Differential Space of Feynman Integrals: Annihilators and $\mathcal{D}$-module

TL;DR

This paper introduces a new algorithm leveraging Griffiths-Dwork reduction and Macaulay matrices to construct differential operators that annihilate Feynman integrals, revealing their $$-module structure and algebraic relations.

Contribution

It presents a novel computational method for deriving annihilators and relations of Feynman integrals using algebraic geometry techniques, applicable to a broad class of integrals.

Findings

The method successfully computes annihilators for Feynman graphs and Witten diagrams.

The holonomic rank matches the de Rham cohomology dimension, suggesting an equivalence.

The approach explicitly avoids surface term contributions in the relations.

Abstract

We present a novel algorithm for constructing differential operators with respect to external variables that annihilate Feynman-like integrals and give rise to the associated $\mathcal{D}$-modules, based on Griffiths-Dwork reduction. By leveraging the Macaulay matrix method, we derive corresponding relations among partial differential operators, including systems of Pfaffian equations and Picard-Fuchs operators. Our computational approach is applicable to twisted period integrals in projective coordinates, and we showcase its application to Feynman graphs and Witten diagrams. The method yields annihilators and their algebraic relations for generic regulator values, explicitly avoiding contributions from surface terms. In the cases examined, we observe that the holonomic rank of the $\mathcal{D}$-modules coincides with the dimension of the corresponding de Rham co-homology groups, indicating an equivalence relation between them, which we propose as a conjecture.