What is SO(14) or SO*(14) Unification? Appeal to 't Hooft?

[marcus]

Rhody raised this question. Garrett tweeted that he had talked with 't Hooft about unification and that 't Hooft likes SO(14).
http://twitter.com/#!/garrettlisi/status/111135200192376832
So what's up with that?

I was interested enough to look around and did not find much (tell me what I missed, if I did overlook some major development with SO(14).)

But I found this by Theo Verwimp regarding SO*(14):
http://iopscience.iop.org/0305-4470/27/8/015

T Verwimp 1994 J. Phys. A: Math. Gen. 27 2773 doi:10.1088/0305-4470/27/8/015
Unification based on SO*(14) Yang-Mills theory: the gauge field Lagrangian
AbstractReferences
Gravity can be described as a gauge field theory where connection and curvature are so(2,3) valued. In the standard gauge field theory for strong and electroweak interactions, corresponding quantities take their value in the su(3)(+)su(2)(+)u(1) algebra. Therefore, unification of gravity with the other fundamental interactions is obtained by using the non-compact simple real Lie algebra so(14) contains/implies so(2,3)(+)su(3)(+)su(2)(+)u(1) as a unifying algebra. Commutation relations for so*(14) are derived in a basis adapted to this subalgebra structure. The so*(14) gauge field defined by a connection one-form on the SO*(14) principal fibre bundle unifies the fundamental interactions in particle physics, gravity included. The 91 components of the connection contain the 10 anti-de Sitter gauge fields, the 12 gauge bosons associated with SU(3)(+)SU(2)(+)U(1), two SU(3) triplets of lepto-quark bosons. An anti-de Sitter five-vector which is also an SU(2) triplet and finally two SU(3) triplets of four-spinors which are also SU(2) doublets. Although so*(14) is a Lie algebra and not a superalgebra, it is a general property of the theory that bosons and fermions can be incorporated in irreducible supermultiplets. The unified gauge field Lagrangian is defined by the Yang-Mills Weil form on the SO*(14) principal bundle.

 

[arivero]

Wilczek and Zee, (Washington U., Seattle) went for SO(18) in
http://prd.aps.org/abstract/PRD/v25/i2/p553_1

 

[Haelfix]

SO(14) is an allowed GUT group (eg it satisfies anomaly constraints), its just less interesting than SO(18).

SO(14) allows 2 generations to be unified in the 64 dimension spinor rep, whereas in SO(18) 3 families will sit in the 256. Consequently the latter is more studied. If you are going to make a big group, might as well fit all 3 families.

I don't know if this is what Garret had in mind.

 

[MTd2]

This paper has only one citation and the paper belongs to the same author. The author now works independently and is retired from http://en.wikipedia.org/wiki/Laborelec, but was associated with the university of Antwerp. Given the proximity of Belgium and Holland and age, it is likely that he knows t`Hooft personally.

The citation comes from the following paragraph, page 3 of :

http://arxiv.org/abs/1006.1614

Anti-de Sitter gauge theory for gravity

Theo Verwimp
(Submitted on 8 Jun 2010)
First a review is given of Riemann-Cartan space-time and Einstein-Cartan gravity. This gives us the necessary tools to handle the SO(2,3) Yang-Mills gauge theory for gravity. New here is the derivation of the conservation laws. Finally possible solutions of the field equations are discussed. They depend on the scale of the de Sitter length.

~The standard theory of strong and electroweak interaction is based on the Lie-algebra su(3)su(2)u(1). Using the so(2,3) algebra for gravity, there is the possibility to unify the fundamental interactions within a Yang-Mills gauge theory [6]. The smallest simple Lie-group that containes all the required subsymmetries is SO*(14). In fact:
so(2,3)su(3)su(2)u(1)so*(6)so*(8)so*(14)so*(16), and so*(16) is a maximum subalgebra of both non-compact real forms of E8. Notice that so(2,3)su(3)su(2)u(1) is the even subspace in a Z2 grading of so*(6)so*(8)." *******************

Notice that he explicitly mentions that this is a subgroup of the real compact E(8) which is what Garret uses, I think. Also, this paper displays a similar theory compared to E(8), but he uses Einstein-Cartan connection, in general form, instead of a superconnection.

 

[arivero]

Shaw and Daghighian http://prd.aps.org/abstract/PRD/v26/i7/p1798_1 argue that SO(14) is an "straightforward extension of E6". So it could be a reasonable path once you notice that E7 and E8 have problems to support chiral fermions.

 

[MTd2]

I don't know if this is what Garret had in mind.

It wasn`t Garret that sugested that, but t`Hooft. I asked Garret on facebook, since my facebook is liked to his tweeter, about why t`Hooft liked that group, but he didn`t answer.

 

[MTd2]

Shaw and Daghighian http://prd.aps.org/abstract/PRD/v26/i7/p1798_1 argue that SO(14) is an "straightforward extension of E6". So it could be a reasonable path once you notice that E7 and E8 have problems to support chiral fermions.

