Twistors and the relationship of TGD to GRT

For year or two ago I ended up with a vision about how twistor approach could generalize to TGD framework. A more explicit realization of twistorialization as lifting of the preferred extremal X⁴ of Kähler action to corresponding 6-D twistor space X⁶ identified as surface in the 12-D product of twistor spaces of M⁴ and CP₂ allowing Kähler structure suggests itself: this makes these spaces completely unique twistorially and seems more or less obvious that the Kähler structure must have profound physical meaning. It turned out that it has: the projection of Kähler form defines the representation of preferred quaternionic imaginary unit needed to assign twistor structure to space-time surface. Almost equally obvious idea is that the lifting of the dynamics for space-time surface to that for its twistor space in the product of twistor spaces of M⁴ and CP₂ must be based on 6-D Kähler action.

Contrary to the original expectations, the twistorial approach is not mere reformulation but leads to a first principle identification of cosmological constant and perhaps also of gravitational constant and to a modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.

There are some new results forcing a profound modification of the recent view about TGD but consistent with the general picture. A more explicit realization of twistorialization as lifting of the preferred extremal X⁴ of Kähler action to corresponding 6-D twistor space X⁶ identified as surface in the 12-D product of twistor spaces of M⁴ and CP₂ allowing Kähler structure suggests itself.

The action principle in 6-D context is also Kähler action, which dimensionally reduces to Kähler action plus cosmological term. This brings in the radii of spheres S² associated with the twistor space of CP₂ presumably determined by CP₂ radius and radius of S² associated with M⁴ twistor space for which an attractive identification is as Planck length, which would be now purely classical parameter. The radius of S² associated with space-time surface is determined by induced metric and is emergent length scale. The normalization of 6-D Kähler action by a scale factor with dimension which is inverse length squared brings in a further length scale closely related to cosmological constant which is also dynamical and has correct sign to explain accelerated expansion of the Universe.

The dimensionally reduced dynamics is a highly non-trivial modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.

In the sequel I will discuss the recent understanding of twistorizalization, which is considerably improved from that in the earlier formulation. I formulate the dimensional reduction of 6-D Kähler action and consider the physical interpretation. After that I proceed to discuss the basic principles behind the recent view about scattering amplitudes as generalized Feynman diagrams.

1. Some mathematical background

First I will try to clarify the mathematical details related to the twistor spaces and how they emerge in the recent context. I do not regard myself as a mathematician in technical sense and I can only hope that the representation based on physical intuition does not contain serious mistakes.

1.1. Imbedding space is twistorially unique

It took roughly 36 years to learn that M⁴ and CP₂ are twistorially unique. Space-times are surfaces in H=M⁴× CP₂. M⁴ and CP₂ are unique 4-manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M⁴ and its Euclidian variant E⁴ allow both twistor space and the twistor space of M⁴ is Minkowskian variant T(M⁴)= SU(2,2)/SU(2,1)× U(1) of 6-D twistor space CP₃= SU(4)/SU(3)× U(1) of E⁴. The twistor space of CP₂ is 6-D T(CP₂)= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin.

This leads to a proposal - just a proposal - for the formulation of TGD in which space-time surfaces X⁴ in H are lifted to twistor spaces X⁶, which are sphere bundles over X⁴ and such that they are surfaces in 12-D product space T(M⁴)× T(CP₂) such the twistor structure of X⁴ are in some sense induced from that of T(M⁴)× T(CP₂). What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (Calabi-Yau manifolds).

1.2 What does twistor structure in Minkowskian signature really mean?

What twistor structure in Minkowskian signature really means geometrically has remained a confusing question for me. The problems associated with the Minkowskian signature of the metric are encountered also in twistor Grassmann approach to scattering amplitudes but are circumvented by performing Wick rotation that is using E⁴ or S⁴ instead of M⁴ and applying algebraic continuation. Also complexification of Minkowksi space for momenta is used. These tricks do not apply now.

Let us try to collect thoughts about what is involved.

1. Instead of M⁴ one considers the conformal compactification M⁴_c of M⁴ identifiable as the boundary of light-cone boundary of 6-D Minkowski space with signature (1,1,-1,-1,-1), whose points differing by scaling are identified. One has a slicing by spheres of signature (-1,-1,-1) and varying radius ρ and these spheres are projectively identified so that one can "fix the gauge" by choosing ρ=ρ₀. Since one has light-cone, the contribution dρ² to the line element vanishes and one obtains ds²= ρ₀²dφ²- ρ²₀ ds²(S³). Conformal compactification means that the scale ρ₀ of the metric is not unique. The scaling of the metric of the twistor space ρ₀². Conformal invariance of the theory saves from problems.
   
