Quantum matrix geometry in the lowest Landau level and higher Landau levels

TL;DR

This paper explores quantum geometries in Landau levels, revealing fuzzy sphere structures in the lowest level and novel nested geometries in higher levels, highlighting a hierarchical relationship linked to quantum anomalies.

Contribution

It introduces a hierarchical quantum geometry framework in Landau models, connecting fuzzy spheres and higher-dimensional nested geometries with monopole backgrounds.

Findings

Fuzzy sphere geometry appears in the lowest Landau level.

Higher Landau levels exhibit nested matrix geometries.

A hierarchical structure links geometries across dimensions.

Abstract

One of the most celebrated works of Professor Madore is the introduction of fuzzy sphere. I briefly review how the fuzzy two-sphere and its higher dimensional cousins are realized in the (spherical) Landau models in non-Abelian monopole backgrounds. For extracting quantum geometry from the Landau models, we evaluate the matrix elements of the coordinates of spheres in the lowest and higher Landau levels. For the lowest Landau level, the matrix geometry is identified as the geometry of fuzzy sphere. Meanwhile for the higher Landau levels, the obtained quantum geometry turns out to be a nested matrix geometry with no classical counterpart. There exists a hierarchical structure between the fuzzy geometries and the monopoles in different dimensions. That dimensional hierarchy signifies a Landau model counterpart of the dimensional ladder of quantum anomaly.