Extended gauge theory and gauged Free Differential Algebras

Recently, Antoniadis, Konitopoulos and Savvidy introduced, in the context of the so-called extended gauge theory, a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by higher degree forms. In this article it is shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can be constructed from an algebraic structure known as Free Differential Algebra. In other words, we show that the above mentioned non abelian gauge theory, where the gauge fields are described by p-forms with p>1, can be obtained by gauging Free Differential Algebras.


I. INTRODUCTION
Higher gauge theory [1][2][3][4][5][6][7][8] is an extension of ordinary gauge theory, where the gauge potentials and their gauge curvatures are higher degree forms. It is believed that higher gauge theories describe the dynamics of higher dimensional extended objects thought to be the basic building blocks of fundamental interactions.
The basic field of the abelian higher gauge theory, originated in supergravity, is a pform gauge potential A whose (p + 1)-form curvature is given by F = dA from which the Lagrangian and the action of the theory can be constructed. This abelian theory is known in the specialized literature as p-form electrodynamics and it is endowed with a local gauge symmetry with the transformation law A → A ′ = A + dϕ for some (p − 1)-form ϕ.
The natural question is: does there exist a non-abelian higher gauge theory? To answer this question it is interesting to remember that the points of a curve have a natural order and the definition of the parallel transport along a given curve indeed makes use of this order. However, for higher dimensional submanifolds such a canonical order is not available.
This lack of natural order led to C. Teitelboim in Ref. [9] to the formulation of a no-go theorem, ruling out the existence of non-abelian gauge theories for extended objects.
As with standard Chern-Simons forms, the secondary forms C (2n+p−1) ChSAS are background-free, quasi-invariant and only locally defined (and therefore defined only up to boundary terms, The purpose of this paper is to show that the invariants introduced in Refs. [1][2][3][4] can be constructed from a gauged free differential algebra. This paper is organized as follows: In Section 2, we briefly review the extended gauge theory developed in Refs. [1][2][3][4][5][6][7][8]. In Section 3, we will make a short review about free differential algebras and their gauging. Section 4 contains the results of the main objective of this work, namely: to show that the algebraic structure known as free differential algebras (FDA), allows to formulate a theory of non-abelian gauge with gauge fields described by p-forms with p ≥ 2 and to prove that the extended invariants found in Refs. [1][2][3][4] can be constructed by gauging free differential algebras. We finish in Section 5 with some final remarks and considerations on future possible developments.
Using the Chern-Weil theorem we can find an explicit expression for the Chern-Simons forms. In fact: let A (0) and A (1) be two one-form gauge connections on a fiber bundle over a (2n + 1)-dimensional base manifold M, and let F (0) and F (1) be the corresponding curvatures. Then, the difference of Pontryagin-Chern forms is exact where is called a transgression (2n Setting (2), we obtain the well known Chern-Simons (2n + 1)-form From the Chern-Weil theorem it is straightforward to show that under gauge transformations the Chern-Simons forms are quasi-invariant. However, it is important to stress that since a connection cannot be globally set to zero unless the bundle (topology) is trivial, Chern-Simons forms turn out to be only locally defined.

B. Non-abelian tensor gauge fields
The idea of extending the Yang-Mills fields to higher rank tensor gauge fields was used in Ref. [1][2][3][4] 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.
1. ChSAS forms in (2n + 2)-dimensions The first series of exact (2n + 3)-forms is given by where is the 3-form field-strength tensor for the 2-rank gauge field where ξ 0 = ξ a T a is a 0-form gauge parameter and ξ 1 = ξ a µ T a ⊗ dx µ is a 1-form gauge parameter.
Using the Chern-Weil theorem, we can find an explicit expression for the Chern-Simons form. In fact: Let A (0) and A (1) be two gauge connection 1-forms, and let F (0) and F (1) be their corresponding curvature 2-forms. Let A be their corresponding curvature 3-forms. Then, the difference 2 ), (7) where 2 , is what we call Antoniadis-Savvidy (AS) transgression form.
Using the procedure followed in the case of Chern-Simons forms, we define the (2n + 2)-ChSAS form as This result is analogous to the usual Chern-Simons form (3), but in even dimensions [8].
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 AS generalization) are locally defined and are off-shell gauge invariant only up to boundary terms (i.e., quasi-invariants).

