Abelian Extensions, Fractional Loop Group and Quantum Fields

Sammanfattning: This thesis deals with the theory of Lie group extensions, Lie conformal algebras and twisted K-theory, in the context of quantum physics. These structures allow for a mathematically precise description of certain aspects of interacting quantum ?eld theories. We review three concrete examples, namely symmetry breaking (or anomalies) in gauge theory, classification of D-brane charges in string theory and the formulation of integrable hierarchies in the language of Poisson vertex algebras. The main results are presented in three appended scienti?c papers.In the ?rst paper we establish, by construction, a criterion for when an in?nite dimensional abelian Lie algebra extension corresponds to a Lie group extension.In the second paper we introduce the fractional loop group LqG, that is the group of maps from a circle to a compact Lie group G, with only a small degree of differentiability q ? R+ in the Sobolev sense. We construct abelian extensions and highest weight modules for the Lie algebra Lqg, and discuss an application to equivariant twisted K-theory on G.In the third paper, we construct a structure of calculus algebra on the Lie conformal algebra complex and provide a more detailed description in the special case of the complex of variational calculus.