Gauge-invariant theories and higher-degree forms

A free differential algebra is generalization of a Lie algebra in which the mathematical structure is extended by including of new Maurer-Cartan equations for higher-degree differential forms. In this article, we propose a generalization of the Chern-Weil theorem for free differential algebras containing only one $p$-form extension. This is achieved through a generalization of the covariant derivative, leading to an extension of the standard formula for Chern-Simons and transgression forms. We also study the possible existence of anomalies originated on this kind of structure. Some properties and particular cases are analyzed.


Introduction
Higher gauge theories are generalizations of the standard gauge theories that involve higher-degree differential forms. In the simplest case, this means introducing not only the usual one-form gauge connection but also a two-form gauge connection and a three-form field-strength, describing the parallel transport along surfaces. It is possible to continue such extension to gauge fields of degree higher than two, describing parallel transport along extended objects. In a higher gauge theory [1], the gauge potentials are locally represented as p-forms, whose corresponding (p + 1)-form gauge curvatures allow the construction of action principles. The corresponding field equations are able to describe the dynamics of extended objects, such as p-branes, in a similar manner in which a standard gauge theory describes the dynamics of point particles. An example of this is found in p-form electrodynamics [2], whose gauge symmetry is described by the invariance under the transformation law A → A ′ = A + dϕ where the abelian gauge field A is a p-form, and ϕ is a (p − 1)-form and the parameter of the transformation. Another example of this type is the rank-2 abelian Kalb-Ramond gauge field [3]. A generalization to non-abelian higher-degree gauge theories has been studied on Refs. [4][5][6]. In particular, in Refs. [7][8][9][10], gauge-invariants forms were found, similar to the usual Chern-Pontryagin densities and their corresponding Chern-Simons (CS) forms for special cases.
From a physical point of view, it is interesting to note that a common feature to loop quantum gravity and string theory is the generalization of point particles to extended objects. It is then interesting to study the possible role that higher gauge theory could take in both frameworks.
In 1980 R. D'Auria, P. Fré and T. Regge [11] found an algebraic structure known as free differential algebra (FDA) or Cartan integrable system that allows formulating supergravity in the superspace in a geometric manner, representing the spacetime as a supermanifold. In 1982, R. D'Auria and P. Fré made use of such structure to unveil a hidden symmetry algebra in eleven-dimensional supergravity, previously constructed by Cremmer, Julia and Scherk [12,13]. On the other hand, from Ref. [14] it is known that the D'Auria-Fré formulation of supergravity is a higher-order geometric formulation of the Cartan supergeometry, where extended algebraic structures replace Poincaré Lie superalgebra.
First-order formulations of supergravity in six or more dimensions have a field content that includes bosonic higher-degree differential forms. Such field presence is a consequence of the consistence requirement of an equal number of bosonic and fermionic degrees of freedom in supersymmetry [15]. Since the field content of these theories cannot be encoded in one-forms dual to the generators of a Lie group, a possible solution is to replace the concept of group manifold used in the formulation of gravity and supergravity theories for a manifold that inherently involves higher degree forms. This led to introduce some mathematical structures in physics such as FDAs. These generalize the Maurer-Cartan equations that describe a Lie algebra but including higher-degree differential forms as potentials. FDAs are the natural generalization of Lie algebras, and since they include Lie algebras as subalgebras, they can be used to describe the field content of higher-dimensional gravity and supergravity theories.
The aim of this work is to use the FDA considered on Refs. [16][17][18][19] to obtain a gauge-invariant density and its corresponding CS form and study the presence of anomalies in this kind of theory. This paper is organized as follows: In Section 2, we briefly review free differential algebras and their gauging, focused on the particular case which will be important in the results of this article. In Section 3, we will propose a definition of covariant derivative that will be necessary for the construction of invariant gauge theories. In Section 4 we introduce a generalization of the gauge invariant density of Lie algebras that includes the p-form of the already mentioned FDA and study the corresponding invariant tensor conditions. Section 5 contains a generalization of the Chern-Weil theorem with explicit expressions for transgression and CS actions for non-abelian gauge theory whose gauge fields are a one-form and a p-form (p ≥ 2). In Section 6 we finish with a study on the existence of gauge anomalies for higher gauge theory, for which is also necessary to introduce new notation and study some mathematical properties about invariant tensors for FDAs. There are also two appendices with some useful properties and an application for gravity.

