A longstanding question has been the origin of preferred padic primes characterizing
elementary particles. I have proposed several explanations and the most convincing hitherto
is related to the algebraic extensions of rationals and padic numbers selecting naturally
preferred primes as those which are ramified for the extension in question.
See the chapter Unified Number Theoretic Vision .

In the previous posting I told about the possibility that string world sheets with area action could be present in TGD at fundamental level with the ratio of hbar G/R^{2} of string tension to the square of CP_{2} radius fixed by quantum criticality. I however found that the assumption that gravitational binding has as correlates strings connecting the bound partonic 2surfaces leads to grave difficulties: the sizes of the gravitationally bound states cannot be much longer than Planck length. This binding mechanism is strongly suggested by AdS/CFT correspondence but perturbative string theory does not allow it.
I proposed that the replacement of h with h_{eff} = n× h= h_{gr}= GMm/v_{0} could resolve the problem. It does not. I soo noticed that the typical size scale of string world sheet scales as h_{gr}^{1/2}, not as h_{gr}= GMm/v_{0} as one might expect. The only reasonable option is that string tension behave as 1/h_{gr}^{2}. In the following I demonstrate that TGD in its basic form and defined by supersymmetrized Kähler action indeed predicts this behavior if string world sheets emerge. They indeed do so number theoretically from the condition of associativity and also from the condition that electromagnetic charge for the spinor modes is welldefined. By the analog of AdS/CFT correspondence the string tension could characterize the action density of magnetic flux tubes associated with the strings and varying string tension would correspond to the effective string tension of the magnetic flux tubes as carriers of magnetic energy (dark energy is identified as magnetic energy in TGD Universe).
Therefore the visit of string theory to TGD Universe remained rather short but it had a purpose: it made completely clear why superstring are not the theory of gravitation and why TGD can be this theory.
Do associativty and commutativity define the laws of physics?
The dimensions of classical number fields appear as dimensions of basic objects in quantum TGD. Imbedding space has dimension 8, spacetime has dimension 4, lightlike 3surfaces are orbits of 2D partonic surfaces. If conformal QFT applies to 2surfaces (this is questionable), onedimensional structures would be the basic objects. The lowest level would correspond to discrete sets of points identifiable as intersections of real and padic spacetime sheets. This suggests that besides padic number fields also classical number fields (reals, complex numbers, quaternions, octonions are involved and the notion of geometry generalizes considerably. In the recent view about quantum TGD the dimensional hierarchy defined by classical number field indeed plays a key role. H=M^{4}× CP_{2} has a number theoretic interpretation and standard model symmetries can be understood number theoretically as symmetries of hyperquaternionic planes of hyperoctonionic space.
The associativity condition A(BC)= (AB)C suggests itself as a fundamental physical law of both classical and quantum physics. Commutativity can be considered as an additional condition. In conformal field theories associativity condition indeed fixes the npoint functions of the theory. At the level of classical TGD spacetime surfaces could be identified as maximal associative (hyperquaternionic) submanifolds of the imbedding space whose points contain a preferred hypercomplex plane M^{2} in their tangent space and the hierarchy finite fieldsrationalsrealscomplex numbersquaternionsoctonions could have direct quantum physical counterpart. This leads to the notion of number theoretic compactification analogous to the dualities of Mtheory: one can interpret spacetime surfaces either as hyperquaternionic 4surfaces of M^{8} or as 4surfaces in M^{4}× CP_{2}. As a matter fact, commutativity in number theoretic sense is a further natural condition and leads to the notion of number theoretic braid naturally as also to direct connection with super string models.
At the level of modified Dirac action the identification of spacetime surface as a hyperquaternionic submanifold of H means that the modified gamma matrices of the spacetime surface defined in terms of canonical momentum currents of Kähler action using octonionic representation for the gamma matrices of H span a hyperquaternionic subspace of hyperoctonions at each point of spacetime surface (hyperoctonions are the subspace of complexified octonions for which imaginary units are octonionic imaginary units multiplied by commutating imaginary unit). Hyperoctonionic representation leads to a proposal for how to extend twistor program to TGD framework .
How to achieve associativity in the fermionic sector?
In the fermionic sector an additional complication emerges. The associativity of the tangent or normal space of the spacetime surface need not be enough to guarantee the associativity at the level of KählerDirac or Dirac equation. The reason is the presence of spinor connection. A possible cure could be the vanishing of the components of spinor connection for two conjugates of quaternionic coordinates combined with holomorphy of the modes.
 The induced spinor connection involves sigma matrices in CP_{2} degrees of freedom, which for the octonionic representation of gamma matrices are proportional to octonion units in Minkowski degrees of freedom. This corresponds to a reduction of tangent space group SO(1,7) to G_{2}. Therefore octonionic Dirac equation identifying Dirac spinors as complexified octonions can lead to nonassociativity even when spacetime surface is associative or coassociative.
 The simplest manner to overcome these problems is to assume that spinors are localized at 2D string world sheets with 1D CP_{2} projection and thus possible only in Minkowskian regions. Induced gauge fields would vanish. String world sheets would be minimal surfaces in M^{4}× D^{1}⊂ M^{4}× CP_{2} and the theory would simplify enormously. String area would give rise to an additional term in the action assigned to the Minkowskian spacetime regions and for vacuum extremals one would have only strings in the first approximation, which conforms with the success of string models and with the intuitive view that vacuum extremals of Kähler action are basic building bricks of manysheeted spacetime. Note that string world sheets would be also symplectic covariants.
Without further conditions gauge potentials would be nonvanishing but one can hope that one can gauge transform them away in associative manner. If not, one can also consider the possibility that CP_{2} projection is geodesic circle S^{1}: symplectic invariance is considerably reduces for this option since symplectic transformations must reduce to rotations in S^{1}.
 The fist heavy objection is that action would contain Newton's constant G as a fundamental dynamical parameter: this is a standard recipe for building a nonrenormalizable theory. The very idea of TGD indeed is that there is only single dimensionless parameter analogous to critical temperature. One can of coure argue that the dimensionless parameter is hbarG/R^{2}, R CP_{2} "radius".
Second heavy objection is that the Euclidian variant of string action exponentially damps out all string world sheets with area larger than hbar G. Note also that the classical energy of Minkowskian string would be gigantic unless the length of string is of order Planck length. For Minkowskian signature the exponent is oscillatory and one can argue that wild oscillations have the same effect.
The hierarchy of Planck constants would allow the replacement hbar→ hbar_{eff} but this is not enough. The area of typical string world sheet would scale as h_{eff} and the size of CD and gravitational Compton lengths of gravitationally bound objects would scale (h_{eff})^{1/2} rather than h_{eff} = GMm/v_{0} which one wants. The only way out of problem is to assume T ∝ (hbar/h_{eff})^{2}. This is however unnatural for genuine area action. Hence it seems that the visit of the basic assumption of superstring theory to TGD remains very short. In any case, if one assumes that string connect gravitationally bound masses, super string models in perturbative description are definitely wrong as physical theories as has of course become clear already from landscape catastrophe.
Is supersymmetrized KählerDirac action enough?
