Is there a duality between associative and co-associative space-time surfaces?

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?

1. Is it possible to map these two kinds of preferred extremals to each other by replacing the tangent space of the space-time surface at given point with its normal space? Could the non-associative normal spaces allow interpretation as an integrable distribution of tangent spaces defining non-associative dual of the space-time surface with integrable distribution of associative normal spaces? This would solve the problem of generating both Minkowskian and Euclidian regions of preferred extremals. Clearly, this would be a generalization of the duality of elementary geometry. Could this duality give rise to a new duality at the level of Yangian symmetries different from that discussed above?

2. Since tangent and normal space have different signatures of the induced metric, the duality should relate Minkowskian and Euclidian regions of space-time – that is interiors and exteriors of lines of scattering diagrams defined by the light-like orbits of partonic 2-surfaces at which the induced 4-D metric becomes degenerate having signature (0,-1,-1,-1), and is neither Minkowskian nor Euclidian or is both (this is a matter of taste).

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.

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