Arrangements and Frobenius like structures

TL;DR

This paper demonstrates that for generic hyperplane arrangements, the Gauss-Manin connection, contravariant form, and algebra of functions form a Frobenius-like structure, with matrix elements derived from derivatives of a potential function.

Contribution

It introduces a Frobenius-like structure on the base of hyperplane arrangements, linking geometric, algebraic, and differential structures in a novel way.

Findings

Matrix elements of Gauss-Manin connection are derivatives of a potential function.

Establishes a Frobenius-like structure on hyperplane arrangement families.

Connects hypergeometric integrals with algebraic and geometric structures.

Abstract

We consider a family of generic weighted arrangements of $n$ hyperplanes in $\C^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the algebra of functions on the critical set of the master function define a Frobenius like structure on the base of the family. As a result of this construction we show that the matrix elements of the linear operators of the Gauss-Manin connection are given by the 2k+1-st derivatives of a single function on the base of the family, the function called the potential of second kind, see formula (6.46).