Could one do without string area in the action and use only KD action, which is in any case forced by the superconformal symmetry? This option I have indeed considered hitherto. KD Dirac equation indeed tends to reduce to a lowerdimensional one: for massless extremals the KD operator is effectively 1dimensional. For cosmic strings this reduction does not however take place. In any case, this leads to ask whether in some cases the solutions of KählerDirac equation are localized at lowerdimensional surfaces of spacetime surface.
 The proposal has indeed been that string world sheets carry vanishing W and possibly even Z fields: in this manner the electromagnetic charge of spinor mode could be welldefined. The vanishing conditions force in the generic case 2dimensionality.
Besides this the canonical momentum currents for Kähler action defining 4 imbedding space vector fields must define an integrable distribution of two planes to give string world sheet. The four canonical momentum currents Π_{k}^{α}= ∂ L_{K}/∂_{∂α hk} identified as imbedding 1forms can have only two linearly independent components parallel to the string world sheet. Also the Frobenius conditions stating that the two 1forms are proportional to gradients of two imbedding space coordinates Φ_{i} defining also coordinates at string world sheet, must be satisfied. These conditions are rather strong and are expected to select some discrete set of string world sheets.
 To construct preferred extremal one should fix the partonic 2surfaces, their lightlike orbits defining boundaries of Euclidian and Minkowskian spacetime regions, and string world sheets. At string world sheets the boundary condition would be that the normal components of canonical momentum currents for Kähler action vanish. This picture brings in mind strong form of holography and this suggests that might make sense and also solution of Einstein equations with point like sources.
 The localization of spinor modes at 2D surfaces would would follow from the welldefinedness of em charge and one could have situation is which the localization does not occur. For instance, covariantly constant righthanded neutrinos spinor modes at cosmic strings are completely delocalized and one can wonder whether one could give up the localization inside wormhole contacts.
 String tension is dynamical and physical intuition suggests that induced metric at string world sheet is replaced by the anticommutator of the KD gamma matrices and by conformal invariance only the conformal equivalence class of this metric would matter and it could be even equivalent with the induced metric. A possible interpretation is that the energy density of Kähler action has a singularity localized at the string world sheet.
Another interpretation that I proposed for years ago but gave up is that in spirit with the TGD analog of AdS/CFT duality the Noether charges for Kähler action can be reduced to integrals over string world sheet having interpretation as area in effective metric. In the case of magnetic flux tubes carrying monopole fluxes and containing a string connecting partonic 2surfaces at its ends this interpretation would be very natural, and string tension would characterize the density of Kähler magnetic energy. String model with dynamical string tension would certainly be a good approximation and string tension would depend on scale of CD.
 There is also an objection. For M^{4} type vacuum extremals one would not obtain any nonvacuum string world sheets carrying fermions but the successes of string model strongly suggest that string world sheets are there. String world sheets would represent a deformation of the vacuum extremal and far from string world sheets one would have vacuum extremal in an excellent approximation. Situation would be analogous to that in general relativity with point particles.
 The hierarchy of conformal symmetry breakings for KD action should make string tension proportional to 1/h_{eff}^{2} with h_{eff}=h_{gr} giving correct gravitational Compton length Λ_{gr}= GM/v_{0} defining the minimal size of CD associated with the system. Why the effective string tension of string world sheet should behave like (hbar/hbar_{eff})^{2}?
The first point to notice is that the effective metric G^{αβ} defined as h^{kl}Π_{k}^{α}Π_{l}^{β}, where the canonical momentum current Π_{k}α=∂ L_{K}/∂_{∂α hk} has dimension 1/L^{2} as required. Kähler action density must be dimensionless and since the induced Kähler form is dimensionless the canonical momentum currents are proportional to 1/α_{K}.
Should one assume that α_{K} is fundamental coupling strength fixed by quantum criticality to α_{K}≈1/137? Or should one regard g_{K}^{2} as fundamental parameter so that one would have 1/α_{K}= hbar_{eff}/4π g_{K}^{2} having spectrum coming as integer multiples (recall the analogy with inverse of critical temperature)?
The latter option is the in spirit with the original idea stating that the increase of h_{eff} reduces the values of the gauge coupling strengths proportional to α_{K} so that perturbation series converges (Universe is theoretician friendly). The nonperturbative states would be critical states. The nondeterminism of Kähler action implying that the 3surfaces at the boundaries of CD can be connected by large number of spacetime sheets forming n conformal equivalence classes. The latter option would give G^{αβ ∝ heff2 and det(G) ∝ 1/heff2 as required.
}
 It must be emphasized that the string tension has interpretation in terms of gravitational coupling on only at the GRT limit of TGD involving the replacement of manysheeted spacetime with single sheeted one. It can have also interpretation as hadronic string tension or effective string tension associated with magnetic flux tubes and telling the density of Kähler magnetic energy per unit length.
Superstring models would describe only the perturbative Planck scale dynamics for emission and absorption of h_{eff}/h=1 on mass shell gravitons whereas the quantum description of bound states would require h_{eff}/n>1 when the masses. Also the effective gravitational constant associated with the strings would differ from G.
The natural condition is that the size scale of string world sheet associated with the flux tube mediating gravitational binding is G(M+m)/v_{0}, By expressing string tension in the form 1/T=n^{2} hbar G_{1}, n=h_{eff}/h, this condition gives hbar G_{1}= hbar^{2}/M_{red}^{2}, M_{red}= Mm/(M+m). The effective Planck length defined by the effective Newton's constant G_{1} analogous to that appearing in string tension is just the Compton length associated with the reduced mass of the system and string tension equals to T= [v_{0}/G(M+m)]^{2} apart from a numerical constant (2G(M+m) is Schwartschild radius for the entire system). Hence the macroscopic stringy description of gravitation in terms of string differs dramatically from the perturbative one. Note that one can also understand why in the Bohr orbit model of Nottale for the planetary system and in its TGD version v_{0} must be by a factor 1/5 smaller for outer planets rather than inner planets.
Are 4D spinor modes consistent with associativity?
The condition that octonionic spinors are equivalent with ordinary spinors looks rather natural but in the case of KählerDirac action the nonassociativity could leak in. One could of course give up the condition that octonionic and ordinary KD equation are equivalent in 4D case. If so, one could see KD action as related to noncommutative and maybe even nonassociative fermion dynamics. Suppose that one does not.
 KD action vanishes by KD equation. Could this save from nonassociativity? If the spinors are localized to string world sheets, one obtains just the standard stringy construction of conformal modes of spinor field. The induce spinor connection would have only the holomorphic component A_{z}. Spinor mode would depend only on z but KD gamma matrix Γ^{z} would annihilate the spinor mode so that KD equation would be satisfied. There are good hopes that the octonionic variant of KD equation is equivalent with that based on ordinary gamma matrices since quaternionic coordinated reduces to complex coordinate, octonionic quaternionic gamma matrices reduce to complex gamma matrices, sigma matrices are effectively absent by holomorphy.
