Deligne's conjecture and mirror symmetry

Abstract

In this paper, we will study the connections between the mirror symmetry of Calabi-Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop a method to compute the Deligne's period for a Calabi-Yau threefold in the mirror family of a one-parameter mirror pair. We will give two examples to show how this method works, and we will express the Deligne's period in terms of the classical periods of the threeform. Using this method, we will compute the Deligne's period of a Calabi-Yau threefold studied in a recent paper by Candelas, de la Ossa, Elmi and van Straten. Based on their numerical results, we will explicitly show that this Calabi-Yau threefold satisfies Deligne's conjecture. A second purpose of this paper is to introduce the Deligne's conjecture to the physics community, and provide further evidence that there might exist interesting connections between physics and number theory.

arXiv comment23 pages

References in corpus (3)

• Tables of Calabi--Yau equations
• Supersymmetric Flux Compactifications and Calabi-Yau Modularity
• Periods of CY $n$-folds and mixed Tate motives, a numerical study

Cited by in corpus (7)

• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Time reversal and $\mathbf{CP}$ invariance in Calabi-Yau compactifications
• Flux Vacua and Modularity for ${\mathbb{Z}}_2$ Symmetric Calabi-Yau Manifolds
• Rank-2 attractors and Deligne's conjecture
• K3 mirror symmetry, Legendre family and Deligne's conjecture for Fermat quartic
• Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants
• Rank-2 attractors and Fermat type CY $n$-folds