- Assume that the solar - and also other gravitational fluxes can be associated with monopole flux tubes which have 2-D M
^{4}projection as a string world sheet. If these flux tubes are defined so that the CP_{2}projection as a homologically non-trivial 2-surface depends on time, Kähler electric field is generated and the flux tube has conserved Kähler electric charge Q_{K}. - The simplest guess for the flux tube carrying Kähler electric field is that the homologically trivial sphere as CP
_{2}projection rotates, not in 1-D sense but in 2-D sense meaning that at a given point of the string world sheet X^{2}⊂ M^{4}it is obtained by a local color rotation of S^{2}at standard position in CP_{2}.A natural interpretation of Q

_{K}would be as a counterpart of gravitational flux. Note that this requires that Kähler electric charges have the same sign. This picture conforms with the finding that space-time surfaces with stationary, spherically symmetric induced metric with non-vanishing gravitational mass have at least some non-vanishing gauge charges. For monopole flux tubes Kähler electric charge is non-vanishing. If the flux tubes are U-shaped, the Kähler electric flux must vanish.The M

^{4}projections of the flux tubes would be counterparts of strings mediating gravitational interaction in AdS/CFT duality and mediate gravitational interaction and with Newtonian view. - How to describe the formation of the planets or smaller structures in this picture? One can regard the radial flux tubes from the Sun as analogs of particles and introduce for them a wave function in the orientational degrees of freedom, say as spherical harmonics with defined angular momentum.
The magnetic bubble would correspond to a flux tube structure tangential to say 2-D sphere around the Sun and attached to the radial flux tube structure by wormhole contacts. This structure carries matter as dark particles (fermions).

A nearly complete collective localization in the orientational degrees of freedom would correspond to a state function reduction involving the reorganization of the gravitational flux tubes to a radial bundle with a definite orientation forcing the tangential flux tube tangle to reduce in size so that it corresponds to the magnetic body of say, planet. This would give rise to the planet after the transformation of dark matter to ordinary matter. Also a localization to a torus-like structure is possible and gives rise to a ring-like structure.

The reduction of quantum coherence to a smaller scale would give rise to smaller structures such as formation of flux tube bundles assignable to mini-planets and even smaller structures as in the case of the Kuiper belt and Oort cloud.

- I have proposed this kind of extremals in the model of honeybee dance (see this), which was inspired by the work of topologist Barbara Shipman (see this), who proposed that honeybee dance reflects the color symmetry of strong interactions. In the standard model this proposal does not make sense but is natural in the TGD framework.
The local color rotation s

^{k}→ g^{k}(s^{l}) is an isometry of CP_{2}and maps the Kähler form J_{kl}ds^{k}∧ ds^{l}and line element of ds^{2}=s_{kl}ds^{k}ds^{l}of the Kähler metric invariant. Using coordinates x^{μ}for X^{2}and s^{k}for S^{2}, the induced Kähler form has the following structure- S
^{2}part is the same as for the standard S^{2}, that is J_{kl}→ ∂_{k}g^{r}J_{rs}(g^{-1}(s))∂_{l}g^{s}= J_{kl}(s). The same formula holds true for the CP_{2}contribution to the induced metric. - X
^{2}part is of the formJ

_{μν}= g_{μ}^{k}(g^{-1}Jg^{-1})_{kl}(s) g_{ν}^{~l}== (∂_{μ}g g^{-1}) J (g^{-1}∂_{ν}g) .The formula resembles the gauge transformation formula.

Here the shorthand notations

g

_{μ}^{k}=∂_{μ}g^{k}(s) ,g

^{k}_{l}(s)= ∂_{l}g^{k},(g

^{-1}g)k_{l}=δ^{k}_{l},have been used.

- The mixed X
^{2}-S^{2}components areJ

_{μl}= g_{μ}^{~k}(g^{-1}J)_{kl}(s) .For the CP

_{2}contribution to the induced metric similar formulas hold true.

- S
- The induced Kähler electric field has both X
^{2}- and S^{2}component and X^{2}component defines the K ''ahler charge assignable to transversal section S^{2}as an electric flux. What is nice is that, although one does not have electric-magnetic duality, the Kähler electric field is very closely related to the Kähler magnetic field. Whether the solution ansatz works without additional conditions on the local color rotation has not been proven.

