The Ising model: from elliptic curves to modular forms and Calabi–Yau equations

Abstract

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributions 's of the susceptibility of the Ising model for n ⩽ 6 are linear differential operators associated with elliptic curves. Beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators Z2, F2, F3, can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-4 globally nilpotent linear differential operator is not reducible to this elliptic curve, modular form scheme. This operator is shown to actually correspond to a natural generalization of this elliptic curve, modular form scheme, with the emergence of a Calabi–Yau equation, corresponding to a selected 4F3 hypergeometric function. This hypergeometric function can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back (square root extension), the corresponding Calabi–Yau fourth order differential operator having a symplectic differential Galois group . The mirror maps and higher order Schwarzian ODEs, associated with this Calabi–Yau ODE, present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms, in particular an exact (isogenies) representation of the generators of the renormalization group, extending the modular group to a symmetry group.