A Note on Superfields and Noncommutative Geometry

We consider the supersymmetric field theories on the noncommutative R using the superspace formalism on the commutative space. The terms depending on the parameter of the noncommutativity Θ are regarded as the interactions. In this way we construct the N = 1 supersymmetric action for the U(N) vector multiplets and chiral multiplets of the fundamental, anti-fundamental and adjoint representations of the gauge group. The action for vector multiplets of the products gauge group and its bi-fundamental matters is also obtained. We discuss the problem of the derivative terms of the auxiliary fields. ∗ E-mail: [email protected] In the past few years there has been much development in our understanding of the dynamics of supersymmetric gauge theories and superstring theories. Among these it has been discovered that the noncommutative gauge theories naturally appear [1, 2] when the D-branes with constant B fields is considered. Recently Seiberg and Witten have argued that the noncommutative gauge theories realized as effective theories on D-branes are equivalent to some ordinary gauge theories [3]. In a single D-brane case, they have shown that the effective action for the D-brane is consistent with the equivalence if all derivative terms are neglected. Furthermore, it has been shown that the D-brane action, including derivative terms, computed in the string theory is consistent with the equivalence if we keep the two derivative terms but neglect the fourth and higher order derivative terms [4, 5, 6]. For deeper understanding for these phenomena, it is natural to investigate the field theories on the noncommutative geometry by the field theoretical approaches. In particular by the perturbative analysis it was found in [7] that the IR effects and UV effects are mixed in the noncommutative field theory. To proceed further it may be important to study the noncommutative field theories with supersymmetry since their actions are highly constrained and we may understand the dynamics of these theories. To obtain the supersymmetric action, superfields on the noncommutative geometry may be desired. Explicit two-dimensional N = 1 noncommutative superspace was obtained in [8]. However, in this note, instead of investigating the noncommutative superspace formalism, we consider the ordinary superspace and superfields [9] and represent the noncommutative field theory using these notions. From the commutative supersymmetric action written by the superfields, we can obtain the noncommutative supersymmetric action by replacing the ordinary product between superfields to the ∗ product defined by the formula f(x) ∗ g(x) = e i 2 Θ ∂ ∂ξi ∂ ∂ζj f(x+ ξ)g(x+ ζ) ∣