^{4}⊂ H=M

^{4}× CP

_{2}. The space-time surfaces are roots of two generalized analytic functions defined in H.

By holography, the space-time surfaces correspond to not completely deterministic orbits of particles as 3-surfaces and are thus analogous to Bohr orbits. This implies zero energy ontology (ZEO) and to the view of quantum TGD as wave mechanics in the space of these Bohr orbits located inside a causal diamond (CD), which form a causal hierarchy. Also the consruction of vertices for particle reactions has evolved dramatically during the last year and one can assign the vertices to partonic 2-surfaces.

M^{8}-H duality is a second key principle of TGD. M^{8}-H duality can be seen a number theoretic analog for momentum-position duality and brings in mind Langlands duality. M^{8} can be identified as octonions when the number-theoretic Minkowski norm is defined as Re(o^{2}). The quaternionic normal space N(y) of y∈ Y^{4}⊂M^{8} having a 2-D commutative complex sub-space is mapped to a point of CP_{2}. Y^{4} has Euclidian signature with respect to Re(o^{2}). The points y\∈ Y^{4} are lifted by a multiplication with a co-quaternionic unit to points of the quaternionic normal space N(y) and mapped to M^{4}⊂ H inversion.

This article discusses the relationship of the holography=holomorphy vision with the number theoretic vision predicting a hierarchy h_{eff}=nh_{0} of effective Planck constants such that n corresponds to the dimension of extension rationals. How could this hierarchy follow from the recent view of M^{8}-H duality. The proposed realization relies on the idea that quantum criticality implies that the two polynomials P_{1},P_{2} defining space-time surfaces as their roots have rational coefficients at the partonic 2-surfaces X^{2} appearing as generalized vertices. Partonic 2-surfaces would represent a stronger form of quantum criticality than generalized holomorphy so that the islands of algebraic extensions from the ocean of complex numbers would be selected. Adelic physics would emerge at quantum criticality and scattering amplitudes would become number-theoretically universal. In particular, the hierarchy of Planck constants and the identificaiton of p-adic primes as ramified primes would emerge as a prediction.

Also a generalization of the theory of analytic functions to the 4-D situation is suggestive. The poles of cuts of analytic functions would correspond to the 2-D partonic surfaces as vertices at which holomorphy fails and 2-D string worlds sheets could correspond to the cuts. This provides a general view of the breaking of the generalized conformal symmetries and their super counterparts as a necessary condition for the non-triviality of the scattering amplitudes.

See the article Holography=holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M^{8}-H duality 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.