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

[kneemo]

While looking back at some gradings for exceptional Lie algebras, I re-discovered an old jewel of a paper (hep-th/9704054) by Itzhak Bars, known for his work on supergravity and the 13-dimensional S-theory. Essentially, Bars argues that a 14-dimensional theory could lie behind non-perturbative string theory and its dualities. He shows how to embed type A,B,C, heterotic and type-I superalgebras covariantly in the framework of 14 dimensions with signature (11,3) and SO(11,3) symmetry and that in lower dimensions one can embed three sets of 32 supercharges as different projections of 64 supercharges, which form three distinct superalgebras.

The SO(11,3) symmetry has also been used by Nesti and Percacci in arXiv:0909.4537 in a GraviGUT model, that has ties with Lisi's E8 theory, as can be seen from the grading of E8(-24)'s algebra: 14*+64*+so(11,3)+R+64+14. The gradings clearly shows the 64 supercharges, (11,3) signature symmetry and some extra G2 symmetry. A supersymmetric form of Lisi's theory might very well be equivalent to Bars' 14-dimensional theory that would contain M-theory and F-theory.

Lisi has also expressed interest in the other non-compact form of E8, namely E8(8) which admits the algebraic grading: 14*+64*+so(7,7)+R+64+14. Itzhak Bars hasn't explored this (7,7) signature theory yet, but in light of E8(C) with grading: 14*+64*+so(14,C)+C+64+14, it should be dual to the SO(11,3) theory in the sense that E8(8) and E8(-24) are just real non-compact forms of E8(C). The complex E8(C) theory, in this sense, would be more fundamental.

Other gradings of E8 have been used extensively in the construction of 57-dimensional extremal black hole charge spaces in D=3 (e.g. E8(C)=1*+56*+E7(C)+C+56+1) (hep-th/0008063). There, the E8(8) and E8(-24) gradings amount to a choice of split-octonion or octonion variables inside the algebra of complexified octonions. The construction of a Jordan C*-algebra and its extended Freudenthal triple system leads to the full E8(C) symmetry, that is ultimately necessary to accommodate solutions where the E7 quartic invariant takes negative values and leads to non-real values for the 57D conformal invariant.

Does anybody have any thoughts on such a 14-dimensional theory?

 

[MathematicalPhysicist]

The final theory will have $$\infty$$ dimensions anyway...

 

[tom.stoer]

What are the physical reasons to go beyond 10 dimensions?

 

[MathematicalPhysicist]

What are the physical reasons to go beyond 10 dimensions?

We shouldn't go beyond 1+3, but mathematically we can do as we wish, and with no empirical evidence who can stop us?

In PDEs we also deal with equations with two time variables and even more, like in the ultrahyperbolic pde.

 

[tom.stoer]

Of course we can. The question is whether it's physics or pure math.

 

[MathematicalPhysicist]

Of course we can. The question is whether it's physics or pure math.

Well, history has it that most pure math is being used in theoretical physics.

So the distinction between the two in my case is quite blur anyway.

 

[tom.stoer]

In history math was related to physics and reality (GR, QM, QFT, ...). This changed with string theory ;-)

 

[kneemo]

What are the physical reasons to go beyond 10 dimensions?

Arguments for M-theory in D=11 (e.g. nonperturbative string dualities, D=11 SUGRA, etc.) are well understood. Vafa's proposal for F-theory in D=12 relies on the observation that the IIB D-string with a U(1) super-Maxwell gauge field on its worldsheet has a critical dimension of 12, as one has to introduce additional ghosts of spin(0,1) which shifts the central charge by -2 giving a total space with signature (10,2).

Sezgin was able to formulate a super Yang-Mills theory in (11,3) but going past 14-dimensions (e.g. signature (12,4)) yields unwanted contributions to the Yang-Mills field equation and fails to be supersymmetric. Note: Sezgin considered spacetime superalgebras containing a single Majorana-Weyl spinor generator with 2^{n+3} real components and all possible bosonic generators that can occur in their anticommutation relation. The (11,3) superalgebra was found to be maximal in the series: (8,0), (9,1), (10,2), (11,3).

