# 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
- License
- Unknown
- External links

## Abstract

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 $\chi^{(n)}$. ...