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M^{8}-H duality and twistor space counterparts of space-time surfaces

It seems that by identifying CP

_{3,h}as the twistor space of M

^{4}, one could develop M

^{8}-H duality to a surprisingly detailed level from the conditions that the dimensional reduction guaranteed by the identification of the twistor spheres takes place and the extensions of rationals associated with the polynomials defining the space-time surfaces at M

^{8}- and twistor space sides are the same. The reason is that minimal surface conditions reduce to holomorphy meaning algebraic conditions involving first partial derivatives in analogy with algebraic conditions at M

^{8}side but involving no derivatives.

- The simplest identification of twistor spheres is by z
_{1}=z_{2}for the complex coordinates of the spheres. One can consider replacing z_{i}by its Möbius transform but by a coordinate change the condition reduces to z_{1}=z_{2}.

- At M
^{8}side one has either RE(P)=0 or IM(P)=0 for octonionic polynomial obtained as continuation of a real polynomial P with rational coefficients giving 4 conditions (RE/IM denotes real/imaginary part in quaternionic sense). The condition guarantees that tangent/normal space is associative.

Since quaternion can be decomposed to a sum of two complex numbers: q= z

_{1}+ Jz_{2}RE(P)=0 correspond to the conditions Re(RE(P))=0 and Im(RE(P))=0. IM(P)=0 in turn reduces to the conditions Re(IM(P))=0 and Im(IM(P))=0.

- The extensions of rationals defined by these polynomial conditions must be the same as at the octonionic side. Also algebraic points must be mapped to algebraic points so that cognitive representations are mapped to cognitive representations. The counterparts of both RE(P)=0 and IM(P)=0 should be satisfied for the polynomials at twistor side defining the same extension of rationals.

- M
^{8}-H duality must map the complex coordinates z_{11}=Re(RE) and z_{12}=Im(RE) (z_{21}=Re(IM) and z_{22}=Im(IM)) at M^{8}side to complex coordinates u_{i1}and u_{i2}with u_{i1}(0)=0 and u_{i2}(0)=0 for i=1 or i=2, at twistor side.

Roots must be mapped to roots in the same extension of rationals, and no new roots are allowed at the twistor side. Hence the map must be linear: u

_{i1}= a_{i}z_{i1}+b_{i}z_{i2}and u_{i2}= c_{i}z_{i1}+d_{i}z_{i2}so that the map for given value of i is characterized by SL(2,Q) matrix (a_{i},b_{i};c_{i},d_{i}).

- These conditions do not yet specify the choices of the coordinates (u
_{i1},u_{i2}) at twistor side. At CP_{2}side the complex coordinates would naturally correspond to Eguchi-Hanson complex coordinates (w_{1},w_{2}) determined apart from color SU(3) rotation as a counterpart of SU(3) as sub-group of automorphisms of octonions.

If the base space of the twistor space CP

_{3,h}of M^{4}is identified as CP_{2,h}, the hyper-complex counterpart of CP_{2}, the analogs of complex coordinates would be (w_{3},w_{4}) with w_{3}hypercomplex and w_{4}complex. A priori one could select the pair (u_{i1},u_{i2}) as any pair (w_{k(i)},w_{l(i)}), k(i)≠ l(i). These choices should give different kinds of extremals: such as CP_{2}type extremals, string like objects, massless extremals, and their deformations.

^{8}?

- The interpretation of the pre-images of these singularities in M
^{8}should be number theoretic and related to the identification of quaternionic imaginary units. One must specify two non-parallel octonionic imaginary units e^{1}and e^{2}to determine the third one as their cross product e^{3}=e^{1}× e^{2}. If e^{1}and e^{2}are parallel at a point of octonionic surface, the cross product vanishes and the dimension of the quaternionic tangent/normal space reduces from D=4 to D=2.

- Could string world sheets/partonic 2-surfaces be images of 2-D surfaces in M
^{8}at which this takes place? The parallelity of the tangent/normal vectors defining imaginary units e_{i}, i=1,2 states that the component of e_{2}orthogonal to e_{1}vanishes. This indeed gives 2 conditions in the space of quaternionic units. Effectively the 4-D space-time surface would degenerate into 2-D at string world sheets and partonic 2-surfacesa as their duals. Note that this condition makes sense in both Euclidian and Minkowskian regions.

- Partonic orbits in turn would correspond surfaces at which the dimension reduces to D=3 by light-likeness - this condition involves signature in an essential manner - and string world sheets would have 1-D boundaries at partonic orbits.

^{8}-H Duality, SUSY, and Twistors.

For a summary of earlier postings see Latest progress in TGD.

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