Globally nilpotent differential operators and the square Ising model

Abstract

We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their russian-doll and direct sum structures. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors (resp. p-curvature nullity) corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genus five, six, . . . , and even a remarkable weight-1 modular form emerging in the three-particle contribution χ (3) of the magnetic susceptibility of the square Ising model. Noticeably, this associated weight-1 modular form is also seen in the factors of the differential operator for another n-fold integral of the Ising class, ΦH , for the staircase polygons counting, and in Apéry's study of ζ(3). G-functions naturally occur as solutions of globally nilpotent operators. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or ∞) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property. PACS: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14KxxGlobally nilpotent operators 2 Key-words: Globally nilpotent operators, p-curvature, G-functions, arithmetic Gevrey series, Form factors of the square Ising model, susceptibility of the Ising model, Fuchsian linear differential equations, moduli space of curves, two-point correlation functions of the lattice Ising model, complete elliptic integrals, scaling limit of the Ising model, apparent singularities, modular forms, Atkin-Lehmer involutions, Fricke involutions, Dedekind eta functions, Weber modular functions, Calabi-Yau manifolds, three-choice polygons, enumerative combinatorics. IntroductionGenerating large series expansions of physical quantities that are quite often defined as n-fold integrals is the bread and butter of lattice statistical mechanics, enumerative combinatorics, and more generally theoretical physics. The n-fold integrals...

Heun functions and diagonals of rational functions

“…It is clear, from the x (1 ±2 1/2 )/4 factors, that the order-two operator L 2 is not even ¶ globally nilpotent [11]. Furthermore, the two series h + + h − and 2 −1/2 · (h + − h − ) are series with rational number coefficients that are not globally bounded series.…”

“…Diag R(x, y, z, w) = 1 + 2 x + 18 x 2 + 164 x 3 + 1810 x 4 + · · · (12) A creative telescoping program [6] gives the order-three linear differential operator annihilating the diagonal (12) of the previous rational function (11):…”

“…Let us revisit example 1, considering, similarly, the diagonal of the rational function of four variables (11), its telescoper L 3 given by (13), and the corresponding Heun solutions (15). We introduce, as previously, a new rational function (for the rational function (11) of four variables R(x, y, z, w)) corresponding to simple involutions of the form t → A/t on the four variables:…”

“…In the first two sections, we examine the Heun functions emerging from diagonals of simple rational functions that fall into the first or second category above, and show how they happen to be related to classical modular forms, or derivatives of classical modular forms, corresponding to pullbacked 2 F 1 hypergeometric functions. These Heun functions have integer coefficient series, (or can be recast as series with integer coefficients [2] after a rescaling of the variable), and are solutions of globally nilpotent [11] linear differential operators: the critical exponents of all the singularities are rational numbers. This leads to define a criterion in Appendix F, that allows to draw up a list of parameters of the Gauss hypergeometric function 2 F 1 ([a, b], [c], x), for which it corresponds to a classical modular form (see section 2).…”

A fast algorithm for computing the characteristic polynomial of the p-curvature

“…This computation is for instance one of the basic steps in algorithms for factoring linear differential operators in characteristic p [30,31,17]. Additional motivations for studying this question come from concrete applications, in combinatorics [6,7] and in statistical physics [2], where the p-curvature serves as an a posteriori certification filter for differential operators obtained by guessing techniques from power series expansions. In such applications, the prime number p may be quite large (thousands, or tens of thousands), since its value is lower bounded by the precision of the power series needed by guessing, which is typically large for operators of large size.…”