_{n}of partonic 2-surfaces with genus g≥ 1 such that factors of n define subgroups of conformal symmetries of Z

_{n}. By the decomposition Z

_{n}=∏

_{p|n}Z

_{p}, where p|n tells that p divides n, this hierarchy corresponds to an hierarchy of increasingly quantum critical systems in modular degrees of freedom. For a given prime p one has a sub-hierarchy Z

_{p}, Z

_{p2}=Z

_{p}× Z

_{p}, etc... such that the moduli at n+1:th level are contained by n:th level. In the similar manner the moduli of Z

_{n}are sub-moduli for each prime factor of n. This mapping of integers to quantum critical systems conforms nicely with the general vision that biological evolution corresponds to the increase of quantum criticality as Planck constant increases.

The group of conformal symmetries could be also non-commutative discrete group having Z_{n} as a subgroup. This inspires a very shortlived conjecture that only the discrete subgroups of SU(2) allowed by Jones inclusions are possible as conformal symmetries of Riemann surfaces having g≥ 1. Besides Z_{n} one could have tedrahedral and icosahedral groups plus cyclic group Z_{2n} with reflection added but not Z_{2n+1} nor the symmetry group of cube. The conjecture is wrong. Consider the orbit of the subgroup of rotational group on standard sphere of E^{3}, put a handle at one of the orbits such that it is invariant under rotations around the axis going through the point, and apply the elements of subgroup. You obtain Riemann surface having the subgroup as its isometries. Hence all subgroups of SU(2) can act as conformal symmetries.

The number theoretically simple ruler-and-compass integers having as factors only first powers of Fermat primes and power of 2 would define a physically preferred sub-hierarchy of quantum criticality for which subsequent levels would correspond to powers of 2: a connection with p-adic length scale hypothesis suggests itself.

Spherical topology is exceptional since in this case the space of conformal moduli is trivial and conformal symmetries correspond to the entire SL(2,C). This would suggest that only the fermions of lowest generation corresponding to the spherical topology are maximally quantum critical. This brings in mind Jones inclusions for which the defining subgroup equals to SU(2) and Jones index equals to M/N =4. In this case all discrete subgroups of SU(2) label the inclusions. These inclusions would correspond to fiber space CP_{2}→ CP_{2}/U(2) consisting of geodesic spheres of CP_{2}. In this case the discrete subgroup might correspond to a selection of a subgroup of SU(2)subset SU(3) acting non-trivially on the geodesic sphere. Cosmic strings X^{2}× Y^{2} subset
M^{4}×CP_{2} having geodesic spheres of CP_{2} as their ends could correspond to this phase dominating the very early cosmology.

For more details see the revised chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" or the chapter Construction of Elementary Particle Vacuum Functionals of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy.

## 2 comments:

Matti, sorry it's off-topic, but where do I find your ideas on LHC predictions? I'm not sure exactly where to look.

Not an off-topic at all. I should actually write chapter about LHC predictions since they are scattered as comments here and there. The chapters about p-adic mass calculations, in particular the fourth chapter p-Adic Particle Massivation: New Physics is about the predictions.

The previous chapter p-Adic Particle Massivation: Hadron physics contains a discussion related to top quark.

I try to briefly sum up here (I hope that I do not forget anything essential).

a) The basic almost-prediction for LHC is a scaled-up copy of hadron physics with QCD Lambda scaled up by factor 512. This corresponds to gluons characterized by Mersenne prime M_89: M_107 characterizes ordinary gluons.

b) Also the bosons labelled by pairs (g_1,g_2) of genera and forming SU(3) multiplet and inducing neutral flavor changing transitions could be there unless the existing data do not force their p-adic mass scale too high. M_61 would be the next Mersenne mass scale and by a factor 2^(23)=about 8000 higher than hadronic mass scale and strange cosmic rays (Centauros,etc) give some support for this copy of hadron physics.

c) Higgs should be there but couple to fermions with a coupling about 1/100 times that in standard model since Higgs gives only small fraction to fermion mass. This might explain why Higgs has not been observed even in the case that it is relatively light.

d) No sparticles should be there: I dare to make this as a prediction. The reason is that they is no counterpart of super-symmetric extension of Poincare invariance in TGD. This is somehow a pity since p-adic length scale would give rise to an elegant breaking of super-symmetry with same universal super-symmetric mass formulas but with different mass scale given by p-adic thermodynamics.

Covariantly constant right handed neutrino spinor (in CP_2) defines a super-symmetry but this relates to super-conformal symmetries which indeed predict exotic states which seem however to be different. As a matter fact, a fundamental parton level one has on mass-shell N=4 type super-conformal symmetry but this is broken for physical states.

e) Hadrons containing fractal scaled up copies of quarks with quarks masses scaled by power of sqrt(2) are predicted and they would be present already in low energy hadron physics and their presence would explain Gell-Mann-Okubo mass formula. There is also evidence for several bumps in top quark mass distribution and this could have explanation in terms fractally scaled versions of various quarks.

f) Dark matter would be low energy phenomenon in TGD framework and not observed in LHC.

g) It is in principle possible that also colored excitations of quarks appear but I do not have clear picture about this.

h) No light blackholes nor big new dimensions are predicted so that everyone can sleep safely in TGD is right;-).

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