What makes the story more amusing is that the structure of (8,0), (9,1), (10,2) and (11,3) signature theories (symmetry and supercharges) is already encoded in non-compact forms of E6, E7 and E8. Namely, one has the Lie algebraic gradings (which can be lifted to subgroups):

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.

Usually, these non-compact forms of E6, E7 and E8 are interpreted as U-duality groups for magic supergravities in D=5,4 and 3 dimensions, respectively. However, the U-duality interpretation makes use of different gradings (e.g. in D=3, one uses E8(-24)=1*+56*+E7(-25)+R+56+1 where E8(-24) acts on a 57 dimensional charge-entropy space).

 

[tom.stoer]

Yes. I agree.

But the experimental, physical support is zero.

 

[mitchell porter]

Eric Weinstein's mystery theory (hyped in the Guardian by Marcus du Sautoy earlier this year, and later presented in a talk at Oxford) is supposed to involve 14 dimensions. Peter Woit wrote: "The metric tensor is a symmetric bilinear form, so 10 components in 4 d. So, you could make a bundle over your 4d spacetime, with 10d fibers given by the symmetric bilinear forms on the tangent space." My theory about his theory was that it was a 64-dimensional spinor coupled to an SO(14) GraviGUT, with the first two generations coming from the spinor and the third generation from somewhere else (like Lisi, Weinstein has to handwave to get three generations).

If Bars' 14-dimensional theory has any reality, you could look for a 3-brane in its spectrum, and then try to obtain a Weinstein-like theory as the worldvolume theory of the 3-brane... But I have a question for kneemo. The conventional stringy wisdom regarding the idea of gravity as a gauge theory (a la LQG, etc) is that it only works in the special case of 2+1 dimensions. Do you have some special take on GraviGUTs which is supposed to be consistent with M-theory?

 

[kneemo]

If Bars' 14-dimensional theory has any reality, you could look for a 3-brane in its spectrum, and then try to obtain a Weinstein-like theory as the worldvolume theory of the 3-brane...

Sezgin suggested there may be a superbrane with (3,3) dimensional worldvolume propagating in (11,3) dimensions. Vafa's (2,2) worldvolume in F-theory would be recovered from this. In F-theory the (10,2) is connected to the usual (9, 1) reformulation via wrapping of a (1,1) part of the (2,2) worldvolume about a compact (1,1) space, leaving a (1,1) string in 10 dimensions - the type IIB string. With this, one can use T-duality to go from IIB to IIA and S-duality to get to M-theory.

Bars argued that the 32 supercharges of IIB and IIA are distinct, as projections of the 64 supercharges coming from the Weyl spinor of SO(11,3), coming from a BPS equation. Each projection branch admits compactifications down to 4D with SO(3, 1) Lorentz symmetry and inherited internal symmetries. The kicker is that in D=4 Bars extends the SO(3,1) symmetry to SO(3,1)xSO(1,1) (or another variant, as one is coming down from (10,2) or (11,3) signature) and gets a novel Kaluza-Klein family generation mechanism from the extra (1,1) coordinates. This mechanism might be useful in the GraviGUT approaches.

 

[arivero]

Happy Xmas.

SO(10) works as the symmetry group of the 9 sphere, so classical Kaluza-Klein for GUT group SO(10) needs only 13 dimensions. I am not sure how many dimensions should a Kaluza Klein with GUT group E6 (which is the greatest group with complex representations as needed for the fermions) have.

 

[arivero]

Happy Xmas.

SO(10) works as the symmetry group of the 9 sphere, so classical Kaluza-Klein for GUT group SO(10) needs only 13 dimensions. I am not sure how many dimensions should a Kaluza Klein with GUT group E6 (which is the greatest group with complex representations as needed for the fermions) have.

 

[kneemo]

My theory about his theory was that it was a 64-dimensional spinor coupled to an SO(14) GraviGUT, with the first two generations coming from the spinor and the third generation from somewhere else (like Lisi, Weinstein has to handwave to get three generations)

You might be right. It was said that Weinstein's theory is a type of 14D Yang-Mills theory. He may very well be using SO(14,C) and a complex 64-dimensional spinor. This is the structure found in the full complexified form of E8. SO(11,3) and SO(7,7) Yang-Mills theories with real 64-dimensional spinors would be special cases of this.

