What's new inTopological Geometrodynamics: an OverviewNote: Newest contributions are at the top! 
Year 2016 
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. 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. For background see the chapter Topological Geometrodynamics: Three Visions. 
Antimatter as dark matter?
It has been found in CERN (see this ) that matter and antimatter atoms have no differences in the energies of their excited states. This is predicted by CPT symmetry. Notice however that CP and T can be separately broken and that this is indeed the case. Kaon is classical example of this in particle physics. Neutral kaon and antikaon behave slightly differently. This finding forces to repeat an old question. Where does the antimatter reside? Or does it exist at all? GUTs predicted that baryon and lepton number are not conserved separately and suggested a solution to the empirical absence of antimatter. GUTs have been however dead for years and there is actually no proposal for the solution of matterantimatter asymmetry in the framework of mainstream theories (actually there are no mainstream theories after the death of superstring theories which also assumed GUTs as low energy limits!). In TGD framework manysheeted spacetime suggests possible solution to the problem. Matter and antimatter are at different spacetime sheets. One possibility is that antimatter corresponds to dark matter in TGD sense that is a phase with h_{eff}=n× h, n=1,2,3,... such that the value of n for antimatter is different from that for visible matter. Matter and antimatter would not have direct interactions and would interact only via classical fields or by emission of say photons by matter (antimatter) suffering a phase transition changing the value of h_{eff} before absorbtion by antimatter (matter). This could be rather rare process. Biophotons could be produced from dark photons by this process and this is assumed in TGD based model of living matter. What the value of n for ordinary visible matter could be? The naive guess is that it is n=1, the smallest possible value. Randell Mills has however claimed the existence of scaled down hydrogen atoms  Mills calls them hydrinos  with ground state binding energy considerably higher than for hydrogen atom. The experimental support for the claim is published in respected journals and the company of Mills is developing a new energy technology based on the energy liberated in the transition to hydrino state. These findings can be understood in TGD framework if one has actually n=6 for visible atoms and n=1, 2, or 3 for hydrinos. Hydrino states would be stabilized in the presence of some catalysts. See this. The model suggests a universal catalyst action. Among other things catalyst action requires that reacting molecule gets energy to overcome the potential barrier making reaction very slow. If an atom  say (dark) hydrogen  in catalyst suffers a phase transition to hydrino (hydrogen with smaller value of h_{eff}/h), it liberates binding energy, and if one of the reactant molecules receives it it can overcome the barrier. After the reaction the energy can be sent back and catalyst hydrino returns to the ordinary hydrogen state. The condition that the dark binding energy is above the thermal energy gives a condition on the value of h_{eff}/h=n as n≤ 32. The size scale of the dark largest allowed dark atom would be about 100 nm, 10 times the thickness of the cell membrane. The notion of phosphate high energy bond is somewhat mysterious concept and manifests as the ability provide energy in ATP to ADP transition. There are claims that there is no such bond. I have spent considerable amount of time to ponder this problem. Could phosphate contain (dark) hydrogen atom able to go to the hydrino state (state with smaller value of h_{eff}/h) and liberate the binding energy? Could the decay ATP to ADP produce the original possibly dark hydrogen? Metabolic energy would be needed to kick it back to ordinary bond in ATP. So: could it be that one has n=6 for stable matter and n is different from this for stable antimatter? Could the small CP breaking cause this? For background see the chapter Classical TGD. 
