Advanced Studies in Pure Mathematics

The KZ system via polydifferentials

Eduard Looijenga

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We show that the KZ system has a topological interpretation in the sense that it may be understood as a variation of complex mixed Hodge structure whose successive pure weight quotients are polarized. This in a sense completes and elucidates work of Schechtman–Varchenko done in the early 1990's. A central ingredient is a new realization of the irreducible highest weight representations of a Lie algebra of Kac–Moody type, namely on an algebra of rational polydifferentials on a countable product of Riemann spheres. We also obtain the kind of properties that in the $\mathfrak{sl} (2)$ case are due to Ramadas and are then known to imply the unitarity of the WZW system in genus zero.

Article information

Arrangements of Hyperplanes — Sapporo 2009, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 189-231

Received: 3 May 2010
Revised: 11 January 2011
First available in Project Euclid: 24 November 2018

Permanent link to this document euclid.aspm/1543085010

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G34: Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) [See also 34Mxx]
Secondary: 14D07: Variation of Hodge structures [See also 32G20]

KZ system polydifferentials highest weight module


Looijenga, Eduard. The KZ system via polydifferentials. Arrangements of Hyperplanes — Sapporo 2009, 189--231, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06210189.

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