Modularity of logarithmic parafermion vertex algebras

The parafermionic cosets Ck = Com( H, Lk ( sl2)) are studied for negative admissible levels k, as are certain infinite-order simple current extensions Bk of Ck. Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to Ck, irreducible Ck -and Bk-modules are obtained from those of Lk ( sl2). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible Bk-modules. The irreducible Ck -and Bk-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the Bk are C2-cofinite vertex operator algebras.