As for relating GraviGUTs to M-theory, for now it seems the D=14 SO(11,3) super Yang-Mills theory with real 64 spinor is the closest match.

 

[kneemo]

I am not sure how many dimensions should a Kaluza Klein with GUT group E6 (which is the greatest group with complex representations as needed for the fermions) have.

E6 contains all the symmetries of the Cayley plane, so a 16+4=20-dimensional classic Kaluza-Klein construction should suffice.

A more modern approach would have E6(C) as a structure group of a nonassociative C*-algebraic bundle. The E6(C) gauge degrees of freedom are then interpreted as inner fluctuations of a nonassociative geometry.

 

[tom.stoer]

I don't want to be impolite, but I am still interested in physical reasons to consider these models. Which physical problems are solved by these models? And what is the physical reason to go beyond 10 or 11 dimensions?

 

[kneemo]

I don't want to be impolite, but I am still interested in physical reasons to consider these models. Which physical problems are solved by these models? And what is the physical reason to go beyond 10 or 11 dimensions?

When reduced to a 4+2 version of the Standard Model, Bars claims the strong CP violation problem of QCD is resolved (without an axion) and the model predicts a dilaton driven electroweak spontaneous symmetry breaking, which has implications for string theory vacuum selection, dark matter composition and inflation.

 

[tom.stoer]

What does "4+2" mean? Is it a two-time formalism?

How is a "4+2" version of the standard model related to the "3+1" version which has been verified experimentally?(within the accessable energy range)

Why do we need a dilaton-driven symmetry breaking instead of the Higgs-mechanism?

As far as I unterstand you correctly, another benefit of this huge mathematical structure is to get rid of the axion, correct?

What about experimentally testable predictions beyond the standard model? Which particles and interactions do we have to find in order to prove the model? Which experiments can disprove the model?

 

[kneemo]

What does "4+2" mean? Is it a two-time formalism?

How is a "4+2" version of the standard model related to the "3+1" version which has been verified experimentally?(within the accessable energy range)

Why do we need a dilaton-driven symmetry breaking instead of the Higgs-mechanism?

As far as I unterstand you correctly, another benefit of this huge mathematical structure is to get rid of the axion, correct?

What about experimentally testable predictions beyond the standard model? Which particles and interactions do we have to find in order to prove the model? Which experiments can disprove the model?

Yes, it's a two-time formalism with an extra Sp(2,R) symmetry that can be gauge fixed down to 3+1 dimensions without leaving behind any Kaluza-Klein type modes or extra components of vector and spinor fields in the extra 1+1 dimensions. An action in d+2 dimensions requires a dilaton (as a singlet of SO(d,2)) in order to achieve two-time gauge symmetry. (Note: The 4+2-dimensional SM has SO(4,2) symmetry, which interestingly is also the isometry group of AdS_5.). The dilaton couples to the Higgs in a purely quartic potential where electroweak symmetry breaking by the Higgs is driven by the vev of the dilaton.

The higher structure coming from 4+2 dimensions prevents the problematic F ∗ F term in QCD from appearing in 3+1 dimensions, thus resolving the strong CP problem without the axion.

Bars' 4+2 SM predicts a Higgs-dilaton mixing. There are regions of parameter space where the mixing is large enough for the rates to diphoton and ZZ to receive observable corrections. Measurements ruling out Higgs-dilaton mixing can disprove Bars' proposed 4+2 model.

 

[tom.stoer]

Thanks. For me this is too much speculative input, but now I can make the relation to established physics.

 

[MTd2]

Usually, these non-compact forms of E6, E7 and E8 are interpreted as U-duality groups for magic supergravities in D=5,4 and 3 dimensions, respectively. However, the U-duality interpretation makes use of different gradings (e.g. in D=3, one uses E8(-24)=1*+56*+E7(-25)+R+56+1 where E8(-24) acts on a 57 dimensional charge-entropy space).

