Nuclear Physics B ( IF 2.5 ) Pub Date : 2020-04-14 , DOI: 10.1016/j.nuclphysb.2020.115012 Kazuki Hasebe
The Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the Landau models and the associated matrix geometries through the Landau level projection. With the monopole harmonics, we explicitly derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau level the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an Pauli-Schrödinger model and the Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.
中文翻译:
SO(5)Landau模型和嵌套Nambu矩阵几何
的 Landau模型是4D量子霍尔效应的数学平台,为模糊四球的物理实现提供了难得的机会。我们提出了对通过Landau水平投影,可以找到Landau模型和相关的矩阵几何。随着 单极谐波,我们可以明确推导任意Landau层上四球体的矩阵几何:在最低Landau层上,矩阵坐标由广义方程给出 满足量子Nambu代数的模糊四球体的γ矩阵,而在更高的Landau层次上,矩阵几何成为嵌套的模糊结构,实现了纯量子几何,而经典几何中没有对应的几何。内部模糊几何结构的讨论是从 Pauli-Schrödinger模型和 Landau模型,我们在其背景非阿贝尔域配置之间揭示了隐藏的奇异轨距转换。相对论的还研究了Landau模型,并阐明了与Berezin-Toeplitz量化的关系。最后,我们将讨论甚至更大尺寸的Landau模型的矩阵几何。