What singularities can correspond to vertices for fermion pair creation?

It is far from clear whether all singularities have an interpretation in terms of exotic smooth structures. The physical criterion would be that the creation of a fermion pair takes place at the defect and that the minimal surface property fails. Fermions can correspond to induced spinor fields and fermion pairs could be created at surfaces of dimension d<4.

1. For closed two-sheeted cosmic strings and monopole flux tubes, which split by reconnection, the interpretation makes sense and means a generalization of the basic vertex for closed strings. These objects can be 2-sheeted as elementary particles in which case the reconnection would occur in the direction of CP₂. If they are single sheeted, the reconnection would occur in the direction of M⁴.
2. 3-D light-like light-partonic orbits appearing as interfaces between Euclidean and Minkowskian space-time regions and as boundaries of space-time surfaces are singularities (see this). Boundary conditions state that the possible flows of conserved charges from the interior go to the partonic orbit so that the divergence of the Chern-Simons-Kähler canonical momentum current coming from instanton term equals to the sum of the normal components of the canonical currents associated with Kähler action and volume term.
   1. Chern-Simons action at the light-like partonic orbit coming from the instanton term is well-defined and finite and field equations should not give rise to a singularity except at partonic 2-surfaces, which have been identified as analogs of vertices at which the partonic 2-surface X² splits to two.
   2. At the light-like partonic orbit 4-metric has a vanishing determinant and is therefore effectively 2-D (the light-like components of g_uv=g_vu of the 4-metric vanish). As a consequence, g₄^1/2 vanishes like L² at the partonic orbit unless some coordinate gradients diverge.

      The canonical momentum currents for the volume action are proportional to the contravariant induced metric appearing in the trace of the second fundamental form diverging like 1/L² and to g₄^1/2 so that they remain finite.

   3. Kähler action contains the contravariant metric twice and is proportional to g₄^1/2. This can give rise to a divergence of type 1/L² unless the boundary conditions make it finite. I have proposed long ago that the electric-magnetic self-duality at the partonic orbit can transform the Kähler action to an instanton term giving Chern-Simons Kähler term. In this case, a separate instanton term would not be needed. In this case everything would be finite at the partonic orbit. Minimal surface property fails in a smooth manner.

      The intuitive picture is that the contributions from the normal currents at the partonic orbit and the Chern-Simons term cancel each other and the partonic orbit cannot play a role of a vertex.

   4. The possible presence of 1/L² divergence could however give rise to a 2-D defect and genuine vertex. If it is identified as a creation of a pair of partonic 2-surfaces, the interpretation in terms of a creation of a fermion pair is possible and could be assigned to the splitting of a monopole flux tube.

      In accordance with the QFT picture, I have considered the possibility that the 2-D vertex could correspond to a branching of a partonic orbit. In the recent picture it would be accompanied by a creation of a fermion pair. The stringy view however suggests that pair creation occurs in the creation of partonic orbits in the splitting of monopole flux tubes. The stringy view is more attractive.

3. I have also proposed that 1-D singularities identifiable as boundaries of string world sheets and identifiable as fermion lines at the partonic orbits are important. The creation of a pair of fermion lines would give rise to the analogs of gauge theory vertices as 0-D singularities. It is however far from clear whether the stringy singularities are actually present and whether they could correspond to exotic smooth structures. One can imagine two options.
   1. There are no string world sheets. Monopole flux tubes can be regarded as deformations of cosmic strings. Instead of strings several monopole flux tubes can emerge from a wormhole contact. For the minimal option, monopole flux tubes, CP₂ type extremals, and massless extremals as counterparts of radiation fields are the basic extremals and the splitting of monopole flux tubes gives rise to vertices as defects of the ordinary smooth structure.
   2. String world sheets appear as singularities of the monopole flux tubes or even more general 4-surfaces and are analogous to wormhole contacts as blow-ups in which a point of X⁴ explodes to CP₂ type extremal. I have indeed proposed that a blow-up at which the points of the string world sheet as surface X²⊂ X⁴ are replaced with a homologically non-trivial 2-surface Y²⊂ CP₂ takes place. Y² could connect two parallel space-time sheets. Could these singularities correspond to defects of exotic smooth structures such that the ends of the string carry fermion number? The vertex for the creation of a pair of fermion and antifermion lines would correspond to a diffeo defect. Note that also these defects could reduce to a splitting of a monopole flux tube so that TGD would generalize the stringy picture.

See the article What gravitons are and could one detect them in TGD Universe? 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.