Frobenius algebras and root systems: the trigonometric case

Abstract

We construct Frobenius structures on the \({\mathbb {C}}^{\times }\)-bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives rise to a class of Frobenius manifolds parameterizing Frobenius algebras in terms of root systems in a trigonometric setting. We then determine their potential functions, which amounts to giving the trigonometric solutions of WDVV equations.