In the following I will consider some questions related to the twistor lift of TGD and end up to a possible vision about general mechanism of CP breaking and generation of matter antimatter asymmetry.
 Can the analog of Kähler form J(M^{4}) assignable to M^{4} suggested by the symmetry between M^{4} and CP_{2} and by number theoretical vision appear in the theory? What would be the physical implications? The basic objection is the loss of Poincare invariance is lost. This can be however avoided by introducing the moduli space for Kähler forms. This moduli space is actually the moduli space of causal diamonds (CDs) forced in any case by zero energy ontology (ZEO) and playing central role in the generalization of quantum measurement theory to a theory of consciousness and in the explanation of the relationship between geometric and subjective time.
Why J(M^{4}) would be needed? J(M^{4}) corresponds to parallel constant electric and magnetic fields in given direction. Constant E and B=E fix directions of quantization axes for energy (rest system) and spin. One implication is transversal localization of imbedding space spinor modes: imbedding space spinor modes are products of harmonic oscillator Gaussians in transversal degrees of freedom very much like quarks inside hadrons.
Also CP breaking is implied by the electric field and the question is whether this could explain the observed CP breaking as appearing already at the level of imbedding space M^{4}× CP_{2}. The estimate for the CP mass splitting of neutral kaon and antikaon is of correct order of magnitude. Whether stationary spherically symmetric metric as minimal surface allows a sensible physical generalization is a killer test for the hypothesis.
 How does gravitational coupling emerge at fundamental level? The answer is obvious: string area action is scaled by 1/G as in string models. The objection is that padic mass calculations suggest that string tension is determined by CP_{2} size R: the analog of string tension appearing in mass formula given by padic mass calculations would be by a factor about 10^{8} smaller than that estimated from string tension. The discrepancy evaporates by noticing that padic mass calculations rely on padic thermodynamics at imbedding space level whereas string world sheets appear at spacetime level.
 Could one regard the localization of spinor modes to string world sheets as a localization to Lagrangian submanifolds of spacetime surface having by definition vanishing induced Kähler form: J(M^{4})+J(CP_{2})=0. Lagrangian submanifolds would be commutative in the sense of Poisson bracket. Could string world sheets be minimal surfaces satisfying J(M^{4})+J(CP_{2})=0. The Lagrangian condition allows also more general solutions  even 4D spacetime surfaces and one obtains analog of brane hierarchy. Could one allow spinor modes also at these analogs of branes. Is Lagrangian condition equivalent with the original condition that induced W boson fields making the em charge of induced spinor modes illdefined vanish and allowing also solution with other dimensions. How Lagrangian property relates to the idea that string world sheets correspond to complex (commutative) surfaces of quaternionic spacetime surface in octonionic imbedding space.
1. Can the Kähler form of M^{4} appear in Kähler action?
I have already earlier considered the question whether the analog of Kähler form assignable to M^{4} could appear in Kähler action. Could one replace the induced Kähler form J(CP_{2}) with the sum J=J(M^{4})+J(CP_{2}) such that the latter term would give rise to a new component of Kähler form both in spacetime interior at the boundaries of string world sheets regarded as pointlike particles? This could be done both in the Kähler action for the interior of X^{4} and also in the topological magnetic flux term ∈t J associated with string world sheet and reducing to a boundary term giving couplings to U(1) gauge potentials A_{μ}(CP_{2}) and A_{μ}(M^{4}) associated with J(CP_{2}) and J(M^{4}). The interpretation of this coupling is an interesting challenge.
Consider first the objections against introducing J(M^{4}) to the Kähler action at imbedding space level.
