### From N=0 to N=2 to N=4

Below a somewhat hastily written letter that I sent to Istvan and Carlos. I have done some editing of typos.

The development have been really fast: super-conformal N has increased in two units per day during the last three days!

N=4 superconformal symmetry would however mean that both M

- N=0 meant the realization that rational conformal field theories and c=1 and c=0 theories would give stringy representations for scattering amplitudes.

- The basic observation of Vafa and Berkovitz is that critical N=2 super-conformal theories extend to N=4 super-conformal theories at criticality. N=2 super-conformal symmetry is implied in leptonic sector by complex covariantly constant right handed neutrinos: quarks are problem. For N=2 critical theories all n-particle scattering amplitudes vanish for particle number n>3 and this has extremely nice interpretation in TGD framework although the result cannot hold true for c≤3 phases described by rational N=2 super-conformal field theories.

- N=4 super-conformal algebras in turn allow an exceptional variant found by Rasmussen for five years ago: this I learned today. The N=4 super-conformal algebra of Rasmussen contains besides Virasoro, 4 super generators G, U(2) Kac-Moody, the counterparts of right handed neutrino spinors of both H=M
^{4}× CP_{2}-chirality, and scalar which cannot however correspond to Higgs as I noticed later. This is a perfect fit in TGD framework. Stringy amplitudes with this symmetry algebra would allow to calculate the predictions of TGD and one expects enormous symmetries in critical phase.

^{4}×CP_{2}spinor chiralities would carry leptonic charge assignments.

- Both quarks and leptons could exist in integer charged free phase and in fractionally charged anyonic phases with quantum phase q different from one.

- Usual fractional quark charges would correspond to the lowest possible Jones inclusion for which fermion number is fractionized to 1/n= 1/3 and em charge becomes fractional and color corresponds to 3-fold covering of M
^{4}points by three CP_{2}points.

- Both leptons and quarks could form hadron like states and n could have all values n≥ 3 meaning entire hierarchy of color confined states with various SU(n):s and other ADE groups. The extremely beautiful topological understanding of color confinement using the generalized view about imbedding space is perhaps the best justificication for this option.
The basic hypothesis hitherto has been that different H-chiralities (quarks and leptons) correspond to fractional and integer em charges. The ontological question is whether I accept that both quarks and leptons can exist in integer charged phase and infinite number of anyonic phases with fractional charges proportional to 1/n, n=3,4,5,.... The quantum biological model already assumes large values of n coming as powers of 2

^{11}for leptons so that internal consistency favors this picture. And most theoreticians would say that it would be idiotic to break these gigantic symmetries just because of standard model based prejudices about what can exist. Even more so after I have populated the universe with quantum coherent dark matter with varying values of hbar!I have been trying to find whether this interpretation is consistent with the existing wisdom.

- One should understand why the free integer charged quarks interact so weakly with ordinary leptons. This is easy to understand: as a matter fact, gauge boson exchanges and even graviton exchange would interfere to zero if H-axial and vectorial bosons have same masses and couplings (already for 10 years ago I realized that gauge bosons can couple to quarks and leptons with same or opposite -signs so that one as H-axial or H-vectorial bosons and the question was why only vectorial would be visible).

- Also the interactions of leptons with genuine quarks in fractionally charged phase come out correctly under very simple assumption.

- If H-axial (say) vector bosons inside hadrons are massless or light, they could give rise to strong interactions so that the explanation for why isospin and hypercharge have interpretation both as weak and strong isospin would emerge. This I realized for a couple years ago but had not yet realized the possibility of H-axial and -vectorial electroweak gauge bosons.

- According to the earlier dualities this description of strong interactions would be dual to a description in terms of SU(3) gluon exchanges.
Putting all this together, I feel that I have right to say that it is more or less done now. The rest is just filling up the details and getting the results to general awareness. The technical side is not problem anymore.

The last section of chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is just now.

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