Updated View about Kähler geometry of WCW
Quantum TGD reduces to a construction of Kähler geometry for what I call the "World of Classical Worlds. It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M4+/-× CP2. The string model type description of gravitation emerges and also the TGD based view about dark matter becomes more precise. String tension is however dynamical rather than pregiven and the hierarchy of Planck constants is necessary in order to understand the formation of gravitationally bound states. Also the proposal that sparticles correspond to dark matter becomes much stronger: sparticles actually are dark variants of particles.
A crucial element of the construction is the assumption that super-symplectic and other super-conformal symmetries having the same structure as 2-D super-conformal groups can be seen a broken gauge symmetries such that sub-algebra with conformal weights coming as n-ples of those for full algebra act as gauge symmetries. In particular, the Noether charges of this algebra vanish for preferred extremals- this would realize the strong form of holography implied by strong form of General Coordinate Invariance. This gives rise to an infinite number of hierarchies of conformal gauge symmetry breakings with levels labelled by integers n(i) such that n(i) divides n(i+1) interpreted as hierarchies of dark matter with levels labelled by the value of Planck constant heff=n× h. These hierarchies define also hierarchies of quantum criticalities and are proposed to give rise to inclusion hierarchies of hyperfinite factors of II1 having interpretation in terms of finite cognitive resolution. These hierarchies would be fundamental for the understanding of living matter.
For details see the article Updated view about Kähler geometry of WCW.
For a summary of earlier postings see Links to the latest progress in TGD.