The Cappelli-Itzykson-Zuber A-D-E Classification

. In 1986 Cappelli, Itzykson and Zuber classified all modular invariant partition functions for the conformal field theories associated to the affine A 1 algebra; they found they fall into an A-D-E pattern. Their proof was difficult and attempts to generalise it to the other affine algebras failed -- in hindsight the reason is that their argument ignored most of the rich structure present. We give here the "modern" proof of their result; it is an order of magnitude simpler and shorter, and much of it has already been extended to all other affine algebras. We conclude with some remarks on the A-D-E pattern appearing in this and other RCFT classifications. 1. The problem One of the more important results in conformal field theory is surely the classification due to Cappelli, Itzykson, and Zuber [3; see also 4] of the genus 1 partition functions for the theories associated to A (1) 1 (which in turn implies the classification of the minimal models). Their list was curious: Kac noticed th..