Friday, December 06, 2013

A 14-dimensional Theory?

With M-theory, F-theory and S-theory in mind, as well as Percacci's SO(3,11) theory, it's fun to look at gradings of exceptional algebras as a starting point for speculation.  The gradings can be lifted to find interesting subgroups of the corresponding exceptional groups.

Some interesting E6 gradings are:

g = E6(6) , 
       II = E6 , 
       II1 = {a1},
       g(0) = so(5,5) + R , 
       g(-1) = M1,2(O')
 g = E6(-26) , 
       II = A2 , 
       II1 = {a1},
       g(0) = so(1,9) + R , 
       g(-1) = M1,2(O)
 g = E6C , 
       II = E6 , 
       II1 = {a1},
       g(0) = so(10)C + C , 
       g(-1) = M1,2(O)C

where we see some nice 10-dimensional space-time signatures arise.

Moving up to 5-gradings, E7 gives an interesting decomposition:

 g = E7(-25), 
        II = C3, 
        II1 = {y1},
        g(0) = so(2,10) + R, 
        dimR g(-1) = 32, 
        dimR g(-2) = 1
where we see a two-time 12-dimensional (2,10) signature arise that reminds us of S-theory constructions with real 32-dimensional Weyl spinor.  S-theory proper is 13-dimensional and one expects to see SO(9,1)xSO(2,1) as well, however.  Alas, E7 also admits the grading:

g = E7(-25), 
        II = E7, 
        II1 = {y2},
        g(0) = so(1,9) + sl(2,R) + R, 
        dimR g(-1) = 32, 
        dimR g(-2) = 10
where the real 32-dimensional Weyl spinor is still present and SO(9,1)xSO(2,1) can be recovered from SO(9,1) and SL(2,R)~Spin(2,1).

Can we go further and see any signs of a higher dimensional theory from gradings of E8?  It appears to be the case:

g = E8(-24), 
        II = F4, 
        II1 = {y4},
        g(0) = so(3,11) + R, 
        dimR g(-1) = 64, 
        dimR g(-2) = 14.

g = E8(8), 
        II = E8, 
        II1 = {y1},
        g(0) = so(7,7) + R, 
        dimR g(-1) = 64, 
        dimR g(-2) = 14
g = E8C, 
        II = E8, 
        II1 = {y1},
        g(0) = so(14,C) + C, 
        dimC g(-1) = 64, 
        dimC g(-2) = 14
There seems to be hints of a 14-dimensional theory lurking in the gradings with a 64-dimensional real or complex spinor.  Viewed from the viewpoint of octonions, E8(8) corresponds to a split-octonion (would be) theory with 14-dimensional (7,7) signature.  E8(-24) arises from the octonions, where the 14-dimensional (3,11) three-time signature is recovered.  The complexified octonion construction gives the E8(C) case with 14-dimensional SO(14,C) symmetry and complex 64-dimensional spinor.

So are the exceptional groups hinting at various higher dimensional theories that go beyond M-theory?  The 5-gradings of E7 and E8 (non-compact and compact) seem to support that hypothesis.

1 comment:

Javier said...

The link is bad (or fallen when I write this at least).