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

Authors
Type
Published Article
Publication Date
Oct 15, 2007
Submission Date
Oct 15, 2007
Identifiers
DOI: 10.3842/SIGMA.2007.099
Source
arXiv
We recall the form factors $f^{(j)}_{N,N}$ 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 $\chi^{(n)}$. ...