Kac-Peterson, Perron-Frobenius, and the Classification of Conformal Field Theories

Dedicated to the memory of Rudelle Hall, teacher and friend 1. Introduction: The classification of conformal field theories. Conformal field theories (CFTs) and related structures have been of considerable value to mathematics , as for instance the work of Witten has shown. This paper is concerned with their classification. Fortunately, the problem has a simple expression in terms of the characters of Kac-Moody algebras (see (1.2) below), and requires no prior knowledge of CFT. Nevertheless , for reasons of motivation, in the following paragraphs we will sketch the definition of CFT. Before discussing this background material, let us quickly state the actual mathematical problem addressed in this paper. The characters of an affine algebra at fixed level k define in a natural way a unitary representation of SL 2 (Z) (see equations (3.3) below). The ultimate classification problem here is to find all matrices M which commute with the matrices of this representation, and which in addition obey relations (1.2b) and (1.2c) – such M are called physical invariants. In this paper we address the subproblem of finding all physical invariants which in addition satisfy (1.3b), where S is the group of all symmetries of the (extended) Coxeter-Dynkin diagram – these M we call ADE 7-type invariants. Almost every physical invariant is expected to be a ADE 7-type invariant. In this paper we develop a program to find all of these for any affine algebra, and apply it to explicitly find them for the algebra A