su(3) su(2) u(1)

This Lie group is the formulation of the Standard Model as a gauge theory with the gauge group SU(3) × SU(2) × U(1) or

/\mathbb{Z}_6

with a couple of fermion fields and a Higgs field, which is a

(1,2)_{\frac{1}{2}}

and/or a

(1,2)_{-\frac{1}{2}}

. SU(3) describes quantum chromodynamics, SU(2) describes the weak interaction* and U(1) describes hypercharge. *Technically speaking, the Z and W bosons are described by a field which is really a linear combination of SU(2) and U(1). See electroweak. There are three families of fermions, each consisting of the representations,

(3,2)_{\frac{1}{6}}

(q for left-handed quark),

(\bar{3},1)_{\frac{1}{3}}

(d^c for the left-handed anti d-quark),

(\bar{3},1)_{-\frac{2}{3}}

(u^c for the left handed up antiquark),

(1,2)_{-\frac{1}{2}}

(l for the left handed leptons),

(1,1)_1

(e^c for the left-handed positron) and

(1,1)_0

(ν^c for the left-handed antineutrino, which is now known to exist. See Neutrino oscillation.). The Higgs field acquires a VEV, resulting in a spontaneous symmetry breaking from

/\mathbb{Z}_2

SU(2)\times U(1)

U(1)_{em}

. Of course, calling the representations things like

(3,2)_{\frac{1}{6}}

is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, it is standard among high energy physicists. Since the homotopy group

/\mathbb{Z}_2}{U(1)_{em}}\right)=0

this model predicts no monopoles associated with the electroweak breaking scale. See Hooft-Polyakov monopole. The Yukawa couplings of the scalar Higgs fields with the fermions produces the fermion masses after the Higgs field acquires a VEV. See also grand unified theory.

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