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

Abstract

The classification of CFTs has an important subproblem, namely classifiying the partition functions for WZW theories. This subproblem is intimately connected to the modular behaviour of the characters of affine algebras. This paper addresses the classification of the largest class of these -- the natural generalization to any affine algebra of the A-series, D-series, and $E_7$-exceptional appearing in the A-D-E classification of Cappelli-Itzykson-Zuber for $A_1^{(1)}$. We give a programme for classifying these for any algebra, and explicitly do this for the most interesting case: $A_r^{(1)}$ at any level k. The exceptionals appear at (r,k)=(1,16), (2,9), (3,8), (4,5), (7,4), (8,3) and (15,2).