Yang-Mills theory for multiplicative Ehresmann connections

Abstract

We develop a twofold generalization of classical Yang-Mills theory, extending it from principal bundles to the setting of possibly non-transitive and non-integrable Lie algebroids. The classical theory is recovered when one considers the Atiyah algebroid of a principal bundle. In our framework, principal bundle connections are replaced by the more general notion of (infinitesimal) multiplicative Ehresmann connections. An action functional for such connections is constructed, now including a curvature 3-form contribution, alongside the usual curvature 2-form term, and the resulting variational problem is naturally constrained by a cohomological condition. We derive the associated Euler-Lagrange equations, and define a class of self-dual solutions (instantons) in both 4 and 5 dimensions. We also show that the solution space is invariant under gauge transformations, and compute its tangent space at a solution. As an important example, we show that our framework produces a Yang-Mills theory for connections on bundle gerbes.