On the identification of finite operator algebras in two-dimensional conformally invariant field theories

We investigate the structure of the linear differential operators whose solutions determine the four point correlations of primary operators in the d=2 conformally invariant SU(2) σ-model with Wess-Zumino term and the d=2 critical statistical systems with central Virasoro charge smaller than one. Factorisation properties of the differential operators are related to a finite closure of the operator algebras. We recover the selection and fusion rules of Fateev, Zamolodchikov and Gepner, Witten for the SU(2) σ-model. It is outlined how the results of the SU(2) model can be used for the identification of closed operator algebras in the statistical model. (orig.)