Topological explanation of family replication phenomenon

One of the basic ideas of TGD approach has been genus-generation correspondence: boundary components of the 3-surface should be carriers of elementary particle numbers and the observed particle families should correspond to various boundary topologies. Last summer meant quite a progress in the understanding of quantum TGD, which forced also the updating of the views about the topological explanation of family replication phenomenon.

With the advent of zero energy ontology the original picture changed somewhat. It is the wormhole throats identified as light-like 3-surfaces at with the induced metric of the space-time surface changes its signature from Minkowskian to Euclidian, which correspond to the light-like orbits of partonic 2-surfaces. One cannot of course exclude the possibility that also boundary components could allow to satisfy boundary conditions without assuming vacuum extremal property of nearby space-time surface. The intersections of the wormhole throats with the light-like boundaries of causal diamonds (CDs) identified as intersections of future and past directed light cones (CD × CP₂ is actually in question but I will speak about CDs) define special partonic 2-surfaces and it is the comformal moduli of these partonic 2-surfaces which appear in the elementary particle vacuum functionals naturally.

The first modification of the original simple picture comes from the identification of physical particles as bound states of pairs of wormhole contacts and from the assumption that for generalized Feynman diagrams stringy trouser vertices are replaced with vertices at which the ends of light-like wormhole throats meet. In this picture the interpretation of the analog of trouser vertex is in terms of propagation of same particle along two different paths. This interpretation is mathematically natural since vertices correspond to 2-manifolds rather than singular 2-manifolds which are just splitting to two disjoint components. Second complication comes from the weak form of electric-magnetic duality forcing to identify physical particles as weak strings with magnetic monopoles at their ends and one should understand also the possible complications caused by this generalization.

These modifications force to consider several options concerning the identification of light fermions and bosons and one can end up with a unique identification only by making some assumptions. Masslessness of all wormhole throats- also those appearing in internal lines- and dynamical SU(3) symmetry for particle generations are attractive general enough assumptions of this kind. This means that bosons and their super-partners correspond to wormhole contacts with fermion and antifermion at the throats of the contact. Free fermions and their superpartners could correspond to CP₂ type vacuum extremals with single wormhole throat. It turns however that dynamical SU(3) symmetry forces to identify massive (and possibly topologically condensed) fermions as (g,g) type wormhole contacts.

Do free fermions correspond to single wormhole throat or (g,g) wormhole?

The original interpretation of genus-generation correspondence was that free fermions correspond to wormhole throats characterized by genus. The idea of SU(3) as a dynamical symmetry suggested that gauge bosons correspond to octet and singlet representations of SU(3). The further idea that all lines of generalized Feynman diagrams are massless poses a strong additional constraint and it is not clear whether this proposal as such survives.

1. Twistorial program assumes that fundamental objects are massless wormhole throats carrying collinearly moving many-fermion states and also bosonic excitations generated by super-symplectic algebra. In the following consideration only purely bosonic and single fermion throats are considered since they are the basic building blocks of physical particles. The reason is that propagators for high excitations behave like p^-n, n the number of fermions associated with the wormhole throat. Therefore single throat allows only spins 0,1/2,1 as elementary particles in the usual sense of the word.

2. The identification of massive fermions (as opposed to free massless fermions) as wormhole contacts follows if one requires that fundamental building blocks are massless since at least two massless throats are required to have a massive state. Therefore the conformal excitations with CP₂ mass scale should be assignable to wormhole contacts also in the case of fermions. As already noticed this is not the end of the story: weak strings are required by the weak form of electric-magnetic duality.

3. If free fermions corresponding to single wormhole throat, topological condensation is an essential element of the formation of stringy states. The topological condensation of fermions by topological sum (fermionic CP₂ type vacuum extremal touches another space-time sheet) suggest (g,0) wormhole contact. Note however that the identification of wormhole throat is as 3-surface at which the signature of the induced metric changes so that this conclusion might be wrong. One can indeed consider also the possibility of (g,g) pairs as an outcome of topological conensation. This is suggested also by the idea that wormhole throats are analogous to string like objects and only this option turns out to be consistent with the BFF vertex based on the requirement of dynamical SU(3) symmetry to be discussed later. The structure of reaction vertices makes it possible to interpret (g,g) pairs as SU(3) triplet. If bosons are obtained as fusion of fermionic and antifermionic throats (touching of corresponding CP₂ type vacuum extremals) they correspond naturally to (g₁,g₂) pairs.