Free differential algebras
The dual formulation of Lie algebras provided by the Maurer-Cartan equations can be naturally extended to p-forms. Let us consider a basis of differential forms Θ A(p) N p=1 defined on a manifold M with N ≥ 2. Each quantity Θ A(p) ∈ Λ p (M ) is a differential p-form and the algebraic index A (p) takes values on different sets depending on the value of p. Since Θ A(p) N p=1 is a basis, the exterior derivative dΘ A(p) can be written in terms of the same basis. This allows to write a set of Maurer-Cartan (MC) equations for a mathematical structure called free differential algebra The coefficients C A(p) B1(p1)···Bn(pn) are called generalized structure constants and are the generalization of the structure constants of Lie algebras to the case of FDAs. The nilpotent condition d 2 Θ A(p) = 0 leads to the corresponding generalized Jacobi identity We will restrict the analysis to a particular case that has been extensively studied in Refs. [16,17], in which the FDA is given by a Lie algebra with only one p-form extension, i.e., depending only on a 1-form µ A and a p-form B i and therefore, reducing (2.1) to a set of two MC equations 1 [16,17] Eq. (2.7) shows how the FDA is an extension of the Lie algebra defined by Eq. (2.6), achieved by the addition of a new MC equation for a p-form potential B whose non-trivial structure is given by the presence of a cocycle For consistency, this (p + 1)-form must be covariantly closed. Otherwise, the second exterior derivative on Eq. (2.7) does not vanish and the generalized Jacobi identity would not hold. Note that, if the cocycle is covariantly exact, i.e., Ω = ∇ϕ, it would be possible to write the Maurer-Cartan equation (2.7) as ∇B i = 0, through a redefinition of the field B i → B i + ϕ. Therefore, in order to have a non-trivial structure for the extended algebra, the cocycle must be covariantly closed but not covariantly exact. This means that, given a Lie algebra, there are as many non-equivalent FDA extensions as Chevalley-Eilenberg cohomology classes the Lie algebra has [11,12].
To define gauge transformations using this algebra, we need to write the complete set of diffeomorphism transformations on the FDA manifold. Such diffeomorphisms are given by the Lie derivatives along all the possible directions on the FDA manifold. We need then to define a regular Lie derivative (as with Lie groups) and an extended one that determines the transformation of the p-form using a (p − 1)-form parameter ε j These derivatives are defined in terms of the contraction operators i ε A tA = ε A i tA , i ε j tj = ε j i tj whose 1 From now on we will omit the wedge product between differential forms.
action is defined in terms of the basis as follows Here, t A are the generators of the Lie algebra described by the Eqs. (2.6) (subalgebra of the FDA). In the same way, t j correspond to a basis of vectors on the FDA manifold in the direction of the p-forms, i.e., t j is dual to B j in the same way in which t A is dual to µ A [18,19]. Applying the regular Lie derivative ℓ ε A tA on the gauge fields µ A and B i we have Applying the extended Lie derivative ℓ ε j tj on the gauge fields µ A and B i we have Eqs. (2.12) -(2.15) contain the complete set of diffeomorphism transformations along all the independent directions of the FDA manifold [18,19]. Both transformation laws depend on the parameters ε A and ε j of the transformations and can be sumarized as follows Note that these transformations also depend on the curvature of the FDA manifold. It is possible to obtain a restricted version of such transformations demanding horizontality of the curvatures in some directions of the FDA manifold, i.e., The curvature forms admit a splitting such that the horizontality conditions can be written as follows In a more convenient way, Eqs. (2.24) and (2.25) can be written as As happens with Lie groups, with these conditions, the diffeomorphisms become gauge transformations Eq. (2.27) corresponds to the usual covariant derivative of the 0-form parameter. The second equation is the natural extension to the case of a p-form. Note that the transformation of µ A is the same that appears in the study of standard gauge theory and it depends only on ε A . On the other side, the transformation of B i depends on ε A and ε i [18,19].

