Potentials of a family of arrangements of hyperplanes and elementary subarrangements

TL;DR

This paper constructs two potential functions for the Frobenius algebra of functions on the critical set of a hyperplane arrangement, revealing a local structure and linking it to Bethe algebras in quantum integrable models.

Contribution

It introduces potential functions of two kinds that fully determine the Frobenius algebra and demonstrates their local nature via elementary subarrangements.

Findings

Potential functions are sums over elementary subarrangements.

Matrix coefficients relate to derivatives of potential functions.

Frobenius algebra is isomorphic to the Bethe algebra of the arrangement.

Abstract

We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in $\C^k$ with normal crossings. We construct two potential functions (of first and second kind) of variables labeled by hyperplanes of the arrangement and prove that the matrix coefficients of the Grothendieck residue bilinear form on the algebra are given by the $2k$-th derivatives of the potential function of first kind and the matrix coefficients of the multiplication operators on the algebra are given by the $(2k+1)$-st derivatives of the potential function of second kind. Thus the two potentials completely determine the Frobenius algebra. The presence of these potentials is a manifestation of a Frobenius like structure similar to the Frobenius manifold structure. We introduce the notion of an elementary subarrangement of an arrangement with normal crossings. It turns out that our potential functions are local in the sense that the potential functions are sums of contributions from elementary subarrangements of the given arrangement. This is a new phenomenon of locality of the Grothendieck residue bilinear form and multiplication on the algebra. It is known that this Frobenius algebra of functions on the critical set is isomorphic to the Bethe algebra of this arrangement. (That Bethe algebra is an analog of the Bethe algebras in the theory of quantum integrable models.) Thus our potential functions describe that Bethe algebra too.