   
2. The Euclidian version of the twistor space of M⁴ corresponds to the twistor space of S⁴ identifiable as CP₃= SU(4)/SU(3)× U(1) identifiable in terms of complex 2+2-spinors. The twistor space of M⁴_c is SU(2,2)/SU(2,1)× U(1) (see this) and can be seen as a kind of algebraic continuation of CP₃=SU(4)/SU(3)× U(1). This space is complex manifold but it is not completely clear to me whether this really guarantees the existence of Kähler structure consistent with the complex structure.
   
   
3. If the Minkowskian variant of the twistor space (rather than only that associated with S⁴) is to have complex structure in the ordinary sense of the word, its metric must have even signature. M⁴_c has signature (1,-1-1,-1) so that the signature of the analog of S² fiber should have signature which is (1,-1) to give even signature (1,-1,1-1,-1,-1) for the twistor space. The sphere S² would be replaced with its non-compact hyperbolic counterpart SO(2,1)/SO(1,1) and has metric signature (1,-1). One cannot assign to it finite size in the usual sense. However, since this space corresponds to hyperboloid t²-x²-y²=-R² (3-D mass shell), one can assign to it finite hyperbolic radius R_H. There are however problems.
   
   
   1. One cannot assign to H² finite size in the usual sense. However, since this space corresponds to hyperboloid t²-x²-y²=-R² (3-D mass shell), one could assign to it finite hyperbolic radius R_H. In dimensional reduction of 6-D Kähler action however the integral over H² gives its area if the restriction of J to H² has square equal to metric as is extremely natural to assume. The area is R_H² times an infinite number and 4-D dimensionally reduced action would have infinite value. At the limit R_H=0 (2-D light-cone boundary) the area vanishes as also the dimensionally reduced action.
      
      
   2. For even signature of twistor space the determinant of the induced 6-metric would be real in both Euclidian and Minkowskian space-time regions. Both Euclidian Minkowskian regions contribute to Kähler function (as was the original wrong assumption using |det(g)^1/2| in volume element). The exponent of Kähler action in Minkowskian regions would not define phase as QFT picture demands.
      
      
   3. This picture is in conflict with the vision about the fiber as space S² of directions defined by antisymmetric forms. The hidden assumption is that one has field of preferred time-like directions n and one considers induced Kähler form at space-like 3-surface with metric signature (-1,-1,-1) with n as time-like normal field.
      

      Could one imagine fixing of space-like direction field defining normals for a slicing by 3-surfaces with metric signature (1,-1,-1)? If so, one would end up with SO(2,1)/SO(1,1) as the fiber characterizing directions of projections of J to this subspace. The slicing by 3-surfaces parallel to the light-like 3-surfaces at the boundaries of Minkowskian and Euclidian space-time regions could indeed do the job. The light-likeness of these 3-surfaces also fits nicely with conformal invariance. The above problems are however enough to guarantee that the lifetime of H² option was rather precisely 24 hours.
      

4. The only alternative, which comes in mind is a hypercomplex generalization of the Kähler structure. This requires that the metric of the Minkowskian twistor space has signature (-1,-1,1,-1,-1,-1). This would give 1 time-like direction and the hypercomplex coordinate would correspond to a sub-space with signature (1,-1). Hypercomplex coordinate can be represented as h=t+iez, i²=-1,e²=-1. Kähler form representing imaginary unit would be replaced with eJ. One would consider sub-spaces of complexified quaternions spanned by real unit and units eI_k, k=1,2,3 as representation of the tangent space of space-time surfaces in Minkowskian regions. This is familiar already from M⁸ duality (see this).
   

   One could regard Minkowskian twistor space as a kind of Wick rotation of the Euclidian twistor space. Hyper-complex numbers do not define number field since for light-like hypercomplex numbers t+iez, t=+/- z do not have finite inverse. Hypercomplex numbers allow a generalization of analytic functions used routinely in physics. Fiber would be sphere S² with metric signature (-1,-1). Cosmological term would be finite and the sign of the cosmological term in the dimensionally reduced action would be positive as required. Also metric determinant would be imaginary as required. At this moment I cannot invent any killer objection against this option.
   

