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Explaining SU(5) GUTs (using the first particle family as an example) in the last SUSY lecture 10, Lenny Susskind mentioned that there are at present no ideas how to combine simultaneously all 3 particle families into a large GUT theory. I somehow dont believe him, suspecting that he just didnt want to talk about this :-P...

So, are there any ideas around how to incorporate all 3 families into a larger structer?

If so, I would appreciate explanations about how it works at a "Demystified" level :-)

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  • $\begingroup$ Did he exactly state "at present"? This is true. Attempts and ideas happened in the past and they are no "around" anymore, just buried in the reference books. $\endgroup$
    – arivero
    Commented Aug 17, 2011 at 0:42
  • $\begingroup$ Somewhat related: physics.stackexchange.com/q/2051/2451 $\endgroup$
    – Qmechanic ♦
    Commented Jan 3, 2012 at 23:44
  • $\begingroup$ @Qmechanics Thanks for the link to this question, I`ll look at it. $\endgroup$
    – Dilaton
    Commented Jan 4, 2012 at 8:14
  • $\begingroup$ Potentially nice explanation why there are 3 particle families involving D-brane scenarios $\endgroup$
    – Dilaton
    Commented Jul 26, 2013 at 23:18
  • $\begingroup$ An orbifolded 10d E8 GUT like this (e.g. in 10d, where the 6 extra dimensions are compactified on an orbifold to dodge the usual problem of no complex E8 representations) naturally has 3 generations. $\endgroup$
    – alexchandel
    Commented Aug 16, 2021 at 19:13

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Jacob Bourjaily derives three Standard Model generations from an E8 singularity here and here. (See comments by Lubos, 1 2 3.)

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  • $\begingroup$ Thanks for these interesting links, I will be busy a while to fully evaluate them :-) $\endgroup$
    – Dilaton
    Commented Aug 17, 2011 at 8:33
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The problem of families in GUT is sometimes referred as an "Horizontal symmetry". There are two lines of work, roughly: those which get a continous symmetry, say SU(3), and then all the gauge malabars, and those which add a discrete symmetry, such as A4. Of course in both cases, a serious GUT should show everything embodied in a larger simple group. E8 has some value because it can go down to E6xSU(3), and E6 can lodge chiral fermions (but then perhaps this SU(3) does not work as it should, in more detailed examination) Other alternatives are just growing up SO(2n) until everything fits... You always have V+A currents you dont want, plus a bag of any of the usual problems in phenomenology.

Zee is the adecuate source to check if you want to look deeper in this topic.

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  • $\begingroup$ Thanks arivero, do You mean Zee`s QFT nutshell or something else? $\endgroup$
    – Dilaton
    Commented Aug 17, 2011 at 8:38
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    $\begingroup$ U r welcome. I mean both, Zee's QFT nutshell, where it has some chapter going beyond SO(10), and Zee's something else, aka "Unity of forces in the universe. , Volume 1". $\endgroup$
    – arivero
    Commented Aug 17, 2011 at 20:27
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SO(8)'s triality can be used to generate three families. It's promising from a GUT perspective.

E6 and SO(18) are the best potential GUT groups containing SO(8) for doing so; of these E6 appears more "seamless". See the following:

  1. Z K Silagadze, SO(8) colour as possible origin of generations, arXiv:hep-ph/9411381v2 (2009).

  2. Y BenTov, A Zee, The origin of families and SO(18) grand unification, arXiv:1505.04312v2 [hep-th] (2016).

  3. H Rubenthaler, The (A_{2},G_{2}) duality in E_{6}, octonions and the triality principle, Transactions of the American Mathematical Society 360, No. 1, 347-367 (2008).

  4. M Ito et al, E_{6} grand unified theory with three generations from heterotic string, arXiv:1012.1690v2 [hep-ph] (2011).

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There has been quite a lot of coverage about Garrett Lisi's Exceptionally Simple Theory of Everything. You can find the preprint here, as well as a SciAm special here.

Lisi's theory involves using $E_8$, which is the largest exceptional Lie algebra in Killing's classification. I'm not aware of the theory appearing in a peer-reviewed journal. I think Lisi claims to also incorporate gravity but, because the group is so large, there are also a lot of new particles that would need explaining away. The Wikipedia article is reasonably fair about the coverage of the original preprint and makes some attempts to explain it with pretty visualizations of $E_8$. The theory is highly controversial and doesn't have widespread acceptance. but it is an "idea around how to incorporate all 3 families into a larger structure"...

There are probably more conventional ideas that use SU(5) or SO(10), but they clearly don't have as good PR people because I'm not aware of any leading theory.

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    $\begingroup$ I don't want to downvote what appears to be an honest answer but proposing crackpotty theories such as Lisi's that contain no real physics is mocking standard GUT theories (which are perhaps not correct but at least physically and mathematically sound). $\endgroup$
    – Marek
    Commented Aug 16, 2011 at 13:25
  • $\begingroup$ To be honest, I thought twice about putting this up as an answer. To the extent that it is an idea that is discussed by scientists (not all of the mainstream was dismissive...), I think it is an answer to the question. But this answer probably doesn't give credit to other SU(5) or SO(10) theories. Like I say, I'm just unaware of them because they haven't been as widely publicized... I'll happily remove my answer if/when its usurped by a more balanced one (or if mods feel its inappropriate to promote stuff like Lisi's). $\endgroup$
    – Warrick
    Commented Aug 16, 2011 at 13:56
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    $\begingroup$ fair enough. Just to be sure though, standard GUT theories are conservative generalizations of the usual SM gauge theories (e.g. SU(3)). In those theories, force particles come from adjoint representations (thus we have eight gluons for SU(3) since dimension of adjoint rep equals dimension of the group) while matter particles have to come from other reps (e.g. from 5 and 10 of SU(5)). But Lisi just mixes all this stuff up in a naive way that makes no sense, adding fermions and bosons, adjoint reps with any other rep, etc... $\endgroup$
    – Marek
    Commented Aug 16, 2011 at 14:07
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    $\begingroup$ @Warrick: Thanks for Your answer, I`m generally open-minded to different approaches ;-). But from Marek s nice explanation, I understand why Lisi s theory can not work; so I cant give an upvote but will leave it as it is ... $\endgroup$
    – Dilaton
    Commented Aug 16, 2011 at 18:02
  • $\begingroup$ Another confusing point of this answer is that Lisi's goal goes beyond GUT. The OP is just asking for theories with gauge group and families. $\endgroup$
    – arivero
    Commented Aug 17, 2011 at 0:26

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