On the Topology of the Symmetry Group of the Standard Model

Abstract

We study the topological structure of the symmetry group of the standard model, $G_{SM}=U(1)\times SU(2)\times SU(3)$. Locally, $G_{SM}\cong S^1\times (S^3)^2\times S^5$. For SU(3), which is an $S^3$ bundle over $S^5$ (and therefore a local product of these spheres) we give a canonical gauge i.e. a canonical set of local trivializations. These formulae give the matrices of SU(3) in terms of points of spheres. Globally, we prove that the characteristic function of SU(3) is the suspension of the Hopf map $h: S^3 \to S^2$. We also study the case of SU(n) for arbitrary $n$, in particular the cases of SU(4), a flavour group, and of SU(5), a candidate group for grand unification. We show that the 2-sphere is also related to the fundamental symmetries of nature due to its relation to $SO^0(3,1)$, the identity component of the Lorentz group, a subgroup of the symmetry group of several gauge theories of gravity.