Modular invariants and twisted equivariant \(K\)-theory. II: Dynkin diagram symmetries. (Q2874240)

This article deepens the relationship between conformal field theory and twisted \(K\)-theory. Conformal field theories are two-dimensional quantum field theories with many symmetries. They consist of two simpler halves, chiral field theories, that are glued together; the glueing is described by the modular invariant. A long-term goal of this article is to understand better which modular invariants come from a conformal field theory.NEWLINENEWLINE\(K\)-theory here also means finer structure such as actual vector bundles, and related theories such as \(K\)-homology and eventually KK-theory.NEWLINENEWLINEFreed, Hopkins and Teleman have identified the Verlinde rings of a group \(G\) with certain equivariant twisted \(K\)-theory groups of \(G\), with \(G\) acting on itself by conjugation. This result may be interpreted in terms of conformal field theory, but only encodes limited information about the chiral halves. More structure of a conformal field theory may be described in terms of an associated subfactor; the structure considered in this article, besides the Verlinde ring, is the full system, the two alpha-induction maps between them, the neutral system, the nimrep and D-brane charges. Here these invariants are described systematically in the language of twisted \(K\)-theory, in the case of general Lie groups. This \(K\)-theoretic approach is crucial, among other things, to prove a conjecture about the nimreps for certain natural modular invariants.NEWLINENEWLINE For Part I see [the authors, Commun. Number Theory Phys. 3, No. 2, 209--296 (2009; Zbl 1182.19003)].