The Knizhnik-Zamolodchikov equations for positive genus, and Krichever-Novikov

Abstract. We give a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous article of the authors (Russ. Math. Surv., (54)(1)). In the present article we construct the conformal blocks and the projectively flat connection on the bundle constituted by them