E8 andCl(8) 

Frank Dodd (Tony) Smith, Jr. - November 2007 -www.valdostamuseum.org/hamsmith/ 

Abstract

Garrett Lisi, in a 6 November 2007 paper 07110770 on the CornellarXiv in hep-th (as of 16 November 2007), proposed "An ExceptionallySimple Theory of Everything" based on the struucture of the248-dimensional exceptional Lie algebra E8. This paper compares hisgeometric structure and physical interpretation with that of myphysics model based on the 256-dimensional Cl(8) Clifford algebra.This paper has three sections:

In this paper I am, for simplicity of exposition, often ignoringsignature (for example, writing Cl(8) for Cl(1,7), etc).

Introduction to Garrett Lisi's E8physics model

Garrett Lisi, in a 6 November 2007 paper 07110770 on the CornellarXiv in hep-th (as of 16 November 2007), said:

"... All fields of the standard model and gravity are unified asan E8 principal bundle connection. ... three generations of fermions[are] related by triality. The interactions and dynamics ...are described by the curvature and action over a four dimensionalbase manifold. ...

The weights of ... 222 elements - corresponding to the quantumnumbers of all gravitational and standard model fields - exactlymatch 222 roots out of the 240 of the largest simple exceptional Liegroup, E8. ... The E8 root system ... with elementary particlesymbols assigned to their associated roots according to Table 9, isshown ...with lines drawn between triality partners ...

 

 

...[In the] plot...[shown below]... of E8 showinga rotation ...[ I have rotated the image about 150 degrees fromGarrett's figure ]...

 

 

[ Note that in the above rotated image someof the 240 E8 vertices are projected to the same point:

... we can find subalgebras by starting with the root system of aLie algebra, rotating it until multiple roots match up on parallellines, and collapsing the root system along these lines to anembedded space of lower dimension - a projection ... the centralcluster of 72 roots ... is the E6 root system, which acts on each ofthe three colored and anti-colored 27 element clusters of theexceptional Jordan algebra ...[ J3(O) ]... the e6 subalgebraof e8 reveals how the fermions and anti-fermions ... are combined...

[A] specific triality matrix [is] chosen to rotatebetween the fermion generations ... It is conceivable that there is amore complicated way of assigning three generations of fermions tothe E8 roots to get standard model quantum numbers for all threegenerations without triality equivalence. ...

The action for everything, [is] chosen by hand to be inagreement with the standard model ...[and to be]... over afour dimensional base manifold. ... All known fields are parts of anE8 principal bundle connection in agreement with the Pati-Salam

SU(2)_L x SU(2)_R x SU(4)

grand unified theory, with a handfull of new fields suggested bythe structure of E8 ...[and]... with gravity included via theMacDowell-Mansouri technique. The theory has no free parameters. Thecoupling constants are unified at high energy, and the cosmologicalconstant and masses arise from the vacuum expectation values of thevarious Higgs fields ...".

 

Geometric Structure comparison withCl(8) physics model

 

 To compare the E8 and Cl(8) physics models, begin with thegraded structure of 256-dimensional Cl(8) and see how each Liealgebra of the series D4 - D5 - E6 - E7 - E8 can be constructed fromparts of Cl(8), noting that

D4 is 28-dimensional rank 4 with a24-vertex root polytope;

D5 is 45-dimensional rank 5 with a40-vertex root polytope;

E6 is 78-dimensional rank 6 with a72-vertex root polytope;

E7 is 133-dimensional rank 7 with a126-vertex root polytope;

E8 is 248-dimensional rank 8 with a240-vertex root polytope.

  

Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1=

= 1 +8 + (24+4)+(27+1+27+1)+(32+27+3+8)+(27+1+27+1)+ (27+1) +8 +1

 

D4 =(24+4)

D5 =D4 +1 +8 +8

E6 =D5 + 32 +1

E7 =E6 +(27+1) +27

E8 =E7 + (27+1)+ (27+3) +(1+27+1) +(27+1)

256-dim Cl(8) - 248-dim E8 =8

 In the above, underline denotes Cartansubalgebra elements that are not represented by root vectors.

