### N=4 SCA as basic symmetry of TGD and the basic mistake of M-theory

N=4 super-conformal algebra (SCA) emerges naturally in TGD framework and is basically due to the covariantly constant right handed neutrinos and super Kac-Moody and super-canonical algebras defining the generalized coset representations. N=4 SCA is the maximal associative SCA and has an interpretation in terms of super-affinization of a complexified quaternion algebra. The (4,4) signature of target space metric characterizing N=4 SCA topological field theory is not a problem since in TGD framework the target space becomes a fictive concept defined by the Cartan algebra. This picture allows to assign the critical dimensions of super string models and M-theory with the fictive target spaces associated with the vertex operator construction.

N=4 super-conformal topological field theory defines in TGD framework a non-trivial physical theory since classical interactions induce correlations between partonic 2-surfaces and CP_{2} type extremals provide a space-time correlate for virtual particles. Both M^{4}× CP_{2} decomposition of the imbedding space and space-time dimension are crucial for the 2+2+2+2 structure of the Cartan algebra, which together with the notion of the configuration space guarantees N=4 super-conformal invariance. Therefore it is not exaggeration to say that the basic structure of TGD is uniquely fixed also by N=4 super-conformal invariance. In the following the interpretation of the critical dimension and critical signature of metric is discussed in some detail.

The basic problem is that the signature of the induced space-time metric cannot be (2,2) which is essential for obtaining the cancellation for N=2 SCA imbedded to N=4 SCA with critical dimension D=8 and signature (4,4). Neither can the metric of imbedding space correspond to the signature (4,4). The (4,4) signature of the target space metric is not so serious limitation as it looks if one is ready to consider the target space appearing in the calculation of N-point functions as a fictive notion.

The resolution of the problems relies on two observations.

- The super Kac-Moody and super-canonical Cartan algebras have dimension D=2 in both M
^{4}=M^{2}×E^{2}and CP_{2}degrees of freedom giving total effective dimension D=2+2+2+2=8.- Super Kac-Moody algebra acts as deformations of partonic 2-surfaces X
^{2}. It consists of supersymmetrized X^{2}-local E^{2}translations completely analogous to transversal deformations of string in M^{4}and supersymmetrized electro-weak Kac-Moody algebra U(2)_{ew}acting on quantum variants of spinors identifiable as super-counterparts of super-symmetrized complexified quaternions. Both Cartan algebras have obviously dimension D=2. - Super-canonical algebra acts as deformations of 3-surfaces and is generated by Hamiltonians in δ M
^{4}_{+/-}× CP_{2}with degenerate symplectic and complex structures made possible by the metric 2-dimensionality of the boundary of the four-dimensional light-cone. It consists of canonical transformations of CP_{2}and (with a suitable choice of gauge) of the canonical transformations of the tangent plane E^{2}of sphere having origin at the tip of δ M^{4}_{+/-}. Also now both Cartan algebras have dimensions D=2.

- Super Kac-Moody algebra acts as deformations of partonic 2-surfaces X
- The generalized coset construction discussed in detailhere allows to assign opposite signatures of metric to super Kac-Moody Cartan algebra and corresponding super-canonical Cartan algebra so that the desired signature (4,4) results. Altogether one has 8-D effective target space with signature (4,4) characterizing N=4 super-conformal topological strings. Hence the number of physical degrees of freedom is D
_{phys}=8 as in super-string theory. Including the non-physical M^{2}degrees of freedom, one has critical dimension D=10. If also the radial degree of freedom associated with δ M^{4}_{+/-}is taken into account, one obtains D=11 as in M-theory.