 J(M^{4}) would would break translational and Lorentz symmetries at the level of imbedding space since J(M^{4}) cannot be Lorentz invariant. For imbedding space spinor modes this term would bring in coupling to the selfdual Kähler form in M^{4}. The simplest choice is A=(A_{t}=z, A_{z}=0,A_{x}=y,A_{y}=0) defining decomposition M^{4} =M^{2}× E^{2}. For Dirac equation in M^{4} one would have free motion in preferred timelike (t,z)plane plane M^{2} in whereas in x and ydirections (E^{2} plane) would one have harmonic oscillator potentials due to the gauge potentials of electric and magnetic fields. One would have something very similar to quark model of hadron: quark momenta would have conserved longitudinal part and nonconserved transversal part. The solution spectrum has scaling invariance Ψ(m^{k})→ Ψ(λ m^{k}) so that there is no preferred scale and the transversal scales scale as 1/E and 1/k_{x}.
 Since J(M^{4}) is not Lorentz invariant Lorentz boosts would produce new M^{2}× E^{2} decomposition. If one assumes above kind of linear gauge as gauge invariance suggests, the choices with fixed second tip of causal diamond (CD) define finitedimensional moduli space SO(3,1)/SO(1,1)× SO(2) having in number theoretic vision an interpretation as a choice of preferred hypercomplex plane and its orthogonal complement. This is the moduli space for hypercomplex structures in M^{4} with the choices of origins parameterized by M^{4}. The introduction of the moduli space would allow to preserve Poincare invariance.
 If one generalizes the condition for Kähler metric to J^{2}(M^{4})=g(M^{4}) fixing the scaling of J, the coupling to A(M^{4}) is also large and suggests problems with the large breaking of Poincare symmetry for the spinor modes of the imbedding space for given moduli. The transversal localization by the selfdual magnetic and electric fields for J(M^{4}) would produce wave packets in transversal degrees of freedom: is this physical?
This moduli space is actually the moduli space introduced for causal diamonds (CDs) in zero energy ontology (ZEO) forced by the finite value of volume action: fixing of the line connecting the tips of CD the Lorentz boost fixing the position for the second tip of CD parametrizes this moduli space apart from division with the group of transformations leaving the planes M^{2} and E^{2} having interpretation a plane defined by lightlike momentum and polarization plane associated with a given CD invariant.
 Why this kind of symmetry breaking for Poincare invariance? A possible explanation proposed already earlier is that quantum measurement involves a selection of quantization axis. This choice necessarily breaks the symmetries and J(M^{4}) would be an imbedding space correlate for the selection of rest frame and quantization axis of spin. This conforms with the fact that CD is interpreted as the perceptive field of conscious entity at imbedding space level: the contents of consciousness would be determined by the superposition of spacetime surfaces inside CD. The choice of J(M^{4}) for CD would select preferred rest system (quantization axis for energy as a line connecting tips of CD) via electric part of J(M^{4}) and quantization axis of spin (via magnetic part of J(M^{4}). The moduli space for CDs would be the space for choices of these particular quantization axis and in each state function reduction would mean a localization in this moduli space. Clearly, this reduction would be higher level reduction and correspond to a decision of experimenter.
To summarize, for J(M^{4})=0 Poincare symmetries are realized at the level of imbedding space but obviously broken slightly by the geometry of CD. The allowance of J(M^{4})≠ 0 implies that both translational and rotational symmetries are reduced for a given CD: the interpretation would be in terms of a choice of quantization axis in state function reduction. They are however lifted to the level of moduli space of CDs and exact in this more abstract sense. This is nothing new: already the introduction of ZEO and CDs force by volume term in action forced by twistor lift of TGD implies the same. Also the view about state function reduction requires wave functions in the moduli space of CDs. This is also essential for understanding how the arrow of geometric time is inherited from that of subjective time in TGD inspired theory of consciousness.
What about the situation at spacetime level?
 The introduction of J(M^{4}) part to Kähler action has nice number theoretic aspects. In particular, J selects the preferred complex and quaternionic subspace of octonionic space of imbedding space. The simplest possibility is that the Kähler action is defined by the Kähler form J(M^{4})+J(CP_{2}).
Since M^{4} and CP_{2} Kähler geometries decouple it should be possible to take the counterpart of Kähler coupling strength in M^{4} to be much larger than in CP_{2} degrees of freedom so that M^{4} Kähler action is a small perturbation and slowly varying as a functional of preferred extremal. This option is however not in accordance with the idea that entire Kähler form is induced.
