V-systems, holonomy Lie algebras and logarithmic vector fields

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.

21 pages, slightly revised version, to appear in IMRN

References in corpus (6)

• The freeness of ideal subarrangements of Weyl arrangements
• Confluence of hypergeometric functions and integrable hydrodynamic type systems
• N=4 Multi-Particle Mechanics, WDVV Equation and Roots
• Flat Bases of Invariant Polynomials and P-matrices of E7 and E8
• Frobenius manifolds associated to Coxeter groups of type E_7 and E_8
• On deformation and classification of V-systems