On Gauge Invariance of Noncommutative Chern-Simons Theories

TL;DR

This paper constructs gauge-invariant noncommutative Chern-Simons actions for disc and double-layer geometries, revealing constraints on their form and proposing a connection to quantum Hall systems.

Contribution

It introduces a gauge-invariant formulation of noncommutative Chern-Simons theories on bounded geometries, including a boundary term and restrictions on parameters, linking to quantum Hall states.

Findings

Gauge invariance constrains NCCS action form on a disc.

A boundary boson field cancels gauge variation at the boundary.

Double NCCS with uniform integer coefficient describes Halperin (kkk) state.

Abstract

Motivated by possible applications to condensed matter systems, in this paper we construct U(N) noncommutative Chern-Simons (NCCS) action for a disc and for a double-layer geometry, respectively. In both cases, gauge invariance severely constrains the form of the NCCS action. In the first case, it is necessary to introduce a group-valued boson field with a non-local chiral boundary action, whose gauge variation cancels that of the bulk action. In the second case, the coefficient matrix $K$ in the double U(N) NCCS action is restricted to be of the form with all the matrix elements being the same integer $k$. We suggest that this double NCCS theory with U(1) gauge group describes the so-called Halperin $(kkk)$ state in a double-layer quantum Hall system. Possible physical consequences are addressed.