Abstract
Recently, we found the supersymmetric counterpart of the spectral triple. When we restrict the representation space to the fermionic functions of matter fields, the counterpart which we name "the triple" reduces to the original spectral triple which defines noncommutative geometry. We see that the fluctuation to the supersymmetric Dirac operator induced by algebra in the triple generates vector supermultiplet which mediates gauge interaction. Following the supersymmetric version of spectral action principle, we calculate the heat kernel expansion of the square of fluctuated Dirac operator and obtain the correct supersymmetric Yang-Mills action with U(N) gauge symmetry.
Highlights
The standard model of high energy physics coupled to gravity was derived on the basis of noncommutative geometry(NCG) by Connes and his co-workers[1, 2, 3]
We introduced the supersymmetric counterpart of the spectral triple which defines NCG on the finite space as well as on the manifold
A vector supermultiplet was introduced as the internal fluctuation to the supersymmetrically extended Dirac operator DM defined on the manifold
Summary
The standard model of high energy physics coupled to gravity was derived on the basis of noncommutative geometry(NCG) by Connes and his co-workers[1, 2, 3]. Obeying the original idea that the Hilbert space of the spectral triple constituted of wave functions of matter fields while gauge fields and Higgs fields were derived from fluctuations of the Dirac operator, a component of the functional space HM in our counterpart was made up by spinor and scalar functions of C∞(M ) which constructed a chiral or an antichiral supermultiplet and represented wave functions of a matter field and its superpartners. In addition to the discussion of the triple defined on the manifold, we should intorduce the counterpart in the finite space denoted by (AF , HF , DF ) in order to incorporate gauge quantum numbers of matter particles and their superpartners and to obtain mass terms of them. We can define an adequate supersymmetric invariant product of elements in HM and obtain bilinear form similar to the first term in (1.2) It represents the action for the chiral and antichiral supermultiplets of matter fields and their superpartners. We will calculate the square of the fluctuated Dirac operator and Seelay-DeWitt coefficients of heat kernel expansion so that we will arrive at the correct action of the supersymmetric Yang-Mills theory
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