Chern-Simons--Antoniadis-Savvidy forms and standard supergravity

TL;DR

This paper constructs a four-dimensional supergravity action based on ChSAS forms and Maxwell superalgebra, introducing a method to decompose the action into bulk and boundary parts reflecting the algebra's structure.

Contribution

It develops a novel approach using FDA decomposition and the Extended Cartan Homotopy Formula to analyze ChSAS supergravity actions and their subspace structures.

Findings

Successfully constructed a supergravity action for Maxwell superalgebra.

Provided a method to separate bulk and boundary contributions in ChSAS actions.

Demonstrated splitting of the Lagrangian according to algebraic subspaces.

Abstract

In the context of the so called the Chern--Simons--Antoniadis--Savvidy (ChSAS) forms, we use the methods for FDA decomposition in 1-forms to construct a four-dimensional ChSAS supergravity action for the Maxwell superalgebra. On the another hand, we use the Extended Cartan Homotopy Formula to find a method that allows the separation of the ChSAS action into bulk and boundary contributions and permits the splitting of the bulk Lagrangian into pieces that reflect the particular subspace structure of the gauge algebra.

Equations

$$A=A_{\mu }\otimes \mathrm{d}{x}^{\mu }=A^a{}_{\mu }{T}_a\otimes \mathrm{d}{x}^{\mu }.$$

$$F=\mathrm{d}A+A^2,H=\mathrm{D}B=\mathrm{d}B+[A,B].$$

$$\mathrm{D}F=0,\mathrm{D}H+[B,F]=0.$$

$$\delta A=\mathrm{D}{\xi }_0,\delta B=\mathrm{D}{\xi }_1+[B,{\xi }_0],$$

$$\delta F$$

$$\delta H$$

$$\left[X,Y\right]=XY-{\left(-1\right)}^{pq}YX,$$

$${\Gamma }_{2n+3}=⟨F^{n}H⟩.$$

$$F_t=\mathrm{d}{A}_t+A_t^2,H_t={\mathrm{D}}_t{B}_t=\mathrm{d}{B}_t+[A_t,B_t],$$

$$\frac{\mathrm{d}}{\mathrm{d}t}{F}_t={\mathrm{D}}_t\Theta ,\frac{\mathrm{d}}{\mathrm{d}t}{H}_t={\mathrm{D}}_t\Phi +\left[\Theta ,B_t\right].$$

$${\Gamma }_{2n+3}^{(1)}-{\Gamma }_{2n+3}^{(0)}=⟨F_1^n{H}_1⟩-⟨F_0^n{H}_0⟩=\mathrm{d}{\mathfrak{T}}^{\left(2n+2\right)}(A_0,B_0;A_1,B_1),$$

$${\mathfrak{T}}^{\left(2n+2\right)}(A_0,B_0;A_1,B_1)={\int }_0^1\mathrm{d}t\left(n⟨F^{n-1}\Theta H_t⟩+⟨F_t^n\Phi ⟩\right)$$

$${\mathfrak{C}}_{\mathrm{ChSAS}}^{\left(2n+2\right)}={\mathfrak{T}}^{\left(2n+2\right)}(A,B;0,0)={\int }_0^1\mathrm{d}t⟨nA{F}_t^{n-1}{H}_t+B{F}_t^n⟩.$$

$${\mathfrak{C}}_{\mathrm{ChSAS}}^{(4)}={\int }_0^1\mathrm{d}t⟨A{H}_t+F_{t}B⟩=⟨FB⟩-\mathrm{d}⟨AB⟩.$$

$$\left[J_{ab},J_{cd}\right]$$

$$\left[J_{ab},P_c\right]$$

$$\left[J_{ab},Z_{cd}\right]$$

$$\left[P_a,Q_{\alpha }\right]$$

$$\left[J_{ab},{\Sigma }_{\alpha }\right]$$

$$\left\{Q_{\alpha },Q_{\beta }\right\}$$

$$\left\{Q_{\alpha },{\Sigma }_{\beta }\right\}$$

$$\left[J_{ab},{\tilde{Z}}_{cd}\right]$$

$$\left[{\tilde{Z}}_{ab},{\tilde{Z}}_{cd}\right]$$

$$=0.$$

$$A=\frac{1}{\ell }{e}^a{P}_a+\frac{1}{2}{\omega }^{ab}{J}_{ab}+\frac{1}{2}{k}^{ab}{Z}_{ab}+\frac{1}{2}{\tilde{k}}^{ab}{\tilde{Z}}_{ab}+\frac{1}{\sqrt{\ell }}{\psi }^{\alpha }{Q}_{\alpha }+\frac{1}{\sqrt{\ell }}{\xi }^{\alpha }{\Sigma }_{\alpha },$$

