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

Abstract

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.

References in corpus (7)

• Instantons and Yang-Mills Flows on Coset Spaces
• Characteristic Classes and Integrable Systems for Simple Lie Groups
• Characteristic Classes and Integrable Systems. General Construction
• Monopoles and Modifications of Bundles over Elliptic Curves
• Correspondence between Calogero-Moser systems and integrable SL(N,$\mathbb{C}$) Euler-Arnold tops
• Quadratic algebras related to elliptic curves
• Three lectures on classical integrable systems and gauge field theories

Cited by in corpus (4)

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