The Ising model and special geometries

Abstract

Abstract. We show that the globally nilpotent G-operators 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 self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators 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 order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group Sp(12, C). The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group SO(21, C).

Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics

“…Caveat: Since we are going to use our tools [25,26,28,29,30] to find (Fuchsian) linear differential operators modulo rather small primes (the first eight primes), one may be facing a problem we do not encounter with our previous studies [25,26] performed with rather large primes (2 15 − 19 = 32749, ... ). Modulo a prime p, any power series with integer coefficients is solution of the linear differential operators θ p − θ, where θ denotes the homogeneous derivative w · d/dw, or much more simply of the operator d p /dw p .…”

“…In fact the series (30) can actually be understood from the previously introduced lacunary series. The series (30) can in fact be seen to be equal, modulo 3 2 , to:…”

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

“…Forty years ago, Wu, Barouch, McCoy and Tracy [4] showed that the full susceptibility of the square-lattice Ising model can be decomposed as the infinite sum of holonomic n-fold integrals [5,6,7,8,9], denoted χ (n) . In the last decade the linear differential operators corresponding to the first χ (n) 's, up to n = 6, were obtained, underlying the role of the elliptic curve parametrization [10], but showing also the emergence of (at least) one Calabi-Yau ODE, and beyond, of linear differential operators with selected differential Galois groups [11,12,13]. A complete description of the singular points of the linear differential operators corresponding to the first few χ (n) 's has also been obtained [6,14,15,16].…”

Diagonals of rational functions and selected differential Galois groups

“…This natural emergence in physics of n-fold integrals that are diagonals of rational functions, such that their associated linear differential operators correspond to selected differential Galois groups, SO(n, C) or Sp(n, C) (or subgroups, like exceptional groups [10]), was illustrated on important problems of lattice statistical mechanics like the n-fold integrals χ (5) and χ (6) of the square Ising model [20], or non-trivial lattice Green functions examples [11].…”

“…F 1 ([1/3, 1/6],[1], 108 x 3 ), some with, at first sight, more involved HeunG function solutions[35] which turn out to be pullbacked 2 F 1 hypergeometric functions, with two possible pullbacks, and, in fact, modular forms[35].The 128 order-four linear differential operators are (non-trivially) homomorphic to their adjoints. They have SO(4, C) differential Galois groups and have a canonical † These two series(20) and(22) are not in Sloane's on-line encyclopedia http://oeis.org.…”