A topological interpretation of the KZ system

TL;DR

This paper provides a topological interpretation of the KZ system, framing it as a variation of complex mixed Hodge structures with polarized pure weight quotients, building on earlier work from the 1990s.

Contribution

It introduces a new realization of Kac-Moody algebra representations on rational polydifferentials, clarifying the topological nature of the KZ system.

Findings

KZ system interpreted as a variation of mixed Hodge structures

New realization of Kac-Moody representations on rational polydifferentials

Elucidates the topological aspects of the KZ system

Abstract

We show that the KZ system has a purely 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.