Monday, November 14, 2022

Could the exotic smooth structures have a physical interpretation in the TGD framework?

The exotic smooth structures appearing for 4-D manifolds, even R4, challenges both the classical GRT and quantum GRT based on path integral approach. They could also cause problems in the TGD framework. I have already considered the possibility that the exotics could be eliminated by what could be called holography of smoothness meaning that the smooth structure for the boundary dictates it for the entire 4-surface. One can however argue that the atlas of charts for 3-D boundary cannot define uniquely the atlas of charges for 4-D surface.

In the TGD framework, exotic smooth structures could also have a physical interpretation. As noticed, the failure of the standard smooth structure can be thought to occur at a point set of dimension zero and correspond to a set of point defects in condensed matter physics. This could have a deep physical meaning.

  1. The space-time surfaces in H=M4× CP2 are images of 4-D surfaces of M8 by M8-H-duality. The proposal is that they reduce to minimal surfaces analogous to soap films spanned by frames. Regions of both Minkowskian and Euclidean signature are predicted and the latter correspond to wormhole contacts represented by CP2 type extremals. The boundary between the Minkowskian and Euclidean region is a light-like 3-surface representing the orbit of partonic 2-surface identified as wormhole throat carrying fermionic lines as boundaries of string world sheets connecting orbits of partonic 2-surfaces.
  2. These fermionic lines are counterparts of the lines of ordinary Feynman graphs, and have ends at the partonic 2-surfaces located at the light-like boundaries of CD and in the interior of the space-time surface. The partonic surfaces, actually a pair of them as opposite throats of wormhole contact, in the interior define topological vertices, at which light-like partonic orbits meet along their ends.
  3. These points should be somehow special. Number theoretically they should correspond points with coordinates in an extension of rationals for a polynomial P defining 4-surface in H and space-time surface in H by M8-H duality. What comes first in mind is that the throats touch each other at these points so that the distance between Minkowskian space-time sheets vanishes. This is analogous to singularities of Fermi surface encountered in topological condensed matter physics: the energy bands touch each other. In TGD, the partonic 2-surfaces at the mass shells of M4 defined by the roots of P are indeed analogs of Fermi surfaces at the level of M4⊂ M8, having interpretation as analog of momentum space.

    Could these points correspond to the defects of the standard smooth structure in X4? Note that the branching at the partonic 2-surface defining a topological vertex implies the local failure of the manifold property. Note that the vertices of an ordinary Feynman diagram imply that it is not a smooth 1-manifold.

  4. Could the interpretation be that the 4-manifold obtained by removing the partonic 2-surface has exotic smooth structure with the defect of ordinary smooth structure assignable to the partonic 2-surface at its end. The situation would be rather similar to that for the representation of exotic R4 as a surface in CP2 with the sphere at infinity removed (see this).
  5. The failure of the cosmic censorship would make possible a pair creation. As explained, the fermionic lines can indeed turn backwards in time by going through the wormhole throat and turn backwards in time. The above picture suggests that this turning occurs only at the singularities at which the partonic throats touch each other. The QFT analog would be as a local vertex for pair creation.
  6. If all fermions at a given boundary of CD have the same sign of energy, fermions which have returned back to the boundary of CD, should correspond to antifermions without a change in the sign of energy. This would make pair creation without fermionic 4-vertices possible.

    If only the total energy has a fixed sign at a given boundary of CD, the returned fermion could have a negative energy and correspond to an annihilation operator. This view is nearer to the QFT picture and the idea that physical states are Galois confined states of virtual fundamental fermions with momentum components, which are algebraic integers. One can also ask whether the reversal of the arrow of time for the fermionic lines could give rise to gravitational quantum computation as proposed here.

See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

Sunday, November 13, 2022

Could the existence of exotic smooth structures pose problems for TGD?

The article of Gabor Etesi (see this) gives a good idea about the physical significance of the existence of exotic smooth structures and how they destroy the cosmic censorship hypothesis (CCH of GRT stating that spacetimes of GRT are globally hyperbolic so that there are no time-like loops.

1. Smooth anomaly

No compact smoothable topological 4-manifold is known, which would allow only a single smooth structure. Even worse, the number of exotics is infinite in every known case! In the case of non-compact smoothable manifolds, which are physically of special interest, there is no obstruction against smoothness and they typically carry an uncountable family of exotic smooth structures.

