The December issue of Scientific American is out, and it has an article by Garrett Lisi and Jim Weatherall about geometry and unification entitled A Geometric Theory of Everything. Much of the article is about the geometry of Lie groups, fiber-bundles and connections that underpins the Standard Model as well as general relativity, and it promotes the idea of searching for a unified theory that would involve embedding the SU(3)xSU(2)xU(1) of the Standard Model and the Spin(3,1) Lorentz group in a larger Lie group.

The similarities between (pseudo)-Riemannian geometry in the “vierbein” formalism where there is a local Spin(3,1) symmetry, and the Standard Model with its local symmetries makes the idea of trying to somehow unify these into a single mathematical structure quite appealing. There’s a long history of such attempts and an extensive literature, sometimes under the name of “graviGUT”s. For a recent example, see here for some recent lectures by Roberto Percacci. The Scientific American article discusses two related unification schemes of this sort, one by Nesti and Percacci that uses SO(3,11), another by Garrett that uses E_{8}. Garrett’s first article about this is here, the latest version here.

While I’m very sympathetic to the idea of trying to put these known local symmetry groups together, in a set-up close to our known formalism for quantizing theories with gauge symmetry, it still seems to me that major obstructions to this have always been and are still there, and I’m skeptical that the ideas about unification mentioned in the Scientific American article are close to success. I find it more likely that some major new ideas about the relationship between internal and space-time symmetry are still needed. But we’ll see, maybe the LHC will find new particles, new dimensions, or explain electroweak symmetry breaking, leading to a clear path forward.

For a really skeptical and hostile take on why these “graviGUT” ideas can’t work, see blog postings here and here by Jacques Distler, and an article here he wrote with Skip Garibaldi. For a recent workshop featuring Lisi, as well as many of the most active mathematicians working on representations of exceptional groups, see here. Some of the talks feature my favorite new mathematical construction, Dirac Cohomology.

One somewhat unusual aspect of Garrett’s work on all this, and of the Scientific American article, is that his discussion of Lie groups puts their maximal torus front and center, as well as the fascinating diagrams you get labeling the weights of various representations under the action of these maximal tori. He has a wonderful fun toy to play with that displays these things, which he calls the Elementary Particle Explorer. I hear that t-shirts will soon be available…

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Update**: T-shirts are available here.

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Garrett, please correct me if I misunderstand. It seems to me that what you are now doing is (using “mod” in your Cartan geometry sense):

Mod E8 / so(4,4) x so(8) to get 8x8_v + 128_s+

which by reducing the 128_s+ half-spinor of D8 into 64_s+ + 64s_s-

(is this done by looking at things in terms of D6 instead of D8?)

to get

E8 / so(4,4) + so(8) = 8x8_v + 8x8_s+ + 8x8_s-

Then you use the fact that both so(4,4) and so(8) have triality among their three 8-dimensional representations

8-vector

8-halfspinor+

8-halfspinor-

to construct a triality automorphism inside E8 among the three 64-dimensional things

8x8_v

8x8_s+

8x8_s-

so 8x8_s+ + 8x8_s- must inherit mirror-image halfspinor structure from both the so(4,4) and the so(8)

You do not identify them with fermion particles and antiparticles as would be natural from so(10) models,

but

you identify them with one fermion generation plus one antigeneration,

and you say that the antigeneration somehow is suppressed.

Does that not destroy the triality symmetry and so render it useless for further things, such as higher generations ?

Also, the 8x8_v must inherit vector structure from both the so(4,4) and the so(8)

and that vector structure includes signature,

so it seems to me to be an inconsistency because

the so(4,4) split signature (4,4)

Is not

the so(8) Euclidean signature.

Can you address these concerns?

Tony

Garrett,

“Daniel:

When I am talking about E8 triality I am talking about the triality outer automorphisms of the so(4,4) and so(8) subalgebras, and the corresponding inner automorphisms of E8.”

But what about the operators that do the outer automorphisms inside 8? Where will they find space inside E8 if everything is already busy with other fields?

Would you give me an example of that in a simpler theory?

Best,

Daniel.

Wolfgang: (Thanks Chris.) Without triality it “works” in the same sense that the SO(10) GUT works — there are many predicted particles, and the three generations are left unexplained. For the cosmological constant and renormalization, the [tex]\frac{4}{3} \Lambda e e[/tex] term might be independent but related to the [tex]\phi^2 e e[/tex] term.

Tony: Your analysis is correct. However, keep in mind that an 8D vector is triality-equivalent to an 8D chiral spinor wrt so(8) or so(4,4).

Daniel: The triality operator is an element of the E8 Lie group, and can be used to gauge-transform between the vector and positive and negative chiral spinor parts of E8. I’m afraid I don’t know of an example.

Chris said:

“i guess he just means that … you can add 2 carbon copy generations and by hand…”

If that’s what he means, then there is even easier solution:

1) Give large vector-like mass to the fermions that Garrett found “in E8” (distler-garibaldi theorem guarantees these fermions are in vector-like representation, so can be given a gauge-invariant mass).