Perhaps t`Hooft had in mind a middleway of linking a known GUT with Garret`s idea. From marcus' first post, quoting the abstract of the paper: "Although so*(14) is a Lie algebra and not a superalgebra, it is a general property of the theory that bosons and fermions can be incorporated in irreducible supermultiplets. The unified gauge field Lagrangian is defined by the Yang-Mills Weil form on the SO*(14) principal bundle." So, it is another reason of why SO(14) is an intermediate structure between E(6) and E(8). Hmm...

 

[mitchell porter]

It's very unlikely that 't Hooft's interest in SO(14) has anything to do with using it as a "GraviGUT" group.

You can break up E8 into SO(14) x SU(2), so I can imagine making an F-theory GUT with SO(14) preons bound by a confining SU(2) interaction. But it's even more unlikely that that is what 't Hooft is thinking. :-)

 

[MTd2]

Why unlikely? I get from the context of Garret's research that they were talking about GraviGut...

 

[rhody]

Rhody raised this question. Garrett tweeted that he had talked with 't Hooft about unification and that 't Hooft likes SO(14).
http://twitter.com/#!/garrettlisi/status/111135200192376832
So what's up with that?

I was interested enough to look around and did not find much (tell me what I missed, if I did overlook some major development with SO(14).)

But I found this by Theo Verwimp regarding SO*(14):
http://iopscience.iop.org/0305-4470/27/8/015

T Verwimp 1994 J. Phys. A: Math. Gen. 27 2773 doi:10.1088/0305-4470/27/8/015
Unification based on SO*(14) Yang-Mills theory: the gauge field Lagrangian
AbstractReferences
Gravity can be described as a gauge field theory where connection and curvature are so(2,3) valued. In the standard gauge field theory for strong and electroweak interactions, corresponding quantities take their value in the su(3)(+)su(2)(+)u(1) algebra. Therefore, unification of gravity with the other fundamental interactions is obtained by using the non-compact simple real Lie algebra so(14) contains/implies so(2,3)(+)su(3)(+)su(2)(+)u(1) as a unifying algebra. Commutation relations for so*(14) are derived in a basis adapted to this subalgebra structure. The so*(14) gauge field defined by a connection one-form on the SO*(14) principal fibre bundle unifies the fundamental interactions in particle physics, gravity included. The 91 components of the connection contain the 10 anti-de Sitter gauge fields, the 12 gauge bosons associated with SU(3)(+)SU(2)(+)U(1), two SU(3) triplets of lepto-quark bosons. An anti-de Sitter five-vector which is also an SU(2) triplet and finally two SU(3) triplets of four-spinors which are also SU(2) doublets. Although so*(14) is a Lie algebra and not a superalgebra, it is a general property of the theory that bosons and fermions can be incorporated in irreducible supermultiplets. The unified gauge field Lagrangian is defined by the Yang-Mills Weil form on the SO*(14) principal bundle.

Thanks Marcus, interesting discussion, even though I am not a physicist. It is interesting to see where the trial of the research leads.

Rhody...

 

[mitchell porter]

Why unlikely? I get from the context of Garret's research that they were talking about GraviGut...

't Hooft has written many papers on gravity but he has never taken an interest in that approach.

Furthermore, I cannot shake the impression that trying to obtain quantum gravity as a quantum gauge theory just doesn't work - and so I wouldn't expect one of the very best physicists in the world to be spending time on the idea. I have to put it that way because I don't have sufficient technical grasp to reproduce the argument independently, and I won't just "argue from authority" (because then others will just cite different authorities). But it is a theme that came up repeatedly in the web debates between string theorists and loop gravity theorists, and which appears occasionally in the literature - that treating gravity as a gauge theory is a fundamentally flawed approach. This appears to be the almost unanimous opinion of people coming from a particle physics background (and therefore, people who know quantum gauge theories well), when they see gravitational physicists trying to develop quantum gravity just using gauge theory.

One of the key issues has to be dynamics. The Ashtekar variables for general relativity, which express everything in terms of a connection rather than a metric, produce a phase space that you might see in a gauge theory, but that's just kinematics. It's just a state space. What I repeatedly hear from the critics of LQG is that it's in defining the dynamics on that space where the theory goes wrong. My vague impression of LQG is that you have old-style LQG, based on Hamiltonian and other constraints, and then you have new-style LQG, based on spin foams and Feynman rules, and that old-style LQG just didn't work (when judged by the existence of a classical limit), while new-style LQG doesn't have any principle to determine the amplitudes for the infinitely many higher-order transitions that can occur in the spin-network. This is why I sometimes say that new-style LQG could work if it somehow used a known working model of quantum gravity, such as Vasiliev gravity, as a guide in determining the spin-foam Feynman rules.

It might be hard to believe that Rovelli, Smolin, Percacci, Krasnov, etc could be pursuing an inherently flawed approach for so many years. So I think there has to be something wrong with my analysis; but it may just be that I have the details wrong. I just have to notice that after at least 20 years' work, the attempt to build a "diffeomorphism-invariant gauge theory" of gravity in a top-down way doesn't seem to have worked, while the approach that might have a future (spin-foam approach) is a bottom-up one defined by the piecemeal postulation of Feynman rules.

