Trinification

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


 * $$SU(3)_C\times SU(3)_L\times SU(3)_R$$

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.

Trinification is a maximal subalgebra of E6, whose matter representation $27$ 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 $$\mathbf{16}\oplus\mathbf{\overline{16}}$$.