 

 

Note that in the above image some of the 240 E8vertices are projected to the same point:

Note further that I am pretty sure of thecolor-identification of the points in the above figure, except forpossible misplacements within the central 72-vertex E6 root vectorpolytope:

 Since an important aspect of both the E8 model and the Cl(8)model is the representation of fermions by spinor-typestructures,

such as by 8-dimensional Spin(8) +half-spinors and 8-dimensionalSpin(8) -half-spinors in the Cl(8) model

and by 128-dimensional Spin(16) half-spinors in the E8 model(based on the identification of 248-dimensional E8 as the sum of the120-dimensional Spin(16) adjoint plus a 128-dimensional Spin(16)half-spinor space)

it is useful to see how spinor-type structures appear in theabove-described structure of E8 by enclosing theire enumeration in[ brackets ]:

Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1=

= 1 +8 + (24+4)+(27+1+27+1)+(32+27+3+8)+(27+1+27+1)+ (27+1) +8 +1

 

D4 =(24+4)

D5 =D4 +1 +8 +8

E6 =D5 +[32] +1

E7 =E6 +([16]+8+3+1)+[16]+8+3

E8 =E7 +([16]+8+3+1)+([16]+8+3+3)+(1+[16]+8+3+1)+([16]+8+3+1)

256-dim Cl(8) - 248-dim E8 = 8

Note that each of the [16] above live in the27-dimensional exceptional Jordan algebra J3(O), which is the algebraof 3x3 Hermitian Octonion matrices

Re(O1) O4 O5 O4* Re(O2) O6 O5* O6* Re(O3)

where Re(Oi) is the real part (just a real number) of the i_thoctonion for i = 1,2,3 and O4, O5, and O6 are full octonions, with O4and O5 corresponding to the 8-dimensional Spin(8) +half-spinors and8-dimensional Spin(8) -half-spinors, and O6 corresponding to the8-dimensional Spin(8) vectors.

The total of [ bracketed ] spinor-type elements = 32+ 6x16 = 8x16 = 128 = dimension of a Spin(16) half-spinor.

 

 There are 7 independent E8 Root Vector Polytopesand so 7 independent E8 Lie algebras

 

Given a fixed 8-dimensional Root Vector Space with Octonionicbasis { 1, i, j, k, e, ie, je, ke }, the coordinates of the 240vertices of an E8 Root Vector Polytope can be written as:

16 related to the Octonionic Basis Vectors and to the 120-8 = 112adjoint Spin(16) Root Vectors

±1, ±i, ±j, ±k, ±e, ±ie, ±je,±ke,

96 containing both a and ae (where a = i,j,k) related to the 120-8= 112 adjoint Spin(16) Root Vectors 

(±1 ±ke ±e ±k)/2 (±i ±j ±ie±je)/2

(±1 ±je ±j ±e)/2 (±ie ±ke ±k±i)/2

(±1 ±e ±ie ±i)/2 (±ke ±k ±je±j)/2

128 containing all i,j,k types (some possibly multiplied by e)related to 128 half-spinors of Spin(16)

(±1 ±ie ±je ±ke)/2 (±e ±i ±j±k)/2

(±1 ±k ±i ±je)/2 (±j ±ie ±ke±e)/2

(±1 ±i ±ke ±j)/2 (±k ±je ±e±ie)/2

(±1 ±j ±k ±ie)/2 (±je ±e ±i±ke)/2

 

They generate a lattice (an E8 Lattice) that spans the8-dimensional Root Vector Space and is consistent with OctonionMultiplication, and they are the coordinates of 240 nearest neighborsto the origin in that E8 Lattice.

