1. Exact solutions of the TGD counterpart of the Chern-Simons action
As I wrote the previous posting I had not yet realized that classical field equations can be solved exactly: the only solutions are those for which CP2 projection of 3-dimensional light-like 3-surface is at most 2-dimensional. Thus the construction of theory at this level reduces to that of constructing light-like 3-surfaces of imbedding space as representation of partons.
2. Solutions of the modified Dirac equation and eigen modes of modified Dirac operator
Also the modified Dirac equation can be solved exactly and solutions can be added to the generalized eigen modes of the modified Dirac operator: this defines the super-conformal symmetries in TGD sense. The hypothesis is that the Dirac determinant defined by the product of the eigenvalues gives rise to the exponent of Kähler action. If true, it would allow to construct the theory solely from the data provided by the light-like partonic 3-surfaces.
The allowance of only the eigenvalues belonging to a particular algebraic extension of rationals defining the extension of p-adic number field could guarantee the finiteness by leaving only finite number of eigenvalues to the spectrum. One would obtain number theoretic hierarchy of physics serving as a correlate for a cognitive hierarchy if one accepts p-adic physics as a physics of cognition and intentionality.
The earlier hypothesis that the super-canonical conformal weights relate in a simple manner to the zeros of Riemann Zeta finds additional support from the more detailed structure of the spectrum of eigen values if the super-canonical conformal weights are identified as eigenvalues of the modified Dirac operator. According to earlier idea, the existence of p-adicization might quantize the eigenvalues rather than boundary conditions or finiteness requirement.
3. Super-conformal symmetries
The basic super-conformal symmetries follow trivially from the almost-topological character of the theory and actually generalize. The light-likeness condition brings in the metric of the imbedding space and manifests itself in eigenvalue spectrum of the modified Dirac operator inducing breaking of the super-conformal invariance and the loss of topological field theory property. Therefore TGD would be as near as possible to a topological quantum field theory and somewhat ironically, the breaking of the super-conformal invariance and the emergence of gauge and gravitational interactions as well as classical Kähler dynamics would follow from light-likeness condition.
4. TGD counterpart of super-space formalism
The precise relationship between super-conformal invariance of TGD and that of super-string models has been quite a stressor since the super-space formalism in the standard form breaks down in TGD. The reason is that Majorana spinors are replaced by Weyl spinors (quarks and leptons correspond to different H-chiralities). That super-conformal invariance makes sense has been clear for a long time but one cannot never exlude the possibility that some subtle error combined with overall optimism could have led to a self deception.
It is now however clear that the super-space formalism exists also in TGD framework but involves a different definition of the Grassmann integral measure. The formalism means the replacement of the radial lightlike coordinate of partonic 2-surface with its super-counter part involving second quantized spinor field consisting of eigenmodes of the modified Dirac operator behaving like Weyl spinors. Only the fermionic fields appear besides imbedding space coordinates as dynamical variables in this case. Super-space formalism gives the modified Dirac action as one integrates over the Grassmann parameters using a modified integration measure
where Γr is the modified gamma matrix associated with the lightlike direction and * denotes Dirac conjugation (it would be nice to have also bar in html symbol repertoire). The modified gamma matrices associated with the other coordinates vanish identically so that super-symmetrization occurs only in the radial direction. Also super-symmetrized canonical and Kac-Moody symmetries are formal symmetries of the Chern-Simons action.
The generalization of the super-space formalism to a higher- dimensional case is obtained by the replacement of the product
with the wedge product
dθ* γ1dθ ×w dθ*γ2dθ ×w...
completely analogous to the bosonic volume element
dx1 ×w dx2×w...
Here ×w symbolizes wedge product. It would be interesting to whether this Grassmann integral allows to construct a physically acceptable super-symmetric field theories based on Weyl- rather than Majorana condition. The requirement that the number of Grassmann parameters given by 2D is the number of spinor components of definite chirality (counting also conjugates) given by 2×2D/2-1 gives critical dimension D=8, which suggest that this kind of quantum field theory might exist and define the quantum field theory limit of TGD.
The last section of chapter Construction of Quantum Theory represents the detailed form of the argument above.