Noncommutative Chern-Simons theory and exotic geometry emerging from the lowest Landau level

TL;DR

This paper reveals how internal geometric modes in the lowest Landau level of fractional quantum Hall systems lead to a noncommutative Chern-Simons theory, connecting quantum geometry with topological field theories and defining key FQH indices.

Contribution

It introduces a topological noncommutative Chern-Simons theory derived from geometric modes in FQH states, generalizing to a noncommutative K-matrix framework.

Findings

Quantum geometry emerges from internal modes in the lowest Landau level.

The geometric field theory is a topological noncommutative Chern-Simons theory.

Guiding center indices are naturally defined within this framework.

Abstract

We relate the collective dynamic internal geometric degrees of freedom to the gauge fluctuations in $\nu=1/m$(m odd) fractional quantum Hall effects. In this way, in the lowest Landau level, a highly nontrivial quantum geometry in two-dimensional guiding center space emerges from these internal geometric modes. Using the Dirac bracket method, we find that this quantum geometric field theory is a topological non-commutative Chern-Simons theory.Topological indices, such as the guiding center angular momentum (also called the shift) and the guiding center spin, which characterize the fractional quantum Hall (FQH) states besides the filling factor, are naturally defined. A noncommutative K-matrix Chern-Simons theory is proposed as a generalization to a large class of Abelian FQH topological orders.