V-systems, holonomy Lie algebras and logarithmic vector fields

TL;DR

This paper explores the relationship between holonomy Lie algebra representations and $ ext{vee}$-systems associated with hyperplane arrangements, proposing conjectures and providing formulas for specific classes.

Contribution

It establishes an equivalence between certain holonomy Lie algebra representations and $ ext{vee}$-systems, and verifies a conjecture about the freeness of hyperplane arrangements for known systems.

Findings

Holonomy Lie algebra representations correspond to $ ext{vee}$-systems.

Confirmed the freeness conjecture for all known $ ext{vee}$-systems.

Derived formulas for deformations and potentials of harmonic $ ext{vee}$-systems.

Abstract

It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated to $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.