Thursday, July 19, 2012

1207.4386 (A. Levin et al.)

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

A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov
We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles 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}, Z(G)), where Z(G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly and prove its flatness for elliptic curves with marked points. For the trivial bundles our construction provides the KZB connection proposed by Felder-Wieczerkowski.
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