Covariant derivative
In order to formulate a gauge theory involving p-forms whose invariance is governed by an FDA, it is necessary to define a covariant derivative that involves all the components of the connection. To find such derivative, it is useful to consider the information provided by the transformation law for the curvatures and the Bianchi identities.
Using Eqs. (2.27) and (2.28) it is possible to prove the following relations Eq. (3.1) is equivalent to the Lie bracket between the 2-form curvature and the 0-form parameter. The natural generalization for the (p + 1)-form curvature is expressed in Eq. (3.2). We can see that R i also transforms homogeneously, i.e., not depending on derivatives of the parameters. On the other hand, starting from the definition of the curvature forms, we can also calculate its exterior derivative to find the Bianchi identities is a 0-form and ε i is a (p − 1)-form, then its covariant derivative is given by ∇ε = (∇ε) On the other hand, from Eqs. (3.3) and (3.4) we can see that if we consider the curvature tensor of the FDA R = R A , R i where R A is a 2-form and R i is a (p + 1)-form, then its covariant derivative is given A possible answer is that these equations define the covariant derivative of every set of differential forms x = x A , x i in the (A, i)-representation. However, there are two problems with this definition: • It does not satisfy the homogeneity condition, i.e., the second covariant derivative ∇ 2 x depends on the exterior derivative of x.
• The gauge curvature does not satisfy δR i = ∇δB i .
At this point, we only know how to take the derivative of ε and R. To solve this caveat, it is necessary to find a general definition that depends on the order of the differential forms of the corresponding set of fields. In the first case we have a set ε A , ε i of (0, p − 1)-forms and in the second one we have set R A , R i of (2, p + 1)-forms. In general, we have arrays of (q, p + q − 1)-forms, and therefore, a general covariant derivative must be defined in terms of q. Let us introduce a (q, p + q − 1)-set of differential forms in the (A, i)-representation of the algebra x = x A , x i . We propose that the covariant derivative of x can be written in components as follows where we have introduced the functions f (q) and g (q) in the terms involving the gauge fields and structure constants. From (3.6) and (3.8) we have Besides, in order to satisfy the homogeneity condition, we need to remove the dependence on the derivatives of x A and x i in the second covariant derivatives of x. Those requirements lead to the following conditions from which we have g (q + 1) = g (q) + 1. (3.14) A valid solution to this equation system is given by f (q) = g (q) = q, giving us the following definition Note that this definition satisfies the homogeneity condition, i.e., the second covariant derivative of x does not depend on the derivatives on x. With this definition, we can write the variation of the gauge curvatures as

Invariant density
Now we postulate an invariant density, analogous to the Chern-Pontryagin invariant of a Lie group. Using combinations of the curvavure forms, the most general q-form that can be written in terms of R A and R i is given by 2 The constants g A1···Ami1···in must be such that χ q is gauge invariant, namely δχ q = 0. The sum runs over all the possible values of m and n such that 2m + (p + 1) n = q, and hence the resulting form is a q-form. Note the coefficients g A1···Ami1···in contain mixed indices A and i. Enforcing the gauge invariance condition δχ q = 0 will constraint the form of such coefficients. Notice that the total variation of χ q under gauge transformations is given by Since the parameters ε A , ε j are independent, the terms proportional to each one have to vanish independently without imposing any extra condition on the fields and curvatures. This allows to split the independent conditions, resulting in the following three equations where the indices with hatÂ andî denote the absence of A and i in the sequence. Eqs.