- The model for the massless extremals (MEs)(see this) assumes that the space-time surface is locally representable as a map M
^{4}→ CP_{2}such that the CP_{2}coordinates are arbitrary functions of coordinates u= k· m and v= ε· m. k is light-like wave vector and ε a polarization vector orthogonal to it. This motivates the term "massless extremal". - If this representation is global, one expects that the space-time surface has a boundary assignable to E
^{2}so that a tube-like structure is obtained. Boundary conditions guaranteeing that isometry charges do not flow out of the boundary must be satisfied. In particular, the boundary must be light-like. These conditions are discussed in detail in(see this). - The color rotating objects could correspond to a situation in which
the color rotation depends on light-like coordinate u only and the solution is such that the map of a region of E
^{2}to CP_{2}to CP_{2}is 2-valued and has S^{2}as an image. Besides S^{2}, also more general complex 2-submanifolds of CP_{2}can be considered. - The key difference between MEs and massless fields of gauge theories is that MEs are characterized by a non-vanishing light-like Kähler current(see this). This must have deep physical implications.
One has Kähler electric charge defined by the standard formula. Kähler electric flux orthogonal to the transversal cross section of ME and has light-like direction instead of space-like direction. One can also calculate the charge also for a section with time-like normal. Could this make it possible for the flux tubes to have Kähler electric flux as analog of gravitational flux? This picture would be consistent with both the Newtonian picture of gravitation mediated by the gravitational flux and the field theory picture of gravitation mediated by massless particles represented by MEs.

- The surface could define a many-sheeted covering of M
^{4}. The conditions for the surface could be formulated as conditions stating that 4 functions of coordinates u,v and CP_{2}coordinates vanish. - The "polarization coordinate" v could depend on the linear coordinates of E
^{2}non-linearly. For instance, it could correspond to a radial coordinate of E^{2}. The polarization would not be linear anymore.A possible restriction on v is that v is a real part of complex analytic function. The surface would possess a 4-D analog of holomorphy in the sense that complex CP

_{2}coordinates are analytic functions of a complex coordinate w of E^{2}and hypercomplex coordinate of M^{2}. Also the coordinate u could be replaced with a "real" part of a hyper-analytic function of M^{4}depending on a light-like coordinate u but this does not seem to change the situation in any way. This is a highly attractive 4-D generalization of the holomorphy of string world sheets. - One can even consider the possibility that the decomposition M
^{4}= M^{2}times E^{2}to longitudinal and transversal spaces could be local so that also the light-like direction would be local. The condition would be that the distribution of the tangent spaces of M^{2}and E^{2}are integrable and defines a 4-surface having slicings to mutually orthogonal 2-D string world sheets and partonic 2-surfaces. This would correspond to what I have christened as Hamilton-Jacobi structure(see this).Physically this would mean the replacement of M

^{2}as a planar analog of a string world sheet with a curved string world sheet in M^{4}. The partonic 2-surface could in turn be interpreted as a many-valued image of a complex 2-surface of CP_{2}in the local E^{2}.

^{2}or of a more general complex 2-manifold Y

^{2}⊂ CP

_{2}depends on the light-like coordinate u=k· m only. The induced Kähler gauge potential depends on u only so that the M

^{2}part of the Kähler electric field would vanish.

The Kähler electric flux would be parallel to E^{2} (or the image of S^{2} in M^{4}) and Kähler electric charge as electric flux could be (but need not be) non-vanishing. This flux would not however be in the direction of the flux tube so that it cannot correspond to gravitational flux.

Since Kähler electric flux would be very closely related to Kähler magnetic flux, an electric analog of the homological Kähler magnetic charge would make sense. This could topologically quantize the Kähler electric charge and also electric charge classically? In the case of CP_{2} type extremals, the self-duality of CP_{2} Kähler form indeed implies this. One would have electric-magnetic duality proposed to hold true in TGD.

See the article Magnetic Bubbles in TGD Universe: Part I or the chapter with the same title.

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

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