One can argue that this is a catastrophe for classical general relativity since smoothness is an essential prerequisite for tensory analysis and partial differential equations. This also destroys hopes that the path integral formulation of quantum gravitation, involving path integral over all possible space-time geometries, could make sense. The term anomaly is certainly well-deserved.

Note however that for 3-geometries appearing as basic objects in Wheeler's superspace approach, the situation is different since for D≤3 there is only a single smooth structure. If one has holography, meaning that 3-geometry dictates 4-geometry, it might be possible to avoid the catastrophe.

The failure of the CCH is the basic message of Etesi's article. Any exotic R4 fails to be globally hyperbolic and Etesi shows that it is possible to construct exact vacuum solutions representing curved space-times which violate the CCH. In other words, GRT is plagued by causal anomalies.

Etesi constructs a vacuum solution of Einstein's equations with a vanishing cosmological constant which is non-flat and could be interpreted as a pure gravitational radiation. This also represents one particular aspect of the energy problem of GRT: solutions with gravitational radiation should not be vacua.

  1. Etesi takes any exotic R4, which has the topology of S3× R and has an exotic smooth structure, which is not a Cartesian product. Etesi maps maps R4 to CP2, which is obtained from C2 by gluing CP1 to it as a maximal ball B3r for which the radial Eguchi-Hanson coordinate approaches infinity: r → &infty;. The exotic smooth structure is induced by this map. The image of the exotic atlas defines atlas. The metric is that of CP2 but SU(3) does not act as smooth isometries anymore.
  2. After this Etesi performs Wick rotation to Minkowskian signature and obtains a vacuum solution of Einstein's equations for any exotic smooth structure of R4.
The question of exotic smoothness is encountered both at the level of embedding space and associated fixed spaces and at the level of space-time surfaces and their 6-D twistor space analogies.

2. Holography of smoothness

In the TGD framework space-time is 4-surface rather than abstract 4-manifold. 4-D general coordinate invariance, assuming that 3-surfaces as generalization of point-like particles are the basic objects, suggests a fully deterministic holography. A small failure of determinism is however possible and expected, and means that space-time surfaces analogous to Bohr orbits become fundamental objects. Could one avoid the smooth anomaly in this framework?

The 8-D embedding space topology induces 4-D topology. My first naive intuition was that the 4-D smooth structure, which I believed to be somehow inducible from that of H=M4× CP2, cannot be exotic so that in TGD physics the exotics could not be realized. But can one really exclude the possibility that the induced smooth structure could be exotic as a 4-D smooth structure?

What does the induction of a differentiable structure really mean? Here my naive expectations turn out to be wrong.

  1. If a sub-manifold S⊂ H can be regarded as an embedding of smooth manifold N to S⊂ H, the embedding N→ S⊂ H induces a smooth structure in S (see this). The problem is that the smooth structure would not be induced from H but from N and for a given 4-D manifold embedded to H one could also have exotic smooth structures. This induction of smooth structure is of course physically adhoc.

    It is not possible to induce the smooth structure from H to sub-manifold. The atlas defining the smooth structure in H cannot define the charts for a sub-manifold (surface). For standard R4 one has only one atlas.

  2. Could M8-H duality help and holography help? One has holography in M8 and this induces holography in H. The 3-surfaces X3 inducing the holography in M8 are parts of mass shells, which are hyperbolic spaces H3⊂ M4⊂ M8. 3-surfaces X3 could be even hyperbolic 3-manifolds as unit cells of tessellations of H3. These hyperbolic manifolds have unique smooth structures as manifolds with dimension D<4.
  3. One can ask whether the smooth structure at the boundary of a manifold could dictate that of the manifold uniquely. One could speak of holography for smoothness.

    The implication would be that exotic smooth 4-manifold cannot have a boundary. Indeed, R4 does not have a boundary. Could this theorem generalize so that 3-surfaces as sub-manifolds of mass shells H3m determined by the polynomials defining the 4-surface in M8 take the role of the boundaries?

