Annihilators and D-modules for Feynman integrals

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 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.

For the studied examples, we observe that the holonomic rank of the D-modules coincides with the dimension of the corresponding de Rham co-homology groups.

We also discuss some general structures of a D-ideal for the banana family.