The KZB equations on Riemann surfaces

Abstract

In this paper, based on the author's lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan--Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at $n$ marked points are given. The covariant derivatives are expressed in terms of ``dynamical $r$-matrices'', a notion borrowed from integrable systems. The case of marked points moving on a fixed Riemann surface is studied more closely. We prove a universal form of the (projective) flatness of the connection: the covariant derivatives commute as differential operators with coefficients in the universal enveloping algebra -- not just when acting on conformal blocks.

29 pages, LaTeX, to appear in the Proceedings of the 1995 les Houches Summer School

References in corpus (2)

• Affine Algebras, Langlands Duality and Bethe Ansatz
• Knizhnik-Zamolodchikov-Bernard equations on Riemann surfaces

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• Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as hierarchy of isomonodromic deformations. Free fields approach
• Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
• Conformal blocks from vertex algebras and their connections on ${\overline{\mathcal{M}}}_{g,n}$
• R-Matrix Structure of Hitchin System in Tyurin Parameterization
• Commuting differential and difference operators associated to complex curves, I
• TASI Lectures on Moonshine
• Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations
• Dynamical $r$-matrices for Hitchin's systems on Schottky curves
• Zhu's Theorem and an algebraic characterization of conformal blocks
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