An Exceptionally Technical Discussion of AESToE

garrett

Tony,
If we rotate the E8 root system until the vertical axis is weak hypercharge, and rotate out the other axes horizontally to separate the roots a bit, it looks like this:
http://deferentialgeometry.org/blog/hyper.jpg
This makes it visible exactly what is meant by "the charge assignments only work correctly for the first generation," with the other two (smaller triangles) related by triality.

Tony Smith

When I count the 13-grading from that image I get:

5 + 6 + 15 + 20 + 30 + 30 + 26 + 30 + 30 + 20 + 15 + 6 + 5

which only add to 238, so I must be miscounting two of them somewhere ???

Anyhow, modulo my error of two, the even graded structure would be

5 + 15 + 30 + 26 + 30 + 15 + 5 = 126-dimensional
(if the missing 2 are even, then 128-dimensional)

and the odd graded structure would be

6 + 20 + 30 + 30 + 20 + 6 = 112-dimensional

so

it seems to me that the odd gradings correspond to the 112 root vectors of the adjoint Spin(16) (120 generators - 8 Cartan subalgebra generators = 112)

and

that the even grading probably have the two I miscounted and are the 128 root vectors corresponding to the half-Spinor of Spin(16).

What bothers me about that is that the fermionic spinor-type things are in the even grading and the bosonic vector/bivector adjoint-type things are in the odd grading,

whereas in Thomas Larsson's 7-grading

8 + 28* + 56 + (sl(8) + 1) + 56* + 28 + 8* = 8 + 28 + 56 + 64 + 56 + 28 + 8

the even grade part is
28 + 64 + 28 = 112 dimensional corresponding to the root vectors of adjoint Spin(16) which seems to represent bosonic vector/bivector stuff
while
the odd grade part is
8 + 56 + 56 + 8 = 128-dimensional corresponding to half-spinor of Spin(16) which seems to represent fermionic spinor-type stuff.

Do you have any thoughts about that ?

Tony Smith

garrett

Tony,
Two gluons overlap. There are many such gradings of E8 -- there may be the same kind of 13 grading along a different direction that gives the even/odd 120/128 split you're after.

sambacisse

Hi Garrett,

I would like to have your comments on Distler's second post, where he shows that the embedding you propose in E8 is not chiral, contrary to what you and others have claimed, and his proposal (in an appendix to his first post) that perhaps all embeddings in E8 are not chiral, hence rendering your idea of embedding the Standard Model and gravity in E8 rather difficult to realize, to say the least. There has been quite a lot of discussions about this on Cosmic Variance (and it's still ongoing: see comments 90 up to now on this post).

I personally feel Distler's argument is fundamental, relatively easy to follow, and seems to be correct, at least up to the level of my knowledge (perhaps I'm making a mistake). Lee has been trying to address it on Cosmic Variance, but hasn't succeeded in finding a mistake or a loophole in it yet. Do you have anything to say about it?

Thanks a lot! :-)

moveon

Please! To my knowledge the only algebras that contain both bosonic and fermionic generators are superalgebras, and E8 is not one of them. How can the commutation relations close into E8?

Tony Smith

moveon said "... the only algebras that contain both bosonic and fermionic generators are superalgebras ...".

No, that is not true.
As Pierre Ramond in hep-th/0112261 said "... exceptional algebras relate tensor and spinor representations of their orthogonal subgroups ...",
and
the exceptional algebra 248-dim E8 contains
120 generators corresponding to the tensor/bosonic part 120-dim adjoint Spin(16)
and
128 generators corresponding to the spinor/fermionic part 128-dim half-spinor Spin(16)
and
their commutation relations do close into E8.

However, Pierre Ramond went on to say in that paper:
"... Spin_Statistics requires them [ the adjoint/bosonic and half-spinor/fermionic ] to be treated differently ...",
so
any model you build with E8 must somehow treat them differently.

For example, you might just construct a Lagrangian into which you put
the 128 half-spinor fermionic generators into a fermion term
and
8 of the 120 bosonic generators into a spacetime base manifold term
and
120-8 = 112 of the 120 bosonic generators into a gauge boson curvature term.

Then you might have disagreement as to how natural (or ad hoc) is such an assignment of parts of E8 to terms in a Lagrangian,
but all should agree that you have "treat[ed] them differently" as required by Spin-Statistics.

