Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: hep-lat
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The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga