# Triality and its uses.

1. Jul 4, 2007

### arivero

After the resuscitation of triality in Baez's commentary of Garrett Lisi work, it seems worthwhile to open a separate thread to discuss the expectatives of using triality to build the models Beyond Standard Model.

To start, I have found this one:

http://arxiv.org/abs/hep-th/0104050 Lee Smolin The exceptional Jordan algebra and the matrix string. It was a bold speculative paper from Lee, naming Baez in the last page and quoting Motl in the references, so perhaps it was the missed opportunity for the internet physicists to become an independent workforce. Tony invokes this paper a couple times in a closed thread at Woit's

In sps, Lubos did sometime a brief comment about triality, it is archived in PF but not in google: https://www.physicsforums.com/showthread.php?p=220265&highlight=triality#post220265 Also, CarB did some remarks about another paper from Lubos as presenting "triple Pauli statistics", https://www.physicsforums.com/showthread.php?p=1299190 and got some attention of our usual suspects :-) even if SU(3) is not so 3'd as SO(8).

Last edited by a moderator: Apr 22, 2017
2. Jul 5, 2007

### arivero

Another example, SU(3) $$\supset$$ SU(2) x U(1), is recalled by T. Pengpan, P. Ramond (funny, I believed to remember it was Polchinski) in http://arxiv.org/abs/hep-th/9808190

Basically, the embedding of SU(2)xU(1) in SU(3) is to put a 1x1 and a 2x2 boxes inside a 3x3 box. In matrix form, there are three ways to do it: the two ones you can see pictorialy, and the one using the corner of the matrix:

$$\left( \begin{array}{ccc} y & & \\ & x & x \\ & x & x \end{array} \right) , \left( \begin{array}{ccc} x & x & \\ x & x & \\ & & y \end{array} \right) , \left( \begin{array}{ccc} x & & x \\ & y & \\ x & & x \end{array} \right)$$

3. Jul 6, 2007

### arivero

In s.p.r, Silagadze invited to check their old papers at hep-ph/9411381 and, extended, http://ccdb4fs.kek.jp/cgi-bin/img_index?9405634 [Broken] It includes all the main ingredients: GUT multiplets, octonions, etc

I keep feeling that if the scheme works by using clifford algebras to build the representations, something of SUSY (and perhaps R-parities) should be relevant.

Last edited by a moderator: May 3, 2017
4. Jul 6, 2007

### arivero

Baez flirts with triality

Or, list of Baez's flirts with triality

http://www.math.ucr.edu/home/baez/week59.html (which quotes week 61!)
http://www.math.ucr.edu/home/baez/twf_ascii/week61 back in 1995

In week 61, a reference to 1) Alex J. Feingold, Igor B. Frenkel, and John F. X. Rees, Spinor construction of vertex operator algebras, triality, and E_8^{(1)}, Contemp. Math. 121, AMS, Providence Rhode Island, ISBN 0-8218-5128-4. Pity this work is not online, at least not for free.

http://www.math.ucr.edu/home/baez/week90.html Sept 1996
http://www.math.ucr.edu/home/baez/week91.html 1996
During the next weeks (93, 95,...) Baez kept visiting other topics on Beyond Standard Model physics.

http://www.math.ucr.edu/home/baez/twf_ascii/week104 in 1997 gets personal and tells about Dixon and Tony Smith. The discussion follows to Bott periodicity in week 105 and extends a bit further in
http://www.math.ucr.edu/home/baez/week106.html

http://www.math.ucr.edu/home/baez/week162.html (december 2000) and
http://www.math.ucr.edu/home/baez/week163.html revisit the octonions from the point of view of projective plane.

http://www.math.ucr.edu/home/baez/week180.html in 2002 meets again the representation theory of SO(8) but triality is here "a distraction" from the main topic of the week.

http://www.math.ucr.edu/home/baez/week191.html , in Jan 2003, reviews some notes from Ramond and from my old advisor, Boya. I'd add that LJ has some more work in the arxiv about the topic (e.g. math-ph/0409077) and he keeps interested on the topic.
http://www.math.ucr.edu/home/baez/week194.html visits the lattices related to octonions etc (see week 95.

Finally, I do not find any other mention of the word triality until this week,
http://www.math.ucr.edu/home/baez/week253.html which has a separate thread

Let me note too
http://www.math.ucr.edu/home/baez/octonions/node7.html
where the SO(8) diagram with vertex in vector, right and left representations is put in parallel with a Feynman diagram for a vector fermion fermion vertex.

Last edited by a moderator: Apr 22, 2017
5. Jul 6, 2007

### arivero

6. Jul 10, 2007

### arivero

Zee

Zee "QFT in a nutshell", chapter VII.7 includes also "A speculation on the origin of familes" based also in triality. He uses the breaking of SO(8) into SO(5).

Incidentally, hep-th/0511086v2 redirects to this chapter too.

