Could one regard space-time surfaces as surfaces in twistor space?

Twistors are used to construct solutions of free wave equations with given spin and self-dual solutions of both YM theories and Einstein's equations . Twistor analyticity plays a key role in the construction of construction of solutions of free field equations. In General Relativity the problem of the twistor approach is that twistor space does not make sense for a general space-time metric . In TGD framework this problem disappears and one can ask how twistors could possibly help to construct preferred extremals. In particular, one can ask whether it might be possible to interpret space-time surfaces as as counterparts of surfaces - not necessarily four-dimensional - in twistor space or in some space naturally related to it. The 12-dimensional space PT×CP₂ indeed emerges as a natural candidate (if something is higher dimensional, the standard association which of string theories corresponds to this dimension and F-theory does the job at this time).

1. How M⁴×CP₂ emerges in twistor context?

The finding that CP₂ emerges naturally in twistor space considerations is rather encouraging.

1. Twistor space allows two kinds of 2-planes in complexified M⁴ known as α- and β-planes and assigned to twistor and its dual . This reflects the fundamental duality of the twistor geometry stating that the points Z of PT label also complex planes (CP₂) of PT via the condition

   Z_aW^a=0.

   To the twistor Z one can assign via twistor equation complex α-plane, which contains only null vectors and correspond to the plane defined by the twistors intersecting at Z.

   For null twistors (5-D sub-space N of PT) satisfying Z^atildeZ_a=0 and identifiable as the space of light-like geodesics of M⁴ α-plane contains single real light-ray. β-planes in turn correspond to dual twistors which define 2-D null plane CP₂ in twistor space via the equation Z_aW^a=0 and containing the point W = tildeZ. Since all lines CP₁ of CP₂ intersect, also they parameterize a 2-D null plane of complexified M⁴. The β-planes defined by the duals of null twistors Z contain single real light-like geodesic and intersection of two CP₂:s defined by two points of line of N define CP₁ coding for a point of M⁴.

2. The natural appearance of CP₂ in twistor context suggests a concrete conjecture concerning the solutions of field equations. Light rays of M⁴ are in 1-1 correspondence with the 5-D space N subset P of null twistors. Compactified M⁴ corresponds to the real projective space PN. The dual of the null twistor Z defines 2-plane CP₂ of PT.

3. This suggests the interpretation of the counterpart of M⁴×CP₂ as a bundle like structure with total space consisting of complex 2-planes CP₂ determined by the points of N. Fiber would be CP₂ and base space 5-D space of light-rays of M⁴. The fact that N does not allow holomorphic structure suggests that one should extend the construction to PT and restrict it to N. The twistor counterparts of space-time surfaces in T would be holomorphic surfaces of PT×CP₂ or possibly of PT_± (twistor analogs of lower and upper complex plane and assignable to positive and negative frequency parts of classical and quantum fields) restricted to N×CP₂.

2. How to identify twistorial surfaces in PT×CP₂ and how to map them to M⁴× CP₂?

The question is whether and how one could construct the correspondence between the points of M⁴ and CP₂ defining space-time surface from a holomorphic correspondence between points of PT and CP₂ restricted to N.

1. The basic constraints are that space-time surfaces with varying values for dimensions of M⁴ and CP₂ projections are possible and that these surfaces should result by a restriction from PT× CP₂ to N× CP₂ followed by a map from N to M⁴ either by selecting some points from the light ray or by identifying entire light rays or their portions as sub-manifolds of X⁴.

2. Quantum classical correspondence would suggest that surfaces holomorphic only in PT₊ or PT_- should be used so that one could say that positive and negative energy states have space-time correlates. This would mean an analogy with the construction of positive and negative energy solutions of free massless fields. The corresponding space-time surfaces would emerge from the lower and upper light-like boundaries of the causal diamond CD.

3. A rather general approach is based on an assignment of a sub-manifold of CP₂ to each light ray in PT_+/- in holomorphic manner that is by n equations of form

   F_i(ξ¹,ξ²,Z)=0 , i=1,...,n≤ 2.

   The dimension of this kind of surface in PT× CP₂ is D=10-2n and equals to 6, 8 or 10 so that a connection or at least analogy with M-theory and branes is suggestive. For n=0 entire CP₂ is assigned with the point Z (CP₂ type vacuum extremals with constant M⁴ coordinates): this is obviously a trivial case. For n=1 8-D manifold is obtained. In the case that Z is expressible as a function of CP₂ coordinates, one could obtain CP₂ type vacuum extremals or their deformations. Cosmic strings could be obtained in the case that there is no Z dependence. For n=4 discrete set of points of CP₂ are assigned with Z and this would correspond to field theory limit, in particular massless extremals. If the dimension of CP₂ projection for fixed Z is n, one must construct 4-n-dimensional subset of M⁴ for given point of CP₂.

4. If one selects a discrete subset of points from each light ray, one must consider a 4-n-dimensional subset of light rays. The selection of points of M⁴ must be carried out in a smooth manner in this set. The light rays of M⁴ with given direction can be parameterized by the points of light-cone boundary having a possible interpretation as a surface from which the light rays emerge (boundary of CD).

5. One could also select entire light rays of portions of them. In this case a 4-n-1-dimensional subset of light rays must be selected. This option could be relevant for the simplest massless extremals representing propagation along light-like geodesics (in a more general case the first option must be considered). The selection of the subset of light rays could correspond to a choice of 4-n-1-dimensional sub-manifold of light-cone boundary identifiable as part of the boundary of CD in this case. In this case one could worry about the intersections of selected light rays. Generically the intersections occur in a discrete set of points of H so that this problem does not seem to be acute. The lines of generalized Feynman diagrams interpreted as space-time surfaces meet at 3-D vertex surfaces and in this case one must pose the condition that CP₂ projections at the 3-D vertices are identical.