Adjoint representation
An interesting case can be found when the p-form B i is also in the adjoint representation of the Lie subalgebra. In such case, the structure constants C i Ak become equivalent to the structure constants of the Lie subalgebra, i.e., To avoid confusion with the indices of the invariant tensor, we introduce a comma to separate the indices corresponding to different sectors of the algebra g A1···Ami1···in → g A1···Am,B1···Bn . Note that the tensor is symmetric on the first set of indices but it still can be symmetric or antisymmetric on the second one depending on the value of p. In this case, the invariant tensor conditions (4.3) -(4.5) take the form Eq. (4.7) is equivalent to the invariant tensor condition for Lie algebras. This makes it easier to find invariant tensors for FDAs; when the p-form is in the adjoint representation of the Lie subalgebra, an invariant tensor of the whole FDA is an invariant tensor of the Lie subalgebra that also satisfies the second and third conditions (4.8) and (4.9).

Chern-Weil theorem
Transgression and CS forms are chosen as Lagrangians for gravitational theories due to their invariance properties under gauge transformations. In particular, transgression forms are completely invariant and allow finding conserved charges. In the second order formalism of gravity, the fundamental field is the spacetime metric g µν and not the Levi-Civita connection Γ λ µν . The Levi-Civita connection is present in the formulation, but it is completely determined by the metric tensor. However, the first-order formalism allows one to consider both the metric and the connection as off-shell independent fields, each encoded in the components of one-forms evaluated in a Lie algebra. Since Chern-Simons and transgression theories are background-free theories that depend on a one-form gauge connection, they are good candidates to be considered gravitational theories, generalizing general relativity and introducing gauge invariance under a certain Lie group. See Refs. [20,21] for detailed reviews on the relation between invariant densities, CS and transgression forms, and physical theories. Gravitational theories in which the Lagrangians are CS forms and the invariance is described by space-time Lie groups, namely Poincaré and (Anti) de-Sitter groups, were proposed in Refs. [22][23][24] for the three-dimensional case. In the last decade of the 20th century, A. Chamseddine extended CS forms to higher dimensions [25,26]. The supersymmetrization of such models was introduced and extensively studied in Refs. [27][28][29][30][31][32].

CS and transgression forms
The relation between CS and transgression forms emerges naturally in the Chern-Weil theorem: Let µ andμ be one-form gauge connections on a 2n + 1 dimensional manifold, evaluated on a Lie algebra. Let R = dµ + µ 2 andR = dμ +μ 2 their corresponding two-form curvatures. Then where Tr denotes the trace over the elements of the Lie algebra on which the curvatures are valuated. The (2n − 1)-form Q 2n−1 (µ,μ) is called transgression form and its explicit expression can be found by introducing a homotopic gauge field µ t =μ + t (µ −μ) with its corresponding homotopic curvature R t = dµ t + µ 2 t depending on a parameter t ∈ [0, 1]. Note that the homotopic parameter interpolates µ t between µ and µ and therefore, also R t betweenR and R. Then, it is possible to write down the transgression (2n − 1)-form as (5.2) By locally settingμ = 0, the transgression form become a CS-form Q 2n−1 (µ) = Q 2n−1 (µ, 0), satisfying the well-known relation Tr R n = dQ 2n−1 (µ) .

(5.3)
Since the connectionμ cannot be globally fixed, the CS form can be only locally well defined.

Extended Chern-Weil theorem
In order to generalize the Chern-Weil theorem to the case of FDAs, let us consider the invariant density from Eq. (4.1) for the homotopic gauge fields µ A t and B i t with t ∈ [0, 1]. Since the homotopic parameter t interpolates between (µ 0 , B 0 ) and (µ 1 , B 1 ), the difference χ q (µ 1 , B 1 ) − χ q (µ 0 , B 0 ) can be written as the following integral Using the definition of covariant derivative for the FDA and performing the derivation inside of the integral, it is possible to show that the difference χ q (µ 1 , B 1 ) − χ q (µ 0 , B 0 ) is an exact form where the (q − 1)-form Q q−1 (µ 1 , µ 0 ) is explicitly given by and R i t are the curvatures corresponding to the homotopic gauge fields. For convenience, we rename B i = µ i , B i t = µ i t and b i = u i . Then we can write µ = µ A , µ i as a single composite field and the same for the corresponding curvature. We can also write χ q µ A t , B i t = χ q µ A t , µ i t = χ q (µ t ). This allows to write the generalization of the Chern-Weil theorem in a compact way that will be useful in future calculations As with the standard transgression forms, if we fix µ 0 = µ A 0 , µ i 0 = (0, 0) in Eq. (5.8), we obtain the generalization of the Chern-Simons form to the case of FDAs. Analogously to the case with Lie algebras, this can be done only locally.