 Whether the proposed ansätze for general solutions make still sense is not clear. In particular, can one still assume that preferred extremals are minimal surfaces? Number theoretical vision strongly suggests  one could even say demands  the effective decoupling of Kähler action and volume term. This would imply the universality of quantum critical dynamics. The solutions would not depend at all on the coupling parameters except through the dependence on boundary conditions. The coupling between the dynamics of Kähler action and volume term would come also from the conservation conditions at lightlike 3surfaces at which the signature of the induced metric changes.
 At spacetime level the field equations get more complex if the M^{4} projection has dimension D(M^{4})>2 and also for D(M^{4})=2 if it carries nonvanishing induced J(M^{4}). One would obtain cosmic strings of form X^{2}× Y^{2} as minimal surface extremals of ordinary Kähler action or X^{2} Lagrangian manifold of M^{4} as also CP_{2} type vacuum extremals and their deformations with M^{4} projection Lagrangian manifold. Thus the differences would not be seen for elementary particle and string like objects. Simplest string worlds sheet for which J(M^{4}) vanishes would correspond to a piece of plane M^{2}.
M^{4} is the simplest minimal surface extremal of Kähler action necessarily involving also J(M^{4}). The action in this case vanishes identically by selfduality (in Euclidian signature selfduality does not imply this). For perturbations of M^{4} such as spherically symmetric stationary metric the contribution of M^{4} Kähler term to the action is expected to be small and the come mainly from cross term mostly and be proportional to the deviation from flat metric. The interpretation in terms of gravitational contribution from M^{4} degrees of freedom could make sense.
 What about massless extremals (MEs)? How the induced metric affects the situation and what properties second fundamental form has? Is it possible to obtain a situation in which the energy momentum tensor T^{αβ} and second fundamental form H^{k}_{αβ} have in common components which are proportional to lightlike vector so that the contraction T^{αβ}H^{k}_{αβ} vanishes?
Minimal surface property would help to satisfy the conditions. By conformal invariance one would expect that the total Kähler action vanishes and that one has J^{α}_{γ}J^{γβ} = a× g^{αβ}+b × k^{α}k^{β}.
These conditions together with lightlikeness of Kähler current guarantee that field equations are satisfied.
In fact, one ends up to consider a generalization of MEs by starting from a generalization of holomorphy. Complex CP_{2} coordinates ξ^{i} would be functions of lightlike M^{2} coordinate u_{+}=k• m, k lightlike vector, and of complex coordinate w for E^{2} orthogonal to M^{2}. Therefore the CP_{2} projection would 3D rather than 2D now.
The second fundamental form has only components of form H^{k}_{u+w}, H^{k}_{u+w*} and H^{k}_{ww}, H^{k}_{w*w*}. The CP_{2} contribution to the induced metric has only components of form Δ g_{u+w}, Δ g_{+w*}, and g_{w*w}. There is also contribution g_{u+u}=1, where v is the lightlike dual of u in plane M^{2}. Contravariant metric can be expanded as a power series for in the deviation (Δ g_{u+w}, Δ g_{u+w*}) of the metric from (g_{u+u}, g_{ww*}). Only components of form g^{u+,ui} and g^{ww*} are obtained and their contractions with the second fundamental form vanish identically since there are no common index pairs with simultaneously nonvanishing components. Hence it seems that MEs generalize!
I have asked earlier whether this construction might generalize for ordinary MEs. One can introduce what I have called HamiltonJacobi structure for M^{4} consisting of locally orthogonal slicings by integrable 2surfaces having tangent space having local decomposition M^{2}_{x}× E^{2}_{x} with lightlike direction depending on point x. An objection is that the direction of lightlike momentum depends on position: this need not be inconsistent with momentum conservation but would imply that the total fourmomentum is not lightlike anymore. Topological condensation for MEs and at MEs could imply this kind modification.
