zbMATH — the first resource for mathematics

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
Full Text: