Border Bases in the Rational Weyl Algebra
Carlos Rodriguez, Anna-Laura Sattelberger
TL;DR
The paper develops border bases for left ideals in the noncommutative rational Weyl algebra $R_n$, extending the commutative border-basis theory to a Pfaffian/D-modules setting. It characterizes border bases in terms of integrability conditions on the associated connection matrices $A_i$ (with $A_i=M_{ abla_i}^ op$), enabling a constructive passage from connection data to cyclic $D$-modules and explicit $D$-ideals. The approach yields practical constructions of integrable connections as Pfaffian systems and provides a framework to classify certain $D$-ideals via Hilbert schemes of points, including constant-coefficient and Frobenius ideals. Applications to physics illustrate how border bases recover and illuminate Pfaffian systems for stringy integrals, sunrise Feynman integrals, and cosmological correlators, as well as considerations of $oldsymbol{eta}$- or $oldsymbol{oldsymbol{ m abla}}$-factorized forms and integrability.
Abstract
Border bases are a generalization of Gröbner bases for polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra. We elaborate on their properties and present algorithms to compute with them. We apply this theory to represent integrable connections as cyclic $D$-modules explicitly. As an application, we visit differential equations behind a stringy, a Feynman as well as a cosmological integral. We also address the classification of particular $D$-ideals of a fixed holonomic rank, namely the case of linear PDEs with constant coefficients as well as Frobenius ideals. Our approach rests on the theory of Hilbert schemes of points in affine space.