 There is also a topological magnetic flux type term for string world sheet. Topological term can be transformed to a boundary term coupling classical particles at the boundary of string world sheet to CP_{2} Kähler gauge potential (added to the equation for a lightlike geodesic line). Now also the coupling to M^{4} gauge potential would be obtained. The condition J(M^{4})+ J(CP_{2})=0 at string world sheets is very attractive manner to identify string world sheets as analogs of Lagrangian manifolds but does not imply the vanishing of the net U(1) couplings at boundary since the induce gauge potentials are in general different.
Also topological term including also M^{4} Kähler magnetic flux for string world sheet contributes also to the modified Dirac equation since the gamma matrices are modified gamma matrices required by superconformal symmetries and defined as contractions of canonical momentum densities with imbedding space gamma matrices. This is true both in spacetime interior, at string world sheets and at their boundaries. CP_{2} (M^{4}) term gives a contribution proportional to CP_{2} (M^{4}) gamma matrices.
At imbedding space level transversal localization would be the outcome and a good guess is that the same happens also now. This is indeed the case for M^{4} defining the simplest extremal. The general interpretation of M^{4} Kähler form could be as a quantum tool for transversal dynamical localization of wave packets in Kähler magnetic and electric fields of M^{4}. Analog for decoherence occurring in transversal degrees of freedom would be in question. Hadron physics could be one application.
How to test this idea?
 It might be possible to kill the idea by showing that one does not obtain spherically symmetric Schwartschild type metric as a minimal surface extremal of generalized Kähler action: these extremals are possible for ordinary Kähler action. For the canonical imbedding of M^{4} field equations are satisfied since energy momentum tensor vanishes identically. For the small deformations the presence of J(M^{4}) would reduce rotational symmetry to cylindrical symmetry.
 J(M^{4}) could make its presence manifest in the physics of righthanded neutrino having no direct couplings to electroweak gauge fields. Mixing with left handed neutrino is however induced by mixing of M^{4} and CP_{2} gamma matrices. The transversal localization of righthanded neutrino in a background, which is a small deformation of M^{4} could serve as an experimental signature.
 CP breaking in hadronic systems is one of the poorly understood aspects of fundamental physics and relates closely to the mysterious matterantimatter asymmetry. The constant electric part of self dual J(M^{4}) implies CP breaking. I have earlier considered the possibility that Kähler electric fields could cause this breaking but this breaking would be local. Second possibility is that matter and antimatter correspond to different values of h_{eff} and are dark relative to each other.
Could J(M^{4}) explain the observed CP breaking as appearing already at the level of imbedding space M^{4}× CP_{2} and could this breaking explain hadronic CP breaking and matter antimatter asymmetry? Could M^{4} part of Kähler electric field induce different h_{eff}/h=n for particles and antiparticles?
To answer these questions one can study Dirac equation at imbedding space level coupled to the gauge potential A(M^{4}) for J(M^{4}).
 The coupling of Kähler form to leptons is 3 times larger than to to quarks as in the case of A(CP_{2}). This would give coupling k=1 for quarks an k=3 for leptons. k corresponds to fermion number which is opposite for fermions and antifermions having therefore opposite values of k at the respective spacetime sheets.
 The potential satisfies ∂_{μ}A^{μ}(M^{4})=0. Let the nonvanishing components of the Kähler gauge potential be (A_{0},A_{z})=ε (x,+/ y). The sign fact ε+/ 1 corresponds to self dual and antiselfdual options, let us assume selfduality as in the case of CP_{2} Kähler form. Scalar d'Alembertian reads as (∂^{μ}∂_{μ}+ A^{μ}A_{μ})Ψ= m^{2} Ψ.
