Intersection theory and canonical differential equations

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.

💡 Research Summary

In this proceedings contribution the authors review recent advances in applying twisted cohomology and intersection theory to the study of canonical differential equations for Feynman integrals. After a brief motivation—dimensional regularisation introduces an ε‑parameter that makes the integrands multivalued, a situation naturally described by twisted cohomology—the paper focuses on the problem of identifying and constructing a canonical basis of master integrals. A canonical basis is defined by three conditions: (i) the differential equations are ε‑factorised, (ii) the matrix of one‑forms has at most simple poles at each singularity, and (iii) the one‑forms are linearly independent as cohomology classes (i.e. up to total derivatives).

The central object is the intersection matrix C(x,ε), defined as a bilinear pairing between a chosen set of master integrands and a dual set (for maximal cuts the dual can be taken by flipping ε→−ε). The matrix satisfies a first‑order differential equation dC = Ω C + C Ωᵀ(−ε), where Ω is the original differential‑equation matrix in the chosen basis. Moreover, the twisted Riemann bilinear relations imply that if the basis satisfies the three canonical conditions, then dC = 0, i.e. C becomes independent of the kinematic variables. This provides a simple diagnostic: a constant intersection matrix signals a canonical basis, while any x‑dependence indicates hidden linear relations among the iterated integrals and thus a non‑canonical choice.

To illustrate the method the authors study a toy family of elliptic integrals \