A possible model for a monopole flux tube carrying an electric flux

The localization of the dark mass should have a classical space-time counterpart at the level of the space-time surface. It should be also consistent with the Newtonian view of gravitation in which gravitational flux as an analog of electric flux is conserved. Also consistency with stringy description of gravitation based on 3→ 4 holography is desirable. This reaises the question whether flux tubes carrying Kähler electric flux are possible and whether one can construct candidates for them as simultaneous extremals of Kähler action and volume action.

1. Assume that the solar - and also other gravitational fluxes can be associated with monopole flux tubes which have 2-D M⁴ projection as a string world sheet. If these flux tubes are defined so that the CP₂ 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.
2. The simplest guess for the flux tube carrying Kähler electric field is that the homologically trivial sphere as CP₂ projection rotates, not in 1-D sense but in 2-D sense meaning that at a given point of the string world sheet X²⊂ M⁴ it is obtained by a local color rotation of S² at standard position in CP₂.

   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 M⁴ 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.

3. 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.

What can one say of the flux tubes carrying Kähler electric field?

1. 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 CP₂ and maps the Kähler form J_klds^k ∧ ds^l and line element of ds²=s_klds^kds^l of the Kähler metric invariant. Using coordinates x^μ for X² and s^k for S², the induced Kähler form has the following structure

   • S² part is the same as for the standard S², 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 CP₂ contribution to the induced metric.
   • X² part is of the form

     J_μν= g_μ^k (g^-1Jg^-1)_kl(s) g_ν^~l == (∂_μg g^-1) J (g^-1∂_νg) .

     The formula resembles the gauge transformation formula.

     Here the shorthand notations

     g_μ^k=∂_μg^k(s) ,

     g^k_l(s)= ∂_l g^k,

     (g^-1g)k_l=δ^k_l ,

     have been used.

   • The mixed X²-S² components are

     J_μl= g_μ^~k (g^-1J)_kl(s) .

     For the CP₂ contribution to the induced metric similar formulas hold true.

2. The induced Kähler electric field has both X² - and S² component and X² component defines the K ''ahler charge assignable to transversal section S² 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.

What could one say about the possible additional conditions on the locally color rotating object?

1. The model for the massless extremals (MEs)(see this) assumes that the space-time surface is locally representable as a map M⁴→ CP₂ such that the CP₂ 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".
2. If this representation is global, one expects that the space-time surface has a boundary assignable to E² 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).
3. 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 E² to CP₂ to CP₂ is 2-valued and has S² as an image. Besides S², also more general complex 2-submanifolds of CP₂ can be considered.
4. 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.

One can consider several generalizations of the solution ansatz motivated by physical intuition but not really proven.

1. The surface could define a many-sheeted covering of M⁴. The conditions for the surface could be formulated as conditions stating that 4 functions of coordinates u,v and CP₂ coordinates vanish.
2. The "polarization coordinate" v could depend on the linear coordinates of E² non-linearly. For instance, it could correspond to a radial coordinate of E². 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 CP₂ coordinates are analytic functions of a complex coordinate w of E² and hypercomplex coordinate of M². Also the coordinate u could be replaced with a "real" part of a hyper-analytic function of M⁴ 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.

3. One can even consider the possibility that the decomposition M⁴= M²times E² 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 M² and E² 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 M² as a planar analog of a string world sheet with a curved string world sheet in M⁴. The partonic 2-surface could in turn be interpreted as a many-valued image of a complex 2-surface of CP₂ in the local E².

In the recent situation, the simplest form of MEs motivates the question that the local color rotation of S² or of a more general complex 2-manifold Y²⊂ CP₂ depends on the light-like coordinate u=k· m only. The induced Kähler gauge potential depends on u only so that the M² part of the Kähler electric field would vanish.

The Kähler electric flux would be parallel to E² (or the image of S² in M⁴) 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 CP₂ type extremals, the self-duality of CP₂ 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.