    The regions of X4⊂ M8 connecting two sub-sequent mass shells would have a unique smooth structure induced by the hyperbolic manifolds H3 at the ends. These smooth structures are unique by D<4 and cannot be exotic. Smooth holography would determine the smooth structure from that for the boundary of 4-surface.

  4. However, the holography for smoothness is argued to fail (see this). Assume a 4-manifold W with 2 different smooth structures. Remove a ball B4 belonging to an open set U and construct a smooth structure at its boundary S3. Assume that this smooth structure can be continued to W. If the continuation is unique, the restrictions of the 2 smooth structures in the complement of B4 would be equivalent but it is argued that they are not.

    The first layman objection is that the two smooth structures of W are equivalent in the complement W-B3 of an arbitrary small ball B3⊂ W but not in the entire W. This would be analogous to coordinate singularity. For instance, a single coordinate chart is enough for a sphere in the complement of an arbitrarily small disk. An exotic smooth structure would be like a local defect in condensed matter physics.

    The second layman objection is that smooth structure, unlike topology, cannot be induced from W to W-B3 but only from W-B3 to W. If one a smooth structure at the boundary S3 is chosen, it determines the smooth structure in the interior as standard smooth structure.

  5. In fact, one could argue that the mere fact the 4-surfaces have boundaries as their ends at the light-like boundaries of CD, implies a unique smooth structure by holography. It is however possible that the mass-shells correspond to discontinuities of derivatives so that the smooth holography decomposes to a piece-wise holography. This would mean that M8-H duality is needed.
Amazingly, if the holography of smoothness holds true, the avoidance of the smooth exotics requires holography and both number theoretical vision and general coordinate invariance of geometric vision predict the holography in the TGD framework. For higher space-time dimensions D>4 one cannot avoid the exotics. Also the number theoretic vision fails for them.

3. Can embedding space and related spaces have exotic smooth structure?

One can worry about the exotic smooth structures possibly associated with the M4, CP2, H=M4× CP2, causal diamond CD=cd× CP2, where cd is the intersection of the future and past directed light-cones of M4, and with M8. One can also worry about the twistor spaces CP3 resp. SU(3)/U(1)× U(1) associated with M4 resp. CP2.

The key assumption of TGD is that all these structures have maximal isometry groups so that they relate very closely to Lie groups, whose unique smooth structures are expected to determine their smooth structures.

  1. The first sigh of relief is that all Lie groups have the standard smooth structure. In particular, exotic R4 does not allow translations and Lorentz transformations as isometries. I dare to conclude that also the symmetric spaces like CP2 and hyperbolic spaces such as Hn= SO(1,n)/SO(n) are non-exotic since they provide a representation of a Lie group as isometries and the smoothness of the Lie group is inherited. This would mean that the charts for the coset space G/H would be obtained from the charts for G by an identification of the points of charts related by action of subgroup H.

    Note that the mass shell H3, as any 3-surface, has a unique smooth structure by its dimension.

  2. Second sigh of relief is that twistor spaces CP3 and SU(3)/U(1)× U(1) have by their isometries and their coset space structure a standard smooth structure.

    In accordance with the vision that the dynamics of fields is geometrized to that of surfaces, the space-time surface is replaced by the analog of twistor space represented by a 6-surface with a structure of S2 bundle with space-time surface X4 as a base-space in the 12-D product of twistor spaces of M4 and CP2 and by its dimension D=6 can have only the standard smooth structure unless it somehow decomposes to (S3× R)× R2. Holography of smoothness would prevent this since it has boundaries because X4 as base space has boundaries at the boundaries of CD.

  3. cd is an intersection of future and past directed light-cones of M4. Future/past directed light-cone could be seen as a subset of M4 and implies standard smooth structure is possible. Coordinate atlas of M4 is restricted to cd and one can use Minkowski coordinates also inside the cd. cd could be also seen as a pile of light-cone boundaries S2× R+ and by its dimension S2× R allows only one smooth structure.
  4. M8 is a subspace of complexified octonions and has the structure of 8-D translation group, which implies standard smooth structure.
The conclusion is that continuous symmetries of the geometry dictate standard smoothness at the level of embedding space and related structures.

