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We give the exact expressions of the partial susceptibilities $chi^{(3)}_d$ and $chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],, [1,1];, z)$ and $_4F_3([1/2,1/2,1/2,1/2],, [1,1,1]; , z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $chi^{(3)}_d$ and $chi^{(4)}_d$. We also give new results for $chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six 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, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.
We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contribution of the susceptibility of the Ising model for $, n le 6$, are operators associated with elliptic curves. Beyond the simplest fact
We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model.
We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers-Wannier duality to anisotropic correlation functio
We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate
We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homolo