Holography = holomorphy principle (H-H) and CP type extremals as a model for wormhole contact

I used a lot of effect in trying to solve what I thought to a technical problem related to the  finding of the roots of (f1,f2) appearing in the Euclidean space-time regions. It took time to realizes that this only an ansatz, which is  less general than H-H  and need not work  for wormhole contacts as deformations of CP2 type extremals (see this).

  1. The first problem is that in Minkowskian regions defining the parallel space-time sheets one has two kinds of solutions for which  hypercomplex coordinate u resp.  its conjugate v appears in fi resp. its conjugate. These should correspond to a single solution and the only way is to consider their  union. The two regions in question have a natural identification as Minkowskian space-time sheets connected by  a wormhole contact with an Euclidean signature of the induced metric.

    At the  surface,  where  the two sheets are glued,   fi must be invariant under conjugation, which for real coefficients of fi requires  u=v  and reality of various complex coordinates or at least that the surface in question is invariant under complex conjugation.

  2.   In Euclidean regions, the realization of holography = holomorphy principle (H-H), using (f1,f2)=(0,0) ansatz assuming that either hypercomplex coordinate   u or v   is a  dynamical variable,  leads to a problem. Either u or v is a complex analytic function f of CP2 coordinates  and its reality implies  Im(f)=0   so that CP2 projection is 3-dimensional, which means the failure of the holomorphy with respect to the CP2 coordinates.  For a moment I thought that Wick rotation might help but this was not the case.
  3. This forces to give up (f1,f2)=(0,0) ansatz and assume only H-H. The original vision was that the Euclidean region as a wormhole contact corresponds to a deformation of a canonically embedded CP2 so that it has a light-like coordinate curve  of u or v as M4 projection. These space-time surfaces are holomorphic so that field equations are satisfied.

    The gluing condition implies constancy condition  v=v0 resp. u=u0   and v resp. u  is replaced with a real CP2 coordinate s(u) resp. s(v).  M4 complex coordinate w can be a function of CP2 coordinates.

  4. The gluing condition for the two sheets requires u0=v0 which for u=m0+m3 and v= m0-m3 gives m0 = 2u0 and m3=0. At the points of this 3-surface) there is an edge at which the coordinate curves for u and v meet: the interpretation could be in terms of an exotic smooth structure (see this, this, and this) as standard smooth structure with a defect to which fermion pair creation or fermion scattering vertex can be assigned. The two sheets are glued together along a 3-surface X3 with 3-D CP2 projection invariant under complex conjugation. The CP2 projection X3 must contain a homologically non-trivial 2-surface since the wormhole contact must carry a monopole flux between the space-time sheets.  
This tentative picture would relate several key ideas of TGD: H-H involving hypercomplex numbers, the notion of light-like partonic orbit, the idea that exotic smooth structures make possible non-trivial scattering theory in 4 dimensional space-time. One can compare this picture with the intuitive phenomenological picture.

See the chapter Holography= holomorphy vision: analogues of elliptic curves and partonic orbits or the article Holography= holomorphy vision and a more precise view of partonic orbits.