Edge States from Defects on the Noncommutative Plane

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.