Mathematics

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. I shall discuss 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.  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.