CONFORMAL BLOCKS ON ELLIPTIC-CURVES AND THE KNIZHNIK-ZAMOLODCHIKOV-BERNARD EQUATIONS

We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as a subbundle of a vecto r bundle of Weyl group invariant vector valued theta functions on a Ca rtan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperpla nes. In some cases, explicit calculations are possible and confirm the conjecture. The Friedan-Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generali zation of the Knizhnik-Zamolodchikov equation.