The Cappelli-Itzykson-Zuber A-D-E Classification
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Bibliographic record
Abstract
Itzykson and Zuber classified all modular invariant partition functions for the conformal field theories associated to the affine A1 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 that their partition functions fall into the AD-E pattern familiar from the finite subgroups of SU2(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\\Gamma 1g, and let S and T be the (n \\Gamma 1) \\Theta (n \\Gamma 1) matrices with entries Sab = r 2n sin(ss abn) ; Tab = exp[ssi a 2 2n] ffia;b: Find all (n \\Gamma 1) \\Theta (n \\Gamma 1) matrices M such that
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it