Some questions related to the twistor lift of TGD

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.

1. Can the analog of Kähler form J(M⁴) assignable to M⁴ suggested by the symmetry between M⁴ and CP₂ 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(M⁴) would be needed? J(M⁴) 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 M⁴× CP₂. The estimate for the CP mass splitting of neutral kaon and anti-kaon 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.
   
   

2. 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 p-adic mass calculations suggest that string tension is determined by CP₂ size R: the analog of string tension appearing in mass formula given by p-adic mass calculations would be by a factor about 10^-8 smaller than that estimated from string tension. The discrepancy evaporates by noticing that p-adic mass calculations rely on p-adic thermodynamics at imbedding space level whereas string world sheets appear at space-time level.
   
   
3. Could one regard the localization of spinor modes to string world sheets as a localization to Lagrangian sub-manifolds of space-time surface having by definition vanishing induced Kähler form: J(M⁴)+J(CP₂)=0. Lagrangian sub-manifolds would be commutative in the sense of Poisson bracket. Could string world sheets be minimal surfaces satisfying J(M⁴)+J(CP₂)=0. The Lagrangian condition allows also more general solutions - even 4-D space-time 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 ill-defined 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 space-time surface in octonionic imbedding space.
   
   

1. Can the Kähler form of M⁴ appear in Kähler action?

I have already earlier considered the question whether the analog of Kähler form assignable to M⁴ could appear in Kähler action. Could one replace the induced Kähler form J(CP₂) with the sum J=J(M⁴)+J(CP₂) such that the latter term would give rise to a new component of Kähler form both in space-time interior at the boundaries of string world sheets regarded as point-like particles? This could be done both in the Kähler action for the interior of X⁴ 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(CP₂) and A_μ(M⁴) associated with J(CP₂) and J(M⁴). The interpretation of this coupling is an interesting challenge.

Consider first the objections against introducing J(M⁴) to the Kähler action at imbedding space level.

1. J(M⁴) would would break translational and Lorentz symmetries at the level of imbedding space since J(M⁴) cannot be Lorentz invariant. For imbedding space spinor modes this term would bring in coupling to the self-dual Kähler form in M⁴. The simplest choice is A=(A_t=z, A_z=0,A_x=y,A_y=0) defining decomposition M⁴ =M²× E². For Dirac equation in M⁴ one would have free motion in preferred time-like (t,z)-plane plane M² in whereas in x- and y-directions (E² 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 non-conserved 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.
   
   
2. Since J(M⁴) is not Lorentz invariant Lorentz boosts would produce new M²× E² decomposition. If one assumes above kind of linear gauge as gauge invariance suggests, the choices with fixed second tip of causal diamond (CD) define finite-dimensional 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 M⁴ with the choices of origins parameterized by M⁴. The introduction of the moduli space would allow to preserve Poincare invariance.
   
   
3. If one generalizes the condition for Kähler metric to J²(M⁴)=-g(M⁴) fixing the scaling of J, the coupling to A(M⁴) 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 self-dual magnetic and electric fields for J(M⁴) 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 M² and E² having interpretation a plane defined by light-like momentum and polarization plane associated with a given CD invariant.
   
   

4. 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(M⁴) 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 space-time surfaces inside CD. The choice of J(M⁴) for CD would select preferred rest system (quantization axis for energy as a line connecting tips of CD) via electric part of J(M⁴) and quantization axis of spin (via magnetic part of J(M⁴). 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(M⁴)=0 Poincare symmetries are realized at the level of imbedding space but obviously broken slightly by the geometry of CD. The allowance of J(M⁴)≠ 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 space-time level?

1. The introduction of J(M⁴) part to Kähler action has nice number theoretic aspects. In particular, J selects the preferred complex and quaternionic sub-space of octonionic space of imbedding space. The simplest possibility is that the Kähler action is defined by the Kähler form J(M⁴)+J(CP₂).
   

   Since M⁴ and CP₂ Kähler geometries decouple it should be possible to take the counterpart of Kähler coupling strength in M⁴ to be much larger than in CP₂ degrees of freedom so that M⁴ 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.
   
   

2. 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 light-like 3-surfaces at which the signature of the induced metric changes.
   
   
3. At space-time level the field equations get more complex if the M⁴ projection has dimension D(M⁴)>2 and also for D(M⁴)=2 if it carries non-vanishing induced J(M⁴). One would obtain cosmic strings of form X²× Y² as minimal surface extremals of ordinary Kähler action or X² Lagrangian manifold of M⁴ as also CP₂ type vacuum extremals and their deformations with M⁴ projection Lagrangian manifold. Thus the differences would not be seen for elementary particle and string like objects. Simplest string worlds sheet for which J(M⁴) vanishes would correspond to a piece of plane M².
   

