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M8-H duality maps the preferred extremals in H to those M4× CP2 and vice versa. The tangent spaces of associative space-time surface in M8 would be quaternionic Minkowski spaces.
What is interesting that one can consider also co-associative space-time surfaces having associative normal space. Could these two kinds of space-time surfaces be dual to each other in some sense? For instance, could information about either of them allow to fix both Minkowskian and Euclidian regions of space-time surfaces or scattering amplitudes. If so then Euclidian/Minkowskian regions would be kind of Leibniz monads, mirror images of each other.
An objection against this duality is that the associated octonionic momenta are co-quaternionic momenta and space-like. However, by multiplying co-quaternionic momentum with a light-like octonionic, one obtains a light-like quaternionic momentum: this due the multiplicativity of the octonionic norm. The space of light-like imaginary units is 6-D sphere but the existence of the preferred M2 central for M8-H duality provides a unique light-like unit.
What this Minkowskian-Euclidian duality could mean for the preferred extremals?
Partonic 2-surfaces and their orbits should be mapped to themselves in this duality. This and also the symmetry between Euclidian Minkowskian regions requires that string world sheets are present also in Euclidian regions and have discrete points as intersections with partonic 2-surfaces shared by both Minkowskian and Euclidian regions.
This duality would be analogous to inversion with respect to the surface of sphere, which is conformal symmetry. Maybe this inversion could be seen as the TGD counterpart of finite-D conformal inversion at the level of space-time surfaces.
There is also an analogy with the method of images used in some 2-D electrostatic problems used to reflect the charge distribution outside conducting surface to its virtual image inside the surface. The 2-D conformal invariance would generalize to its 4-D quaterionic counterpart.
For a summary of earlier postings see Latest progress in TGD.
Articles and other material related to TGD.