However, in Garrett's 13-grading decomposition of the 240 root vectors of E8

5 + 6 + 15 + 20 + 30 + 30 + 28 + 30 + 30 + 20 + 15 + 6 + 5

some of the graded parts contain both bosonic terms and fermionic terms,
for example the central 28 has both circles (bosons) and triangles (leptons and quarks),
which has led Thomas Larsson to complain (on Cosmic Variance):
"... both fermions and bosons belong to the same E8 multiplet. This is surely plain wrong. ...".

I think that the point of Thomas Larsson is that
the model must treat the fermions and bosons differently to satisfy Spin-Statistics
so
the fermionic generators must be put into some part of the model where the bosonic generators are not put
so
if you decompose the generators into multiplets some of which contain both fermionic and bosonic generators (as in Garrett's 13-grading decomposition) then you are not respecting your multiplets when you, from a given multiplet, put some of them into a fermionic part of the model and some of them into a bosonic part of the model.

This is not merely an objection of ad hoc assignments of generators to parts of the model,
it is an objection that the assignments do not respect the chosen decomposition into multiplets.

Tony Smith

PS - It is possible to choose a decomposition that does keep the bosonic and fermionic generators separate, the simplest being 64 + 120 + 64
where the 120 is bosonic and the 64+64 = 128 is fermionic.

moveon

Oh yes, this is of course very well known since ages. But those tensor and spinor rep generators are all bosonic, and close into the usual E8 commutator relations. My point is, apparently still not appreciated, that if some of the generators are made fermionic (as it happens for superalgebras), then they cannot produce the E8 commutation relations (and jacobi identities etc) any more. The opposite seems to be claimed here all over, so I'd like to see, how. Please prove this by writing them down!

And if the E8 commutation relations are not there, there is no E8 to talk about. There is "somewhat" more to E8 than a drawing of the projection of its polytope...

John G

This from Tony's website might be good to look at (I'm sure Tony can say more if needed):

http://www.valdostamuseum.org/hamsmi...eStdModel.html

Thomas Larsson

One needs to distinguish between spin and statistics.

There are two types of statistics: fermions, which anticommute and obey Pauli's exclusion principle, and bosons, which commute.

There are also two types of spin: spinors, which have half-integer spin, and tensors and vectors, which have integer spin.

The spin-statistics theorem asserts that physical fermions always have half-integer spin and physical bosons have integer spin. But this is non-trivial and surprisingly difficult to prove. In contrast, BRST ghosts are fermions with integer spin, and therefore unphysical. Physical and unphysical fermions are not the same.

What is quite easy to prove is that statistics is conserved, i.e.

[boson, boson] = boson
[boson, fermion] = fermion
{fermion, fermion} = boson.

People like Lee, Peter and Bee know this, of course, and it must be obvious that putting both bosons and fermions into the same E8 multiplet violates this fundamental principle. That they don't emphasize this simple fact but instead complain about manners is something that I find surprising and quite disappointing.

Tony Smith

Here is what I hope is a concrete example of what I think that Thomas Larsson is saying (please feel free to correct my errors):

If you were to (not what Garrett did) make a physics model by decomposing E8 according to its e17 5-grading:

g(-2) = 14-dim physically being spacetime transformations
g(-1) = 64-dim physically being fermion antiparticles
g(0) = so(7,7)+R = 92-dim physically being gauge bosons
g(+1) = 64-dim physically being fermion particles
g(+2) = 14-dim physically being spacetime transformations

then that would be consistent with spin-statistics because
the products fermion(-1) times fermion(+1) would be gauge bosons(-1+1=0)
the products of gauge bosons(0) times gauge bosons(0) would be gauge bosons(0+0=0)
the products of gauge bosons(0) times fermions(-1) would be fermions(0-1=-1)
the products of gauge bosons(0) times fermions(+1) would be fermions(0+1=+1)

etc

The point is that if you have fermions and bosons mixed up together in the same part of the graded decomposition, you do not get good spin-statistics,
but
it is possible to decompose in a way that you do get good spin-statistics
and
that is something that should be taken into account in model-building.

