V-systems, holonomy Lie algebras and logarithmic vector fields
FeiginMikhail V.
VeselovAlexander
2016
It is shown that the description of certain class of representations
of the holonomy Lie algebra g Δ associated to hyperplane arrangement is Δ essentially equivalent to the classification of V-systems associated to Δ. The flat
sections of the corresponding V-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane
arrangement of any V-system is free in Saito's sense and show this
for all known V-systems and for a special class of V-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 V-systems.