The Algebra & Geometry of Modern Physics

In a series of lectures, the applicability of the structure of a principal bundle with connection in the context of field theory will be demonstrated. First, the definition of a principal bundle with a structural groupoid and a compatible connection will be presented, with view to establishing a natural description of field theories with some of their global symmetries rendered local (or "gauged"). The concept of equivariant descent of a geometric object (such as, e.g., a tensor, a bundle or a higher structure) will be introduced and illustrated through the example of an equivariant bundle over a base acted upon by a group. Finally, a general gauge principle will be formulated for field theories defined in terms of lagrangean densities, and subsequently concretised in the physically interesting setting of the dynamics of topologically charged objects in external fields. This will, in particular, lead us to an algebroidal and cohomological classification of obstructions to the gauging, aka gauge anomalies.