## Advanced Studies in Pure Mathematics

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

### The KZ system via polydifferentials

#### 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 Ramadas and are then known to imply the unitarity of the WZW system in genus zero.

#### Article information

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

**Dates**

Received: 3 May 2010

Revised: 11 January 2011

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543085010

**Digital Object Identifier**

doi:10.2969/aspm/06210189

**Mathematical Reviews number (MathSciNet)**

MR2933798

**Zentralblatt MATH identifier**

1260.32004

**Subjects**

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]

**Keywords**

KZ system polydifferentials highest weight module

#### Citation

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. https://projecteuclid.org/euclid.aspm/1543085010