@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.}, 
}