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

TL;DR

This paper uses holonomic functions to analyze singularities in anisotropic Ising model integrals, revealing elliptic and Calabi-Yau structures that depend on anisotropy, advancing understanding of complex singularity manifolds.

Contribution

It demonstrates the effectiveness of holonomic functions in identifying singular manifolds in complex integrals related to the anisotropic Ising model, linking them to elliptic and Calabi-Yau geometries.

Findings

Nickelian singularities form a two-parameter family of elliptic curves.

Non-Nickelian singularities for χ^{(3)} and χ^{(4)} are rational or elliptic curves.

Singular curves depend on the anisotropy of the Ising model.

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.