1.3 What the induction of twistor structure could mean?

To proceed one must make explicit the definition of twistor space. The 2-D fiber S² consists of antisymmetric tensors of X⁴ which can be taken to be self-dual or anti-self-dual by taking any antisymmetric form and by adding to its plus/minus its dual. Each tensor of this kind defines a direction - point of S². These points can be also regarded as quaternionic imaginary units. One has a natural metric in S² defined by the X⁴ inner product for antisymmetric tensors: this inner product depends on space-time metric. Kähler action density is example of a norm defined by this inner product in the special case that the antisymmetric tensor is induced Kähler form. Induced Kähler form defines a preferred imaginary unit and is needed to define the imaginary part ω(X,Y)= ig(X,-JY) of hermitian form h= h+iω.

Consider now what the induction of twistor structure could mean.

1. The induction procedure for Kähler structure of 12-D twistor space T requires that the induced metric and Kähler form of the base space X⁴ of X⁶ obtained from T is the same as that obtained by inducing from H=M⁴× CP₂. Since the Kähler structure and metric of T is lift from H this seems obvious. Projection would compensate the lift.
   
   
2. This is not yet enough. The Kähler structure and metric of F projected from T must be same as those lifted from X⁴. The connection between metric and ω implies that this condition for Kähler form is enough. The antisymmetric Kähler forms in fiber obtained in these two manners co-incide. Since Kähler form has only one component in 2-D case, one obtains single constraint condition giving a commutative diagram stating that the direct projection to F equals with the projection to the base followed by a lift to fiber. The resulting induced Kähler form is not covariantly constant but in fiber F one has J²=-g.
   

   As a matter of fact, this condition might be trivially satisfied as a consequence of the bundle structure of twistor space. The Kähler form from S²× S² can be projected to S² associated with X⁴ and by bundle projection to a two-form in X⁴. The intuitive guess - which might be of course wrong - is that this 2-form must be same as that obtained by projecting the Kähler form of CP₂ to X⁴. If so then the bundle structure would be essential but what does it really mean?
   
   

3. Intuitively it seems clear that X⁶ must decompose locally to a product X⁴× S² in some sense. This is true if the metric and Kähler form reduce to direct sums of contributions from the tangent spaces of X⁴ and S². This guarantees that 6-D Kähler action decomposes to a sum of 4-D Kähler action and Kähler action for S².
   

   This could be however too strong a condition. Dimensional reduction occurs in Kaluza-Klein theories and in this case the metric can have also components between tangent spaces of the fiber and base being interpreted as gauge potentials. This suggests that one should formulate the condition in terms of the matrix T↔ g^αμg^βν-g^ανg^βμ defining the norm of the induced Kähler form giving rise to Kähler action. T maps Kähler form J↔ J_αβ to a contravariant tensor J_c↔ J^αβ and should have the property that J_c(X⁴) (J_c( S²)) does not depend on J( S²) (J(X⁴)).
   

   One should take into account also the self-duality of the form defining the imaginary unit. In X⁴ the form S=J+/- *J is self-dual/anti-self dual and would define twistorial imaginary unit since its square equals to -g representing the negative of the real unit. This would suggest that 4-D Kähler action is effectively replaced with (J+/- *J)∧(J+/- *J)/2 =J^*∧J +/- J∧J, where *J is the Hodge dual defined in terms of 4-D permutation tensor ε. The second term is topological term (Abelian instanton term) and does not contribute to field equations. This in turn would mean that it is the tensor T+/- ε for which one can demand that S_c(X⁴) (S_c(S²)) does not depend on S(S²) (S(X⁴)).
   

2. Surprise: twistorial dynamics does not reduce to a trivial reformulation of the dynamics of Kähler action

I have thought that twistorialization classically means only an alternative formulation of TGD. This is definitely not the case as the explicit study demonstrated. Twistor formulation of TGD is in terms of of 6-D twistor spaces T(X⁴) of space-time surfaces X⁴⊂ M⁴× CP₂ in 12-dimensional product T=T(M⁴)× T(CP₂) of 6-D twistor spaces of T(M⁴) of M⁴ and T(CP₂) of CP₂. The induced Kähler form in X⁴ defines the quaternionic imaginary unit defining twistor structure: how stupid that I realized it only now! I experienced during single night many other "How stupid I have been" experiences.