   M⁴ is the simplest minimal surface extremal of Kähler action necessarily involving also J(M⁴). The action in this case vanishes identically by self-duality (in Euclidian signature self-duality does not imply this). For perturbations of M⁴ such as spherically symmetric stationary metric the contribution of M⁴ 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 M⁴ degrees of freedom could make sense.
   
   

4. 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 light-like vector so that the contraction T^αβH^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_γJ^γβ = a× g^αβ+b × k^αk^β.
   These conditions together with light-likeness 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 CP₂ coordinates ξⁱ would be functions of light-like M² coordinate u₊=k• m, k light-like vector, and of complex coordinate w for E² orthogonal to M². Therefore the CP₂ projection would 3-D rather than 2-D 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 CP₂ 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 light-like dual of u in plane M². 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₊,u_i and g^ww* are obtained and their contractions with the second fundamental form vanish identically since there are no common index pairs with simultaneously non-vanishing 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 Hamilton-Jacobi structure for M⁴ consisting of locally orthogonal slicings by integrable 2-surfaces having tangent space having local decomposition M²_x× E²_x with light-like direction depending on point x. An objection is that the direction of light-like momentum depends on position: this need not be inconsistent with momentum conservation but would imply that the total four-momentum is not light-like anymore. Topological condensation for MEs and at MEs could imply this kind modification.
   
   

5. 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 CP₂ Kähler gauge potential (added to the equation for a light-like geodesic line). Now also the coupling to M⁴ gauge potential would be obtained. The condition J(M⁴)+ J(CP₂)=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 M⁴ Kähler magnetic flux for string world sheet contributes also to the modified Dirac equation since the gamma matrices are modified gamma matrices required by super-conformal symmetries and defined as contractions of canonical momentum densities with imbedding space gamma matrices. This is true both in space-time interior, at string world sheets and at their boundaries. CP₂ (M⁴) term gives a contribution proportional to CP₂ (M⁴) 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 M⁴ defining the simplest extremal. The general interpretation of M⁴ Kähler form could be as a quantum tool for transversal dynamical localization of wave packets in Kähler magnetic and electric fields of M⁴. Analog for decoherence occurring in transversal degrees of freedom would be in question. Hadron physics could be one application.
   
   

How to test this idea?

1. 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 M⁴ field equations are satisfied since energy momentum tensor vanishes identically. For the small deformations the presence of J(M⁴) would reduce rotational symmetry to cylindrical symmetry.
   
   
2. J(M⁴) could make its presence manifest in the physics of right-handed neutrino having no direct couplings to electroweak gauge fields. Mixing with left handed neutrino is however induced by mixing of M⁴ and CP₂ gamma matrices. The transversal localization of right-handed neutrino in a background, which is a small deformation of M⁴ could serve as an experimental signature.
   
   
3. CP breaking in hadronic systems is one of the poorly understood aspects of fundamental physics and relates closely to the mysterious matter-antimatter asymmetry. The constant electric part of self dual J(M⁴) 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(M⁴) explain the observed CP breaking as appearing already at the level of imbedding space M⁴× CP₂ and could this breaking explain hadronic CP breaking and matter anti-matter asymmetry? Could M⁴ 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(M⁴) for J(M⁴).

1. The coupling of Kähler form to leptons is 3 times larger than to to quarks as in the case of A(CP₂). 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 space-time sheets.
   
   
2. The potential satisfies ∂_μA^μ(M⁴)=0. Let the non-vanishing components of the Kähler gauge potential be (A₀,A_z)=ε (x,+/- y). The sign fact ε+/- 1 corresponds to self dual and antiself-dual options, let us assume self-duality as in the case of CP₂ Kähler form. Scalar d'Alembertian reads as (∂^μ∂_μ+ A^μA_μ)Ψ= -m² Ψ.
   
   
3. Assuming momentum eigenstate in time and z-direction (plane M²), one obtains by separation of variables (H₁+H₂)Ψ= (E^-m²-k_z²)Ψ. H_x= -∂_x²+k²x² and H_y= -∂_y²+k²y²) are oscillator Hamiltonians. The spectrum is of H_x+H_y is given by k_T²= (n₁+n₂+1)2^1/2|k| and one obtains E²=m² +k_z² +k_T². This contribution is CP invariant and same for fermions and anti-fermions. 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 m² is increased 2^1/2|k| hbar²/L², L= 1 m if standard convention for metric is used. For other conventions the numerical value of CP₂ radius is scale by L/L_new. L must correspond to some physical scale assignable to particle: secondary p-adic length scale is the natural identification.
   