Gauge anomalies
The presence of anomalies in a theory is due to the breaking of classical symmetries in the quantization process. The chiral anomaly, introduced in Refs [ [33][34][35][36] appears in gauge theories that interact with Weyl fermions. The U A (1) or abelian anomaly is given by the divergence of the classically conserved current, and it is proportional to the Chern-Pontryagin 4-form where µ ν = µ A ν T A is a gauge connection valuated on the Lie algebra of an internal Lie group with generators t A [37]. By introducing the one-form µ = µ A µ dx µ t A and the exterior derivative d = dx µ ∂ µ it is possible to write the divergence (6.1) in terms of differential forms and the Hodge operator where Q 3 (µ) is the standard CS 3-form.
On the other side, the so called non-abelian anomaly is given by the covariant divergence of a nonabelian current J µA or, in terms of differential forms In general, it is not possible to write the right-hand side of Eqs. (6.4) in terms of gauge fields and field strengths. However, an interesting result due to B. Zumino in Refs. [37,38] shows that the non-abelian gauge anomaly can be derived from the gauge-variation of the CS form on Eq. (6.2), which is porportional to the exterior derivative of certain two-form Q 1 2 (ε, µ) depending on the gauge fields and the 0-form parameter of the transformation ε = ε A T A δQ 3 (µ) = dQ 1 2 (ε, µ) . In the same way, the non-abelian anomaly in a (2n − 2)-dimensional spacetime can derived from the 2ndimensional abelian anomaly on Eq. (6.6) by introducing a set of Lorentz-scalar fields ε A and computing the gauge variation of the (2n − 1)-dimensional CS form The (2n − 2)-form Q 1 2n−2 (ε, µ) has the following integral representation [37,38] Q 1 2n−2 (ε, µ) = n(n − 1) where Str denotes the symmetrized trace over the algebraic elements.

Extended anomalies
Recently, it was found that it is possible to find gauge-invariant densities in the context of higher gauge theory 3 . Moreover, in Ref. [7,8] it was studied the existence of gauge anomalies generated by such invariants. In this section we study the existence of anomalies, starting from the extended CS forms introduced in previous sections. Let us consider the following gauge field consisting on one-forms µ A and p-forms µ i . The corresponding curvature is given by a (2, p + 1)-form R = R A , R i . As we have seen, a general field X can be decomposed as a set X = X A , X i , where X A is a r-form and X i is ar-form withr = r + p − 1. In such case we say that the array X is of degree r. Let X be an array of degree x and Y be an array of degree y. We introduce the following products between two arrays where the components of Z are given by Note that Eqs. (6.11) and (6.12) imply that Z A is a (x + y)-form and Z i is a (x + y + p − 1)-form. Let us consider now the arrays X r and Y s of degree x r and y s respectively. We introduce the product between p + 1 arrays with the following components On the other hand, as we will see later, it is convenient to introduce a compact notation for the FDA 3 See for instance Refs. [4-6, 9, 10] invariant tensor. We denote This bracket separates the element before and after the semicolon, being the first ones evaluated in the Lie subalgebra and the latter on the extended sector. It is easy to check the following (anti)symmetry rules . . . , X r , X r+1 , . . . ; Y 1 , . . . , Y n = (−1) xrxr+1 . . . , X r+1 , X r , . . . ; Y 1 , . . . , Y n , (6.16) Now we recall the invariant tensor conditions (4.3) -(4.5). Multiplying the first invariant tensor condition Using the new notation for the invariant tensor and the product given in Eqs. (6.11) and (6.12) we obtain the following identity In the same way, introducing a new set of arbitrary arrays Θ 1 , . . . , Θ p and multiplying (4.4) and (4.5) by Θ B1 · · · Θ Bp X A1 · · · X Am+1 Y i2 · · · Y in and Θ j X A1 · · · X Am+1 Y i2 · · · Y in respectively we obtain the second and third invariant tensor conditions in terms of the new notation The purpose of this notation is to be able to find a gauge anomaly from the extended CS form in a more compact way. To achieve this, it is necessary to find an expression for the gauge variation of Q (q−1) (µ) in terms of an exact form as with Lie groups δQ q−1 (µ) = dω 1 q−2 (ε, µ) . (6.24) It is also convenient to separate the independent variations with respect to the parameters ε A and ε i . Since the standard and extended transformations are independent, this leaves to two different generalizations of the gauge anomaly from Eq. (6.8). Let us begin with the extended variation, i.e., the one proportional to ε i .

