Could a 14-Dimensional Theory Unify String Theories and Dualities?

[torsten]

I came across a paper hep-ph/9411381 suggesting the use of three SO(10)'s hitting distinct 27's of E6, in order to properly respect triality. The use of three 27's seems to take one back to the maximal subgroup E6xSU(3)/(Z/3Z) of E8, giving the decomposition: 248=(8,1)+(1,78)+(3,27)+(3*,27*). E6's 78 acts on 27 and preserves its cubic form i.e., determinant. This decomposition is pretty popular in heterotic compactifications where one looks for Calabi-Yau manifolds with Euler characteristic χ=± 6, so that the generations come from |χ|/2, leading to a three-generation E6-model. The other maximal subgroup E7xSU(2)/(-1,-1) is also interesting, with E8 decomposition 248=(3,1)+(1,133)+(2,56). The 56's are two copies of the Freudenthal triple system with structure 56=27+27+1+1, that are usually used as charge spaces for D=4, N=8 SUGRA extremal black holes. E7's 133 acts on a 56 and preserves its quartic form. ...

Dear kneemo and MdT2,
only some remarks to your interesting discussion. It is interesting that most people think in higher dimensions for groups like E6 or E8. But I made the experience that many relations also exists for low dimensional manifolds (like 2-, 3- or 4-manifolds) (where I'm a kind of specialist).
So let me mention some aspects:
1. SO(4,2) could be the symmetry group of 6-dim space (with two time coordinates) but at the same time it is the conformal group (including translations) of the 4-dimensional Minkowski space famous in the 60s wher one discussed the conformal group to understand the strong force.
2. Lie groups are characterized (via its Lie algebra) by the root system forming a discrete object (a polytope). Also via the Dynkin diagram one obtains also a simple graph.
3. For instance the Dynakin diagram of the E8 can be used to construct a closed 4-manifold (which does not carry any smooth structure) or to construct a 4-manifold with boundary (the Poincare sphere).
3. In my work (MdT2 mentioned the link) I made also this expierence. So I considered the Yang-Mills action which is a sum of quadratic curvature components having values in the Cartan subalgebra (via the Casimir operators). Therefore I obtained the correct groups because I got the correct number of quadratic curvatures.
4. In M theory there is also a mysterious relation (found by Cumrun Vafa, Amer Iqbal, and Andrew Neitzke in 2001) between the charges of M theory on a torus and the so-called del Pezzo surface (a special class of 4-manifolds). The main observation is that the large diffeomorphisms of del Pezzo surfaces match the Weyl group of the U-duality group of the corresponding compactification of M-theory. The elements of the second homology of the del Pezzo surfaces are mapped to various BPS objects of different dimensions in M-theory. The complex projective plane P2(C) is related to M-theory in 11 dimensions. When k points are blown-up, the del Pezzo surface describes M-theory on a k-torus, and the exceptional del Pezzo surface, namely P1(C) × P1(C), is connected with type IIB string theory in 10 dimensions.
5. The Cayley-Salmon theorem states that a smooth cubic surface over an algebraically closed field contains 27 straight lines. These can be characterized independently of the embedding into projective space as the rational lines with self-intersection number −1, or in other words the −1-curves on the surface. The 27 lines can be identified with the 27 possible charges of M-theory on a six-dimensional torus (6 momenta; 15 membranes; 6 fivebranes) and the group E6 then naturally acts as the U-duality group. (another form of mysterious duality)
You see there is also interesting relations between 4-manifolds and higher Lie groups like E6 or E8.
In my opinion higher-dimensional spaces are not necessary.

 

[kneemo]

Indeed every smooth cubic in CP^3 is isomorphic to CP^2 blown up at six points. Hence, M-theory on T^6.

 

[MTd2]

Dear kneemo and MdT2,
only some remarks to your interesting discussion. It is interesting that most people think in higher dimensions for groups like E6 or E8.

I am considering fiber bundles. S7, in the case of S4 is a total space.

This total space is a total space and S8 is a kind of internal space of particles, where S15 is the total space of all of this. I will explain myself better later.

 

[MTd2]

So, I will put part of what I was discussing with Tony Smith. He's got to say this about the interpretation of exotic structure from superior dimensional hopf fibrations. It's not exactly quoted since I changed my mind about a few things.

"Here is a physical interpretation of exotic structures:

Exotic structures correspond to spinors which correspond to fermions.

S7 = SO(8)
SO(8) is 28-dim bivector of Clifford Algebra Cl(8)
Cl(8) has 8-dim + half-spinors and 8-dim -half-spinors
How many different ways can you construct a spinor structure on S7 ?
You have 8 choices for one of the half-spinors.
For the other half-spinor, you have to make an antisymmetric choice
because choice A B looks the same as choice B A
so there are 8 x 7 / 2 = 28 different spin structures = differentiable structures (27 of the 28 are exotic[My observation is that 1 of them is equivalent to vacuum]).

S15 = SO(16))
SO(16) is 120-dim bivector of Clifford Algebra Cl(16)
Cl(16) has 128-dim + half-spinors and 128-dim -half-spinors
How many different ways can you construct a spinor structure on S7 ?
You have 128 choices for one of the half-spinors.
For the other half-spinor, you have to make an antisymmetric choice
because choice A B looks the same as choice B A
so there are 128 x 127 / 2 = 8,128 different spin structures = differentiable structures
but
at the S15 level each of those 8,128 has a mirror image structure that is distinct
so
the true total number of differentiable structures is 128 x 127 = 16,256The half-spinor part of E8 (which is inside Cl(16) is 128-dim
64 dim = 8 components of 8 fermion particles
the other 64 of the 128 = 8 components of 8 fermion antiparticles."

This is in accordance to my proposal that all physics comes from exotic structures which exist in the long sequence of Hopf fibration.

S15-S7-S8 is a sequence which generate particles.
S7 - S3 -S4 is a sequence which is reponsible for quantization, including of space time. For that I need to confirm that S4 is uncountable infinite. Torsten found the gauge structure of the standard model, in S4. But not the fermions.

I'd say, that S7 provides the fermons, but without charge, spin, colors and timeless ( SM has 12 dimensions, which gives 27- 15 total particles). 3 neutrinos, 3 leptons, 6 quarks. 3 fermions are missing. Maybe the right handed neutrinos.

 

[kneemo]

Mysterious duality was mentioned, where D=11 M-theory corresponds to CP^1. Toric grometric constructions were given up to three blown up points, i.e., M-theory on T^3. In the toric construction of CP^2, one builds a simplex with boundary given by circles fibered over three intervals. The circle radii vanish at each vertex and we have a U(1) acting on the circle fibers. Taken together, each each fibered interval is a sphere, S^2.

H. Sati has proposed that M-theory could have hidden OP^2 Cayley plane fibers. If one proceeds with a toric construction, as in the case of CP^2, we no longer have circles fibered over intervals, but rather 7-spheres fibered over the three intervals.

 

[arivero]

I'd say, that S7 provides the fermons, but without charge, spin, colors and timeless ( SM has 12 dimensions, which gives 27- 15 total particles). 3 neutrinos, 3 leptons, 6 quarks. 3 fermions are missing. Maybe the right handed neutrinos.

Are you speaking of the same paper than above, or another one, here?

 

[MTd2]

Are you speaking of the same paper than above, or another one, here?

Out of my head :) But I am changing my mind about these stuff as I talk, by email, to Tony and Torsten.

 

[MTd2]

H. we no longer have circles fibered over intervals, but rather 7-spheres fibered over the three intervals.

I did not notice this. What do you mean by 3 intervals? Z3 orbifold?

 

[arivero]

Out of my head :) But I am changing my mind about these stuff as I talk, by email, to Tony and Torsten.

Do not forget a old rummiation of myself, then: the intriguing relationship between CP2 and S4, one being a "branched covering under complex conjugation" of the other. Whatever it means. Atiyah liked to remark this.

