Modular invariants and twisted equivariant K-theory

The modular invariant partition functions of conformal field theory (CFT) have a rich interpretation within von Neumann algebras (subfactors), which has led to the development of structures such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction, etc. Modular categorical interpretations for these have followed. More recently, Freed-Hopkins-Teleman have expressed the Verlinde ring of conformal field theories associated to loop groups as twisted equivariant K-theory. For the generic families of modular invariants (i.e. those associated to Dynkin diagram symmetries), we build on Freed-Hopkins-Teleman to provide a $K$-theoretic framework for other CFT structures, namely the full system, nimrep, alpha-induction, D-brane charges and charge-groups, etc. We also study conformal embeddings and the E7 modular invariant of SU(2), as well as some families of finite groups. This new K-theoretic framework allows us to simplify and extend the less transparent, more ad hoc descriptions of these structures obtained within CFT using loop group representation theory