SU(3)XSU(2)XU(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 <math>[SU(3)\times SU(2)\times U(1)]/\mathbb{Z}_6<math> with a couple of fermion fields and a Higgs field, which is a <math>(1,2)_{\frac{1}{2}}<math> and/or a <math>(1,2)_{-\frac{1}{2}}<math>. 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, <math>(3,2)_{\frac{1}{6}}<math> (q for left-handed quark), <math>(\bar{3},1)_{\frac{1}{3}}<math> (dc for the left-handed anti d-quark), <math>(\bar{3},1)_{-\frac{2}{3}}<math>(uc for the left handed up antiquark), <math>(1,2)_{-\frac{1}{2}}<math> (l for the left handed leptons), <math>(1,1)_1<math>(ec for the left-handed positron) and <math>(1,1)_0<math>(νc 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 <math>[SU(2)\times U(1)]/\mathbb{Z}_2<math> or <math>SU(2)\times U(1)<math> to <math>U(1)_{em}<math>.
Of course, calling the representations things like <math>(3,2)_{\frac{1}{6}}<math> 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
- <math>\pi_2\left(\frac{[SU(2)\times U(1)]/\mathbb{Z}_2}{U(1)_{em}}\right)=0<math>
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.
