Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)

TL;DR

This paper explores the properties of certain linear differential operators related to lattice Green functions and Calabi-Yau operators, revealing their special geometric nature and homomorphism to their adjoints, with implications for physics and algebraic geometry.

Contribution

It establishes that irreducible, globally nilpotent operators homomorphic to their adjoints have symmetric or exterior squares with rational solutions, defining a new 'Special Geometry' class.

Findings

Operators are globally nilpotent and homomorphic to their adjoints.

Symmetric or exterior squares of these operators have rational solutions.

Many physics-related integrals are associated with such operators.

Abstract

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups. We show that these irreducible operators are not only globally nilpotent, but are such that they are homomorphic to their (formal) adjoints. Considering these operators, or, sometimes, equivalent operators, we show that they are also such that, either their symmetric square or their exterior square, have a rational solution. This is a general result: an irreducible linear differential operator homomorphic to its (formal) adjoint is necessarily such that either its symmetric square, or its exterior square has a rational solution, and this situation corresponds to the occurrence of a special differential Galois group. We thus define the notion of being "Special Geometry" for a linear differential operator if it is irreducible, globally nilpotent, and such that it is homomorphic to its (formal) adjoint. Since many Derived From Geometry n-fold integrals ("Periods") occurring in physics, are seen to be diagonals of rational functions, we address several examples of (minimal order) operators annihilating diagonals of rational functions, and remark that they also seem to be, systematically, associated with irreducible factors homomorphic to their adjoint.