2 00 1 Chern - Simons action in noncommutative space

We derive the noncommutative Chern-Simons action induced by Dirac fermions coupled to a background gauge field, for the fundamental , antifundamental, and the adjoint representation. We discuss properties of the noncommutative Chern-Simons action showing in particular that the Seiberg-Witten formula maps it into the standard commutative Chern-Simons action. Recent results in noncommutative geometry and string theory [1]-[3], revealed the interest on its own right for studying different field theories like Yang-Mills, λφ 4 , QED, Chern-Simons, Wess-Zumino theories and two-dimensional models, in noncommutative space [4]-[24]. In this respect, it is the purpose of this paper to analyse different aspects of the noncommutative Chern-Simons (CS) action. First, we discuss how the parity anomaly in a 2 + 1 massive fermion model induces a Chern-Simons term (as originally observed in [11] for the massless case). Then, we discuss relevant properties of the non-commutative CS action, its relation with the chiral Wess-Zumino-Witten model and its dependence on the noncommutative parameter θ µν. Let us start by establishing our conventions. The *-product for fields is defined by (ˆ f * ˆ g)(x) = e i 2 θµν ∂ ξµ ∂ ζνˆf(x + ξ)ˆ g(x + ζ) ξ=ζ=0 (1)