 

[MTd2]

Mitchell, take a look at my post, #4, https://www.physicsforums.com/showpost.php?p=3495618&postcount=4, t'Hooft could be suggesting him a theory from a friend. From what Garrett says, anyway, it seems that t'Hooft favors that idea over E8, not that it is the best one in his opinion.

 

[lpetrich]

I decided to check out the Lie-algebra decompositions with my Lie-algebra software. It works with the complex forms, treating SO(5) as equivalent to SO(3,2), for instance.


SO(14) -> SO(5) * SO(9)

Vector: 14 -> (5,1) + (1,9) -- vector + vector
Adjoint: 91 -> (10,1) + (1,36) + (5,9) -- adjoints + vector*vector
Spinor: 64, 64* -> (4,16) -- spinor*spinor

Each subalgebra has only one spinor instead of 2 conjugate ones, which is what the original one has.


SO(9) -> SU(3) * SU(2) * U(1) -- demote root 3 of B4: o - o - (o) - *

Vector: 9 -> (3,1,1) + (3*,1,-1) + (1,3,0)
Adjoint: 36 -> (8,1,0) + (1,3,0) + (0,0,0) + (3,3,1) + (3*,3,-1) + (3,1,-2) + (3*,1,2)
Spinor: 16 -> (3,2,-1/2) + (3*,2,1/2) + (1,2,3/2) + (1,2,-3/2)
The spinor breaks down into conjugate parts.


This does not fit very well with the Standard Model. The elementary are in these reps:
Left-handed: (3,2,1/6) + (3*,1,-2/3) + (3*,1,1/3) + (1,2,-1/2) + (1,1,0) + (1,1,1)
Right-handed: (3*,2,-1/6) + (3,1,2/3) + (3,1,-1/3) + (1,2,1/2) + (1,1,0) + (1,1,-1)
The MSSM Higgses are in these reps: (1,2,1/2) and (1,2,-1/2)

So each spinor of SO(14) makes SM weak-isospin doublets and SM "mirror WIS doublets", but not SM WIS singlets.

If T Verwimp gets the Standard Model EF's and Higgses out of this model, how does he do it?

 

[lpetrich]

Shaw and Daghighian http://prd.aps.org/abstract/PRD/v26/i7/p1798_1 argue that SO(14) is an "straightforward extension of E6". So it could be a reasonable path once you notice that E7 and E8 have problems to support chiral fermions.

SO(14) a straightforward extension of E6? That has to be something other than being a superalgebra, since E6 cannot be a subalgebra of SO(14), as far as I can tell.

However, SO(14) breaks down into SO(4) * SO(10) or SU(2) * SU(2) * SO(10):

vector: 14 -> (2,2,1) + (1,1,10) -- vector + vector
adjoint: 91 -> (3,1,1) + (1,3,1) + (1,1,45) + (2,2,10) -- adjoints + vector*vector
spinor: 64 -> (2,0,16) + (0,2,16*) -- spinor*spinor
spinor: 64* -> (2,0,16*) + (0,2,16) -- spinor*spinor

So if SO(14) 64 is left-handed and 64* right-handed, they will produce 2 of both left-handed and right-handed SO(10) 16 and 16*. That's also rather unsatisfactory, since the Standard Model wants 3 left-handed 16's, and 3 right-handed 16*'s.

 

[arivero]

SO(14) is also interesting because it is the group of the 13-sphere. I'd be grateful if someone can review here the properties of such sphere, but let me note at least Arnold "Theorem H", that S13 is the quotient space of the four-dimensional quaternionic space by the automorphism of the algebra of the quaternions, after a factorisation by the quaternionic involution.

 

[lpetrich]

I tried searching for nonlinear automorphisms of the quaternion multiplication group, and even with a lot of effort with Mathematica, I'm not sure that I've ruled them out. But linear ones are easy to find with Mathematica -- they are 3D rotations, in the group SO(3). A quaternion-multiplication automorphism can be constructed by conjugation: element a -> g.a.g^-1 for some element g, making it an inner automorphism.

Back to the 4-quaternion result, is it true in general? For N quaternions with the automorphisms factored out, does the result reside on a (4N-2)-dimensional sphere?

Let's advance to the next-in-sequence proposed algebra, SO(16). It breaks down to SO(6) * SO(10) as
vector: 16 -> (6,1) + (1,10) -- vectors
adjoint: 120 -> (15,1) + (1,45) + (6,10) -- adjoints + vector*vector
spinor: 128 -> (4,16) + (4*,16*) -- spinor*spinor
spinor: 128' -> (4,16*) + (4*,16) -- spinor*spinor
The 128 and 128' are both real and self-conjugate; they are not conjugate to each other.
Also, SO(6) ~ SU(4)

This easily contains the Standard Model, but at the expense of having 4 generations and "mirror reps" and starting from one self-conjugate spinor for each chirality.