S. Hassani

The Fuchsian differential equation of the square lattice Ising model χ (3) susceptibility

Using an expansion method in the variables xi that appear in (n ? 1)-dimensional integrals representing the n-particle contribution to the Ising square lattice model susceptibility ?, we generate a long series of coefficients for the three-particle contribution ?(3), using an N4 polynomial time algorithm. We give the Fuchsian differential equation of order 7 for ?(3) that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.

Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ (4)

We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighbouring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution (χ (3) and χ (4) ) to the magnetic susceptibility of the square lattice Ising model. We deduce all the critical behaviours of the solutions χ (3) and χ (4) , as well as the asymptotic behaviour of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic singularities of the Fuchsian ODE associated with χ (3) are not singularities of the particular solution χ (3) itself. We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for χ (3) (and χ (4) ) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for χ (3) (and χ (4) ), whose analysis is just sketched here. As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibility χ at high temperature (respectively low temperature) and the first two terms χ (1) and χ (3) (respectively χ (2) and χ (4) ).

Globally nilpotent differential operators and the square Ising model

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 ?-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 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 factorized 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 ?(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, ?(3)H, for the staircase polygons counting, and in Ap?rys study of ?(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.

The diagonal Ising susceptibility

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions ?(1)d and ?(2)d of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ?(5)d, but only modulo a given prime. We use these exact linear differential equations to show that not only the Russian-doll structure but also the direct sum structure on the linear differential operators for the n-particle contributions ?(n)d are quite directly inherited from the direct sum structure on the form factors f(n). We show that the nth particle contributions ?(n)d have their singularities at roots of unity. These singularities become dense on the unit circle |sinh?2Ev/kTsinh?2Eh/kT| = 1 as n ? ?.

The Ising model: from elliptic curves to modular forms and Calabi–Yau equations

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributions s of the susceptibility of the Ising model for n ≤ 6 are linear differential operators 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 Z2, F2, F3, can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-4 globally nilpotent linear differential operator is not reducible to this elliptic curve, modular form scheme. This operator is shown to actually correspond to a natural generalization of this elliptic curve, modular form scheme, with the emergence of a Calabi–Yau equation, corresponding to a selected 4F3 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 . 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 to a symmetry group.

Singularities of n-fold integrals of the Ising class and the theory of elliptic curves

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the

Square lattice Ising model \tilde{\chi }^{(5)} ODE in exact arithmetic

n

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi–Yau equations

-particle contributions

Renormalization, isogenies, and rational symmetries of differential equations

\chi^{(n)}

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equation satisfied by these multiple integrals for