Loop Groups, K-theory and Noncommutative Geometry

We describe the representation theory of loop groups in
terms of K-theory and noncommutative geometry. For any simply
connected compact Lie group G and any positive integer level l we
consider a natural noncommutative universal algebra whose 0th K-group
can be identified with abelian group generated by the level l
positive-energy representations of the loop group LG.
Moreover, for any of these representations, we define a spectral
triple in the sense of A. Connes and compute the corresponding index
pairing with K-theory. As a result, these spectral triples give rise
to a complete noncommutative geometric invariant for the
representation theory of LG at fixed level l. The construction is
based on the supersymmetric conformal field theory models associated
with LG and it can be generalized, in the setting of conformal nets,
to many other rational chiral conformal field theory models including
loop groups model associated to non-simply connected compact Lie
groups, coset models and the moonshine conformal field theory. (Based
on a joint work with Robin Hillier)