  We extend naturally the spectral triple which define noncommutative geometry
(NCG) in order to incorporate supersymmetry and obtain supersymmetric Dirac
operator D_M which acts on Minkowskian manifold. Inversely, we can consider the
projection which reducts D_M to $\not{D}$, the Dirac operator of the original
spectral triple. We investigate properties of the Dirac operator, some of which
are inherited from the original Dirac operator. Z/2 grading and real structure
are also supersymmetrically extended. Using supersymmetric invariant product,
the kinematic terms of chiral and antichiral supermultiplets which represent
the wave functions of matter particles and their superpartners are provided by
D_M. Considering the fluctuation given by elements of the algebra to the
extended Dirac operator, we can expect to obtain vector supermultiplet which
includes gauge field and to obtain super Yang-Mills theory according to the
supersymmetric version of spectral action principle.

High Energy Physics - Theory