# Moscow Mathematical Journal

Volume 19, Issue 1, January–March 2019  pp. 153–180.

Potentials of a Family of Arrangements of Hyperplanes and Elementary Subarrangements

Authors:  Andrew Prudhom (1) and Alexander Varchenko (1)
Author institution:(1) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Summary:

We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in ℂk 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. This 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.

2010 Math. Subj. Class. 32S22, 53D45, 14N20.

Keywords: Arrangement of hyperplanes, Grothedieck residue form, master function, critical set.