See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

Thursday, November 10, 2022

Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD

Gary Ehlenberger sent a highly interesting commentary related to smooth structures in R4 discussed in the article of Gompf (see this) and more generally to exotics smoothness discussed from the point of view of mathematical physics in the book of Asselman-Maluga and Brans (see this). I am grateful for these links for Gary. The intersection form of 4-manifold (see this) characterizing partially its 2-homology is a central notion.

The role of intersection forms in TGD

I am not a topologist but I had two good reasons to get interested on intersection forms.

  1. In the TGD framework (see this), the intersection form describes the intersections of string world sheets and partonic 2-surfaces and therefore is of direct physical interest (see this and this).
  2. Knots have an important role in TGD. The 1-homology of the knot complement characterizes the knot. Time evolution defines a knot cobordism as a 2-surface consisting of knotted string world sheets and partonic 2-surfaces. A natural guess is that the 2-homology for the 4-D complement of this cobordism characterizes the knot cobordism. Also 2-knots are possible in 4-D space-time and a natural guess is that knot cobordism defines a 2-knot.

    The intersection form for the complement for cobordism as a way to classify these two-knots is therefore highly interesting in the TGD framework. One can also ask what the counterpart for the opening of a 1-knot by repeatedly modifying the knot diagram could mean in the case of 2-knots and what its physical meaning could be in the TGD Universe. Could this opening or more general knot-cobordism of 2-knot take place in zero energy ontology (ZEO) (see this, this and this) as a sequence of discrete quantum jumps leading from the initial 2-knot to the final one.

Why exotic smooth structures are not possible in TGD?

The article of Gabor Etesi (see this) gives a good idea about the physical significance of the existence of exotic smooth structures (see the book and the article). They mean a mathematical catastrophe for both classical relativity and for the quantization of general relativity based on path integral formulation.

The first naive guess was that the exotic smooth structures are not possible in TGD but it turned out that this is not trivially true. The reason is that the smooth structure of the space-time surface is not induced from that of H unlike topology. One could induce smooth structure by assuming it given for the space-time surface so that exotics would be possible. This would however bring an ad hoc element to TGD. This raises the question of how it is induced.

  1. This led to the idea of a holography of smoothness, which means that the smooth structure at the boundary of the manifold determines the smooth structure in the interior. Suppose that the holography of smoothness holds true. In ZEO, space-time surfaces indeed have 3-D ends with a unique smooth structure at the light-like boundaries of the causal diamond CD= cd× CP2 ⊂ H=M4× CP2, where cd is defined in terms of the intersection of future and past directed light-cones of M4. One could say that the absence of exotics implies that D=4 is the maximal dimension of space-time.
  2. The differentiable structure for X4⊂ M8, obtained by the smooth holography, could be induced to X4⊂ H by M8-H-duality. Second possibility is based on the map of mass shell hyperboloids to light-cone proper time a=constant hyperboloids of H belonging to the space-time surfaces and to a holography applied to these.
  3. There is however an objection against holography of smoothness (see this). In the last section of the article, I develop a counter argument against the objection. It states that the exotic smooth structures reduce to the ordinary one in a complement of a set consisting of arbitrarily small balls so that local defects are the condensed matter analogy for an exotic smooth structure.
See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8−H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

Saturday, November 05, 2022

Finite Fields and TGD

TGD involves geometric and number theoretic physics as complementary views of physics. Almost all basic number fields: rationals and their algebraic extensions, p-adic number fields and their extensions, reals, complex number fields, quaternions, and octonions play a fundamental role in the number theoretical vision of TGD.

Even a hierarchy of infinite primes and corresponding number fields appears. At the first level of the hierarchy of infinite primes, the integer coefficients of a polynomial Q defining infinite prime have no common prime factors. P=Q hypothesis states that the polynomial P defining space-time surface is identical with a polynomial Q defining infinite prime at the first level of hierarchy.

However, finite fields, which appear naturally as approximations of p-dic number fields, have not yet gained the expected preferred status as atoms of the number theoretic Universe. Also additional constraints on polynomials P are suggested by physical intuition.

Here the notions of prime polynomial and concept of infinite prime come to rescue. Prime polynomial P with prime order n=p and integer coefficients smaller than p can be regarded as a polynomial in a finite field. The proposal is that all physically allowed polynomials are constructible as functional composites of prime polynomials satisfying P=Q condition.

See the article Finite Fields and TGD.

