**Holonomy of the Ising model form factors and correlation functions**

**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 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.