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Extension of Chern-Simons forms

We investigate metric independent, gauge invariant and closed forms in the generalized YM theory. These forms are polynomial on the corresponding fields strength tensors - curvature forms and are analogous to the Pontryagin-Chern densities in the YM gauge theory. The corresponding secondary characteristic classes have been expressed in integral form in analogy with the Chern-Simons form. Because they are not unique, the secondary forms can be dramatically simplified by the addition of properly chosen differentials of one-step-lower-order forms. Their gauge variation can also be found yielding the potential anomalies in the gauge field theory.

In recent articles [19,20,21] the authors found closed invariant forms similar to the Pontryagin-Chern-Simons forms in non-Abelian tensor gauge field theory [22,23,24]. The first series of closed invariant forms are defined in D = 2n + 4 dimensions and are given by the expression where the corresponding secondary (2n + 3)-form ψ 2n+3 is in D = 2n + 3 dimensions ψ 2n+3 = Str(A 3 , G n ) (1.7) 1 L a are the generators of the Lie algebra. 2 In this article we shall use the symmetrized trace Str(A 1 , A 2 , ..., A n ) ≡ 1 n! (i1,...,in) (1.5) where the sum is over all permutations. Its properties are described in the Appendix B of the article [10].
All forms Φ 2n+4 , Ξ 2n+6 and Υ 2n+8 are analogous to the Pontryagin-Chern-Simons densities P 2n in the YM gauge theory (1.1) in the sense that they are gauge invariant, closed and metric independent.
Our aim is to investigate this rich class of topological invariants of extended gauge theory as well as to find out potential gauge anomalies performing transgressions analogous to (1.1) and (1.3): (1.10) Therefore we shall perform the following transgressions: We shall find explicit expressions for these primary invariants in terms of higher order polynomials of the curvature forms on a vector bundle. The most difficult challenge will be the evaluation and differentiation of the very complicated noncommutative polynomial expressions as well as the search of the most simple expressions for the secondary forms. The secondary forms are not uniquely defined. Indeed, the secondary form ψ 2n+3 is defined modulo the exterior derivative of an arbitrary (2n + 2)-form ψ 2n+3 ∽ ψ 2n+3 + dα 2n+2 , the form φ 2n+5 modulo the exterior derivative of a (2n + 4)form φ 2n+5 ∽ φ 2n+5 + dβ 2n+4 and the form ρ 2n+7 modulo the exterior derivative of a (2n + 6)-form ρ 2n+7 ∽ ρ 2n+7 + dγ 2n+6 . When the difference of two closed forms is an exact form, they are said to be cohomologous to each other. Therefore the problem is to find out the most simple representatives in the set of equivalence classes. Conveniently chosen exact forms will dramatically simplify the expressions. These problems will be solved by using properties of symmetrized traces (1.5) defined in [10].
In Section 2 we shall present a general construction and analysis of the primary forms Φ 2n+4 and Ξ 2n+6 , their secondary forms ψ 2n+3 and φ 2n+5 and the corresponding anomalies represented by ψ 1 2n+2 and φ 1 2n+4 . The material of this section is not completely new, but the alternative derivation in terms of symmetrized traces will allow to extend the results to the higher-dimensional forms Υ 2n+8 .
In Section 3 we shall derive the explicit expressions for the primary form Υ 2n+8 , written in terms of the symmetrized traces (1.5), starting from the low-dimensional forms listed in [21]. Next, we shall find the secondary forms ρ 2n+7 and the corresponding gauge anomalies ρ 1 2n+6 associated with each of the independent gauge transformations δ ξ , δ ζ 2 , δ ζ 4 , δ ζ 6 . In the conclusion we summarize the primary and secondary invariant forms constructed in the article. In the Appendix we present useful formulas for the gauge transformations of the fields, the corresponding Bianchi identities and a one-parameter deformation of fields generalizing deformation of [6,7].

Gauge and Metric Independent Forms
We shall start by deriving the already known results for the form Φ 2n+4 using the properties of the symmetrized traces. This approach will allow to extend the derivation to more complicated cases.
