John Baez and his student John Huerta have an article in Scientific American this month about octonions in string theory and M-theory. Peter Woit has given it a bit of a cynical review describing it as hype. The defence from John in the comments is worth reading. Here is a bit of what he says

*“So, don’t try to make it sound like an obscure yawn-inducing technicality about “some supersymmetry algebra working out a certain way in a certain dimension”. It’s a shocking and bizarre fact, which hits you in the face as soon as you start trying to learn about superstrings. It’s a fact that I’d been curious about for years. So when John Huerta finally made it really clear, it seemed worth explaining — in detail in some math papers, and in a popularized way in Scientific American.”*

The article entitled “The Strangest Numbers in String Theory” is about an early observation from the study of superstring theories that the four division algebras are related to four classical formulations of superstring theory in 3, 4, 6 and 10 dimensions. The four division algebras are the reals, complex numbers, quaternions and octonions with dimensions 1,2,4 and 8 and the dimensions of the superstring theories are 2 dimensions higher in each case.

Many of our readers will be familiar with the internet writings of John Baez where he has described these things so I wont attempt to cover any details. His student John Huerta has just finished off his thesis in which he clarifies these observations using higher dimensional lie algebras. The results extend to one dimension higher when strings are replaced by membranes. In the quantum theory only the highest dimensional versions related to the octonions hold up consistently giving us the 10 dimensional superstring theories and M-theory in 11 dimensions. Of course this is not complete since we still don’t know what the full formulation for M-theory is. Even these higher dimensional observations are not new, see for example Mike Duffs brane-scan from 1987 where the relationships were already plotted out. This new work clarifies these results using the concept of 3-lie-algebras from n-category theory.

The Scientific American version does not go into great detail but is a very well written introduction to the ideas. If you don’t have access to it don’t worry, John says he will be allowed to post an online version after a month or so. You can also explore what has been posted already starting here which is more advanced than the article but still very pedagogical.

Personally I find these algebraic ideas for M-theory very enticing. It is a major goal to formulate a complete non-perturbative version of string theory that encompasses all its forms and I think the purely algebraic approach is the best line of attack. It is especially intriguing that the octonions have such a direct relationship to the dimensions in which these theories work, but ultimately the algebraic structures we need to understand it fully are probably much more complex.

The work of Mike Duff and his collaborators which brings in the algebra of hyperdeterminants and qubits to understand a slightly different role of the octonions in string theory is one of the areas to follow. This work brings in the duality algebras found in string theory black holes. I know that several of our regular commenters are very familiar with this already so I need not give more details. Indeed it is the fact that the same algebraic structures keep appearing in different contexts that is so intriguing, yet so confusing, as if we are missing some principle of unification that relates these things.

Dimensions of classical number fields are fundamental ones and the idea that they are crucial for physics is deep.

However, the un-necessary belief that one must have N=1 SUSY and Majorana spinors leads to D=10 instead of D=2 in D=4 in D=8 and eventually the attempts to save the theory lead to D=11 and to the landscape misery. As if this were not enough, then comes the sad discovery that the predicted kind of SUSY is not observed at LHC. But still we want to believe in superstrings;-).

Couldn’t we for a fleeting moment just shake all those names from our shoulders, use our own brains, and consider starting from something much much simpler ;-). Just those classical number fields?

[...] also Philip Gibbs, “Octonions in String Theory,” viXra log, April 29, [...]

The Baez-Huerta theory involves 3 and 4 cycles. The 3-cycle is a rule on fields (ψ*ψ)ψ =/= 0 then ψ(ψ*ψ)ψ = 0 is a cohomology. The 4-cycle takes this into a spinor-vector rule with the product , which involves an antisymmetric system of elements which defines a hyperdeterminant. The result is this defines a Clifford algebra CL_{10,1} = R[32]. So this in a rule of thumb is where we add 3 to each of the Cayley numbers. I refer you to a paper by http://arxiv.org/abs/1003.3436 Baez and Huerta for this how 3 and 4 cycles determine Clifford algebras on Cayley numbers, with Clifford dimensions 3, 4, 7, and 11. We have is a nice system in 10 dimensions, which is dual to something in 16 = 8+8 dimensions. In group theory this is SO(10) and E_8xE_8, where the last part is the infamous heterotic string, or closed string which carries 24 field elements that contain the “graviton.” The SO(10) is our more well behaved (well except for renormalization) open string theory (eg type II) which describes things like the nuclear interaction. We also have this 4-cycle stuff, which pops us up one dimension and completes in some low energy approximation this thing we call M-theory.