 One can consider also 4D situation (maybe inside wormhole contacts). Could some form of quaternion holomorphy allow to realize the KD equation just as in the case of super string models by replacing complex coordinate and its conjugate with quaternion and its 3 conjugates. Only two quaternion conjugates would appear in the spinor mode and the corresponding quaternionic gamma matrices would annihilate the spinor mode. It is essential that in a suitable gauge the spinor connection has nonvanishing components only for two quaternion conjugate coordinates. As a special case one would have a situation in which only one quaternion coordinate appears in the solution. Depending on the character of quaternionion holomorphy the modes would be labelled by one or two integers identifiable as conformal weights.
Even if these octonionic 4D modes exists (as one expects in the case of cosmic strings), it is far from clear whether the description in terms of them is equivalent with the description using KD equation based ordinary gamma matrices. The algebraic structure however raises hopes about this. The quaternion coordinate can be represented as sum of two complex coordinates as q=z_{1}+Jz_{2} and the dependence on two quaternion conjugates corresponds to the dependence on two complex coordinates z_{1},z_{2}. The condition that two quaternion complexified gammas annihilate the spinors is equivalent with the corresponding condition for Dirac equation formulated using 2 complex coordinates. This for wormhole contacts. The possible generalization of this condition to Minkowskian regions would be in terms HamiltonJacobi structure.
Note that for cosmic strings of form X^{2}× Y^{2}⊂ M^{4}× CP_{2} the associativity condition for S^{2} sigma matrix and without assuming localization demands that the commutator of Y^{2} imaginary units is proportional to the imaginary unit assignable to X^{2} which however depends on point of X^{2}. This condition seems to imply correlation between Y^{2} and S^{2} which does not look physical.
Summary
To summarize, the minimal and mathematically most optimistic conclusion is that KählerDirac action is indeed enough to understand gravitational binding without giving up the associativity of the fermionic dynamics. Conformal spinor dynamics would be associative if the spinor modes are localized at string world sheets with vanishing W (and maybe also Z) fields guaranteeing welldefinedness of em charge and carrying canonical momentum currents parallel to them. It is not quite clear whether string world sheets are present also inside wormhole contacts: for CP_{2} type vacuum extremals the Dirac equation would give only righthanded neutrino as a solution (could they give rise to N=2 SUSY?).
Associativity does not favor fermionic modes in the interior of spacetime surface unless they represent righthanded neutrinos for which mixing with lefthanded neutrinos does not occur: hence the idea about interior modes of fermions as giving rise to SUSY is dead whereas the original idea about partonic oscillator operator algebra as SUSY algebra is well and alive. Evolution can be seen as a generation of gravitationally bound states of increasing size demanding the gradual increase of h__{eff} implying generation of quantum coherence even in astrophysical scales.
The construction of preferred extremals would realize strong form of holography. By conformal symmetry the effective metric at string world sheet could be conformally equivalent with the induced metric at string world sheets. Dynamical string tension would be proportional to hbar/h_{eff}^{2} due to the proportionality α_{K}∝ 1/h_{eff} and predict correctly the size scales of gravitationally bound states for h_{gr}=h_{eff}=GMm/v_{0}. Gravitational constant would be a prediction of the theory and be expressible in terms of α_{K} and R^{2} and hbar_{eff} (G∝ R^{2}/g_{K}^{2}).
In fact, all bound states  elementary particles as pairs of wormhole contacts, hadronic strings, nuclei, molecules, etc.  are described in the same manner quantum mechanically. This is of course nothing new since magnetic flux tubes
associated with the strings provide a universal model for interactions in TGD Universe. This also conforms with the TGD counterpart of AdS/CFT duality.
See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

I decided to take here from a previous posting an argument allowing to conclude that super string models are unable to describe macroscopic gravitation involving formation of gravitationally bound states. Therefore superstrings models cannot have desired macroscopic limit and are simply wrong. This is of course reflected also by the landscape catastrophe meaning that the theory ceases to be a theory in macroscopic scales. The failure is not only at the level of superstring models: it is at the level of quantum theory itself. Instead of single value of Planck constant one must allow a hierarchy of Planck constants predicted by TGD. My sincere hope is that this message could gradually leak through the iron curtain to the ears of the super string gurus.
Superstring action has bosonic part proportional to string area. The proportionality constant is string tension proportional
to 1/hbar G and is gigantic. One expects only strings of length of order Planck length be of significance.
It is now clear that also in TGD the action in Minkowskian regions contains a string area. In Minkowskian regions of
spacetime strings dominate the dynamics in an excellent approximation and the naive expectation is that string theory should give an excellent description of the situation.
String tension would be proportional to 1/hbar G and this however raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required nonperturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.
In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.
I have however proposed that gravitons  at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has h_{eff}= h_{gr}=GMm/v_{0}, where v_{0}/c<1 (v_{0} has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v_{0}/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v_{0}/G^{2}Mm. For v_{0}= GMm/hbar, which remains below unity for Mm/m_{Pl}^{2} one would have h_{gr}/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water)  probably not an accident  would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. h_{gr}=h_{eff} hypothesis is indeed central in TGD based view about living matter.
To conclude, it seems that superstring theory with single value of Planck constant cannot give rise to macroscopic gravitationally bound matter and would be therefore simply wrong much better than to be notevenwrong.
See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

TGD differs in several respects from quantum field theories and string models. The basic mathematical difference is that the mathematically poorly defined notion of path integral is replaced with the mathematically welldefined notion of functional integral defined by the Kähler function defining Kähler metric for WCW ("world of classical worlds"). Apart from quantum jump, quantum TGD is essentially theory of classical WCW spinor fields with WCW spinors represented as fermionic Fock states. One can say that Einstein's geometrization of physics program is generalized to the level of quantum theory.
It has been clear from the beginning that the gigantic superconformal symmetries generalizing ordinary superconformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and spacetime degrees of freedom (or between Minkowskian and Euclidian spacetime regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anticommutators of WCW gamma matrices identifiable as superconformal Noether supercharges for the symplectic algebra assignable to δ M^{4}_{+/}× CP_{2}. The string model description of gravitation emerges and also the TGD based view about dark matter becomes more precise.
Kähler function, Kähler action, and connection with string models
The definition of Kähler function in terms of Kähler action is possible because spacetime regions can have also Euclidian signature of induced metric. Euclidian regions with 4D CP_{2} projection  wormhole contacts  are identified as lines of generalized Feynman diagrams  spacetime correlates for basic building bricks of elementary particles. Kähler action from Minkowskian regions is imaginary and gives to the functional integrand a phase factor crucial for quantum field theoretic interpretation. The basic challenges are the precise specification of Kähler function of "world of classical worlds" (WCW) and Kähler metric.
There are two approaches concerning the definition of Kähler metric: the conjecture analogous to AdS/CFT duality is that these approaches are mathematically equivalent.
 The Kähler function defining Kähler metric can be identified as Kähler action for spacetime regions with Euclidian signature for a preferred extremal containing 3surface as the ends of the spacetime surfaces inside causal diamond (CD). Minkowskian spacetime regions give to Kähler action an imaginary contribution interpreted as the counterpart of quantum field theoretic action. The exponent of Kähler function defines functional integral in WCW. WCW metric is dictated by the Euclidian regions of spacetime with 4D CP_{2} projection.