ChSAS forms in (2n + 3)-dimensions
The second series of invariant forms is defined in 2n + 4 dimensions and is given by where the corresponding (2n + 3)-form C ChSAS is defined in terms of the 4-form F 4 = dA 3 + [A, A 3 ] field-strength tensor for the rank-3 gauge field A 3 . In fact, following the procedure shown in the above subsection, we define the (2n + 3)-ChSAS form as

III. FREE DIFFERENTIAL ALGEBRAS
In this section, we will make a short review on free differential algebras and their gauging [11][12][13][14].
The dual formulation of Lie algebras provided by the Maurer-Cartan equations [13] can be naturally extended to p-forms (p > 1) . Let's consider an arbitrary manifold M and a basis of exterior forms Θ A 1 (p 1 ) , Θ A 2 (p 2 ) , . . . , Θ An(pn) defined on M, labeled by the index The external derivative dΘ A(p) can be expressed as a combination of the elements of the base, which leads to write a generalized Maurer-Cartan equation of the following type [11][12][13][14] where the coefficients C A(p) B 1 (p 1 )···Bn(pn) are called generalized structure constants. The symmetry of these constants in the lower index is induced by the permutation of the forms Θ A(p) in the product wedge and are different from zero only if p 1 + p 2 + · · · + p n = p + 1.
Here, the number N is equal to p max + 1, where p max is the highest degree in the set Θ A(p) .
One can say that Eq. (16) is a generalized Maurer-Cartan equation and that it describes a FDA if and only if the integrability condition d 2 Θ A(p) = 0 follows automatically from (16).
Explicitly, the condition for (16) to be a FDA is given by This equation is just the analogue of the Jacobi identities of an ordinary Lie algebra. It is very instructive to have a look at the most general form of a FDA as it emerges from theorems of Sullivan. From Ref. [13] we know that: (i) a FDA is called "minimal algebra" when it is true that C A(p) B(p+1) = 0. This means that all forms appearing in the expansion of dΘ A(p) have at most degree p, being the degree (p + 1) ruled out; (ii) a FDA is called a contractible algebra when the only form appearing in the expansion of dΘ A(p) has degree Sullivan's fundamental theorem: The most general free differential algebra is the direct sum of a contractible algebra with a minimal algebra.
A. Gauging free differential algebras Physical applications of FDA require a generalization of the concepts of soft 1-forms and curvatures introduced gauging Maurer-Cartan equations [12][13][14].
If we apply the exterior derivative to both sides of Eq. (20), we obtain a generalization of the Bianchi identity [13] In complete analogy to what one does in ordinary group theory, we say that the left side of (21) defines the covariant derivative ∇ of an adjoint set of (p + 1)-forms. With this definition, the Bianchi identity (21) just states that the covariant derivative of the curvature set F A(p+1) is zero as it happens for ordinary groups.

IV. EXTENDED GAUGE THEORY AND GAUGED FDA
Let us now consider the explicit form of the equations (20,21). In the case of a minimal FDA, the explicit form of equations (20,21) for p = 1, 2, 3, 5, 7, 9, is given in Appendices A and B respectively. Here we will list, using the nomenclature of Refs. [1][2][3][4], only the equations we will use later. In fact, from (79) we can see that, if we restrict ourselves to the case of an FDA whose structure constants satisfy the condition C BC correspond to the structure constants of a Lie algebra , then the equations (78, 79) can be written in the form (see Appendix A) In the same way, for the equation (80) we find (see Appendix B) DF = 0, We will consider this condition in the rest of this paper.
It should be noted that equations (22) and (23) match those found in Refs. [1][2][3][4], except for numerical coefficients. However, they coincide exactly after an appropriate transformation of the gauge fields (see Appendix F).

)-forms gauge parameters and let
A B 1 (p 1 ) , . . . , A Bn(pn) be a set of p-forms gauge potentials labeled by an index B and by the degree p. Under a gauge transformation, the gauge potential transforms as In the case of a minimal FDA, the explicit form of equation (24) for n = 2 and p = 1, 2, 3, 5, 7, 9, is given in Appendix C. From (83) we can see that δA = Dλ,

B. Gauge transformations for curvatures
Following the definition of the usual gauge theory, we have so that In the case of a minimal FDA, the explicit form of equation (27) for p = 1, . . . , 9 is given Appendix D. When a FDA has structure constants that satisfy the condition C BC , we find that the equations (85) can be written in the form (see Appendix D) The equations (25,28) match those found in Refs. [1][2][3][4], after an appropriate redefinition of the gauge fields (see Appendix F).

V. EXTENDED INVARIANTS
In this section it is shown that the extended invariants found by Antoniadis and Savvidy in Refs. [1][2][3][4] can be constructed from a gauged free differential algebra.

