Mirror symmetry, mixed motives and $\zeta(3)$

TL;DR

This paper applies mirror symmetry to arithmetic geometry, computing mixed Hodge structures related to Calabi-Yau threefolds, revealing a connection between mirror symmetry, motivic structures, and special values of the zeta function.

Contribution

It demonstrates how mirror symmetry computations can provide evidence for the motivic origin of certain mixed Hodge structures involving , linking geometry and number theory.

Findings

The limit mixed Hodge structure contains an extension involving .

The extension class relates to the prepotential and .

Connections between mirror symmetry and the Hodge conjecture for mixed Tate motives are established.

Abstract

In this paper, we present an application of mirror symmetry to arithmetic geometry. The main result is the computation of the period of a mixed Hodge structure, which lends evidence to its expected motivic origin. More precisely, given a mirror pair $(M,W)$ of Calabi-Yau threefolds, the prepotential of the complexified Kahler moduli space of $M$ admits an expansion with a constant term that is frequently of the form $$-3\, \chi (M) \,\zeta(3)/(2 \pi i)^3+r,$$ where $r \in \mathbb{Q}$ and $\chi(M)$ is the Euler characteristic of $M$. We focus on the mirror pairs for which the deformation space of the mirror threefold $W$ forms part of a one-parameter algebraic family $W_{\varphi}$ defined over $\mathbb{Q}$ and the large complex structure limit is a rational point. Assuming a version of the mirror conjecture, we compute the limit mixed Hodge structure on $H^3(W_{\varphi})$ at the large complex structure limit. It turns out to have a direct summand expressible as an extension of $\mathbb{Q}(-3)$ by $\mathbb{Q}(0)$ whose isomorphism class can be computed in terms of the prepotential of $M$, and hence, involves $\zeta(3)$. By way of Ayoub's works on the motivic nearby cycle functor, this reveals in precise form a connection between mirror symmetry and a variant of the Hodge conjecture for mixed Tate motives.