Just a lot of numerology. If kneemo sees some pattern here, I would love your observation since Marni, which is a person I greatly admires, admires you

It would be interesting a (4,4) brane (It seems he uses a 3,3 for 14 dimensions). It also looks like a generalized twistor space, that is H^2, instead C^2.

Setting the number of dimensions to 15, you'd perhaps could say this is an S15, with S8 with fiber S7.

 

[kneemo]

Just a lot of numerology. If kneemo sees some pattern here, I would love your observation since Marni, which is a person I greatly admires, admires you

It would be interesting a (4,4) brane (It seems he uses a 3,3 for 14 dimensions). It also looks like a generalized twistor space, that is H^2, instead C^2.

Setting the number of dimensions to 15, you'd perhaps could say this is an S15, with S8 with fiber S7.

The observation is that spinors and rotation groups for (9,1), (10,2), (11,3) theories are encoded in non-compact real forms of the E6, E7 and E8 exceptional groups, respectively.

In the E8 case, as Lisi has noted, one can also recover a (12,4) signature and real 128-dimensional spinor. This structure would arise from a 3-grading of the Lie algebra for E8(-24), namely 128*+so(12,4)+128. This would point to a possible D=16 theory beyond D=14.

So far, Bars has only found superalgebras up to (11,3), however. At the superalgebra level this allows an embedding of M-theory, F-theory and S-theory into a 14-dimensional (11,3) signature theory. Sezgin tried to find a super Yang-Mills theory beyond D=14 (11,3) but ran into some issues. There's likely a way to push the formalism into (12,4)-signature. To my knowledge it just hasn't been done yet.

The octonionic Hopf fibration S^15 -> S^8 (S^7 fiber) you mention is definitely interesting. On one side you have an SO(16) for the S^15 and an SO(9) for S^8. The inclusion of SO(9) in SO(16) can be seen as the 16-dimensional spinor rep of SO(9) in the 16-dimensional vector rep of SO(16).

The higher map from S^23 -> OP^2 gives OP^1~S^8 lines in the Cayley plane. SO(9) contains those isometries (in F4) that fix a point in the plane. There are three equivalent embeddings of SO(9) in F4, each fixing one of the three orthogonal idempotent points in the plane. Ramond has suggested this SO(9) be identified with the lightcone little group in D=11. If that's done, how does one interpret the three copies of SO(9) acting on the 16-dimensional OP^2?

There are also split Hopf maps coming from the split forms of the division algebras (C_s, H_s, O_s): H^(2,1) -> H^(1,1), H^(4,3) -> H^(2,2) and H^(8,7) -> H^(4,4). The split octonion case will probably interest you most as it maps from split octonion 2-space O_s^2 to the (4,4) signature hyperboloid, a generalized twistor space.

 

[MTd2]

that's done, how does one interpret the three copies of SO(9) acting on the 16-dimensional OP^2?

http://www.math.okayama-u.ac.jp/mjo..._38/mjou_38_197.pdf?origin=publication_detail

As for the split division algebra, it came to my mind this.

S7 have 27 exotic spheres. But that's because the different non mapping between the bundle S3? What about the base S4? ( Side note: What if is infinite due S4? Perhaps it's a key to solve the S4 diffeomorphism case?)

If you know the work from Torsten, he gets a clue of the SO(3)XSO(2)XU(1) gauge structure of the standard model by classifying the torus bundles used to construct the complexity of the exotic smoothness. Maybe the S3 part as above is related to the particles?

Now, all of the above would be included in the fiber of S^8, which would give triality. So, that would give 3 generations. And space time in S^8?

I found a clue here:

http://en.wikipedia.org/wiki/Eight-dimensional_space

"Whether the real universe in which we live is somehow eight-dimensional is a topic that is debated and explored in several branches of physics, including astrophysics and particle physics. For example, the biquaternion algebra, which is based on C4, a four-dimensional space over the field of complex numbers, can be used to represent to the theory of special relativity."

But if we are talking about generating particles from spacetime disturbances, we'd need also the S^7 fiber, thus, the description you gave above would correspond to a generalized twistor spce.

