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

TL;DR

This paper investigates the fusion rules of sl^(2)-symmetric conformal field theories at level k=-1/2, revealing their logarithmic structure and indecomposable modules, with implications for related models like the c=-2 triplet and ghost systems.

Contribution

It provides a detailed analysis of fusion rules involving relaxed modules at fractional level k=-1/2, identifying indecomposable modules and their structure, and clarifies the logarithmic nature of these theories.

Findings

Fusion closes on irreducible and spectral flow modules, not relaxed highest weight modules.

Fusion of relaxed modules yields indecomposable modules with non-diagonalisable Virasoro action.

No logarithmic couplings (beta-invariants) are present in the indecomposable modules.

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.