The basic question concerns the attribute "preferred". Physically the preferred extremal is analogous to Bohr orbit. What is the mathematical meaning of preferred extremal of Kähler action? The latest step of progress is the realization that the vanishing of generalized conformal charges for the ends of the spacetime surface fixes the preferred extremals to high extent and is nothing but classical counterpart for generalized Virasoro and KacMoody conditions.
 Fermions are also needed. The welldefinedness of electromagnetic charge led to the hypothesis that spinors are restricted to string world sheets. It has become also clear that string world sheets are most naturally minimal surfaces with 1D CP_{2} projection (this brings in gravitational constant) and that Kähler action in Minkowskian regions involves also the string area (, which does not contribute to Kähler function) giving the entire action in the case of M^{4} type vacuum extremals with vanishing Kähler form. Hence vacuum extremals might serve as an excellent approximation for the sheets of the manysheeted spacetime in Minkowskian spacetime regions.
 Second manner to define Kähler metric is as anticommutators of WCW gamma matrices identified as supersymplectic Noether charges for the Dirac action for induced spinors with string tension proportional to the inverse of Newton's constant. These charges are associated with the 1D spacelike ends of string world sheets connecting the wormhole throats. WCW metric contains contributions from the spinor modes associated with various string world sheets connecting the partonic 2surfaces associated with the 3surface.
It is clear that the information carried by WCW metric about 3surface is rather limited and that the larger the number of string world sheets, the larger the information. This conforms with strong form of holography and the notion of measurement resolution as a property of quantums state. Clearly. Duality means that Kähler function is determined either by spacetime dynamics inside Euclidian wormhole contacts or by the dynamics of fermionic strings in Minkowskian regions outside wormhole contacts. This duality brings strongly in mind AdS/CFT duality. One could also speak about fermionic emergence since Kähler function is dictated by the Kähler metric part from a real part of gradient of holomorphic function: a possible identification of the exponent of Kähler function is as Dirac determinant.
Realization of superconformal symmetries
The detailed realization of various superconformal symmetries has been also a long standing problem but
recent progress leads to very beautiful overall view.
 Superconformal symmetry requires that Dirac action for string world sheets is accompanied by string world sheet area as part of bosonic action. String world sheets are implied and can be present only in Minkowskian regions if one demands that octonionic and ordinary representations of induced spinor structure are equivalent (this requires vanishing of induced spinor curvature to achieve associativity in turn implying that CP_{2} projection is 1D). Note that 1dimensionality of CP_{2} projection is symplectically invariant property. Neither string world sheet area nor Kähler action is invariant under symplectic transformations. This is necessary for having nontrivial Kähler metric. Whether WCW really possesses supersymplectic isometries remains an open problem.
 Superconformal symmetry also demands that Kähler action is accompanied by what I call KählerDirac action with gamma matrices defined by the contractions of the canonical momentum currents with imbedding spacegamma matrices. Hence also induced spinor fields in the spacetime interior must be present. Indeed, inside wormhole contacts KählerDirac equation reducing to CP_{2} Dirac equation for CP_{2} vacuum extremals dictates the fermionic dynamics.
Strong form of holography implied by strong form of general coordinate invariance strongly suggests that superconformal invariance in the interior of the spacetime surface is a broken gauge invariance in the sense that the superconformal charges for a subalgebra with conformal weights vanishing modulo some integer n vanish. The proposal is that n corresponds to the effective Planck constant as h_{eff}/h=n. For string world sheets superconformal symmetries are not gauge symmetries and strings dominate in good approximation the fermionic dynamics.
Interior dynamics for fermions, the role of vacuum extremals, dark matter, and SUSY
The key role of CP_{2}type and M^{4}type vacuum extremals has been rather obvious from the beginning but the detailed understanding has been lacking. Both kinds of extremals are invariant under symplectic transformations of δ M^{4}× CP_{2}, which inspires the idea that they give rise to isometries of WCW. The deformations CP_{2}type extremals correspond to lines of generalized Feynman diagrams. M^{4} type vacuum extremals in turn are excellent candidates for the building bricks of manysheeted spacetime giving rise to GRT spacetime as approximation. For M^{4} type vacuum extremals CP_{2} projection is (at most 2D) Lagrangian manifold so that the induced Kähler form vanishes and the action is fourthorder in small deformations. This implies the breakdown of the path integral approach and of canonical quantization, which led to the notion of WCW.
If the action in Minkowskian regions contains also string area, the situation changes dramatically since strings dominate the dynamics in excellent approximation and string theory should give an excellent description of the situation: this of course conforms with the dominance of gravitation.
String tension would be proportional to 1/hbar G and this raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required nonperturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.
In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.
I have however proposed that gravitons  at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has h_{eff}= h_{gr}=GMm/v_{0}, where v_{0}/c<1 (v_{0} has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v_{0}/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v_{0}/G^{2}Mm. For v_{0}= GMm/hbar, which remains below unity for Mm/m_{Pl}^{2} one would have h_{gr}/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water)  probably not an accident  would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. h_{gr}=h_{eff} hypothesis is indeed central in TGD based view about living matter.
In this framework superstring theory with single value of Planck constant would not give rise to macroscopic
gravitationally bound matter and would be thus simply wrong.
If one assumes that for nonstandard values of Planck constant only nmultiples of superconformal algebra in interior annihilate the physical states, interior conformal gauge degrees of freedom become partly dynamical. The identification of dark matter as macroscopic quantum phases labeled by h_{eff}/h=n conforms with this.
The emergence of dark matter corresponds to the emergence of interior dynamics via breaking of superconformal symmetry. The induced spinor fields in the interior of flux tubes obeying Kähler Dirac action should be highly relevant for the understanding of dark matter. The assumption that dark particles have essentially same masses as ordinary particles suggests that dark fermions correspond to induced spinor fields at both string world sheets and in the spacetime interior: the spinor fields in the interior would be responsible for the long range correlations characterizing h_{eff}/h=n. Magnetic flux tubes carrying dark matter are key entities in TGD inspired quantum biology. Massless extremals represent second class of M^{4} type nonvacuum extremals.
This view forces once again to ask whether spacetime SUSY is present in TGD and how it is realized. With a motivation coming from the observation that the mass scales of particles and sparticles most naturally have the same padic mass scale as particles in TGD Universe I have proposed that sparticles might be dark in TGD sense. The above argument leads to ask whether the dark variants of particles correspond to states in which one has ordinary fermion at string world sheet and 4D fermion in the spacetime interior so that dark matter in TGD sense would almost by definition correspond to sparticles!
See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

In Thinking Allowed Original there was a link to a talk by James Sully and having the title Geometry of Compression. I must admit that I understood very little about the talk. My not so educated guess is however that information is compressed: UV or IR cutoff eliminating entanglement in short length scales and describing its presence in terms of density matrix  that is thermodynamically  is another manner to say it. The TGD inspired proposal for the interpretation of the inclusions of hyperfinite factors of type II_{1} (HFFs) is in spirit with this.
