Dimensional Reduction of Dirac Operator

We elaborate an explicit example of dimensional reduction of the free massless Dirac operator with SU(3)-symmetry, defined on 12-dimensional manifold, which is the total space of principle SU(3)-bundle over 4-dimensional manifold. It turns out that after the dimensional reduction we obtain the “usual” Dirac operator, defined on 4-dimensional pseudo-Riemannian (non-flat) manifold but with mass term, acting on two spinor SU(3)-octets in the presence of gauge field with structure group SU(3) and source term depending on the stress tensor of the gauge field. The group of symmetry SU(3) is chosen because it is a classifying group for the Standard model and in the same time is complicated enough to demonstrate all the details in the general case. We pay attention and point out the crucial moments in the procedure of the dimensional reduction, where the new structures (gauge field with structure group SU(3), its stress tenzor, two spinor SU(3)-octets and mass term) arise.