Knizhnik–Zamolodchikov equations for positive genus and Krichever–Novikov algebras

In this paper a global operator approach to the Wess–Zumino–Witten–Novikov theory for compact Riemann surfaces of arbitrary genus with marked points is developed. The term ‘global’ here means that Krichever–Novikov algebras of gauge and conformal symmetries (that is, algebras of global symmetries) are used instead of loop algebras and Virasoro algebras (which are local in this context). The basic elements of this global approach are described in a previous paper of the authors (Russ. Math. Surveys 54:1 (1999)). The present paper gives a construction of the conformal blocks and of a projectively flat connection on the bundle formed by them.