Extended variations
The gauge variations of the gauge fields and curvatures parametrized by the (p − 1)-form ε i are given by Using the generalized Jacobi identity, it is direct to prove that the variation of the corresponding homotopic curvatures are The variation of the CS form is then given by This allows to remove the terms including brackets between µ and dε and write the gauge variation as follows The next step is to use the generalized Bianchi identities, the definition of the homotopic curvatures and Eqs. (6.19) -(6.21) to obtain the following relations We use relations (6.34) -(6.39) to write the extended variation of Q (q−1) (µ) in terms of the total derivatives ∂/∂t and d Note that the first term vanishes while the second and third terms are exact forms. This means that we can write the gauge variation of the Chern-Simons form (6.23) in terms of a (q − 2)-form proportional to ε i which generalizes the expression for the non-abelian gauge anomaly where , ε . (6.42)

Standard variations
Let us consider now the standard variations of gauge fields and curvatures. This case presents some differences because, in order to obtain the anomaly term, we need to write the CS form in terms of new homotopic curvatures. Let us introduce some definitions before we proceed on this.
Given a degree-1 array µ = µ A , µ i of gauge fields, we have a corresponding degree-2 array of curvatures R = R A , R i . In terms of µ and R we define the derivative operator [37] Using the Jacobi identities we can check that this definition verifies the nilpotent condition d 2 = 0. Now we consider an arbitrary variation on µ and R. With respect to that variation we introduce the homotopy operator ℓ such that its action on the arrays µ and R is given by By direct inspection it can be shown that the homotopy operator ℓ satisfy the following anticommuting relations (ℓd + dℓ) µ = δµ, (ℓd + dℓ) R = δR. (6.46) Now we introduce new homotopic gauge fields. The one-form µ A is parametrized as usual. However, the p-form µ i is parametrized with a power of the parameter, such that µ t still interpolates between µ 0 and µ 1 along a convenient trajectory in the parametric space If we consider a variation along the parameter t, then it is possible to define a homotopic operator ℓ t with respect to such variation satisfying the relation ℓd + dℓ = d t Integrating the homotopic operator between 0 and 1 we find By applying the left-hand side of (6.49) into χ q (µ t ) we use Stokes' theorem and recover in this way the Chern-Weil theorem where [ [16][17][18][19] and its main feature is the presence of a Lie subalgebra and only one p-form extension through the inclussion of a non-trivial cocycle, representative of a Chevalley-Eilenberg cohomology class. Explicit expressions for transgression and CS forms whose integral representation is given by Eq. (5.8) and possible gauge anomalies given by Eqs. (6.42) and (6.70) were also found. The procedure to find those expressions is analogous to the one described in Refs. [37,38] in the calculation of non-trivial chiral anomalies. This is, by starting from the invariant density, obtain an integral expression for the corresponding CS form and finally, through its gauge variation, obtain the anomaly term. The main difference with respect to the standard case is the presence of extended indices. Since every expression contains both type of indices A and i, being the first in the Lie subalgebra and the latter in the extended sector of the FDA, it is not possible to use the standard mathematical identities of group theory. In particular, the FDAs invariant tensor g A1···Ami1···in does not satisfy the invariant tensor conditions of its Lie subalgebra unless n = 0. To perform the calculations, it is therefore necessary to use its definition to obtain the generalized properties to which the invariant tensor obeys (6.19) -(6.21). Eqs. (6.42) and (6.70) are the generalization of the non-abelian anomaly (6.8), including non only one form gauge fields but also a p-form. Such extension of the field content in the theory is made in the same way in which the gauge symmetry is extended from a Lie algebra to an FDA through the inclusion of a non-trivial cocycle. Since the extension of the anomaly term has been found in a geometrical framework, its physical meaning must be separately studied in order to understand its possible consistency with the breaking of classical symmetries in QFT.
