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

We recall the form factors $f_{N,N}^{(j)}$ 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 ${\chi }^{(n)}$. …