The spacetime counterpart for the compression would be in TGD framework discretization. Discretizations using rational points (or points in algebraic extensions of rationals) make sense also padically and thus satisfy number theoretic universality. Discretization would be defined in terms of intersection (rational or in algebraic extension of rationals) of real and padic surfaces. At the level of "world of classical worlds" the discretization would correspond to  say  surfaces defined in terms of
polynomials, whose coefficients are rational or in some algebraic extension of rationals. Pinary UV and IR cutoffs are involved too. The notion of padic manifold allows to interpret rthe padic variants of spacetime surfaces as cognitive representations of real spacetime surfaces.
Finite measurement resolution does not allow state function reduction reducing entanglement totally. In TGD framework also negentropic entanglement stable under Negentropy Maximixation Principle (NMP) is possible. For HFFs the projection into single ray of Hilbert space is indeed impossible: the reduction takes always to infiniteD subspace.
The visit to the URL was however not in vain. There was a link to an article discussing the geometrization of entanglement entropy inspired by the AdS/CFT hypothesis.
Quantum classical correspondence is basic guiding principle of TGD and suggests that entanglement entropy should indeed have spacetime correlate, which would be the analog of HawkingBekenstein entropy.
Generalization of AdS/CFT to TGD context
AdS/CFT generalizes to TGD context in nontrivial manner. There are two alternative interpretations, which both could make sense. These interpretations are not mutually exclusive. The first interpretation makes sense at the level of "world of classical worlds" (WCW) with symplectic algebra and extended conformal algebra associated with δ M^{4}_{+/} replacing ordinary conformal and KacMoody algebras. Second interpretation at the level of spacetime surface with the extended conformal algebras of the lightlikes orbits of partonic 2surfaces replacing the conformal algebra of boundary of AdS^{n}.
1. First interpretation
For the first interpretation 2D conformal invariance is generalised to 4D conformal invariance relying crucially on the 4dimensionality of spacetime surfaces and Minkowski space.
 One has an extension of the conformal invariance provided by the symplectic transformations of δ CD× CP_{2} for which Lie algebra has the structure of conformal algebra with radial lightlike coordinate of δ M^{4}_{+} replacing complex coordinate z.
 One could see the counterpart of AdS_{n} as imbedding space H=M^{4} × CP_{2} completely unique by twistorial considerations and from the condition that standard model symmetries are obtained and its causal diamonds defined as subsets CD×
CP_{2}, where CD is an intersection of future and past directed lightcones. I will use the shorthand CD for CD× CP_{2}. Strings in AdS^{5}× S^{5} are replaced with spacetime surfaces inside 8D CD.
 For this interpretation 8D CD replaces the 10D spacetime AdS^{5}× S^{5}. 7D lightlike boundaries of CD correspond to the boundary of say AdS^{5}, which is 4D Minkowski space so that zero energy ontology (ZEO) allows rather natural formulation of the generalization of AdS/CFT correspondence since the positive and negative energy parts of zero energy states are localized at the boundaries of CD.
2. Second interpretation
For the second interpretation relies on the observation that string world sheets as carriers of induced spinor fields emerge in TGD framework from the condition that electromagnetic charge is welldefined for the modes of induced spinor field.
 One could see the 4D spacetime surfaces X^{4} as counterparts of AdS^{4}. The boundary of AdS^{4} is replaced in this picture with 3surfaces at the ends of spacetime surface at opposite boundaries of CD and by strong form of holography the
union of partonic 2surfaces defining the intersections of the 3D boundaries between Euclidian and Minkowskian regions of spacetime surface with the boundaries of CD. Strong form of holography in TGD is very much like ordinary holography.
 Note that one has a dimensional hierarchy: the ends of the boundaries of string world sheets at boundaries of CD as pointlike partices, boundaries as fermion number carrying lines, string world sheets, lightlike orbits of partonic 2surfaces, 4surfaces, imbedding space M^{4}× CP_{2}. Clearly the situation is more complex than for AdS/CFT correspondence.
 One can restrict the consideration to 3D submanifolds X^{3} at either boundary of causal diamond (CD): the ends of spacetime surface. In fact, the position of the other boundary is not welldefined since one has superposition of CDs with only one boundary fixed to be piece of lightcone boundary. The delocalization of the other boundary is essential for the understanding of the arrow of time. The state function reductions at fixed boundary leave positive energy part (say) of the zero energy state at that boundary invariant (in positive energy ontology entire state would remain
unchanged) but affect the states associated with opposite boundaries forming a superposition which also changes between reduction: this is analog for unitary time evolution. The average for the distance between tips of CDs in the superposition increases and gives rise to the flow of time.
 One wants an expression for the entanglement entropy between X^{3} and its partner. Bekenstein area law allows to guess the general expression for the entanglement entropy: for the proposal discussed in the article the entropy would be the area of the boundary of X^{3} divided by gravitational constant: S= A/4G. In TGD framework gravitational constant might be replaced by the square of CP_{2} radius apart from numerical constant. How gravitational constant emerges in TGD framework is not completely understood although one can deduce for it an estimate using dimensional analyses. In any case, gravitational constant is a parameter which characterizes GRT limit of TGD in which manysheeted spacetime is in long scales replaced with a piece of Minkowski space such that the classical gravitational fields and gauge potentials for
sheets are summed. The physics behind this relies on the generalization of linear superposition of fields: the effects of different spacetime sheets particle touching them sum up rather than fields.
 The counterpart for the boundary of X^{3} appearing in the proposal for the geometrization of the entanglement entropy naturally corresponds to partonic 2surface or a collection of them if strong form of holography holds true.
With what kind of systems 3surfaces can entangle?
With what system X^{3} is entangled/can entangle? There are several options to consider and they could correspond to the two TGD variants for the AdS/CFT correspondence.
 X^{3} could correspond to a wormhole contact with Euclidian signature of induced metric. The entanglement would be between it and the exterior region with Minkowskian signature of the induced metric.
 X^{3} could correspond to single sheet of spacetime surface connected by wormhole contacts to a larger spacetime sheet defining its environment. More precisely, X^{3} and its complement would be obtained by throwing away the wormhole contacts with Euclidian signature of induce metric. Entanglement would be between these regions. In the generalization of the formula
S= A/4hbar G
area A would be replaced by the total area of partonic 2surfaces and G perhaps with CP_{2} length scale squared.
 In ZEO the entanglement could also correspond to timelike entanglement between the 3D ends of the spacetime surface at opposite lightlike boundaries of CD. Mmatrix, which can be seen as the analog of thermal Smatrix, decomposes to a product of hermitian square root of density matrix and unitary Smatrix and this hermitian matrix could also define padic thermodynamics. Note that in ZEO quantum theory can be regarded as square root of thermodynamics.
Minimal surface property is not favored in TGD framework
Minimal surface property for the 3surfaces X^{3} at the ends of spacetime surface looks at first glance strange but a proper generalization of this condition makes sense if one assumes strong form of holography. Strong form of holography realizes General Coordinate Invariance (GCI) in strong sense meaning that lightlike parton orbits and spacelike 3surfaces at the ends of spacetime surfaces are equivalent physically. As a consequence, partonic 2surfaces and their 4D tangent space data must code for the quantum dynamics.
