From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve

TL;DR

This paper explores the mathematical structures underlying the Ising model's correlation functions, revealing connections to elliptic curves, differential equations, and integrals, and introduces new potential singularities in the model's susceptibility.

Contribution

It uncovers the differential equations and elliptic curve connections of Ising model form factors and introduces multiple integrals that suggest new singularities in the susceptibility.

Findings

Differential operators decompose into symmetric powers of elliptic integrals

Scaling limits alter the differential operator structure

New singularities for susceptibility contributions are identified

Abstract

We recall the form factors $ f^{(j)}_{N,N}$ corresponding to the $\lambda$-extension $C(N,N; \lambda)$ of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a ``Russian-doll'' nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral $E$). The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure, the ``scaled'' linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the $n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for $n=1,2,3,4$ and, only modulo a prime, for $n=5$ and 6, thus providing a large set of (possible) new singularities of the $\chi^{(n)}$. ...