Fractional Loop Group and Twisted K-Theory

TL;DR

This paper explores the structure of abelian extensions of q-differentiable loop groups, constructs highest weight modules, extends to supersymmetric models, and discusses applications to twisted K-theory, advancing understanding of higher-dimensional current algebras.

Contribution

It introduces a generalization of central extensions to q-differentiable loop groups and extends module constructions to supersymmetric models, linking to twisted K-theory.

Findings

Constructed highest weight modules for the Lie algebra.

Extended the construction to supersymmetric Wess-Zumino-Witten models.

Discussed applications to twisted K-theory on G.

Abstract

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.