2) Add 3 chiral generations “by hand”.

Problem solved! No need for triality or other fancy mechanism.

Garrett said:

“Without triality it “works” in the same sense that the SO(10) GUT works — there are many predicted particles, and the three generations are left unexplained.”

Unexplained, or unexistent? If you don’t allow to add fermions “by hand”, then I would say “unexistent.” If you allow to add fermions by hand (as in SO(10) GUT), then “unexplained” is more accurate, but then why big fight with distler-garibaldi, who explain how to get rid of unwanted (and now unnecessary) fermions originally from E8?

“For the cosmological constant and renormalization, the $\frac{4}{3} \Lambda e e$ term might be independent but related to the $\phi^2 e e$ term.”

What does “independent but related mean”? Are these independent terms in the action, or do they come from expanding a single term in fluctuations about VEV?

Garrett and Daniel, the simplest example of inherited triality within a larger group containing so(8) is 52-dim F4 which is made up of:

28-dim so(8) D4

8-dim vector 8_v

8-dim +halfspinor 8_s+

8-dim -half-spinor 8_s-

Pierre Ramond describes F4 (mostly from the point of view of superstring theory, but the basic math structures are the same regardless of physics point of view) in

hep-th/0112261 and hep-th/0301050. Please read the papers for a lot of very interesting details that may be relevant to E8 physics models. Here, in this comment, I will only mention a general F4 quote and a quote about fermions, bosons, and spacetime dimension.

In the earlier paper 0112261 Ramond says:

“… the triality of the … little group so(8) … links its tensor and spinor representations via a Z3 symmetry …

The exceptional group F4 is the smallest which realizes this triality explicitly …”.

In the later paper 0301050 Ramond says:

“… In four dimensions, fermions and bosons are naturally differentiated, as fermions have half-integer helicities while the boson helicities are integers. …

In d+1 spacetime dimensions, fermions transform as spin representations of the transverse little group so(d-1), while bosons are transverse tensors.

As a result … In 9+1 dimensions, the little group is so(8), with its unique triality property according to which bosons and fermions are group-theoretically equivalent …

This triality is explicit in the F4 [ containing ] so(8) decomposition …”.

In short, by Ramond’s 0301050, the nice fermion-boson correspondence in E8 physics does NOT work with a 4-dimensional spacetime, but requires a 10-dim spacetime as in superstring theory (or at least an 8-dim spacetime if you reduce it to the so(8) vector structure),

so

Garrett’s E8 over 4-dim spacetime structure will NOT inherit the nice fermion-boson correspondence and WILL violate spin-statistics

(unless its spacetime structure is enlarged to the 10-dim of string theory or at least to the 8-dim Klauza-Klein of Batakis).

Tony

PS – Note that an 8-dim Kaluza-Klein can be extended to 10-dim

by extending its 4-dim Minkowski part to a 6-dim Conformal spacetime.

“I find it more likely that some major new ideas about the relationship between internal and space-time symmetry are still needed.”

Don’t we have them already in noncommutative geometry ? (According to Alain Connes, NCG allows for breaking the chains of the Coleman-Mandula theorem). I wonder if non-commutative geometry plays any role in Lisi’s E8 model, given its success ? (E. g. the derivation of the full-fledged Lagrangian of the standard model).

It seems Garrett has lost interest in answering questions about his theory. But I’ll ask this one anyway.

Nesti-Percacci study SO(3,11) theory. Their SO(3,11) gauge fields form a subset of E8 gauge fields. And their frame field (1-form with values in the 14 of SO(3,11) ) also sits in E8.

So one might expect that their theory forms a subsector of yours. Is that correct? If so, do you agree with their analysis of that subsector? If not, where would you say their analysis is incorrect?

Wolfgang:

Yes, if one is less ambitious, then there are many ways to make things work by hand. But I am not so interested in that. I want to figure out how the three generations derive naturally from geometry, and I think triality in E8 may do that.

I suspect they might be independent terms that could arise in the perturbative expansion when doing the renormalization. But I’m just guessing. It would be really nice to also have a geometric understanding of why the action is what it is, which might shed some light on this question.

Tony:

Funny you should mention F4… that’s what I’m working on right now, as a warm up. The triality automorphisms of so(4,4), and corresponding automorphisms of the F4 Lie algebra, are tricky, but fascinating! You make an interesting point about 8D vs 4D spin-statistics; I’ll ponder that.

“But I am not so interested in that. I want to figure out how the three generations derive naturally from geometry, and I think triality in E8 may do that.”

If one is going to add fermions “by hand”, is hard to see any advantage of your “E8” theory over Nesti-Percacci SO(3,11). So, yes, your triality idea seems to be “last best hope” for E8 theory.

“I suspect they might be independent terms that could arise in the perturbative expansion when doing the renormalization.”

In your paper, they come from expanding a single term in the action about VEV, so it is hard to see how that could be true.

On the other hand, Nesti-Percacci (this is why I have been looking at their paper) have a very different analysis of this subsector of your theory.

Could it be that their analysis is the correct one?