Conserved vector current hypothesis (CVC) and partially conserved axial current hypothesis (PCAC) are essential elements of oldfashioned hadron physics and hold true also in the standard model.
 The simplest 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
J_{L/R}= J+/ I
is nonvanishing. For c≠ 1, both J_{L} and J_{R} are nonvanishing. Since both right and lefthanded weak currents exist, c≠ 1 seems to be a plausible option.
By quantum classical correspondence, these currents serve as spacetime correlates for the left and righthanded 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 spacetime coordinates. The conservation of J forces conservation of I if the condition ∂_{α}cI^{α}=0 is true. This gives a nontrivial condition only in regions with 4D 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_{K} have both M^{4} part and CP_{2} parts
and Kähler action K, A_{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 of I must vanish:
∂_{α}I^{α}=0 .
A possible interpretation is in terms of the 8D variant of twistorialization by twistor lift requiring masslessness in an 8D sense.
PCAC states that the divergence of the axial current is nonvanishing. This is not in conflict with the conservation of the total instanton current I. PCAC corresponds to the nonconservation I(CP_{2}), whose nonconservation is compensated by that of I(M^{4}).
 For regions with at most 3D 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 2D 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 4D. This phase is interpreted as a chaotic phase and by the nonconservation of electroweak axial currents could correspond to a massive phase.
CP_{2} type extremals have 4D projection and for them Kähler current and instanton current vanish identically so that also they correspond to massless phase (M^{4} projection is lightlike). Could CP_{2} type extremals allow deformations with 4D M^{4} projection (DEs)?
The wormhole throat between spacetime region with Minkowskian signature of the induced metric and CP_{2} type extremal (wormhole contact) with Euclidian signature is lightlike and the 4metric is effectively 3D. It is not clear whether this allows 4D 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 lightlike curve also to DE. For spacetime surfaces determined by polynomials (cosmological constant Λ>0), this curve consists of pieces which are lightlike geodesics.
Also real analytic functions (Λ=0) can be considered and they would allow a continuous lightlike curve, whose definition boils down to Virasoro conditions. In both cases, the zigzag motion with lightvelocity 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} ChernSimons term assigned with the 3D lightlike partonic orbit. M^{4} ChernSimons term can be interpreted as a boundary term due to the nonvanishing divergence of I(M^{4}) so that a connection with two views about massivation is obtained. Does the ChernSimons term come from the Euclidean or Minkowskian region?
I have proposed two models for the generation of matterantimatter asymmetry. In both models, CP breaking by M^{4} Kähler form is essential. Classical electric field induces CP breaking. CP takes selfdual (E,B) to antiselfdual (E,B) and selfduality 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 matterantimatter 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 3quark 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 matterantimatter asymmetry be induced by the interior of DE alone or by its interaction with the Minkowskian spacetime region outside DE.
In the standard model also charged weak currents are allowed. Does TGD allow their spacetime counterparts? CP_{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 spacetime analog of current algebra.
See the chapter Comparing the Berry phase model of superconductivity with the TGD based model or the article with the same title.
