A superspace formulation of SUSY in NCG with spectral action

The aim of this paper is to present a possible framework for incorporating a superspace formulation of supersymmetry into the formalism of noncommutative geometry `a la Alain Connes. In analogy with the almost-commutative (AC) manifold construction of field theory, a base space is taken to be the superspace ${\mathbb{R}}^{3\mathrm{\mid }2}$, with associated 2-point (Grassmann valued) finite space. The data of the spectral triple is explored, including decorations, i.e. algebra, Hilbert space, grading, real structure, and Dirac operator. The gauge fields arising from the inner fluctuations of the Dirac operator are computed. And both the SUSY invariant spectral action and fermionic actions are calculated.