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

Abstract. In 1986 Cappelli, Itzykson and Zuber classied all modular invariant partition functions for the conformal eld theories associated to the ane A 1 algebra; they found they fall into an A-D-E pattern. Their proof was dicult and attempts to generalise it to the other ane 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 ane algebras. We conclude with some remarks on the A-D-E pattern appearing in this and other RCFT classications. 1. The problem One of the more important results in conformal eld theory is surely the classication 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 classication of the minimal models). Their list was curious: Kac noticed that their partition functions fall into the A-D-E pattern familiar from the nite subgroups of SU 2 (C), simple singularities, simply-laced Lie algebras, subfactors with index < 4, etc. See e.g. [9]. The problem can be phrased as follows. Fix any integer n 3. Let P = f1; 2; : : : ; n 1g, and let S and T be the (n 1) (n 1) matrices with entries S a