The Ising correlation $C(M,N)$ for $ u=-k$

We present Painlev{e} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $ C(M,N)$ with $M leq N $ in the special case $ u = -k$ where $ u = , sinh 2E_h/k_BT/sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M $ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $ M+N$ even or odd. These four different non-linear ODEs are also valid for $M ge N$ when $ u = -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $ u = , -k$ condition, a remarkable phenomenon of a Painleve VI sigma function being the sum of four Painleve VI sigma functions having the same Okamoto parameters. We show in this $ u = , -k$ case for $ T < T_c $ and also $ T > T_c$, that $ C(M,N)$ with $ M leq N $ is given as an $ N times N$ Toeplitz determinant.

Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)

We show that the results we had obtained on diagonals of nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked $ _2F_1$ hypergeometric functions, can be obtained, much more efficiently, calculating the $ j$-invariant of an elliptic curve canonically associated with the denominator of the rational functions. In the case where creative telescoping yields pullbacked $ _2F_1$ hypergeometric functions, we generalize this result to other families of rational functions in three, and even more than three, variables. We also generalise this result to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, foliation in elliptic curves. We also extend these results to rational functions in three variables when the denominator is associated with a {em genus-two curve such that its Jacobian is a split Jacobian} corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked $ _2F_1$ hypergeometric solutions, the denominator corresponding to an algebraic variety being not simply foliated in elliptic curves, but having a selected elliptic curve in the variety explaining the pullbacked $ _2F_1$ hypergeometric solution.

Heun functions and diagonals of rational functions (unabridged version)

We provide a set of diagonals of simple rational functions of three and four variables that are squares of Heun functions. These Heun functions obtained through creative telescoping, turn out to be either pullbacked $_2F_1$ hypergeometric functions and in fact classical modular forms. We also obtain Heun functions that are Shimura curves as solutions of telescopers of rational functions.

Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2F1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2F1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2F1 hypergeometric functions and modular forms.

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

We show that the n-fold integrals $chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the Ising class, or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of function generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x=0, solutions of these linear differential equations Derived From Geometry are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

We show that the n-fold integrals $chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the Ising class, or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations Derived From Geometry are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christols conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.

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

We give the exact expressions of the partial susceptibilities $chi^{(3)}_d$ and $chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],, [1,1];, z)$ and $_4F_3([1/2,1/2,1/2,1/2],, [1,1,1]; , z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $chi^{(3)}_d$ and $chi^{(4)}_d$. We also give new results for $chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only Derived from Geometry (globally nilpotent), but actually correspond to Special Geometry (homomorphic to their formal adjoint). This raises the question of seeing if these special geometry Ising-operators, are special ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.

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 contribution of the susceptibility of the Ising model for $, n le 6$, are operators associated with elliptic curves. Beyond the simplest 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 can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-four globally nilpotent operator is not reducible to this elliptic curve, modular forms scheme. It is shown to actually correspond to a natural generalization of this elliptic curve, modular forms scheme, with the emergence of a Calabi-Yau equation, corresponding to a selected $_4F_3$ hypergeometric function which can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back, the corresponding Calabi-Yau fourth-order differential operator having a symplectic differential Galois group SP(4,C). The associated mirror maps and higher order Schwarzian ODEs has an exact (isogenies) representation of the generators of the renormalization group, extending the modular group SL(2,Z) to a GL(2, Z) symmetry group.

The saga of the Ising susceptibility

We review developments made since 1959 in the search for a closed form for the susceptibility of the Ising model. The expressions for the form factors in terms of the nome $q$ and the modulus $k$ are compared and contrasted. The $lambda$ generalized correlations $C(M,N;lambda)$ are defined and explicitly computed in terms of theta functions for $M=N=0,1$.

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

We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}(Phi^{(5)}) = E^{(5)}$, where $Phi^{(5)} =tilde{chi}^{(5)}-tilde{chi}^{(3)}/2+tilde{chi}^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(rm left)} cdot L_{12}^{(rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $tilde{chi}^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $tilde{chi}^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {bf 41} 455202 (2008)), we find that $tilde{chi}^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.