Topological gravity and Wess-Zumino-Witten term

It is shown that the action for topological gravity in even dimensions is, except by a multiplicative constant, a gauged Wess-Zumino-Witten Term.

The connection between this even-dimensional structure and the Chern-Simons gravity theories suggest that this mechanism could be regarded as an alternative to compactification or dimensional reduction.
On the other hand, in ref. [1], Chamseddine constructed topological actions for gravity in all dimensions. The odd-dimensional theories are based on the Chern-Simons forms. Evendimensional theories use, in addition to the gauge fields, a scalar field φ a in the fundamental representation of the gauge group.
In this work it is shown that the action for topological gravity in even dimensions found by Chamseddine in ref. [1] is a gWZW.
This article is organized as follows. In section II, we review some aspects of the topological gravity theory, the so called Stelle-West formalism and of the gWZW term. In section III, it is shown that the action for the topological gravity studied in ref. [1] corresponds to a gWZW term. Section IV concludes the work with some comments and conclusions. The details of some calculations are summarized in an Appendix A and B.  [5], [6]. These actions were constructed from the product of n field strengths, F ab , and a scalar field φ a in the fundamental representation of the gauge group.
In (1 + 1)-dimensions the action is given by [6] and in (2n − 1) + 1 dimensions the corresponding action can be written in the form [6], [1] where F ab = dA ab + A ac A b c and A is a one-form gauge connection. This action was obtained from a Chern-Simons form using a dimensional reduction method.
In ref. [21] a related approach to this problem was discussed. In this reference it was shown that the action (2) can be obtained from the (2n + 1)-dimensional Chern-Simons gravity genuinely invariant under the Poincaré group with suitable boundary conditions. Now we will show that the action (2) corresponds to a gWZW term.

B. The Stelle-West-formalism
The basic idea of the Stelle-West formalism is founded on the non-linear realizations studied in refs. [22], [23], [24]. Following these references, we consider a Lie group G and its stability subgroup H. The Lie group G has n generators. Let us call {X i } n−d i=1 the generators of H. We shall assume that the remaining generators {Y l } d l=1 are chosen so that they form a representation of H. In other words, the commutator [X i , Y l ] should be a linear combination of Y l alone. If the elements of G/H are denoted by z, and if the independent fields needed to parametrise z, are denoted by φ l , i.e., the φ l parametrize the coset space G/H, then a group element g ∈ G can be uniquely represented in the form g = zh where h is an element When G is the group associated to the AdS Lie algebra [P a , P b ] = m 2 J ab ; [J ab , P c ] = (η bc P a − η ac P b ) ; [J ab , J cd ] = (η ac J bd − η bc J ad + η bd J ac − η ad J bc ) whose generators are P a , J ab and if the subalgebra H is the Lorentz algebra SO(3, 1) whose generators are J ab , then [25] (see also [26], [27], [28], [29] Using the commutation relation of the AdS algebra, we find that the nonlinear fields V a and W ab are given by Taking the limit m → 0 in the commutation relation of the AdS Lie algebra and in (4) and (5) we find that the AdS Lie algebra takes the form of the Poincare Lie algebra where now the nonlinear fields are given by C. The gauged Wess-Zumino-Witten Term

Chern Simons Form and WZ Term
Consider the gauge transformed field [30] with V = dgg −1 and the transformed curvature where g = g(x) denotes the gauge element. Let us choose a homotopy such as with A t = tA, t ∈ [0, 1]. The corresponding homotopic curvature is with Both homotopies (9) and (10) interpolate continuously between A g t=1 = A g , F g t=1 = F g and ) Applying the Cartan homotopy formula (B7) to a Chern-Simons form(B13) containing the homotopies (9) and (10), we have From (B13) we can see that the gauge transformed Chern-Simons term is given by [30] Q 2n+1 (A g , F g ) = (n + 1) so that = (n + 1) Analogously, from (14) we can see that with so that To calculate (k 01 d + dk 01 ) Q 2n+1 (A g so that Defining the 2n-form we find that the Cartan homotopy formula (13) takes the form The second term in the r.h.s. of (24) corresponds to the so called Wess-Zumino term and since it represents a winding number, it will be total derivative, unless G has non-trivial homotopy group π 3 (G) and large gauge transformations are performed.
We now consider the terms Q 2n+1 (V, 0) and α 2n . From (16) which corresponds to the generalization of the Wess-Zumino term.

