Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi–Yau equations

Abstract

We give the exact expressions of the partial susceptibilities χ(3)d and χ(4)d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi–Yau ODEs, and more specifically, 3F2([1/3, 2/3, 3/2], [1, 1]; z) and 4F3([1/2, 1/2, 1/2, 1/2], [1, 1, 1]; z) hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for χ(3)d and χ(4)d. We also give new results for χ(5)d. We see, in particular, the emergence of a remarkable order-6 operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the n-fold integrals of the Ising model are not only ‘derived from geometry’ (globally nilpotent), but actually correspond to ‘special geometry’ (homomorphic to their formal adjoint). This raises the question of seeing if these ‘special geometry’ Ising operators are ‘special’ ones, reducing, in fact systematically, to (selected, k-balanced, ...) q + 1Fq hypergeometric functions, or correspond to the more general solutions of Calabi–Yau equations.