Gauge-invariant theories and higher-degree forms

TL;DR

This paper extends the Chern-Weil theorem to free differential algebras with higher-degree forms, providing new formulas for Chern-Simons and transgression forms and exploring potential anomalies.

Contribution

It introduces a generalized Chern-Weil theorem for free differential algebras with one p-form extension, expanding the mathematical framework for gauge theories.

Findings

Generalized covariant derivative for free differential algebras

Extended formulas for Chern-Simons and transgression forms

Analysis of potential anomalies in this structure

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.