7. Jul 12, 2007

### arivero

It seems there is a rule "quaternionist" + "composites"+ "generations" = "octonian" but Adler is a counterexample. He concentrates a lot in quaternion theory but he does not hurry to generalize to octonions, nor to disgress on triality. Hmm.

8. Jul 13, 2007

### arivero

Georgi's neutrality

H Georgi book, Lie Algebras in Particle Physics, was published in 1982 and it mentions all the issues but in very neutral way. The word "triality" is not invoked, but a full subchapter is titled "Fun with SO(8)".

There is also a last chapter that the first edition titles equal to the whole book (thus untitled) and the second one, even more neutral, refers as "odds and ends", or something so. It containts a subchapter Exceptional Algebras, which speaks of "octonians", and a short subchapter on Anomalies, that mentions the special case of SO(6) as being the only non anomaly free of its series (because it is equivalent to SU(4) and all the SU(n >=3 ) are not anomaly-free.).

9. Jul 14, 2007

### CarlB

The Jordan algebra paper by Lee Smolin reminds me of some of Michael Rios' more recent papers, some of which cite Lee. For example:

http://arxiv.org/abs/hep-th/0703238
http://arxiv.org/abs/hep-th/0503017

The above is over my head but it is supposedly connected in some way to the things I play with.

By the way, I've just learned that wordpress (i.e. see http://carlbrannen.wordpress.com/ ) allows LaTex in its blogs and comments. They also allow private blogs where one can have more than one poster (by invitation only). The whole thing seems perfect for doing research. The only thing I could ask for beyond what they have is a method of making posts invisible on a post by post basis. Hmmm.

10. Aug 12, 2007

### arivero

11. Aug 14, 2007

### kneemo

Yes, these three embeddings actually have a geometrical interpretation. To see this most clearly, it's convenient to use the projection operator construction of $$\mathbb{CP}^2$$, where points are identified with operators (see Conway pg. 143). Since we're working with 3x3's, such projection operators via suitable automorphisms, can be made to look like these three:
$$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) , \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) , \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)$$

It's a quick check to see these matrices satisfy P^2=P and tr(P)=1, hence are primitive idempotents. These idempotents form an (orthonormal) projective basis for $$\mathbb{CP}^2$$, and each SU(2)xU(1) embedding inside SU(3) will leave one of the $$\mathbb{CP}^2$$ basis idempotents invariant, while transforming the other two. By combining two or more different SU(2)xU(1) embeddings, i.e. UVPV*U*, we recover a general SU(3) transformation, much like we do for Euler angles and SO(3). So triality, in the sense of embeddings of maximal subgroups, can be geometrically interpreted as the three ways we can transform the basis of a projective plane.

This construction also works for the higher projective planes $$\mathbb{HP}^2$$ and $$\mathbb{OP}^2$$, where the corresponding subgroups are Sp(4)xSp(2)~SO(5)xSO(3) and SO(9) for the groups Sp(6) and F4, respectively. The SO(9) case is relevant to Smolin's model, and more recently, to BPS black holes (hep-th/0512296).

More practically, the primitive idempotent construction is related to Carl's Lepton Masses work, where each lepton generation corresponds to a different projective basis element for $$\mathbb{CP}^2$$ and the masses to their corresponding eigenvalues. Therefore, using Carl's circulant Hermitian matrix construction, it can be shown that triality and SU(2)xU(1) embeddings in SU(3) map the three lepton generations to the projective basis for $$\mathbb{CP}^2$$.

Last edited: Aug 15, 2007
12. Aug 19, 2007

### Dcase

Hi kneemo, arivero, CarlB,

The Conway reference, On Quaternions and Octonions, is now on my reading list.

Lieven le bruyn, MoonshineMath [NeverEndingBooks-v2], discusses work by Conway.

"Topics to be covered here include : finite simple groups, curves, modular forms, modular subgroups and braid groups, games, representation theory, noncommutative geometry"

http://www.neverendingbooks.org/

Richard Borcherds was a Conway student.
Ed Witten is now doing work with the 'Moonshine Monster'.

13. Aug 22, 2007

### arivero

On the same topic, plus trialities, the paper of JM Evans that I mentioned in the 3 4 6 10 thread is available in the internet, from the KEK server, and it scores as a "must read" too.

14. Aug 24, 2007

### arivero

Last edited by a moderator: May 3, 2017
15. Sep 20, 2011

### arivero

Code (Text):
o
/
/
/
o-------o             this is SO(8)
\
\
\
o
Code (Text):
o

o-------o             this is SU(3)xSU(2)xSU(2)

o
There is some triality in the ways to embed a SU(2) inside SU(3), but also in the ways to embed SU(3) in SO(8). Note that you can not embed the full [LR] standard model group into SO(8), same that you can not embed SU(3) O----O into SO(5) O====O.