Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

TL;DR

This paper demonstrates that certain n-fold integrals from the Ising model and combinatorics are diagonals of rational functions, leading to globally bounded series expansions with integer coefficients, and explores their relation to modularity.

Contribution

It establishes that Ising integrals and related functions are diagonals of rational functions, connecting geometric solutions of differential equations to integrality and modularity properties.

Findings

Ising n-fold integrals are diagonals of rational functions.

Solutions to Calabi-Yau and Picard-Fuchs ODEs are diagonals of rational functions.

Generating functions with nested binomial sums are diagonals of rational functions.

Abstract

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of function generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x=0, solutions of these linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.