Geometry of Gauge Fields 1102-681

Gauge fields play a fundamental role in physics: they provide tools to describe interaction carrying bosons. The electromagnetic field was the first to be recognized as a gauge field; later, there appeared Yang-Mills fields, associated with non-abelian gauge groups and it became clear that gravitation can also be considered as a gauge field. Connections on principal bundles provide the mathematical structures necessary to describe these fields; their topological properties explain the non-trivial gauge configurations that are associated with magnetic monopoles and instantons. Of fundamental importance is the mechanism of symmetry breaking; its geometry is described by a reduction of the principal bundle with a connection.

Program:
1. Remarks on the history of the subject; the notion of gauge transformations was introduced by Weyl (1918) in his attempt to construct a unified theory of gravitation and electromagnetism. Dirac (1931) introduced quantization of magnetic charges. The Yang-Mills theory (1954) was based on SU(2) as the gauge group. From 1964 on there is further development of the theory based on the discovery of spontaneous symmetry breaking that allows vector bosons to acquire mass.
2. Elements of the classical theory of fields: variational principles, their invariance and Noether theorems; conservation laws and constraints; the Cauchy problem.
3. Elements of the theory of connections on principal bundles; curvature and Bianchi identities; holonomy groups, reduction of bundles and symmetry breaking. Gauge potentials as connections. Symmetries of gauge configurations. Cartan connections.
4. Characteristic classes. The Weil homomorphism and invariant polynomials. Chern, Pontryagin and Euler classes. Magnetic charges and instantons.
5. Relativistic gravitation can be considered as a gauge theory. The Einstein-Cartan theory.

Description by Prof. Andrzej Trautman, June 2007