Fusion in Fractional Level sl^(2)-Theories with k=-1/2

Abstract

The fusion rules of conformal field theories admitting an sl^(2)-symmetry at level k=-1/2 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 u^(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=-2 triplet model and the implications for the beta gamma ghost system are briefly discussed.

33 pages, 8 figures; v2 - added a ref and deleted a paragraph from the conclusions

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• Perturbing the symmetric orbifold from the worldsheet
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• Unitary and non-unitary $N=2$ minimal models
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• Modularity of Bershadsky-Polyakov minimal models
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Representations of the Nappi--Witten vertex operator algebra
• Quasi-Characters in $\widehat{su(2)}$ Current Algebra at Fractional Levels
• Staggered modules of $N=2$ superconformal minimal models