Arrangements, KZ Systems and Lie Algebra Homology

Summary

To Terry Wall, for his 60th birthday.

Introduction

The Knizhnik–Zamolodchikov equations were originally defined in terms of a local system associated to tuples of finite dimensional irreducible representations of SU₂, but were soon afterwards generalized to a Kac–Moody setting. The natural question that arises is whether these local systems admit a topological interpretation. A paper by Varchenko–Schechtman [8] comes close to answering this affirmatively and it is this article and related work that we intend to survey here.

Our presentation deviates at certain points from the original sources. First, we felt it worthwhile to introduce the notion of a Knizhnik–Zamolodchikov system (in (1.4)), whose value is enhanced by the simple criterion (1.5). Such systems also occur in the theory of root systems (both in a linear and in an exponential setting) and we took the occasion to discuss them briefly from our point of view. A KZ system leads to what is perhaps the most natural class of local systems on hyperplane complements endowed with a given extension over the whole space as vector bundle. That may already be sufficient reason for this notion to merit a more thorough investigation than we can give here.

Second, we treated the cohomology of hyperplane complements (due Arnol'd, Brieskorn and Orlik–Solomon) using the methods of sheaf theory, an approach we advocated on an earlier occasion for its effectiveness.