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We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB)equations related to the WZW-theory corresponding to the adjoint $G$-bundles ofdifferent topological types over complex curves $\Sigma_{g,n}$ of genus $g$with $n$ marked points. The bundles are defined by their characteristic classes- elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is acenter of the simple complex Lie group $G$. The KZB equations are thehorizontality condition for the projectively flat connection (the KZBconnection) defined on the bundle of conformal blocks over the moduli space ofcurves. The space of conformal blocks has been known to be decomposed into afew sectors corresponding to the characteristic classes of the underlyingbundles. The KZB connection preserves these sectors. In this paper we constructthe connection explicitly for elliptic curves with marked points and prove itsflatness.

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles