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.

1. Questions about SCS in TGD framework

This work was inspired by questions not related to N=2 SCS, and it is good to consider first these questions.

1. 1 Could the super-conformal generators have conformal weights given by poles of fermionic zeta?

The conjecture (see this) is that the conformal weights for the generators super-symplectic representation correspond to the negatives of h= -ks_k of the poles s_k fermionic partition function ζ_F(ks)=ζ(ks)/ζ(2ks) defining fermionic partition function. Here k is constant, whose value must be fixed from the condition that the spectrum is physical. ζ(ks) defines bosonic partition function for particles whos energies are given by log(p), p prime. These partition functions require complex temperature but is completely sensible in Zero Energy Ontology (ZEO), where thermodynamics is replaced with its complex square root.

For non-trivial zeros 2ks=1/2+iy of ζ(2ks) s would correspond pole s= (1/2+iy)/2k of z_F(ks). The corresponding conformal weights would be h=(-1/2-iy)/2k. For trivial zeros 2ks=-2n, n=1,2,.. s=-n/k would correspond to conformal weights h=n/k>0. Conformal confinement is assumed meaning that the sum of imaginary parts of of generators creating the state vanishes.

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.

1.2 What could be the origin of negative ground state conformal weights?

Super-symplectic conformal symmetries are realized at light-cone boundary and various Hamiltonians defined analogs of Kac-Moody generators are proportional functions f(r_M)H_J,m H_A, where H_J,m correspond to spherical harmonics at the 2-sphere R_M=constant and H_A is color partial wave in CP₂, f(r_M) is a partial wave in radial light-like coordinate which is eigenstate of scaling operator L₀=r_Md/dR_M and has the form (r_M/r₀)^-h, where h is conformal weight which must be of form h=-1/2+iy.

To get plane wave normalization for the amplitudes

(r_M/r₀) ^h=(r_M/r₀)^-1/2exp(iyx) ,

x=log(r_M/r₀) ,

one must assume h=-1/2+iy. Together with the invariant integration measure dr_M this gives for the inner product of two conformal plane waves exp(iy_ix), x=log(r_M/r₀) the desired expression ∫ exp[iy₁-y₂)x] dx= δ(y₁-y₂), where dx= dr_M/r_M is scaling invariance integration measure. This is just the usual inner product of plane waves labelled by momenta y_i.

If r_M/r₀ can be identified as a coordinate along fermionic string (this need not be always the case) one can interpret it as real or imaginary part of a hypercomplex coordinate at string world sheet and continue these wave functions to the entire string world sheets. This would be very elegant realization of conformal invariance.

1.3. How to relate degenerate representations with h>0 to the massless states constructed from tachyonic ground states with negative conformal weight?

This realization would however suggest that there must be also an interpretation in which ground states with negative conformal weight h_vac=-k/2 are replaced with ground states having vanishing conformal weights h_vac=0 as in minimal SCAs and what is regarded as massless states have conformal weights h= -h_vac>0 of the lowest physical state in minimal SCAs.

One could indeed start directly from the scaling invariant measure dr_M/r_M rather than allowing it to emerge from dr_M. This would require in the case of p-adic mass calculations that has representations satisfying Virasoro conditions for weight h=-h_vac>0. p-Adic mass squared would be now shifted downwards and proportional to L₀+h_vac. There seems to be no fundamental reason preventing this interpretation. One can also modify scaling generator L₀ by an additive constant term and this does not affect the value of c. This operation corresponds to replacing basis {zⁿ} with basis {z^n+1/2}.

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.

2. Questions about N=2 SCS

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.

2.1 Inherent problems of N=2 SCS

N=2 SCS has some severe inherent problems.

1. 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 M⁴ to have this signature and the degrees of freedom of the sphere S² are indeed frozen.
2. 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.

2.2 Can one really apply N=2 SCFTs to TGD?

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 S²). Both these algebras have conformal structure with respect to the light-like radial coordinate r_M and conformal algebra also with respect to the complex coordinate of S². Symplectic algebra replaces finite-dimensional Lie algebra as the analog of Kac-Moody algebra. Also light-like orbits of partonic 2-surfaces possess this SCA but now Kac-Moody algebra is defined by isometries of imbedding space. String world sheets possess an ordinary SCA assignable to isometries of the imbedding space. An attractive interpretation is that r_M at light-cone boundary corresponds to a coordinate along fermionic string extendable to a hypercomplex coordinate at string world sheet.

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 II₁, which is central for quantum TGD. The hierarchy of Planck constants assignable to the hierarchy of isomorphic sub-algebras of the super-symplectic and related algebras suggest interpretation in terms of ADE hierarchy a rather detailed view about a hierarchy of conformal field theories and even the identification of primary fields in terms of critical deformations.

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.

1. 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^kγ_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^kγ_k but in Zero Energy Ontology (ZEO) can be taken to be proportional to n^kγ_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.

2. 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 M₈₉ 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.