 Assuming momentum eigenstate in time and zdirection (plane M^{2}), one obtains by separation of variables (H_{1}+H_{2})Ψ= (E^{}m^{2}k_{z}^{2})Ψ. H_{x}= ∂_{x}^{2}+k^{2}x^{2} and H_{y}= ∂_{y}^{2}+k^{2}y^{2}) are oscillator Hamiltonians. The spectrum is of H_{x}+H_{y} is given by k_{T}^{2}= (n_{1}+n_{2}+1)2^{1/2}k and one obtains E^{2}=m^{2} +k_{z}^{2} +k_{T}^{2}. This contribution is CP invariant and same for fermions and antifermions. The special feature is the presence of zero point transversal momentum. It is not possible to have a particle, which would be completely at rest. One can also say that m^{2} is increased 2^{1/2}k hbar^{2}/L^{2}, L= 1 m if standard convention for metric is used. For other conventions the numerical value of CP_{2} radius is scale by L/L_{new}. L must correspond to some physical scale assignable to particle: secondary padic length scale is the natural identification.
 Spinor d'Alembertian contains also dipole moment term kX=J^{muν}Σ_{μν} giving a contribution, which depends on the sign of k: E^{2}=m^{2} +k_{z}^{2} +k_{T}^{2}+ kX. The term is sum of magnetic and electric dipole moment terms. The coupling k changes sign in CP operation and be of opposite sign for fermions and antifermions. One has a breaking of CP for given spin state. The dependence of X on spin state gives a test for the theory and also for the predicted CP breaking.
 Scaling covariance allows in principle all values L. To estimate the size of the effect one must fix the length scale L. CP_{2} size has only different value using L as unit and in flat background it does not matter. L should correspond to the size scale of the CD associated with particle. The secondary padic length scale of fermion defining also the size scale of its magnetic body is a natural guess so that Δ E^{2}/E^{2}≈ 2Δ E/E≈ Δ m/m ∼ 2/p^{1/2}, p≈ 2^{k} would hold true. This mass splitting is very small. For weak bosons having k=89 the mass splitting would be of order 3× 10^{4} eV. For small values of p at ultrahigh energies the scale of CP breaking is larger, which conforms with the idea that matterantimatterasymmetry has emerged in very early cosmology.
The recent experiment found that the mass difference Δ m/m for proton and antiproton satisfies Δ m <69× 10^{12}m ≈ 6.9× 10^{2} eV (see this) so that this gives no constraints. Kaonantikaon mass difference is estimated to be about 3.5× 10^{6} eV (see this). This would correspond to a padic length scale k=96. Top quark is mainly responsible for the mixing of neutral kaon and its antiparticle in the model of based on loops involving decay to virtual quark pairs. The estimate from padic mass calculations for top quark mass scale is k=94 so that the order of magnitude estimate has correct of order of magnitude (being by factor 4 too large). This is an encouraging sign.
How the mass splitting of neutral kaons would result? In quark model kaon and antikaon can be regarded as sdbbar and dsbar pairs. The net spins vanishes but the mass splitting due to electric moment dipole moment term X is nonvanishing due to the different sign of coupling k. The sign of the mass splitting is also opposite for kaon and antikaon.
 One can also consider the modified Dirac equation for canonically imbedded M^{4} which is simplest preferred extremal. The coupling to J(M^{4}) to modified Dirac equation in spacetime interior with gamma matrices replaced with modified gamma matrices are obtained as contractions of canonical momentum currents with M^{4} gamma matrices. Completely analogous phenomenon happens for CP_{2} type extremals. T^{αβ}=0 so that the modified gamma comes from J^{αβ} J^{k}_{~l} ∂_{β}m^{l}γ_{k}. These give just ordinary gamma matrices so that the two Dirac equations are identical.
2. About string like objects
String like objects and partonic 2surfaces carry the information about quantum states and about spacetime surfaces as preferred extremals if strong form of holography (SH) holds true. SH has of course some variants. The weakest variant states that fundamental information carrying objects are metrically 2D. The lightlike 3surfaces separating spacetime regions with Minkowskian and Euclidian signature of the induced metric are indeed metrically 2D, and could thus carry information about quantum state.
