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Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8).

SO(8) is unique among the simple Lie groups in that its Dynkin diagram (below) (D4 under the Dynkin classification) possesses a three-fold symmetry. This gives rise to a surprising feature of Spin(8) known as triality. Related to this is the fact that the two spinor representations, as well as the fundamental vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality automorphism of Spin(8) lives in the outer automorphism group of Spin(8) which is isomorphic to the symmetric group $S_3$ that permutes these three representations.

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What are physics applications of $SO(8)$ and Spin(8) triality?

  1. For example, one of physics applications of $SO(8)$ and Spin(8) triality is that, in the classifications of interacting fermionic topological phases protected by global symmetries, the 1+1D BDI Time-Reversal invariant Topological Superconductor and 2+1D $Z_2$-Ising-symmetric Topological Superconductor have $\mathbb{Z}_8$ classifications (see a related post here), that can be deduced from adding non-trivial four-fermion interaction terms respect the $SO(8)$ and Spin(8) triality, see for example the Appendix A of this web version (free access).

There may be other examples, other applications in physics(?). In particular, I have an impression that one can use $SO(8)$ and Spin(8) triality in string theory (but may not count as real-world physics) or possibly in disorder condensed matter system or nuclear energy spectrum. If that is true, could one explain how does the $SO(8)$ and Spin(8) triality come in there? In the previous example, I give, the concept of non-perturbative ('t Hooft) anomaly matching is implicit hidden there in the strongly-coupled higher order Majorana interactions. Do we see something similar or different concepts of the $SO(8)$ and Spin(8) triality applications?

[Citations/References are encouraged, but some explanations are required.]

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closed as too broad by ACuriousMind Sep 29 '17 at 12:07

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Please note that questions that ask for an open-ended list of examples are off-topic here. We want questions with a potentially uniquely identifiable correct answer, lists are a poor fit for the SE model. $\endgroup$ – ACuriousMind Sep 29 '17 at 12:08
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    $\begingroup$ @ACuriousMind, I am sorry for that. But I am sure that you know it is not an open-ended list so far in our knowledge. I can barely come up with another example. I would say possibly only string theory or some disorder system, or Kondo effect may have a similar use. As a reasonable moderator, one would know that this imposes the strict policy on this potential nice question. I can delete my question after all if this is what the moderator tries to do for this site. $\endgroup$ – wonderich Sep 30 '17 at 18:55
  • $\begingroup$ Many questions can be put on hold based on what you described, like this: physics.stackexchange.com/q/294299 or this physics.stackexchange.com/q/61241 $\endgroup$ – wonderich Sep 30 '17 at 18:58
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    $\begingroup$ I suggest to make your question more narrow, it is too worthy for a deletion. Try to find a compromise between what you want to know, and what you can here ask. Sometimes it requires to part your question into multiple ones, or a significant re-formulation. You may ask for help on the meta or on the chat, I think everybody will understand that this content is too worthy here to be removed and you will got a lot of help. (Ext: I asked it on the site chatroom) $\endgroup$ – peterh Oct 1 '17 at 0:28
  • $\begingroup$ @peterh, thanks, I add new precise statement to make it re-opened: "I have an impression that one can use $SO(8)$ and Spin(8) triality in particular in string theory (but may not count as real-world physics) or possibly in disorder condensed matter system or nuclear energy spectrum. If that is true, could one explain how does the $SO(8)$ and Spin(8) triality come in? [Citations/References are encouraged, but some explanations are required.]" $\endgroup$ – wonderich Oct 1 '17 at 1:43