Rank-2 attractors and Fermat type CY $n$-folds

The Fermat type Calabi-Yau $n$-fold, denoted by $\mathscr{F}_n$, is the hypersurface of $\mathbb{P}^{n+1}$ defined by $\sum_{i=0}^{n+1}x_i^{n+2}=0$, which is the smooth fiber over the Fermat point $\psi=0$ of the Fermat pencil $$\sum_{i=0}^{n+1} x^{n+2}_i -(n+2)\, \psi\, \prod_{i=0}^{n+1} x_i =0.$$ The nowhere vanishing holomorphic $n$-form on $\mathscr{F}_n$ defines an $n+1$ dimensional sub-Hodge structure of $(H^n(\mathscr{F}_n,\mathbb{Q}),F_p)$. 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 $\leq 2$, among which is a direct summand $\mathbf{H}^n_{a,1}$ whose Hodge decomposition is $$\mathbf{H}^n_{a,1}=H^{n,0}(\mathscr{F}_n) \oplus H^{0,n}(\mathscr{F}_n).$$ 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 $\psi=0$ is of the form $$t|_{\psi=0}=\frac{1}{2}+\xi \,i,$$ where $\xi$ is a real algebraic number that intuitively depends on the integer $n+2$. Furthermore, we have also numerically found that the quotient $c^+(\mathbf{H}^n_{a,1})/c^-(\mathbf{H}^n_{a,1})$ of the Deligne's periods of $\mathbf{H}^n_{a,1}$ 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 $\mathbf{H}^4_{a,1}$ satisfies the prediction of Deligne's conjecture.