains some comments and the conclusions.
The even-dimensional theories use, in addition to the gauge fields, a scalar multiplet in the fundamental representation of the gauge group. To form a group invariant 2n-form, the n-product of the field strength is not enough, but will require in addition a scalar field φ a in the fundamental representation. The 2n-dimensional action is then where F ab = dA ab + A ac A b c . This topological gravity has interesting applications, for example in 1+1-dimensions it allows one to describe Liouville's theory for gravity from a local Lagrangian.

B. Lovelock gravity theory
In Ref. [4], [5] was proved that the Lovelock lagrangian [6], [7] can be written as where α p are arbitrary constants and L (p) is given by L (p) = ε a 1 a 2 ······a d R a 1 a 2 ····R a 2p−1 a 2p e a 2p+1 · · · ·e a d with R ab = dω ab + ω a c ω cb . In ref. [8] was shown that requiring that the equations of motion uniquely determine the dynamics for as many components of the independent fields as possible fixes the α p coefficients in terms of the gravitational and cosmological constants. For d = 2n the coefficients are α p = α 0 (2γ) p n p , and the action (3) takes a Born-Infeld-like form. With these coefficients, the LL action is invariant only under local Lorentz rotations. For d = 2n − 1, the coefficients become where α 0 = κ dl d−1 , γ = −sgn(Λ) l 2 2 , and, for any dimension d, l is a length parameter related to the cosmological constant by Λ = ±(d − 1)(d − 2)/2l 2 . With these coefficients (4), the vielbein and the spin connection may be accommodated into a connection for the AdS group, allowing for the lagrangian (3) to become the Chern-Simons form in d = 2n + 1 dimensions, whose exterior derivative is the Euler topological invariant in d = 2n dimensions.

C. Lovelock gravity theory invariant under Poincaré group
In Refs. [9], [10], [11] it was shown that the Stelle-West formalism [12], which is an application of the theory of nonlinear realizations to gravity, permits constructing an action for Lanczos-Lovelock gravity theory genuinely invariant under the AdS group. In fact, a truly AdS-invariant action for even as well as for odd dimensions was constructed in ref.
[10] using the Stelle-West formalism [12] for non-linear gauge theories. The action for this theory is where Here φ a corresponds to the so-called "(A)dS coordinate" which parametrizes the coset space SOη (D + 1) /SO η (D), and z = φ/l. This coordinate carries no dynamics, as any value that we pick for it is equivalent to a gauge choice breaking the symmetry from (A)dS down to the Lorentz group. This is best seen in the light that the equations of motion for the action (5) are the same as those for ordinary LL gravity, with e a and ω ab replaced by V a and W ab .
The fields V a and W ab are called non-linear vielbein and spin connection, respectively, and they take up all the relevant information in the Stelle-West formalism.
From (5) we can see that, when one picks the physical gauge φ a = 0, the theory becomes indistinguishable from the usual one, and the AdS symmetry is broken down to the Lorentz group. However, a very interesting exception to this rule occurs in odd dimensions when the coefficients α p (4) are chosen. In this case, and for any value of φ a , it is possible to show that the Euler-Chern-Simons action written with e a and ω ab differs from that written with V a and W ab by a boundary term. As a matter of fact, the defining relation for the non-linear fields V a and W ab given in [12], represents a gauge transformation for the linear connection A = 1 2 iω ab J ab − ie a P a , which can be written in the form A →Ã = g −1 (d + A) g, where g = e −iφ a Pa andÃ = 1 2 iW ab J ab − iV a P a . This means that the linear and non-linear curvatures F = dA + A 2 andF = dÃ +Ã 2 are related byF = g −1 Fg. Just as the usual Euler-Chern-Simons lagrangian, the odd-dimensional non-linear lagrangian, with the special choice of coefficients given in eq. (4), satisfies dL and hence we see that both lagrangians may locally differ only by a total derivative. The same arguments lead to the conclusion that, in general, any Chern-Simons lagrangian written with non-linear fields, which is genuinelly invariant, differs from the usual one by a total derivative.
It is direct to show that in the limit l −→ ∞ we obtain a Lovelock gravity theory genuinelly invariant under the Poincaré group: where now V a = e a + D ω φ a = dφ a + ω a b φ b + e a and R ab = dω ab + ω a c ω cb , with W ab = ω ab , and φ a corresponds to the so-called "Poincaré coordinate". The fields φ a , e a , ω ab under local Poincaré translations change as δφ a = −ρ a ; δe a = κ a b e b ; δω ab = −Dκ ab .

III. TOPOLOGICAL GRAVITY FROM CHERN-SIMONS GRAVITY
In this section we show that the topological action for gravity in 2n-dimensions [2] can be obtained from (2n+1)-dimensional Chern-Simons gravity genuinelly invariant under the Poincaré group.

The Lanczos-Lovelock action genuinelly invariant under the Poincaré group is given by
Introducing the non-linear gauge fields V a into (8) we obtain where we have used the Bianchi identity DR ab = 0. So that From (11) we can see that the whole dependence of the coset field φ a is on the surface term and that the form of the surface term exactly coincides with the form of the evendimensional action for even-dimensional topological gravity. However, the surface term cannot be directly considered as an action principle for the boundary, because the dynamics of the boundary is determined by the dynamics of the Bulk. We will show that, for solutions with suitable boundary conditions, it is possible to obtain the dynamics for the even-dimensional topological action. It must be noticed that the coset field φ a , which is associated with a non-linear realization of the Poincaré group, appears in the action (11) in a geometrically natural form.

A. Invariance of the action
We show now that the action (11) we obtain where we have used the Bianchi identity DR ab = 0. This means that
From (18) we can see that the boundary condition associated with δφ a is identically satisfied due to the validity of equation (19)  into (18) we obtain the following condition: Therefore, the dynamics of the boundary will be characterized by the following set of equations: which can be obtained from the action principle This action correspond to topological gravity of ref. [2].
The condition e a | ∂M = 0 can be imposed due to the fact that e a is a part of a connection, namely A = i 2 ω ab J ab − ie a P a . To have vielbeins and therefore non-invertible metrics in the boundary of a manifolf means that the boundary of a manifold defines a singularity in the metric sector of the theory. However, from the point of view of the gauge structure, this does not represent any singularity due to the fact that the vielbein can be annulled because it is part of a gauge connection.
This shows that configurations with non invertible vielbeins can play an important role in the structure of the theory. Configurations with singularities on the boundary are not new (ref. [13]). They correspond to natural configurations of the gauge theories for gravity.
Finally it is interesting to notice that now the geometric origin of the φ a field is clear, due to the fact that this one is a coset field associated with non-linear realizations of the Poincaré group.

IV. COMMENTS
We have shown in this work that the topological action for gravity in 2n-dimensions, introduced in ref. [2], can be obtained from the Chern-Simons gravity in (2n + 1)-dimensions genuinely invariant under the Poincaré group. The 2n-dimensional topological gravity is described by the dynamics of the boundary of the (2n + 1) Chern-Simons gravity theory with suitable boundary conditions. The boundary of the manifold defines a singular hypersurface in the metric sector of the theory. The singularity appears only when we consider configurations with metric invertibles. However, this singularity is not an intrinsic singularity of the theory due to the fact that the vielbein is a part of a gauge connection.
The dynamics on the boundary of a (2n + 1)-dimensional manifold is described by the field equations of the topological gravity of ref. [2].
The field φ a , which is necessary to construct this type of topological gravity in even dimensions [2], is identified by the coset field associated with non-linear realizations of the Poincare group ISO(2n, 1). This shows a clear geometric interpretation of this field originally introduced "ad-hoc".