Twistor structure of space-time surface is induced to the allowed 6-D surfaces of T(H), which as twistor spaces T(X4) must have fiber space structure with S2 as fiber and space-time surface X4 as base. The Kähler form of T(H) expressible as a direct sum
J(T(H)= J(T(M4))⊕ J(T(CP2)
induces as its projection the analog of Kähler form in the region of T(X4) considered.
There are physical motivations (CP breaking, matter antimatter symmetry, the well-definedness of em charge) to consider the possibility that also M4 has a non-trivial symplectic/Kähler form of M4 obtained as a generalization of ordinary symplectic/Kähler form (see this). This requires the decomposition M4=M2× E2 such that M2 has hypercomplex structure and E2 complex structures.
This decomposition might be even local with the tangent spaces M2(x) and E2(x) integrating to locally orthogonal 2-surfaces. These decomposition would define what I have called Hamilton-Jacobi structure (see this). This would give rise to a moduli space of M4 Kähler forms allowing besides covariantly constant self-dual Kähler forms with decomposition (m0,m3) and (m1, m2) also more general self-dual closed Kähler forms assignable to integrable local decompositions. One example is spherically symmetric stationary self-dual Kähler form corresponding to the decomposition (m0,rM) and (θ,φ) suggested by the need to get spherically symmetric minimal surface solutions of field equations. Also the decomposition of Robertson-Walker coordinates to (a,r) and (θ,π) assignable to light-cone M4+ can be considered.
The moduli space giving rise to the decomposition of WCW to sectors would be finite-dimensional if the integrable 2-surfaces defined by the decompositions correspond to orbits of subgroups of the isometry group of M4 or CD. This would allow planes of M4, and radial half-planes and spheres of M4 in spherical Minkowski coordinates and of M4+ in Robertson-Walker coordinates. These decomposition could relate to the choices of measured quantum numbers inducing symmetry breaking to the subgroups in question. These choices would chose a sector of WCW (see this) and would define quantum counterpart for a choice of quantization axes as distinct from ordinary state function reduction with chosen quantization axes.