Noncommutative Yang-Mills Theory

We investigate asymptotic behaviors of the strong coupling limit in the N = 2 supersymmetric non-commutative Yang-Mills theory. The strong coupling behavior is quite different from the commutative one since the non-commutative dual U(1) theory is asymptotic free, although the monodoromy is the same as that of the ordinary theory. Singularities are produced by infinitely heavy monopoles and dyons. Nonperturbative corrections may be determined by holomorphy. 1 S-duality of Non-commutative U(1) gauge theory In this note we consider the non-perturbative aspect of non-commutative Yang-Mills theory making good use of supersymmetry and duality. We also use perturbative analyses done in Refs.[1, 2, 3]. Strong coupling region can be analyzed by using duality. Holomorphy puts severe constraints on the funcitonal form of the prepotential. We show a strong coupling behavior which is in contrast with the analysis in Ref.[4]. We would like to determine the low energy coupling constant and the θ parameter. First, we review the S-duality of non-commutative U(1) gauge theory.[5, 6] In Ref.[7] they give a field redefinition between fields in ordinary theory and in non-commutative theory. Let us suppose the lagrangian of the non-commutative U(1) gauge theory. We denote the field of the non-commutative theory by putting hat like Âμ. We perform a field redifinition from the gauge field Âμ to the ordinary one Aμ according to Ref.[7]. In order to perform S-duality we introduce (dual) auxiliary field Bμ for imposing the Bianchi identity dF ≡ 0. Then, eliminating the field strength F and performing a renormalization transformation fromBμ back to B̂μ, we end up with a non-commutative U(1) gauge theory. This is a dual description of the initial non-commutative theory. In this description we find the S-duality relation[5] gD = 1/g, θDij = − g 2 ǫijklθkl. (1.1) This argument holds in order by order of θ, and there is no full order treatment for the non-commutative S-duality. In this paper we assume, or believe, the relevance of this S-duality in quantum field theory. In Ref.[6] it is conjectured that strongly coupled spatially non-commutative N = 4 Yang-Mills theory is dual to to a weakly coupled non-commutative open string (NCOS) theory. One may think that this duality will hold for the N = 2 theory after some modification of the theory. In the NCOS theory the effective Regge slope parameter is given by the non-commutative parameter as α eff = θ/(2π). Then, the NCOS theory reduces to a non-commutative Yang-Mills theory when the Yang-Mills non-perturvative scale Λ is smaller than 1/α eff . In this situation the above S-dualities[5, 6] will be essentially the same.