Explicit formulas for the vertices in TGD

It is now intuitively clear that the vertices at the fundamental level should be just the standard model vertices assignable to the 3-D singularities of the divergence of the fermionic current. Also the idea that the vertices should have an interpretation as defects of standard smooth structures to which exotic smooth structure (see this, this, and this) can be assigned (see this, this, this). The difficult challenge (to me at least) is to deduce the vertices in a convincing way. The option discussed below seems to be the most promising of the options considered hitherto.

Are the vertices due to the non-conservation of fermion current associated with the induced/modified Dirac action

One can start from the standard model view of vertices as an intuitive guide line.

1. The singularities should give the emission vertices for Higgs and electroweak gauge bosons. What is new is that electroweak gauge bosons have an interpretation as gluons with the weak gauge group identified as the holonomy group U(2) identifiable as a subgroup of color group SU(3). Strong interactions correspond to the isometries of SU(3) and electroweak interactions to the holonomies assignable to the CP₂ spinor degrees of freedom. Generalized Higgs corresponds to the trace of the second fundamental form having a 3-D singularity at the vertex having interpretation as 8-D local acceleration. This means that the classical action has no role as far as vertices are considered.
2. The interaction vertices emerge from the induced/modified Dirac action. By the modified Dirac equation, this action vanishes almost everywhere. The Dirac action reduces to a divergence of the fermion current and this suggests that at the singularity this divergence is non-vanishing. This conforms with the view that a fermion pair is created and fermion line terms backwards in time. At the 3-D singularity the divergence of the fermion current should have a delta function divergence making possible non-vanishing various vertices.
3. The vertex for the generalized Higgs is not a problem. The trace of the quantity D_μ(Γ^μg₄^1/2) is proportional to the trace of the seocnd fundamental form and gives 3-D delta function for the vertex of the annihilation of Higgs to fermion-antifermion pair.
4. How to obtain electroweak vertices and the possible vertex related to the M⁴ Kähler gauge potential? The electroweak vertices should come from a 3-D delta function singularity X³ of the Dirac action density

Ψ(-D^←_μ Γ^μ -Γ^μD_μ)Ψg₄^1/2).

The problem is that the components of the induced spinor connection A have only a step function like discontinuity rather than the desired delta function singularity. The desired singularity should be equal to the difference A₊-A_- of the gauge potentials at the two sides of the singularity, which is invariant under gauge transformations if their action is the same at the two sides. Vertex would be defined by this difference rather than vector potential as in perturbative gauge theories.

This can be achieved if the spinor fields have a discontinuity at X³, which gives rise to the difference A₊-A_- under the action of derivatives to the spinor field at X³. This is achieved if the gauge potentials A₊ and A_- are related by a gauge transformation g in the holonomy group. This gives A₊= A_- +dgg^-1 giving dg= (A₊-A_-)g. The spinor fields are discontinuous at the singularity and related by Ψ₊= gΨ_-. The derivative dΨ₊ is given by dΨ₊ = d(gΨ_-)= dgΨ_- + gdΨ_- = (A₊-A_-)gΨ_-+ gdΨ_-. The difference of gauge potentials at the singularity gives rise to the desired vertex.

For this option, the vertices are universal and do not depend on classical action for the induced Dirac equation. Also for the modified Dirac equation the electroweak vertices are determined by a gauge transformation in the holonomy group of CP₂. The conservation of the isometry currents for the classical action implies that the Higgs vertex as the divergence of the induced/modified Gamma matrices are expressible in terms of current assignable to the remaining parts of the action. If the classical action is a mere volume action, the conservation of classical isometry currents does not allow singularities. Therefore Kähler action plus volume term strongly suggested by the twistor lift is the minimal option.

The relation to the exotic smooth structures

The vertices should be non-trivial for the 3-D singularities at which the minimal surface property of the space-time surface fails and which corresponds to the defects for the standard smooth structure, which transform it to exotic smooth structure cite{bmat/exoticR4,exoticsmooth,exoticanomaly}. What does this really mean, is far from clear. I have discussed this problem already earlier (see this, this, this) but I am not satisfied with the view.

The edge of the fermion line at the singularity means the breaking of standard smooth structure at which the light-like hypercomplex coordinate changes from u to its hypercomplex conjugate v. The derivatives of the embedding space coordinates as functions of u resp. v at the singularity are infinite for the standard smooth structure. Also the induced Dirac spinors are discontinuous and related by a gauge transformation at the two sides of the singularity if the above argument is correct.

For the exotic smooth structure, the derivatives are continuous at the singularity. Also the vertices should remain the same in the exotic smooth structure. The only reasonable identification of the vertices is as regions at which the exotic smooth structure fails to reduce to the standard smooth structure. Could one introduce a 3-D term to the Dirac action additional term localized to the singular surfaces of the standard smooth structure to guarantee that the non-vanishing divergence of the ordinary Dirac action is compensated by this additional term giving rise to the standard model vertices for the Dirac action.

I have proposed the assignment of Chern-Simons-Kähler action and its fermionic counterpart to the 3-D light-like partonic orbits and a similar term can be considered also now. The modified Dirac action for the Kähler Chern-Simons term would contain the standard model couplings to spinors.

See the article Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/09/explicit-formulas-for-vertices-in-tgd.html