Classical dynamics is determined by 6-D variant of Kähler action with coefficient 1/L² having dimensions of inverse length squared. Since twistor space is bundle, a dimensional reduction of 6-D Kähler action to 4-D Kähler action plus a term analogous to cosmological term - space-time volume - takes place so that dynamics reduces to 4-D dynamics also now. Here one must be careful: this happens provided the radius of F associated with X⁴ does not depend on point of X⁴. The emergence of cosmological term was however completely unexpected: again "How stupid I have been" experience. The scales of the spheres and the condition that the 6-D action is dimensionless bring in 3 fundamental length scales!

2.1 New scales emerge

The twistorial dynamics gives to several new scales with rather obvious interpretation. The new fundamental constants that emerge are the radius of hyperbolic sphere associated with T(M⁴) and of sphere associated with T(CP₂). The radius of the fiber associated with X⁴ is not a fundamental constant but determined by the induced metric. By above argument the fiber is sphere for Euclidian signature and hyperbolic sphere for Minkowskian signature.

1. For CP₂ twistor space the radius of S² must be apart from numerical constant equal to CP₂ radius R. For M⁴ the simplest assumption is that also now the radius for S²(M⁴ equals to R(M⁴=R so that Planck length would not emerge from fundamental theory classically. Second option is that it does and one has R(M⁴=l_P.
   
   
2. If the signature of S²(M⁴) is (-1,-1) both Minkowskian and Euclidian space-time regions have S²(X⁴) with the same signature (-1,-1)/. The radius R_D of S²(X⁴) is dynamically determined.
   
   
   

Recall first how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.

1. The first term is essentially the ordinary Kähler action multiplied by the area of S²(X⁴), which is compensated by the length scale, which can be taken to be the area 4π R²(M⁴) of S²(M⁴). This makes sense for winding numbers (w₁,w₂)=(n,0) meaning that S²(CP₂) is effectively absent but S²(M⁴) is present.
   
   
2. Second term is the analog of Kähler action assignable assignable to the projection of S²(M⁴) Kähler form. The corresponding Kähler coupling strength α_K (M⁴) is huge - so huge that one has
   

   α_K (M⁴)4π R²(M⁴)== L² ,
   

   where 1/L² is of the order of cosmological constant and thus of the order of the size of the recent Universe. α_K(M⁴) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as α_K(M⁴) ∝ p≈ 2^k, p prime, k prime.
   
   

3. The Kähler form assignable to M⁴ is not assumed to contribute to the action since it does not contribute to spinor connection of M⁴. One can of course ask whether it could be present. For canonically imbedded M⁴ self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If α_K(M⁴) is given by above expression, then this contribution is extremely small.
   
   

Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M⁴) and T(CP₂) but with different values of α_K if one has (w₁,w₂)=(n,0). Also other w₂≠ 0 is possible but corresponds to gigantic cosmological constant.

Given the parameter L² as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive.

1. 6-D Kähler action has dimensions of length squared and one must scale it by a dimensional constant: call it 1/L². L is a fundamental scale and in dimensional reduction it gives rise to cosmological constant. Cosmological constant Λ is defined in terms of vacuum energy density as Λ =8π Gρ_vac can have two interpretations. Λ can correspond to a modification of Einstein-Hilbert action or - as now - to an additional term in the action for matter. In the latter case positive Λ means negative pressure explaining the observed accelerating expansion. It is actually easy to deduce the sign of Λ.
   

   1/L² multiplies both Kähler action - F^ijF_ij (∝ E²-B² in Minkowskian signature). The energy density is positive. For Kähler action the sign of the multiplier must be positive so that 1/L² is positive. The volume term is fiber space part of action having same form as Kähler action. It gives a positive contribution to the energy density and negative contribution to the pressure.
   

   In Λ= 8π Gρ_vac one would have ρ_vac=π/L²R_D² as integral of the -F^ijF_ij over S² given the π/R_D² (no guarantee about correctness of numerical constants). This gives Λ= 8π²G/L²R_D². Λ is positive and the sign is same as as required by accelerated cosmic expansion. Note that super string models predict wrong sign for Λ. Λ is also dynamical since it depends on R_D, which is dynamical. One has 1/L² =kΛ, k=8π²G/R_D² apart from numerical factors.
   

   The value of L of deduced from Euclidian and Minkowskian regions in this formal manner need not be same. Since the GRT limit of TGD describes space-time sheets with Minkowskian signature, the formula seems to be applicable only in Minkowskian regions. Again one can argue that one cannot exclude Euclidian space-time sheets of even macroscopic size and blackholes and even ordinary concept matter would represent this kind of structures.
   