Minimal surface cosmology
Before the discovery of the twistor lift TGD inspired cosmology has been based on the assumption that vacuum extremals provide a good estimate for the solutions of Einstein's equations at GRT limit of TGD . One can find imbeddings of RobertsonWalker type metrics as vacuum extremals and the general finding is that the cosmological with supercritical and critical mass density have finite duration after which the mass density becomes infinite: this period of course ends before this. The interpretation would be in terms of the emergence of new spacetime sheet at which matter represented by smaller spacetime sheets suffers topological condensation. The only parameter characterizing critical cosmologies is their duration. Critical (overcritical) cosmologies having SO3× E^{3} (SO(4)) as isometry group is the duration and the CP_{2} projection at homologically trivial geodesic sphere S^{2}: the condition that the contribution from S^{2} to g_{rr} component transforms hyperbolic 3metric to that of E^{3} or S^{3} metric fixes these cosmologies almost completely. Subcritical cosmologies have onedimensional CP_{2} projection. Do RobertsonWalker cosmologies have minimal surface representatives? Recall that minimal surface equations read as D_{α}(g^{αβ} ∂_{β}h^{k}g^{1/2})= ∂_{α}[g^{αβ} ∂_{β}h^{k} g^{1/2}] + {_{α}^{k}_{m}} g^{αβ} ∂_{β}h^{m} g^{1/2}=0 , {_{α}^{k}_{m}} ={_{l} ^{k}_{m}} ∂_{α}h^{l} . Subcritical minimal surface cosmologies would correspond to X^{4}⊂ M^{4}× S^{1}. The natural coordinates are RobertsonWalker coordinates, which coincide with lightcone coordinates (a=[(m^{0})^{2}r^{2}_{M}]^{1/2}, r= r_{M}/a,θ, φ) for lightcone M^{4}_{+}. They are related to spherical Minkowski coordinates (m^{0},r_{M},θ,φ) by (m^{0}=a(1+r^{2})^{1/2}, r_{M}= ar). β =r_{M}/m^{0}=r/(1+r^{2})^{1/20},r_{M}). r corresponds to the Lorentz factor r= γ β=β/(1β^{2})^{1/2} The metric of M^{4}_{+} is given by the diagonal form [g_{aa}=1, g_{rr}=a^{2}/(1+r^{2}), g_{θθ}= a^{2}r^{2}, g_{φφ}= a^{2}r^{2}sin^{2}(θ)]. One can use the coordinates of M^{4}_{+} also for X^{4}. The ansatz for the minimal surface reads is Φ= f(a). For f(a)=constant one obtains just the flat M^{4}_{+}. In nontrivial case one has g_{aa}= 1R^{2} (df/da)^{2}. The g^{aa} component of the metric becomes now g^{aa}=1/(1R^{2}(df/da)^{2}). Metric determinant is scaled by g_{aa}^{1/2} =1 → (1R^{2}(df/da)^{2}^{1/2}. Otherwise the field equations are same as for M^{4}_{+}. Little calculation shows that they are not satisfied unless one as g_{aa}=1. Also the minimal surface imbeddings of critical and overcritical cosmologies are impossible. The reason is that the criticality alone fixes these cosmologies almost uniquely and this is too much for allowing minimal surface property. Thus one can have only the trivial cosmology M^{4}_{+} carrying dark energy density as a minimal surface solution! This obviously raises several questions.
See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title. 
LIGO blackhole anomaly and minimal surface model for star
The TGD inspired model of star as a minimal surface with stationary spherically symmetric metric suggests strongly that the analog of blackhole metric as two horizons. The outer horizon is analogous to Scwartschild horizon in the sense that the roles of time coordinate and radial coordinate change. Radial metric component vanishes at Scwartschild horizon rather than divergence. Below the inner horizon the metric has Eucldian signature. Is there any empirical evidence for the existence of two horizons? There is evidence that the formation of the recently found LIGO blackhole (discussed from TGD view point in is not fully consistent with the GRT based model (see this). There are some indications that LIGO blackhole has a boundary layer such that the gravitational radiation is reflected forth and back between the inner and outer boundaries of the layer. In the proposed model the upper boundary would not be totally reflecting so that gravitational radiation leaks out and gave rise to echoes at times .1 sec, .2 sec, and .3 sec. It is perhaps worth of noticied that time scale .1 sec corresponds to the secondary padic time scale of electron (characterized by Mersenne prime M_{127}= 2^{127}1). If the minimal surface solution indeed has