On axioms of Frobenius like structure in the theory of arrangements

Abstract

A Frobenius manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points of the manifold such that the structure constants of multiplication are given by third derivatives of a potential function on the manifold with respect to flat coordinates. In this paper we present a modification of that notion coming from the theory of arrangements of hyperplanes. Namely, given natural numbers $n>k$, we have a flat $n$-dimensional manifold and a vector space $V$ with a nondegenerate symmetric bilinear form and an algebra structure on $V$, depending on points of the manifold, such that the structure constants of multiplication are given by $2k+1$-st derivatives of a potential function on the manifold with respect to flat coordinates. We call such a structure a {\it Frobenius like structure}. Such a structure arises when one has a family of arrangements of $n$ affine hyperplanes in $\C^k$ depending on parameters so that the hyperplanes move parallely to themselves when the parameters change. In that case a Frobenius like structure arises on the base $\C^n$ of the family.