Edge States from Defects on the Noncommutative Plane

TL;DR

This paper explores how boundary states emerge from defects in a noncommutative plane, revealing their connection to commutative boundaries and algebraic deformations, with implications for Chern-Simons theory.

Contribution

It demonstrates the recovery of boundary states from noncommutative defects and links algebraic deformations to geometric boundaries in the commutative limit.

Findings

Boundary states are recovered from noncommutative defects.

Defects act as sources in Chern-Simons theory, deforming the $w_ abla$ algebra.

The undeformed $w_ abla$ algebra localizes near the puncture in the commutative limit.

Abstract

We illustrate how boundary states are recovered when going from a noncommutative manifold to a commutative one with a boundary. Our example is the noncommutative plane with a defect, whose commutative limit was found to be a punctured plane - so here the boundary is one point. Defects were introduced by removing states from the standard harmonic oscillator Hilbert space. For Chern-Simons theory, the defect acts as a source, which was found to be associated with a nonlinear deformation of the $w_\infty$ algebra. The undeformed $w_\infty$ algebra is recovered in the commutative limit, and here we show that its spatial support is in a tiny region near the puncture.