How Does RCHO Unification Relate to Normed Division Algebras?

[MTd2]

What composites?

 

[arivero]

What composites?

I told aboit it above in #50, and it is my thread in Independent Research. Not relevant hera except as motivation perhaps, it is a trick of my own: I kept thinking about the QCD string, and one day I noticed that the famous prerequisite for supersymmetry, to have the same number of scalar states and fermionic states, actually works for the particles bound with the QCD string if and only if you have three generations and a fast decaying top quark (so that it does not bind in the extreme of the string).

It is even madder than the maddest idea from supertring theory, but later I found that Schwarz himself was considering it in 1970 for six or seven months, and even published it in the Physical Review -so at least a referee considered it as a valid argument-. But he only considered susy between quarks and gluons, ie diquarks at most, and in the early seventies there was only one generation and half, everything light and well bound, so there was no grounds to follow this path.

Still, I kept to it always that I find susy: I think of muons and diquarks as if they were the susy partners, with the qcd string, of fundamental fermions. I prefer this way to the orthodox think, and it helps, because the link susy-triality-division algebras is very strong, so to close the eyes is not an option.

 

[sneutron]

Sorry, Arivero, if you'd got bored with this thread...

You started with a Furey paper that cited FDT Smith's 1993 paper, then posted a link to Smith's home page. This contains a lot of stuff - most of no interest to me, whatsoever, but also much post-1993 including quite a lot of calculations that seem to be credibly close to the mark.

I'd be very grateful if you could spare the time to give an opinion on Smith's approach, as he doesn't seem to have been explicitly considered in this thread (though I'm too layman to have recognised it, perhaps).

Paul

 

[arivero]

Well, Paul, yes it is true that we have not touched Smith's in this thread, it is because I see it more as a pioneer - and an excellent bibliographer - than an active modelist. He did a lot of modelling in the nineties, went to congresses, showed his job, etc. With no results because, mainly and regrettably, the success of a model depends not on the physical content but on how useful it is for other researchers. And partly, because he was coming a bit late to the GUT pursuit, and people was surely tired of hundreds of models here and there. In the last fifteen years, of course, a lot of problems come with the bad interaction with establishment, arXiv and even web hosting... I think that in this time he has kept fine tuning his model -not sure if a good thing- and doing an excellent bibliographer task, but if he has got advances or he is stagnated somewhere, I can not perceive it.

 

[sneutron]

Thanks for that, Sir. Could someone help with another of my layman probs?

Ghosts are usually thought to be wreckers in string theories. Do they pair with positive mass partners with identical quantum numbers? If so, what would be wrong with taking "reality" as the sum-of-squares mass (as in QM real/complex)?

 

[arivero]

Peter remarks on rcho http://www.math.columbia.edu/~woit/wordpress/?p=3665

 

[S.Daedalus]

Something I've been wondering about since this thread came up originally: why does none of the 'physics-from-division-algebras' frameworks (like the one by Furey discussed in here, or Dixon's, Dray/Manogue, etc.) seem to contain any hint of supersymmetry? It seems that when you look at supersymmetry, division algebras stare right back at you, but when you look at division algebras, supersymmetry appears much more coy...

 

[S.Daedalus]

So, I've been thinking about, and reading up on the history of, this RCHO business on and off for a few months, and I'd like to take a new look at it...

First of all, Geoffrey Dixon has a new book out ('Division Algebras, Lattices, Physics, Windmill Tilting'), but I haven't gotten my hands on it (yet).

Also, it's been said that it's hard to get the Higgs in the division algebra approach, but it seems that it was actually there, in some sense, in one of the earliest attempts (which I don't think has been mentioned in this thread so far) by Finkelstein, Jauch, Schiminovich and Speiser: they used a 'http://jmp.aip.org/resource/1/jmapaq/v4/i6/p788_s1?isAuthorized=no', i.e. invariance under transformations of the form $\Psi\to q\Psi q^{-1}$, and find a massless boson (photon), and two additional massive, charged vector bosons ($W^{±}$). Tony Smith calls it:

the first paper [...] that used Quaternionic SU(2) symmetry to describe the mechanism whereby two charged SU(2) bosons get mass, and the electromagnetic field is unified with the SU(2) bosons. Their paper effectively did the "Higgs Mechanism" before Higgs, and did ElectroWeak Unification before Glashow,Salam, and Weinberg

They don't find the $Z^0$, perhaps related to the fact that they only use SU(2), rather than the full U(1)xSU(2) (i.e. $ℂ\otimes \mathbb{H}$).

