## Could N=2 super-conformal algebra be relevant for TGD?The concrete realization of the super-conformal symmetry (SCS) in TGD framework has remained poorly understood. In particular, the question how SCS relates to super-conformal field theories (SCFTs) has remained an open question. The most general super-conformal algebra assignable to string world sheets by strong form of holography has N equal to the number of 4+4 =8 spin states of leptonic and quark type fundamental spinors but the space-time SUSY is badly broken for it. Covariant constancy of the generating spinor modes is replaced with holomorphy - kind of "half covariant constancy". I have considered earlier a proposal that N=4 SCA could be realized in TGD framework but given up this idea. Right-handed neutrino and antineutrino are excellent candidates for generating N=2 SCS with a minimal breaking of the corresponding space-time SUSY. Covariant constant neutrino is an excellent candidate for the generator of N=2 SCS. The possibility of this SCS in TGD framework will be considered in the sequel.
This work was inspired by questions not related to N=2 SCS, and it is good to consider first these questions.
The conjecture (see this) is that the conformal weights for the generators super-symplectic representation correspond to the negatives of h= -ks
For non-trivial zeros 2ks=1/2+iy of ζ(2ks) s would correspond pole s= (1/2+iy)/2k of z What can one say about the value of k? The pole of ζ(ks) at s=1/k would correspond to pole and conformal weight h=-1/k. For k=1 the trivial conformal weights would be positive integers h=1,2,...: this certainly makes sense. This gives for the real part for non-trivial conformal weights h=-1/4. By conformal confinement both pole and its conjugate belong to the state so that this contribution to conformal weight is negative half integers: this is consistent with the facts about super-conformal representations. For the ground state of super-conformal representation the conformal weight for conformally confined state would be h=- K/2. In p-adic mass calculations one would have K=6 (see this) . The negative ground state conformal weights of particles look strange but p-adic mass calculations require that the ground state conformal weights of particles are negative: h=-3 is required.
Super-symplectic conformal symmetries are realized at light-cone boundary and various Hamiltonians defined analogs of Kac-Moody generators are proportional functions f(r To get plane wave normalization for the amplitudes
(r
x=log(r
one must assume h=-1/2+iy. Together with the invariant integration measure dr
If r
This realization would however suggest that there must be also an interpretation in which ground states with negative conformal weight h
One could indeed start directly from the scaling invariant measure dr What makes this interpretation worth of discussing is that the entire machinery of conformal field theories with non-vanishing central charge and non-vanishing but positive ground state conformal weight becomes accessible allowing to determine not only the spectrum for these theories but also to determine the partition functions and even to construct n-point functions in turn serving as basic building bricks of S-matrix elements (see this) . ADE classification of these CFTs in turn suggests at connection with the inclusions of hyperfinite factors and hierarchy of Planck constants. The fractal hierarchy of broken conformal symmetries with sub-algebra defining gauge algebra isomorphic to entire algebra would give rise to dynamic symmetries and inclusions for HFFs suggest that ADE groups define Kac-Moody type symmetry algebras for the non-gauge part of the symmetry algebra.
N=2 SCFTs has some inherent problems. For instance, it has been claimed that they reduce to topological QFTs. Whether N=2 can be applied in TGD framework is questionable: they have critical space-time dimension D=4 but since the required metric signature of space-time is wrong.
N=2 SCS has some severe inherent problems. - N=2 SCS has critical space-time dimension D=4, which is extremely nice. On the other, N=2 requires
that space-time should have complex structure and thus metric signature (4,0), (0,4) or (2,2) rather than Minkowski signature. Similar problem is encountered in twistorialization and TGD proposal is Hamilton-Jacobi structure (se the appendix of (see this), which is hybrid of hypercomplex structure and Kähler structure. There is also an old proposal by Pope et al (see this) that one can obtain by a procedure analogous to dimensional reduction N=2 SCS from a 6-D theory with signature (3,3). The lifting of Kähler action to twistor space level allows the twistor space of M
^{4}to have this signature and the degrees of freedom of the sphere S^{2}are indeed frozen. - There is also an argument by Eguchi that N=2 SCFTs reduce under some conditions to mere topological QFTs (see this). This looks bad but there is a more refined argument that N=2 SCFT transforms to a topological CFT only by a suitable twist (see this). This is a highly attractive feature since TGD can be indeed regarded as almost topological QFT. For instance, Kähler action in Minkowskian regions could reduce to Chern-Simons term for a very general solution ansatz. Only the volume term having interpretation in terms of cosmological constant (see this) (extremely small in recent cosmology) would not allow this kind of reduction. The topological description of particle reactions based on generalized Feynman diagrams identifiable in terms of space-time regions with Euclidian signature of the induced metric would allow to build n-point functions in the fermionic sector as those of a free field theory. Topological QFT in bosonic degrees of freedom would correspond naturally to the braiding of fermion lines.
