Geometric Engineering and Almost Mathieu Operator

Abstract

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 $R^2=e^{\beta }$.

3 pages