Minimum Supersymmetric Standard Model on the Noncommutative Geometry

Abstract

We have obtained the supersymmetric extension of spectral triple which specify a noncommutative geometry(NCG). We assume that the functional space H constitutes of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model(MSSM). We introduce the internal fluctuations to the Dirac operator on the manifold as well as on the finite space by elements of the algebra A in the triple. So, we obtain not only the vector supermultiplets which meditate SU(3)xSU(2)xU(1)_Y gauge degrees of freedom but also Higgs supermultiplets which appear in MSSM on the same standpoint. Accoding 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 coeffients give the correct action of MSSM. We also verify that the relation between coupling constants of $SU(3)$,$SU(2)$ and $U(1)_Y$ is same as that of SU(5) unification theory.