Ap\'ery extensions

TL;DR

This paper explores the connection between Apéry numbers of Fano varieties and higher cycle classes on Landau-Ginzburg models, proposing a new motivic framework and providing detailed calculations for mirror models of Fano threefolds.

Contribution

It introduces an Apéry motive and links Apéry numbers to regulators of higher cycles, advancing the understanding of their arithmetic and geometric properties.

Findings

Constructed an Apéry motive with mixed Hodge structure containing extension classes

Linked Apéry numbers to regulators of higher algebraic K-theory classes

Performed detailed calculations for LG-models mirror to Fano threefolds

Abstract

The Ap\'ery numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau-Ginzburg models -- and thus, in particular, as periods. We also construct an ``Ap\'ery motive'', whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard-Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG-models mirror to several Fano threefolds. By describing the ``elementary'' Ap\'ery numbers in terms of regulators of higher cycles (i.e., algebraic $K$-theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG-models are modular families of $K3$ surfaces, and the distinction between multiples of $\zeta(2)$ and $\zeta(3)$ (or $(2\pi\mathbf{i})^3$) translates ultimately into one between algebraic $K_1$ and $K_3$ of the family.