Does it make any sense to you?

 

[kneemo]

that's done, how does one interpret the three copies of SO(9) acting on the 16-dimensional OP^2?

Imagine three distinct points in the plane OP^2, where given two points, there is an S^8 "line" passing through them. (In a toric geometry framework, one can think of an S^7 fibered over an interval defined by any given two points) This gives three S^8 lines, pairwise intersecting, yielding a triangle in the (octonionic) plane. An SO(9) transformation of this triangle fixes a vertex of the triangle and transforms the other two vertices. As points of OP^2 are usually represented by projection matrices with trace equal to one, SO(9) preserves the trace of the points with each transformation. If we don't care about preserving the trace, defining points as just rank one Hermitian matrices over O, SO(9,1) will fix a point of the triangle.

The image above is for the real case but it's handy for a visualization of the situation in O^3.

In the usual twistor correspondence, a point x in real Minkowski space defines a set of null rays, defining the null cone at that point. The rays can be mapped to points of a CP^1 projective line in projective twistor space CP^3. We can transform the CP^1 with SL(2,C)~SO(3,1). In 9+1 dimensions, we can map the null rays to an S^8 and transform it with SL(2,O)~SO(9,1).

So given three pairwise intersecting S^8's in the plane OP^2, we would have three lightcones that each copy of SO(9) in F4 transforms. This is a manifestation of triality. These lightcones pairwise share a null ray, which in the projective plane gives lines pairwise intersecting at a point. A full F4 transformation occurs in 27-dimensions so the picture resembles bosonic M-theory more than conventional M-theory in D=11.

Cohl Furey used CxO to represent a generation of fermions. Defining Hermitian matrices over CxO, makes use of three copies that transform in complex 27-dimensional space. There is a projective plane in this case as well, (CxO)P^2, but makes use of SO(10) in E6(C) to fix a point.

 

[MTd2]

CXO, being O seems to recall me of sedenions. The transformations happen in 27 dimensions, but it seems they are constrained in 16 dimensions, which means there are 9 degrees of freedom, so, we are back to S^8, that is, the fiber.

Given that this S^8 is a kind of connection, maybe we are talking about the connection which is working like a momentum space. Well, I am saying that this intersection to form some kind of koide relation. Well, this relation will of null rays will satisfy a koide relation.

As you can see here http://arxiv.org/pdf/1212.3182.pdf , there is some sort of Koide relation related to SL(2,O). Besides, it seems that the triple generation, looking at the functorial arrows, in page 9 and 10, it seems that the triple generation is related to SL(3,O) being the double cover of SO(9,1).

You mention "If that's done, how does one interpret the three copies of SO(9) acting on the 16-dimensional OP^2? " I think the issue here is that the 16 dimensional space is the result of the dimension constrain due SL(3,0) being a covering group.

Also, since the we are talking about topology, note also that S^7, have 28 non diffeormorphic spheres, while S8 has 2. So, using 1 S7 and 1 S8 for the metric connection, the other 27 S7, with a freedom of 16, can be use to connect particles, in the other S27 can be used to connect particles from the three copies in the other S8 real space.

Note also that in the end S7 and S8 are being used both to represent a more "physical situation", being somewhat equivalent in this case.

To get more realistic, we'd go to one fiber down S7-S3-S4, where S3 should be first dealt in the context of normal twistor, and S4 as in the case of exotic smoothness, which is not solved yet, but should be seen as quantization of gravity itself (PL=Diff in 4d)

Also, notice that with 16d constrained, maybe we can build 2 copies of that E8 polytope Garrett lisi uses to represent his particles. I don't remember his construction. Would that correspond to left-right symmetry, or spin up down?

Well, at least with what I wrote above, if it can be made to make sense, you'd get 3 generations appealing to algebraic topology instead of coming with some triality, somewhat forcefully.

 

[kneemo]

Also, notice that with 16d constrained, maybe we can build 2 copies of that E8 polytope Garrett lisi uses to represent his particles. I don't remember his construction. Would that correspond to left-right symmetry, or spin up down?