None of those 240 nearest neighbors are of the form(±1±i±j±k±e±ie±je±ke)/2(conventional light-cone form). They all appear in the next layer outfrom the origin of that E8 Lattice, at radius sqrt 2, which layercontains in all 2160 vertices.

The E8 set out above has 16 vertices of the form ±a and 224vertices of the general form (±a ±b ±c ±d)/2.

Since there are ( 8 choose 4 ) = 8x7x6x5 / 1x2x3x4 = 70 sets of 4components and 2^4 = 16 sets of signs, there are 70 x 16 = 1,120possible 4-component vertices,

only 224 of which are used in the E8 set out above.

The other 1120 - 224 = 896 vertices are found in other independentRoot Vector Polytopes of type E8.

As it turns out, there are 7 independent Root Vector Polytopes /Lattices of type E8, denoted E8_1, E8_2, E8_3, E8_4, E8_5, E8_6,E8_7. Some of them have vertices in commmon, but they are alldistinct.

For more details, see my web pages at www.valdostamuseum.org/hamsmith/E8.htmland www.valdostamuseum.org/hamsmith/E6E7E8clif.html

All of the 7 independent Root Vector Polytope Lie algebras E8_icorrespond to E8 Lattices consistent with Octonion Multiplication,and the the 7 Lie algebras / Lattices / Root Vector Polytopes E8_iare related to each other as the 7 Octonion imaginariesi,j,k,e,ie,je,ke.

There are 480 different Octonion Multiplication rules, but all 7!permutations of the 7 Octonion imaginaries do not give distinctOctonion Multiplications since Octonion Multiplication has symmetrygroup PSL(2,7) = of order 168.

For more details, see my web page at www.valdostamuseum.org/hamsmith/480op.html

The 240 - 72 =168 vertices outside the central E6

[ Notethat in the above image the central point of each of the 6 clusterscorresponding to the 27-dimensional exceptional Jordan algebra J3(O)is a projection of 3 vertices are projected, so that each clusterrepresents 27 vertices.]

 

correspond to 6 copies of 27-dimensional J3(O) plus a singlevertex.

In the Cl(8) model, each of the 6 copies of J3(O)+1 is a link tothe other 6 E8_i Lie algebras, and the fact that the 6 copies ofJ3(O)+1 have 168 elements may be related to the symmetry of OctonionMultiplication of the 7 Octonion imaginaries, corresponding to the 7Lie algebras E8_i,

and

the physical use of the 7 independent E8_i in the Cl(8) model isin constructing D8 branes in a Bosonic String Theory with Fermionscoming from Orbifolding, but without conventional supersymmetry ( seeCERN preprint CERN-CDS-EXT-2004-031 and my web page at www.valdostamuseum.org/hamsmith/stringbraneStdModel.html).

The 168-element symmetry is also related to the KleinQuartic .

 



Physical Interpretations


Garrett Lisi describes the physics of the E8 model as

"... The action for everything, [is] chosen by hand to bein agreement with the standard model ...


The Cl(8) model Lagrangian is entirely made up of components fromthe Cl(8) Cifford algebra with graded structure

Cl(8) = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256 = 16x16= (8+8)(8+8)

Dimensional reduction of physical spacetime to 4 dimensions comesfrom freezing out at low (our present experimental) energies of apreferred quaternionic subspace of the octonionic 8-dimensionalspacetime, and the process produces:


How do the E8 and Cl(8) models differ ?

Base Manifold

The E8 base manifold is an ad hoc 4-dimensional addition to248-dimensional E8,

while

the Cl(8) base manifold is a natural part (8-dimensional vectorspace) of 256-dimensional Cl(8).

One possibility might be to let the E8 base manifold go to 8dimensions, as in the 4+4 = 8 dimensional Kaluza-Klein Spacetimedescribed by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105,which was successful with respect to the Standard Model gauge bosonsand was not generally accepted because of difficulties of addingfermions. Perhaps the boson-fermion structure of E8 could solve thosedifficulties of the Batakis model. By combining the 248 E8 dimensionswith the 8 Batakis Kaluza-Klein spacetime dimensions, you could formthe 256 dimensional Cl(8) Clifford algebra, whose structure could beused as a guide for naturally constructing a Lagrangian that couldreproduce the successes of the Standard Model and Gravity based onMacDowell-Mansouri.