The paper 'Octonionic Structures in Particle Physics' by Gürsey gives some fascinating insight into the history of the subject, in particular how octonionic extensions of quantum mechanics were originally looked into in order to incorporate the new phenomena of nuclear physics before the advent of gauge theory. Some other interesting papers (some of which will probably have been mentioned already):
'Remark on the Algebra of Interactions' - A. Pais (an early one, 1961)
'Octonionic Quark Confinement' - H. Ruegg (related to the idea that the non-associativity of the octonions ensures the unobservability of quarks; in a sense, a special imaginary unit is picked out, which gives us our usual quantum mechanics, and has the side effects of breaking $G_2$ to SU(3), and bringing $SL(2,\mathbb{O}) \simeq SO(9,1)$ down to $SO(3,1)$. Under this SU(3), the split-octonion units $u_0 = \frac{1}{2}(e_0+ie_7)$ and $u_0^* = \frac{1}{2}(e_0-ie_7)$ transform as a singlet and antisinglet, while $u_i = \frac{1}{2}(e_i+ie_7)$ and $u_i^* = \frac{1}{2}(e_i-ie_7)$ (i = 1,2,3) transform as a triplet and antitriplet -- a lepton and quark, together with antiparticles. (This is of course the familiar Günaydin-Gürsey scheme.) So this gives us 'half a generation' of fermions living in Minkowski space.)
'SO(8) Color as a Possible Origin of Generations' - Z.K. Silagadze (discusses a possible extension of the above to incorporate a full generation, then uses SO(8)'s triality to find the observed three.)
'Algebraic Realization of Quark-Diquark Supersymmetry' - S. Catto (related to my question above; 'composite' SUSY from octonionic color algebra -- arivero, I think, is familiar with this)
'Quaternion Higgs and the Electroweak Gauge Group' - DeLeo, Rotelli (another look at Higgsing from a div. alg. perspective)
'Derivation of the Standard Model' - Dixon (I think this is the first paper where Dixon lays out his model, or its basics, completely; of course, the full treatment is available in his first book (most of which I don't understand, unfortunately))
'Algebraic Approach to the Quark Problem' - Casalbuoni et al (a somewhat alternative approach to that of Günaydin/Gürsey, explaining the quark confinement through the realization that in a path-integral quantized version of their theory, only color singlet states can propagate)

There's many more, but this post is already getting rather lengthy, and there's one last, perhaps somewhat too out-there, thing that I've come across that I wanted to share. In entanglement theory, there's an interesting connection between two- and three-qubit entanglement and the Hopf fibrations, and hence, the division algebras, laid out in the paper 'Geometry of the Three-Qubit State, Entanglement, and Division Algebras' by Bernevig and Chen, and summarized in the slides to this talk by Chen. The idea is, basically, that the Hopf map is sensitive to the entanglement properties of two- and three-qubit systems: the state space of one qubit is given by the first Hopf fibration, $S^1 \hookrightarrow S^3 \to S^2$, where the $S^3$ is the qubit state space, the $S^1$ fiber is the global phase, and $S^2$ is the Bloch sphere. Analogously, the state spaces of two- and three-qubit systems, $S^7$ and $S^{15}$ can be related to the second and third Hopf fibrations, $S^3 \hookrightarrow S^7 \to S^4$ and $S^7 \hookrightarrow S^{15} \to S^8$. Of course, there are no more Hopf fibrations after that, because there are no more division algebras beyond the octonions. In each case, the fiber is the unit sphere of the complexes, quaternions, or octonions respectively, while the base spaces are the respective projective lines. This map is entanglement sensitive in the sense that if, for instance, the three-qubit stat is biseparable, it maps only into the complex subspace of the octonionic projective line. Furthermore, each base space can be seen as containing one qubit, plus the entanglement degrees of freedom, while the rest of the state (two qubits for the three qubit case, one for two qubits) lives in the fiber.

