Self-duality from twisted cohomology

TL;DR

This paper links the concept of self-duality in differential equations of maximal cuts to twisted cohomology, showing that the intersection matrix becomes constant and revealing a natural rational symmetry in the system.

Contribution

It establishes a connection between self-duality in differential equations and twisted cohomology, introducing a Lie algebra representation and basis-independent rational symmetry.

Findings

The cohomology intersection matrix is constant when equations are in canonical form.

A Lie algebra representation can be associated with epsilon-factorized systems.

The natural symmetry of the differential equations is rational and basis-independent.

Abstract

Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an {\epsilon}-factorised differential equation is persymmetric. It was observed that the rotation to this special basis may introduce a Galois symmetry relating different integrals. We argue that the proposed notion of self-duality for maximal cuts stems from a very natural notion of self-duality from twisted cohomology. Our main result is that, if the differential equations and their duals are simultaneously brought into canonical form, the cohomology intersection matrix is a constant. Furthermore, we show that one can associate quite generically a Lie algebra representation to an {\epsilon}-factorised system. For maximal cuts, this representation is irreducible and self-dual. The constant intersection matrix can be interpreted as expressing the equivalence of this representation and its dual, which in turn results in constraints for the differential equation matrix. Unlike the earlier proposal, the most natural symmetry of the differential equation matrix is defined entirely over the rational numbers and is independent of the basis choice.