A note on the non-commutative Chern-Simons model on manifolds with boundary

TL;DR

This paper investigates the classical and quantum aspects of non-commutative U(1) Chern-Simons theory on manifolds with boundaries, introducing a new *-product and exploring implications for boundary states and quantum Hall systems.

Contribution

It introduces a novel *-product adapted to regions with boundaries and analyzes the resulting gauge group and boundary states in non-commutative Chern-Simons theory.

Findings

Introduction of a *-product for boundary regions

Complexification of the gauge group due to boundary effects

Identification of boundary-dependent correlation functions

Abstract

We study field theories defined in regions of the spatial non-commutative (NC) plane with a boundary present delimiting them, concentrating in particular on the U(1) NC Chern-Simons theory on the upper half plane. We find that classical consistency and gauge invariance lead necessary to the introduction of $K_0$-space of square integrable functions null together with all their derivatives at the origin. Furthermore the requirement of closure of $K_0$ under the *-product leads to the introduction of a novel notion of the *-product itself in regions where a boundary is present, that in turn yields the complexification of the gauge group and to consider chiral waves in one sense or other. The canonical quantization of the theory is sketched identifying the physical states and the physical operators. These last ones include ordinary NC Wilson lines starting and ending on the boundary that yield correlation functions depending on points on the one-dimensional boundary. We finally extend the definition of the *-product to a strip and comment on possible relevance of these results to finite Quantum Hall systems.