Intersection theory and canonical differential equations

TL;DR

The work reframes canonical differential equations for Feynman integrals in the language of twisted cohomology and uses the intersection matrix $\boldsymbol{C}(\boldsymbol{x},\varepsilon)$ to diagnose basis properties. It shows that, in an $\varepsilon$-factorized setting, $\boldsymbol{C}$ is constant for a canonical basis and can reveal hidden linear relations among iterated integrals, enabling a principled discrimination between bases. A procedure based on $\boldsymbol{\Delta}$ and the constancy of $\boldsymbol{C}$ allows one to fix $\varepsilon$-functions in the final rotation and to derive polynomial relations among them, thus simplifying the canonical form. The approach is illustrated on a simple maximal-cut elliptic example and promises broader impact for simplifying high-loop Feynman integral computations beyond $\mathrm{dlog}$-type bases.

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.