Rank-2 attractors and Fermat type CY n-folds

2020/05/14 by Yang, Wenzhe
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2005.06722

Abstract

The Fermat type Calabi-Yau n-fold, denoted by \mathscrF_n, is the hypersurface of ℙⁿ⁺¹ defined by ∑_i=0ⁿ⁺¹x_iⁿ⁺²=0, which is the smooth fiber over the Fermat point ψ=0 of the Fermat pencil ∑_i=0ⁿ⁺¹ xⁿ⁺²_i -(n+2) ψ ∏_i=0ⁿ⁺¹ x_i =0. The nowhere vanishing holomorphic n-form on \mathscrF_n defines an n+1 dimensional sub-Hodge structure of (Hⁿ(\mathscrF_n,ℚ),F_p). In this paper, we will formulate a conjecture which says that this n+1 dimensional sub-Hodge structure splits completely into the direct sum of pure Hodge structures with dimensions ≤ 2, among which is a direct summand Hⁿ_a,1 whose Hodge decomposition is Hⁿ_a,1=H^n,0(\mathscrF_n) ⊕ H^0,n(\mathscrF_n). Using numerical methods, we are able to explicitly construct such a split for the cases where n=3,4,6, while we also construct a partial split for the cases where n=8,10. For n=3,4,6,8,10, we have numerically found that the value of the mirror map t for the Fermat pencil at the Fermat point ψ=0 is of the form t|_ψ=0=(1)/(2)+ξ i, where ξ is a real algebraic number that intuitively depends on the integer n+2. Furthermore, we have also numerically found that the quotient c⁺(Hⁿ_a,1)/c^-(Hⁿ_a,1) of the Deligne's periods of Hⁿ_a,1 is an algebraic number for the cases where n=3,4,6,8,10, and in fact we will formulate a stronger conjecture generalizing this observation. We will also show that H⁴_a,1 satisfies the prediction of Deligne's conjecture.

Related

• Periods of CY n-folds and mixed Tate motives, a numerical study
• Calabi-Yau attractor varieties and degeneration of Hodge structure
• More on G-flux and General Hodge Cycles on the Fermat Sextic
• Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants
• Hodge numbers for orbifolds of Calabi-Yau threefolds Ferma type and the Roan pairs