The Electro-weak and Color Bundles for the Standard Model in a Gravitation Field

It is known that the Standard Model describing all of the currently known elementary particles is based on the U(1)×SU(2)×SU(3) symmetry. In order to implement this symmetry on the ground of a non-flat space-time manifold one should introduce three special bundles. Some aspects of the mathematical theory of these bundles are studied in this paper. 1. The Dirac bundle and its basic fields. Spin is a common property of all elementary particles. It is described in terms of the Dirac bundle DM , where M is a space-time manifold. The Dirac bundle is a four-dimensional complex bundle over the four-dimensional real manifold M . The detailed description of this bundle can be found in [1], [2], and [3]. In this section we shall remind in brief some of its properties. The base manifold M of the Dirac bundle DM is equipped with the following three geometric structures which are well-known in General Relativity: (1) a pseudo-Euclidean Minkowski-type metric g; (2) an orientation; (3) a polarization (see details in [4]). The Dirac bundle DM is linked to the above three structures through frame pairs. The most popular frame pairs are listed in the following table: Canonically orthonormal chiral frames → Positively polarized right orthonormal frames P -reverse anti-chiral frames → Positively polarized left orthonormal frames T -reverse anti-chiral frames → Negatively polarized right orthonormal frames PT -reverse chiral frames → Negatively polarized left orthonormal frames (1.1) (see details in [1] and [2]). Each of the frames in the right hand side of the table (1.1) is an ordered set of four vector fields Υ0, Υ1, Υ2, Υ3 linearly independent 2000 Mathematics Subject Classification. 81T20, 81V05, 81V10, 81V15, 53A45. Typeset by AMS-TEX