From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve
We recall the form factors f(j)N,N corresponding to the l-extension C(N,N; l) 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 l...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2007 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2007
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147201 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve / S. Boukraa, S. Hassani, Jean-Marie Maillard, N. Zenine // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 130 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862618163163693056 |
|---|---|
| author | Boukraa, S. Hassani, S. Maillard, Jean-Marie Zenine, N. |
| author_facet | Boukraa, S. Hassani, S. Maillard, Jean-Marie Zenine, N. |
| citation_txt | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve / S. Boukraa, S. Hassani, Jean-Marie Maillard, N. Zenine // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 130 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We recall the form factors f(j)N,N corresponding to the l-extension C(N,N; l) 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 c(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 x(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 + 4w² = 0, that occurs in the linear differential equation of x⁽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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| first_indexed | 2025-12-07T13:13:41Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147201 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:13:41Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Boukraa, S. Hassani, S. Maillard, Jean-Marie Zenine, N. 2019-02-13T19:01:24Z 2019-02-13T19:01:24Z 2007 From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve / S. Boukraa, S. Hassani, Jean-Marie Maillard, N. Zenine // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 130 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 34M55; 47E05; 81Qxx; 32G34; 34Lxx; 34Mxx; 14Kxx https://nasplib.isofts.kiev.ua/handle/123456789/147201 We recall the form factors f(j)N,N corresponding to the l-extension C(N,N; l) 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 c(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 x(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 + 4w² = 0, that occurs in the linear differential equation of x⁽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. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). We have deserved great benefit from discussions on various aspects of this work with F. Chyzak, G. Delfino, S. Fischler, P. Flajolet, A.J. Guttmann, M. Harris, I. Jensen, L. Merel, G. Mussardo, B. Nickel, J.H.H. Perk, B. Salvy, C.A. Tracy and N. Witte. We thank A. Bostan for a search of linear ODEs modulo primes with one of his magma program. We thank one of the three referees for very usefull comments. We acknowledge a CNRS/PICS financial support. One of us (NZ) would like to acknowledge kind hospitality at the LPTMC where part of this work has been completed. One of us (JMM) thanks the MASCOS (Melbourne) where part of this work was performed. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve Article published earlier |
| spellingShingle | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve Boukraa, S. Hassani, S. Maillard, Jean-Marie Zenine, N. |
| title | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve |
| title_full | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve |
| title_fullStr | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve |
| title_full_unstemmed | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve |
| title_short | From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve |
| title_sort | from holonomy of the ising model form factors to n-fold integrals and the theory of elliptic curve |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147201 |
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