The KZB equations on Riemann surfaces

1996/09/18 by Giovanni Felder, Felder, Giovanni · 5 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #hep-th

paper · pdf · doi:10.48550/arxiv.hep-th/9609153

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

arxiv created 1996/09/18 · arxiv updated 2009/11/30

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.

Cited by

• TASI Lectures on Moonshine
• Zhu's Theorem and an algebraic characterization of conformal blocks
• Dynamical r-matrices for Hitchin's systems on Schottky curves
• Hecke modifications of conformal blocks outside the critical level
• Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as hierarchy of isomonodromic deformations. Free fields approach

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