Characters and fusion rules for \(W\)-algebras via quantized Drinfeld- Sokolov reduction
Realization of \(W\)-algebras \(W(\bar{\mathfrak{g}})\), associated to any simple finite-dimensional Lie algebra \(\bar{\mathfrak{g}}\), appearing as a result of quantization of the Drinfeld-Sokolov reduction, is used. This approach allows to construct a functor \(F\) from the category of positive energy representations of the affine algebra \(\mathfrak g\) to the category of positive energy representations of the algebra \(W(\bar{\mathfrak{g}})\). This functor is applied to admissible (conjecturally = all modular invariant) representations of \(\mathfrak g\) of fractional levels. The functor \(F\) is also used in evaluating characters of \(W(\bar{\mathfrak{g}})\)-modules. The calculations give many informations about conformal field theory with \(W(\bar{\mathfrak{g}})\)-symmetry. In particular, Verlinde's formula for the modular transform of \(W(\bar{\mathfrak{g}})\)-characters to describe the fusion algebra of the minimal \(W(\bar{\mathfrak{g}})\)-models is applied. Because the application of Verlinde's arguments to affine characters at a fractional level can lead to negative fusion coefficients, the use of the functor \(F\) is essential in this case.
scientific article; zbMATH DE number 95413
scientific article