$$F=\frac{1}{\ell }{\hat{T}}^a{P}_a+\frac{1}{2}{R}^{ab}{J}_{ab}+\frac{1}{2}{f}^{ab}{Z}_{ab}+\frac{1}{2}{\tilde{f}}^{ab}{\tilde{Z}}_{ab}+\frac{1}{\sqrt{\ell }}{\Psi }^{\alpha }{Q}_{\alpha }+\frac{1}{\sqrt{\ell }}{\Xi }^{\beta }{\Sigma }_{\beta },$$

$${\hat{T}}^a$$

$$R^{ab}$$

$$f^{ab}$$

$${\tilde{f}}^{ab}$$

$${\Psi }^{\alpha }$$

$${\Xi }^{\beta }$$

$$B=B^a{P}_a+\frac{1}{2}{B}^{ab}{J}_{ab}+\frac{1}{2}{\beta }^{ab}{Z}_{ab}+\frac{1}{2}{\tilde{\beta }}^{ab}{\tilde{Z}}_{ab}+{\lambda }^{\alpha }{Q}_{\alpha }+{\chi }^{\alpha }{\Sigma }_{\alpha },$$

$$H=H^a{P}_a+\frac{1}{2}{H}^{ab}{J}_{ab}+\frac{1}{2}{\Theta }^{ab}{Z}_{ab}+\frac{1}{2}{\tilde{\Theta }}^{ab}{\tilde{Z}}_{ab}+{\tilde{H}}^{\alpha }{Q}_{\alpha }+{\mathcal{H}}^{\alpha }{\Sigma }_{\alpha },$$

$$H^a$$

$$H^{ab}$$

$${\Theta }^{ab}$$

$${\tilde{\Theta }}^{ab}$$

$${\tilde{H}}^{\alpha }$$

$${\mathcal{H}}^{\alpha }$$

Full text

I Introduction

In Refs. savv6 ; savv7 ; savv8 ; savv9 Antoniadis, Konitopoulos and Savvidy introduced a procedure to construct gauge invariant, background-free gauge forms.  The integrals of these forms over the corresponding space-time coordinates provides new topological actions  that we have called Chern–Simons–Antoniadis–Savvidy (ChSAS) actions, which generalize the usual Chern–Simons theory. Of special interest are those which can be constructed in even dimensions.

Using calculation methods for Free Differential Algebra (FDA) that allow the decomposition of $p$-forms in 1-forms developed in Ref. dauria and applied in Refs. azcar ; salg ; ims , we construct a four-dimensional ChSAS supergravity action for Maxwell superalgebra.

Following Ref. irs  it is found, in the context of the so called the ChSAS theory,  a subspace separation method for the Lagrangian. The method is based on the iterative use of the generalized Extended Cartan Homotopy Formula, and allows one to separate the action in bulk and boundary contributions, and systematically split the Lagrangian in appropriate reflection of the subspace structure of the gauge algebra. In order to apply the method, one must regard ChSAS forms as a particular case of more general objects known as generalized transgression forms.

This work is organized as follows: in Section 2, we briefly review the principal aspects of transgression and Chern–Simons–Antoniadis–Savvidy forms. In section 3, we use the calculation methods for Free Differential Algebra (FDA) that allow the decomposition of $p$-forms in 1-forms developed in Ref. dauria and applied in Refs. azcar ; salg ; ims to construct a four-dimensional Chern–Simons–Antoniadis–Savvidy supergravity action for Maxwell superalgebra. Section 4 presents the generalized Extended Cartan Homotopy Formula and shows how a subspace separation method that allows for a deeper understanding of the ChSAS Lagrangian can be built upon it. We finish in Section 5 with conclusions and some considerations on future possible developments.

