Selection rules of canonical differential equations from Intersection theory

TL;DR

This paper introduces a method using intersection theory and dual forms in relative cohomology to derive selection rules for canonical differential equations, linking their structure to the pole properties of master integrands.

Contribution

It develops a novel approach employing intersection numbers and cohomology to determine the structure and pole conditions of canonical differential equations.

Findings

Provides a systematic way to derive selection rules

Connects pole structure with intersection theory

Enhances understanding of differential equations in Feynman integrals

Abstract

The matrix of canonical differential equations consists of the 1-$\mathrm{d}\log$-form coefficients obtained by projecting ($n$+1)-$\mathrm{d}\log$-forms onto $n$-$\mathrm{d}\log$-form master integrands. With dual form in relative cohomology, the intersection number can be used to achieve the projection and provide the selection rules for canonical differential equations, which relate to the pole structure of the $\mathrm{d}\log$ master integrands.