** 1. Large N limit of gauge theories and series of Jones inclusions**

The large N limit of SU(N) gauge field theories has as definite resemblance with the series of Jones inclusions with the integer n≥ 3 characterizing the quantum phase q=exp(iπ/n) and the order of the maximal cyclic subgroup of the subgroup of SU(2) defining the inclusion. Recall that all ADE groups except D_{2n+1} and E_{7} are allowed (SU(2) is excluded since it would correspond to n=2).

The limiting procedure keeps the value of g^{2}N fixed. Rather remarkably, this is equivalent with keeping α N constant but assuming hbar to scale as n=N. Thus the quantization of Planck constants would provide a physical laboratory for the testing of large N limit.

The observation suggesting a description of YM theories in terms of closed strings is that Feynman diagrams can be interpreted as being imbedded at closed 2-surfaces of minimal genus guaranteing that the internal lines meet except in vertices. The contribution of genus g diagrams is proportional to N^{g-1} at the large N limit. The interpretation in terms of closed partonic 2-surfaces is highly suggestive and the N^{g-1} should come from the multiple covering property of CP_{2} by N M^{4}-points (or vice versa) with the finite subgroup of G of SU(2) defining the Jones inclusion and acting as symmetries of the surface.

** 2. Analogy between stacks of branes and multiple coverings of M ^{4} and CP_{2}**

An important aspect of AdS/CFT dualities is a prediction of an infinite hierarchy of gauge groups, which as such is as interesting as the claimed dualities. The prediction relies on the notion Dp-branes. Dp-branes are p+1-dimensional surfaces of the target space at which the ends of open strings can end. In the simplest situation one considers N parallel p-branes at the limit when the distances between branes characterized by an expectation value of Higgs fields approach zero to obtain what is called N-stack of branes. There are N^{2} different strings connecting the branes and the heuristic idea is that they correspond to gauge bosons of U(N) gauge theory. Note that the requirement that AdS/CFT dualities exist forces the introduction of branes and the optimistic interpretation is that a non-perturbative effect of still unknown M-theory is in question. In the limit of an ideal stack one assumes that U(N) gauge theory at the brane representing the stack is obtained. The branes must also carry a p-form defining gauge potential for a closed p+1-form. This Ramond charge is quantized and its value equals to N.

Consider now the group G_{a}× G_{b} in SL(2,C)× SU(2) defining double Jones inclusion and implying the scalings hbar(M^{4})→ n(G_{b})× hbar(M^{4}) and hbar(CP_{2})→ n(G_{a})× hbar(CP_{2}). These space-time surfaces define n(G_{a})-fold multiple coverings of CP_{2} and n(G_{b})-fold multiple coverings of M^{4}. In CP_{2} degrees of freedom the collection of G_{b}-related partonic 2-surfaces (/3-surfaces/4-surfaces) is highly analogous to the stack of branes. In M^{4} degrees of freedom the stack of copies of surface typically correspond to along a circle (A_{n},D_{2n} or at vertices of tedrahedron or isosahedron.

In TGD framework strings are not needed to define gauge fields. The group algebra of G realized as discrete plane waves at G-orbit gives rise to representations of G. The hypothesis supported by few examples is that these additional degrees of freedom allow to construct multiplets of the gauge group assignable to the ADE diagram characterizing the inclusion.

The detailed comparison with AdS/CFT corresponds provides non-trivial insights. For instance for G=SU(2) inclusions which correspond to Kac-Moody group associated with extended ADE diagram defining the inclusion, Ramond charge corresponds to nontrivial homology (magnetic charge) in CP_{2} or in δ M^{4}_{+/-}.

The last section of chapter Construction of Quantum Theory of "Towards S-Matrix" represents the detailed construction in its recent form.

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