Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 095, 37 pages      arXiv:1207.4386

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

Andrey M. Levin a, b, Mikhail A. Olshanetsky b, Andrey V. Smirnov b, c and Andrei V. Zotov b
a) Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow, 117312, Russia
b) Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia
c) Department of Mathematics, Columbia University, New York, NY 10027, USA

Received July 14, 2012, in final form November 29, 2012; Published online December 10, 2012

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different 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 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 for elliptic curves with marked points and prove its flatness.

Key words: integrable system; KZB equation; Hitchin system; characteristic class.

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