Abstract
It is pointed out that in the Kac-Moody algebras fulfilling the fermionization criterion of Goddard, Nahm and Olive and having a non-minimal value of the central charge k, only a proper subset of the allowed unitary highest weight representations can actually be encoded in a free fermion theory. These truly fermionizable representations are selected by a very specific non-regular embedding of the fermionizable Kac-Moody algebra into the lowest level SO (N F ) Kac-Moody algebra, N F being both the number of fermions and the dimension of the GNO symmetric space. This embedding is a particular case of the embeddings considered by Bais and Bouwknegt and by Schellekens and Warner, for which the Virasoro central charge of the subgroup is equal to that of the group. Furthermore, these fermionizable representations span an orbit of the modular group always leading to a non trivial modular invariant partition function.
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