English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibilityof the Ising model, as well as various other n-fold integrals of the "Isingclass", or n-fold integrals from enumerative combinatorics, like lattice Greenfunctions, are actually diagonals of rational functions. As a consequence, thepower series expansions of these solutions of linear differential equations"Derived From Geometry" are globally bounded, which means that, after just onerescaling of the expansion variable, they can be cast into series expansionswith integer coefficients. Besides, in a more enumerative combinatoricscontext, we show that generating functions whose coefficients are expressed interms of nested sums of products of binomial terms can also be shown to bediagonals of rational functions. We give a large set of results illustratingthe fact that the unique analytical solution of Calabi-Yau ODEs, and moregenerally of MUM ODEs, is, almost always, diagonal of rational functions. Werevisit Christol's conjecture that globally bounded series of G-operators arenecessarily diagonals of rational functions. We provide a large set of examplesof globally bounded series, or series with integer coefficients, associatedwith modular forms, or Hadamard product of modular forms, or associated withCalabi-Yau ODEs, underlying the concept of modularity. We finally address thequestion of the relations between the notion of integrality (series withinteger coefficients, or, more generally, globally bounded series) and themodularity (in particular integrality of the Taylor coefficients of mirrormap), introducing new representations of Yukawa couplings.

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity