Higgs, SuperStrings and Kaluza Klein

kneemo

What you mention is the way certain non-compact exceptional groups are seen from the perspective of M-theory on T^k (k-torus). But such non-compact groups only refer to the split real forms, e.g. E6(6), E7(7), E8(8). The other real forms such as E6(-26), E7(-25) and E8(-24) are not yet described in M-theory and are called the U-duality groups of the magic supergravity theories in D=5, D=4 and D=3, respectively. However, maybe there is a "dual" way to view the dimensionality, as the gradings of the exceptional Lie algebras suggests.

For example, consider the following grading of E8(-24):

g = E8(-24), g(0) = E7(-25) + R, dimR g(-1) = 56, dimR g(-2) = 1.

This grading is usually interpreted with the g(-1) part being the 56-dimensional Freudenthal triple system (black hole charge space in D=4), which is acted on by the E7(-25) (D=4 U-duality group). In the corresponding D=3 magic supergravity E8(-24) acts on the 56+1-dimensional "charge-entropy" space of the extremal black hole.

Yet, as mentioned before, there is also the 5-grading of E8(-24) with components:

g = E8(-24), g(0) = so(3,11) + R, dimR g(-1) = 64, dimR g(-2) = 14.

For this one, I don't yet know the interpretation, but the grading is suggestive of a 14-dimensional theory with 64-component spinors.

The two different 5-gradings, are akin to two different "slicings" of E8(-24), giving rise to a "black hole frame" in the first 5-grading with dimR g(-1)=56 and the "spinor frame" in the second. Morphisms between these gradings would give rise to a type of "duality" between the frames which, individually, appear to describe different physical systems.

arivero

Yes I see. Reading the original papers there was a lot of excitation about all this, perhaps because it was the first time where they were able to show/use the exceptional structures. Still, I think that it is distraction; a lot of it must be tautological with all the other octonionic justifications of the existence of M-theory (the brane scan, etc) and part of it could be just the KK version of a catalogue of subgroup branchings and breakings as we reduce dimensions. Some of it could, in the long rung, explain the pieces of the SM that are not explained by Kaluza Klein: the yukawas, CKM, perhaps even part of the higgs structure. I would not try to use them as QFT gauge symmetries... but ok, you have not suggested any use at all, so perhaps you agree with me

kneemo

To see the role of the non-compact exceptional groups in a geometric context, it's better to look at a simpler example.

Consider the complex projective plane, CP^2, for which SU(3) acts via isometries. If we would like to transform points in CP^2, and only care about preserving collinearity, we can use SL(3,C) transformations. In the Jordan algebraic context, SL(3,C) is the group of determinant preserving transformations of the Jordan algebra of 3x3 Hermitian matrices over C, J(3,C).

If we consider collineations (line preserving transformations) that fix a point in CP^2, these same transformations are those that preserve the determinant of 2x2 Hermitian matrices in the Jordan algebra J(2,C). These transformations lie in SL(2,C) and the homomorphism onto SO(3,1) follows pretty quickly once J(2,C) is identified with Minkowski spacetime such that the determinant is the squared length in (3,1) spacetime.

The relation between OP^2 and E6(-26) is basically just the octonionic version of that of CP^2 and SL(3,C), where SO(9,1) rather than SO(3,1) transformations fix a point in the plane. Using the split octonions, one has E6(6) with SO(5,5) transformation fixing a point in the "split" Cayley plane. The complex octonion (i.e. bioctonions) case yields E6(C) acting on the complexified Cayley plane (i.e. the bioctonionic projective plane) with SO(10,C) transformations fixing a point.

So, geometrically, E6 behaves more as a conformal group than an isometry group. Its use is more likely to be in the study of scattering amplitudes that are functions defined on the product of multiple copies of OP^2 (and its split and complex forms), with a copy for each particle.

arivero

I can not tell that I quite follow all the argument, but I think I globally agree. Of course there is more structure in field theory, and string theory, that just the obvious lagrangian. Practicioners of QFT, such as Predrag Cvitanovic have always had the risk of become obsessed with the extra symmetries of a lot of calculations. And given that the Standard Model Kaluza Klein space is basically a variation of the seven sphere (one where S4 has somehow collapsed to CP2) a role for octonions and then the exceptional groups is to be expected here.

kneemo

I was referring to an octonionic version of the usual twistor-string approach initiated by Witten (hep-th/0312171) where instead of scattering particles on copies of CP^3 with (d=4 complexified) conformal group SL(4,C), one studies scattering amplitudes on copies of OP^2, invariant under the group E6(-26).