Indeed, the form can be represented in terms of a symmetrized trace as [20,21] Φ 2n+4 = Str G 4 , G n (2.1) so that its gauge invariance with respect to the standard-scalar gauge transformations δ ξ and the tensor gauge transformations δ ζ 2 can be easily checked: On the last steps we used the identity (B.10) of the Appendix B of the article [10]. Our next step is to check that Φ 2n+4 is a closed form. Indeed, where we used the Bianchi identity DG = 0. To find out the secondary form we shall use a oneparameter deformation of the gauge potentials: [6,20,21] A t = tA, A 3t = tA 3 , A 5t = tA 5 , A 7t = tA 7 , A 9t = tA 9 , ...., defined in the Appendix (5.6), (5.7), take the derivative and employ (B.13) of [10]: Integrating the above equation over the parameter t in the interval [0, 1] we get the integral representation of the secondary form This secondary form is not unique. It can be modified by the addition of the differential of a one-steplower-order form dα 2n+2 . Two closed forms which differ by an exact form are said to be cohomologous to each other. The integral on the right hand side of the equation looks complicated, but if we add a properly chosen exact form dα 2n+2 , then it will be dramatically simplified. Let us take it in the following form: Then we get: where D t G t = 0 4 . Using the relations (5.2), (5.7) and the integral representation (2.4), we have: (2.5) Thus, the secondary form gets the following compact form The secondary forms (2.4) and (2.6) are representatives of the same cohomology class, because their difference is an exact form dα 2n+2 , but, as one can see (2.6), has a much more simple expression.
Using (5.2), (5.5) we can verify that dψ 2n+3 = Φ 2n+4 : The form (2.6) allows to find the potential anomalies of the theory, by the following transgression steps. The gauge invariance of the primary form Φ 2n+4 means that δΦ 2n+4 = d(δψ 2n+3 ) = 0. By employing the Poincare's lemma it follows that δψ 2n+3 = dψ 1 2n+2 , where ψ 1 2n+2 is the potential anomaly. Thus, in order to proceed we have to calculate the gauge variation of the secondary form with respect to the gauge transformations δ ξ and δ ζ 2 . We have 5 that is, the secondary form is gauge invariant with respect to standard-scalar gauge transformations δ ξ and therefore there are no gauge anomalies associated with the scalar gauge transformations. But with respect to the tensor gauge transformations δ ζ 2 there are anomalies Therefore the anomaly is In summary, we have the expressions (2.1) for primary form Φ 2n+4 , the expression (2.6) for the secondary form ψ 2n+3 and (2.9) for the anomaly.
We shall now move to the next primary form Ξ 2n+6 , which can be written in terms of symmetrized traces as [21]: Each term of this expression is independently gauge invariant. Indeed, and for the second term we shall get where in the last two equations we again used (B.10) of [10]. However only the sum of these terms is a closed form. We can check the closeness of the form Ξ 2n+6 by taking the exterior derivative: On the first step we used (B.13) of [10] and on the second -the Bianchi identities DG = 0. On the last step, the terms in the big brace as well as the last term are zero because of (B.10) of [10]. Again, according to Poincaré's lemma, this equation implies that Ξ 2n+6 can be locally written as an exterior derivative of a certain (2n +5)-form. In order to find that form we need to differentiate Ξ 2n+6 over the deformation parameter t. We have: On the second, third and last steps we used the relations (5.7) and (B.13), (B.10) of [10] respectively.
Integrating the above equation over the parameter t in the interval [0, 1] we shall get the following integral representation of the secondary form: As we already mentioned above, the secondary form is not unique and it can be modified by the addition of the differential of a one-step-lower-order form dβ 2n+4 (1.11). As in the case of ψ 2n+3 , we will use this freedom in order to simplify our result. Adding dβ 2n+4 , where we get: where one should use the Bianchi identities D t G t = 0 and (B.13) of [10]. Next, with the aid of (5.2) and (5.7) we get, One can see that the first, the third and fifth terms cancel with the first two terms of φ 2n+5 (2.14).
With the aid of (B.9) of [10], the second and the forth terms combine to give Finally using the equation tD t A 3 = G 4t and (5.2) we shall get the following expression for φ 2n+5 : Hence, after the integration we get By comparing the representations of the secondary form φ 2n+5 in (2.14) and in (2.15) it becomes clear that the last expression is much more simple and transparent. Let us verify that the exterior derivative of the above form leads us back to Ξ 2n+6 .
where on the second step the second and third terms cancel because of (B.10) of [10].
In order to find out the potential anomalies we have to calculate the gauge variation of the secondary form φ 2n+5 with respect to the scalar, rank-2 and rank-4 gauge parameters. We have Thus, there are no anomalies in the standard gauge symmetry. But there are potential anomalies in the higher-rank gauge symmetries. Indeed, Hence the anomalies are: In summary, we have the expressions (2.10) for primary form Ξ 2n+6 , the expression (2.15) for the secondary form φ 2n+5 and (2.18) for the anomalies.
In the recent article [21] the authors found the following exact, metric independent forms, linear in G 8 : In order to find the general expression for the forms linear in G 8 let us first find the next form Υ 14 .
For that let us consider the linear combination of all possible rank-14 Lorentz invariant traces which can be constructed in terms of field strength tensors: The two terms with the coefficients c 6 = c 8 can be dropped since they compose the form Ξ 14 (2.10).
