Thursday, September 22, 2016

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

What happens to the extremals of Kähler action when volume term 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.

2. 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.

3. p-Adically Λ behaves like 1/p as also string tension. Could hadronic string tension be understood also in terms of cosmological constant in hadronic p-adic length scale for strings if one assumes that cosmological constant for given space-time sheet is determined by its p-adic length scale?

The so called Maxwell phase which would correspond to small perturbations of M4 is also possible for 4-D Kähler action. For the twistor lift the volume term makes this phase possible. Maxwell phase is highly interesting since it corresponds to the intuitive view about what QFT limit of TGD could be.
1. The field equations are a generalization of massless field equations for fields identifiable as CP2 coordinates and with a coupling to the deviation of the induced metric from M4 metric. It representes very weak perturbation. Hence the linearized field equations are expected to be an excellent approximation. The general challenge would be however the construction of exact solutions. One should also understand the conditions defining preferred extremals and stating that most of symplectic Noether charges vanish at the ends of space-time surface about boundaries of CD.

2. Maxwell phase is the TGD analog for the perturbative phase of gauge theories. The smallness of the cosmological constant in cosmic length scales would make the perturbative approach useless in the path integral formulation. In TGD approach the path integral is replaced by functional integral involving also a phase but also now the small value of cosmological constant is a problem in long length scales. As proposed, the hierarchy of Planck constants would provide the solution to the problem.

3. The value of cosmological constant behaving like Λ ∝ 1/p as the function of p-adic prime could be in short p-adic length scales large enough to allow a converging perturbative expansion in Maxwellian phase. This would conform with the idea that Planck constant has its ordinary value in short p-adic length scales.

4. Does Maxwell phase allow extremals for which the CP2 projection is 2-D Lagrangian manifold - say a perturbation of a minimal Lagrangian manifold? This perturbation could be seen also as an alternative view about thickened minimal Lagrangian string allowing also M4 coordinates as local coordinates. If the projection is homologically trivial geodesic sphere this is the case. Note that solutions representable as maps M4→ CP2 are also possible for homologically non-trivial geodesic sphere and involve now also the induced Kähler form.

5. The simplest deformations of canonically imbedded M4 are of form Φ= k• m, where Φ is an angle coordinate of geodesic sphere. The induced metric in M4 coordinates reads as gkl= mkl-R2kkkl and is flat and in suitably scaled space-time coordinates reduces to Minkowski metric or its Euclidian counterpart. kk is proportional to classical four-momentum assignable to the dark energy. The four-momentum is given by

Pk = A× hbar kk ,

A=[Vol(X3)/L4Λ] × (1+2x/1+x) ,

x= R2k2 .

Here kk is dimensionless since the the coordinates mk are regarded as dimensionless.

6. There are interesting questions related to the singularities forced by the compactness of CP2. Eguchi-Hanson coordinates (r,θ,Φ,Ψ) (see this) allow to get grasp about what could happen.

For the cyclic coordinates Ψ and Φ periodicity conditions allow to get rid of singularities. One can however have n-fold coverings of M4 also now.

(r,θ) correspond to canonical momentum type canonical coordinates. Both of them correspond to angle variables (r/(1+r2)1/2 is essentially sine function). It is convenient to express the solution in terms of trigonometric functions of these angle variables. The value of the trigonometric function can go out of its range [-1,1] at certain 3-surface so that the solution ceases to be well-defined. The intersections of these surfaces for r and θ are 2-D surfaces. Many-sheeted space-time suggests a possible manner to circumvent the problem by gluing two solutions along the 3-D surfaces at which the singularities for either variable appear. These surfaces could also correspond to the ends of the space-time surface at the boundaries of CD or to the light-like orbits of the partonic 2-surfaces.

Could string world sheets and partonic 2-surfaces correspond to the singular 2-surfaces at which both angle variables go out of their allowed range. If so, 2-D singularities would code for data as assumed in strong form of holography (SH). SH brings strongly in mind analytic functions for which also singularities code for the data. Quaternionic analyticity which makes sense would indeed suggest that co-dimension 2 singularities code for the functions in absence of 3-D counterpart of cuts (light-like 3-surfaces?)

7. A more general picture might look like follows. Basic objects come in two classes. Surfaces X2× Y2, for which Y2 is either homologically non-trivial complex minimal 2-surface of CP2 of Lagrangian minimal surface. The perturbations of these two surfaces would also produce preferred extremals, which look locally like perturbations of M4. Quaternionic analyticity might be shared by both solution types. Singularities force many-sheetedness and strong form of holography.

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.

4. 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.