Now, if a discrete operation relates CP2 and S4, should this discrete operation also relate the fibering of S4 by S3 to build S7, with some fibering of CP2 by S3. Note that CP2 has the isometry group SU(3), while S3 has isometry SU(2)xSU(2) and any deformation to lens spaces, or to S2xS1, should have isometry group SU(2)xU(1). On the other hand, S7 has isometry group SO(8). This discrete "branched covering" could reduce S7 to a manifold with the symetry of the standard model, and at the same time justify the existence of chirality.

SO(8) has 28 generators. The SM-like group SU(3)xSU(2)xSU(2) has 8+3+3=14. Somewhere hidden in this forest we have a discrete symmetry that kills one half of the generators but builds a group which is not a subgroup of the former.

 

[MTd2]

For me, and probably Torsten, all that exists are smooth (and so triangulable) manifolds. Every thing else comes from obstructions to fluxes on 4-manifolds.

BTW, there is a very simple way to get the chirality. We're trying to verify this.

I will try to see that OP2 fibration the knemo mentioned. I like special things, and among the infinite projection groups, it seems OP2 is the unique special projection. This should have a deep meaning.

There are others, obtained by group divisors.

http://en.wikipedia.org/wiki/Freudenthal_magic_square#Rosenfeld_projective_planes

So, OP2, should be related to G2, following the trend of others groups.

As we can see, we should expect something in 8 dimensions, since we all see powers of 2. An not quite Octonion like, since the first case is already F4 is already the group of symmetries of Octonions. Looking at this table, we have:

http://en.wikipedia.org/wiki/Riemannian_symmetric_space

We have:

G_2/SO(4) Space of subalgebras of the octonion algebra O which are isomorphic to the quaternion algebra H.

For this reason the riemanian symmetric space and the Rosenfeld projective space have a sort of unique cross over in this aspect.

 

[Jim Kata]

I have a stupid question. In the books I've read it always said beyond twelve dimensions you got fields with helicities higher than 2 and it was unclear how to couple such fields to matter. I also remember it being said that supersymmetry could not exist in dimensions larger than eleven. So what is all of this?

 

[John G]

Rather than:

E6(-26)=8*+16*+so(8)+R+R+16+8

E6(-26)=M1,2(O)*+so(1,9)+R+M1,2(O)

E7(-25)=1*+32*+so(10,2)+R+32+1

E8(-24)=14*+64*+so(11,3)+R+64+14.

This from Tony Smith I think is better if you like Tony and Lisi think Triality is important:

Start with D4 = Spin(8):

28 = 28 + 0 + 0 + 0 + 0 + 0 + 0

Add spinors and vector to get F4:

52 = 28 + 8 + 8 + 8 + 0 + 0 + 0

Now, "complexify" the 8+8+8 part of F4 to get E6:

78 = 28 + 16 + 16 + 16 + 1 + 0 + 1

Then, "quaternionify" the 8+8+8 part of F4 to get E7:

133 = 28 + 32 + 32 + 32 + 3 + 3 + 3

Finally, "octonionify" the 8+8+8 part of F4 to get E8:

248 = 28 + 64 + 64 + 64 + 7 + 14 + 7

At E6 you can orbifold with the spinors to get fermions and via Tony you have an F4 8+8+8 Triality plus 2 (F4 to E6) = 26-dim bosonic string (no supersymmetry needed). Going to E7 and E8 gives a bosonic M and F theories (related to a Lee Smolin idea for bosonic M-theory).

It's actually better for Tony's model to think in LQG terms with Cl(8) used as the spin foam for the discrete spacetime (Tony mentioned this long ago in a sci.physics.research discussion with John Baez). It's an 8-dim Kaluza Klein-like spacetime since Tony has the 8-dim F4 vector as the spacetime.

Going up to E8 gives you the "octonifying" which Heisenberg Hamiltonion quantization-wise gives you 8 momentum operators to go with the 8 position operators (8x8=64-dim E8 vectors). This 8x8 vector spacetime kind of takes the place of Lisi's 4x4 Frame. The advantage of an 8-dim Kaluxa Klein instead of just Lisi's 4-dim spacetime is the normal Kaluza-Klein idea for handling the Standard Model charges (Electroweak and Color).

