In physics, the trinification model is a Grand Unified Theory proposed by Alvaro De Rújula, Howard Georgi and Sheldon Glashow in 1984.
Details
It states that the gauge group is either
:<math>SU(3)_C\times SU(3)_L\times SU(3)_R</math>
or
:<math>[SU(3)_C\times SU(3)_L\times SU(3)_R]/\mathbb{Z}_3</math>;
and that the fermions form three families, each consisting of the representations: <math>\mathbf Q=(3,\bar{3},1)</math>, <math>\mathbf Q^c=(\bar{3},1,3)</math>, and <math>\mathbf L=(1,3,\bar{3})</math>. 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 <math>(1,3,\bar{3})</math> and maybe also a <math>(1,\bar{3},3)</math> scalar field called the Higgs field which acquires a vacuum expectation value. This results in a spontaneous symmetry breaking from
:<math>SU(3)_L\times SU(3)_R</math> to <math>[SU(2)\times U(1)]/\mathbb{Z}_2</math>.
The fermions branch (see restricted representation) as
:<math>(3,\bar{3},1)\rightarrow(3,2)_{\frac{1}{6\oplus(3,1)_{-\frac{1}{3</math>,
:<math>(\bar{3},1,3)\rightarrow 2\,(\bar{3},1)_{\frac{1}{3\oplus(\bar{3},1)_{-\frac{2}{3</math>,
:<math>(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</math>,
and the gauge bosons as
:<math>(8,1,1)\rightarrow(8,1)_0</math>,
:<math>(1,8,1)\rightarrow(1,3)_0\oplus(1,2)_{\frac{1}{2\oplus(1,2)_{-\frac{1}{2\oplus(1,1)_0</math>,
:<math>(1,1,8)\rightarrow 4\,(1,1)_0\oplus 2\,(1,1)_1\oplus 2\,(1,1)_{-1}</math>.
Note that there are two Majorana neutrinos per generation (which is consistent with neutrino oscillations). Also, each generation has a pair of triplets <math>(3,1)_{-\frac{1}{3</math> and <math>(\bar{3},1)_{\frac{1}{3</math>, and doublets <math>(1,2)_{\frac{1}{2</math> and <math>(1,2)_{-\frac{1}{2</math>, which decouple at the GUT breaking scale due to the couplings
:<math>(1,3,\bar{3})_H(3,\bar{3},1)(\bar{3},1,3)</math>
and
:<math>(1,3,\bar{3})_H(1,3,\bar{3})(1,3,\bar{3})</math>.
Note that calling representations things like <math>(3,\bar{3},1)</math> 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
:<math>\pi_2\left(\frac{SU(3)\times SU(3)}{[SU(2)\times U(1)]/\mathbb{Z}_2}\right)=\mathbb{Z}</math>,
this model predicts 't Hooft–Polyakov magnetic monopoles.
The trinification symmetry Lie algebra
<math>\mathfrak{su}(3)_C \oplus \mathfrak{su}(3)_L \oplus \mathfrak{su}(3)_R</math> is a maximal subalgebra of E<sub>6</sub>, whose matter representation has exactly the same representation and unifies the <math>(3,3,1)\oplus(\bar{3},\bar{3},1)\oplus(1,\bar{3},3)</math> fields. E<sub>6</sub> adds 54 gauge bosons, 30 it shares with SO(10), the other 24 to complete its <math>\mathbf{16}\oplus\mathbf{\overline{16</math>.