Tony Smith

PS - Sorry for burying stuff like fermion(-1) times fermion(-1) giving spacetime(-2) into an "etc" (sort of like spinor x spinor = vector) but in this comment I am just trying to make a point and not build a complete model here.

garrett

sambacisse,
The issue is a bit more complicated than it appears because of how the real representations are mixed together in exceptional groups into complex representation spaces, relying on an inherent complex structure. This sort of thing is described halfway through John Baez's TWF253 for the case of E6. When describing so(3,1) reps in terms of sl(2,c) this is further complicated, and when swapping in conjugated anti-fermions it's more complicated still -- because one has to be clear in each step which complex structure one is conjugating with respect to. I thought I had this figured out several years ago, but I don't like to make statements about complicated things without having slowly worked through them in detail. So I've stayed out of the arguments. Of course, I can say that the worst case scenario is that one might have to use a complex E8.

moveon,
Tony addressed this a bit, and I'll try to summarize the specific case in the paper. The E8 Lie algebra may be naturally decomposed into a D4+D4 subalgebra, and everything else. In terms of the number of elements, this decomposition is:
(28+28)+64+64+64
which I don't consider a "grading," but it relates to gradings. The important thing is the Lie brackets. If we label the D4+D4 elements "bosons," and the rest "fermions," the brackets are as Thomas Larsson has helpfully described. Now, if the E8 symmetry is broken such that the "fermion" part of the Lie algebra is pure gauge, then that part of the connection may be replaced by Lie algebra valued Grassmann fields. We end up with a D4+D4 valued connection 1-form field, [tex]\underline{H}_1+\underline{H}_2[/tex], and three other fields, the first of which is the first generation fermions, [tex]\Psi[/tex], which are Grassmann valued E8 Lie algebra elements. Because of the structure of E8, the Lie brackets between these give the fundamental action:
[tex][\underline{H}_1+\underline{H}_2,\Psi] = \underline{H}_1 \Psi - \Psi \underline{H}_2[/tex]
The brackets between two [tex]\Psi[/tex]'s are in D4+D4, but these terms vanish in the action. Notice that there is no symmetry here relating the fermions to bosons. That symmetry was destroyed when we broke the E8 symmetry by adding the terms we did to the action. I did that by hand in my paper, and Lee talks about how that can happen dynamically in his. There is a cute trick in the BRST literature whereby these objects can be formally added in a generalized connection:
[tex]\underline{H}_1+\underline{H}_2+\Psi[/tex]
Since I like cute math tricks, I used it -- allowing all fields to be written as parts of this "superconnection," with the dynamics coming from its generalized curvature.

Tony Smith

So, it seems to me that:

1 - Garrett has shown that his physical identifications of E8 generators are consistent with spin-statistics;

2 - Garrett is not claiming that any BRST ghost-fermions-with-integer-spin are physical,
but
is just using one of the technical "math tricks" from BRST literature in order to construct his "superconnection" containing both gauge boson curvature terms and curvature terms derived from spinor/fermions;

3 - Garrett has explicitly broken full E8 symmetry so that it is irrelevant whether or not Garrett's physics stuff (whether it is Pati-Salam or not) fits inside E8,
so that Jacques Distler's arguments about it not fitting inside E8 are irrelevant.

4 - However, just as Jacques Distler's comments were useful in seeing that E8(8) might be more useful than E8(-24),
it may be that his comments about Pati-Salam vs. the Standard Model might also be useful indicators that Garrett's model should perhaps be put directly in terms of the minimal Standard Model than in terms of Pati-Salam.

Tony Smith

PS - If I had to guess, I would guess that Garrett used Pati-Salam because he thought that it was an established particle physics model, and its use would make his E8 model more acceptable to conventional physicists.
Since it has turned out otherwise, maybe just using the plain vanilla minimal Standard Model plus MacDowell-Mansouri gravity might be a way to go.

PPS - It is unfortunate that a "food-fight" atmosphere has obscured much of the sensible physics in discussions on some parts of the web, and I would like to say that I very much appreciate the moderate (in more meanings than one) atmosphere here on Physics Forums. Such moderation-in-climate does not come about without moderation-in-the-other-sense, and that takes effort, which I appreciate very much.

moveon

Garret,

OK so let me translate this in my language.. your superconnection does not take values in the Lie algebra of E8 as some generators are fermionic (they square to zero, eg).
Therefore the curvature, or field strength does not take values in all of E8, but in D4+D4 only. The full commutation relations of E8 are therefore not non-trivially realized. So in what sense then does E8 play a role? It seems that the purpose of your E8 is to organize, as a bookkeeping device, the fermionic part of the spectrum in terms of the coset E8/(D4+D4), as far as their quantum numbers are concerned.

