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

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.