Now this is quite a surprising way for the division algebras to turn up in entanglement! In particular, this appears to allow us to consider a two-qubit state as a single quaternionic qubit, and similarly, a three-qubit state as a single octonionic qubit (essentially via the Cayley-Dickson construction: if two qubits are parametrized by the complex numbers $\alpha_k = a_k + ib_k$, k = 1...4, then it can be parametrized by the quaternions $q_1 = (a_1 + ib_1) + j(a_2 + ib_2)$ and $q_2 = (a_3 + ib_3) + j(a_4 + ib_4)$, where $j^2=-1$ and $|q_1|^2+|q_2|^2=1$; an analogous construction works for the three-qubit case). (The connection between Hopf fibrations and qubits over division algebras was also noticed in the paper 'Extremal Black Holes as Qudits', by M. Rios who I think posts here occasionally.)

But this brings us right to the quaternionic and octonionic extensions of quantum mechanics discussed earlier! So, could there be a connection? (Probably not, but it's just the kind of '...but what if?'-thing that sometimes goes through my head at night...) In particular, since I've always liked the 'spacetime is made of qubits'-idea from Weizsäcker's ur-theory, to look for 'inner space' in entanglement between urs seems somehow appealing to me... But I realize this is just far-out speculation.

In any case, this has gotten somewhat lengthy, so thanks to anyone who persisted to this point; I'd be very glad for any comments, especially elucidations of things I don't quite grasp yet (my background is not in particle physics, so this is not really my home field...).

 

[friend]

But this brings us right to the quaternionic and octonionic extensions of quantum mechanics discussed earlier! So, could there be a connection? (Probably not, but it's just the kind of '...but what if?'-thing that sometimes goes through my head at night...) In particular, since I've always liked the 'spacetime is made of qubits'-idea from Weizsäcker's ur-theory, to look for 'inner space' in entanglement between urs seems somehow appealing to me... But I realize this is just far-out speculation.

To be complete, is there any way to derive GR from the division algebras?

 

[S.Daedalus]

To be complete, is there any way to derive GR from the division algebras?

I only know of the paper 'Non-Associativity as Gravity' by Dorofeev, which I don't think is particularly convincing. Personally, and somewhat off-topic, I consider the recent paper by Jacobson, 'Gravitation and Vacuum Entanglement Entropy', which identifies entanglement entropy with the Bekenstein-Hawking entropy (under certain assumptions, such as its non-divergence), and then uses Jacobson's old thermodynamic argument to get the EFEs, much more promising...

 

[arivero]

So, what happens with 4-qubits etc? I would expect it to be formulated in the usual terms of Spinorial Chessboard and Bott periodicity. The peculiar thing of division algebras in Spinors is, as you have remarked, that they beget SUSY. Is there some similar property peculiar to 2 and 3 qubits?

On a different theme, I do not know of a relevant role for the third hoft fibration, with S15 sitting there. Of course it hints of SO(16) and then some of the string theory symmetries, but the standard model group seems to travel well just with the second fibration, S7, halving it so that the basis is not S4 but CP2 (there is a concept there, "branched covering", for which I would welcome an octonionic or quaternionic formulation). Also, thinking on GUT groups such as SO(10) and SO(14), it could be interesting to ask more of the S9 and S13 spheres.

 

[mitchell porter]

On a different theme, I do not know of a relevant role for the third hopf fibration, with S15 sitting there.

It could be something to do with how you get 11-dimensional M-theory from 26-dimensional bosonic string theory.

 

[S.Daedalus]

So, what happens with 4-qubits etc?

You get the (one-loop) partition function for bosonic strings...

However, there's a qualitative difference between three- and four-qubit (or generally n > 3) entanglement: while for three qubits, you have four SLOCC-equivalence classes (two kinds of genuine three-partite entanglement, biseparable states, and fully separable states), for four qubits, there's already infinitely many. (Although some say it's nine, in accordance with the famed prediction from string theory, but those are kind of meta-classes, each depending on a continuous parameter.) So that's something peculiar to two and three qubits, but doesn't seem related to SUSY in any way...