A problem might be that the 8 extra-from-E8 dimensions of Cl(8)seem to be structurally part of the 70-dimensional 4-vector part ofCl(8) and not in the 8-dimensional vector part of Cl(8).

Fermion Generations

The triality generation mechanism of the E8 model is substantiallydifferent from the dimensional reduction mechanism of the Cl(8)model, but Garrett Lisi does say that "... It is conceivable thatthere is a more complicated way [ than the E8 model triality] of assigning three generations of fermions ...".

Use of Triality

The Cl(8) model uses Spin(8) Triality to establish a subtlesupersymmetry between Fermions (8-dimensional +half-spinors and8-dimensional -half-spinors) and Bosons (bivectors constructed from8-dimensional vectors),

while

The E8 model uses Triality to get 3 generations of Fermions.

Relation to String Theory

The Cl(8) model uses the 7 independent E8_i in constructing D8branes in a Bosonic String Theory with Fermions coming fromOrbifolding, but without conventional supersymmetry ( see CERNpreprint CERN-CDS-EXT-2004-031 and my web page at www.valdostamuseum.org/hamsmith/stringbraneStdModel.html),

while Garrett Lisi said in a comment on Bee's blog entry about hisE8 model "... LQE8 theory or whatever one wants to call it would be acompetitive ToE, vying with strings ...".

 

Calculation of Particle Masses, Force Strengths,etc

The Cl(8) model has a highly developed system of calculatingparticle masses and force strengths, etc.  For an outline, seemy web page at www.valdostamuseum.org/hamsmith/July2006Update.html#octuninflation.

The Cl(8) model also permits calculation of

Details are on my web site at www.valdostamuseum.org/hamsmith/

 

 

How are the E8 and Cl(8) models similar ?

Relation to the Standard Model

Both the E8 and Cl(8) models produce the non-supersymmetricStandard Model.

Gravity

Both the E8 and Cl(8) models use the MacDowell-Mansouri mechanismto produce Gravity.

Fermions and Bosons

Both the E8 and Cl(8) models use spinor-type representations ofSpin groups that are naturally within the defining structures of E8and Cl(8).

The Spin(16) of E8 and the Spin(8) of Cl(8) are related asfollows: consider the graded structure of Cl(8)

1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256

and that by periodicity Cl(16) = Cl(8) (x) Cl(8) so that the 120SO(16) bivectors of Cl(16) should be made up of three parts:

to get the 28+28+64 = 120-dim SO(16) bivector algebra of Cl(16) =Cl(8) (x) Cl(8).

Relation to Spin Foam

Garrett Lisi said in comments on Bee's blog entry about his E8model, and in his paper 07110770, "... The primary LQG program is asearch for a successful method for describing a backgroundindependent, quantum theory of a connection -- the connection ofgravity. What this E8 theory does is include all fields in a largeconnection. So, combined, LQE8 theory or whatever one wants to callit would be a competitive ToE, vying with strings ... ...

e8 = f4 + g2 + 26 x 7

... The 26 is the traceless [ part of the ] exceptionalJordan algebra ... the central cluster of 72 roots in ...[ the E8root vector polytope ]... is the E6 root system, which acts oneach of the three colored and anti-colored 27 element clusters of theexceptional Jordan algebra ...

In a spin network the edges carry representation labels forgravity and gauge fields, while the vertices (nodes) are labeled bynumbers for the fermions. Since, in a sense, all the fields in thistheory I [ Garrett Lisi ] presented are E8 gauge fields,including the fermions, this would mean that all the representationlabels would be on the edges ...".

