The superspace representation of super Yang-Mills theory on noncommutative geometry

A few years ago, we found the supersymmetric (SUSY) counterpart of the spectral triple which specified noncommutative geometry (NCG). Based on "the triple," we considered the SUSY version of the spectral action principle and had derived the action of super Yang-Mills theory, minimal supersymmetric standard model, and supergravity. In these theories, we used vector notation in order to express a chiral or an anti-chiral matter superfield. We also represented the NCG algebra and the Dirac operator by matrices which operated on the space of the matter field. In this paper, we represent the triple in the superspace coordinate system (x(mu), theta, (theta) over bar). We also introduce " extracting operators" and a new definition of the supertrace so that we can also investigate the square of the Dirac operator on the Minkowskian manifold in the superspace. We finally reconstruct the superYang-Mills theory on NCG in the superspace coordinates in which we are familiar with describing SUSY theories.