I think this is best extended into the Jordan matrix algebra. This naturally includes supersymmetry, with three copies of the octonions, one for vectors and the other two for spinor field, which can be conjugates of each other. The exceptional group G_2 is the automorphism on the octonions O, or equivalently that F_4xG_2 defines a centralizer on E_8. The fibration G_2 –> S^7 is completed with SO(8), where the three O’s satisfy the triality condition in SO(8). The G_2 fixes a vector basis in S^7 according to the triality condition on vectors V \in J^3(O) and spinors θ in O, t:Vxθ_1xθ_2 –> R. The triality group is spin(8) and a subgroup spin(7) will fix a vector in V and a spinor in θ_1. To fix a vector in spin(7) the transitive action of spin(7) on the 7-sphere with spin(7)/G_2 = S^7 with dimensions

dim(G_2) = dim(spin(7)) – dim(S^7) = 21 – 7 = 14.

The G_2 group in a sense fixes a frame on the octonions, and has features similar to a gauge group. The double covering so(O) ~= so(8) and the inclusion g_2 \subset spin(8) determines the homomorphism g_2 hook–> spin(8) –> so(O). The 1-1 inclusion of g_2 in so(O) maps a 14 dimensional group into a 28 dimensional group. So as the F_4 diagonalizes the J^3(O) in a Fruedenthal system (hyperdeterminants etc) this connects with the 3-qubit entanglement system of Duff et al.

[...] y aquí): Peter Woit, “This Week’s Hype,” Not Even Wrong, April 28th, 2011; Philip Gibbs, “Octonions in String Theory,” viXra log, April 29, 2011; Lubos Motl, “John Baez, octonions, and string theory,” The [...]

A interesting point of the 7-sphere is that it seems to hint us a way to go down, once we have used the division algebra ladder to go up to D=11. And the argument also relates to division algebras. Everyone knows that S7 is, just because of them, a Hopf fibration, S3 –> S7 —> S4. It is less known that the relationship of S4 with the quaternions allow for a discrete conjugation, making S4 a branched covering by CP2. And the final touch, consider fiber bundles M of the kind S3 –> M –> CP2. It is posible to build a family where the group of isometries is still the group of the trivial bundle S3 x CP2. And this group is… (SU(2)xSU(2)) x (SU(3))

One has to be a bit careful with octonions. The first difficulty is that E_8 is not self adjoint. The subgroup E_6 is such and it makes nice gauge group, and is a reasonable gadget to work with within SYM theories for a GUT, where one can get a 27-dimensional irrep for the transformations of Dirac field and bosons. The Baez-Huerta result here illustrates a connection between the Cayley number 1, 2, 4, and 8 with the SO(2,1), SO(3,1), SO(5,1), and SO(9,1), such as SL_2(O) ~ SO(9,1). Also E_8xE8 has the same content (number of roots etc) as SO(32) for a type I string theory. However, the two are not isomorphic

Of course who is to say that nature is most fundamentally a SYM theory? If octonions play a fundamental role it suggests that physics has some fundamentally different structure, one which may be more general than what we ordinarily consider. The heterotic string theory is the grin of a Cheshire cat that we have not identified properly as yet. It might turn out that physics is fundamentally nonassociative, but as yet we do not know what this means physically or in an S-matrix setting.