The terms with coefficients c 10 and c 11 can also be dropped since they represent the forms Φ 14 (2.1) and P 14 (1.1) respectively. Using the Bianchi identities (5.5) one can calculate the derivatives of the following terms: and see that the following combination is a closed form: Hence, and it can be written in terms of symmetric trace. The rest of the terms with the coefficients c 1 , c 2 , c 9 do not comprise any closed form.
One can check that the Υ 14 is gauge invariant. In terms of the standard gauge parameter we get: In an analogous way one can easily prove that Υ 14 is invariant under the transformations of the higher-tensor gauge parameters ζ 2 , ζ 4 , ζ 6 .
Having in hand the series of forms Υ 10,12,14 one can guess a general expression for Υ 2n+8 and check that it fulfills all the required properties. We suggest the following general form for Υ 2n+8 : Analogously, one can check that each term of Υ 2n+8 is separately invariant under the variation over the higher-tensor gauge parameters. The last two calculations clearly demonstrate the power of the use of the symmetrized traces and of their properties.
Taking the exterior derivative of Υ 2n+8 one can become convinced that it is a closed form: Again, according to Poincaré's lemma, this equation implies that Υ 2n+8 can be locally written as an exterior derivative of a certain (2n+7)-form ρ 2n+7 . In order to find that form we need to differentiate Υ 2n+8 over the deformation parameter t, as we did in the previous section: In some of the terms we can extract the covariant exterior derivatives outside the symmetrized traces: As one can see, some of the terms are written as exterior derivatives. We shall collect them in the formula below and then combine the rest of the terms in square brackets: The terms in the square brackets vanish thanks to the identity (B.10). Therefore we have the following integral representation for the secondary form: As we already discussed in the introduction and in the previous section, the secondary forms are defined modulo exact forms and in the given case up to (2n+7)-form ρ 2n+7 ∽ ρ 2n+7 +dγ 2n+6 . Therefore we have to choose an appropriate candidate for γ 2n+6 . It appears that to simplify the result the exterior derivative of the following form should be subtracted: Subtracting the first two terms of the dγ 2n+6 from ρ 2n+7 we will get: Subtracting now the last three terms of the dγ 2n+6 , we will get: Let us check that the exterior derivative of the simplified secondary form gives us back the primary form Υ 2n+8 . We have, dρ 2n+7 = Str DA 7 , G n + nStr DG 6 , A 3 , G n−1 + nStr G 6 , DA 3 , G n−1 + +2nStr DG 4 , A 5 , G n−1 + 2nStr G 4 , DA 5 , G n−1 + Due to (B.10) of [10] the first part of the second term cancels with the sixth one, the second part of the second term cancels with the forth one, and the third term cancels with the ninth one. The secondary form allows to find the potential anomalies of the theory by performing the transgression steps. Thus in order to find out potential anomalies we have to calculate the gauge variation of the secondary form ρ 2n+7 with respect to the scalar, rank-2, rank-4 and rank-6 gauge parameters: where the identity (B.13) of [10] was used. There are no anomalies in the standard gauge symmetry.
In summary, we have the expressions (3.6) for primary form Υ 2n+8 , the expression (3.10) for the secondary form ρ 2n+6 and (3.15) for the anomalies.
In this article we are interested in enumerating and classifying metric independent, gauge invariant and closed forms in generalized YM theory. The forms that we constructed are defined in various dimensions, are based on non-Abelian tensor gauge fields and are polynomial on the corresponding fields strength tensors -curvature forms. All these forms Φ 2n+4 , Ξ 2n+6 and Υ 2n+8 are analogous to the Pontryagin-Chern-Simons densities P 2n in YM gauge theory (1.1). They are closed forms, but not globally exact.
The secondary characteristic classes ψ 2n+3 , φ 2n+5 and ρ 2n+7 have been expressed in integral form (2.4),(2.14) and (3.9) in analogy with the Chern-Simons form (1.4). The secondary forms are not unique, because they can be modified by the addition of the differential of a one-step-lower-order forms. By adding the properly chosen exact forms (2.6), (2.15) and (3.10) respectively to the secondary forms ψ 2n+3 , φ 2n+5 and ρ 2n+7 , we are led to much more simple expressions. The gauge variation of the secondary forms can also be found: (2.9), (2.18) and (3.15) yielding the potential anomalies in gauge field theory. The above general considerations should be supplemented by an explicit calculation of loop diagrams involving chiral fermions. The argument in favor of the existence of these potential anomalies is based on the fact that they fulfill Wess-Zumino consistency conditions [4,6,7,8,9,18]. The integrals of these forms over the corresponding space-time coordinates provide us with new topological Lagrangians [21] and with a generalization of the Chern-Simons quantum field theory [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39].
At the same time, these densities constructed on a high-dimensional manifold have their own value. Their integrals represent global geometric invariants suggesting the existence of new topological characterization of the manifolds [6,7,10,16,17].