Notes on Selection Rules of Canonical Differential Equations and Relative Cohomology

TL;DR

This paper explains the structure of canonical differential equations' coefficient matrices using intersection theory and relative cohomology, simplifying computations and reducing redundancy in the regulator method.

Contribution

It introduces a novel projection method for coefficient matrices of differential equations using intersection numbers and relative cohomology, streamlining calculations.

Findings

Provides a simple computation method for coefficient matrices.

Reduces redundancy in the regulator method.

Connects differential equations with relative cohomology techniques.

Abstract

We give an explanation of the $\mathrm{d}\log$-form of the coefficient matrix of canonical differential equations using the projection of ($n$+1)-$\mathrm{d}\log$ forms onto $n$-$\mathrm{d}\log$ forms. This projection is done using the leading-order formula for intersection numbers. This formula gives a simple way to compute the coefficient matrix. When combined with the relative twisted cohomology, redundancy in computation using the regulator method can be avoided.