The lambda extensions of the Ising correlation functions C(M, N)

Abstract

We revisit, with a pedagogical heuristic motivation, the lambda extension of the low-temperature row correlation functions C(M,N) of the two-dimensional Ising model. In particular, using these one-parameter series to understand the deformation theory around selected values of $\lambda$, namely $\lambda =\cos (\pi m/n)$ with m and n integers, we show that these series yield perturbation coefficients, generalizing form factors, that are D-finite functions. As a by-product these exact results provide an infinite number of highly non-trivial identities on the complete elliptic integrals of the first and second kind. These results underline the fundamental role of Jacobi theta functions and Jacobi forms, the previous D-finite functions being (relatively simple) rational functions of Jacobi theta functions, when rewritten in terms of the nome of elliptic functions.

arXiv comment35 pages

References in corpus (24)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Square lattice Ising model susceptibility: Series expansion method and differential equation for ${\chi }^{(3)}$
• Ising model susceptibility: Fuchsian differential equation for ${\chi }^{(4)}$ and its factorization properties
• The diagonal Ising susceptibility
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Form factor expansion of the row and diagonal correlation functions of the two dimensional Ising model
• Holonomy of the Ising model form factors
• High order Fuchsian equations for the square lattice Ising model: ${\chi }^{(6)}$
• Mirror Maps, Modular Relations and Hypergeometric Series II
• Globally nilpotent differential operators and the square Ising model
• On Christol's conjecture
• Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations
• Painleve versus Fuchs
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
• Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
• Heun functions and diagonals of rational functions (unabridged version)
• The saga of the Ising susceptibility
• Fuchs versus Painlevé
• The Ising correlation $C(M,N)$ for $\nu =-k$
• Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine
• Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)
• Connection formulas for the lambda generalized Ising correlation functions