Loop groups and noncommutative geometry

TL;DR

This paper explores the representation theory of loop groups through the lens of noncommutative geometry and K-theory, constructing spectral triples linked to various conformal field theory models.

Contribution

It introduces a novel construction of spectral triples for loop group representations, extending to multiple rational chiral conformal field theories.

Findings

Spectral triples are constructed for loop group representations.

The approach applies to coset and moonshine conformal nets.

Provides a new geometric perspective on conformal field theories.

Abstract

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level l projective unitary positive-energy representations of any given loop group $LG$. The construction is based on certain supersymmetric conformal field theory models associated with LG in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.