- ReQ(o)=0 and ImQ(P)=0 allow M4 and its complement as associative/co-associative subspaces of Oc. The roots P=0 for the complexified octonionic polynomials satisfy two conditions X=0 and Y=0. They are 6-D brane-like entities X6c having real projection X6r ("real" means that the number theoretic complex valued octonion norm squared is real valued).
Posing the condition ImQ(P)=0 gives a complex surface X2c at which the number theoretic norm squared defines complex norm squared, which is not physically acceptable. The real projection X2r with real norm is 2-D.
The condition ReQ(P)=0 gives complex surface X3c which has 3-D real projection X3r, which should be completed to a real surface X4r by holography- perhaps by giving up the condition Y=0 and taking real projection X4r of the resulting 4-D surface X4c.
- The cold shower came as I learned that 4-D associative sub-manifolds of quaternion spaces are geodesic manifolds and thus trivial. Co-associativity is however possible since any distribution of associative normal spaces integrates to a sub-manifold. Typically these sub-manifolds are minimal surfaces, which conforms with the physical intuitions. Therefore the surface X4r given by holography should be co-associative. By the same argument space-time surface contains string world sheets and partonic 2-surfaces as co-complex surfaces.
- The key observation is that G2 as the automorphism group of octonions respects the co-associativity of the 4-D real sub-basis of octonions. Therefore a local G2 (or even G2,c ) gauge transformation applied to a 4-D co-associative sub-space Oc gives a co-associative four-surface as a real projection if the boundary conditions stating that X4r has as its ends surfaces X3r at X6r.
An open question is whether this approach is equivalent to giving up Y=0 conditions so that octonion analyticity would correspond to G2 gauge transformation: this would realize the original idea about octonion analyticity. If this surface contains a co-complex 2-surface as a string world sheet, the conditions making possible to map X4r to H by M8-H duality are satisfied and there is no need for a separate holography in H. There is no objection against this option and it would replaces SH with much strong number theoretic holography fixing space-time region from the roots of a real polynomial. One could say that classical TGD is an exactly solvable theory.
- Remarkably, the group SU(3)c⊂ G2,c has interpretation as a complexified color group and the map defining space-time surface defines a trivial gauge field in SU(3)c wheras the connection in SU(3) is non-trivial. Color confinement could mean geometrically that SU(3)c reduces to SU(3) at large distances: could it be simpler!
This picture conforms with the H-picture in which gluon gauge potentials are identified as color gauge potentials. Note that at QFT limit the gauge potentials are replaced by their sums over parallel space-time sheets to give gauge fields as the space-time sheets are approximated with a single region of Minkowski space.
- Minkowski signature turns out to be the only possible option for X4r. Also the phenomenological picture based on co-assiative space-time sheets, light-like 3-surfaces, string world sheets and partonic 2-surfaces, and wormhole contacts carrying monopole flux emerges.