3D Current Algebra and Twisted K Theory

TL;DR

This paper explores extending the realization of equivariant twisted K theory classes from loop groups to higher 3D current algebras, using algebraic constructions that mimic Fredholm operators but do not form genuine operators.

Contribution

It introduces a partial extension of the current algebra framework to 3D algebras, using algebraic expressions instead of true operators, advancing the understanding of higher-dimensional K theory.

Findings

Partial success in extending to 3D current algebra

Algebraic expressions transform correctly under gauge transformations

No true Fredholm operators obtained, only algebraic forms

Abstract

Equivariant twisted K theory classes on compact Lie groups $G$ can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra $LG$ using a supersymmetric Wess-Zumino-Witten model. The aim of the present article is to extend the construction to higher loop algebras using an abelian extension of a $3D$ current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to $3D$ gauge transformations but do not define true Hilbert space operators.