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What is the simplest explanation on why is E6 gauge group more favored as a group for Grand unified theory builders, while E8 is not? What about other exceptional groups? Which of them originate from the string theory in the most natural way, and why?

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Ha ha @Newman that is a funny title (even if it has no question mark) ;-) – Dilaton Jan 3 '12 at 23:36
Related: – Qmechanic Jan 3 '12 at 23:41
The title was kind of cute :-) but we're on a push to make question titles into actual questions, so I edited it. – David Z Jan 4 '12 at 0:24
OK, no problems. Actually it was not my intention to make it funny, it just turned that way spontaneously. – Newman Jan 4 '12 at 0:27
what was the title :p ? Thanks for asking this question anyways. I learned from it. – stupidity Jun 9 '12 at 20:28
up vote 9 down vote accepted

Saying that $E_6$ is "favored" over $E_8$ in GUT model building is a big understatement.

There can't be any grand unified theory with an $E_8$ gauge group because $E_8$ has no complex representations, i.e. representations that are inequivalent from their complex conjugates. The existence of complex representations is a necessary condition for the theory to contain chiral fermions, i.e. Dirac fermions whose left-handed components carry different quantum numbers (and interactions) than the right-handed components. One may also say that complex representations are needed for the violation of C, P, and CP.

$E_6$ is the only one among five exceptional groups that has any complex representations. It's related to the fundamental ${\bf 27}$ or antifundamental $\overline{{\bf 27}}$ representation of the group which are interchanged by an outer automorphism of $E_6$, a symmetry that boils down to the ${\mathbb Z}_2$ symmetry of its Dynkin diagram. $E_6$ is the only exceptional Lie group with a nontrivial symmetry of the Dynkin diagram.

All other exceptional Lie groups, namely $G_2, F_4, E_7, E_8$, only have real representations, a fact that can be seen by looking at their real fundamental representations, too. The spectrum of gauge theories using these groups would be inevitably left-right symmetric, and therefore experimentally excluded. Patterns about particles such as "neutrinos have to be left-handed" would be impossible.

Despite the comments above, $E_6$ is a subgroup of $E_8$. So in string theory, it is possible to break $E_8$ (a key group e.g. in heterotic string theory) to $E_6$ by stringy effects, e.g. by nontrivial configurations of the $E_8$ gauge field as a function of the extra (compact) dimensions in $E_8\times E_8$ heterotic string theory. Spontaneous breaking of $E_8$ by field-theoretical methods (Higgs mechanism) is no good because it would only produce real representations of $E_6$ again. In string theory, $E_6$, a viable GUT group, may emerge as a subgroup of $E_8$ (or $E_7$). $G_2$ and $F_4$ are too small to be relevant for GUT model building.

All papers that claim to build viable models of particle physics from an $E_8$ field theory are pseudoscientific gibberish, denying elementary features of the groups in which the known quantum fields transform. In the case of Garrett Lisi's paper, the absence of complex representations is the main point of the paper by Garibaldi and Distler.

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Lubos, thanks for clear answer. There are some models in which parity is restored at some high energy scale (left-right symmetry , so if this is really the case, would it mean that we no longer need groups having complex representations, or is it the case that even in left-right models fermions live in complex representations? – Newman Jan 4 '12 at 18:49
Dear @Newman - sorry for this 4-year delay. ;-) I just got a link to this page yesterday again. The left-right-symmetric models restore the left-right symmetry at higher energies so there are also couplings to right-handed leptons etc. But this left-right symmetry not only reverts the space but also exchanges two factors in the gauge group, like SU(2) left and SU(2) right. There must still exist a correlation between the fundamental/antifundamental type of a representation of a gauge group; and the handedness under the Lorentz group. And the gauge groups without cplx reps juust can't do it! – Luboš Motl Nov 29 '15 at 6:44
So even in models with the left-right symmetry, when they're embedded into a simple group, the group must be like SO(10) or E6 etc., something that has complex reps inequivalent to their complex conjugates. A subgroup of SO(10) may have things like SU(2) x SU(2) which only have pseudoreal and real reps. But the exchange of the two SU(2)'s replaces the complex conjugation (and is combined with the left-right flip of space). The exchange is only possible if there are at least 2 factors - if the gauge group isn't simple (grand unified). If there's nothing to exchange, you need cplx reps. – Luboš Motl Nov 29 '15 at 6:46

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