Last night I was thinking about possible future project in TGD. The construction of scattering amplitudes has been the dream impossible that has driven me for decades. Maybe the understanding of fermionic M^{8}H duality provides the needed additional conceptual tools.
 M^{8} picture looks simple. Spacetime surfaces in M^{8} can be constructed from real polynomials with real (rational) coefficients, actually knowledge of their roots is enough. Discrete data  roots of the polynomial! determines spacetime surface as associative or coassociative region! Besides this one must pose additional condition selecting 2D string world sheets and 3D lightlike surfaces as orbits of partonic 2surfaces. These would define strong form of holography (SH) allowing to map spacetime surfaces in M^{8} to M^{4}×CP_{2}.
 Could SH generalize to the level of scattering amplitudes expressible in terms of npoint functions of CFT?! Could the n points correspond to the roots of the polynomial defining spacetime region!
Algebraic continuation to quaternion valued scattering amplitudes analogous to that giving spacetime sheets from the data coded SH should be the key idea. Their moduli squared are real  this led to the emergence of Minkowski metric for complexified octonions/quaternions) would give the real scattering rates: this is enough! This would mean a number theoretic generalization of quantum theory.
 One can start from complex numbers and string world sheets/partonic 2surfaces. Conformal field theories (CFTs) in 2D play fundamental role in the construction of scattering string theories and in modelling 2D statistical systems. In TGD 2D surfaces (2D at least metrically) code for information about spacetime surface by strong holography (SH) .
Are CFTs at partonic 2surfaces and string world sheets the basic building bricks? Could 2D conformal invariance dictate the data needed to construct the scattering amplitudes for given spacetime region defined by causal diamond (CD) taking the role of sphere S^{2} in CFTs. Could the generalization for metrically 2D lightlike 3surfaces be needed at the level of "world of classical worlds" (WCW) when states are superpositions of spacetime surfaces, preferred extremals?
The challenge is to develop a concrete number theoretic hierarchy for scattering amplitudes: R→C→Q→O  actually their complexifications.
 In the case of fermions one can start from 1D data at lightlike boundaries LB of string world sheets at lightlike orbits of partonic 2surfaces. Fermionic propagators assignable to LB would be coded by 2D Minkowskian QFT in manner analogous to that in twistor Grassmann approach. npoint vertices would be expressible in terms of Euclidian npoint functions for partonic 2surfaces: the latter element would be new as compared to QFTs since pointlike vertex is replaced with partonic 2surface.
 The fusion (product?) of these Minkowskian and Euclidian CFT entities corresponding to different realization of complex numbers as subfield of quaternions would give rise to 4D quaternionic valued scattering amplitudes for given spacetime sheet. Most importantly: there moduli squared are real! A generalization of quantum theory (CFT) from complex numbers to quaternions (quaternionic "CFT").
 What about several spacetime sheets? Could one allow fusion of different quaternionic scattering amplitudes corresponding to different quaternionic subspaces of complexified octonions to get octonionvalued nonassociative scattering amplitudes. Again scattering rates would be real. A further generalization of quantum theory?
There is also the challenge to relate M^{8} and Hpictures at the level of WCW. The formulation of physics in terms of WCW geometry leads to the hypothesis that WCW Kähler geometry is determined by Kähler function identified as the 4D action resulting by dimensional reduction of 6D surfaces in the product of twistor spaces of M^{4} and CP_{2} to twistor bundles having S^{2} as fiber and spacetime surface X^{4}⊂ H as base. The 6D Kähler action reduces to the sum of 4D Kähler action and volume term having interpretation in terms of cosmological constant.
The question is whether the Kähler function  an essentially geometric notion  can have a counterpart at the level of M^{8}.
 SH suggests that the Kähler function identified in the proposed manner can be expressed by using 2D data or at least metrically 2D data (lightlike partonic orbits and lightlike boundaries of CD). Note that each WCW would correspond to a particular CD.
 Since 2D conformal symmetry is involved, one expects also modular invariance meaning that WCW Kähler function is modular invariant, so that they have the same value for X^{4}⊂ H for which partonic 2surfaces have induced metric in the same conformal equivalence class.
 Also the analogs of KacMoody type symmetries would be realized as symmetries of Kähler function. The algebra of supersymplectic symmetries of the lightcone boundary can be regarded as an analog of KacMoody algebra. Lightcone boundary has topology S^{2}× R_{+}, where R_{+} corresponds to radial lightlike ray parameterized by radial lightlike coordinate r. Super symplectic transformations of S^{2}× CP_{2} depend on the lightlike radial coordinate r, which is analogous to the complex coordinate z for he KacMoody algebras.
The infinitesimal supersymplectic transformations form algebra SSA with generators proportional to powers r^{n} . The KacMoody invariance for physical states generalizes to a hierarchy of similar invariances. There is infinite fractal hierarchy of subalgebras SSA_{n}⊂ SSA with conformal weights coming as nmultiples of those for SSA. For physical states SSA_{n} and [SSA_{n},SSA] would act as gauge symmetries. They would leave invariant also Kähler function in the sector WCW_{n} defined by n. This would define a hierarchy of sub WCWs of the WCW assignable to given CD.
The sector WCW_{n} could correspond to extensions of rationals with dimension n, and one would have inclusion hierarchies consisting of sequences of n_{i} with n_{i} dividing n_{i+1}. These inclusion hierarchies would naturally correspond to those for hyperfinite factors of type II_{1}.
See the chapter ZEO and matrices or the article Fermionic variant of M^{8}H duality.
