- The simplest identification of twistor spheres is by z1=z2 for the complex coordinates of the spheres. One can consider replacing zi by its Möbius transform but by a coordinate change the condition reduces to z1=z2.
- At M8 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= z1 + Jz2 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.
- M8-H duality must map the complex coordinates z11=Re(RE) and z12=Im(RE) (z21=Re(IM) and z22=Im(IM)) at M8 side to complex coordinates ui1 and ui2 with ui1(0)=0 and ui2(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: ui1= aizi1+bizi2 and ui2= cizi1+dizi2 so that the map for given value of i is characterized by SL(2,Q) matrix (ai,bi;ci,di).
- These conditions do not yet specify the choices of the coordinates (ui1,ui2) at twistor side. At CP2 side the complex coordinates would naturally correspond to Eguchi-Hanson complex coordinates (w1,w2) 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 CP3,h of M4 is identified as CP2,h, the hyper-complex counterpart of CP2, the analogs of complex coordinates would be (w3,w4) with w3 hypercomplex and w4 complex. A priori one could select the pair (ui1,ui2) as any pair (wk(i),wl(i)), k(i)≠ l(i). These choices should give different kinds of extremals: such as CP2 type extremals, string like objects, massless extremals, and their deformations.
- The interpretation of the pre-images of these singularities in M8 should be number theoretic and related to the identification of quaternionic imaginary units. One must specify two non-parallel octonionic imaginary units e1 and e2 to determine the third one as their cross product e3=e1× e2. If e1 and e2 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 M8 at which this takes place? The parallelity of the tangent/normal vectors defining imaginary units ei, i=1,2 states that the component of e2 orthogonal to e1 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.
For a summary of earlier postings see Latest progress in TGD.