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Tony Smith
Dec14-07, 10:18 PM
P: 40
In keeping with this being "a technical discussion",
here is something that I asked in response to a comment by Thomas Larsson over on Cosmic Variance in Sean's post "Garrett Lisi’s Theory of Everything!":

Could Garrett Lisi’s model be understood in terms of a 7-grading of e8 that was described in a sci physics research thread Re: Structures preserved by e8, in which Thomas Larsson said:

“… … e_8 also seems to admit a 7-grading,
g = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3,
of the form

e_8 = 8 + 28* + 56 + (sl(8) + 1) + 56* + 28 + 8* .

…[in]… the above god-given 7-grading of e_8 … g_-3 is identified with spacetime translations and one would therefore get that spacetime has 8 dimensions rather than 11. …”.

So, if you used g-3 for an 8-dim Kaluza-Klein spacetime,
could you see the 28* and 28 as the two copies of D4 used by Garrett Lisi to get MacDowell-Mansouri gravity from one and the Standard Model gauge bosons from the other
see the central sl(8)+1 being related to transformations of the 8-dim spacetime
(actually being a 64-dim thing that is substantially 8×8* ).

The even part of the grading would then be the 112 elements
28* + 8×8* + 28
the odd part of the grading would then be the 128 elements
8 + 56 + 56* + 8*
If the 8 and 8* are used for 8-dim Kaluza-Klein spacetime
could the 56 + 56* be used for fermion particles and antiparticles ?

Even if the above assignment needs improvement,
my basic question is

could Thomas Larsson’s 7-grading of e8 be useful in making Garrett Lisi’s model a realistic description of physics ?

Tony Smith

PS - My personal favorite interpretation of the e8 7-grading is a bit different from what I described above, but I altered it to fit Thomas Larsson's explicit idea that the 8 should correspond to a spacetime.