Cartan homotopy formula and transgression form
Applying the Cartan homotopy formula (B7) to the Chern-Simons form (B13) we have [30] where (27) we can see that where is known as the transgression form.
we find that (28) takes the form where

Gauged Wess-Zumino-Witten Term
If A 1 is related to A 0 by a gauge transformation and if A 1 , A 0 are denoted by A g , A respectively, we can write From (24) and (33) we can see that the transgression form for two gauge equivalent connections correspond to the gWZW term In the particular case n = 1, i.e., in the (2 + 1)−dimensional case, we have that α 2 takes the form where we have used (B1) and (B2). On the other hand, from equation (32) we see that A) is given by From (26)

III. TOPOLOGICAL GRAVITY AS A GAUGED WESS-ZUMINO-WITTEN TERM
In this section we show that even-dimensional topological gravity is a gWZW term.
A. Topological Gravity in (1 + 1)-dimensions From (3) we can see that the nonlinear and the linear connection, A Z and A = e + ω respectively, are related by a gauge transformation given by where z = e −φ a Pa and A Z = 1 2 W ab J ab + V a P a = V + W . This means that the linear and nonlinear curvatures F Z and F are related by In the (2 + 1)−dimensional case, the only non-vanishing component is given by so that the Wess-Zumino term (26) vanishes Hence, the gWZW term (37) takes the form and defines a Lagrangian in (1 + 1) −dimensions. Consider first the term VA which can be rewritten as where we have used the identity ε abc ω ab ω c d φ d = 2ε abc ω a d ω db φ c . On the other hand, the term A z A is given by Substituting (42) and (41) in (40) we obtain which proves that the action for Topological gravity in (1+1)-dimensions, found in ref. [1], [6], is a gWZW term given by,
If A z and A are given by A = e a P a + 1 2 ω ab J ab = e+ω and A Z = V a P a + 1 2 W ab J ab = V +W, where V a = e a + D ω φ a and W ab = ω ab , then Q 2n+1 (A Z , F Z ) is given by Introducing (48) and (49) in (47) we have (51) using the Bianchi identity DR ab = 0 we can write which proves that the action for Topological gravity in 2n−dimensions, found in Ref. [1], [6], is a gWZW term given by

IV. COMMENTS
We have shown in this work that the action for topological gravity in 2n-dimensions, introduced in ref.
[1] [6], is a gauged Wess-Zumino-Witten term. This means that the 2ndimensional topological gravity is described by the dynamics of the boundary of a (2n + 1) Chern-Simons gravity.
The field φ a , which is necessary to construct this type of topological gravity in even dimensions [1], 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 in an "ad-hoc" manner.
This work was supported in part by FONDECYT Grants N 0 1130653 and by Universidad  (A, A) on the other hand, is an invariant differential form whose exterior derivative is the difference of two Chern classes. It generalizes the Chern-Simons form and has the additional advantage that it is gauge invariant.
To obtain the Lagrangian for (2n + 1)-dimensional Chern-Simons gravity we use the so called Triangle equation [31] Q 2n+1 (A,Ā) = Q 2n+1 (A,Ã) − Q 2n+1 (Ā,Ã) − dQ 2n (A,Ā,Ã) withÃ = 0, and the method of separation in subspaces. Let us recall that the method of separation in subspaces consists of the following steps [14], [12]: The first step is to decompose the algebra into subspaces. In our case G = V 1 ⊕ V 2 , where V 1 corresponds to the Lorentz subalgebra generated by {J ab } and V 2 corresponds to the subspace spanned by {P a }. The second step is to write the connection as a sum of pieces valued in each subspace.
This means A = a 1 + a 2 , where a 1 = ω and a 2 = e. The third step is to use the triangular equation withÃ = 0,Ā = ω and A = ω + e.
which implies for any polynomial S in A t and F t Defining the homotopy operator by i.e., as the t-integrated version of the derivation l t , we find that integrating equation (B5) with respect to t we arrive at the Cartan homotopy formula [30] S(A 1 , F 1 ) − S(A 0 , F 0 ) = (k 01 d + dk 01 ) S(A t , F t ).
In the particular case where the arbitrary polynomial S(A t , F t ) is an invariant polynomial P n+1 (F t ) = F n+1 t we have dP n+1 (F t ) = 0 and the Cartan homotopy formula takes the form P n+1 (F 1 ) − P n+1 (F 0 ) = dk 01 P n+1 (F t ).
Since k 01 P n+1 (F t ) = 1 0 l t P n+1 (F t ) = (n + 1) we find that the Cartan homotopy formula supplies the Chern-Weil theorem: Finally, choosing the case A 0 = 0, A 1 = A the homotopy operator k 01 is usually denoted by k, i.e.,