   

2. L is not size scale of any fundamental geometric object. This suggests that L is analogous to α_K and has value spectrum dictated by p-adic length scale hypothesis. In fact, one can introduce the ratio of ε=R²/L² as a dimensionless parameter analogous to coupling strength what it indeed is in field equations. If so, L could have different values in Minkowskian and Euclidian regions.
   
   
3. I have earlier proposed that R_U==(1/Λ)^1/2 is essentially the p-adic length scale L_p ∝ p^1/2= 2^k/2, p≈ 2^k, k prime, characterizing the cosmology at given time and satisfies R_U∝ a meaning that vacuum energy density is piecewise constant but on the average decreases as 1/a², a cosmic time defined by light-cone proper time. A more natural hypothesis is that L satisfies this condition and in turn implies similar behavior or R_U. p-Adic length scales would be the critical values of L so that also p-adic length scale hypothesis would emerge from quantum critical dynamics! This conforms with the hypothesis about the value spectrum of α_K labelled in the same manner (see this).
   
   
4. At GRT limit the magnetic energy of the flux tubes gives rise to an average contribution to energy momentum tensor, which effectively corresponds to negative pressure for which the expansion of the Universe accelerates. It would seem that both contributions could explain accelerating expansion. If the dynamics for Kähler action and volume term are coupled, one would expect same orders of magnitude for negative pressure and energy density - kind of equipartition of energy.
   
   

Consider first the scales emerging from GRT picture. R_U ∼ (1/Λ1/2∼ 10²⁶ m = 10 Gly is not far from the recent size of the Universe defined as c× t ∼ 13.8 Gly. The derived size scale L₁==(R_U× l_P)^1/2 is of the order of L₁=.5× 10^-4 meters, the size of neuron. Perhaps this is not an accident. To make life of the reader easier I have collected the basic numbers to the following table.

m(CP2)≈ 5.7× 1014 GeV	mP=2.435 × 1018 GeV	R(CP2)/lP≈ 4.1× 103
RU= 10 Gy	t= 13.8 Gy	L1= (lPRU)1/2=.5 × 10-4

Let us consider now some quantitative estimates. R(X⁴) depends on homotopy equivalence classes of the maps from S²(X⁴)→ S²(M⁴) and S²(X⁴)→ S²(CP₂) - that is winding numbers w_i , i=0,2 for these maps. The simplest situations correspond to the winding numbers (w₁,w₂)=(1,0) and (w₁,w₂)=(0,1) . For (w₁,w₂)=(1,0) M⁴ contribution to the metric of S²(X⁴) dominates and one has R(X⁴)≈ R(M⁴) . For R(M⁴)=l_P so Planck length would define a fundamental length and Planck mass and Newton's constant would be quantal parameters. For (w₁,w₂)=(0,1) the radius of sphere would satisfy R_D≈ R ( CP₂ size): now also Planck length would be quantal parameter.

Consider next additional scales emerging from TGD picture.

1. One has L = ( 2^3/2π l_P/R_D)× R_U. In Minkowskian regions with R_H=l_P this would give L = 8.9× R_U: there is no obvious interpretation for this number. If one takes the formula seriously in Euclidian regions one obtains the estimate L=29 Mly. The size scale of large voids varies from about 36 Mly to 450 Mly (see this).
   
   
2. Consider next the derived size scale L₂=(L× l_P)^1/2 = [L/R_U]^1/2 × L₁ = [2^3/2π l_P/R_D]^1/2× L₁. For R_D=l_P one has L₂ ≈ 3L₁. For R_D=R making sense in Euclidian regions, this is of the order of size of neutrino Compton length: 3 μm, the size of cellular nucleus and rather near to the p-adic length scale L(167)= 2.6 m, corresponds to the largest miracle Gaussian Mersennes associated with k=151,157,163,167 defining length scales in the range between cell membrane thickness and the size of cellular nucleus. Perhaps these are co-incidences are not accidental. Biology is something so fundamental that fundamental length scale of biology should appear in the fundamental physics.
   

The formulas and predictions for different options are summarized by the following table.

Option	L=[23/2π lP/RD]× RU	L2=(LlP)1/2 = [23/2π lP/RD]1/2× L1
RD= R	29 Mly	≈ 3 μ m
RD=lP	8.9RU	≈ 3L1=1.5× 10-4 m

In the case of M⁴ the radius of S² cannot be fixed it remains unclear whether Planck length scale is fundamental constant or whether it emerges.

