Edge Currents in Non-commutative Chern-simons Theory from a New Matrix Model

This paper discusses the formulation of the non-commutative Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard ∗products are not correct for such manifolds, the standard non-commutative CS theory is not also appropriate here. Instead we formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work which 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 approach 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 Moyal plane. The resultant matrix CS model has a fuzzy edge. It becomes the required sharp edge when N and η → ∞ in a suitable sense. The non-commutative CS theory on the strip is defined by this limiting procedure. After performing the canonical constraint analysis of the matrix theory, we find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kaç-Moody algebra. Our study shows that there are (η+1) abelian charges(observables) given by the matrix elements (Âi)N−1 N−1 and (Âi)nm (where n orm ≥ 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 N th level. We show that all non-abelian edge observables except these three can be constructed from the (η + 1) abelian charges above. Using some of the results of this analysis we discuss in detail the limit where this matrix model approximates the CS theory on the infinite strip. Finally, we include a short section containing our comments on the commutative limit of our model, where we also give a closed formula for the central charge of the Kaç-Moody-like algebra of the non-commutative CS theory in terms of the diagonal coherent state matrix elements of operators and star products. Regular Associate, The Abdus Salam ICTP, Trieste, Italy E-mails: [email protected], [email protected], [email protected],