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

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.