- The ansatz, which realizes the Beltrami hypothesis, states that the vectorial Kähler current J equals apart from sign c=+/- 1 to instanton current I, which is axial current:
J=+/- I .

The condition states that only the left or right handed current chiral defined as

L

_{L/R}= J+/- Iis non-vanishing. For c≠ 1, both J

_{L}and J_{R}are non-vanishing. Since both right- and left-handed weak currents exist, c≠ 1 seems to be a plausible option.By quantum classical correspondence, these currents serve as space-time correlates for the left- and right-handed fermion currents of the standard model. Note however that induced gamma matrices differ from those of M

^{4}: for instance, they are not covariantly constant but defines a current with divergence which vanishes by field equations. - A more general condition would allow c to depend on space-time coordinates. The conservation of J forces conservation of I if the condition ∂
_{α}cI^{α}=0 is true. This gives a non-trivial condition only in regions with 4-D CP_{2}and M^{4}projections. - The twistor lift of TGD requires that also M
^{4}has Kähler structure. Therefore J and I and corresponding Kähler gauge potential A have both M^{4}part and CP_{2}parts and Kähler action K, J_{K}, J and I are sums of M^{4}and CP_{2}parts:A

_{K}= A(M^{4})+A(CP_{2}),

J_{K}=J_{K}(M^{4})+J_{K}(CP_{2}) ,

K = K(M^{4})+K(CP_{2}) ,

J =J(M^{4})+J(CP_{2}) ,

I= I(M^{4})+I(CP_{2}) .

Only the divergence for the sum I of M

^{4}and CP_{2}parts of the instanton currents must vanish:∂

_{α}I^{α}=0 .A possible interpretation is in terms of the 8-D variant of twistorialization by twistor lift requiring masslessness in an 8-D sense.

PCAC states that the divergence of the axial current is non-vanishing. This is not in conflict with the conservation of the total instanton current I. PCAC corresponds to the non-conservation I(CP

_{2}), whose non-conservation is compensated by that of I(M^{4}). - For regions with at most 3-D M
^{4}- and CP_{2}projections, the M^{4}- and CP_{2}instanton currents have identically vanishing divergence. In these regions the conservation of I is not lost if c has both signs. c could be also position dependent and even differ for I(M^{4}) and I(CP_{2}) in these regions.D

_{α}I^{α}=0 is true for the known extremals. For the simplest CP_{2}type extremals and for extremals with 2-D CP_{2}projection, I itself vanishes. Therefore parity violation is not possible in these regions. This would suggest that these regions correspond to a massless phase. - D
_{α}I^{α}≠ 0 is possible only if both M^{4}and CP_{2}projections are 4-D. This phase is interpreted as a chaotic phase and by the non-conservation of electroweak axial currents could correspond to a massive phase.CP

_{2}type extremals have 4-D projection and for them Kähler current and instanton current vanish identically so that also they correspond to massless phase (M^{4}projection is light-like). Could CP_{2}type extremals allow deformations with 4-D M^{4}projection (DEs)?The wormhole throat between space-time region with Minkowskian signature of the induced metric and CP

_{2}type extremal (wormhole contact) with Euclidian signature is light-like and the 4-metric is effectively 3-D. It is not clear whether this allows 4-D M^{4}projection in the interior of DE. - The geometric model for massivation based on zitterbewegung of DE provides additional insight. M
^{8}-H duality allows to assign a light-like curve also to DE. For space-time surfaces determined by polynomials (cosmological constant Λ>0), this curve consists of pieces which are light-like geodesics.Also real analytic functions (Λ=0) can be considered and they would allow a continuous light-like curve, whose definition boils down to Virasoro conditions. In both cases, the zigzag motion with light-velocity would give rise to velocity v<c in long length scales having interpretation in terms of massivation.

The interaction with J(M

^{4}) would be essential for the generation of momentum due to the M^{4}Chern-Simons term assigned with the 3-D light-like partonic orbit. M^{4}Chern-Simons term can be interpreted as a boundary term due to the non-vanishing divergence of I(M^{4}) so that a connection with two views about massivation is obtained. Does the Chern-Simons term come from the Euclidean or Minkowskian region?

^{4}Kähler form is essential. Classical electric field induces CP breaking. CP takes self-dual (E,B) to anti-self-dual (-E,B) and self-duality of J(M

^{4}) does not allow CP as a symmetry.

- In the first model the electric part of J(M
^{4}) would induce a small CP breaking inside cosmic strings thickened to flux tubes inducing in turn small matter-antimatter asymmetry outside cosmic strings. After annihilation this would leave only matter outside the cosmic strings. - In the simplest variant of TGD only quarks are fundamental particles and leptons are their local composites in CP
_{2}scale. Both quarks and antiquarks are possible but antiquarks would combine leptons as almost local 3-quark composites and presumably realized CP_{2}type extremals with the 3 antiquarks associated with the partonic orbit. I should vanish identically for the DEs representing quarks and leptons but not for antiquarks and antileptons.Could the number of DEs with vanishing I be smaller for antiquarks than for quarks by CP breaking and could this induce leptonization of antiquarks and favor baryons instead of antileptons? Could matter-antimatter asymmetry be induced by the interior of DE alone or by its interaction with the Minkowskian space-time region outside DE.

_{2}allows quaternionic structure in the sense that the conformally invariant Weyl tensor has besides W

_{3}=J(CP

_{2}) also charged components W

_{+/-}, which are however not covariantly constant. One can assign to W

_{+/-}analogs of Kähler currents as covariant divergences and also the analogs of instanton currents. These currents could realize a classical space-time analog of current algebra.

See the article Comparing the Berry phase model of super-conductivity with the TGD based model or the chapter with the same title.

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