Motives and mirror symmetry for Calabi-Yau orbifolds

TL;DR

This paper explores the topological mirror symmetry of Calabi-Yau orbifolds derived from Fermat hypersurfaces, introducing Fermat motives to interpret mirror symmetry phenomena through point counting methods.

Contribution

It introduces Fermat motives for Calabi-Yau orbifolds and establishes a motivic interpretation of topological mirror symmetry at the Fermat point.

Findings

Established a one-to-one correspondence between monomial classes and Fermat motives.

Computed the number of rational points over finite fields using two different methods.

Provided detailed examples illustrating the motivic interpretation of mirror symmetry.

Abstract

We consider certain families of Calabi-Yau orbifolds and their mirror partners constructed from Fermat hypersurfaces in weighted projective 4-spaces. Our focus is the topological mirror symmetry. There are at least three known ingredients to describe the topological mirror symmetry, namely, integral vertices in reflexive polytopes, monomials in graded polynomial rings (with some group actions), and periods (and Picard-Fuchs differential equations). In this paper we will introduce Fermat motives associated to these Calabi-Yau orbifolds and then use them to give motivic interpretation of the topological mirror symmetry phenomenon between mirror pairs of Calabi-Yau orbifolds. We establish, at the Fermat (the Landau-Ginzburg) point in the moduli space, the one-to-one correspondence between the monomial classes and Fermat motives. This is done by computing the number of ${\bf F}_q$-rational points on our Calabi-Yau orbifolds over ${\bf F}_q$ in two different ways: Weil's algebraic number theoretic method involving Jacobi (Gauss) sums, and Dwork's $p$-adic analytic method involving Dwork characters and Gauss sums. We will discuss specific examples in detail.