Idea before its time: space-time surfaces as Kähler calibrated surfaces

When ideas stop flowing, it is best to stay calm and do something practical. Updating of books or homepage is not rocket science but gives a feeling that one is doing something useful. I realized that 7 books have grown so that they have about thousand pages and decided to divide them to two pieces: the result is that the number of books grew to the magic number 24.

This led to the updating of the introductions of books. I have the habit of writing introductions so that they reflect the latest overall view - books themselves contain older archeological layers and inconsistencies are unavoidable. Also at this time I experienced several not merely pleasant surprises.

A pleasant surprise was that the discrete coupling contant evolution predicted by TGD implying the vanishing of loop corrections, simplifying twistorial scattering amplitudes and their recursion formulas dramatically, and also implying that scattering amplitudes reduce to sums of resonance contributions. I realized that this is nothing but the Veneziano duality, which served as starting point of dual resonance models leading to string picture and later to super string theories.

This suggests a new insight possibly allowing to get out of the dead end of super string models. What would be the really deep thing would be the sum over resonances picture. The continuous cuts are obtained only approximately at the limit when the density of poles becomes large enough.

In string model picture this is not possible since one cannot obtain anything resembling gauge theories. In TGD framework ot seems however possible to circumvent all the objections that I have managed to discover. The first crucial element is that in TGD also classical conserved quantities can be complex (finite width for resonances needed for unitarity). Second crucial element is that string tension has discrete spectrum reducing to that for cosmological constant.

A surprise that looked unpleasant at first was the finding that I had talked about so called calibrations of sub-manifolds as something potentially important for TGD and later forgotten the whole idea! A closer examination however demonstrated that I had ended up with the analog of this notion completely independently later as the idea that preferred extremals are minimal surfaces apart form 2-D singular surfaces, where there would be exchange of Noether charges between Kähler and volume degrees of freedom.

1. The original idea that I forgot too soon was that the notion of calibration (see this) generalizes and could be relevant for TGD. A calibration in Riemann manifold M means the existence of a k-form φ in M such that for any orientable k-D sub-manifold the integral of φ over M equals to its k-volume in the induced metric. One can say that metric k-volume reduces to homological k-volume.
   

   Calibrated k-manifolds are minimal surfaces in their homology class since the variation of the integral of φ is identically vanishing. Kähler calibration is induced by the k^th power of Kähler form and defines calibrated sub-manifold of real dimension 2k. Calibrated sub-manifolds are in this case precisely the complex sub-manifolds. In the case of CP₂ they would be complex curves (2-surfaces) as has become clear.
   
   

2. By the Minkowskian signature of M⁴ metric, the generalization of calibrated sub-manifold so that it would apply in M⁴× CP₂ is non-trivial. Twistor lift of TGD however forces to introduce the generalization of Kähler form in M⁴ (responsible for CP breaking and matter antimatter asymmetry) and calibrated manifolds in this case would be naturally analogs of string world sheets and partonic 2-surfaces as minimal surfaces. Cosmic strings are Cartesian products of string world sheets and complex curves of CP₂. Calibrated manifolds, which do not reduce to Cartesian products of string world sheets and complex surfaces of CP₂ should also exist and are minimal surfaces.
   

   One can also have 2-D calibrated surfaces and they could correspond to string world sheets and partonic 2-surfaces which also play key role in TGD. Even discrete points assignable to partonic 2-surfaces and representing fundamental fermions play a key role and would trivially correspond to calibrated surfaces.
   
   

3. Much later I ended up with the identification of preferred extremals as minimal surfaces by totally different route without realizing the possible connection with the generalized calibrations. Twistor lift and the notion of quantum criticality led to the proposal that preferred extremals for the twistor lift of Kähler action containing also volume term are minimal surfaces. Preferred extremals would be separately minimal surfaces and extrema of Kähler action and generalization of complex structure to what I called Hamilton-Jacobi structure would be an essential element. Quantum criticality outside singular surfaces would be realized as decoupling of the two parts of the action. May be all preferred extremals be regarded as calibrated in generalized sense.
   

   If so, the dynamics of preferred extremals would define a homology theory in the sense that each homology class would contain single preferred extremal. TGD would define a generalized topological quantum field theory with conserved∈dexNoether charge Noether charges (in particular rest energy) serving as generalized topological invariants having extremum in the set of topologically equivalent 3-surfaces.
   
   

4. The experience with CP₂ would suggest that the Kähler structure of M⁴ defining the counterpart of form φ is unique. There is however infinite number of different closed self-dual Kähler forms of M⁴ defining what I have called Hamilton-Jacobi structures. These forms can have subgroups of Poincare group as symmetries. For instance, magnetic flux tubes correspond to given cylindrically symmetry Kähler form. The problem disappears as one realizes that Kähler structures characterize families of preferred extremals rather than M⁴.
   
   

If the notion of calibration indeed generalizes, one ends up with the same outcome - preferred extremals as minimal surfaces with 2-D string world sheets and partonic 2-surfaces as singularities - from many different directions.

1. Quantum criticality requires that dynamics does not depend on coupling parameters so that extremals must be separately extremals of both volume term and Kähler action and therefore minimal surfaces for which these degrees of freedom decouple except at singular 2-surfaces where the necessary transfer of Noether charges between two degrees of freedom takes place at these. One ends up with string picture but strings alone are of course not enough. For instance, the dynamical string tension is determined by the dynamics for the twistor lift.
   
   
2. Almost topological QFT picture implies the same outcome: topological QFT property fails only at the string world sheets.
   
   
3. Discrete coupling constant evolution, vanishing of loop corrections, and number theoretical condition that scattering amplitudes make sense also in p-adic number fields, requires a representation of scattering amplitudes as sum over resonances realized in terms of string world sheets.
   
   
4. In the standard QFT picture about scattering incoming states are solutions of free massless field equations and interaction regions the fields have currents as sources. This picture is realized by the twistor lift of TGD in which the volume action corresponds to geodesic length and Kähler action to Maxwell action and coupling corresponds to a transfer of Noether charges between volume and Kähler degrees of freedom. Massless modes are represented by minimal surfaces arriving inside causal diamond (CD) and minimal surface property fails in the scattering region consisting of string world sheets.
   
   
5. Twistor lift forces M⁴ to have generalize Kähler form and this in turn strongly suggests a generalization of the notion of calibration. At physics side the implication is the understanding of CP breaking and matter anti-matter asymmetry.
   
   
6. M⁸-H duality requires that the dynamics of space-time surfaces in H is equivalent with the algebraic dynamics in M⁸. The effective reduction to almost topological dynamics implied by the minimal surface property implies this. String world sheets (partonic 2-surfaces) in H would be images of complex (co-complex sub-manifolds) of X⁴⊂ M⁸ in H. This should allows to understand why the partial derivatives of imbedding space coordinates can be discontinuous at these edges/folds but there is no flow between interior and singular surface implying that string world sheets are minimal surfaces (so that one has conformal invariance).
   
   
   

See the chapter The Recent View about Twistorialization in TGD Framework or the article More about the construction of scattering amplitudes in TGD framework.

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

Articles and other material related to TGD.