Bicoloured torus loop groups

TL;DR

This paper introduces bicoloured torus loop groups, a new generalization of traditional loop groups, and develops their central extensions and representation theory, expanding the mathematical framework related to conformal field theories.

Contribution

It defines bicoloured torus loop groups, constructs their central extensions, and classifies their positive energy representations, extending the theory of loop groups to a new setting.

Findings

Bicoloured torus loop groups are well-defined and analogous to ordinary loop groups.

Central extensions of these groups by U(1) are constructed and studied.

Irreducible positive energy representations are classified for these extensions.

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 $\mathrm U(1)$ 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.