Jean-Marie Maillard (CNRS & University Pierre and Marie Curie, Paris, France)

Holonomy of the Ising model form factors and correlation functions

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 l, the j-particle contributions,  f(j)N,N, in the form factor expansion. The corresponding  l extension of the two-point diagonal correlation function,  C(N,N; l), is shown, for arbitrary l, to be a solution of the sigma form of the Painlevé VI equation introduced by Jimbo and Miwa in their isomonodromic approach to the Ising model. Fuchsian linear differential equations for the form factors  f(j)N,N are obtained for  j  9  and shown to have both a "Russian doll" nesting, and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral  E. From this, we show that each  f(j)N,N is unexpectedly simple, being expressed polynomially in terms of the elliptic integrals  E and  K. In contrast, we exhibit three mathematical objects, built from these form factors  f(j)N,N, which break the direct sum of symmetric powers decomposition, with its associated polynomial expressions. First we show that the scaling limit of these differential operators, and form factors, breaks the direct sum structure but not the "Russian doll" structure. Secondly, we show that the previous  l-extension of two-point diagonal correlation functions,  C(N,N;  l) are, for singled-out values  l =  cos(pm/n) (m,  n integers), also solutions of Fuchsian linear differential equations. These solutions of Painlevé VI are not polynomial in  E and  K but are actually algebraic functions, being associated with modular curves. Finally we introduce another infinite sum of the  f(j)N,N, the "diagonal susceptibility", which also breaks the direct sum structure but not the "Russian doll" structure.