Abstract
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.
Highlights
Abelian Kalb-Ramond gauge field [3]
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 higherdegree differential forms
In this article we have introduced a gauge invariant density for a particular free differential algebra (FDA), analogous to the Chern-Pontryagin topological invariant for Lie algebras
Summary
The dual formulation of Lie algebras provided by the Maurer-Cartan equations can be naturally extended to p-forms. Since ΘA(p) N is a basis, the exterior derivative dΘA(p) can be written in terms p=1 of the same basis This allows to write a set of Maurer-Cartan (MC) equations for a mathematical structure called free differential algebra dΘA(p). 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. (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 εA and εj of the transformations and can be sumarized as follows δμA = dεA + CBAC μBεC + 2RABC εBμC + εj RAj ,. The transformation of Bi depends on εA and εi [18, 19]
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