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I have a few questions about triality for the representations of $SU(3)$.

(I have seen the wikipedia page, but it does not make the connection with physics.)

  • What is triality, how can you compute it from the dynkin labels?
  • What has triality to do with the fractional charges of quarks?
  • Why does the fact that $\overline{\underline{6}}$ has the same triality as $\underline{3}$ implies that we cannot have quarks in the irreps of the subgroup $SU(3)\times G_2$ of $E_6$?

These questions arises from the follows paragraph in the text "Group theory for unified model building" by Slanksy (on page 98):

The $\underline{27}$ of $E_6$ branches to $(\underline{3},\underline{7}) + (\underline{\overline{6}},\underline{1})$ of its maximal subgroup $SU(3)\times G_2$. If $SU(3)$ were color with $U^\text{em}(1)$ in $G_2$, then the $(\overline{\underline{6}}^c,1)$ states would be neutral. Since $\underline{6}^c$ has the same triality as $\underline{3}^c$, it is impossible to maintain the usual quark charges.

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As very well known, the root lattice $Q$ of a semisimple Lie group is a sub-lattice of its weight lattice $P$ because the roots are the weights of the adjoint representation. The congruence class of a weight (which is the generalization of the triality in $SU(3)$) is its representative in the coset $P/Q$. (Slansky calls it "congruency class"). In this construction the rules of conservation of triality are obvious. The center of the group is the character group $\mathcal{X}(P/Q)$, i.e., the group of one dimensional representations. Thus the triality can be viewed as a discrete charge of the center of the gauge group.

1) Slansky gives a few examples of formulas of the congruence class of $SU(N)$, $E_6$ etc. as a linear combination of the weight components on the top of page 37 of his review. The following thesis by:Lenka Ha’kova’ contains the formulas for the full Cartan classification on page $42$.

2) The relation of the triality and the electric charge follows from the empirical "triality rule" based on the fact that all known baryons belong to (the flavor) representations corresponding to Young tableaux whose number of boxes is a multiple of 3, thus must belong to a multiple of 3 tensor products of the quark representation, thus must be of triality zero. The electric charges of these Baryons are integer multiples of the electron's charge $e$, thus the charges of the constituent quarks must be multiples of $ \frac{1}{3} e$.

Magnetic monopoles provide an explanation of the triality rule and the fractionalization of the quark charges. Please see for example the explanation by Preskill :(based on the work of Goddard Nuyts and Olive) The explanation is based on:

1) The magnetic charge generator $M$ is a Linear combination of the generators of the broken gauge group (for example $SU(5)$).

2) The magnetic charge generator must satisfy a generalized Dirac quantization condition $e^{2 i \pi M} = I$

3) The center of the broken gauge group remains unbroken because it is a subgroup of unbroken U(1) subgroup.

These three conditions force a normalization of the generators such that a triality zero state must be zero for integer charges which is exactly what is observed experimentally.

It is worthwhile to mention that the set of all possible magnetic charges generate the magnetic dual of the gauge group which is the basis of the famous Electric-magnetic duality.

The answer to the third question is that if such a symmetry breaking existed then the triality 1 representation $\underline{\overline{6}}$ would be of electric charge 0 thus breaking the empirical triality rule.

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