From quantum field theory to gerbes and twisted K-theory

A breakdown of classical symmetries (gauge or reparametrization) can occur in a system of fermions in external fields. The topology of this mechanism is understood in terms of families index theory in the path integral formalism. In the Hamiltonian approach the families index theory leads to odd cohomology classes as obstructions to covariant quantization. In particlular, the 3-cohomology class known as the Dixmier-Douady class plays a central role; it gives a topological classification of gerbes over the configuration space of the quantum system.

I give a review how the quantum mechanical symmetry breaking leads to the concept of gerbe in terms of concerete examples from gauge theory. Also, more recent developments leading to constructions of twisted K theory classes on a manifold from a field theory model will be covered.