On a different theme, I do not know of a relevant role for the third hoft fibration, with S15 sitting there.

I thought maybe it's just that you need bioctonions for one full generation (a single octonion -- or split octonion -- in the Günaydin/Gürsey scheme incorporates only one flavor). There's another interesting paper I didn't mention earlier, 'Freudenthal Triple Classification of Three Qubit Entanglement' by Borsten et al., which collects the qubits into the Freudenthal triple $\mathfrak{M}(J)=\mathbb{C}\oplus\mathbb{C}\oplus J \oplus J$ over the Jordan algebra $J =\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}$, and identifies the entanglement classes with the rank of its elements; but those elements are of the form of (complex) Zorn matrices, i.e. bioctonions. Not sure if it means anything, but it's kinda neat.

But maybe we should just look at the base spaces: the $S^8$ gives the octonions (or the octonion projective line), the fiber, $S^7$, is again fibered with base space $S^4$, the quaternion projective line, its fiber in turn gives the complex, then the real line -- which kinda reminds of the tensor algebra $T=\mathbb{R}\otimes\mathbb{C} \otimes \mathbb{H} \otimes \mathbb{O}$ Dixon and Furey use in their schemes...

On another note, I read somewhere (though I don't recall where) that the original octonionic/quaternionic quantum mechanics scheme fell out of favor for some reason (and certainly, it seems like it was pursued somewhat less than I would have thought it should have been), does anybody know why that might have been? I mean, there's of course the tensor product troubles etc., but is there a known reason schemes like those can't work?

 

[arivero]

On another note, I read somewhere (though I don't recall where) that the original octonionic/quaternionic quantum mechanics scheme fell out of favor for some reason

I think to remember that the introduction to Adler's book discusses this point. Regretly it is not in my local library.

The whole issue of quaternions and how they become tainted by political fight between academical schools in the XIXth century is already part of the history of mathematics. See e.g. Felix Klein treatise about this period; his personal remarks in the last part of chapter IV are very explicit.

 

[S.Daedalus]

I think to remember that the introduction to Adler's book discusses this point. Regretly it is not in my local library.

Our university library has it, so I'll have a look, thanks for the pointer!

About that third Hopf map: the decomposition is such that of an entangled triplet, two qubits live in the $S^7$ fiber, while the third lives in the $S^8$ base, together with the additional degrees of freedom coming from the entanglement (if there's no entanglement, the whole construction collapses -- as one would expect -- to $S^2 \times S^2 \times S^2$, i.e. three single-qubit state spaces). So this third qubit is somehow 'augmented' by the entanglement to an octonionic being -- lepton/quark as per Günaydin/Gürsey -- living in 9+1 dimensions (because of the connection between $\mathbb{O}P^1$ and $SL(2,\mathbb{O})\cong SO(9,1)$). That's probably a bit much free association I realize, but I think there's a story to be told here, even though I don't yet fully see it.

 

[S.Daedalus]

I've skimmed Adler's book, and he points out that octonion quantum mechanics really only exists in the case of the exceptional Jordan algebra $J_3(\mathbb{O})$, describing a single quantum system over the Moufang plane $\mathbb{O}P^2$ as constructed by Günaydin, Prion, and Ruegg. (The book also backs up -- and might be the source of -- Tony Smith's statement son the quaternion electroweak paper referenced above by Finkelstein et al.)

Also, I've stumbled across an approach that I think of as 'RCHO in disguise', propagated by Greg Trayling, which has been briefly mentioned before on this forum here and https://www.physicsforums.com/showthread.php?t=82187. There's only two papers on this, 'A Geometric Approach to the Standard Model' and 'A Geometric Basis for the Standard Model Gauge Group', the latter of which is the more extensive one.

