We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these resu
lts can be seen as a consequence of the fact that, modulo 2^r, one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2^r, and, consequently, satisfy exact functional equations modulo 2^r. We sketch a possible physical interpretation of these functional equations modulo 2^r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.
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, correspon
d 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.
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 actua
lly 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.
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.
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 fact
ors 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.
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 $M=N=0,1$.
This paper deals with $tilde{chi}^{(6)}$, the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for $tilde{chi}^{(6)}$. The length of the series is suffi
cient to produce the corresponding Fuchsian linear differential equation (modulo a prime). We obtain the Fuchsian linear differential equation that annihilates the depleted series $Phi^{(6)}=tilde{chi}^{(6)} - {2 over 3} tilde{chi}^{(4)} + {2 over 45} tilde{chi}^{(2)}$. The factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper. The depleted differential operator is shown to have a structure similar to the corresponding operator for $tilde{chi}^{(5)}$. It splits into factors of smaller orders, with the left-most factor of order six being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral $E$. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.
We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization group.
We consider some two-dimensional birational transformations. One of them is a birational deformation of the Henon map. For some of these birational mappings, the post critical set (i.e. the iterates of the critical set) is infinite and we show that t
his gives straightforwardly the algebraic covariant curves of the transformation when they exist. These covariant curves are used to build the preserved meromorphic two-form. One may have also an infinite post critical set yielding a covariant curve which is not algebraic (transcendent). For two of the birational mappings considered, the post critical set is not infinite and we claim that there is no algebraic covariant curve and no preserved meromorphic two-form. For these two mappings with non infinite post critical sets, attracting sets occur and we show that they pass the usual tests (Lyapunov exponents and the fractal dimension) for being strange attractors. The strange attractor of one of these two mappings is unbounded.
We consider the Fuchsian linear differential equation obtained (modulo a prime) for $tilde{chi}^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the c
orresponding linear differential operator from calculations using just a single prime. A particular linear combination of $tilde{chi}^{(1)}$ and $tilde{chi}^{(3)}$ can be removed from $tilde{chi}^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the depleted differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $tilde{chi}^{(3)}$ and $tilde{chi}^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $tilde{chi}^{(n)}$s.
