Modularity of logarithmic parafermion vertex algebras

The parafermionic cosets (mathsf {C}_{k} = {text {Com}} ( mathsf {H} , mathsf {L}_{k}(mathfrak {sl}_{2}) )) are studied for negative admissible levels k, as are certain infinite-order simple current extensions (mathsf {B}_{k}) of (mathsf {C}_{k}). Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to (mathsf {C}_{k}), irreducible (mathsf {C}_{k})- and (mathsf {B}_{k})-modules are obtained from those of (mathsf {L}_{k}(mathfrak {sl}_{2})). 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 (mathsf {B}_{k})-modules. The irreducible (mathsf {C}_{k})- and (mathsf {B}_{k})-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 (mathsf {B}_{k}) are (C_2)-cofinite vertex operator algebras.