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.

35 pages

References in corpus (9)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• The diagonal Ising susceptibility
• 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
• Heun functions and diagonals of rational functions (unabridged version)
• Fuchs versus Painlevé
• The Ising correlation $C(M,N)$ for $\nu =-k$
• Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)