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

Thursday, November 03, 2022

VO2 can remember like a brain

The following comments were inspired by a popular article (see this) with the title "Scientists accidentally discover a material that can 'remember' like a brain". These materials can remember the history of its physical stimuli. The findings are described in the article "Electrical control of glass-like dynamics in vanadium dioxide for data storage and processing" published in Nature (see see this).

The team from the Ecole Polytechnique Federale de Lausanne (EPFL) in Switzerland did this discovery while researching insulator-metal phase transitions of vanadium dioxide (VO2), a compound used in electronics.

  1. PhD student Mohammad Samizadeh Nikoo was trying to figure out how long it takes for VO2 to make a phase transition from insulating to conducting phase under "incubation" by a stimulation by a radio frequency pulse of 10 μs duration and voltage amplitude V= 2.1 V. Note that the Wikipedia article talks about semiconductor-metal transition. The voltage pulse indeed acted like a voltage in a semiconductor.
  2. As the current heated the sample it caused a local phase transition to metallic state in VO2. The induced current moved across the material, following a path until it exited on the other side. A conducting filament connecting the ends of the device was generated by a percolation type process.
  3. Once the current had passed, the material exhibited an insulating state but after incubation time tinc, which was tinc∼ .1 μs for the first pulse, it became conducting. This state lasted at least 10,000 seconds.

    After applying a second electrical current during the experiment, it was observed that tinc appeared to be directly related to its history and was shorter than for the first incubation period .1 μs. The VO2 seemed to remember the first phase transition and anticipate the next. One could say that the system learned from experience.

Before trying to understand the finding in the TGD framework, it is good to list some basic facts about vanadium and vanadium-oxide VO2 or Vanadium(IV) oxide (see this).
  1. Vanadium is a transition metal, which has valence shells d3s2. It is known that the valence electrons of transition metals can mysteriously disappear, for instance in heating (see this). The TGD interpretation (see this) would be that heating provides energy making it possible to transform ordinary valence electrons to dark valence electrons with a higher value of heff and higher energy. In the recent case, the voltage pulses could have the same effect.
  2. VO2 forms a solid lattice of V4+ ions. There are two lattice forms: the monoclinic semiconductor below Tc=340 K and the tetragonal metallic form above Tc. In the monoclinic form, the V4+ ions form pairs along the c axis, leading to alternate short and long V-V distances of 2.65 Angström and 3.12 Angström. In the tetragonal form, the V-V distance is 2.96 Angström. Therefore size of the unit cell for the monoclinic form is 2 times larger than for the tetragonal form. At Tc IMT takes place. The optical band gap of VO2 in the low-temperature monoclinic phase is about 0.7 eV.
  3. Remarkably, the metallic VO2 contradicts the Wiedemann Franz law, which states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature. The thermal conductivity that could be attributed to electron movement was 10 per cent of the amount predicted by the Wiedemann Franz law. That the conductivity is 10 times higher than expected, suggests that the mechanism of conductivity is not the usual one.

    Semiconductor property below Tc suggests that a local phase transition modifying the lattice structure from monoclinic to tetragonal takes place at the current path in the incubation.

One can try to understand the chemistry and unconventional conductivity of VO2 in the TGD framework.
  1. Vanadium could give 4 valence electrons to O2: 3 electrons d3:sta and one from s2. In the TGD Universe, the second electron from s2 could become dark and go to the bond between V4+ ions in the VO2 lattice and take the role of conduction electron.
  2. This could explain the non-conventional character of conductivity. In the semiconductor phase, an electric voltage pulse or some other perturbation, such as impurity atoms or heating, can provide the energy needed to increase the value of heff. Electric conductivity could be due to the transformation of electrons to dark electrons possibly forming Cooper pairs at the flux tube pairs connecting V4+ ions or their pairs. The current would run along the flux tubes as a dark current.
  3. In a semi-conducting (insulating) state, the flux tube pairs connecting V4+ ions would be relatively short. The voltage pulse inducing a local metallic state could provide the energy needed to increase heff and thus the quantum coherence scale. This would be accompanied by a reconnection of the short flux tube pairs to longer flux tube pairs serving as bridges along which the dark current could run.

