Abstract

In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne’s conjecture on the special values of L-functions. We will also formulate several open questions concerning the potential connections between attractors in string theory and number theory.

Highlights

  • Is very difficult to find examples of rank-2 attractors, even by numerical methods

  • In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits

  • We will look at the interesting connections between rank-2 attractors in string theory and Deligne’s conjecture on the special values of L-functions

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Summary

The rank-2 attractors

Let us first introduce some properties of the attractor mechanism that will be needed in this paper, while the readers could consult the paper [15] for a more rigorous treatment. We will use the more formal mathematical language, i.e. pure Hodge structure, to explain what is a rank-2 attractor. The Hodge filtration Fφp varies holomorphically with respect to the variable φ, which defines a holomorphic vector bundle over the smooth locus of the family (1.1) [14]. A point φ ∈ P1 is called an attractor point if there exists a nonzero charge vector γ ∈. A point φ ∈ P1 is called a rank-2 attractor point if there exist two linearly independent charge vectors γ1 and γ2 such that their Hodge decompositions satisfy γ1 = γ1(3,0) + γ1(0,3), γ2 = γ2(3,0) + γ2(0,3). If Xφ is a rank-2 attractor, the two charge vectors γ1 and γ2 will define a two dimensional sub-Hodge structure MaBtt, whose Hodge type is (3, 0) + (0, 3). As the name has suggested, MeBll is closely related to the theory of elliptic curves

The pure motives and Deligne’s conjecture
The arithmetic properties of rank-2 attractors
Mirror symmetry and the attractor equation
Picard-Fuchs equation
Mirror symmetry
The attractor equation
An example of the rank-2 attractor
Attractor equations
The zeta functions and L-function of the rank-2 attractor
The algebraic de Rham cohomology of the rank-2 attractor
The involution induced by complex conjugation
The split of the algebraic de Rham cohomology
The verification of Deligne’s conjecture
The attractive sub-motive
The elliptic sub-motive
The lattice and the complex elliptic curve
Conclusions and further prospects
Full Text
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