Towards General Relativity as a generalized Yang-Mills theory
As an application of the generalized principal bundle theory to covariant Lagrangian field theories, we aim at the development of an instance of generalized gauge theories, with the prospect of a unifying language for Yang-Mills theories and General Relativity. After reviewing the basic definitions of Lie group fiber bundles and generalized principal bundles, we provide horizontal lift and local characterizations of Lie group fiber bundle connections and generalized principal connections. Subsequently, we consider in the framework of classical field theories the kinematics and dynamics sides of generalized principal connections, which lead to a proposed notion of generalized Yang-Mills theories. Finally, we show how vector bundles are examples of generalized principal bundles and that a generalized principal connection on a vector bundle is an affine connection given in terms of basic soldering forms. We are able to recover, under appropriate assumptions, the Vielbein formulation of (vacuum) General Relativity in this setting, hinting at a (generalized) gauge theory of gravity.