Garrett Lisi and Jacques Distler: debates revived — Part II

The text below was extracted from the original post by Distler over at the n-Category Café:

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[Begin of Distler's comment excerpt]

This is my summary of Lisi’s programme (at least, as best I have been able to understand it).

1. Choose an embedding of Spin(3,1)×SU(3)×SU(2)×U(1) in (noncompact) E 8. The generators of the Lie algebra, e 8, then transform as some representation of this subgroup.

* In particular, the action of the center of SL(2,ℂ)≃Spin(3,1) gives a ℤ 2 grading on e 8. The generators of e 8 which transform in spinorial representations of Spin(3,1) are “odd”; the generators which transform in tensorial representations are “even.”

* There is a similar ℤ 2 grading on the fields of any QFT: fermions are “odd” and bosons are “even.” The spin-statistics theorem requires that these be the same grading.

* While Lisi say that he doesn’t want to envoke a ℤ 2 grading, one is clearly physically required, and mathematically provided by the aforementioned embedding of Spin(3,1). He might as well say that he doesn’t want to speak in prose.

2. Use this ℤ 2 grading to build a Schreiber superconnection. The bosonic fields transform as 1-forms with values in various tensor representations of Spin(3,1); the fermionic fields transform as 0-forms in spinor representations of Spin(3,1).

* Lisi says that he isn’t using a Schreiber superconnection. Instead he’s doing ‘standard’ BRST. I can’t make head or tails of his usage of the term “BRST.” In the end, to each generator of e 8, he associates either a bosonic or a fermionic field. Spin-statistics dictates that he do this in a fashion compatible with the ℤ 2 grading. Which is to say that his fields comprise a Schreiber superconnection. Protestations to the contrary he, again, seems to be speaking in prose.

3. Use this Schreiber superconnection to build an action.

4. Quantize that action.

5. Try to extract some quasi-realistic physics from it.

Unfortunately, the construction falls down at step 1.

* Lisi wants there to be 192 odd generators, with respect to some embedding of Spin(3,1). This, of course, is impossible.

* Moreover, in his paper, Lisi embeds Spin(3,1)×SU(3)×SU(2)×U(1) via a D 4×D 4 subgroup of E 8. I classified all such embedding. They all lead (via the above prescription) to a non-chiral fermion spectrum. The closest one can come to the Standard Model spectrum of fermions is to get 1 generation and 1 anti-generation.

* This, in fact, is completely general. Any embedding SL(2,ℂ)×SU(3)×SU(2)×U(1)↪E 8 yields a nonchiral spectrum of fermions, with — at best — a generation and an anti-generation of Standard Model particles.

None of these statements is particularly hard to prove. In fact, once you know that there’s no ℤ 2 grading of e 8 with more than 128 odd generators, you know that it’s impossible to accommodate 3 generations. The best you could get is 2, but even that proves not to be possible.

That said, there is something kinda cool about the elements of the construction:

1. An embedding of Spin(d−1,1) in G gives a ℤ 2 grading on 𝔤.

2. Using the corresponding Schreiber superconnection, one naturally gets a theory with fermions, corresponding to the odd generators of 𝔤, transforming as spinors Spin(d−1,1).

It would be mildly interesting to see what sort of actions one could build with this construction.

[End of Distler's comment excerpt]
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Although that site is perfectly adequate for rigorous discussions on the matter using mathematical language, I leave here a welcome space for comments on the above intrepretation by Distler in layman terms.

I have previously posted over at n-Cat café the following:

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Do I understand correctly that the “other stuff” that sits in the odd part is considered important for Distler (it is the “anti-generation” which for him is one of the points that would make the whole approach doomed to be incorrect), whereas for Lisi the “other stuff” – whatever it is – can be worked out, eventually avoiding a possible invalidation? Is this the point of tension?

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I have also posted the following remark (slightly edited):

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If I understand it correctly, it is agreeded on both parts that Lisi’s model as a whole results in a non-chiral spectrum (net number of generations = 0). Furthermore, Distler appears to have shown that there are no decompositions of E8 allowing the inclusion of the 3 SM generations. (Does Lisi agree with the latter?)

So, I was wondering – is it really all there is to be concerning the use of E8 (or any other group, for what is worth)?

I mean, on speculative grounds, is it possible that simply using the group “as it is” is not the whole story, but actually one could gain more room for analysis or insight by seeing the group from a different “perspective”?

What I have in mind here comes from something I was reading superficially about, groups of polynomial growth and the work of Gromov. Does E8 have any relation to such groups? If so, would it be possible to prove whether the “net # gen = 0” feature shown by Distler for the E8 is preserved (or not) when considering related groups of polynomial growth (if that is possible at all), in which the group is “seen from infinity”?
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I keep these posts here for personal record.

[Part I of the present post here.]