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The level two and three modular invariants of \(SU(n)\). (English) Zbl 0871.17019

Summary: The author explicitly classifies all modular invariant partition functions for \(A_r^{(1)}\) at levels 2 and 3. Previously, these were known only for level 1. Level 2 exceptions exist at \(r=9\), 15, and 27; level 3 exceptions exist at \(r=4\), 8, and 20. One of these is new, but the others were all anticipated by the ‘rank-level duality’ relating \(A_r^{(1)}\) level \(k\) and \(A_{k-1}^{(1)}\) level \(r+1\). The main recent result which this Letter rests on is the classification of ‘\({\mathcal A} {\mathcal D} {\mathcal E}_7\)-type invariants’.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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