arXiv subjects

S. Boukraa

Publications and source records attributed to S. Boukraa.

At least 19 recordsLinked to original sources

arXiv2022

Symmetries of non-linear ODEs: lambda extensions of the Ising correlations

This paper provides several illustrations of the numerous remarkable properties of the lambda-extensions of the two-point correlation functions of the Ising model, sheding some light on the non-linear ODEs of the Painlev\'e type. We first show that this concept also exists for the factors of the two-point correlation functions focusing, for pedagogical reasons, on two examples namely C(0,5) and C(2,5) at $\nu = -k$. We then display, in a learn-by-example approach, some of the puzzling properties and structures of these lambda-extensions: for an infinite set of (algebraic) values of $ \lambda$ these power series become algebraic functions, and for a finite set of (rational) values of lambda they become D-finite functions, more precisely polynomials (of different degrees) in the complete elliptic integrals of the first and second kind K and E. For generic values of $ \lambda$ these power series are not D-finite, they are differentially algebraic. For an infinite number of other (rational) values of $ \lambda$ these power series are globally bounded series, thus providing an example of an infinite number of globally bounded differentially algebraic series. Finally, taking the example of a product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlev\'e type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions. The question of their possible reduction, after complicated transformations, to Okamoto sigma forms of Painlev\'e VI remains an extremely difficult challenge.

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arXiv2022

The lambda extensions of the Ising correlation functions C(M, N)

We revisit, with a pedagogical heuristic motivation, the lambda extension of the low-temperature row correlation functions C(M,N) of the two-dimensional Ising model. In particular, using these one-parameter series to understand the deformation theory around selected values of $\lambda$, namely $\lambda = \cos(\pi \, m/n)$ with m and n integers, we show that these series yield perturbation coefficients, generalizing form factors, that are D-finite functions. As a by-product these exact results provide an infinite number of highly non-trivial identities on the complete elliptic integrals of the first and second kind. These results underline the fundamental role of Jacobi theta functions and Jacobi forms, the previous D-finite functions being (relatively simple) rational functions of Jacobi theta functions, when rewritten in terms of the nome of elliptic functions.

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arXiv2022

Factorization of Ising correlations C(M,N) for $ \nu= \, -k$ and M+N odd, $M \le N$, $T < T_c$ and their lambda extensions

We study the factorizations of Ising low-temperature correlations C(M,N) for $\nu=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlev\'e VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlev\'e VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlev\'e VI equations which generalizes the factorization of the correlations C(M,N) to an additive decomposition of the corresponding sigma's solutions of the Okamoto sigma-form of Painlev\'e VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy-Widom (Painlev\'e V) relations between the sum and difference of sigma's to this Painlev\'e VI framework.

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arXiv2020

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

We present Painlev{é} 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 $ν= -k$ where $ν= \, \sinh 2E_h/k_BT/\sinh 2E_v/k_BT$. 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 $ ν= -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $ν= \, -k$ condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this $ν= \, -k$ 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.

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arXiv2020

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.

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arXiv2019

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.

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arXiv2018

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.

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arXiv2015

Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics

We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2 +2 \cos(jπ/n)$, the holonomic status of the q= 4 being unclear. We focus on the analysis of the q= 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q=4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or algebraic, functions of diagonals of rational functions.

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arXiv2015

Diagonals of rational functions and selected differential Galois groups

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we analyze exhaustively three sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. We impose the constraints that the degree of the denominators in each variable is at most one, and the coefficients of the monomials are 0 or $ \pm 1$, so that the analysis can be exhaustive. We find the minimal linear differential operators annihilating the diagonals of these rational functions of three, four, five and six variables. We find that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We discuss the conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.

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arXiv2014

Canonical decomposition of linear differential operators with selected differential Galois groups

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.

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arXiv2014

The Ising model and Special Geometries

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals $χ^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators we already know to be Derived from Geometry, are special globally nilpotent operators: they correspond to "Special Geometries". Beyond the small order factor operators (occurring in the linear differential operators associated with $ χ^{(5)}$ and $ χ^{(6)}$), and, in particular, those associated with modular forms, we focus on the quite large order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group $ Sp(12, \mathbb{C})$. The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group $ SO(21, \mathbb{C})$.

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arXiv2013

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

We show that the n-fold integrals $χ^{(n)}$ 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.

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arXiv2012

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

We show that the n-fold integrals $χ^{(n)}$ 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 Christol's 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.

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arXiv2012

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

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $ χ^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $ χ^{(3)}$ and $ χ^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.

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arXiv2011

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

We give the exact expressions of the partial susceptibilities $χ^{(3)}_d$ and $χ^{(4)}_d$ 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 $χ^{(3)}_d$ and $χ^{(4)}_d$. We also give new results for $χ^{(5)}_d$. 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.

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arXiv2010

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.

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arXiv2010

Square lattice Ising model $\tildeχ^{(5)}$ 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$L_{24}(Φ^{(5)}) = E^{(5)}$, where $Φ^{(5)} =\tildeχ^{(5)}-\tildeχ^{(3)}/2+\tildeχ^{(1)}/120$ 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 $\tildeχ^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $\tildeχ^{(5)}$ 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 $\tildeχ^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.

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arXiv2010

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 $λ$ generalized correlations $C(M,N;λ)$ are defined and explicitly computed in terms of theta functions for $M=N=0,1$.

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