Fusion in fractional level slˆ(2)-theories with k=−12

The fusion rules of conformal field theories admitting an slˆ(2)-symmetry at level k=−12 are studied. It is shown that the fusion closes on the set of irreducible highest weight modules and their images under spectral flow, but not when “highest weight” is replaced with “relaxed highest weight”. The fusion of the relaxed modules, necessary for a well-defined uˆ(1)-coset, gives two families of indecomposable modules on which the Virasoro zero-mode acts non-diagonalisably. This confirms the logarithmic nature of the associated theories. The structures of the indecomposable modules are completely determined as staggered modules and it is shown that there are no logarithmic couplings (beta-invariants). The relation to the fusion ring of the c=−2c=−2 triplet model and the implications for the βγ ghost system are briefly discussed.