two horizons and a layer like structure between them, one might at least see the trouble of killing the idea that it could give rise to repeated reflections of gravitational radiation. The proposed model (see this) assumes that the inner horizon is Schwarstchild horizon. TGD would however suggests that the outer horizon is the TGD counterpart of Schwartschild horizon. It could have different radius since it would not be a singularity of g_{rr} (g_{tt}/g_{rr} would be finite at r_{S} which need not be r_{S}=2GM now). At r_{S} the tangent space of the spacetime surface would become effectively 2dimensional: could this be interpreted in terms of strong holography (SH)? One should understand why it takes rather long time T=.1 seconds for radiation to travel forth and back the distance L= r_{S}r_{E} between the horizons. The maximal signal velocity is reduced for the lightlike geodesics of the spacetime surface but the reduction should be rather large for L∼ 20 km (say). The effective lightvelocity is measured by the coordinate time Δ t= Δ m^{0}+ h(r_{S})h(r_{E}) needed to travel the distance from r_{E} to r_{S}. The Minkowski time Δ m^{0}_{+} would be the from null geodesic property and m^{0}= t+ h(r) Δ m^{0}_{+} =Δ t h(r_{S})+h(r_{E}) , Δ t = ∫_{rE}^{rS}(g_{rr}/g_{tt})^{1/2} dr== ∫_{rE}^{rS} dr/c_{#} . The time needed to travel forth and back does not depend on h and would be given by Δ m^{0} =2Δ t =2∫_{rE}^{rS}dr/c_{#} . This time cannot be shorter than the minimal time (r_{S}r_{E})/c along lightlike geodesic of M^{4} since lightlike geodesics at spacetime surface are in general timelike curves in M^{4}. Since .1 sec corresponds to about 3× 10^{4} km, the average value of c_{#} should be for L= 20 km (just a rough guess) of order c_{#}∼ 2^{11}c in the interval [r_{E},r_{S}]. As noticed, T=.1 sec is also the secondary padic time assignable to electron labelled by the Mersenne prime M_{127}. Since g_{rr} vanishes at r_{E} one has c_{#}→ ∞. c_{#} is finite at r_{S}. There is an intriguing connection with the notion of gravitational Planck constant. The formula for gravitational Planck constant given by h_{gr}= GMm/v_{0} characterizing the magnetic bodies topologically for mass m topologically condensed at gravitational magnetic flux tube emanating from large mass M. The interpretation of the velocity parameter v_{0} has remained open. Could v_{0} correspond to the average value of c_{#}? For inner planets one has v_{0}≈ 2^{11} so that the the order of magnitude is same as for the the estimate for c_{#}. See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title. 
Minimal surface counterpart of ReissnerNordstöm solution
Occarm's razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations at three levels. The level of "world of classical worlds" (WCW) defined by the space of 3surfaces endowed with Kähler structure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein's geometrization program applied to quantum theory. Second level is spacetime level. Spacetime surfaces correspond to preferred extremals of Käction in M^{4}× CP_{2}. The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2surfaces and preferred extremals are minimal surface extremals of Kähler action so that the classical dynamics in spacetime interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee welldefinedness of em charge. At imbedding space level the modes of imbedding space spinor fields define ground states of supersymplectic representations and appear in QFTGRT limit. GRT involves postNewtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. 
How to build TGD spacetime from legos?
TGD predicts shocking simplicity of both quantal and classical dynamics at spacetime level. Could one imagine a construction of more complex geometric objects from basic building bricks  spacetime legos? Let us list the basic ideas.
What could be the simplest surfaces of this kind  legos?
Geodesic minimal surfaces with vanishing induced gauge fields Consider first static objects with 1D CP_{2} projection having thus vanishing induced gauge fields. These objects are of form M^{1}× X^{3}, X^{3}⊂ E^{3}× CP_{2}. M^{1} corresponds to timelike or possible lightlike geodesic (for CP_{2} type extremals). I will consider mostly Minkowskian spacetime regions in the following.
