Absence of the Holographic Principle in Noncommutative Chern-Simons Theory

TL;DR

This paper investigates how noncommutativity affects edge observables in Chern-Simons theory, revealing the absence of the holographic principle and showing that boundary observables become bulk observables, with implications for the theory's Hamiltonian formulation.

Contribution

It demonstrates that noncommutativity causes edge observables to move into the bulk and shows the breakdown of the Hamiltonian formulation, suggesting modifications to the Moyal star product and exploring matrix model alternatives.

Findings

Edge observables move into the bulk with noncommutativity.

Hamiltonian formulation breaks down at all orders of noncommutativity.

Matrix models can describe the theory via Wilson lines.

Abstract

We examine noncommutative Chern-Simons theory on a bounded spatial domain. We argue that upon `turning on' the noncommutativity, the edge observables, which characterized the commutative theory, move into the bulk. We show this to lowest order in the noncommutativity parameter appearing in the Moyal star product. If one includes all orders, the Hamiltonian formulation of the gauge theory ceases to exist, indicating that the Moyal star product must be modified in the presence of a boundary. Alternative descriptions are matrix models. We examine one such model, obtained by a simple truncation of Chern-Simons theory on the noncommutative plane, and express its observables in terms of Wilson lines.