The Ising model and special geometries

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals χ(n) of the magnetic susceptibility of the Ising model (n 6) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms is equivalent to the feature of their symmetric squares, or their exterior squares, having rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators that we already know to be derived from geometry are special globally nilpotent operators: they correspond to 'special geometries'. Beyond the small order factor operators (occurring in the linear differential operators associated with χ(5) and χ(6)), and, in particular, those associated with modular forms, we focus on the quite large order-12 and order-23 operators. We show that the order-12 operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group . The order-23 operator is shown to factorize into an order-2 operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group .