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

TL;DR

This paper derives exact expressions for diagonal susceptibilities in the Ising model using modular forms and Calabi-Yau equations, revealing their deep geometric structure and connection to hypergeometric functions.

Contribution

It provides explicit formulas for susceptibilities in terms of hypergeometric functions and uncovers their underlying 'special geometry' properties, linking Ising model integrals to Calabi-Yau equations.

Findings

Exact expressions for $oldsymbol{ ext{chi}^{(3)}_d}$ and $oldsymbol{ ext{chi}^{(4)}_d}$ in terms of hypergeometric functions.

Identification of a remarkable order-six differential operator with special geometric properties.

New results for $oldsymbol{ ext{chi}^{(5)}_d}$ and insights into the geometric nature of Ising integrals.

Abstract

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.