Globally nilpotent differential operators and the square Ising model

TL;DR

This paper explores the differential operators related to the square Ising model's integrals, revealing their deep geometric origins and properties like global nilpotence and zero p-curvature, linked to algebraic structures such as elliptic and modular curves.

Contribution

It identifies the globally nilpotent nature of these operators and connects their factorized parts to specific algebraic geometric structures, including elliptic and modular curves, and a modular form.

Findings

Differential operators are globally nilpotent or have zero p-curvature.

Factors of these operators correspond to elliptic and modular curves.

A modular form emerges in the three-particle susceptibility contribution.

Abstract

We recall various multiple integrals related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their lambda-extensions. These integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, and even a remarkable weight-1 modular form emerging in the three-particle contribution $ \chi^{(3)}$ of the magnetic susceptibility of the square Ising model. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or $ \infty$) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.