Lawrence B. Crowell, or any other expert, how about writing an introductory overview about that stuff (octonions, clifford algebras, triality, E8, etc.) discussed here? For us average mortals who know about SO(3),SU(2),SU(3) and a little about roots and weights, but don’t know where to turn to when learning about these beautiful mathematical structures mentioned here. (Yes, there is wiki and John Baez, but I need more..) Could not anybody do the world a favor just work out step by step how triality works and what’s so magical about SO(8)/Spin(8)? Please. Tony Zee writes in his QFT book that SO(8) is perhaps the most beautiful of all groups. But nowhere in the libary or in the www can I find someone who takes the time to explain it carefully to the average student. Why is that so? thanks

SO(8) has a D_4 Dynkin diagram that is a 3-fold system of O’s at 2π3 radian apart and connected to a “hub” O. The center of SO(8) is Z_2 and for spin(8) ~ SO(8) it is Z_2xZ_2. Triality then acts on spin(8)/Z_2. The automorphism group of spin(8) acts on this center and is a semidirect product with the symmetric group S_3 ~ GL(2,2).

http://scgp.stonybrook.edu/presentations/20101103_Atiyah_-_From_Algebraic_Geometry_to_Physics.pdf

Atiyah’s magic square – figure linking SU(x)with E-groups.

Is space-time ―granular at very small scale?

http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.3176v1.pdf

The idea is to introduce in a natural geometric way operators which

shift (i.e. retard or advance) the basic operators of mathematical physics. This includes the Dirac, Maxwell and Ricci operators (occurring in the Einstein equations of GR). Of course, this has a quantum-

mechanical aspect and involves Planck’s constant hbar.

Dreams

In the broad light of day mathematicians check

their equations and their proofs, leaving no stone

unturned in their search for rigour. But, at night,

under the full moon, they dream, they float among

the stars and wonder at the miracle of the heavens.

They are inspired. Without dreams there is no art,

no mathematics, no life.

The physical role of octonions is very uncertain at this point. It might be that the ambiguity in the order of multiplication, or its associativity, is involved with just this that Atiyah and Moore discuss here. Of course this is what makes musing over octonions rather fun, it is precisely their mysteriousness which makes them most interesting. My suspicion is that octonions do play some foundational role in physics. It might be at the heart of understanding the heterotic string.

Hi Lawrence,

There seem to be advantages and disadvantages to both E6 and E8.

The disadvantage of E8 is its lack of complex representations, and its resultant inability to describe CP violation. IMHO, the logical solution is to introduce a dual purely imaginary E8*. This allows fermions to “live” in one Gosset lattice, and bosons to “live” in the dual Gosset lattice. Now we have the equivalent of 496 real degrees-of-freedom, and we can build a “C” algebra by “twisting” a pair of “R” algebras together (we can write a+ib as (a,b) and continue with our calculations).

The disadvantage of E6 is that its dual has a different order. E6 order is 78, and E6* order is 48. My models place bosons and fermions in dual lattices. To satisfy that requirement and Supersymmetry (equal number of degrees-of-freedom between bosons and fermions), then we need E6(78 fermions)xE6*(48 bosons)xE6(78 sfermions)xE6*(48 bosinos) with 252 complex degrees-of-freedom equivalent to 504 real degrees-of-freedom.

Personally, I don’t see a huge advantage of one over the other at this stage. Both E6 and E8 get a little bit messy, both root systems include icosahedral symmetries (E8 240 real roots = 2×120, and E6xE6* 120 complex roots = 2×120) although the 5-fold C_5 rotational symmetry is clearly broken in E6xE6*. But E8 has slightly fewer degrees-of-freedom, and E8 has its isometries and similarities with the Gosset lattice and Octonions. Thus, my preference is E8xE8*. I think I understand why Lubos prefers E6, and he may be right – to a point…

One can decompose E_8 into E_6xSU(3), so if E_8 plays some role in physics the E_6xSU(3) would be the gauge theory or a 78 rep of fermions. The dual E*_6xSU(3) is 48, and the two together are 120. However, the convex hull of this root system is a polytope in 6 dimensions, and the 120-cell is in 4. So I do not see any obvious map between these two. I think that E_6 contains “standard physics.” This is where the stuff we know and love lives. E_8 or E_8xE_8 is something else. The problem is that we do not know what octonionic physics means. There may be some substratum to nature which has this strange nonassociative structure, but as yet there is no physical interpretation for that.

Cheers LC