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

Details The Cappelli-Itzykson-Zuber A-D-E classification The Cappelli-Itzykson-Zuber A-D-E classification

1999-02-09

arxiv.org/abs/math/9902064v2

Rev.Math.Phys. 12 (2000) 739-748

Mathematics

Quantum Algebra

10 pages, plain tex; an historical remark on A-D-E corrected, 2 references added, several new comments added to conclusion

Scientific paper

10.1142/S0129055X00000265

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 mathematical 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.

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