How to handle the interfaces between Minkowskian and Euclidean regions of space-time surface?

The treatment of  the dynamics at the   interfaces X³ between Minkowskian and Euclidean regions X³  of the  space-time surface identified as light-like partonic orbits has turned out to be a difficult technical problem. By holomorphy as a realization of generalized holography, the 4-metric at X³  degenerates to 2-D effective Euclidean metric apart from 2-D delta function singularities  X² at which the holomorphy fails but    the metric is  4-D.  

One must treat both the  bosonic and fermionic situations. There are two options for the treatment of the interface dynamics.

1.  The  interface X³ is  regarded as an independent dynamic unit.  The earlier approaches rely on this assumption.  By the light-likeness of X³, C-S-K action is the only possible option. The problem with U(1) gauge  invariance disappears if C-S-K action is identified as a total divergence emerging from  the instanton term for Kähler action.

One can assign to  the instanton term a corresponding contribution to the modified Dirac action at X³.  It however seems that the instanton term associated with the 4-D modified Dirac action does not reduce to a total divergence  allowing to  localize  it a X³.

In this approach, conservation laws require that the normal components of the canonical momentum currents from the Minkowskian and Euclidean sides  add up to the divergence of the canonical momentum currents associated with the C-S-K action.

In order to obtain the counterpart of Einstein’s equations  the metric must be effectively 2-D also at X² so that det(g₂)=0 is true although holomorphy fails. It seems that   one must assume induced, rather than modified, gamma matrices (effectively reducing to the induced ones at X³  outside X²) since for the latter option the gravitational vertex would vanish by the field equations.

The situation is very delicate and I cannot claim  that I understand it sufficiently. It seems that the edge of the partonic orbit due to the turning of the fermion line and involving hypercomplex conjugation is essential.

The orientations of the tangent spaces at the two sides are different. The induced metric at   the Minkowskian side  would become 4-D.  At the Euclidean side it could be Euclidean and   even metrically 2-D. The following overview of the symmetry breaking through the generation of 2-D  singularities is suggestive.  Masslessess and holomorphy are violated via the  generation of the analog of Higgs expectation at the vertices. The use of the  induced gamma matrices violates supersymmetry  guaranteed by the use of the modified gamma matrices   but only at the vertices.

There is however an objection. The use of the induced gammas in the modified Dirac equation seems necessary although the non-vanishing of H^k seems to violate  the hermiticity at the  vertices. Can the turning of the fermion line and the exotic smooth structure allow to get rid of this problem?

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.