Well, at least with what I wrote above, if it can be made to make sense, you'd get 3 generations appealing to algebraic topology instead of coming with some triality, somewhat forcefully.

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.

E7's 133 can be split as 133=5x26+3, where 26 is the traceless part of 27 and 3 is the SU(2) automorphism group of the quaternions. Acting on a 27, we get derivations of the form: X'=[A,X,B]+i(XoL)+j(XoM)+k(XoN), where A,B,L,M,N are traceless matrices from the five 26's and i, j and k are quaternionic units.

E8's 248 can be similarly split as 248=9x26+14, where 14 is the G2 automorphism group of the octonions. Acting on a 27 is done with derivations of the form: X'=[A,X,B]+e1(XoL)+e2(XoM)+e3(XoN)+...+e7(XoS), where the e's are octonionic units.

So if one likes, one can hit a 27 directly with E7 and E8 but after doing so, one is taken to either a 108=3x32+3x4 or a 216=3x64+3x8 dimensional matrix algebra (3x3 Hermitian matrices over quateroctonions or octooctonions). The 216 has elements with three 64's on the off-diagonal and their conjugates, with three octonion diagonals. This algebraic structure (which isn't a Jordan algebra) might be useful in a model like Lisi's which uses a 64 for a single generation. The corresponding plane is (OxO)P^2, having 64 dimensional lines. The Cayley plane OP^2 is the "real part" of this plane, with its S^8 lines.

What's interesting about such planes is that they can be given a spectral interpretation, taking one into noncommutative and nonassociative geometry. 3x3 matrix algebras, from the spectral point of view, are algebras of functions over three points. (OxO)P^2's three 128-dimensional charts correspond to three "fuzzy" points. This is an example of how to connect noncommutative geometry to the usual algebraic geometry.

Connes' model takes the particle generations as input in his model, so it's not derived from a deeper algebraic structure. By deeper structure, I mean, for instance an algebraic or geometric restriction such as the topological obstruction (due to nonassociativity) for getting any higher projective space over the octonions. For example, the cohomology ring of OP^n must be Z[h]/(h^n). However, there is no space with cohomology ring Z[x]/(x^n) (with n>3) unless x has degree 2 or 4, e.g. CP^n and HP^n are well defined for n>3 (see proof on pg. 498 here). Hence, OP^2 is maximal in the sequence of OP^n that exist. Complexifying gives (CxO)P^2, which barely gives a projective plane and maximality is inherited. E6(C) contains all the symmetries of this plane.

Ideally, it would be nice to say one has three particle generations due to an algebraic topological obstruction and that the Standard Model gauge symmetry descends from the symmetry group of a (charge) space that is maximal, as allowed by the same obstruction. If this is the case, it would explain why E6 works so well in GUT models.

 

[MTd2]

Well, don't you think the paper http://arxiv.org/pdf/1212.3182.pdf shows an algebraic obstruction? Also notice that the functors that goes to SL3(3,O) are labeled akin braid group B3, each.

Anyway, I am not really talking about GUT, but an overall unifications. It seems that the preon level, the braidings, happen at the deepest level (the total spaces, S15, S7, S3,S1, being in decreasing order of "deepness"). I am not familiar with the names.

It's like a stairway, where the total space of the above is the bundle of the above.

Also note as in I said above:

"Also, since the we are talking about topology, note also that S^7, have 28 non diffeormorphic spheres, while S8 has 2. So, using 1 S7 and 1 S8 for the metric connection, the other 27 S7, with a freedom of 16, can be use to connect particles, in the other S27 can be used to connect particles from the three copies in the other S8 real space.

Note also that in the end S7 and S8 are being used both to represent a more "physical situation", being somewhat equivalent in this case."

So, the deeper the level, the harder is to tell apart "space" and "particles". Going very down the levels, that is, S1 → S1 with fiber S0, we can see that each exotic sphere is a space which separates the trajectory for each unique kind of particle.