II **Chern–Simons–Antoniadis–Savvidy forms **in $\left(2n+2\right)$-dimensions

The idea of extending the Yang–Mills fields to higher rank tensor gauge fields was used in Ref. savv6 in order to construct gauge invariant and metric independent forms in higher dimensions. These forms are analogous to the Pontryagin–Chern forms in Yang–Mills gauge theory. These results were generalized in Refs. savv7 ; savv8 ; savv9 , where the authors found closed invariant forms similar to the Pontryagin–Chern forms in non-abelian tensor gauge field theory. These forms are based on non-abelian tensor gauge fields and are polynomials on the corresponding curvature forms.

A Lie algebra valued $1$-form connection $A$ can be written making more or less explicit the dependence on the Lie algebra generator basis $T_{a}$ or the basis of 1-forms $\mathrm{d}x^{\mu}$,

[TABLE]

The same is true for the $2$-form $B$ gauge potential $B=\frac{1}{2}B_{\mu\nu}\otimes\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\frac{1}{2}B^{a}{}_{\mu\nu}T_{a}\otimes\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$. The corresponding 2-form and 3-form “curvatures” are given by $F=\frac{1}{2!}F_{\mu\nu}\otimes\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$ and $H=\frac{1}{3!}H_{\mu\nu\lambda}\otimes\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}\mathrm{d}x^{\lambda}$ respectively, where

[TABLE]

The curvatures $F$ and $H$ satisfy the Bianchi identities

[TABLE]

The infinitesimal, non-abelian gauge transformations for the generalized gauge fields are given by

[TABLE]

where $\xi_{0}$ and $\xi_{1}$ are a [math]-form and a $1$-form gauge parameters respectively savv6 . Under these gauge transformations, the curvatures transform as savv7 ,

[TABLE]

It may be of interest to note that we have used the definition of the conmutator for differential forms given by

[TABLE]

where $X$ is a $p$-form and $Y$ is a $q$-form.

In Refs. savv6 ; savv7 there were found closed invariant forms similar to the Pontryagin–Chern forms in non-abelian tensor gauge field theory.  In particular, it was found that there exists a gauge invariant metric-independent invariant in $\left(2n+3\right)$-dimensional space-time

[TABLE]

Chern–Weil theorem in the $\left(2n+2\right)$-dimensional case: the theorem ingredients are: $(i)$ Two Lie-algebra valued, $1$-forms connections $A_{0}$ and $A_{1}$. Their curvatures are given by, $F_{0}=\mathrm{d}A_{0}+A_{0}^{2}$ and $F_{1}=\mathrm{d}A_{1}+A_{1}^{2}$, respectively.$\ (ii)$ Two Lie-algebra valued, generalized $2$-forms gauge fields $B_{0}$ and $B_{1}$. Their generalized curvatures are given by $H_{0}=\mathrm{d}B_{0}+[A_{0},B_{0}]$ and $H_{1}=\mathrm{d}B_{1}+[A_{1},B_{1}]$ respectively $(iii)$ In terms of these fundamental ingredients, it is possible to define the differences \Theta=A_{1}-A_{0}\and $\Phi=B_{1}-B_{0}$, and the interpolating connections $A_{t}=A_{0}+t\Theta$ and $B_{t}=B_{0}+t\Phi$ with $0\leq t\leq 1$**. **Their corresponding curvatures are given by

[TABLE]

which satisfy the conditions

[TABLE]

Theorem ims : Let $A_{0}$ and $A_{1}$ be two gauge connection $1$-forms, and let $F_{0}$ and $F_{1}$ be their corresponding $2$-forms curvature.  Let $B_{0}$ and $B_{1}$ be two gauge connection 2-forms and let $H_{0}$ and $H_{1}$ be their corresponding curvature $3$-forms. Then, the difference $\Gamma_{2n+3}^{(1)}-\Gamma_{2n+3}^{(0)}$ is an exact form

[TABLE]

where

[TABLE]

is what we call “Antoniadis–Savvidy transgression form”. A proof can be found in Ref. ims . Following the same procedure followed in the case of the Chern–Simons forms, we define the $\left(2n+2\right)$-Chern–Simons–Antoniadis–Savvidy form as

[TABLE]

This result agrees with the expression found by Antoniadis and Savvidy in Refs. savv6 ; savv7 .  From eq. **(11), **we have for the $n=1$ case savv7 ,

[TABLE]

It is interesting to notice that transgression forms (both, standard ones and the above generalization) are defined globally on the spacetime basis manifold of the principal bundle, and are off-shell gauge invariant. Chern–Simons forms (both, standard ones and the Antoniadis–Savvidy generalization) are locally defined and are off-shell gauge invariant only up to boundary terms (i.e., quasi-invariants).

