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Holonomic functions of several complex variables and singularities of anisotropic Ising \(n\)-fold integrals. (English) Zbl 1257.82017

Summary: Focusing on examples associated with holonomic functions, we try to bring new ideas on how to look at phase transitions, for which the critical manifolds are not points but curves depending on a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in the most general mathematical framework, be too complex, or simply could not be defined. In a learn-by-example approach, considering several Picard-Fuchs systems of two-variables “above” Calabi-Yau ODEs, associated with double hypergeometric series, we show that D-finite (holonomic) functions are actually a good framework for finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the \(n\)-fold integrals \(\chi^{(n)}\), corresponding to the \(n\)-particle decomposition of the magnetic susceptibility of the anisotropic square Ising model. In this anisotropic case, we revisit a set of so-called Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find the first set of non-Nickelian singularities for \(\chi^{(3)}\) and \(\chi^{(4)}\), that also turns out to be rational or elliptic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model, or, equivalently, that they depend on the spectral parameter of the model. This has important consequences on the physical nature of the anisotropic \(\chi^{(n)}\)s which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and that of Calabi-Yau manifolds on such problems. We also address the question of singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility \(\chi \).

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
47E05 General theory of ordinary differential operators
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
35Q82 PDEs in connection with statistical mechanics
14K30 Picard schemes, higher Jacobians
82B26 Phase transitions (general) in equilibrium statistical mechanics
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