Holonomy of the Ising model form factors

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, $f_{N,N}^{(j)}$. 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{é} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $f_{N,N}^{(j)}$ 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 $f_{N,N}^{(j)}$ 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 Painlevé VI are actually algebraic functions, being associated with modular curves.

39 pages

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Cited by in corpus (21)

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• High order Fuchsian equations for the square lattice Ising model: ${\tilde{\chi }}^{(5)}$
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• The diagonal Ising susceptibility
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Form factor expansion of the row and diagonal correlation functions of the two dimensional Ising model
• A Fuchsian matrix differential equation for Selberg correlation integrals
• Form factor expansions in the 2D Ising model and Painlevé VI
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Resurgence, Painleve Equations and Conformal Blocks
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
• Fuchs versus Painlevé
• The Ising correlation $C(M,N)$ for $\nu =-k$
• Factorization of Ising correlations C(M,N) for $\nu =-k$ and M+N odd, $M\le N$, $T<T_c$ and their lambda extensions
• Scaling functions in the square Ising model
• Factorization of the Ising model form factors
• The Ising model and Special Geometries
• The lambda extensions of the Ising correlation functions C(M, N)
• Connection formulas for the lambda generalized Ising correlation functions