The Ising model and special geometries
Abstract
We show that the globally nilpotent Goperators corresponding to the factors of the linear differential operators annihilating the multifold integrals χ^{(n)} of the magnetic susceptibility of the Ising model (n ⩽ 6) are homomorphic to their adjoint. This property of being selfadjoint up to operator homomorphisms is equivalent to the feature of their symmetric squares, or their exterior squares, having rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This selfadjoint (up to operator equivalence) property means that the factor operators that we already know to be derived from geometry are special globally nilpotent operators: they correspond to ‘special geometries’. Beyond the small order factor operators (occurring in the linear differential operators associated with χ^{(5)} and χ^{(6)}), and, in particular, those associated with modular forms, we focus on the quite large order12 and order23 operators. We show that the order12 operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group Sp(12, \, {C}). The order23 operator is shown to factorize into an order2 operator and an order21 operator. The symmetric square of this order21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group SO(21, \, {C}).
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 June 2014
 DOI:
 10.1088/17518113/47/22/225204
 arXiv:
 arXiv:1402.6291
 Bibcode:
 2014JPhA...47v5204B
 Keywords:

 Mathematical Physics;
 34M55;
 47E05;
 81Qxx;
 32G34;
 34Lxx;
 34Mxx;
 14Kxx;
 G.1.7
 EPrint:
 33 pages