About the identification of Kac Moody algebra and corresponding Virasoro algebraIt is relatively straightforward to deduce the detailed form of the TGD countetpart of KacMoody algebra identified as X^{3}local infinitesimal transformations of H__{+/}=M^{4}_{+/}× CP_{2} respecting the lightlikeness of partonic 3surface X^{3}. This involves the identification of KacMoody transformations and corresponding supergenerators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and NS algebra is needed. Especially interesting is the relationship with the supercanonical algebra consisting of canonical transformations of δ M^{4}_{+/}× CP_{2}. 1. Bosonic part of the algebra The bosonic part of KacMoody algebra can be identified as symmetries respecting the lightlikeness of the partonic 3surface X^{3} in H=M^{4}_{+/}× CP_{2}. The educated guess is that a subset of X^{3}local diffeomorphisms of is in question. The allowed infinitesimeal transformation of this kind must reduce to a conformal transformation of the induced metric plus diffeomorphism of X^{3}. The explicit study of the conditions allows to conclude that conformal transformations of M^{4}_{+/} and isometries of CP_{2} made local with respect to X^{3} satisfy the defining conditions. Choosing special coordinates for X^{3} one finds that the vector fields defining the transformations must be orthogonal of the lightlike direction of X^{3}. The resulting partial differential equations fix the infinitesimal diffeomorphism of X^{3} once the functions appearing in KacMoody generator are fixed. The functions appearing in generators can be chosen to proportional to powers of the radial coordinate multiplied by functions of transversal coordinates whose dynamics is dictated by consistency conditions The resulting algebra is essentially 3dimensional and therefore much larger than ordinary KacMoody algebra. One can identify the counterpart of ordinary KacMoody algebra as a subalgebra for which generators are in oneone correspondence with the powers of the lightlike coordinate assignable to X^{3}. This algebra corresponds to the stringy subalgebra E^{2}× SO(2)×SU(3) if one selects the preferred coordinate of M^{4} as a lightlike coordinate assignable to the lightlike ray of δ M^{4}+/ defining orbifold structure in M^{4}_{+/} ("massless" case) and E^{3}× SO(3)×SU(3) if the preferred coordinate is M^{4} time coordinate (massive case). The local transformation in the preferred direction is not free but fixed by the condition that KacMoody transformation does not affect the value of the lightlike coordinate of X^{3}. This is completely analogous to the nondynamical character of longitudinal degrees of freedom of KacMoody algebra in string models. The algebra decomposes into a direct sum of subspaces left invariant by KacMoody algebra and one has a structure analogous to that defining coset space structure (say SU(3)/U(2)). This feature means that the space of physical states is much larger than in string models and Kac Moody algebra of string models takes the role of the little algebra in the representations of Poincare group. Mackey's construction of induced representations should generalized to this situation. Just as in the case of supercanonical algebra, the Noether charges assignable to the KacMoody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to onedimensional integrals if the SO(2)× SU(3) quantum numbers of the generator vanish. The intepretation is that this algebra leaves invariant various quantization axes and acts as symmetries of quantum measurement situation. 2. Fermionic sector The zero modes and generalized eigen spinors of the modified Dirac equation define the counterparts of Ramond and NS type super generators. The hypothesis inspired by number theoretical conjectures related to Riemann Zeta is that the eigenvalues of the generalized eigen modes associated with ground states correspond to nontrivial zeros of zeta. Also nontrivial eigenvalues must be considered.
3. Super Virasoro algebras The identification of SKM Virasoro algebra as that associated with radial diffeomorphisms is obvious and this algebra replaces the usual Virasoro algebra associated with the complex coordinate of partonic 2surface X^{2} in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for padic thermodynamics. The commutators of super canonical and super KacMoody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Fourmomentum does not appear in the expressions for the Virasoro generators and mass squared is identified as padic thermal expectation value of conformal weight. There are no problems with Lorentz invariance. One can wonder about the role of ordinary conformal transformations assignable to the partonic 2surface X^{2}. The stringy quantization implies the reduction of this part of algebra to algebraically 1D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X^{3} unlike radial conformal weights. TGD analog of superconformal symmetries of condensed matter physics rather than stringy superconformal symmetry would be in question. The last section of the chapter The Evolution of Quantum TGD gives a more detailed summary of the recent picture.