Also note that the structure of the algebras, beginning with sedenions, is repetitive. So, I think that we can restrict all our cases with S15 at the most. Another thing worth of note, and that I did not make clear, it is that there are several levels of obstructions, since a topology can be triangulable eq. or not, smooth eq. or not, top. eq. or not.

Note also, that the second deeper level is that one that contains S4 as a base, and is the one which triang. = smooth, but with different cases, but which there is only 1 top case.. No one knows how many different cases there are, but surely, there are, for other 4 dim cases, infinite cases with unique top, but uncoutable inf. cases of different smooth cases. Note that in the paper here:

http://arxiv.org/pdf/1006.2230

The groups of the standard model are identified with flows in an S4 sphere, in the form of gauge groups. So, the S7 of the deepest level, contains particles which that can fixed or moved in this flow (my idea now).

The rules for the movement are going to be given, by S3 total space = S3 in hopf fibration, which contains the celestial sphere S2, of which each particle can have causal relation.

I think these are ideas, but I am trying to diversify the type of topology restriction, which seems hopf fibration is specially rich, can you find anything in the literature?

BTW, the number of exotic spheres of S15 is 16256, which is 2^7(2^7 - 1). Is that something special?

 

[MTd2]

BTW, the number of exotic spheres of S15 is 16256, which is 2^7(2^7 - 1). Is that something special?

I found it something. That is SO(N/2) the group of symmetries of the worldsheet of the tadpole free bosonic string, that is SO(8128).

 

[kneemo]

Note also that in the end S7 and S8 are being used both to represent a more "physical situation", being somewhat equivalent in this case.
...
So, the S7 of the deepest level, contains particles which that can fixed or moved in this flow (my idea now).

In type I string theory, the Hopf maps were used to classify the types of branes that can end on a 9-brane (http://arxiv.org/abs/hep-th/0606216). The tachyon vev gives a map from $$S^{k-1} \rightarrow S^{2k-1}$$ for k=1,2,4,8.

In a super Yang-Mills framework, the highest Hopf map is interpreted as unit fermions ψ (in S^7) fibered over S^8, the space of lightlike lines of octonionic superparticles.

There is also the map $$M^7\hookrightarrow S^{23}\rightarrow \mathbb{OP}^2$$ where it has been argued that M^7 is not diffeomorphic to S^7, but (as an exotic 7-sphere) should be homeomorphic to S^7 given the two fibrations involving OP^2 and the Poincare conjecture.

 

[MTd2]

I think the concept of gauge, particles and symmetries not make sense at the smallest scales. For example, when you see those graphs of coupling constants getting closer and closer at high energies, I think to myself that is too precise to be true. I think even the concept of forces get fuzzier. But fuzzier not in the sense Heisenberg Uncertainty Principle or non commutative geometry, but in the sense of flux dynamics.

Let me state in this way, I won't be able to express my self very well, certainly, but in a very small, Plankian dimensions, the state of affairs should resemble a kind of sprinkle, fluidic dust, where every dust is an "event". This is similar to the pictures of events in space time in the beginning of the book "Gravitation" So these structures are a way of break between domains of continuous fluxes.

So, the the fundamental hopf fibration is S7-S3-S4, but all is induced from S4. S3 being the way event/dust see each other. When they don't see, you have a particle, that is obstruction. So, out of the 28 exotic spheres in S7, 1 is the freeway, the vacuum, the other 27, the particles. I am not sure how these particles are extended towards the total known or to be found, but, the hopf fibrations from S2-S3-S1 should induce further differentiation. But I am not sure.

As for the SYM framework, I don't know what to do with that. For example, and S8 sphere:But I'd like to point you out something. In the book "The Wild World of 4-Manifolds", by scorpan, see the diagram

SU(2) -- (/rho) -- Spin(4) -- (/rho) -- SU(2)
                        |
                    SO(4)

I am trying to follow this paper http://arxiv.org/pdf/1212.3182.pdf

                SO(8)
                   |
                   \/                                
                  f4
                   |
                  \/
SO(8) -- (f4) -- >SO(8) <-- (f4) -- SO(8)
                  |
                  \/
                 E6

Each SO(8) leads to a particle, anti particle and majorana particle.

 

[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.