TGD version of SCA is gigantic as compared to the ordinary SCA. This SCA involves super-symplectic algebra associated with metrically 2-dimensional light-cone boundary (light-like boundaries of causal diamonds) and the corresponding extended conformal algebra (light-like boundary is metrically sphere S N=8 SCS seems to be the most natural candidate for SCS behind TGD: all fermion spin states would correspond to generators of this symmetry. Since the modes generating the symmetry are however only half-covariantly constant (holomorphic) this SUSY is badly broken at space-time level and the minimal breaking occurs for N=2 SCS generated by right-handed neutrino and antineutrino. The key motivation for the application of minimal N=2 SCFTs to TGD is that SCAs for them have a non-vanishing central charge c and vacuum weight h≥ 0 and the degenerate character of ground state allows to deduce differential equations for n-point functions so that these theories are exactly solvable. It would be extremely nice is scattering amplitudes were basically determined by n-point functions for minimal SCFTs.
A further motivation comes from the following insight. ADE classification of N=2 SCFTs is extremely powerful result and there is connection with the hierarchy of inclusions of hyperfinite factors of type II The application N=2 SCFTs in TGD framework can be however challenged. The problem caused by the negative value of vacuum conformal weight has been already discussed but there are also other problems. - One can argue that covariantly constant right-handed neutrino - call it ν
_{R}- defines a pure gauge super-symmetry and it has taken along time to decide whether this is the case or not. Taking at face value the lacking evidence for space-time SUSY from LHC would be easy but too light-hearted manner to get rid of the problem.Could it be that at space-time level covariantly constant right-handed neutrino (ν _{R}) and its antiparticle (ν*_{R}) generates pure gauge symmetry so that the resulting sfermions correspond to zero norm states? The oscillator operators for ν_{R}at imbedding space level have commutator proportional to p^{k}γ_{k}vanishing at the limit of vanishing massless four-momentum. This would imply that they generate sfermions as zero norm states. This argument is however formulated at the level of imbedding space: induced spinor modes reside at string world sheets and covariant constancy is replaced by holomorphy!At the level of induced spinor modes located at string world sheets the situation is indeed different. The anti-commutators are not proportional to p ^{k}γ_{k}but in Zero Energy Ontology (ZEO) can be taken to be proportional to n^{k}γ_{k}where n_{k}is light-like vector dual to the light-like radial vector of the point of the light-like boundary of causal diamond CD (part of light-one boundary) considered. Therefore also constant ν_{R}and ν*_{R}are allowed as non-zero norm states and the 3 sfermions are physical particles. Both ZEO and strong form of holography (SH) would play crucial role in making the SCS dynamical symmetry. - Second objection is that LHC has failed to detect sparticles. In TGD framework this objection cannot be taken seriously. The breaking of N=2 SUSY would be most naturally realized as different p-adic length scales for particle and sparticle. The mass formula would be the same apart from different p-adic mass scale. Sparticles could emerge at short p-adic length scale than those studied at LHC (labelled by Mersenne primes M
_{89}and M_{G,79}= (1+i)^{79}).One the other hand, one could argue that since covariantly constant right-handed neutrino has no electroweak-, color- nor gravitational interactions, its addition to the state should not change its mass. Again the point is however that one considers only neutrinos at string world sheet so that covariant constancy is replaced with holomorphy and all modes of right-handed neutrino are involved. Kähler Dirac equation brings in mixing of left and right-handed neutrinos serving as signature for massivation in turn leading to SUSY breaking. One can of course ask whether the p-adic mass scales could be identical after all. Could the sparticles be dark having non-standard value of Planck constant h _{eff}=n× h and be created only at quantum criticality (see this).
For details see the new chapter Could N=2 Super-Conformal Algebra Be Relevant For TGD? or the article with the same title. |