The minimum supersymmetric standard model on noncommutative geometry

We have obtained the supersymmetric extension of a spectral triple that specifies a noncommutative geometry. We assume that the functional space H consists of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model (MSSM). We introduce the internal fluctuations of the Dirac operator on the finite space as well as on the manifold by elements of the algebra A in the triple. So, we obtain not only the vector supermultiplets that mediate SU(3) circle times SU(2) circle times U(1)(Y) gauge degrees of freedom but also Higgs supermultiplets that appear in the MSSM from the same standpoint. According to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley-DeWitt coefficients give the correct action of the vector and Higgs supermultiplets. We also verify that the relation between the coupling constants of SU(3), SU(2), and U(1)(Y) is same as that of SU(5) unification theory.