We study the vertex algebras associated with modular invariant representations of affine Kac–Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph’s characteristic varieties. We show that an irreducible highest weight representation of a nontwisted affine Kac–Moody algebra at an admissible level is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamović and Milas on the rationality of admissible affine vertex algebras in the category .
"Rationality of admissible affine vertex algebras in the category ." Duke Math. J. 165 (1) 67 - 93, 15 January 2016. https://doi.org/10.1215/00127094-3165113