The discovery of number theoretical compactification has meant a dramatic increase in the understanding of quantum TGD. There are two manners to undestand the theory.
- Number theoretic view: Space-time surfaces can be regarded as hyper-quaternionic 4-surfaces in 8-dimensional hyper-octonionic space HO.
- Physics view: Space-time surfaces can be seen as 4-surfaces in 8-D space M^4xCP_2 minimizing the so called Kähler action which essentially means that they minimize their non-commutativity measured by Lagrangian density of Kähler action.
These views seem to be complementary, and at this moment the very existence of this duality (conjecture of course) is what has the strongest implications. A lot remains to do in order to see whether the conjecture is indeed correct and what it really implies. At this moment I am trying to find whether this duality, very reminiscent of various M-theory dualities, is internally consistent.
One of the possible implications is the possibility to interpret TGD also as a kind of string theory, not in the usual sense of the world, but as a generalization of so called topological quantum field theories, where the notion of braids is central. Whether this duality is completely general or holds true only for selected space-time surfaces, such as space-time sheets corresponding to maxima of Kähler function (most probable space-times) or space-time sheets representing asymptotic behavior, is an open question.
I have explained the duality in earlier posts and do not go to the details here. Suffice it so say that so called Wess-Zumino-Witten action for group G_2, a group which as a Lie group is completely exceptional, and acts as the automorphism group of octonions, suggests itself as a characterizer of the dynamics of these strings. G_2 has group SU(3) as maximal subgroup and can be said to leave these strings invariant. The interpretation is as the color group and G_2/SU(3) coset theory is the natural guess for the dynamics. SU(3) takes indeed the role of color gauge group.
The so called primary fields of the theory correspond to two color singlets, triplet and antitriplet and the natural guess is that they relate to leptons and quarks. Indeed in the H picture the basic fields are lepton and quark fields and all other particles are constructed from leptonic and quark like excitations.
The beauty of this approach is that QCD might be replaced with an exactly solvable conformal field theory allowing also to deduce how correlation functions change in hyper-octonion analytic transformations affecting space-time surface. There are however also objections against this picture.
a) The basic objection is that G_2 Kac-Moody algebra contains triplet and anti-triplet generators and triplet generators commute to anti-triplet. It is hard to imagine any sensible physical interpretation for these lepto-quark generators, whose commutation relations break the conservation of lepton and quark number.
The point is however that triplet generators affect e_1, and thus S^6 coordinates and also the SU(3) subgroup acting as isotropy group changes. Thus correlation functions involving these currents are not physically meaningful. Indeed, in G/H coset theory only the H Kac-Moody currents appear naturally in correlation functions since the construction involves functional integral only over
H connections.
b) If 14-dimensional adjoint representation of G_2 would appear as primary field, also 3 and \overline{3} lepto-quark like states for which baryon and lepton number are not conserved would appear in the spectrum. This is in conflict with H picture.
The choice k=1 for Kac-Moody central charge provides however a unique manner to circumvent this difficulty. Integrability condition for the highest weight representations allows for a given value of k only the highest weights \lambda_R satisfying Tr(\phi \lambda_R)\leq k, where \phi is the highest root for Lie-algebra. Since the highest root has length squared 2, adjoint representation is not possible as a highest weight representation for k=1 WZW model, and the primary fields of G_2 model are singlet and 7-plet corresponding to the hyper-octonionic spinor field and defining in an obvious manner the primary fields 1+3+\overline{3} of G_2/SU(3) coset model.
Fusion rules for 1\oplus 7 correspond to octonionic multiplication. The absence of G_2 gluons saves from lepto-quark like bosons, and the absence of SU(3) gluons can be interpreted as HO counterpart for the fact that all particles, in particular gluons, can be regarded bound states of fermions and anti-fermions in TGD Universe.
This picture conforms also with the claims that 3+\overline{3} part of G_2 algebra does not allow vertex operator construction whereas SU(3) allows the construction in terms of two free bosonic fields. These fields would naturally correspond to the two X^4 directions transversal to the string orbit defined by 1 and e_1. One could say that strings in X^4 are able to represent color Kac-Moody algebra and that SU(3)is inherent to 4-dimensional space-time.
c) The third objection is that conformal field theory correlation functions obeying simple scaling laws are not consistent with the exponentially decreasing correlation functions suggested by color confinement. A resolution of the paradox could be based on the role of classical gravitation. At light-like causal determinants the time-like component g_{tt} of the induced metric vanishes meaning that classical gravitational field is very strong. Hence also the normal component g_{nn} of the induced metric is expected to become very large so that hadron would look like the interior of black hole. A finite X^4 proper time for reaching the outer boundary of the hadronic surface can correspond to a very long M^4 time and the finite M^4 distance from the boundary can mean very long distance along hadronic space-time surface. Hence quarks and gluons can behave as almost free particles when viewed from hadronic space-time sheet but look confined when seen from imbedding space. If the hyper-quaternionic coordinates appearing in the correlation functions correspond to internal coordinate of the space-time surface, the correlation functions when expressed in terms of M^4 coordinates can look confining.
For more details see the chapter
TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts.
Matti Pitkanen
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