What about minimal surfaces and geodesic submanifolds carrying nonvanishing gauge fields  in particular em field (Kähler form identifiable as U(1) gauge field for weak hypercharge vanishes and thus also its contribution to em field)? Now one must use 2D geodesic spheres of CP_{2} combined with 1D geodesic lines of E^{2}. Actually both homologically nontrivial resp. trivial geodesic spheres S^{2}_{I} resp. S^{2}_{II} can be used so that also nonvanishing Kähler forms are obtained. The basic legos are now D× S^{2}_{i}, i=I,II and they can be combined with the basic legos constructed above. These legos correspond to two kinds of magnetic flux tubes in the ideal infinitely thin limit. There are good reasons to expected that these infinitely thin flux tubes can be thickened by deforming them in E^{3} directions orthogonal to D. These structures could be used as basic building bricks assignable to the edges of the tensor networks in TGD. Static minimal surfaces, which are not geodesic submanifolds One can consider also more complex static basic building bricks by allowing bricks which are not anymore geodesic submanifolds. The simplest static minimal surfaces are form M^{1}× X^{2}× S^{1}, S^{1} ⊂ CP_{2} a geodesic line and X^{2} minimal surface in E^{3}. Could these structures represent higher level of selforganization emerging in living systems? Could the flexible network formed by living cells correspond to a structure involving more general minimal surfaces  also nonstatic ones  as basic building bricks? The Wikipedia article about minimal surfaces in E^{3} suggests the role of minimal surface for instance in biochemistry (see this). The surfaces with constant positive curvature do not allow imbedding as minimal surfaces in E^{3}. Corals provide an example of surface consisting of pieces of 2D hyperbolic space H^{2} immersed in E^{3} (see this). Minimal surfaces have negative curvature as also H^{2} but minimal surface immersions of H^{2} do not exist. Note that pieces of H^{2} have natural imbedding to E^{3} realized as lightone proper time constant surface but this is not a solution to the problem. Does this mean that the proposal fails?
Dynamical minimal surfaces: how spacetime manages to engineer itself? At even higher level of selforganization emerge dynamical minimal surfaces. Here string world sheets as minimal surfaces represent basic example about a building block of type X^{2}× S^{2}_{i}. As a matter fact, S^{2} can be replaced with complex submanifold of CP_{2}. One can also ask about how to perform this building process. Also massless extremals (MEs) representing TGD view about topologically quantized classical radiation fields are minimal surfaces but now the induced Kähler form is nonvanishing. MEs can be also Lagrangian surfaces and seem to play fundamental role in morphogenesis and morphostasis as a generalization of Chladni mechanism. One might say that they represent the tools to assign material and magnetic flux tube structures at the nodal surfaces of MEs. MEs are the tools of spacetime engineering. Here manysheetedness is essential for having the TGD counterparts of standing waves. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. 
Can one apply Occam's razor as a general purpose debunking argument to TGD?Occarm's razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations at three levels. The level of "world of classical worlds" (WCW) defined by the space of 3surfaces endowed with Kählerstructure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein's geometrization program applied to quantum theory. Second level is spacetime level. Spacetime surfaces correspond to preferred extremals of Kähler action in M^{4}× CP_{2}. The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2surfaces and preferred extremals are minimal surface extremals ofKähler action so that the classical dynamics in spacetime interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee welldefinedness of em charge. At imbedding space level the modes of imbedding space spinor fields define ground states of supersymplectic representations and appear in QFTGRT limit. GRT involves postNewtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. 
