Bicoloured torus loop groups

Abstract

For every finite dimensional Lie group one can consider the group of all smooth loops on it, called its loop group. Such loop groups have long been studied for, among other reasons, their relations to conformal field theories and topological quantum field theories. In this thesis we introduce a new generalisation of the loop groups associated to tori, which we name bicoloured torus loop groups. An element of such a group consists mainly of two paths, each lying on two (possibly different) fixed tori. The definition of the group additionally imposes a constraint on the endpoints of these paths with respect to each other. We study these bicoloured torus loop groups by demonstrating that they have a theory analogous to that of ordinary torus loop groups. We namely construct certain central extensions of them by the group U(1) and prove that they have many properties in common with specific, known, central extensions of ordinary loop groups. Notably, we are able to construct and classify the irreducible, positive energy representations of these extensions.