III **Chern–Simons–Antoniadis–Savvidy form for Maxwell

superalgebra**

Now we will use this construction for the particular case of the Maxwell superalgebra, in order to show the connection between eq. (12) and supergravity in $D=4$.

III.1 $\mathit{sM}_{4}$ Maxwell superalgebra

The minimal Maxwell superalgebra $\mathit{sM}_{4}$ in $D=4$ is an algebra whose generators $\{P_{a},J_{ab},Z_{ab},\tilde{Z}_{ab},Q_{\alpha},\Sigma_{\alpha}\}$ satisfy the following commutation relation eve1 ; azc

[TABLE]

This algebra can be found by an $\mathrm{S}$-expansion of $\mathfrak{osp}(4/1)$ superalgebra.** eve1 ; azc **.

In order to write down a four dimensional ChSAS** **action, we start by expressing the gauge fields $A$ and $B$ at the base of Maxwell superalgebra.

To interpret the gauge field associated with a traslational generator $P_{a}$ as the vielbein, one is forced to introduce a length scale $\ell$ in the theory. Since one can always chooses Lie algebra generators $T_{A}$ to be dimensionless as well, the one-form connection fields $A=A_{\mu}^{A}T_{A}\mathrm{d}x^{\mu}$ must also be dimensionless. However, the vielbein $e^{a}=e_{\text{ \ }\mu}^{a}\mathrm{d}x^{\mu}$ must have dimensions of length if it is related to the spacetime metric $g_{\mu\nu}$ through the usual equation $g_{\mu\nu}=e_{\text{ \ }\mu}^{a}e_{\text{ \ }\nu}^{b}\eta_{ab}.$ This means that the “true” gauge field must be of the form $e^{a}/\ell$, with $\ell$ a length parameter. Therefore, following Refs. Cha89 , Cha90 , the one-form gauge field $A$ is given by

[TABLE]

where $e^{a}$ is identified as the $1$**-form vierbein, $\omega^{ab}$ is the $1$-form spin **connection, $k^{ab}$and $\tilde{k}^{ab}$ are extra antisymmetric bosonic $1$-form fields, and $\psi^{\alpha}$, $\xi^{\alpha}$ are fermionic $1$-form fields. The corresponding $2$-form curvature is given by

[TABLE]

with

[TABLE]

For the $2$-form $B$, we can write

[TABLE]

where $B^{a},\;B^{ab},\;\beta^{ab},$ $\tilde{\beta}^{ab},$ $\lambda^{\alpha},$ $\chi^{\alpha}$ are $2$-forms that we must determine. The corresponding** $3$**-form curvature is given by

[TABLE]

with,

[TABLE]

The problem now is to express the form $B$ defined by the equations  (13-17) in terms of the one-forms $\{e^{a},\omega^{ab},k^{ab},\tilde{k}^{ab},\psi^{\alpha},\xi^{\alpha}\}$ of the Maxwell superalgebra. To express the $2$-forms $\left\{B^{a},B^{ab},\beta^{ab},\tilde{\beta}^{ab},\lambda,\chi\right\}$ as the wedge product of the $1$-forms $\{e^{a},\omega^{ab},k^{ab},\tilde{k}^{ab},\psi^{\alpha},\xi^{\alpha}\}$ we follow a procedure developed in Refs. dauria ; azcar . Imposing the ansatz

[TABLE]

where $a_{1},\ldots,a_{6},b_{1},\ldots,b_{9},c_{1},\ldots,c_{9},d_{1},\ldots,d_{9},$ $f_{1},\ldots,f_{8},g_{1},\ldots,g_{8}$ are arbitrary constants, and introducing eqs. (18-23) in eqs. (13-17), when $H^{a}=H^{ab}=\Theta^{ab}=\tilde{\Theta}^{ab}=\tilde{H}^{\alpha}=\mathcal{H}^{\alpha}=0$, we find

[TABLE]

The fields given by eqs. (24-29) represent the most general solution that can be built from the fields $\{e^{a},\omega^{ab},k^{ab},\tilde{k}^{ab},\psi^{\alpha},\xi^{\alpha}\}$. Any choice of the constants represent a solution to the FDA.