Both the E8 model and the Cl(8) model contain structures of the27-dimensional exceptional Jordan algebra J3(O) and, consequently,its 26-dimensional traceless part J3(O)o.

John Baez said on sci.physics.research on 16 Jan 2001 ( here Ichange some notation ): "... The quotient of Lie algebras e6 / f4 isa vector space that can be naturally identified with J3(O)o. That'sreally cool! But the quotient of Lie groups E6 / F4 is what mattersfor the spin foam models, and this is a bit "curvier" - it has anatural metric that's not flat. They are closely related, however:e6/f4 can be viewed as a tangent space of E6 / F4. ...". John Baezthen quoted my earlier post saying that I "... would like very muchto be told how such a construction goes, because in my opinion such aJ3(O)o spin foam model should lead to, not just quantum gravity, buta Theory Of Everything. ..." and John Baez replied to that saying:"... Shhh! That's supposed to be secret. :-) Yes, of course somethinglike this is my goal, but I'm not eager to count my chickens beforethey are hatched, nor take my ideas to market while they're stillhalf-baked. ...". So, although unification of LQC spin foams with E8containing copies of 26-dimensional J3(O)o may be (as John Baezindicated) not yet fully baked, it is an idea that knowledgeablepeople such as John Baez have as a goal.

To use the E8 model for a spin foam, it might be useful to ignorethe ad hoc 4-dimensional base manifold and to use only E8 structureby

and seeing what dimensionality emerges from such a network.

To use the Cl(8) model for a spin foam, basic structures might beE6, Spin(8) adjoint, vector, and half-spinors, and J3(O) with E8lattices related to D8 branes, along with 4-dimensional FeynmanCheckerboard structures ( see my web page at www.valdostamuseum.org/hamsmith/USGRFckb.html).

The resulting spin foams may turn out to be similar, sinceSpin(16) and Spin(8) are related through the real Cliffordperiodicity tensor product factorization

Cl(16) = Cl(8) (x) Cl(8)

 

 


Coleman-Mandula

Garrett Lisi said (in comments to Bee's blog entry about the E8model) that a "... condition ...[of]... the Coleman-Mandulatheorem ... is that there needs to be a Poincare' subgroup. There isno Poincare' subgroup in this E8 theory. ... The G = E8 I[Garrett Lisi] am using does not contain a subgroup locallyisomorphic to the Poincare group, it contains the subgroup SO(4,1) --the symmetry group of deSitter spacetime. ...this theorem does notapply in this case. ...".

However,

Steven Weinberg showed at pages 12-22 of his book The QuantumTheory of Fields, Vol. III (Cambridge 2000) that Coleman-Mandula isnot restricted to the Poincare Group, but extends to the ConformalGroup as well.

Since the Conformal Group SO(4,2) contains Garrett's de SitterSO(4,1) as a subgroup, it seems to me that it is incorrect to claimthat use of deSitter SO(4,1) means that Coleman-Mandula "... does notapply ..." to the E8 model.

However, it may be that an E8 model could be (for other reasons)consistent with Coleman-Mandula.

With respect to Coleman-Mandula (particularly with respect tofermions) it is useful to consider what Bee said: "... the fiveexceptional Lie-groups have the remarkable property that the adjointaction of a subgroup is the fundamental subgroup action on otherparts of the group. This then offers the possibility to arrange both,the fermions as well as the gauge fields, in the Lie algebra and rootdiagram of a single group ..." and what Steven Weinberg said at pages382-384 of his book The Quantum Theory of Fields, Vol. III (Cambridge2000):

"... The proof of the Coleman-Mandula theorem ... makes it clear that the list of possible bosonic symmetry generators is essentially the same in d greater than 2 spacetime dimensions as in four spacetime dimensions:

... there are only the momentum d-vector Pu, a Lorentz generator Juv = -Jvu ( with u and v here running over the values 1, 2, ... , d-1, 0 ), and various Lorentz scalar 'charges' ...

the fermionic symmetry generators furnish a representation of the homogeneous Lorentz group ... or, strictly speaking, of its covering group Spin(d-1,1). ...