Basically, the bid is to get the standard model from the Clifford algebra $C\ell_7$, introducing four additional spatial dimensions. Nevertheless, I think this is related, in particular, to Dixon's approach: firstly, $C\ell_7 = \mathbb{R}[8]\oplus\mathbb{R}[8] = \mathbb{C}\otimes\mathbb{R}[8] \cong \mathbb{C}\otimes\mathbb{O}_L$ (where the 'L' denotes left-action), whose spinor space is just $\mathbb{C}\otimes\mathbb{O}$ (cf. Dixon's 'Division Algebras; Spinors; Idempotens; The Algebraic Structure of Reality', which contains a nice presentation of his model (and its link to the Hopf fibrations!)). However, use is also made of $C\ell_3 \cong \mathbb{C}\otimes\mathbb{H}_L$, introducing all our favourite players after all. There's however some awkwardness in the treatment of the right-handed neutrino, which has to be artificially suppressed in order to get the right structure.

 

[arivero]

I am happy to know that Adler had some valuable info; I really was doing a partly blind shot, as I had read it in 2006 last time.

Jordan algebra seems to have some role, yep. It does appear also when extending the idea of the relationship between SUSY and division algebras, I think that some paper of German Sierra is really about it. And of course Jordan algebras have a deep history of its own, in the context of quantum mechanics, foundations, etc.

In a private comunication from someone (perhaps M Porter? Other?), I have been told about seeing octonions as a set of 8 roots, 7 of them imaginary, the other the trivial 1, and then arguing that this 7+1 decomposition could be used to explain why 12 out of 96 states of the particle spectrum (ie 1 out of 8) have peculiar mass properties. Perhaps the 1 is to be related to the 12 neutrino states, perhapt to the 12 top quark states.

Final rumiation, I have already mentioned it in this tread, and a lot elsewhere, but perhaps not enough in this one: Michael Atiyah, Jurgen Berndt http://arxiv.org/abs/math/0206135 should be the tool to explain why colour is SU(3) and not SO(5), and the contexts for it is either RHCO or Hoft (-like) fibrations with S4 (and CP2, resp) base spaces.

 

[mitchell porter]

Yes, that was me... Baez and Huerta have a connection between the imaginary split octonions and the group G2. Someone tell Gordon Kane!

 

[MTd2]

I don't think he will listen! He is now claiming that SUSY particles to be found around 50TeV...

 

[S.Daedalus]

I am happy to know that Adler had some valuable info; I really was doing a partly blind shot, as I had read it in 2006 last time.

I had the book on my radar, your recommendation bumped it to the top of the list, and it's certainly very interesting, though I'm not sure I buy into all of it. He proposes some Harari-Shupe like preon model, which I've decided I'm not a great fan of, and I'm also not sure about the idea of a quaternion quantum mechanics underlying complex QM. Though it's interesting that the S-matrix is complex, asymptotically at least -- perhaps one could think of this as a mechanism for dimensional reduction, i.e. macroscopic experimenters only 'see' the 3+1 dimensional world associated with the complex numbers, instead of the quaternionic 5+1 (in my own vague ruminations, I have supposed that this role is played by the fact that quantum correlations only get weaker by admixture of states -- i.e. genuine tri- or bipartite entanglement generally doesn't survive to the macroscopic level, effectively reducing octonions and 9+1 dimensional space time to complex numbers and 3+1 dimensions...).

In a private comunication from someone (perhaps M Porter? Other?), I have been told about seeing octonions as a set of 8 roots, 7 of them imaginary, the other the trivial 1, and then arguing that this 7+1 decomposition could be used to explain why 12 out of 96 states of the particle spectrum (ie 1 out of 8) have peculiar mass properties. Perhaps the 1 is to be related to the 12 neutrino states, perhapt to the 12 top quark states.

Interesting, but how is mass related to the octonion roots?

Final rumiation, I have already mentioned it in this tread, and a lot elsewhere, but perhaps not enough in this one: Michael Atiyah, Jurgen Berndt http://arxiv.org/abs/math/0206135 should be the tool to explain why colour is SU(3) and not SO(5), and the contexts for it is either RHCO or Hoft (-like) fibrations with S4 (and CP2, resp) base spaces.

