The superspace representation of Super Yang-Mills theory on NCG

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 matter field. In this paper, we represent the triple in the superspace coordinate system $(x^\mu,\theta,\bar{\theta})$. We also introduce "extracting operators" and the 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 re-construct the super Yang--Mills theory on NCG in the superspace coordinate to which we are familiar to describe SUSY theories.