    One can also consider U-shaped closed flux tubes associated with V4+ ions or ion pairs, which reconnect in IMT to longer flux tubes. The mechanism would be very similar to that proposed for the transition to high temperature superconductivity (see this, this, and this).

Experimenters suggest a glass type behavior.
  1. Spin glass corresponds to the existence of a very large number of free energy minima in the energy landscape implying breaking of ergodicity. A system consisting of regions with varying direction of magnetization is the basic example of spin glass. In the recent case, decomposition to metallic and insulating regions could define the spin glass.
  2. TGD predicts the possibility of spin glass type behavior and leads to a model for spin glasses (see this). The quantum counterpart of spin glass behavior would be realized in terms of monopole flux tube structures (magnetic bodies) carrying dark phases of the radinary particles such as electrons serving as current carries in the metallic phase.The length of the flux tube pair would be one critical parameter near Tc. Quantum criticality against the change of heff increasing the length of the flux tube pair by reconnection would make the system very sensitive to perturbations.
  3. These phases are highly sensitive to external perturbations and represent in TGD inspired theory of consciousness higher levels with longer quantum coherence scale and number theoretical complexity measured by the dimension n= heff/h0 of the extension having interpretation as a kind of IQ. These phases would receive sensory information from lower levels of the hierarchy with smaller values of n and control them.

    The large number of free energy minima as a correlate for number theoretical complexity would make possible the representation of "sensory" information as "memories".

See the article TGD and Condensed Matter or the chapter with the same title.

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

Articles related to TGD.

Wednesday, November 02, 2022

More anomalies related to the standard model of galaxy formation

Various anomalies associated with the ΛCDM model assuming that dark matter forms halos and with the general view of galaxy formation have been accumulating rapidly during years. The MOND model assumes no dark mass but modifies Newton's law of gravitation and is inconsistent with the Equivalence Principle. In TGD, the halo is replaced with a cosmic string and the Equivalence Principle and Newtonian gravitation survive. Both MOND and TGD can handle these anomalies because there is no dark mass halo.

In this article, three new anomalies disfavoring ΛCDM but consistent with MOND and TGD are discussed. There are too many thin disk galaxies, dwarf galaxies do not have dark matter halos, and tidal tails associated with star clusters are asymmetric.

See the article More anomalies related to the standard model of galaxy formation

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

Articles related to TGD.<

Too many thin disk galaxies

The title of the article "Breaking Cosmology: Too Many Disk Galaxies – “A Significant Discrepancy Between Prediction and Reality” (see this) describes quite well the situation in the cosmology. The views of the dynamics of galaxies seem to be wrong.

In the current study (see this, Pavel Kroupa’s doctoral student, Moritz Haslbauer, led an international research group to investigate the evolution of the universe using the latest supercomputer simulations. The calculations are based on the Standard Model of Cosmology; they show which galaxies should have formed by today if this theory were correct. The researchers then compared their results with what is currently probably the most accurate observational data of the real Universe visible from Earth.

It is found that the fraction of disk galaxies is much larger than predicted. This suggests that the morphology of disk galaxies is very slowly changing and mergers of galaxies, favoured by dark matter halos, are not so important as though in the dynamics of galaxies. The cold dark matter scenario predicts spherical halos, which does not fit well with the large fraction of disk galaxies. The MOND approach is favored because there are no dark matter halos favoring spherical galaxies and mergers.

In the TGD framework (see this, this, and this), dark matter or rather, dark energy would be associated with what I call cosmic strings so that halos are absent. Cosmic strings are extremely massive and create a 1/ρ transversal gravitational field, which explains the flat velocity spectrum of distant stars automatically.

The orbits of stars are helical since there is free motion in the direction of a long string. This strongly favors the formation of disk galaxies with the plane of the disk orthogonal to the string and correlation between the normals of the disks along the long cosmic string. In accordance with the findings, the concentration of matter at the galactic plane is very natural in the TGD framework.

The intersections of the string-like objects moving at 3-surface are topologically unavoidable and one can ask whether the galaxies are formed as two cosmic string intersect and the resulting perturbation induces their thickening leading to the transformation of the dark energy of the cosmic string to ordinary matter. This would be an analogy for the decay of an inflaton field to ordinary matter.

See the article More anomalies related to the standard model of galaxy formation.

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

Articles related to TGD.<