2.2 What about extremals of the dimensionally reduced 6-D Kähler action?

It seems that the basic wisdom about extremals of Kähler action remains unaffected and the motivations for WCW are not lost. What is new is that the removal of vacuum degeneracy is forced by twistorial action.

1. All extremals, which are either vacuum extremals or minimal surfaces remain extremals. In fact, all extremals that I know. For minimal surfaces the dynamics of the volume term and 4-D Kähler action separate and field equations for them are separately satisfied. The vacuum degeneracy motivating the introduction of WCW is preserved. The induced Kähler form vanishes for vacuum extremals and the imaginary unit of twistor space is ill-defined. Hence vacuum extremals cannot belong to WCW. This correspond to the vanishing of WCW metric for vacuum extremals.
   
   
2. For non-minimal surfaces Kähler coupling strength does not disappear from the field equations and appears as a genuine coupling very much like in classical field theories. Minimal surface equations are a generalization of wave equation and Kähler action would define analogs of source terms. Field equations would state that the total isometry currents are conserved. It is not clear whether other than minimal surfaces are possible, I have even conjectured that all preferred extremals are always minimal surfaces having the property that being holomorphic they are almost universal extremals for general coordinate invariant actions.
   
   
3. Thermodynamical analogy might help in the attempts to interpret. Quantum TGD in zero energy ontology (ZEO) corresponds formally to a complex square root of thermodynamics. Kähler action can be identified as a complexified analog of free energy. Complexification follows both from the fact that g^1/2 is real/imaginary in Euclidian/Minkowskian space-time regions. Complex values are also implied by the proposed identification of the values of Kähler coupling strength in terms of zeros and pole of Riemann zeta in turn identifiable as poles of the so called fermionic zeta defining number theoretic partition function for fermions (see this). The thermodynamical for Kähler action with volume term is Gibbs free energy G= F-TS= E-TS+PV playing key role in chemistry.
   
   
4. The boundary conditions at the ends of space-time surfaces at boundaries of CD generalize appropriately and symmetries of WCW remain as such. At light-like boundaries between Minkowskian and Euclidian regions boundary conditions must be generalized. In Minkowkian regions volume can be very large but only the Euclidian regions contribute to Kähler function so that vacuum functional can be non-vanishing for arbitrarily large space-time surfaces since exponent of Minkowskian Kähler action is a phase factor.
   
   
5. One can worry about almost topological QFT property. Although Kähler action from Minkowskian regions at least would reduce to Chern-Simons terms with rather general assumptions about preferred extremals, the extremely small cosmological term does not. Could one say that cosmological constant term is responsible for "almost"?
   

   It is interesting that the volume of manifold serves in algebraic geometry as topological invariant for hyperbolic manifolds, which look locally like hyperbolic spaces H_n=SO(n,1)/SO(n). See also the article Volumes of hyperbolic manifolds and mixed Tate motives. Now one would have n=4. It is probably too much to hope that space-time surfaces would be hyperbolic manifolds. In any case, by the extreme uniqueness of the preferred extremal property expressed by strong form of holography the volume of space-time surface could also now serve as topological invariant in some sense as I have earlier proposed. What is intriguing is that AdS_n appearing in AdS/CFT correspondence is Lorentzian analogue H_n.
   

To sum up, the twistor lift of the dynamics of Kähler action allows to understand the origin of Planck length and cosmological constant. Here the earlier picture has been incomplete. Also the size scale of large voids and two fundamental biological length scales appear. p-Adic length scale hypothesis is realized in terms of the scaling factor of the 6-D Käler action defining giving rise to a dimensionless coupling constant. What is most remarkable that since only M⁴ and CP₂ allow twistor space with Kähler structure, TGD is completely unique in twistor formulation.

Addition: I have modified the posting. I had a short one night-relationship with the attractive looking idea that the fiber of Minkowskian twistor space could be hyperbolic sphere H² with Minkowskian signature of metric. I however realized my mistake and returned back to S² from my nightly wanderings and did my best to delete all traces about what happened. I deeply regret my digression. I was also fluctuating between two alternatives for the signature of S². Totally timelike (3 time-like dimensions altogether!) or totally space- like. The latter option turned out to be correct: I have been correct about un-physicality of 3 time-like dimensions. A further issue was whether Planck length is fundamental length or whether it emerges: it seems to be fundamental length.

See the the article From Principles to Diagrams or the chapter From Principles to Diagrams of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.