The anticommutators of the fermionic symmetry generators with each other are bosonic symmetry generators, and therefore must be a linear combination of the Pu, Juv, and various conserved scalars. ...

the general fermionic symmetry generator must transform according to the fundamental spinor representations of the Lorentz group ... and not in higher spinor representations, such as those obtained by adding vector indices to a spinor. ...".

In short, Weinberg's book at pages 382-384 says that the importantthing about Coleman-Mandula is that fermions in a unified model must"... transform according to the fundamental spinor representations ofthe Lorentz group ... or, strictly speaking, of its covering groupSpin(d-1,1). ..." where d is the dimension of spacetime in themodel.

As I said in that comment, E8 is the sum of the adjointrepresentation and a half-spinor representation of Spin(16), so,

if the E8 model is built with respect to Lorentz, spinor, etcrepresentations based on Spin(16) consistently with Weinberg's work,,then the E8 model may be consistent with Coleman-Mandula.

The Cl(8) model is, due to compliance with the criteria set out byWeinberg, consistent with Coleman-Mandula ( see my web page atwww.valdostamuseum.org/hamsmith/d4d5e6hist2.html#ColemanMandula)

 

 


Appendix

With respect to the figure

 

As to the first and third circles from the center, which have 12points, to each of which 2 vertices are projected, so that eachcircle represents 24 vertices, the four circles together appear (inthe projection)

to have a total of 12 + 12 + 12 + 12 = 48 vertices, but a furtherrotation taken from an E8 rotation movie by Garrett Lisi (as is theimage immediately above)

shows that in fact the configuration has 16 + 40 + 16 = 48 + 12 +12 = 72 vertices, the correct number for the E6 root vectorpolytope.

As to the central point of each of the 6 clusters corresponding tothe 27-dimensional exceptional Jordan algebra J3(O), to each of which3 vertices are projected, so that each cluster represents 27vertices, the two concentric 12-vertex circles and the center pointappear (in the projection)

to have a total of 12 + 12 + 1 = 25 vertices, but a furtherrotation taken from an E8 rotation movie by Garrett Lisi (as is theimage immediately above)

shows that in fact the configuration has 1 + 16 + 10 = 12 + 12 + 3= 27 vertices, the correct number for the dimension of theexceptional Jordan algebra J3(O).

 

 


 

Tony Smith post on Mauitian re: Jacques Distler, GarrettLisi, and E8(8)

