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

Abstract

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contribution of the susceptibility of the Ising model for $n\le 6$, are operators "associated with elliptic curves". Beyond the simplest 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 can actually be interpreted as modular forms of the elliptic curve of the Ising model. A last order-four globally nilpotent operator is not reducible to this elliptic curve, modular forms scheme. It is shown to actually correspond to a natural generalization of this elliptic curve, modular forms scheme, with the emergence of a Calabi-Yau equation, corresponding to a selected ${}_4{F}_3$ hypergeometric function which can also be seen as a Hadamard product of the complete elliptic integral K, with a remarkably simple algebraic pull-back, the corresponding Calabi-Yau fourth-order differential operator having a symplectic differential Galois group SP(4,C). The associated mirror maps and higher order Schwarzian ODEs has 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.

46 pages

References in corpus (7)

• On Rationally Parametrized Modular Equations
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Holonomy of the Ising model form factors
• Globally nilpotent differential operators and the square Ising model
• Landau singularities and singularities of holonomic integrals of the Ising class
• Fuchs versus Painlevé

Cited by in corpus (21)

• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• The $Z$-invariant Ising model via dimers
• Diagonals of rational functions and selected differential Galois groups
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Lattice Green Functions: the seven-dimensional face-centred cubic lattice
• Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
• The sixth Painleve transcendent and uniformization of algebraic curves
• Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
• Computing the Lie algebra of the differential Galois group: the reducible case
• Canonical decomposition of linear differential operators with selected differential Galois groups
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
• Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)
• Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)
• The lambda extensions of the Ising correlation functions C(M, N)
• The Ising model and Special Geometries
• Factorization of the Ising model form factors
• Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)
• Automata and the susceptibility of the square lattice Ising model modulo powers of primes