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 µ , θ,θ). 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.Here the first term stands for the matter action and ψ is a fermionic field which belongs to H 0 . The second term represents the bosonic part which depends only on the spectrum of the squared Dirac operator P =D 2 0 and f (x) is an auxiliary smooth function on a fourdimensional compact Riemannian manifold without boundary [8].The SM has some defects, in particular, has the hierarchy problem. It is known that the problem is perfectly remedied by introducing supersymmetry [9]. In order to incorporate the supersymmetry to particle models on concepts of NCG, we have obtained "the triple" (H, A, D) extended from the spectral triple on the flat Riemannian manifold and verified its supersymmetry [10,11]. Here, H is the functional space which consists of chiral and antichiral supermultiplets that correspond to spinorial and scalar wave functions of C ∞ (M ). The triple however does not satisfy the axioms of NCG. For an example, the commutator [D, a] is not bounded for the extended Dirac operator D and an arbitrary element a ∈ A, because D includes d'Alembertian which appears in the Klein-Gordon equation. So, the triple does not produce a new NCG. When we limit the domain H to the space of the spinorial wave functions H 0 , the triple reduces to the spectral triple and the whole theory also reduces to the original one.We also found the internal fluctuation of the Dirac operator which produced vector supermultiplets with gauge degrees of freedom, supersymmetric invariant product of elements in H and the supersymmetric version of the spectral action principle. Using these components, we obtained the kinetic and mass terms of the matter particle interacted with gauge fields. We also investigated the square of the fluctuated Dirac operator and Seeley-DeWitt coefficients of heat kernel expansion, so that we arrived at the action of supersymmetric Yang-Mills theory and that of the minimal supersymmetric standard model [11,12].In the above construction of supersymmetric theories on NCG, a chiral or an antichiral superfield, an element of the functional space H M in the ...