This is linked to the "breaking" of E8. There are different notions of a symmetry being broken. Usually in particle physics a symmetry is spontaneously broken, which means it is "still there" albeit non-linearly realized. It reflects itself in terms of Ward identities of the low energy effective theory.
There is an energy scale above which the symmetry is restored and the theory is in an "unbroken phase". So one may speak of an "underlying" symmetry.

In contrast, you write a theory where there is no E8 symmetry to begin with (ie, its commutation relations are not fully realized) and there is no energy scale above which it is restored. So calling it "breaking" may be misleading...it is just not there. It is a bit like saying the standard model has monster group symmetry, although most of it is broken.



To 1- .... they are not the generators of E8. They are the generators of some superalgebra whose bosonic piece is D4+D4.


To 2- .... it seems to me that the claim was that that the standard model spectrum can be organized in terms of E8/(D4+D4) (rather, of the relevant non-compact real forms). That has been shown by Distler not to be the case.


I would thus advise to look for superalgebras instead of E8. There exist even exceptional ones; they have been classified by Katz, and a useful ref is hep-th/9607161. Choosing one with D4+D4 as its bosonic piece (and a suitable real form) may be more successful. Also, superalgebras are consistent with Coleman-Mandula (that's why supergravity works).

mitchell porter

moveon, thanks for that; a very very illuminating comment.

John G

The full E8 symmetry would seem to be E8/D8, I personally am more familiar with E8/E7xSU(2) and so on down the A-D-E series but maybe one can do something with E8/D8. The D4+D4 part seems after symmetry breaking so one should not expect any E8/(D4+D4) physics.

Tony Smith

moveon "... advise[s] to look for superalgebras instead of E8. There exist even exceptional ones; they have been classified by Katz, and a useful ref is hep-th/9607161. Choosing one with D4+D4 as its bosonic piece (and a suitable real form) may be more successful. ...".

hep-th9607161 is indeed a nice reference. Thanks for it. However (please correct me where I am wrong) when I look at it for exceptional Lie superalgebras, I see only three:
F(4) which is 40-dimensional;
G(3) which is 31-dimensional; and
D(2,1;a) which is 17-dimensional,
so
none of them are large enough to contain 28+28=56-dimensional D4+D4.

From Table III on page 13, it seems that the only one with a Dm bosonic part is
D(m,n) which has bosonic part Dm (+) Cn
which the describe on page 37 as being "... osp(2m|2n) ...[ which ]... has as even [ bosonic ] part the Lie algebra so(2m) (+) sp(2n) ...".

osp(2m|2n) is the basis for supergravity and, in his book Supersymmetry (Cambridge 1986 at page 113), Peter G. O. Freund says "... In extended supergravity of type N the largest internal nonabelian gauge group is O(N), corresponding to a gauged osp(N|4) ... The largest nonabelian gauge symmetry is O(8) ...".

So, since the sp(4) in Freund's notation, which is sp(2) in some other notations accounts for gravity and therefore for one of the D4,
you have the O(8) for the other D4,
so
it seems to me that N=8 supergravity is the only superalgebra based model that could reasonably be seen as fitting something like Garrett's D4 + D4 model-making scheme.

As Freund discusses in some detail in chapter 23, N = 8 supergravity and concludes "... all this makes the ultimate absence of a compelling and realistic spectrum all the more frustrating. ...".

In chapter 26, Freund discusses the related 11-dimensonal supergravity, but as far as I know there has been no satisfactory realistic 11-dim supergravity or N=8 supergravity model.

Therefore, to work with D4 + D4 it seems to me that you must abandon superalgebras because they either do not have it or have not been shown to work (despite much effort),
and that ordinary exceptional Lie algebras, which have both bosonic and spinor parts, are a useful place to look for building models,
and
that Garrett has done a good job of seeing how the root vector generators of E8 can be assigned physically realistic roles in constructing a useful physics model, and therefore is worth a substantial amount of research effort (comparable to that spent so far on supergravity).

Tony Smith

garrett

moveon,
Your translation is interesting, but all fields in the paper are valued in the Lie algebra of E8. I'm not yet certain that the first generation doesn't work in real E8, because of the unusual complex structure employed -- but even if it doesn't work, complex E8 would.