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Globally nilpotent differential operators and the square Ising model. (English) Zbl 1167.82005

Various multiple integrals with one parameter are recalled. Examples are the $n$-particle contribution of the magnetic susceptibility of the isotropic square Ising model, lattice form factors and two-point correlation functions, or examples from enumerative combinatorics. 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 growth property on the coefficients of a solution series. These differential operators are selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are globally nilpotent, or, sometimes, even have zero $p$-curvature. In some examples from enumerative combinatorics, quantities are not naturally expressed as $n$-fold integrals of an algebraic integrand. The discovery of the global nilpotence of the corresponding minimal order Fuchsian linear differential operator has to be seen as a strong indication that they can be expressed as $n$-fold integrals of an algebraic integrand. Other quantities, like the $n$-particle contributions to the magnetic susceptibility of the square Ising model are defined as such $n$-fold integrals. The integrand of the algebraic function can be chosen continuous and single valued on the torus of integration: they are a family of periods. In this last case, the purpose of the paper is not to give another proof of global nilpotence, but to understand, how these differential factors ’succeed’ to be globally nilpotent. Throughout all the examples displayed in the paper the (minimal order) linear differential operators of quite large order (for instance order 23, 33, 50) actually factorize into products, or direct sums and products, of linear differential operators of smaller orders (up to four). Focusing on the factorized parts of all these operators, it is found that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genius five, six, ..., and even a remarkable weight-1 modular form emerging in the three-particle contribution 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 operators for another $n$-fold integral of the Ising class, for the staircase polygons counting and in Apery’s study of $\xi \left(3\right)$. $G$-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 infinity) 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.

The following systematic study of $n$-fold integrals of algebraic expressions, encountered in theoretical physics, is proposed: first generate large series expansions of these $n$-fold integrals to find out the linear differential operators that annihilate these series, then get the minimal order differential operators, then factorize and LCLM-factorize, as much as possible, these minimal order differential operators, then examine the irreducible factors to see if they are not equivalent to symmetric powers of smaller order operators, then calculate the corresponding $p$-curvatures of all these smaller order irreducible operators to see, if they have zero curvature, or if they are globally nilpotent, and, finally, examine all these smaller order irreducible operators to find out, if they correspond to hypergeometric functions up to a rational pullback.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 34M55 Painlevé and other special equations; classification, hierarchies 47E05 Ordinary differential operators 81Qxx General mathematical topics and methods in quantum theory 32G34 Moduli and deformations for ordinary differential equations 34Lxx Ordinary differential operators 34Mxx Differential equations in the complex domain 14Kxx Abelian varieties and schemes