The mathematical realization is in terms of conformal symmetries accompanying the symplectic symmetries of δ M^{4}_{+/}× CP_{2} and conformal transformations of the lightlike partonic orbit. The generalizations of ordinary conformal algebras correspond to conformal algebra, KacMoody algebra at the lightlike parton orbits and to symplectic transformations δ M^{4}× CP_{2} acting as isometries of WCW and
having conformal structure with respect to the lightlike radial coordinate plus conformal transformations of δ M^{4}_{+/}, which is metrically 2dimensional and allows extended conformal symmetries.
 If the conformal realization of the strong form of holography works, conformal transformations act at quantum level as gauge symmetries in the sense that generators with novanishing conformal weight are zero or generate zero norm states. Conformal degeneracy can be eliminated by fixing the gauge somehow. Classical conformal gauge conditions analogous to Virasoro and KacMoody conditions satisfied by the 3surfaces at the ends of CD are natural in this respect. Similar conditions would hold true for the lightlike partonic orbits at which the signature of the induced metric changes.
 What is also completely new is the hierarchy of conformal symmetry breakings associated with the hierarchy of Planck constants h_{eff}/h=n. The deformations of the 3surfaces which correspond to nonvanishing conformal weight in algebra or any subalgebra with conformal weights vanishing modulo n give rise to vanishing classical charges and thus do not affect the value of the Kähler action.
The inclusion hierarchies of conformal subalgebras are assumed to correspond to those for hyperfinite factors. There is obviously a precise analogy with quantal conformal invariance conditions for Virasoro algebra and KacMoody algebra. There is also hierarchy of inclusions which corresponds to hierarchy of measurement resolutions. An attractive interpretation is that singular conformal transformations relate to each other the states for broken conformal symmetry. Infinitesimal transformations for symmetry broken phase would carry fractional conformal weights coming as multiples of 1/n.
 Conformal gauge conditions need not reduce to minimal surface conditions holding true for all variations.
 Note that Kähler action reduces to ChernSimons term at the ends of CD if weak form of electric magnetic duality holds true. The conformal charges at the ends of CD cannot however reduce to ChernSimons charges by this condition since only the charges associated with CP_{2} degrees of freedom would be nontrivial.
Technicalities
The generalisation of the conjecture about surface area proportionality of entropy to TGD context looks rather straightforward but is physically highly nontrivial. There are however some technicalities involved.
 In TGD framework it is not quite clear whether
 G still appears in the formula or
 whether G should be replaced with the square R^{2} of CP_{2} radius to give
S= A/4π R^{2}
apart from numerical constant.
For option a) one must include Planck constant explicitly to the formula to give S= A/4h_{eff}G: the entropy would decrease as h_{eff}=n× h increases. The condition h_{eff}=h_{gr}= GM^{2}/v_{0} would give S= v_{0}/c<1. The entropy using b) would be by a factor of order 10^{5} smaller and would not depend on the value of h_{eff} at all. It will be found that padic mass calculations lead to entropy allowing to circumvent these problems.
 There is also the question about the identification of the area A. For blackhole A would be determined by Schwartschild radius r_{S}= 2GM depending on mass only. In TGD framework one has several candidates.
 The area of partonic 2surface is an obvious first guess. One cannot however expect that the area of partonic 2surface is constant. Could conformal gauge fixing fixes the 3surfaces highly uniquely. Ordinary conformal invariance for partonic 2surface does not however seem to be consistent with the fixing of the area of partonic 2surface since conformal transformations do not preserve area.
 Could the area of partonic 2surface be replaced with the area of the boundary of spacetime sheet at which particle is topologically condensed and has size scale of order Compton length? This option looks the most feasible one on basis of padic mass calculations as will be found.
pAdic variant of BekensteinHawking law
When the 3surface corresponds to elementary particle, a direct connection with padic thermodynamics suggests itself and allows to answer the questions above. pAdic thermodynamics could be interpreted as a description of the entanglement with environment. In ZEO the entanglement could also correspond to timelike entanglement between the 3D ends of the spacetime surface at opposite lightlike boundaries of CD. Mmatrix, which can be seen as the analog of thermal Smatrix, decomposes to a product of hermitian square root of density matrix and unitary Smatrix and this hermitian matrix could also define padic thermodynamics.
 pAdic thermodynamics would not be for energy but for mass squared (or scaling generator L_{0}) would describe the entanglement of the particle with environment defined by the larger spacetime sheet. Conformal weights would comes as positive powers of integers (p^{L}_{0} would replace exp(H/T) to guarantee the number theoretical existence and convergence of the Boltzmann weight: note that conformal invariance that is integer spectrum of L_{0} is also essential).
 The interactions with environment would excite very massive CP_{2} mass scale excitations (mass scale is about 10^{4} times Planck mass) of the particle and give it thermal mass squared identifiable as the observed mass squared. The Boltzmann weights would be extremely small having padic norm about 1/p^{n}, p the padic prime: M_{127}=2^{1271 for electron.
}
 I have proposed earlier padic entropy as a padic counterpart of BekensteinHawking entropy. S= (R^{2}/hbar^{2})× M^{2} holds true identically apart from numerical constant. Note that one could interpret R^{2}M/hbar as the counterpart of Schwartschild radius. Note that this radius is proportional to 1/p^{1/2} so that the area A would correspond to the area defined by Compton length. This is in accordance with the third option.
What is the spacetime correlate for negentropic entanglement?
The new element brought in by TGD framework is that number theoretic entanglement entropy is negative for negentropic entanglement assignable to unitary entanglement and NMP states that this negentropy increases. Since entropy is essentially number of energy degenerate states, a good guess is that the number n=h_{eff}/h of spacetime sheets associated with h_{eff} defines the negentropy. An attractive spacetime correlate for the negentropic entanglement is braiding. Braiding defines unitary Smatrix between the states at the ends of braid and this entanglement is negentropic. This entanglement gives also rise to topological quantum computation.
See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article Surface area as geometric representation of entanglement entropy?.

Classical TGD involves several key questions waiting for clearcut answers.
 The notion of preferred extremal emerges naturally in positive energy ontology, where Kähler metric assigns a unique (apart from gauge symmetries) preferred extremal to given 3surface at M^{4} time= constant section of imbedding space H=M^{4}× CP_{2}. This would quantize the initial values of the time derivatives of imbedding coordinates and this could correspond to the Bohr orbitology in quantum mechanics.
 In zero energy ontology (ZEO) initial conditions are replaced by boundary conditions. One fixes only the 3surfaces at the opposite boundaries of CD and in an ideal situation there would exist a unique spacetime surface connecting them. One must however notice that the existence of lightlike wormhole throat orbits at which the signature of the induced metric changes (det(g_{4})=0) its signature might change the situation. Does the attribute "preferred" become obsolete and does one
lose the beautiful Bohr orbitology which looks intuitively compelling and would realize quantum classical correspondence?
 Intuitively it has become clear that the generalization of superconformal symmetries by replacing 2D manifold with metrically 2D but topologically 3D lightlike boundary of causal diamond makes sense. Generalized superconformal symmetries should apply also to the wormhole throat orbits which are also metrically 2D and for which conformal symmetries respect detg(g_{4})=0 condition. Quantum classical correspondence demands that the generalized superconfornal invariance has classical counterpart. How could this classical counterpart be realized?
