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

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 alarge set of (possible) new singularities of the $\chi^{(n)}$. We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition $1+3w+4 w^2=0$, that occurs in the linear differential equation of $\chi^{(3)}$, actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion.