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(infinity) algebra. The undeformed w(infinity) algebra is recovered in the commutative limit, and here we show that its spatial support is in a tiny region near the puncture.