   
4. Spinor d'Alembertian contains also dipole moment term kX=J^muνΣ_μν giving a contribution, which depends on the sign of k: E²=m² +k_z² +k_T²+ 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 anti-fermions. 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.
   
   
5. Scaling covariance allows in principle all values L. To estimate the size of the effect one must fix the length scale L. CP₂ 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 p-adic length scale of fermion defining also the size scale of its magnetic body is a natural guess so that Δ E²/E²≈ 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 matter-antimatter-asymmetry 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^-12m ≈ 6.9× 10^-2 eV (see this) so that this gives no constraints. Kaon-antikaon mass difference is estimated to be about 3.5× 10^-6 eV (see this). This would correspond to a p-adic 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 p-adic 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 non-vanishing due to the different sign of coupling k. The sign of the mass splitting is also opposite for kaon and antikaon.
   
   

6. One can also consider the modified Dirac equation for canonically imbedded M⁴ which is simplest preferred extremal. The coupling to J(M⁴) to modified Dirac equation in space-time interior with gamma matrices replaced with modified gamma matrices are obtained as contractions of canonical momentum currents with M⁴ gamma matrices. Completely analogous phenomenon happens for CP₂ type extremals. T^αβ=0 so that the modified gamma comes from J^αβ J^k_~l ∂_βm^lγ_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 2-surfaces carry the information about quantum states and about space-time 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 2-D. The light-like 3-surfaces separating space-time regions with Minkowskian and Euclidian signature of the induced metric are indeed metrically 2-D, 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 co-incide at partonic 2-surfaces at the ends of space-time 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 light-like 3-surfaces and give rise to covering space structure. The action is trivial at partonic 2-surfaces should be trivial if one wants singular covering: this would mean that discretizations of partonic 2-surfaces 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 well-defined. This condition can be satisfied in certain situations also for the entire space-time 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 higher-D objects can carry modes of induced spinor field Could the analogs of Lagrangian sub-manifolds of X⁴ ⊂ M⁴× CP₂ satisfying J(M⁴)+J(CP₂)=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_P² 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 M⁴ replaced with its geometric twistor space M⁴× S² 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 M⁴ Kähler form should appear in field equations. 1/G could naturally appear in the string tension for string world sheets as string models suggest. p-Adic mass calculations identify the analog of string tension as something of order of magnitude of 1/R². This identification comes from the fact that the ground states of super-conformal representations correspond to imbedding space spinor modes, which are solutions of Dirac equation in M⁴× CP₂. This argument is rather convincing and allows to expect that the p-adic 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 Non-commutative 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 non-commutativity 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 space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time 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 M⁴ coordinates m^k would look something like [m^k, m^l] = l_P²J^kl, where l_P is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form J^kl.

1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p,q]= hbar J^pq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of J_pq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian sub-manifolds 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 M⁴ to general space-time: linear M⁴ coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H=M⁴× CP₂: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M⁴ makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP₂ has naturally symplectic form.

Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p₁,p₂,q₁,q₂) so that one must use either (p₁,p₂) or (q₁,q₂) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum J_μν(M⁴)+ J_μν(CP₂) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M⁴ and CP₂ are possible: they would correspond to homologically trivial cosmic strings X²× Y²⊂ M⁴× CP₂, 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): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M⁴ × CP₂ symplectic structure to space-time 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 M⁴ 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 non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces.

The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M⁴)+J(CP₂)=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M⁴)+J(CP₂)=0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M⁴)+J(CP₂)=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X²× Y² would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced Kähler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion 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(M⁴) +J(CP₂)=0 using generalized symplectic transformations of M⁴ and CP₂ 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 M² with coordinates (t,z) and homologically non-trivial geodesic sphere S² of CP₂ with coordinates (u= cos(Θ),Φ). One has J(M⁴)_tz=1 and J_uΦ= 1. Identify string world sheet via map (u,Φ)= (kz,ω t) from M² to S². The induced CP₂ Kahler form is J(CP₂)_tz= kω. kω=-1 guarantees J(M⁴) +J(CP₂)=0. The strings have necessarily finite length from L=1/k≤ z≤ L. One can perform symplectic transformations of CP₂ and symplectic transformations of M⁴ 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. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces 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(M⁴) is present implying a universal transversal localization of imbedding space spinor harmonics and the modes of spinor fields in the interior of X⁴: this could perhaps relate to somewhat mysterious de-coherence interaction producing locality and to CP breaking and matter-antimatter asymmetry. The moduli space for M⁴ 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 article Some questions related to the twistor lift of TGD or the chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? of "Towards M-matrix".

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

Articles and other material related to TGD.