M⁸-H duality and twistor space counterparts of space-time surfaces

It seems that by identifying CP_3,h as the twistor space of M⁴, one could develop M⁸-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 M⁸- 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 M⁸ side but involving no derivatives.

1. The simplest identification of twistor spheres is by z₁=z₂ 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 z₁=z₂.
2. 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.

3. 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.
4. 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 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_iz_i1+b_iz_i2 and u_i2= c_iz_i1+d_iz_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).

5. These conditions do not yet specify the choices of the coordinates (u_i1,u_i2) 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 CP_3,h of M⁴ is identified as CP_2,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 (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 CP₂ type extremals, string like objects, massless extremals, and their deformations.

1. 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.
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 e_i, 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.
3. 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.