I think I don't understand this stuff well enough to comment much... Perhaps there's some relation to the non-compact Hopf maps defined using the split-algebras (see here)?...

Yes, that was me... Baez and Huerta have a connection between the imaginary split octonions and the group G2. Someone tell Gordon Kane!

Oh, that one slipped past me! I'm usually on the lookout for Baez' stuff, so thanks for the pointer...

 

[arivero]

Interesting, but how is mass related to the octonion roots?

No idea. Naively, when one has a natural scale and some masses with are near zero respect to such scale, then one is supposed to search for a symmetry that protects such zero masses. What is intriguing is that in the standard model we have two different 7+1 (or 84+12) scenarions: respect to the Dirac mass scale of neutrino, all the others are almost zero. And respect to the mass scale of the top quark, the same. So the most elegant solution could be, instead needing to decide for one or another, to have a duality. Thus I was inclined to look into the M2-brane M-5 brane duality, because its source is a tensor with 84 components. Does such duality (which is simply the Pascal Triangle equality between (9 2+1) and (9 5+1)) has some parallel in octonions?

(yep, "scenarions" is a typo, but a funny one)

 

[friend]

Is there any way to extract a real number from the quaternions and octonions like there is for complex numbers? In complex numbers we can multiply by the complex conjugate to get a real number. Is there an analogous procedure for quaternions and octonions?

 

[arivero]

Is there any way to extract a real number from the quaternions and octonions like there is for complex numbers? In complex numbers we can multiply by the complex conjugate to get a real number. Is there an analogous procedure for quaternions and octonions?

Sure; they are normed division algebras.

 

[arivero]

I noticed this old one from Nahm. No surprises, but interesting ancient reference http://inspirehep.net/record/129766?ln=es

 

[kneemo]

Now this is quite a surprising way for the division algebras to turn up in entanglement! In particular, this appears to allow us to consider a two-qubit state as a single quaternionic qubit, and similarly, a three-qubit state as a single octonionic qubit ... an analogous construction works for the three-qubit case). (The connection between Hopf fibrations and qubits over division algebras was also noticed in the paper 'Extremal Black Holes as Qudits', by M. Rios who I think posts here occasionally.)

Yes, extremal black hole paper also deals with the split-composition algebras, which show up in toroidal M-theory compactifications. The basic picture is that M-theory acquires the symmetries of geometries (projective, symplectic, metasymplectic, etc.) over the split-octonions upon compactification down to d=6, 5, 4 and 3 dimensions. These symmetries give the U-duality groups for the corresponding supergravity theories (which includes the U-duality group E7(7) for N=8 supergravity in the d=4 case).

The black hole/qudit paper covers the d=6,5 cases where the U-duality groups SO(5,5) and E6(6) provide SLOCC gates/transformations for split-octonion qubit and qutrits. Two states are defined as SLOCC equivalent if there is a non-vanishing probability to convert one into the other (and back) via LOCC (local quantum operations assisted by classical communication). Geometrically, SO(5,5) and E6(6) are determinant preserving collineation transformations acting on the (2x2 and 3x3 Hermitian matrix) black hole charge spaces in d=6,5. The transformations are generally not isometries (i.e., not always unitary), but do preserve rank, hence preserve the entropy and fraction of supersymmetry of each black hole.

But this brings us right to the quaternionic and octonionic extensions of quantum mechanics discussed earlier! So, could there be a connection? (Probably not, but it's just the kind of '...but what if?'-thing that sometimes goes through my head at night...) In particular, since I've always liked the 'spacetime is made of qubits'-idea from Weizsäcker's ur-theory, to look for 'inner space' in entanglement between urs seems somehow appealing to me... But I realize this is just far-out speculation.

Yes, there is a connection. Toroidally compactified M-theory and N=8 supergravity make use of (split) quaternionic and octonionic extensions of quantum mechanics. Moreover, if M-theory in d=11 does have hidden Cayley plane fibers arXiv:0807.4899, then M-theory becomes a d=27 theory inherently equipped with a 16-dimensional (over ℝ) octonionic qutrit state space.

 

[S.Daedalus]

Thanks for the replies, sorry for taking so long to get back -- they force me to do actual work occasionally, but now I've got a bit of time for some division algebra fun again!

