Holonomy of the Ising model form factors

TL;DR

This paper presents a new integral expansion for the Ising model's two-point correlation function, linking it to Painlevé VI equations and elliptic integrals, revealing deep algebraic and differential structures.

Contribution

It introduces an alternative exponential and form factor expansion for the Ising correlation function, connecting it to Painlevé VI and elliptic integrals, with detailed differential operator analysis.

Findings

The correlation function satisfies Painlevé VI equations.

Form factors are expressed as polynomials in elliptic integrals.

Differential operators exhibit a nested 'Russian doll' structure.

Abstract

We study the Ising model two-point diagonal correlation function $ C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $\lambda$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding $ \lambda$ extension of the two-point diagonal correlation function, $ C(N,N; \lambda)$, is shown, for arbitrary $\lambda$, to be a solution of the sigma form of the Painlev{\'e} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $ f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll'' nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $ E$. Each $ f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $ E$ and $ K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll'' structure. The previous $ \lambda$-extensions, $ C(N,N; \lambda)$ are, for singled-out values $ \lambda= \cos(\pi m/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painlev\'e VI are actually algebraic functions, being associated with modular curves.