Resonant local systems on complements of discriminantal arrangements and sl_2 representations

TL;DR

This paper explores the skew-symmetric cohomology of complements of discriminantal arrangements with local systems linked to sl_2 representations, revealing a connection between cohomology dimensions and critical points of a master function.

Contribution

It provides a calculation of skew-symmetric cohomology for discriminantal arrangements with local systems from sl_2 representation theory, linking cohomology to critical points of the master function.

Findings

Nontrivial skew-symmetric cohomology occurs when the master function has nonisolated critical points.

The dimension of this cohomology equals the number of critical lines in generic cases.

Critical points form a union of lines in symmetric coordinates.

Abstract

We calculate the skew-symmetric cohomology of the complement of a discriminantal hyperplane arrangement with coefficients in local systems arising in the context of the representation theory of the Lie algebra sl_2. For a discriminantal arrangement in C^k, the skew-symmetric cohomology is nontrivial in dimension k-1 precisely when the "master function" which defines the local system on the complement has nonisolated critical points. In symmetric coordinates, the critical set is a union of lines. Generically, the dimension of this nontrivial skew-symmetric cohomology group is equal to the number of critical lines.