The construction of Kähler geometry of WCW ("world of classical worlds") is fundamental to TGD program. I ended up with the idea about physics as WCW geometry around 1985 and made a breakthrough around 1990, when I realized that Kähler function for WCW could correspond to Kähler action for its preferred extremals defining the analogs of Bohr orbits so that classical theory with Bohr rules would become an exact part of quantum theory and path integral would be replaced with genuine integral over WCW. The motivating construction was that for loop spaces leading to a unique Kähler geometry. The geometry for the space of 3D objects is even more complex than that for loops and the vision still is that the geometry of WCW is unique from the mere existence of Riemann connection.
The basic idea is that WCW is union of symmetric spaces G/H labelled by zero modes which do not contribute to the WCW metric. There have been many open questions and it seems the details of the earlier approach must be modified at the level of detailed identifications and interpretations. What is satisfying that the overall coherence of the picture has increased dramatically and connections with string model and applications of TGD as WCW geometry to particle physics are now very concrete.
 A longstanding question has been whether one could assign Equivalence Principle (EP) with the coset representation formed by the superVirasoro representation assigned to G and H in such a manner that the four momenta associated with the representations and identified as inertial and gravitational fourmomenta would be identical. This does not seem to be the case. The recent view will be that EP reduces to the view that the classical four momentum associated with Kähler action is equivalent with that assignable to modified Dirac action supersymmetrically related to Kähler action: quantum classical correspondence (QCC) would be in question. Also strong form of general coordinate invariance implying strong form of holography in turn implying that the supersymplectic representations assignable to spacelike and lightlike 3surfaces are equivalent could imply EP with gravitational and inertial fourmomenta assigned to these two representations.
 The detailed identification of groups G and H and corresponding algebras has been a longstanding problem. Symplectic algebra associated with δM^{4}_{+/}× CP_{2} (δM^{4}_{+/} is lightcone boundary  or more precisely, with the boundary of causal diamond (CD) defined as Cartesian product of CP_{2} with intersection of future and past direct light cones of M^{4} has KacMoody type structure with lightlike radial coordinate replacing complex coordinate z. Virasoro algebra would correspond to radial diffeomorphisms.
I have also introduced KacMoody algebra assigned to the isometries and localized with respect to internal coordinates of the lightlike 3surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian and which serve as natural correlates for elementary particles (in very general sense!). This kind of localization by force could be however argued to be rather ad hoc as opposed to the inherent localization of the symplectic algebra containing the symplectic algebra of isometries as subalgebra. It turns out that one obtains direct sum of representations of symplectic algebra and KacMoody algebra of isometries naturally as required by the success of padic mass calculations.
 The dynamics of Kähler action is not visible in the earlier construction. The construction also expressed WCW Hamiltonians as 2D integrals over partonic 2surfaces. Although strong form of general coordinate invariance (GCI) implies strong form of holography meaning that partonic 2surfaces and their 4D tangent space data should code for quantum physics, this kind of outcome seems too strong. The progress in the understanding of the solutions of modified Dirac equation led however to the conclusion that spinor modes other than righthanded neutrino are localized at string world sheets with strings connecting different partonic 2surfaces.
This leads to a modification of earlier construction in which WCW superHamiltonians were essentially 2D flux integrals. Now they are 2D flux integrals with superHamiltonian replaced Noether super charged for the deformations in G and obtained by integrating over string at each point of partonic 2surface. Each spinor mode gives rise to super current and the modes of righthanded neutrino and other fermions differ in an essential manner. Righthanded neutrino would correspond to symplectic algebra and other modes to the KacMoody algebra and one obtains the crucial 5 tensor factors of Super Virasoro required by padic mass calculations.
The matrix elements of WCW metric between Killing vectors are expressible as anticommutators of superHamiltonians identifiable as contractions of WCW gamma matrices with these vectors and give Poisson brackets of corresponding Hamiltonians. The anticommutation relates of induced spinor fields are dictated by this condition. Everything is 3dimensional although one expects that symplectic transformations localized within interior of X^{3} act as gauge symmetries so that in this sense effective 2dimensionality is achieved. The components of WCW metric are labelled by standard model quantum numbers so that the connection with physics is extremely intimate.
 An open question in the earlier visions was whether finite measurement resolution is realized as discretization at the level of fundamental dynamics. This would mean that only certain string world sheets from the slicing by string world sheets and partonic 2surfaces are possible. The requirement that anticommutations are consistent suggests that string world sheets correspond to surfaces for which Kähler magnetic field is constant along string in well defined sense (J_{μν}ε^{μν}g^{1/2} remains constant along string). It however turns that by a suitable choice of coordinates of 3surface one can guarantee that this quantity is constant so that no additional constraint results.
See the new chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds or the
article with the same title.