III.2 Chern-Simons-Antoniadis-Savvidy form

Using the invariant tensor found in Ref. eve1

[TABLE]

being $\alpha_{0}$, $\alpha_{2}$ and $\alpha_{4}$ dimensionless arbitrary independent constants, the Chern–Simons–Antoniadis–Savvidy Lagrangian $\mathcal{L}_{\mathrm{ChSAS}}^{(4)}\equiv\mathfrak{C}_{\mathrm{ChSAS}}^{(4)}$ is explicitly given by

[TABLE]

Using the FDA expansion given by eqs.** (24-29), the **Chern–Simons–Antoniadis–Savvidy Lagrangian for the Maxwell algebra takes the form

[TABLE]

From eq. (31) we can see that if $c_{9}=d_{8}=f_{1}=f_{7}=g_{2}=g_{8}=0$, which are conditions  consistent with the equations (18-23) and (13-17), we have that $\mathcal{L}_{\mathrm{ChSAS}}^{(4)}$ is given by

[TABLE]

Here it is necessary to notice that:

(a) The first two terms contains the Einstein–Hilbert and the Rarita-Schwinger terms given by $\epsilon_{abcde}R^{ab}e^{c}e^{d}e^{e}$  and $\Psi\gamma_{5}e_{a}\gamma^{a}\psi$ respectively.

(b) The following terms could be interpreted as non-linear couplings between the bosonic and fermionic ”matter” fields $\tilde{k}^{ab}$, $\xi$, the Rarita-Schwinger field $\psi$ and the curvature, where the parameter $l$ can be considered as a kind of coupling constant.

From eq. (32), we can see that when** $l\ll 1\mathbf{,}$** the Chern–Simons–Antoniadis–Savvidy Lagrangian for the Maxwell superalgebra is given by

[TABLE]

where we can see that the Chern–Simons–Savvidy Lagrangian reproduces, except for numerical coefficients, the Lagrangian for standard supergravity.

It is perhaps interesting to note that the conmutation relation $[P_{a},P_{b}]=Z_{ab}$ depends on the $Z_{ab}$ generators. The consequences on the Lagrangian of this non-zero bracket are related to the gauge field $k^{ab}$ associated to $Z_{ab}$. If we not consider the $k^{ab}$ dependence, or if we take a limit on the theory in which the gauge field $k^{ab}$ effects are not included, then it is not surprising that the curvature term in the above Lagrangian looks like the standard gravity Lagrangian.

IV The Extended Cartan Homotopy Formula** in $\left(2n+2\right)$**-dimensions

The Extended Cartan Homotopy Formula **(ECHF) **reads mss

[TABLE]

where, in this case, $\pi$ represents a polynomial in the forms $\left\{A_{t},B_{t},F_{t},H_{t},\mathrm{d}_{t}A_{t},\mathrm{d}_{t}F_{t}\right\}$ which is also an $m$-form on $M$ and a $q$-form on $T_{r+1}$, with $m\geq p$ and $p+q=r$. The exterior derivatives on $M$ and $T_{r+1}$ are denoted respectively by d and dt. The operator $l_{t}$, called homotopy derivation, maps differential forms on $M$ and $T_{r+1}$ according to

[TABLE]

and it satisfies Leibniz’s rule as well as $\mathrm{d}$ as $\mathrm{d}_{t}$. In our case, we will consider the polinomial $\pi=\left\langle F_{t}^{n}H_{t}\right\rangle$. This choice has the three following properties: (i) $\pi$ is $M$-closed, i.e., $\mathrm{d}\pi=0$, (ii) $\pi$ is a 0-form on $T_{r+1}$, and (iii) $\pi$ is a $\left(2n+3\right)$-form on $M$. The allowed values for $p$ are $p=0,\ldots,2n+3$. The ECHF reduces in this case to

[TABLE]

Since the three operators $\mathrm{d}$, $\mathrm{d}_{t}$ and $l_{t}$ define a graded algebra given by mss

[TABLE]

we have that the action of $l_{t}$ on $\{A_{t}$, $B_{t}$, $F_{t}$, $H_{t}$, $\mathrm{d}_{t}A_{t}$, $\mathrm{d}_{t}F_{t}\}$ reads mss

[TABLE]

while the action of $l_{t}$ on $B_{t}$ and $H_{t}$ must be determined.

