Holography = holomorphy principle (H-H) and CP2 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 f_i 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,   f_i must be invariant under conjugation, which for real coefficients of f_i 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 article Holography= holomorphy vision and a more precise view of partonic orbits or the chapter Holography= holomorphy vision: analogues of elliptic curves and partonic orbits.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.