By gauging the algebra we consider non-zero curvatures. For convenience we denote the 2-form curvature components as R A = R a , R ab , F ab and, since B i is in the adjoint representation, we denote the 4-form curvature as R i → H A = h a , h ab , H ab . This allows us to propose the following gauge invariant 6-form In this case, the invariant tensor conditions (4.7) -(4.9) are given by As we have seen, in this case, the condition (A.9) means that g AB is also an invariant tensor of the Lie subalgebra. The second condition becomes equivalent to the first one, while the third condition is given by Since g AB is an invariant tensor of the Maxwell algebra, we propose the usual rank-2 invariant tensor 4 g ab,cd = α 0 (η ac η bd − η ad η bc ) , (A.13) g ab,[cd] = α 1 (η ac η bd − η ad η bc ) , (A.14) where α 0 , α 1 and α 2 are arbitrary constants. However, this tensor still has to verify Eq. (A.11) in order to be an invariant tensor of the whole FDA. By imposing that condition, we find α 1 = α 2 = 0. The rank-2 invariant tensor of Maxwell-FDA is then given by (A.13). From (5.8) we can write down a Chern-Simons action given by However, in order to obtain a simpler expression, it is convenient to introduce a new set of gauge fields and use the triangle relation given by Eq. (B.9) in Appendix B. We defineμ = μ A ,B A whose components are given as followsμ Then we can write the CS form (A.17) in terms of a transgression form, another CS form and total derivatives Q 5 (µ) = Q 5 (µ,μ) + Q 5 (μ) + total derivative. (A.20) From Eq. (5.8) we know that the explicit expression for Q 5 (µ,μ) is In this case, the homotopic gauge fields are given by µ t =μ + t (µ −μ) where indexwise µ t is given by On the other hand, the CS form Q 5 (μ) is given by with the homotopic gauge fieldμ t = tμ = tμ A , tB B andR t = R A t ,H B t being its corresponding curvature. Using the rank-3 invariant tensor of the Maxwell algebra g ab,cd,e = ǫ abcde and the rank-2 invariant tensor (A.13) for the extended sector, we obtain the following CS form

B Subspace separation method
Extended CS and transgression forms satisfy similar invariance conditions that their standard versions. There is also a triangle relation between them that can be explicitly found using the extended Cartan homotopy formula (ECHF) [20,21,39].
Let us consider a set of r + 2 composite gauge connections µ J = µ A J , µ i J r+1 J=0 defined on a fiber bundle over M and a (r + 1) dimensional simplex T r+1 with r + 2 parameters t J ∈ [0, 1], satisfying the constraint J t J = 1. It is possible to defined an homotopic connection µ t = J t J µ J whose components transform according (2.27, 2.28). The ECHF is given by i.e., ℓ t increases the order of the differential form on T r+1 and decreases the order on M while it satisfies Leibniz's rule as well as d and d t . Note that the ECHF is different for any value of s. However, its allowed values are s = 0, ...q. As happens in the case studied in Section 6, the operators d, d t and ℓ t define a graded algebra given by [39]  In this case, we consider the closed polynomial χ = χ q (µ t ), reducing the ECHF to By setting s = 0, the homotopic connection is given by µ t = µ A t , µ i t = µ 0 + t (µ 1 − µ 0 ) and then Eq. (B.6) reproduces the generalized Chern-Weil theorem