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

Abstract

Dedicated to Fa Yuen Wu on the occasion of his 80th birthday.Abstract. 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 actually finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We, then, analyse the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the n-fold integrals χ (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 a first set of non-Nickelian singularities for χ (3) and χ (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, or, equivalently, that they depend on the spectral parameter of model. This has important consequences on the physical nature of the anisotropic χ (n) 's which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We also address the question of the singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility χ.

Automata and the susceptibility of the square lattice Ising model modulo powers of primes

“…For the low and high-temperature series forχ in the variable w (see (6), (7), (8), (9), ...), we have obtained similar results and functional equations modulo q = 2, 4, 8, 16, 32, 64. These series, and the corresponding functional equations, are given in Appendix A.…”

“…The natural emergence of diagonals of rational functions in an extremely large set of lattice statistical mechanics and enumerative combinatorics models, has been emphasised and explained in [18]. That paper explains why a large class of functions describing lattice models that can be expressed as n-fold integrals of an algebraic ¶ ‡ One takes the derivative with respect to the nome q (see equation (6) in [41]).…”

“…L +χ (6) L is seen to be zero modulo 2, 4, 8, 16, 32, 64, 128, 256. The scenario seems to be that one cannot distinguish between the series forχ L and that for a finite sum likeχ L are given in [23], up to n = 12. This scenario has been checked and found to hold up toχ (12) L .…”

“…Forty years ago, Wu, Barouch, McCoy and Tracy [4] showed that the full susceptibility of the square-lattice Ising model can be decomposed as the infinite sum of holonomic n-fold integrals [5,6,7,8,9], denoted χ (n) . In the last decade the linear differential operators corresponding to the first χ (n) 's, up to n = 6, were obtained, underlying the role of the elliptic curve parametrization [10], but showing also the emergence of (at least) one Calabi-Yau ODE, and beyond, of linear differential operators with selected differential Galois groups [11,12,13].…”

The anisotropic Ising correlations as elliptic integrals: duality and differential equations

“…All the results given in this paper underline the fundamental role of the complete elliptic integral of the third kind in order to have a clean, clear-cut description of the anisotropic Ising model [31]. Along this line, we have been able to extend Ghosh's representation of the Kramers-Wannier duality [24] on the complete elliptic integral of the first and second kind, to a representation of the Kramers-Wannier duality to complete elliptic integral of the third kind.…”