From Dynkin diagram symmetries to fixed point structures

TL;DR

This paper explores how automorphisms of Dynkin diagrams induce automorphisms of Kac-Moody algebras, linking their fixed point structures to orbit Lie algebras and twining characters, with implications for conformal field theory.

Contribution

It describes the relationship between Dynkin diagram automorphisms, orbit Lie algebras, and twining characters, providing new insights into fixed point resolution in conformal field theory.

Findings

Twining characters equal to orbit Lie algebra characters

Orbit Lie algebras describe fixed point structures

Automorphisms induce algebra automorphisms and module maps

Abstract

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra induces an automorphism of the algebra and a mapping between its highest weight modules. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the `orbit Lie algebra'. In particular, the generating function for the trace of the map on modules, the `twining character', is equal to a character of the orbit Lie algebra. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.