The Ising model: from elliptic curves to modular forms and Calabi–Yau equations

Abstract

Abstract. We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributionsχ (n) 's of the susceptibility of the Ising model for n ≤ 6, are linear differential operators "associated with elliptic curves". Beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators Z 2 , F 2 , F 3 ,L 3 can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-four globally nilpotent linear differential operator is not reducible to this elliptic curve, modular forms scheme. This operator is shown to actually correspond to a natural generalization of this elliptic curve, modular forms scheme, with the emergence of a CalabiYau equation, corresponding to a selected 4 F 3 hypergeometric function. This hypergeometric function can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back (square root extension), the corresponding Calabi-Yau fourth-order differential operator having a symplectic differential Galois group SP (4, C). The mirror maps and higher order Schwarzian ODEs, associated with this Calabi-Yau ODE, present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms, in particular an exact (isogenies) representation of the generators of the renormalization group, extending the modular group SL(2, Z) to a GL(2, Z) symmetry group.

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…Imposing the Calabi-Yau condition [29,30] in the case of an order-four linear differential operator gives:…”

“…This condition is invariant by pullback and conjugation. Provided the Schwarzian condition (29) with W (x) given by (30) is satisfied, this symmetric Calabi-Yau condition alone is not sufficient to have L p 4 = L c 4 . Similarly to what we saw with the Calabi-Yau condition (32), would a supplementary condition to the symmetric Calabi-Yau condition be sufficient to have L p 4 = L c 4 ?…”

Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function Coefficients

“…We first treat the introductory example of [10], and then an example from [9] arising in statistical physics. We show how the corresponding systems can be desingularized by "rational" transformations at irreducible polynomials of degrees 3 and 4 respectively.…”

“…Example 6 (The Ising Model, [9]). Given The system is not desingularizable at any of the other polynomials.…”

“…Our method can, in particular, be applied to the companion system of any linear differential equation with arbitrary order n. We thus have an alternative method to the standard methods for removing apparent singularities of linear differential operators. However, it is also interesting by its own since first-order linear differential systems with apparent singularities arise naturally in applications (see, e.g., [15,16,9] and references therein, for applications within Feynman integrals and statistical physics). Moreover, such a desingularization can serve numerical methods.…”