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New Findings Related to the Number Theoretical View of TGD
Abstract
The geometric vision of TGD is rather well-understood but there is still a lot of fog in the number theoretic vision:
(1) There are uncertainties related to the interpretation of the 4-surfaces in M8 what the analogy with space-time surface in H = M4 x CP2 time evolution of 3-surface in H could mean physically?
(2) The detailed realization of M8-H duality involves uncertainties: in particular, how the complexification of M8 to M8c can be consistent with the reality of M4 ⊂ H.
(3) The formulation of the number theoretic holography with dynamics based on associativity involves open questions. The polynomial P determining the 4-surface in M8 doesn't fix the 3-surfaces at mass shells corresponding to its roots. Quantum classical correspondence suggests the coding of fermionic momenta to the geometric properties of 3-D surfaces: How could this be achieved?
(4) How unique is the choice of 3-D surfaces at the mass shells H3m ⊂ M4 ⊂ M8 and whether a strong form of holography as almost 2 → 4 holography could be realized and make this choice highly unique.
These and many other questions motivated this article and led to the observation that the model geometries used in the classification of 3-manifolds seem to be rather closely related to the known space-time surfaces extremizing practically any general coordinate invariant action constructible in terms of the induced geometry. The 4-surfaces in M8 would define coupling constant evolutions for quantum states as analogs of and mappable to time evolutions at the level of H and obeying conservation laws associated with the dual conformal invariance analogous to that in twistor approach. The momenta of fundamental fermions in the quantum state would be coded by the cusp singularities of 3-surfaces at the mass shells of M8 and also its image in H provided by M8-H duality. One can consider the possibility of 2 → 3 holography in which the boundaries of fundamental region of H3/Γ is 2-D hyperbolic space H3/Γ so that TGD could to high degree reduce to algebraic geometry.