An attractive possibility is that this information is basically topological. For instance, the value of Planck constant h_{eff}=n× h would tell the number sheets of the singular covering defining this surface such that the sheets coincide at partonic 2surfaces at the ends of spacetime surface at boundaries of CD. In the following some questions related to string world sheets are considered. The information could be also number theoretical. Galois group for the algebraic extension of rationals defining particular adelic physics would transform to each other the number theoretic discretizations of lightlike 3surfaces and give rise to covering space structure. The action is trivial at partonic 2surfaces should be trivial if one wants singular covering: this would mean that discretizations of partonic 2surfaces consist of rational points. h_{eff}/h=n could in this case be a factor of the order of Galois group.
The original observation was that string world sheets should carry vanishing W boson fields in order that the em charge for the modes of the induced spinor field is welldefined. This condition can be satisfied in certain situations also for the entire spacetime surface. This raises several questions. What is the fundamental condition forcing the restriction of the spinor modes to string world sheets  or more generally, to surface of given dimension? Is this restriction dynamical. Can one have an analog of brane hierarchy in which also higherD objects can carry modes of induced spinor field Could the analogs of Lagrangian submanifolds of X^{4} ⊂ M^{4}× CP_{2} satisfying J(M^{4})+J(CP_{2})=0 define string world sheets and their variants with varying dimension? The additional condition would be minimal surface property.
2.1 How does the gravitational coupling emerge?
The appearance of G=l_{P}^{2} has coupling constant remained for a long time actually somewhat of a mystery in TGD. l_{P} defines the radius of the twistor sphere of M^{4} replaced with its geometric twistor space M^{4}× S^{2} in twistor lift. G makes itself visible via the coefficients ρ_{vac}= 8π Λ/G volume term but not directly and if preferred extremals are minimal surface extremals of Kähler action ρ_{vac} makes itself visible only via boundary conditions. How G appears as coupling constant?
Somehow the M^{4} Kähler form should appear in field equations. 1/G could naturally appear in the string tension for string world sheets as string models suggest. pAdic mass calculations identify the analog of string tension as something of order of magnitude of 1/R^{2}. This identification comes from the fact that the ground states of superconformal representations correspond to imbedding space spinor modes, which are solutions of Dirac equation in M^{4}× CP_{2}. This argument is rather convincing and allows to expect that the padic mass scale is not determined by string tension and it can be chosen to be of order 1/G just as in string models.
2.2 Noncommutative imbedding space and strong form of holography
The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of noncommutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI).
Quantum group theorists have studied the idea that spacetime coordinates are noncommutative and tried to construct quantum field theories with noncommutative spacetime coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor J_{kl} and uncertainty relation in linear M^{4} coordinates m^{k} would look something like [m^{k}, m^{l}] = l_{P}^{2}J^{kl}, where l_{P} is Planck length. This would be a direct generalization of noncommutativity for momenta and coordinates expressed in terms of symplectic form J^{kl}.
1+1D case serves as a simple example. The noncommutativity of p and q forces to use either p or q. Noncommutativity condition reads as [p,q]= hbar J^{pq} and is quantum counterpart for classical Poisson bracket. Noncommutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian submanifold to which the projection of J_{pq} vanishes: coordinates become commutative in this submanifold. This condition can be formulated purely classically: wave function is defined in Lagrangian submanifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it.
GCI poses however a problem if one wants to generalize quantum group approach from M^{4} to general spacetime: linear M^{4} coordinates assignable to Liealgebra of translations as isometries do not generalize. In TGD spacetime is surface in imbedding space H=M^{4}× CP_{2}: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as spacetime coordinates. The analog of symplectic structure J for M^{4} makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP_{2} has naturally symplectic form.
Could it be that the coordinates for spacetime surface are in some sense analogous to symplectic coordinates (p_{1},p_{2},q_{1},q_{2}) so that one must use either (p_{1},p_{2}) or (q_{1},q_{2}) providing coordinates for a Lagrangian submanifold. This would mean selecting a Lagrangian submanifold of spacetime surface? Could one require that the sum J_{μν}(M^{4})+ J_{μν}(CP_{2}) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2surfaces. In special case also higherD surfaces  even 4D surfaces as products of Lagrangian 2manifolds for M^{4} and CP_{2} are possible: they would correspond to homologically trivial cosmic strings X^{2}× Y^{2}⊂ M^{4}× CP_{2}, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term.
