It states that the
gauge group is either :SU(3)_C\times SU(3)_L\times SU(3)_R or :[SU(3)_C\times SU(3)_L\times SU(3)_R]/\mathbb{Z}_3; and that the fermions form three families, each consisting of the
representations: \mathbf Q=(3,\bar{3},1), \mathbf Q^c=(\bar{3},1,3), and \mathbf L=(1,3,\bar{3}). The
L includes a hypothetical
right-handed neutrino, which may account for observed
neutrino masses (see
neutrino oscillations), and a similar sterile "flavon." There is also a (1,3,\bar{3}) and maybe also a (1,\bar{3},3)
scalar field called the
Higgs field which acquires a
vacuum expectation value. This results in a
spontaneous symmetry breaking from :SU(3)_L\times SU(3)_R to [SU(2)\times U(1)]/\mathbb{Z}_2. The fermions branch (see
restricted representation) as :(3,\bar{3},1)\rightarrow(3,2)_{\frac{1}{6}}\oplus(3,1)_{-\frac{1}{3}}, :(\bar{3},1,3)\rightarrow 2\,(\bar{3},1)_{\frac{1}{3}}\oplus(\bar{3},1)_{-\frac{2}{3}}, :(1,3,\bar{3})\rightarrow 2\,(1,2)_{-\frac{1}{2}}\oplus(1,2)_{\frac{1}{2}}\oplus2\,(1,1)_0\oplus(1,1)_1, and the gauge bosons as :(8,1,1)\rightarrow(8,1)_0, :(1,8,1)\rightarrow(1,3)_0\oplus(1,2)_{\frac{1}{2}}\oplus(1,2)_{-\frac{1}{2}}\oplus(1,1)_0, :(1,1,8)\rightarrow 4\,(1,1)_0\oplus 2\,(1,1)_1\oplus 2\,(1,1)_{-1}. Note that there are two
Majorana neutrinos per
generation (which is consistent with
neutrino oscillations). Also, each generation has a pair of
triplets (3,1)_{-\frac{1}{3}} and (\bar{3},1)_{\frac{1}{3}}, and doublets (1,2)_{\frac{1}{2}} and (1,2)_{-\frac{1}{2}}, which decouple at the GUT breaking scale due to the couplings :(1,3,\bar{3})_H(3,\bar{3},1)(\bar{3},1,3) and :(1,3,\bar{3})_H(1,3,\bar{3})(1,3,\bar{3}). Note that calling
representations things like (3,\bar{3},1) and (8,1,1) is purely a physicist's convention, not a mathematician's, where representations are either labelled by
Young tableaux or
Dynkin diagrams with numbers on their vertices, but it is standard among GUT theorists. Since the
homotopy group :\pi_2\left(\frac{SU(3)\times SU(3)}{[SU(2)\times U(1)]/\mathbb{Z}_2}\right)=\mathbb{Z}, this model predicts
't Hooft–Polyakov magnetic monopoles. The trinification symmetry
Lie algebra \mathfrak{su}(3)_C \oplus \mathfrak{su}(3)_L \oplus \mathfrak{su}(3)_R is a maximal subalgebra of
E6, whose matter representation has exactly the same representation and unifies the (3,3,1)\oplus(\bar{3},\bar{3},1)\oplus(1,\bar{3},3) fields. E6 adds 54
gauge bosons, 30 it shares with
SO(10), the other 24 to complete its \mathbf{16}\oplus\mathbf{\overline{16}}. ==References==