6. The use of light rays as the basic building bricks in the construction of space-time surfaces would be the space-time counterpart for the idea that light ray momentum eigen states are more fundamental than momentum eigen states.

M⁸-H duality is Kähler isometry in the sense that both induced metric and induced Kähler form are identical in M⁸ and M⁴× CP₂ representations of the space-time surface. In the recent case this would mean that the metric induced to the space-time surface by the selection of the subset of light-rays in N and subsets of points at them has the same property. This might be true trivially in the recent case.

3. How to code the basic parameters of preferred extremals in terms of twistors?

One can proceed by trying to code what is known about preferred extremals to the twistor language.

1. A very large class of preferred extremals assigns to a given point of X⁴ two light-like vectors U and V of M⁴ and two polarization vectors defining the tangent vectors of the coordinate lines of Hamilton-Jacobi coordinates of M⁴. As already noticed, given null-twistor defines via λ and tildeμ two light-like directions V and U and twistor equation defines M⁴ coordinate m apart from a shift in the direction of V. The polarization vectors ε_i in turn can be defined in terms of U and V. λ=μ corresponds to a degenerate case in which U and V are conjugate light-like vectors in plane M² and polarization vector is also light-like. This could correspond to the situation for CP₂ type vacuum extremals. For the simplest massless extremals light-like vector U is constant and the solution depends on U and transverse polarization ε vector only. More generally, massless extremals depend only on two M⁴ coordinates defined by U coordinate and the coordinate varying in the direction of local polarization vector ε.

2. Integrable distribution of these light-like vectors and polarization vectors is required. This means that these vectors are gradients of corresponding Hamilton-Jacobi coordinate variables. This poses conditions on the selection of the subset of light rays and the selection of M⁴ points at them. Hyper-quaternionic and co-hyper-quaternionic surfaces of M⁸ are also defined by fixing an integrable distribution of 4-D tangent planes, which are parameterized by points of CP₂ provided one can assign to the tangent plane M²(x) either as a sub-space or via the assignment of light-like tangent vector of x.
3. Positive (negative) helicity polarization vector can be constructed by taking besides λ arbitrary spinor μ_a and defining

   ε_aa'= λ_a tildeμ_a'/[tildeλ,tildeμ] ,

   [tildeλ,tilde&mu]= ε_a'b'λ^a'μ^b'

   for negative helicity and

   ε_aa'= μ_a tildeλ_a'/<λ,μ> ,
   <lambda;&mu >= ε_abλ^aμ^b

   for positive helicity. Real polarization vectors correspond to sums and differences of these vectors. In the recent case a natural identification of μ would be as the second light-like vector defining point of m. One should select one light-like vector and one real polarization vector at each point and find the corresponding Hamilton-Jacobi coordinates. These vectors could also code for directions of tangents of coordinate curves in transversal degrees of freedom.

The proposed construction seems to be consistent with the proposed lifting of preferred extremals representable as a graph of some map M⁴→ CP₂ to surfaces in twistor space. What was done in one variant of the construction was to assign to the light-like tangent vectors U and V spinors tildeμ and λ assuming that twistor equation gives the M⁴ projection m of the point of X⁴(X³_l). This is the inverse of the process carried out in the recent construction and would give CP₂ coordinates as functions of the twistor variable in a 4-D subset of N determined by the lifting of the space-time surface. The facts that tangent vectors U and V are determined only apart from overall scaling factor and the fact that twistor is determined up to a phase, imply that projective twistor space PT is in question. This excludes the interpretation of the phase of the twistor as a local Kähler magnetic flux. The next steps would be extension to entire N and a further continuation to holomorphic field in PT or PT_±.

To summarize, although these arguments are far from final or convincing and are bound to reflect my own rather meager understanding of twistors, they encourage to think that twistors are indeed natural approach in TGD framework. If the recent picture is correct, they code only for a distribution of tangent vectors of M⁴ projection and one must select both a subset of light rays and a set of M⁴ points from each light-ray in order to construct the space-time surface. What remains open is how to solve the integrability conditions and show that solutions of field equations are in question. The possibility to characterize preferred extremal property in terms of holomorphy and integrability conditions would mean analogy with both free field equations in M⁴ and minimal surfaces. For known extremals holomorphy in fact guarantees the extremal property.

4. Hyper-quaternionic and co-hyper-quaternionic surfaces and twistor duality

In TGD framework space-time surface decomposes into two kinds of regions corresponding to hyper-quaternionic and co-hyper-quaternionic regions of the space-time surface in M⁸ (hyper-quaternionic regions were considered in preceding arguments). The regions of space-time with M⁴ (Euclidian) signature of metric are identified tentatively as the counterparts of hyper-quaternionic (co-hyper-quaternionic) space-time regions. Pieces CP₂ type vacuum extremals representing generalized Feynman diagrams and having light-like random curve as M⁴ projection represent the basic example here. Also these space-time regions should have any twistorial counterpart and one can indeed assign to M⁴ projection of CP₂ type vacuum extremal a spinor λ as its tangent vector and spinor μ via twistor equation once M⁴ projection is known.

The first guess would the correspondence hyper-quaternionic ↔ α and co-hyper-quaternionic ↔ β. Previous arguments in turn suggest that hyper-quaternionic space-time surfaces are mapped to surfaces for which two null twistors are assigned with given point of M⁴ whereas co-hyper-quaternionic space-time surfaces are mapped to the surfaces for which only single twistor corresponds to a given M⁴ point.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".