[Paper Review] Motives and mirror symmetry for Calabi-Yau orbifolds

Shabnam Nargis Kadir, Noriko Yui|ArXiv.org|Jun 28, 2006

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between monomials in the mirror map and Fermat motives for Calabi–Yau orbifolds constructed from Fermat hypersurfaces in weighted projective 4-spaces. Using Weil's and Dwork's methods to compute Frobenius traces over finite fields, it provides a motivic interpretation of topological mirror symmetry at the Fermat (Landau–Ginzburg) point, showing that Fermat motives are invariant under the mirror map and that extra monomials at conifold points correspond to Tate motives.

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.

Motivation & Objective

• To provide a motivic interpretation of topological mirror symmetry for Calabi–Yau orbifolds constructed from Fermat hypersurfaces in weighted projective 4-spaces.
• To define and compute Fermat motives via algebraic correspondences and cohomological realizations, focusing on the Fermat point in moduli space.
• To establish a one-to-one correspondence between monomial classes and Fermat motives, linking mirror symmetry to motivic structures.
• To analyze the role of extra monomials at conifold points and their relation to Tate motives through zeta-function computations.
• To extend the mirror symmetry correspondence to orbifold quotients by finite abelian groups, particularly in the context of $p$-adic cohomology and $L$-series.

Proposed method

• Defining Fermat motives using Grothendieck's theory of motives, via algebraic correspondences and projectors, following Shioda and Manin.
• Constructing Calabi–Yau orbifolds as quotients of Fermat hypersurfaces in weighted projective 4-spaces by finite abelian groups, followed by crepant resolution of singularities.
• Applying Weil's method of counting $\mathbb{F}_q$-rational points using Gauss and Jacobi sums to compute zeta functions.
• Using Dwork's $p$-adic analytic method involving Dwork characters and Gauss sums to compute the same point counts, enabling comparison of $L$-series.
• Establishing the monomial–motive correspondence via cohomological realizations of motives and matching them with monomials in the mirror map.
• Analyzing specific examples with $m=5$ and $m=10$, computing Hodge numbers, Euler characteristics, and monodromy at conifold points to verify the correspondence.

Experimental results

Research questions

• RQ1 How can topological mirror symmetry for Calabi–Yau orbifolds be interpreted in terms of motives?
• RQ2 What is the precise correspondence between monomials in the mirror map and Fermat motives at the Fermat point?
• RQ3 How do extra monomials at conifold points relate to motivic structures, particularly Tate motives?
• RQ4 Can the monomial–motive correspondence be extended to orbifold quotients by subgroups of the duality group $G$ and $\hat{G}$?
• RQ5 What is the role of $p$-adic cohomology in realizing the motivic mirror correspondence beyond the Fermat point?

Key findings

• A one-to-one correspondence is established between monomial classes and Fermat motives at the Fermat point in moduli space, with motives invariant under the mirror map.
• For $m=5$, $Q=(1,1,1,1,1)$, the rigid Calabi–Yau threefold with $h^{2,1}=0$ and $B_3=2$ is the only one with a rigid weight motive, corresponding to the attractive Calabi–Yau threefold.
• At conifold points like $\psi=1$, extra monomials such as $(4,0,3,2,1)$ of degree 10 and multiplicity 24 appear, which are associated with Tate motives, as confirmed by zeta-function computations.
• The zeta functions computed via Weil's and Dwork's methods agree, validating the motivic decomposition and showing that the $L$-series factorizes into motivic components.
• For $m=10$, $Q=(1,1,1,2,5)$, the mirror pairs of orbifolds exhibit symmetric Hodge numbers under the duality $H \leftrightarrow \hat{H}$, confirming the mirror symmetry conjecture.
• The monodromy at conifold points is linked to the appearance of Tate motives, and the extra monomials do not arise from lower-dimensional Fermat surfaces of degree 5.