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Topology of the symmetry group of the standard model. (English) Zbl 0966.81029
Summary: We study the topological structure of the symmetry group of the standard model, \(G_{SM}= \text{U}(1)\times \text{SU}(2)\times \text{SU}(3)\). Locally, \(G_{SM}\cong S^1\times (S^3)^2\times S^5\). For \(\text{SU}(3)\), 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 formulas give explicitly the matrices of \(\text{SU}(3)\) without using the Lie algebra (Gell-Mann matrices). Globally, we prove that the characteristic function of \(SU(3)\) is the suspension of the Hopf map \(S^3@>h>> S^2\). We also study the case of \(\text{SU}(n)\) for arbitrary \(n\), in particular the cases of \(\text{SU}(4)\), a flavor group, and of \(\text{SU}(5)\), a candidate group of grand unification. We show that the 2-sphere is also related to the fundamental symmetries of nature due to its relation to \(\text{SO}^0(3,1)\), the identity component of the Lorentz group, a subgroup of the symmetry group of several gauge theories of gravity.

MSC:
81R40 Symmetry breaking in quantum theory
81V22 Unified quantum theories
81T13 Yang-Mills and other gauge theories in quantum field theory
22E70 Applications of Lie groups to the sciences; explicit representations
81V15 Weak interaction in quantum theory
81V17 Gravitational interaction in quantum theory
83C45 Quantization of the gravitational field
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