@article{BBHHMWZ11,
author = {Bostan, A. and Boukraa, S. and Hassani, S. and van Hoeij,
M. and Maillard, J.-M. and Weil, J.-A. and Zenine, N.},
title = {The {I}sing model: from elliptic curves to modular forms
and {C}alabi-{Y}au equations},
journal = {Journal of Physics A: Mathematical and Theoretical},
volume = {44},
number = {4},
year = {2011},
pages = {44pp},
doi = {10.1088/1751-8113/44/4/045204},
arxiv = {abs/1007.0535},
abstract = {We show that almost all the linear differential operators
factors obtained in the analysis of the $\, n$-particle contributions $\,
{\tilde \chi}^{(n)}$'s of the susceptibility of the Ising model for $\, n
\le 6$, are linear differential operators ``{\em 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 $\, Z_2$, $\, F_2$, $\, F_3$, $\, {\tilde L}_3$ can
actually be interpreted as {\em modular forms} of the elliptic curve of
the Ising model. A last order-four globally nilpotent linear differential
operator is not reducible to this elliptic curve, modular forms scheme.
This operator is shown to actually correspond to a natural generalization
of this elliptic curve, modular forms scheme, with the emergence of a
Calabi-Yau equation, corresponding to a selected $_4F_3$ 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 $\, SP(4, \, \mathbb{C})$. 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
$SL(2, \, \mathbb{Z})$ to a $GL(2, \, \mathbb{Z})$ symmetry group.},
}