A. Chern-Pontryagin invariants
Let A = A a T a be a 1-form connection evaluated in the Lie algebra g of the group G and let F = F a T a = dA + A 2 be its corresponding 2-form curvature. The Chern-Pontryagin topological invariant in 2n + 2 dimensions is given by [15] where the bracket · · · is a symmetric multilinear form that represents an appropriately normalized trace over the algebra defined by

B. Generalized Chern-Pontryagin invariants
Let's consider now the generalization of the Chern-Pontryagin topological invariant to the case where Lie algebra g is replaced by a free differential algebra. Let be a set of p-forms field intensities. It is possible to construct topological invariants analogous to the Chern-Pontryagin invariant as follows where for each order of the formP, the sum runs over all possible combinations.
b. Term with p 1 = · · · = p n−1 = 2 and p n = p n+1 = 4 : In this case we have that, according to the law of permutations, there must exist n(n + 1)/2 terms of the form so thatP This means that the corresponding extended Chern-Pontryagin invariant is given bỹ which can be write as Using the nomenclature used in Refs. [1][2][3][4] we can write Now let us now prove that the expression (48) is, in addition to being gauge invariant, a closed form. The variation of P 2n+6 is given by Using (23) we have and using the well known identity [16] where each Λ i is a d i -form and Θ is an arbitrary d Θ -form, we have which proves that the form P 2n+6 is closed. This means that P 2n+6 = dC 2n+5 where, following the usual procedure, we find 5. Case p 1 + · · · + p n+1 = 2n + 8 In this case we will choice three combinations which will be analyze separately.

VI. CONCLUDING REMARKS
In this article we have shown that the so-called ChSAS invariants [1][2][3][4] can be constructed from a algebraic structure known as gauged free differential algebras. The series of exact (2n + p)-forms is given by where each F q+1 is a (q + 1)-form field-strength for the rank-q gauge field A q which depends also on other gauge fields A r with r < q. The corresponding secondary (2n + p)-form C (2n+p) are also defined in terms of such gauge fields in the following way If we consider the n = 2 case in the definition of C (2n+7) , we find that the 11-dimensional ChSAS form is given by From here we can see that the second term has the same form as a term that appears in the CJS supergravity [17], whose action is given by where Σ a 1 ···ar : and the * symbol denotes the Hodge operator. In fact, if one sets the metric and gravitino field to zero, 11 dimensional supergravity [17] is reduced to a Chern-Simons like theory based on a three form A whose action is where F 4 is a 4-form and M 11 is an eleven dimensional manifold. This result allows us to conjecture that it would be possible to construct a theory of 11-dimensional Chern-Simons supergravity using a procedure similar to that shown in Ref. [8], which contains or ends at some limit in standard 11-dimensional supergravity theory [17].
where from p = 3 we have considered only odd-order gauge fields. Note that we have considered a FDA whose structure constants satisfy the condition C A(q+r−1) B(q)C(r) for any r < q. These equations can be written as Appendix B The explicit form of the equation (21) for n = 2 and p = 1, 2, 3, 5, 7, 9 is where from p = 3 we have considered only odd-order gauge fields. These equations can be written as Appendix C The explicit form of the equation (24) for n = 2 and p = 1, 2, 3, 5, 7, 9 is given by where from p = 3 we have considered only odd-order gauge fields. These equations can be written as Appendix D The explicit form of the equation (27) for n = 2 and p = 1, 2, 3, 5, 7, 9 is given by where, from p = 3 we have considered only odd-order gauge fields. These equations can be written as In this appendix we show that (85) correspond to homogeneous transformations.
A. Gauge transformation of the rank-2 field strength tensor F 2 Introducing the first equations of (83) in the first equation of (85) we have Using the nomenclature of Refs. [1][2][3][4], this equation takes the form B. Gauge transformation of the rank-4 field strength tensor F 4 Introducing the first and third equations of (83) in the second equation of (85), we have Using the nomenclature of Refs. [1][2][3][4], this equation takes the form C. Gauge transformation of the rank-4 field strength tensor F 6 Introducing the first, third and fifth equations of (83) in the third equation of (85), we have Using the nomenclature of Refs. [1][2][3][4] this equation takes the form so that D. Gauge transformation of the field strength tensor F 8 Introducing the first, third, fifth and seventh equations of (83) in the fourth equation of (85), we have Using the nomenclature of Refs. [1][2][3][4] Using the nomenclature of Refs. [1][2][3][4] The next step is to consider that all the structure constants of the FDA (90) can be written in terms of the structure constants C A B 1 B 2 of a Lie algebra. This allows us to write the Eq.