Particular cases of (35) with $\pi$ given by (6) which we review below, reproduce both the Chern–Weil Theorem and the Triangle Formula. In fact, when $p=0,$ we find that Eq.(35) takes the form,

[TABLE]

where $\pi=\left\langle F_{t}^{n}H_{t}\right\rangle$ and $A_{t}=A_{0}+t\Theta,$ $B_{t}=B_{0}+t\Phi$. The left side of (38) is given by

[TABLE]

while for the right side we have

[TABLE]

so that

[TABLE]

From the Chern–Weil theorem and $B_{t}=B_{0}+t\Phi$, we see that

[TABLE]

so that $l_{t}H_{t}=\mathrm{d}_{t}B_{t}.$ On the other hand, we have

[TABLE]

[TABLE]

and therefore $\left\langle F_{t}^{n},\mathrm{D}_{t}l_{t}B_{t}\right\rangle=\mathrm{d}\left\langle F_{t}^{n},l_{t}B_{t}\right\rangle=0$ and $l_{t}B_{t}=0.$ Summarizing, we can write $l_{t}B_{t}=0$ and $l_{t}H_{t}=\mathrm{d}_{t}B_{t}.$

IV.1 The subspace separation method

In this subsection we will show that the subspace separation method developed in Ref. irs can be generalized to the case of Chern–Simons–Antoniadis–Savvidy formalism. This means that the so called triangle equation (46) splits the transgression form $\mathfrak{T}^{\left(2n+2\right)}\left(A_{2},B_{2};A_{0},B_{0}\right)$ into the sum of two transgressions forms depending on an intermediate connection $A_{1},B_{1}$ plus a exact form  $\mathfrak{T}^{\left(2n+1\right)}\left(A_{2},B_{2};A_{1},B_{1};A_{0},B_{0}\right)$ shown in Eq. (43).

When $p=1$ we have that Eq. (35) is given by

[TABLE]

where

[TABLE]

The left side of (39) corresponds to an integral along the boundary of the simplex $T_{2}=\left(A_{2},B_{2};A_{1},B_{1};A_{0},B_{0}\right):$

[TABLE]

The right side of (39) is given by

[TABLE]

where

[TABLE]

In (43) we have introduced dummy parameters $t=1-t^{0}$ and $s=t^{2}$, in terms of which $A_{t}$ reads

[TABLE]

Thus we have that the triangle equation is given by

[TABLE]

or alternatively

[TABLE]

We would like to stress that use of the Extended Cartan Homotopy Formula has allowed us to pinpoint the exact form of the boundary contribution  $\mathfrak{T}^{\left(2n+1\right)}\left(A_{2},B_{2};A_{1},B_{1};A_{0},B_{0}\right).$

Note that if we choose $A_{0}=0$ and $B_{0}=0$ we obtain an expression that relates the form Antoniadi–Savvidy transgression form to two Chern-Simons-Savvidy forms and a total derivative

[TABLE]

V Concluding Remarks

In this Letter some features of the $\left(2n+2\right)$-dimensional transgressions and Chern–Simons–Antoniadis–Savvidy forms used as Lagrangians for supergravity theories were briefly reviewed. The 2–form field $B$ can be discomposed in terms of components of the 1-form $A$. It is performed in a self–consistent way by considering the generalization of Maurer–Cartan approach to forms of higher order, i.e., free differential algebras, and by following the procedure used in Refs. dauria ; azcar ; salg ; ims . The final result is a four-dimensional supergravity action, which is gauge quasi–invariant under the Maxwell superalgebra. These 4-dimensional results shown that an interesting problem is to extract physical information from the $\left(2n+2\right)$-dimensional Lagrangian (11). A crucial step in this direction is the separation of the Lagrangian in a way that reflects the inner subspace structure of the gauge algebra. This is specially interesting in the case of higher-dimensional supergravity, where superalgebras come naturally split into distinct subspaces.  Examples of the use of the method within the transgression/Chern–Simons–Antoniadis–Savvidy framework will be studied elsewhere.

Acknowledgement 1

This work was supported in part by FONDECYT grants 1130653 and 1150719 from the Government of Chile. One of the authors (SS) was suported by grants 21140490 from CONICYT (National Commission for Scientific and Technological Re-search, spanish initials) and from Universidad de Concepción, Chile.