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What happens to the extremals of Kähler action when volume term is introduced?

Thursday, September 22, 2016 3:06
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What happens to the extremals of Kähler action when volume term having interpretation in terms of cosmological constant is introduced?

  1. The known non-vacuum extremals such as massless extremals (topological light rays) and cosmic strings are minimal surfaces so that they remain extremals and only the classical Noether charges receive an additional volume term. In particular, string tension is modified by the volume term. Homologically non-trivial cosmic strings are of form X2× Y2, where X2⊂ M4 is minimal surface and Y2⊂ CP2 is complex 2-surface and therefore also minimal surface.

  2. Vacuum degeneracy is in general lifted and only those vacuum extremals, which are minimal surfaces survive as extremals.

For CP2 type vacuum extremals the roles of M4 and CP2 are changed. M4 projection is light-like curve, and can be expressed as mk=fk(s) with light-likeness conditions reducing to Virasoro conditions. These surfaces are isometric to CP2 and have same Kähler and symplectic structures as CP2 itself. What is new as compared to GRT is that the induced metric has Euclidian signature. The interpretation is as lines of generalized scattering diagrams. The addition of the volume term forces the random light-like curve to be light-like geodesic and the action becomes the volume of CP2 in the normalization provided by cosmological constant. What looks strange is that the volume of any CP2 type vacuum extremals equals to CP2 volume but only the extremal with light-like geodesic as M4 projection is extremal of volume term.

Consider next vacuum extremals, which have vanishing induced Kähler form and are thus have CP2 projection belonging to at most 2-D Lagrangian manifold of CP2.

  1. Vacuum extremals with 2-D projections to CP2 and M4 are possible and are of form X2× Y2, X2 arbitrary 2-surface and Y2 a Lagrangian manifold. Volume term forces X2 to be a minimal surface and Y2 is Lagrangian minimal surface unless the minimal surface property destroys the Lagrangian character.

If the Lagrangian sub-manifold is homologically trivial geodesic sphere, one obtains string like objects with string tension determined by the cosmological constant alone.

Do more general 2-D Lagrangian minimal surfaces than geodesic sphere exist? For general Kähler manifold there are obstructions but for Kähler-Einstein manifolds such as CP2, these obstructions vanish (see this ). The case of CP2 is also discussed in the slides “On Lagrangian minimal surfaces on the complex projective plane” (see this). The discussion is very technical and demonstrates that Lagrangian minimal surfaces with all genera exist. In some cases these surfaces can be also lifted to twistor space of CP2.

  • More general vacuum extremals have 4-D M4 projection. Could the minimal surface condition for 4-D M4 projection force a deformation spoiling the Lagrangian property? The physically motivated expectation is that string like objects give as deformations magnetic flux tubes for which string is thicknened so that it has a 2-D cross section. This would suggest that the deformations of string like objects X2× Y2, where Y2 is Lagrangian minimal surface, give rise to homologically trivial magnetic flux tubes. In this case Kähler magnetic field would vanish but the spinor connection of CP2 would give rise to induced magnetic field reducing to some U(1) subgroup of U(2). In particular, electromagnetic magnetic field could be present.

    Cosmological constant is expected to obey p-adic evolution and in very early cosmology the volume term becomes large. What are the implications for the vacuum extremals representing Robertson-Walker metrics having arbitrary 1-D CP2 projection?

    1. The TGD inspired cosmology involves primordial phase during a gas of cosmic strings in M4 with 2-D M4 projection dominates. The value of cosmological constant at that period could be fixed from the condition that homologically trivial and non-trivial cosmic strings have the same value of string tension. After this period follows the analog of inflationary period when cosmic strings condense are the emerging 4-D space-time surfaces with 4-D M4 projection and the M4 projections of cosmic strings are thickened. A fractal structure with cosmic strings topologically condensed at thicker cosmic strings suggests itself.

    2. GRT cosmology is obtained as an approximation of the many-sheeted cosmology as the sheets of the many-sheeted space-time are replaced with region of M4, whose metric is replaced with Minkowski metric plus the sum of deformations from Minkowski metric for the sheet. The vacuum extremals with 4-D M4 projection and arbitrary 1-D projection could serve as an approximation for this GRT cosmology. Note however that this representability is not required by basic principles.
    3. For cosmological solutions with 1-D CP2 projection minimal surface property forces the CP2 projection to belong to a geodesic circle S1. Denote the angle coordinate of S1 by Φ and its radius by R. For the future directed light-cone M4+ use the Robertson-Walker coordinates (a=(m02-rM2)1/2, r=arM, θ, φ), where (m0, rM, θ, φ) are spherical Minkowski coordinates. The metric of M4+ is that of empty cosmology and given by ds2 = da2-a22, where Ω2 denotes the line element of hyperbolic 3-space identifiable as the surface a=constant.
  • One can can write the ansatz as a map from M4+ to S1 given by Φ= f(a) . One has gaa=1→ gaa= 1-R2(df/da)2. The field equations are minimal surface equations and the only non-trivial equation is associated with Φ and reads d2f/da2=0 giving Φ= ω a, where ω is analogous to angular velocity. The metric corresponds to a cosmology for which mass density goes as 1/a2 and the gravitational mass of comoving volume (in GRT sense) behaves is proportional to a and vanishes at the limit of Big Bang smoothed to “Silent whisper amplified to rather big bang for the critical cosmology for which the 3-curvature vanishes. This cosmology is proposed to results at the limit when the cosmic temperature approaches Hagedorn temperature.

  • The TGD counterpart for inflationary cosmology corresponds to a cosmology for which CP2 projection is homologically trivial geodesic sphere S2 (presumably also more general Lagrangian (minimal) manifolds are allowed). This cosmology is vacuum extremal of Kähler action. The metric is unique apart from a parameter defining the duration of this period serving as the TGD counterpart for inflationary period during which the gas of string like objects condensed at space-time surfaces with 4-D M4 projection. This cosmology could serve as an approximate representation for the corresponding GRT cosmology.
  • The form of this solution is completely fixed from the condition that the induced metric of a=constant section is transformed from hyperbolic metric to Euclidian metric. It should be easy to check whether this condition is consistent with the minimal surface property.

    See the chapter From Principles to diagrams of “Towards M-Matrix” or the article How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?.

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

    Articles and other material related to TGD.

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