Gauge Theory for Fiber Bundles

Gauge theory usually investigates the space of principal connections on a principal fiber bundle (P, p, M, G) and its orbit space under the action of the gauge group (called the moduli space), which is the group of all principal bundle automorphisms of P which cover the identity on the base space M. It is the arena for the Yang-Mills-Higgs equations which allows (with structure group U (1) × SU (2)) a satisfactory unified description of electromagnetic and weak interactions, which was developed by Glashow, Salam, and Weinberg. This electro-weak theory predicted the existence of massive vector particles (the intermediate bosons W + , W − , and Z), whose experimental verification renewed the interest of physicists in gauge theories. On the mathematical side the investigation of self dual and anti self SU (2)-connections on 4-manifolds and of their image in the orbit space led to stunning topological results in the topology of 4-manifolds by Donaldson. The moduli space of anti self dual connections can be completed and reworked into a 5 dimensional manifold which is a bordism between the base manifold of the bundle and a simple manifold which depends only on the Poincaré duality form in the second dimensional homology space. Combined with results of Freedman this led to the discovery of exotic differential structures on R 4 and on (supposedly all but S 4) compact algebraic surfaces. In his codification of a principal connection [Ehresmann, 1951] began with a more general notion of connection on a general fiber bundle (E, p, M, S) with standard fiber S. This was called an Ehresmann connection by some authors, we will call it just a connection. It consists of the specification of a complement to the vertical bundle in a differen-tiable way, called the horizontal distribution. One can conveniently describe such a connection as a one form on the total space E with values in the vertical bundle, whose kernel is the horizontal distribution. When combined with another venerable notion , the Frölicher-Nijenhuis bracket for vector valued differential forms (see [Frölicher-Nijenhuis, 1956]) one obtains a very convenient way to describe curvature and Bianchi identity for such connections. Parallel transport along curves in the base space is defined only locally, but one may show that each bundle admits complete connections, whose parallel transport is globally defined (theorem 9.10). For such connections one can define holonomy groups and holonomy Lie algebras and …