24 November 2007

 Garrett, this is a long comment (see my PS at the end), but the short version is: It seems to me that Jacques Distler did not refute your E8 model (only a possible use of triality) and that his analysis, along with your comments, show how the E8 model can be OK. Here are details: Distler said in his blog: "... E8(8) includes Spin(16) as a maximal compact subgroup ...In E8(8), the 248 decomposes as 248 = 120 + 128...We would like to find an embedding ofG = SL(2,C) x SU(3)xSU(2)xU(1) in ... E8 ...SL(2,C) = Spin(3,1)o is the connected part of the Lorentz Group, the "gauge group" in the MacDwoell-Mansouri formulation of gravity. ..."and you [Garrett] replied "... The G is embedded in a D4 x D4 subgroup of E8. ...g = so(3,1) + su(2) + u(1) + su(3) ... is in a so(7,1) + so(8) of e8 via the Pati-Salam, left-right symmetric model, g' = so(3,1) + su(2)_L + su(2)_R + su(4) The so(3,1) + su(2)_L + su(2)_R is in so(7,1), the su(4) is in so(8) ..." and Distler said "... Spin(7,1) x Spin(8) ... is a subgroup of E8(8) ...". So, an E8(8) model should be OK with the following interpretation: e8(8) = 120 + 128 120 = spin(16) = spin(8) + 64 + spin(7,1) = 28 + 64 + 28 128 = 64 + 64 wheregravity comes from spin(3,1) MacDowell-Mansouriand the Standard Model comes from Pati-Salam su(4)_c + su(2)_L + su(2)_Rspin(8) includes su(4) that reduces to su(3)_cspin(7,1) includes so(3,1) + su(2)_L + su(2)_R where the so(3,1) of MacDowell-Mansouri gravity is the little group, or local isotropy group, of 4-dim spacetime M4 described by the symmetric space G / Spin(3,1) where G can be anti-desitter or deSitterand where the su(2)_L + su(2)_R reduces to su(2)_L + u(1) which is the little group  of a 4-dim internal symmetry space CP2 = SU(3) / SU(2)xU(1)NOTE that the Lie groups of the spin(7,1) Lie algebra form the little groups of an 8-dim M4 x CP2 Kaluza-Klein space (as used by Batakis in his 1986 paper Class. Quantum Grav. 3 (1986) L99-L105). As to the global groups of M4 x CP2, they are in the spin(8) that includes the su(4): the su(3) gives the global group SU(3) in CP2 = SU(3) / SU(2)xU(1) (as used by Batakis)and the 4-dimensional deSitter or anti-deSitter rotations of G / Spin(3,1) should be a part of 6-dim twistor-related CP3 = SU(4) / SU(3)xU(1) What about the one 64 in the 120 and the two 64 + 64 in the 128 of E8(8) ? Each of the 64 should be of the form 8x8. Generalizing the spacetime algebra approach of Hestenes, let, in each of the three 64, one of the 8 represent 8 Dirac Gammas of the 8-dim K-K space. Denote it by 8_GSince the 64 in the 120 is in the adjoint of Spin(16), its 8x8 should be correspond to the 8-dim vector K-K space, so denote the 64 in 120 by 8_v x 8_GSince the 64 + 64 in the 128 is in a spinor space of Spin(16), its 8x8 + 8x8 should be correspond to fermions ( 8 particles and 8 antiparticles)so denote the 128 64 + 64 in 128 by 8_f+ x 8_G and 8+f- x 8_G There should be a Spin(8)-type triality among the three 64 things8_v x 8_G8_f+ x 8_G8+f- x 8_GThe above E8(8) structure describes Gravity, the Standard Model, 4-dim physical spacetime, a 4-dim K-K space, and first-generation fermions, as well as 8 Gammas of an 8-dim Dirac equation. If the second and third generation fermions come from combinatorics of fermions living partly in 4-dim physical spacetime and partly in 4-dim K-K space, then you get all three generations. My opinion is that: a spin foam can be constructed by putting the 120 of E8(8) on links and the 128 of E8(8) on vertices, and using Jordan algebra structure related to the 27-dim exceptional Jordan algebra J3(O); and particle masses and force strengths come from ratios of geometric volumes in the spirit of Armand Wyer; neutrinos are tree-level massless, with masses coming from corrections; Dark Energy : Dark Matter : Ordinary Matter ratio comes from the structure of the twistor stuff in the spin(8) of the 120.Tony SmithPS - Sorry for the long post on mauitian. If you want me to not put such stuff here, please just let me know. 

 

Here are the 240 vertices of the E8(8) root vectorpolytope:

 

The Spin(16) adjoint 120 =Spin(7,1) + 8_v x8_G + Spin(8) =28 + 64 +28

with 112 = 24 +64 + 24root vector vertices.

The Spin(16) half-spinor 128 = 8_f+x 8_G + 8_f- x 8_G =64 +64

Note that in the above image some of the 240 E8(8) vertices areprojected to the same point:

Note also that here (to be consistent with Garrett Lisi'snotation) I am using a notation Spin(7,1) that differs from what Iusually use, which is Spin(1,7).

Note also that (in my usual notation) Spin(1,7) comes from Cl(1,7)which the 16x16 real matrix algebra and so is isomorphic to Cl(0,8) =Cl(8).