Reconstruction of the standard model in a generalized differential geometry based on the real structure

Abstract.

The standard model is reconstructed in a generalized differential geometry (GDG) based on the idea of a real structure as proposed by Coquereaux et al. and Connes. The GDG considered in this article is a kind of non-commutative geometry (NCG) on the discrete space that successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory. Here, a GDG is a direct generalization of the differential geometry on an ordinary continuous manifold to the product space of this manifold with a discrete manifold. In a GDG, a one-form basis $\chi$ on the discrete space is incorporated in addition to the one-form basis ${\mathrm{d}}x^\mu$ on Minkowski space, rather than $\gamma^5$ as in Connes's original work. Although the Lagrangians obtained in this way are the same as those obtained in our previous formulation of GDG, the basic formalism becomes very simply and clear.