Intersection theory and canonical differential equations

TL;DR

This paper explores how intersection theory and twisted cohomology can be used to understand and simplify the canonical differential equations of Feynman integrals, revealing hidden dependencies and relations among transcendental functions.

Contribution

It introduces the use of intersection matrices to analyze the canonical basis and derive relations, advancing the understanding of Feynman integral differential equations.

Findings

Intersection matrix detects hidden linear dependencies.

Simplifies the rotation to the canonical basis.

Facilitates derivation of relations among transcendental functions.

Abstract

In these proceedings we will review recent progress in applying ideas from the mathematical framework of twisted cohomology to the study of canonical differential equations for Feynman integrals. Firstly, we will show how the intersection matrix can shed some light on the nature of the canonical basis of a Feynman integral family, a concept still not fully understood in the general case. In particular we will show how the intersection matrix can detect hidden linear dependencies of the iterated integrals resulting from an $\eps$-factorized differential equation, which are difficult to find otherwise. Furthermore, we will explain how the intersection matrix can help in deriving (polynomial) relations between the transcendental functions occurring in the rotation to the canonical basis. This allows us to simplify the rotation, and furthermore leads to simplifications in the final result. The discussion we be kept as light as possible, focusing on a simple running example and deferring the technical details to the original publications.