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

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.

💡 Research Summary

The paper extends the Knizhnik‑Zamolodchikov‑Bernard (KZB) formalism of the Wess‑Zumino‑Witten (WZW) conformal field theory to the case of non‑trivial principal (G)‑bundles over a complex curve (\Sigma_{g,n}). While the classical KZB equations are formulated for bundles whose structure group has a trivial center, here the authors consider a simple complex Lie group (G) whose center (\mathcal Z(G)) is non‑trivial. Principal adjoint bundles are then classified by the discrete characteristic class \