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May 29, 2020

Level bounds for exotic quantum subgroups

The canonical source of modular tensor categories are quantum groups at (certain) roots of unity, which correspond to the conformal nets which Wassermann and others associated to loop groups, or the vertex operator algebras which are associated to the affine Lie algebras at positive integer level (roughly speaking, the level is the order of the root of unity). The classification of quantum subgroups is essentially the classification of conformal field theories associated to these quantum groups. The early 2000’s were tantalised by announcements by Ocneanu of the existence of an upper bound for the level of those quantum groups possessing exceptional quantum subgroups. Unfortunately, details were not forthcoming. But a breakthrough happened in 2017 when Schopieray made this explicit for the rank 2 cases: e.g. he found about 18 million potentially exceptional levels for G_2. In my talk I’ll explain that combining Ocneanu-Schopieray’s idea with Galois considerations yields much smaller bounds (e.g. for G_2 it gives only 2 potentially exceptional levels, and for e.g. sl(7) there are only 36 such levels). I’ll then use this bound to give the classification of quantum subgroups for all ranks up to and including 5.