Free Differential Algebras: Their Use in Field Theory and Dual Formulation

TL;DR

This paper explores how free differential algebras extend Lie algebra structures to include antisymmetric tensor fields, providing a geometric and dual formulation framework for gauge theories with p-form potentials.

Contribution

It introduces an algebra of FDA transformations with extended Lie derivatives, establishing their closure and dual formulation, thus offering a new geometric perspective on gauge theories with antisymmetric tensors.

Findings

FDA transformations form a closed algebra.

The dual formulation of FDA is established.

Gauge invariance is extended to p-form potentials.

Abstract

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials ($p > 1$). We study here the algebra of FDA transformations. To every p-form in the FDA we associate an extended Lie derivative $\ell$ generating a corresponding ``gauge" transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's.