From path representations to global morphisms for a class of minimal models

We construct global observable algebras and global DHR morphisms for the Vi rasoro minimal models with central charge c(2, q), q odd. To this end, we p ass from the irreducible highest weight modules to path representations, wh ich involve fusion graphs of the c(2, q) models. The paths have an interpre tation in terms of quasi-particles which capture some structure of non-conf ormal perturbations of the c(2, q) models. The path algebras associated to the path spaces serve as algebras of bounded observables. Global morphisms which implement the superselection sectors are constructed using quantum sy mmetries: We argue that there is a canonical semi-simple quantum symmetry a lgebra for each quasi-rational CFT, in particular for the c(2, q) models. T hese symmetry algebras act naturally on the path spaces, which allows to de fine a global field algebra and covariant multiplets therein.