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 and math 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).

More precisely, the Spin(8) triality specifies the symmetry allowed interation terms for the following systems: a chiral $p_x+i p_y$ with a anti-chiral $p_x-i p_y$ superconductor. Combine a chiral $p_x+i p_y$ with a anti-chiral $p_x-i p_y$ superconductor, what we obtain is a Topological Superconductor respect to $Z_2$-Ising global symmetry as well as a $Z_2^f$-fermionic parity symmetry. So it is a 2+1D $Z_2 \times Z_2^f$-Topological Superconductor. It turns out that stacking from 1 to 8 layers of such $Z_2 \times Z_2^f$-Topological Superconductor ($p_x+i p_y/p_x-i p_y$), you can get 8 distinct classes (and at most 8, mod 8 classes) of TQFTs. They are labeled by $\nu \in \mathbb{Z}_8$ classes of 2+1D fermionic spin-TQFTs:

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This $\mathbb{Z}_8$ is related to the 3rd spin cobordism group $\Omega^{3,spin}_{tor}(B\mathbb{Z}_2,U(1))=\mathbb{Z}_8$.

There may be other examples, other applications in physics and in mathematics(?). 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.]

p.s. This is the modified ellaborated version of a unfortunately closed question from Phys.SE.

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    $\begingroup$ I find the term "$SO(8)$ triality misleading, because $SO(8)$ is only one of three quotients of $\mathrm{Spin}(8)$ that are permuted under triality. If you go down to $PSO(8)$, then triality acts again. $\endgroup$ – Sebastian Goette Oct 7 '17 at 12:35

Seiberg and Witten showed that the $\mathcal{N}=2$ supersymmetric SU(2) gauge theory with $N_f=4$ flavor is endowed with SO(8) flavor symmetry, and it enjoys SO(8) triality.

Later, Gaiotto's construction makes it more manifest through degenerations of 4-point punctured sphere. (See AGT Figure 1.)

Note Added. (Oct 8 2017)

I forgot to include one more important application of the triality to superstring theory. (I remember this when I was preparing string theory course today.)

It is known that there are two formulations of superstring theories:

  • The Ramond–Neveu–Schwarz (RNS) formalism that is supersymmetric on the world sheet viewpoint.
  • The Green-Schwarz (GS) formulation which has ten-dimensional space-time supersymmetry.

By quantizing each formulation in the lightcone gauge, one can show that the GS and RNS formulations are equivalent under SO(8) triality. (Green-Schwarz-Witten Chapter 5.2 or Becker-Becker-Schwarz Chapter 5.)

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    $\begingroup$ The Remann theta relation can be understood through the triality. There are already good answers here. $\endgroup$ – Satoshi Nawata Oct 7 '17 at 18:23

If you have doubts about string theory ("may not count as real-world physics"), then I imagine M-theory may be a step too far for you, but in the recent Twisted Cohomotopy implies M-Theory anomaly cancellation the triality of Spin(8) is used extensively.


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