# Geometric Engineering and Almost Mathieu Operator

## TL;DR

This paper explores the spectral properties of a geometric engineering model derived from string theory, revealing phase transitions in the spectrum related to the almost Mathieu operator and quantum Hall phenomena.

## Contribution

It connects string theory geometric models with spectral analysis of the almost Mathieu operator, identifying phase transitions in the spectrum.

## Key findings

- Spectrum is absolutely continuous for R^2<1
- Spectrum is singular continuous for 1≤R^2<e^β
- Spectrum is pure point for R^2>e^β

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