For the 64+64-dim E8 spinors, you have the 8 component x 8 fermion creation-annihilation operators (one 64 being matter and the other 64 antimatter). Tony thus has one generation instead Lisi's 3 but Tony avoids Lisi's vector fermion problem. To get three generations is related to Tony having the conformal group dilaton Higgs VEV. The cubic Casimir/quantum number added by using the D3 conformal group instead of Lisi's D2 Lorentz group gives the three generations.

Tony like Lisi breaks his E8 into two D4s which still allows the required 8 Casimirs/quantum numbers. For one D4 Tony has D3 gravity plus a U(1) quantum propagator phase (with its trivial Casimir I think) while Lisi has a D2 gravity plus D2 Pati-Salam electroweak. For the other D4 Tony has SU(3)xSU(2)xU(1) aka A2xA1XA0 for the Standard Model (no Pati-Salam). Lisi for the other D4 has SU(3) for strong and a U(1) for Pati-Salam and an unknown new U(1).

Looking at the Casimirs/quantum numbers looks like they both have a couple for SU(3) charges, a couple for electroweak charges and a couple for gravity spins. For Tony the other two are for the generations and quantum propagator phase while the other two for Lisi is the Pati-Salam extra plus an extra new unknown one.

SU(3) aka A2 is a subalgebra of D4 so its OK for people who like to break symmetry with a subalgebra. D3 and D2 obviously both work OK with D4 too. Breaking symmetry leaves some left over D4 slots to work with. Lisi gets his frame from them and some new color-like bosons as well as the Pati-Salam extras. Smith uses the extras for Kaluza-Klein/quantization related duals. Basically the gravity bosons are repeated in the Standard Model D4 for operators working with the Kaluza-Klein space and the Standard Model bosons are repeated in the Gravity D4 for operators working with the large physical 4-dim spacetime.

Both Tony and Lisi (and Torsten) like characterizing via root systems so I'm Going to show Tony's D3, D4, and F4 vector, F4 spinor, F4 other spinor. E8 just superimposes 8 vertices where the F4 vector-spinor-spinor vertices are. The plots are done with Lisi's particle explorer but the labels are for F4 organizational behavior plots not F4 (E8) physics.

D3:

https://scontent-b-lax.xx.fbcdn.net/hphotos-prn2/t1.0-9/10262013_775666652466286_6598460957319489612_n.jpg

D4 additions to D3:

https://scontent-a-lax.xx.fbcdn.net/hphotos-prn2/t1.0-9/10255858_775666862466265_4406155279939892875_n.jpg

F4 Vector:

https://scontent-a-lax.xx.fbcdn.net/hphotos-ash3/t1.0-9/p417x417/10177227_775668212466130_190013459400982508_n.jpg

F4 Spinor:

https://scontent-a-lax.xx.fbcdn.net/hphotos-frc3/t1.0-9/10300117_775668522466099_7249309497727109339_n.jpg

F4 other Spinor:

https://scontent-a-lax.xx.fbcdn.net/hphotos-prn2/t1.0-9/p417x417/10292131_775668675799417_1100529043650058105_n.jpg

 

[kneemo]

At E6 you can orbifold with the spinors to get fermions and via Tony you have an F4 8+8+8 Triality plus 2 (F4 to E6) = 26-dim bosonic string (no supersymmetry needed). Going to E7 and E8 gives a bosonic M and F theories...

My point was that certain D=8,10,12,14 Yang-Mills theories have an algebraic structure seen in certain gradings of the exceptional algebras.

There are other gradings, sure, such as the "extremal black hole" gradings. These other gradings reflect the structure of Jordan algebras and their Freudenthal triple systems.

None of the gradings is more important than the others, but rather shows that Smith and Lisi's approaches are likely "dual" to S-theory its 14-dimensional extension.