Thus I was inclined to look into the M2-brane M-5 brane duality, because its source is a tensor with 84 components. Does such duality (which is simply the Pascal Triangle equality between (9 2+1) and (9 5+1)) has some parallel in octonions?

Well, this M-theory stuff is a bit of a learning curve for me, but perhaps you might find something in the works of Francesco Toppan, who has looked into it from an octonionic perspective, in particular maybe this[/PLAIN] paper:

The generalized supersymmetries admitting abelian bosonic tensorial central charges are classified in accordance with their division algebra structure [...]. It is shown in particular that in D=11 dimensions, the $M$-superalgebra admits a consistent octonionic formulation, involving 52 real bosonic generators (in place of the 528 of the standard $M$-superalgebra). The octonionic $M5$ (super-5-brane) sector coincides with the octonionic $M1$ and $M2$ sectors [...].

OK, now for some numerology: the 84 and 7 x 12 makes one think of Hurwitz surfaces, in particular the Klein quartic. Any Hurwitz surface is a Riemann surface that has the maximal amount of symmetries for its genus, 84*(g-1), and the Klein quartic is the minimal Hurwitz surface with g = 3. As usual, Baez has a nice and detailed explanation of its features (complete with pictures and images by sci-fi writer Greg Egan). Its symmetries can be understood as first a 7-element group coming from twisting around the seven triangles that meet at each of its vertices, and then the whole thing is essentially tetrahedral, yielding another 12-element group for a total of 84; additionally, you can twist the thing inside out, yielding a total of 168 symmetries.

And what's more, its symmetry group is PSL(2,7), which is isomorphic to PSL(3,2), which is nothing else that the group of symmetries of the projective plane over the 2-element field, otherwise known as the Fano plane -- i.e. it basically acts on the imaginary octonion multiplication tables!

(There's also a lot of stuff about this on Tony Smith's pages, who among other things http://www.valdostamuseum.org/hamsmith/KleinQP.pdf of physics in the Klein quartic...)

Now, whether this actually means anything, I've no idea...

Yes, there is a connection. Toroidally compactified M-theory and N=8 supergravity make use of (split) quaternionic and octonionic extensions of quantum mechanics. Moreover, if M-theory in d=11 does have hidden Cayley plane fibers arXiv:0807.4899, then M-theory becomes a d=27 theory inherently equipped with a 16-dimensional (over ℝ) octonionic qutrit state space.

Ah, I'm glad you joined the discussion! I've been hoping to understand this whole black hole/qubit stuff better, but as I said, much of M/string theory is a bit above my paygrade, and I don't really have much time for digging into it as much as I would want to. I'll have a look at the paper you mention, and I'm interested exactly in what way quaternionic/octonionic QM turns up in M-theory (of course, if you have OP2 bundles, you think of the exceptional Jordan algebra), so I'd be happy if you have some pointers there (literature etc.)...

More in my line of thinking, I've been looking at four qubit entanglement, which I originally thought didn't play a role because there's no fourth Hopf map, but it turns out you can still define an entanglement-sensitive mapping involving the sedenions (!) which works at least in some cases (or at least that's the claim of the paper 'Hopf Fibration and Quantum Entanglement in Qubit Systems' by P.A. Pinilla and J.R. Luthra). This may give me a 'sedenion-spinor' description of entangled four-qubit systems, $\psi_4 \in \mathbb{S}^2\cong \mathbb{H}\otimes\mathbb{O}$, which seems enough for a generation of fermions, and which is exactly the kind of thing Katsusada Morita uses to derive the standard model (or more precisely, its left-right symmetric extension, see for instance his 'http://ptp.ipap.jp/link?PTP/68/2159' and 'http://ptp.ipap.jp/link?PTP/66/1519'). His model is very similar to Dixon's -- the latter also gets two su(2)'s, but interprets one as a spatial symmetry (see his 'Division Algebras; Spinors; Idempotents; the Algebraic Structure of Reality').

OK, this is all I've time for right now -- thanks for everybody's comments, I'm learning a lot!