Twistor lift and the reduction of field equations by SH to holomorphy

It has become clear that twistorialization has very nice physical consequences. But what is the deep mathematical reason for twistorialization? Understanding this might allow to gain new insights about construction of scattering amplitudes with space-time surface serving as analogs of twistor diatrams.

Penrose's original motivation for twistorilization was to reduce field equations for massless fields to holomorphy conditions for their lifts to the twistor bundle. Very roughly, one can say that the value of massless field in space-time is determined by the values of the twistor lift of the field over the twistor sphere and helicity of the massless modes reduces to cohomology and the values of conformal weights of the field mode so that the description applies to all spins.

I want to find the general solution of field equations associated with the Kähler action lifted to 6-D Kähler action. Also one would like to understand strong form of holography (SH). In TGD fields in space-time are are replaced with the imbedding of space-time as 4-surface to H. Twistor lift imbeds the twistor space of the space-time surface as 6-surface into the product of twistor spaces of M⁴ and CP₂. Following Penrose, these imbeddings should be holomorphic in some sense.

Twistor lift T(H) means that M⁴ and CP₂ are replaced with their 6-D twistor spaces.

1. If S² for M⁴ has 2 time-like dimensions one has 3+3 dimensions, and one can speak about hyper-complex variants of holomorphic functions with time-like and space-like coordinate paired for all three hypercomplex coordinates. For the Minkowskian regions of the space-time surface X⁴ the situation is the same.
   
   
2. For T(CP₂) Euclidian signature of twistor sphere guarantees this and one has 3 complex coordinates corresponding to those of S² and CP₂. One can also now also pair two real coordinates of S² with two coordinates of CP₂ to get two complex coordinates. For the Euclidian regions of the space-time surface the situation is the same.
   

Consider now what the general solution could look like. Let us continue to use the shorthand notations S²₁= S²(X⁴); S²₂= S²(CP₂);S²₃= S²(M⁴).

1. Consider first solution of type (1,0) so that coordinates of S²₂ are constant. One has holomorphy in hypercomplex sense (light-like coordinate t-z and t+z correspond to hypercomplex coordinates).
   
   
   1. The general map T(X⁴) to T(M⁴) should be holomorphic in hyper-complex sense. S²₁ is in turn identified with S²₃ by isometry realized in real coordinates. This could be also seen as holomorphy but with different imaginary unit. One has analytical continuation of the map S²₁→ S²₃ to a holomorphic map. Holomorphy might allows to achieve this rather uniquely. The continued coordinates of S²₁ correspond to the coordinates assignable with the integrable surface defined by E²(x) for local M²(x)× E²(x) decomposition of the local tangent space of X⁴. Similar condition holds true for T(M⁴). This leaves only M²(x) as dynamical degrees of freedom. Therefore one has only one holomorphic function defined by 1-D data at the surface determined by the integrable distribution of M²(x) remains. The 1-D data could correspond to the boundary of the string world sheet.
      
      
   2. The general map T(X⁴) to T(CP₂) cannot satisfy holomorphy in hyper-complex sense. One can however provide the integrable distribution of E²(x) with complex structure and map it holomorphically to CP₂. The map is defined by 1-D data.
      
      
   3. Altogether, 2-D data determine the map determining space-time surface. These two 1-D data correspond to 2-D data given at string world sheet: one would have SH.
      
      
2. What about solutions of type (0,1) making sense in Euclidian region of space-time? One has ordinary holomorphy in CP₂ sector.
   
   
   1. The simplest picture is a direct translation of that for Minkowskian regions. The map S²₁→ S²₂ is an isometry regarded as an identification of real coordinates but could be also regarded as holomorphy with different imaginary unit. The real coordinates can be analytically continued to complex coordinates on both sides, and their imaginary parts define coordinates for a distribution of transversal Euclidian spaces E²₂(x) on X⁴ side and E²(x) on M⁴ side. This leaves 1-D data.
      
      
   2. What about the map to T(M⁴)? It is possible to map the integrable distribution E²₂(x) to the corresponding distribution for T(M⁴) holomorphically in the ordinary sense of the word. One has 1-D data. Altogether one has 2-D data and SH and partonic 2-surfaces could carry these data. One has SH again.
      
      
3. The above construction works also for the solutions of type (1,1), which might make sense in Euclidian regions of space-time. It is however essential that the spheres S²₂ and S²₃ have real coordinates.
   
   

SH thus would thus emerge automatically from the twistor lift and holomorphy in the proposed sense.

1. Two possible complex units appear in the process. This suggests a connection with quaternion analytic functions suggested as an alternative manner to solve the field equations. Space-time surface as associative (quaterionic) or co-associate (co-quaternionic) surface is a further solution ansatz.
   

   Also the integrable decompositions M²(x)× E²(x) resp. E²₁(x)× E²₂(x) for Minkowskian resp. Euclidian space-time regions are highly suggestive and would correspond to a foliation by string wold sheets and partonic 2-surfaces. This expectation conforms with the number theoretically motivated conjectures.
   
   

2. The foliation gives good hopes that the action indeed reduces to an effective action consisting of an area term plus topological magnetic flux term for a suitably chosen stringy 2-surfaces and partonic 2-surfaces. One should understand whether one must choose the string world sheets to be Lagrangian surfaces for the Kähler form including also M⁴ term. Minimal surface condition could select the Lagrangian string world sheet, which should also carry vanishing classical W fields in order that spinors modes can be eigenstates of em charge.
   

   The points representing intersections of string world sheets with partonic 2-surfaces defining punctures would represent positions of fermions at partonic 2-surfaces at the boundaries of CD and these positions should be able to vary. Should one allow also non-Lagrangian string world sheets or does the space-time surface depend on the choice of the punctures carrying fermion number (quantum classical correspondence)?
   
   

3. The alternative option is that any choice produces of the preferred 2-surfaces produces the same scattering amplitudes. Does this mean that the string world sheet area is a constant for the foliation - perhaps too strong a condition - or could the topological flux term compensate for the change of the area?
   

   The selection of string world sheets and partonic 2-surfaces could indeed be also only a gauge choice. I have considered this option earlier and proposed that it reduces to a symmetry identifiable as U(1) gauge symmetry for Kähler function of WCW allowing addition to it of a real part of complex function of WCW complex coordinates to Kähler action. The additional term in the Kähler action would compensate for the change if string world sheet action in SH. For complex Kähler action it could mean the addition of the entire complex function.
   
   

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article Some questions related to the twistor lift of TGD.

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

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