Critical set of the master function and characteristic variety of the associated Gauss-Manin differential equations

TL;DR

This paper analyzes the characteristic variety of Gauss-Manin differential equations linked to hypergeometric integrals, revealing its structure as Laurent polynomial zero sets and establishing an isomorphism with the algebra of functions on the critical set.

Contribution

It introduces a novel description of the characteristic variety using Laurent polynomials and constructs an isomorphism connecting differential equations with algebraic functions.

Findings

Characteristic variety described as Laurent polynomial zero set

Constructed an isomorphism between differential equations and algebra of functions

Defined an integral structure and combinatorial connection on the algebra

Abstract

We consider a weighted family of $n$ parallelly transported hyperplanes in a $k$-dimensioinal affine space and describe the characteristic variety of the Gauss-Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr$(k,n)$. The Laurent polynomials are in involution. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss-Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras.