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arXiv · 1207.1784

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

S. Boukraa·S. Hassani·J-M. Maillard
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Abstract

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 $ \chi^{(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 $ \chi^{(3)}$ and $ \chi^{(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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S. Boukraa, S. Hassani, J-M. Maillard. 2012-07-07. Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals. https://doi.org/10.1088/1751-8113%2F45%2F49%2F494010

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