  The type IIA string theory on a non-compact Calabi-Yau geometry known as the
local $\mathbb{P}^{1} \times \mathbb{P}^{1}$ gives rise to five-dimensional N
=1 supersymmetric SU(2) gauge theory compactified on a circle, known as
geometric engineering. So it is necessary to study the $\mathbb{P}^{1} \times
\mathbb{P}^{1}$ in details. Since the spectrum of the local $\mathbb{P}^{1}
\times \mathbb{P}^{1}$ can be written as
$E=R^{2}\left(\mathrm{e}^{p}+\mathrm{e}^{-p}\right)+\mathrm{e}^{x}+\mathrm{e}^{-x}$,
then by the result of almost Mathieu operator, we show that: (1) when
$R^{2}<1$, the spectrum is absolutely continuous which meanings the medium is
conductor. (2) when $1\le R^{2}<e^{\beta}$, the spectrum is singular continuous
known as quantum Hall effect. (3) when $R^{2}>e^{\beta}$, the spectrum is
almost surely pure point and exhibits Anderson localization. In other words,
there are two phase transition points which one is $R^{2}=1$ and the other one
is $R^{2}=e^{\beta}$.
