McKay correspondence states that the McKay graphs for the irreducible representations (irreps) of finite subgroups of G⊂ SU(2) characterizing their fusion algebra is given by extended Dynkin diagram of ADE type Lie group. Minimal conformal models with SU(2) Kac-Moody algebra (KMA) allow a classification by the same diagrams as fusion algebras of primary fields. The resolution of the singularities of complex algebraic surfaces in C3 by blowing implies the emergence of complex lines CP1. The intersection matrix for the CP1s is Dynkin diagram of ADE type Lie group. These results are highly inspiring concerning adelic TGD.
- The appearance of Dynkin diagrams in the classification of minimal conformal field theories (CFTs) inspires the conjecture that in adelic physics Galois groups Gal or semidirect products of Gal with a discrete subgroup G of automorphism group SO(3) (having SU(2) as double covering!) classifies TGD generalizations of minimal CFTs. Also discrete subgroups of octonionic automorphism group can be considered. The fusion algebra of irreps of Gal would define also the fusion algebra for KMA for the counterparts of minimal fields. This would provide deep insights to the general structure of adelic physics.
- One cannot avoid the question whether the extended ADE diagram could code for a dynamical symmetry of a minimal CFT or its modification? If the Gal singlets formed from the primary fields of minimal model define primary fields in Cartan algebra of ADE type KMA, then standard free field construction would give the charged KMA generators. In TGD framework this conjecture generalizes.
- A further conjecture is that the singularities of space-time surface imbedded as 4-surface in its 6-D twistor bundle with twistor sphere as fiber could be classified by McKay graph of Gal. The singular intersection of the Euclidian and Minkowskian regions of space-time surface is especially interesting: the twistor spheres at the common points defining light-like partonic orbits need not be same but have intersections with intersection matrix given by McKay graph for Gal. The basic information about adelic CFT would be coded by the general character of singularities for the twistor bundle.
- In TGD also singularities in which the group Gal is reduced to its subgroup Gal/H, where H is normal group are possible and would correspond to phase transition reducing the value of Planck constant. What happens in these phase transitions to single particle states would be dictated by the decomposition of representations of Gal to those of Gal/H and transition matrix elements could be evaluated.
See the new chapter Are higher structures needed in the categorification of TGD? or the article Could McKay correspondence generalize in TGD framework?.