Singularities of<i>n</i>-fold integrals of the Ising class and the theory of elliptic curves

Boukraa, S.; Hassani, S.; Maillard, J.‐M.; Zenine, N.

doi:10.1088/1751-8113/40/39/003

Cited by 28 publications

(277 citation statements)

References 45 publications

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“…Φ (6) H ). In [30] it was shown how this comes about and how it generalizes. For this, we had to solve the Landau conditions [30] for the n-fold integrals (7.5).…”

Section: The Linear Odes Of the Multiple Integrals φ (N)mentioning

confidence: 98%

“…, 6, that we compare with the singularities obtained from the Landau conditions. We have shown [30] that the set of singularities associated with the ODEs of the multiple integrals Φ (n) H reduce to the singularities of the ODEs associated with a finite number of one-dimensional integrals. Section 9 deals with the complex multiplication for the elliptic curves related to the singularities given by the zeros of the quadratic polynomial 1 + 3w + 4w 2 .…”

Section: Other N-fold Integrals Linked To the Susceptibility Of The Imentioning

confidence: 99%

“…H are obtained by expanding in the variables x i and performing the integration (see Appendix A of [30]). One obtains…”

Section: Other N-fold Integrals Linked To the Susceptibility Of The Imentioning

confidence: 99%

“…We thus see the occurrence of additional non-trivial complex selected values of the modulus k, beyond the well-known values k = 1, 0, ∞, corresponding to degeneration of the elliptic curve into a rational curve, and physically, to the critical Ising model and to (high-low temperature) trivializations of the model. Of course, when extending (9.8) to complex values, one can be concerned about keeping track of the sign of k 1 in (9.8) in front of the square root √ k. Reference [30] provides a similar fixed point calculation for (9.8) extended to complex values, but for a representation of (9.8) in term of the modular j-function. Such calculations single out the remarkable integer value j = (−15) 3 , which is known to be one of the nine Heegner numbers (see [30]).…”

Section: Complex Multiplication Of Elliptic Curves and F Ixed Points mentioning

confidence: 99%

“…Of course, when extending (9.8) to complex values, one can be concerned about keeping track of the sign of k 1 in (9.8) in front of the square root √ k. Reference [30] provides a similar fixed point calculation for (9.8) extended to complex values, but for a representation of (9.8) in term of the modular j-function. Such calculations single out the remarkable integer value j = (−15) 3 , which is known to be one of the nine Heegner numbers (see [30]). It is important to note that this representation of (9.8) in term of the modular j-function is the well-known fundamental modular curve symmetric in j and j 1 (see [30,51,75])…”

Section: Complex Multiplication Of Elliptic Curves and F Ixed Points mentioning

confidence: 99%

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From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa

Hassani²,

Maillard³

et al. 2007

SIGMA

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Abstract. We recall the form factors f (j) N,N corresponding to the λ-extension C(N, N ; λ) of the two-point diagonal correlation function of the Ising model on the square lattice and their associated linear differential equations which exhibit both a "Russian-doll" nesting, and a decomposition of the linear differential operators as a direct sum of operators (equivalent to symmetric powers of the differential operator of the complete elliptic integral E). The scaling limit of these differential operators breaks the direct sum structure but not the "Russian doll" structure, the "scaled" linear differential operators being no longer Fuchsian. We then introduce some multiple integrals of the Ising class expected to have the same singularities as the singularities of the n-particle contributions χ (n) to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equations satisfied by these multiple integrals for n = 1, 2, 3, 4 and, only modulo a prime, for n = 5 and 6, thus providing a large set of (possible) new singularities of the χ (n) . We get the location of these singularities by solving the Landau conditions. We discuss the mathematical, as well as physical, interpretation of these new singularities. Among the singularities found, we underline the fact that the quadratic polynomial condition 1 + 3w + 4w 2 = 0, that occurs in the linear differential equation of χ (3) , actually corresponds to the occurrence of complex multiplication for elliptic curves. The interpretation of complex multiplication for elliptic curves as complex fixed points of generators of the exact renormalization group is sketched. The other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting a geometric interpretation in terms of more general (motivic) mathematical structures beyond the theory of elliptic curves. The scaling limit of the (lattice off-critical) structures as a confluent limit of regular singularities is discussed in the conclusion.

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“…Φ (6) H ). In [30] it was shown how this comes about and how it generalizes. For this, we had to solve the Landau conditions [30] for the n-fold integrals (7.5).…”

Section: The Linear Odes Of the Multiple Integrals φ (N)mentioning

confidence: 98%

Section: Other N-fold Integrals Linked To the Susceptibility Of The Imentioning

confidence: 99%

“…H are obtained by expanding in the variables x i and performing the integration (see Appendix A of [30]). One obtains…”

Section: Other N-fold Integrals Linked To the Susceptibility Of The Imentioning

confidence: 99%

Section: Complex Multiplication Of Elliptic Curves and F Ixed Points mentioning

confidence: 99%

Section: Complex Multiplication Of Elliptic Curves and F Ixed Points mentioning

confidence: 99%

See 3 more Smart Citations

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Boukraa

Hassani²,

Maillard³

et al. 2007

SIGMA

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Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Boukraa¹,

Guttmann²,

Hassani³

et al. 2008

J. Phys. A: Math. Theor.

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249

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Abstract. We calculate very long low-and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ (5) and six-particle contribution χ (6) . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. The series for χ (low-and high-temperature regime), χ (5) and χ (6) are now extended to 2000 terms. In addition, for χ (5) , 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime.A diff-Padé analysis of the 2000 terms series for χ (5) and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of "additional" singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the "physical" χ (5) and χ (6) , that is to say the series-solutions of the ODE's which are analytic at w = 0.Furthermore, analysis of the long series for χ (5) (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ (n) , n ≥ 7. The exponents at all these singularities are also seen to be rational numbers.We also present a mechanism of resummation of the logarithmic singularities of the χ (n) leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility χ.

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Globally nilpotent differential operators and the square Ising model

Bostan¹,

Boukraa²,

Hassani³

et al. 2009

J. Phys. A: Math. Theor.

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232

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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 λ-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 very 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 factorised 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 χ (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, ΦH , for the staircase polygons counting, and in Apéry's study of ζ(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. PACS: 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx AMS Classification scheme numbers: 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14KxxGlobally nilpotent operators 2 Key-words: Globally nilpotent operators, p-curvature, G-functions, arithmetic Gevrey series, Form factors of the square Ising model, susceptibility of the Ising model, Fuchsian linear differential equations, moduli space of curves, two-point correlation functions of the lattice Ising model, complete elliptic integrals, scaling limit of the Ising model, apparent singularities, modular forms, Atkin-Lehmer involutions, Fricke involutions, Dedekind eta functions, Weber modular functions, Calabi-Yau manifolds, three-choice polygons, enumerative combinatorics. IntroductionGenerating large series expansions of physical quantities that are quite often defined as n-fold integrals is the bread and butter of lattice statistical mechanics, enumerative combinatorics, and more generally theoretical physics. The n-fold integrals...

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

Cited by 28 publications

References 45 publications

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curves

Experimental mathematics on the magnetic susceptibility of the square lattice Ising model

Globally nilpotent differential operators and the square Ising model

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