Critizing the view about elementary particlesThe concrete model for elementary particles has developed gradually during years and is by no means final. In the recent model elementary particle corresponds to a pair of wormhole contacts and monopole flux runs between the throats of of the two contacts at the two spacetime sheets and through the contacts between spacetime sheets. The first criticism relates to twistor lift of TGD. In the case of Kähler action the wormhole contacts correspond to deformations for pieces of CP_{2} type vacuum extremals for which the 1D M^{4} projection is lightlike random curve. Twistor lift adds to Kähler action a volume term proportional to cosmological constant and forces the vacuum extremal to be a minimal surface carrying nonvanishing lightlike momentum (this is of course very natural): one could call this surface CP_{2} extremal. This implies that M^{4} projection is lightlike geodesic: this is physically rather natural. Twistor lift leads to a loss of the proposed spacetime correlate of massivation used also to justify padic thermodynamics: the average velocity for a lightlike random curve is smaller than maximal signal velocity  this would be a clear classical signal for massivation. One could however conjecture that the M^{4} projection for the lightlike boundaries of string world sheets becomes lightlike geodesic of M^{4}× CP_{2} instead lightlike geodesic of M^{4} and that this serves as the correlate for the massivation in 4D sense. Second criticism is that I have not considered in detail what the monopole flux hypothesis really means at the level of detail. Since the monopole flux is due to the CP_{2} topology, there must be a closed 2surface which carries this flux. This implies that the flux tube cannot have boundaries at larger spacetime surface but one has just the flux tube which closed cross section obtained as a deformation of a cosmic string like object X^{2}× Y^{2}, where X^{2} is minimal surface in M^{4} and Y^{2} a complex surface of CP_{2} characterized by genus. Deformation would have 4D M^{4} projection instead of 2D string world sheet. Note: One can also consider objects for which the flux is not monopole flux: in this case one would have deformations of surfaces of type X^{2}× S^{2}, S^{2} homologically trivial geodesic sphere: these are nonvacuum extremals for the twistor lift of Kähler action (volume term). The net magnetic flux would vanish  as a matter fact, the induced Kähler form would vanish identically for the simplest situation. These objects might serve as correlates for gravitons since the induced metric is the only field degree of freedom. One could also have nonvanishing fluxes for flux tubes with disklike cross section. If this is the case, the elementary particles would be much simpler than I have though hitherto.
If only Minkowskian portions are present, particles could be seen as pairs of open fermionic strings and the counterparts of open string vertices would be possible besides reconnection of closed strings. For this option one can also consider single fermionic open strings connecting wormhole contacts: now possible flux tube would not carry monopole flux. For background see Particle Massivation in TGD Universe. 
Does GRT really allow gravitational radiation?In Facebook discussion Niklas Grebäck mentioned Weyl tensor and I learned something that I should have noticed long time ago. Wikipedia article lists the basic properties of Weyl tensor as the traceless part of curvature tensor, call it R. Weyl tensor C is vanishing for conformally flat spacetimes. In dimensions D=2,3 Weyl tensor vanishes identically so that they are always conformally flat: this obviously makes the dimension D=3 for space very special. Interestingly, one can have nonflat spacetimes with nonvanishing Weyl tensor but the vanishing Schouten/Ricci/Einstein tensor and thus also with vanishing energy momentum tensor. The rest of curvature tensor R can be expressed in terms of so called KulkarniNomizu product P• g of Schouten tensor P and metric tensor g: R=C+P• g, which can be also transformed to a definition of Weyl tensor using the definition of curvature tensor in terms of Christoffel symbols as the fundamental definition. KulkarniNomizu product • is defined as tensor product of two 2tensors with symmetrization with respect to first and second index pairs plus antisymmetrization with respect to second and fourth indices. Schouten tensor P is expressible as a combination of Ricci tensor Ric defined by the trace of R with respect to the first two indices and metric tensor g multiplied by curvature scalar s (rather than R in order to use index free notation without confusion with the curvature tensor). The expression reads as P= 1/(D2)×[Ric(s/2(D1))×g] . Note that the coefficients of Ric and g differ from those for Einstein tensor. Ricci tensor and Einstein tensor are proportional to energy momentum tensor by Einstein equations relate to the part. Weyl tensor is assigned with gravitational radiation in GRT. What I see as a serious interpretational problem is that by Einstein's equations gravitational radiation would carry no energy and momentum in absence of matter. One could argue that there are no free gravitons in GRT if this interpretation is adopted! This could be seen as a further argument against GRT besides the problems with the notions of energy and momentum: I had not realized this earlier. Interestingly, in TGD framework so called massless extremals (MEs) (see this and this) are foursurfaces, which are extremals of Kähler action, have Weyl tensor equal to curvature tensor and therefore would have interpretation in terms of gravitons. Now these extremals are however nonvacuum extremals.
See the article Does GRT really allow gravitational radiation?. For background see the chapter Classical TGD. 
Three reasons for the localization of induced spinor fields at string world sheetsThere are now three good reasons for the modes of the induced spinor fields to be localized to 2D string world sheets and partonic 2surfaces  in fact, to the boundaries of string world sheets at them defining fermionic world lines. I list these three good reasons in the same order as I became aware of them.
For details see the chapter Why TGD and What TGD is?.
