Supersymmetric Dirac Operator on the Noncommutative Geometry

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.