 Holography is one key aspect of TGD and mean that 3surfaces dictate everything. In positive energy ontology the content w of this statement would be rather obvious and reduce to Bohr orbitology but in ZEO situation is different. On the other hand, TGD strongly suggests strong form of holography based stating that partonic 2surfaces (the ends of wormhole throat orbits at boundaries of CD) and tangent space data at them code for quantum physics of TGD. General coordinate invariance would be realied in strong sense: one could formulate the theory either in terms of spacelike 3surfaces at the ends of CD or in terms of lightlike wormhole throat orbits. This would realize Bohr orbitology also in ZEO by reducing the boundary conditions to those at partonic 2surfaces. How to realize this explicitly at the level of field equations? This has been the challenge.
Answering questions is extremely useful activity. During last years Hamed has posed continually questions related to the basic TGD. At this time Hamed asked about the derivation of field equations of TGD. In "simple" field theories involving some polynomial nonlinearities the deduction of field equations is of course totally trivial process but in the extremely nonlinear geometric framework of TGD situation is quite different.
While answering the questions I made what I immediately dare to call a breakthrough discovery in the mathematical understanding of TGD. To put it concisely: one can assume that the variations at the lightlike boundaries of CD vanish for all conformal variations which are not isometries. For isometries the contributions from the ends of CD cancel each other so that the corresponding variations need not vanish separately at boundaries of CD! This is extremely simple and profound fact. This would be nothing but the realisation of the analogs of conformal symmetries classically and give precise content for the notion of preferred external, Bohr orbitology, and strong form of holography. And the condition makes sense only in ZEO!
I attach below the answers to the questions of Hamed almost as such apart from slight editing and little additions, reorganization, and correction of typos.
The physical interpretation of the canonical momentum current
Hamed asked about the physical meaning of T^{n}_{k}== ∂ L/∂(∂_{n} h^{k})  normal components of canonical momentum labelled by the label k of imbedding space coordinates  it is good to start from the physical meaning of a more general vector field
T^{α}_{k} == ∂ L/∂(∂_{α} h^{k})
with both imbedding space indices k and spacetime indices α  canonical momentum currents. L refers to Kähler action.
 One can start from the analogy with Newton's equations derived from action
principle (Lagrangian). Now the analogs are the partial derivatives ∂ L/∂(dx^{k}/dt). For a particle in potential one obtains just the momentum. Therefore the term canonical momentum current/density: one has kind of momentum current for each imbedding space coordinate.
 By contracting with generators of imbedding space isometries (Poincare and color) one indeed obtains conserved currents associated with isometries by Noether's theorem:
j^{A α}= T^{α}_{k}j^{Ak} .
By field equations the divergences of these currents vanish and one obtains conserved charged classical fourmomentum and color charges:
D_{α} T^{A α}=0 .
 The normal component of conserved current must vanish at spacelike boundaries if one has such
T^{An}=0
if one has boundaries with Minkowskian signature of induced metric. Now one has wormhole throat orbits which are not genuine boundaries albeit analogous to them and one must be very careful. The quantity T^{n}_{k} determines the values of normal components of currents and must vanish at possible spacelike boundaries.
Note that in TGD field equations reduce to the conservation of isometry currents as in hydrodynamics where basic equations are just conservation laws.
The basic steps in the derivation of field equations
First a general recipe for deriving field equations from Kähler action  or any action as a matter of fact.
 At the first step one writes an expression of the variation of the Kähler action as sum of variations with respect to the induced metric g and induced Kähler form J. The partial derivatives in question are energy momentum tensor and contravariant Kähler form.
 After this the variations of g and J are expressed in terms of variations of imbedding space coordinates, which are the primary dynamical variables.
 The integral defining the variation can be decomposed to a total divergence plus a term vanishing for extremals for all variations: this gives the field equations. Total divergence term gives a boundary term and it vanishes by boundary conditions if the boundaries in question have timelike direction.
If the boundary is spacelike, the situation is more delicate in TGD framework: this will be considered in the sequel. In TGD situation is also delicate also because the lightlike 3surfaces which are common boundaries of regions with Minkowskian or Euclidian signature of the induced metric are not ordinary topological boundaries. Therefore a careful treatment of both cases is required in order to not to miss important physics.
Expressing this summary more explicitly, the variation of the Kahler action with respect to the gradients of the imbedding space coordinates reduces to an integral of
T^{α}_{k} ∂_{α}δ h^{k} + (∂ L/∂ h^{k}) δ h^{k} .
The latter term comes only from the dependence of the imbedding space metric
and Kähler form on imbedding space coordinates. One can use a simple trick. Assume
that they do not depend at all on imbedding space coordinates, derive field equations,
and replaced partial derivatives by covariant derivatives at the end. Covariant
derivative means covariance with respect to both spacetime and imbedding space vector
indices for the tensorial quantities involved. The trick works because imbedding space
metric and Kähler form are covariantly constant quantities.
The integral of T^{α}_{k} ∂_{α}δ h^{k} decomposes to two parts.
 The first term, whose vanishing gives rise to field equations, is integral of
D_{α} T^{α}_{k} δ h^{k} .
 The second term is integral of
∂_{α} (T^{α}_{k} δ h^{k}) .
This term reduces as a total divergence to a 3D surface integral over the boundary of the region of fixed signature of the induced metric consisting of the ends of CD and wormhole throat orbits (boundary of region with fixed signature of induced metric). This term vanishes if the normal components T^{n}_{k} of canonical momentum currents vanishes at the boundary like region.
In the sequel the boundary terms are discussed explicitly and it will be found that
their treatment indeed involves highly nontrivial physics.
Boundary conditions at boundaries of CD
In positive energy ontology one would formulate boundary conditions as initial conditions by fixing both the 3surface and associated canonical momentum densities at either end of CD (positions and momenta of particles in mechanics). This would bring asymmetry between boundaries of CD.
In TGD framework one must carefully consider the boundary conditions at the boundaries of CDs. What is clear that the timelike boundary contributions from the boundaries of CD to the variation must vanish.
 This is true if the variations are assumed to vanish at the ends of CD. This might be however too strong a condition.
 One cannot demand vanishing of T^{t}_{k} (t refers to time coordinate as normal coordinate) since this would give only vacuum extremals. One could however require quantum classical correspondence for any Cartan subalgebra of isometries, whose elements define maximal set of isometry generators. The eigenvalues of quantal variants of isometry charge assignable to second quantized induced spinors at the ends of spacetime surface are equal to the classical charges. Is this actually formulation of Equivalence Principle, is not quite clear to me.
While writing this a completely new idea popped to my mind. What if one poses the vanishing of the boundary terms at boundaries of CDs as additional boundary conditions for all variations except isometries? Or perhaps for all conformal variations (conformal in TGD sense)? This would not imply vanishing of isometry charges since the variations coming from the opposite ends of CD cancel each other! It soon became clear that this would allow to meet all the challenges listed in the beginning!
 These conditions would realize Bohr orbitology also to ZEO approach and define what "preferred extremal" means.
