Isingn-fold integrals as diagonals of rational functions and integrality of series expansions

Abstract

Abstract. We show that the n-fold integrals χ (n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of functions generalising algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x = 0, solutions of these linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi-Yau ODEs, and, more generally, Picard-Fuchs linear ODEs with solutions of maximal weights, are always diagonal of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.This paper is the short version of the larger (100 pages) version [19], available on http: // arxiv. org/ abs/ 1211. 6031 , where all the detailed proofs are given and where a larger set of examples is displayed.

On Christol’s conjecture

“…x), or one could take any one of the three permuted versions: [4], also see [11] p. 19. † See [23] for a proof that 3 F 2 ([1/9, 4/9, 7/9], [2/3, 1], x) cannot be written as a Hadamard product.…”

“…Because of the crucial role played by diagonals of rational functions in physics [3,4], Christol's conjecture is an important open problem.…”

“…We have shown that the hypergeometric series 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x) appearing in [4,11] are diagonals of rational functions. We did so by first finding two algebraic functions whose diagonals were given by these two hypergeometric functions, and through an algorithm outlined in the paper [17], we were able to recover the rational functions whose diagonals are given by these two 3 F 2 hypergeometric functions.…”

“…For example, the globally ¶ Any diagonal of an algebraic function in n variables can be rewritten as the diagonal of a rational function in 2n variables: see [17]. † See [3,4]…”

“…But even in this case, a lot of the 3 F 2 functions are easily seen to be diagonals of a rational function. Suppose that a 3 ], x ,…”