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Edge Currents in Non commutative Chern Simons Theory

This paper discusses the formulation of the non-commutativ e Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard ∗-products are not correct for suc h manifolds, the standard non-commutativ e CS theory is not also appropriate here. Instead w e formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work whic h has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approac h are operators supported on a subspace of finite dimension N + η of the Hilbert space of eigenstates of a simple harmonic oscillator with N , η ∈ Z + and N 6= 0. This oscillator is associated with the underlying Mo yal plane. The resultan t matrix CS model has a fuzzy edge. It becomes the required sharp edge when N and η → ∞ in a suitable sense. The non-commutativ e CS theory on the strip is defined b y this limiting procedure. After performing the canonical constrain t analysis of the matrix theory , w e find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Mo ody algebra. Our study shows that there are ( η + 1) 2 abelian charges(observables) given b y the matrix elements ( Aˆ i ) N − 1 N − 1 and ( Aˆ i )nm (where n or m ≥ N ) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the Nth level. We observ e that all non-abelian edge observables except these three can b e constructed from the ( η +1) 2 abelian charges ab o ve. Using some of the results of this analysis w e discuss in detail the limit where this matrix model approximates the CS theory on the infinite strip.