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The KZ system via polydifferentials. (English) Zbl 1260.32004

Terao, Hiroaki (ed.) et al., Arrangements of hyperplanes. Proceedings of the 2nd Mathematical Society of Japan-Seasonal Institute, MSJ-SI, Sapporo, Japan, August 1–13, 2009. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-67-9/hbk). Advanced Studies in Pure Mathematics 62, 189-231 (2012).
Authors’ abstract: “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 T. R. Ramadas [Ann. Math. (2) 169, No. 1, 1–39 (2009; Zbl 1167.32011)] and are then known to imply the unitarity of the WZW system in genus zero.”
For the entire collection see [Zbl 1242.14003].

MSC:

32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
14D07 Variation of Hodge structures (algebro-geometric aspects)

Citations:

Zbl 1167.32011
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