But why this kind of restriction? In TGD one has strong form of holography (SH): 2D string world sheets and partonic 2surfaces code for data determining classical and quantum evolution. Could this projection of M^{4} × CP_{2} symplectic structure to spacetime surface allow an elegant mathematical realization of SH and bring in the Planck length l_{P} defining the radius of twistor sphere associated with the twistor space of M^{4} in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the nonuniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2D surfaces.
The analog of brane hierarchy for the localization of spinors  spacetime surfaces; string world sheets and partonic 2surfaces; boundaries of string world sheets  is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian submanifolds of spacetime in the sense that J(M^{4})+J(CP_{2})=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M^{4})+J(CP_{2})=0 at them. The vanishing of induced W boson fields is needed to guarantee welldefined em charge at string world sheets and that also this condition allow also 4D solutions besides 2D generic solutions. This condition is physically obvious but mathematically not wellunderstood: could the condition J(M^{4})+J(CP_{2})=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X^{2}× Y^{2} would allow 4D spinor modes. If the lightlike 3surface defining boundary between Minkowskian and Euclidian spacetime regions is Lagrangian surface, the total induced Kähler form ChernSimons term would vanish. The 4D canonical momentum currents would however have nonvanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of spacetime supersymmetries could be interpreted as addition of higherD righthanded neutrino modes to the 1fermion states assigned with the boundaries of string world sheets.
It is relatively easy to construct an infinite family of Lagrangian string world sheets satisfying J(M^{4}) +J(CP_{2})=0 using generalized symplectic transformations of M^{4} and CP_{2} as Hamiltonian flows to generate new ones from a given Lagrangian string world sheets. One must pose minimal surface property as a separate condition. Consider a piece of M^{2} with coordinates (t,z) and homologically nontrivial geodesic sphere S^{2} of CP_{2} with coordinates (u= cos(Θ),Φ). One has J(M^{4})_{tz}=1 and J_{uΦ}= 1. Identify string world sheet via map (u,Φ)= (kz,ω t) from M^{2} to S^{2}. The induced CP_{2} Kahler form is J(CP_{2})_{tz}= kω. kω=1 guarantees J(M^{4}) +J(CP_{2})=0. The strings have necessarily finite length from L=1/k≤ z≤ L. One can perform symplectic transformations of CP_{2} and symplectic transformations of M^{4} to obtain new string world sheets. In general these are not minimal surfaces and this condition would select some preferred string world sheets.
An alternative  but of course not necessarily equivalent  attempt to formulate this picture would be in terms of number theoretic vision. Spacetime surfaces would be associative or coassociative depending on whether tangent space or normal space in imbedding space is associative  that is quaternionic. These two conditions would reduce spacetime dynamics to associativity and commutativity conditions. String world sheets and partonic 2surfaces would correspond to maximal commutative or cocommutative submanifolds of imbedding space. Commutativity (cocommutativity) would mean that tangent space (normal space as a submanifold of spacetime surface) has complex tangent space at each point and that these tangent spaces integrate to 2surface. SH would mean that data at these 2surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2surfaces intersecting partonic 2surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD.
To sum up, one cannot exclude the possibility that J(M^{4}) is present implying a universal transversal localization of imbedding space spinor harmonics and the modes of spinor fields in the interior of X^{4}: this could perhaps relate to somewhat mysterious decoherence interaction producing locality and to CP breaking and matterantimatter asymmetry. The moduli space for M^{4} Kähler structures proposed by number theoretic considerations would save from the loss of Poincare invariance and the number theoretic vision based on quaternionic and octonionic structure would have rather concrete realization. This moduli space would only extend the notion of "world of classical worlds" (WCW).
For background see the chapter Questions related to the twistor lift of TGD or the article with the same title.