In other words, there may exist a 14-dimensional extension of M-theory that is "dual" to Lisi's theory. This Yang-Mills theory is (still barely) supersymmetric.

Of course, this matching of rotation groups and spinors occurs for real non-compact forms of the exceptional algebras, while the split forms would give another interesting set of theories. Both non-compact forms are in the complex forms of the exceptional groups, and in that sense the complex case is more fundamental.

The same complexification is also necessary in the case of extremal black hole gradings.

At this time, Duff et al are studying the M-algebra for M-theory, expressed in octonionic form. They use 4x4 octonionic matrices and four component octonionic spinors.

 

[arivero]

At this time, Duff et al are studying the M-algebra for M-theory, expressed in octonionic form. They use 4x4 octonionic matrices and four component octonionic spinors.

http://arxiv.org/abs/1309.0546 http://arxiv.org/abs/1402.4649 is it?

 

[kneemo]

http://arxiv.org/abs/1309.0546 http://arxiv.org/abs/1402.4649 is it?

Yes.

This extends the 2003 work of Francesco Toppan http://arxiv.org/abs/hep-th/0307070. When I gave a talk back in 2006 on the 3x3 eigenvalue problem, Francesco asked me if the results hold for 4x4. The problems with 4x4 are: 1. Herm(4,O) does not give a Jordan algebra 2. OP^3 does not exist.

So we need a 32-dimensional octonionic spinor without going to 4-components. Easy, just complexify the octonions and use a 2-component bi-octonion spinor. (CxO)P^2 does exist and Herm(3,CxO) gives the exceptional Jordan C*-algebra. So maybe the most natural M-algebra is based on bi-octonions. If so, we have triality, a projective space, a spectral geometry and an exceptional quantum mechanics as tools to further probe M-theory.

 

[John G]

Obviously Lisi and Smith themselves wouldn't be interested in spinor fermions via supercharges but I have run into a couple people interested in Tony's ideas in a superstring context. Tony may not be interested in a 14 64 so(7,7)+1 64 14 grading for a T-duality inspired 32x2=64-dim D7 spinor fermions via supercharges but he does relate that grading to the 7,1 and 1,7 signatures for his two D4s and Tony does use T-duality in a bosonic string context.

 

[kneemo]

Obviously Lisi and Smith themselves wouldn't be interested in spinor fermions via supercharges but I have run into a couple people interested in Tony's ideas in a superstring context. Tony may not be interested in a 14 64 so(7,7)+1 64 14 grading for a T-duality inspired 32x2=64-dim D7 spinor fermions via supercharges but he does relate that grading to the 7,1 and 1,7 signatures for his two D4s and Tony does use T-duality in a bosonic string context.

The E8(8) grading:

14*+64*+so(7,7)+R+64+14

is a split-octonion grading.

The E8(-24) grading:

14*+64*+so(3,11)+R+64+14

is an octonion grading.

Both the split-octonions and octonions are contained in the complexified octonions i.e., bioctonions.

The corresponding E8(C) bioctonion grading is:

14*+64*+so(14,C)+C+64+14

The extremal black hole grading of E8(C) is:

1*+56*+E7(C)+C+56+1

The charge space for such a black hole is 57-dimensional. E8(C) acts on this space non-linearly.

It's better not to try interpret these structures in conventional stringy terms. It's smarter to first formulate the M-algebra with bioctonions, extend it to 27-complex dimensions, then interpret it in D=11 M-theory. This should be done pretty soon. I'd say, within three months.

 

[arivero]

Yes.

This extends the 2003 work of Francesco Toppan http://arxiv.org/abs/hep-th/0307070.

I see that the team of Duff has keeping doing some work on Division Algebras:

http://arxiv.org/find/hep-th/1/au:+Hughes_L/0/1/0/all/0/1

Five papers up to now.

Perhaps also related, the update of Furey http://arxiv.org/abs/arXiv:1405.4601 We already named him in other threars, particularly RCHO: https://www.physicsforums.com/showthread.php?t=376804

 

[nuclearhead]

Anything with more than one time dimension seems highly dubious to me.