 The conditions would be very much like superVirasoro conditions stating that super conformal generators with nonvanishing conformal weight annihilate states or create zero norm states but no conditions are posed on generators with vanishing conformal weight (now isometries). One could indeed assume only deformations, which are local isometries assignable to the generalised conformal algebra of the δ
M^{4}_{+/}× CP_{2}. For arbitrary variations one would not require the vanishing. This could be the long sought for precise formulation of superconformal invariance at the level of classical field equations!
It is enough co consider the weaker conditions that the conformal charges defined as integrals of corresponding Noether currents vanish. These conditions would be direct equivalents of quantal conditions.
 The natural interpretation would be as a fixing of conformal gauge. This fixing would be motivated by the fact that WCW Kähler metric must possess isometries associated with the conformal algebra and can depend only on the tangent data at partonic 2surfaces as became clear already for more than two decades ago. An alternative, nonpractical option would be to allow all 3surfaces at the ends of CD: this would lead to the problem of eliminating the analog of the volume of gauge
group from the functional integral.
 The conditions would also define precisely the notion of holography and its reduction to strong form of holography in which partonic 2surfaces and their tangent space data code for the dynamics.
Needless to say, the modification of this approach could make sense also at partonic orbits.
Isometry charges are complex
One must be careful also at the lightlike 3surfaces (orbits of wormhole throats) at which the induced metric changes its signature.
 Should one assume that det(g_{4})^{1/2} is imaginary in Minkowskian and real in Euclidian region? For Kähler action this is sensible and Euclidian region would give a real negative contribution giving rise to exponent of Kähler function of WCW ("world of classical worlds") making the functional integral convergent. Minkowskian regions would give imaginary contribution to the exponent causing interference effects absolutely essential in quantum field theory. This contribution would correspond to Morse function for WCW.
The implication would be that the classical fourmomenta in Euclidian/Minkowskian regions are imaginary/real. What could the interpretation be? Should one accept as a fact that fourmomenta are complex.
 Twistor approach to TGD is now in quite good shape. M^{4}× CP_{2} is the unique choice is one requires that the Cartesian factors allow twistor space with Kähler structure and classical TGD allows twistor formulation.
In the recent formulation the fundamental fermions are assumed to propagate with lightlike momenta along wormhole throats. At gauge theory limit particles must have massless or massive fourmomenta. One can however also consider the possibility of complex massless momenta and in the standard twistor approach on mass shell massless particles appearing in graphs indeed have complex momenta. These complex momenta should by quantum classical correspondence correspond directly to classical complex momenta.
 A funny question popping in mind is whether the massivation of particles could be such that the momenta remain massless in complex sense! The complex variant of lightlikeness condition would be
p^{2}_{Re}= p^{2}_{Im} , p_{Re}• p_{Im}=0 .
Could one interpret p^{2}_{Im} as the mass squared of the particle? Or could p^{2}_{Im} code for the decay width of an unstable particle?
Boundary conditions at the wormhole throat orbits and connection with quantum criticality and hierarchy of Planck constants defining dark matter hierarchy
The contributions from the orbits of wormhole throats are singular since the contravariant form of the induced metric develops components which are infinite (det(g_{4})=0). The contributions are real at Euclidian side of throat orbit and imaginary at the Minkowskian side so that they must be treated as independently.
 One can consider the possibility that under rather general conditions the normal components T^{n}_{k}det(g_{4}) ^{1/2} approach to zero at partonic orbits since det(g_{4}) is vanishing. Note however the appearance of contravariant appearing twice as index raising operator in Kähler action. If so, the vanishing of T^{n}_{k}det(g_{4}) ^{1/2} need not fix completely the "boundary" conditions. In fact, I assign to the wormhole throat orbits conformal gauge symmetries so that just this is expected on physical grounds.
 Generalized conformal invariance would suggest that the variations defined as integrals of T^{n}_{k}det(g_{4}) ^{1/2}δ h^{k} vanish in a nontrivial manner for the conformal algebra associated with the lightlike wormhole throats with deformations respecting det(g_{4})=0 condition. Also the variations defined by infinitesimal isometries (zero conformal weight sector) should vanish since otherwise one would lose the conservation laws for isometry charges. The conditions for isometries might reduce to T^{n}_{k}det(g_{4}) ^{1/2}→ 0 at partonic orbits. Also now the interpretaton would be in terms of fixing of conformal gauge.
 Even T^{n}_{k}det(g_{4}) ^{1/2}=0 condition need not fix the partonic orbit completely. The Gribov ambiguity meaning that gauge conditions do not fix uniquely the gauge potential could have counterpart in TGD framework. It could be that there are several conformally nonequivalent spacetime surfaces connecting 3surfaces at the opposite ends of CD.
If so, the boundary values at wormhole throats orbits could matter to some degree: very natural in boundary value problem thinking but new in initial value thinking. This would conform with the nondeterminism of Kähler action implying criticality and the possibility that the 3surfaces at the ends of CD are connected by several spacetime surfaces which are physically nonequivalent.
The hierarchy of Planck constants assigned to dark matter, quantum criticality and even criticality indeed relies on the assumption that h_{eff}=n× h corresponds to nfold coverings having n spacetime sheets which coincide at the ends of CD and that conformal symmetries act on the sheets as gauge symmetries. One would have as Gribov copies n conformal equivalence classes of wormhole throat orbits and corresponding spacetime surfaces. Depending on whether one fixes the conformal gauge one has n equivalence classes of spacetime surfaces or just one representative from each conformal equivalent class.
 There is also the question about the correspondence with the weak form of electric magnetic duality. This duality plus the condition that j^{α}A_{α}=0 in the interior of spacetime surface imply the reduction of Kähler action to ChernSimons terms. This would suggest that the boundary variation of the Kähler action reduces to that for ChernSimons action which is indeed welldefined for lightlike 3surfaces.
If so, the gauge fixing would reduce to variational equations for ChernSimons action! A weaker condition is that classical conformal charges vanish. This would give a nice connection to the vision about TGD as almost topological QFT. In TGD framework these conditions do not imply the vanishing of
Kähler form at boundaries. The conditions are satisfied if the CP_{2} projection of the partonic orbit is 2D: the reason is that ChernSimons term vanishes identically in this case.
 A further intuitively natural hypothesis is that there is a breaking of conformal symmetry: only the generators of conformal subalgebra with conformal weight multiple of n act as gauge symmetries. This would give infinite hierarchies of breakings of conformal symmetry interpreted in terms of criticality: in the hierarchy the integers n_{i} would satisfy n_{i} divides n_{i+1}.
Similar degeneracy would be associated with the spacelike ends at CD boundaries and I have considered the possibility that the integer n appearing in h_{eff} has decomposition n=n_{1}n_{2} corresponding to the degeneracies associated with the two kinds of boundaries. Alternatively, one could have just n=n_{1}=n_{2} from the condition that the two conformal symmetries are 3dimensional manifestations of single 4D analog of conformal symmetry.
As should have become clear, the derivation of field equations in TGD framework is not just an application of a formal recipe as in field theories and a lot of nontrivial physics is involved!
See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action.

