Twistor lift for 2-D objects

TGD involves also 2-D objects - partonic 2-surfaces and string world sheets in an essential manner and strong form of holography (SH) states that these objects carry the information about quantum states. This does not mean that the dynamics would reduce to that for string like objects since it is essential that these objects are sub-manifolds of space-time surface. String world sheets carry induced spinor fields and it seems that these are crucial for understanding elementary particles. There are several questions to be answered.

1. Are fermionic fields localized to 2-surfaces? The generalization superconformal symmetry fixing both the bosonic and fermion parts of the action requires that also the interior of space-time carries induced spinor field. Their interpretation is not quite clear: could they perhaps give rise to a supersymmetry? The condition of super-symmetry fixes the analog of massless Dirac action uniquely for both string world sheets and interior of space-time surface. The situation is not clear for the partonic 2-surfaces, which are in special role physically in zero energy ontology (ZEO). There is an infinite number of conserved super currents associated with the modes of the modified Dirac operator defining fermionic super generators. This leads to quantum classical correspondence stating that the eigenvalues of Cartan generators for the fermionic representations of Noether charges are equal to corresponding classical Noether charges defined by the space-time dynamics.
   
   
2. A long-standing question has been whether stringlike objects and partonic 2-surfaces are fundamental dynamical objects or whether they emerge only at the level of effective action. M⁸-H duality states that space-time surfaces M⁸ picture are associative in the sense that either tangent or normal space of space-time surface at any vpoint is associative and therefore quaternionic.
   

   Number theoretic vision suggests that also 2-D objects are fundamental. In number theoretic vision also commutative sub-manifolds of space-time surfaces having induced quaternionic structure reducing to commutative (complex) structure are very natural. Either the tangent space or normal space of 2-surface can be commutative and this gives rise to string world sheets and partonic 2-surfaces as duals of each other just as space-time surfaces have regions for which either tangent spaces or normal spaces are associative (these correspond to regions of space-time with Minkowskian resp. Euclidian signatures of the induced metric). The reduction of the theory to mere string theory is not possible since partonic two surfaces have commutative normal space (partonic 2-surfaces) as part of the tangent space of space-time surface.
   
   

3. What action one should assign with the 2-D objects. The action should be assigned to string world sheets and partonic 2-surfaces representing vertices but the assignent of action with partonic 2-surfaces at the ends of CD does not look natural since they are in the role of initial values. The first guess for the action is as area action. Fermionic action would be fixed uniquely in terms of modified gamma matrices reducing to induced gamma matrices.
   

   Also space-time surfaces in the simplest scenario are minimal surfaces except for a discrete set of singular points at which there is energy transfer between Kähler action and volume term. Something similar is expected also in 2-D case: there must also second part in the action and transfer of Noether changes between the two parts in this set of points.
   

   These points have an identification as point-like particles carrying fermion number and located at partonic 2-surfaces at boundaries of causal diamond (CD) or defining topological vertices so that a classical space-time correlates for twistor diagrams emerge.
   

   Since particles in twistor approaches are associated with the ends of string boundaries at the ends of light-like orbits of partonic 2-surfaces at boundaries of causal diamond (CD), the exceptional points for both space-time surface and string world sheet would correspond to the intersections of string world sheets and partonic 2-surfaces defining also counterparts of vertices.
   
   

Twistor lift provides a first principle approach to the action assignable to the 2-D surfaces.

1. The simplest possibility is that one has also now a Kähler action but now for 4-D space-time surface in the product of twistor spaces of M⁴ and CP₂ dimensional reduced to Cartesian product of twistor sphere S² and 2-D surface representing string world sheet. The assigment of action to partonic 2-surface at the boundary of CD or 2-D generalization of vertex does not look feasible. 4-D Kähler action would be dimensionally reduced to 2-D form and area term.
   
   
2. Field equations contain two terms coming from the variation with respect to the induced metric and Kähler form respectively. The terms coming from the variation with respect to metric vanishes for minimal surfaces since energy momentum tensor is proportional to the induced metric. The term coming from the variation with respect to induced Kähler form need not vanish for minimal surfaces unless there are additional conditions.
   

   The term is of the same form as in 4-D case, which case this term vanishes for holomorphic solutions and also for all known extremals, and there are excellent reasons that this is true also in 2-D case. It therefore seems that minimal surfaces are in question except for discrete set of points as in 4-D case: this conforms with universality forced by quantum criticality stating that Kähler coupling constant disappears from dynamics except in this discrete set of points.
   

   In accordance with SH, this set of points at which minimal surface property fails would define also the correponding points for space-time surface itself. This singularity could mean breakdown of holomorphy, perhaps analogs of poles for analytic functions are in question. One cannot of course exclude the possibility that the boundaries of string world sheets defining orbits of fundamental fermions are analogous to cuts for holomorphic functions.
   
   

3. 2-D minimal surfaces in space-time are also minimal surfaces in imbedding space since the induction from space-time to 2-surface can be also thought of as an induction from imbedding space. SH suggest that space-time as 4-D surface is determined by fixing the 2-D minimal surfaces and finding space-time surface containing them. This space-time surface need not always exist, and one of the key ideas about cognition is that in p-adic case the possibility of p-adic pseudo-constants allows the existence of p-adic space-time surfaces always but that in real case this is not always the case: what is imaginable is not necessarily realizable.
   

   At the level of M⁸ the condition that the coefficients of a polynomial determining the space-time surface are in extension of rationals is very powerful condition and might prevent the extension. As a matter fact, SH becomes at the level of M⁸ even stronger: discrete set of points naturally identifiable as the set of singular points and thus poles of analytic function basically would determine the space-time surface. If fermion lines correspond to cuts, this super-strong form of SH would weaken. For polynomials considerws here cuts are however not possible and they should be generated in the map from H to M⁴× CP₂.
   
   

For details and background see the article Some Questions Related to the Twistor Lift of TGD or the chapter of "Towards M-matrix" with the same title.

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

Articles and other material related to TGD.