4. p-Adic mass calculations distinguish between fermions and bosons and the identification of fermions and bosons should be consistent with this difference. The maximal p-adic temperature T=1 for fermions could relate to the weakness of the interaction of the fermionic wormhole throat with the wormhole throat resulting in topological condensation. This wormhole throat would however carry momentum and 3-momentum would in general be non-parallel to that of the fermion, most naturally in the opposite direction.

   p-Adic mass calculations suggest strongly that for bosons p-adic temperature T=1/n, n>1, so that thermodynamical contribution to the mass squared is negligible. The low p-adic temperature could be due to the strong interaction between fermionic and antifermionic wormhole throat leading to the "freezing" of the conformal degrees of freedom related to the relative motion of wormhole throats.

5. The weak form of electric-magnetic duality forces second wormhole throat with opposite magnetic charge and the light-like momenta could sum up to massive momentum. In this case string tension corresponds to electroweak length scale. Therefore p-adic thermodynamics must be assigned to wormhole contacts and these appear as basic units connected by Kähler magnetic flux tube pairs at the two space-time sheets involved. Weak stringy degrees of freedom are however expected to give additional contribution to the mass, perhaps by modifying the ground state conformal weight. A nice implication is that all elementary particles -not only gravitons- correspond to pairs of wormhole throats connected by magnetic flux tubes to form "weak strings". This has obvious implications at LHC.

Dynamical SU(3) fixes the identification of fermions and bosons and fundamental interaction vertices

For 3 light fermion families SU(3) suggests itself as a dynamical symmetry with fermions in fundamental N=3-dimensional representation and N× N=9 bosons in the adjoint representation and singlet representation. The known gauge bosons have same couplings to fermionic families so that they must correspond to the singlet representation. The first challenge is to understand whether it is possible to have dynamical SU(3) at the level of fundamental reaction vertices.

This is a highly non-trivial constraint. For instance, the vertices in which n wormhole throats with same (g₁,g₂) glued along the ends of lines are not consistent with this symmetry. The splitting of the fermionic worm-hole contacts before the proper vertices for throats might however allow the realization of dynamical SU(3). The condition of SU(3) symmetry combined with the requirement that virtual lines resulting also in the splitting of wormhole contacts are always massless, leads to the conclusion that massive fermions correspond to (g,g) type wormhole contacts transforming naturally like SU(3) triplet. This picture conformsl with the identification of free fermions as throats but not with the naive expectation that their topological condensation gives rise to (g,0) wormhole contact.

The argument leading to these conclusions runs as follows.

1. The question is what basic reaction vertices are allowed by dynamical SU(3) symmetry. FFB vertices are in principle all that is needed and they should obey the dynamical symmetry. The meeting of entire wormhole contacts along their ends is certainly not possible. The splitting of fermionic wormhole contacts before the vertices might be however consistent with SU(3) symmetry. This would give two a pair of 3-vertices at which three wormhole lines meet along partonic 2-surfaces (rather than along 3-D wormhole contacts).

2. Note first that crossing gives all possible reaction vertices of this kind from F(g₁)Fbar(g₂)→ B(g₁,g₂) annihilation vertex, which is relatively easy to visualize. In this reaction F(g₁) and Fbar(g₂) wormhole contacts split first. If one requires that all wormhole throats involved are massless, the two wormhole throats resulting in splitting and carrying no fermion number must carry light-like momentum so that they cannot just disappear. The ends of the wormhole throats of the boson must glued together with the end of the fermionic wormhole throat and its companion generated in the splitting of the wormhole. This means that fermionic wormhole first splits and the resulting throats meet at the partonic 2-surface.

   This requires that topologically condensed fermions correspond to (g,g) pairs rather than (g,0) pairs. The reaction mechanism allows the interpretation of (g,g) pairs as a triplet of dynamical SU(3). The fundamental vertices would be just the splitting of wormhole contact and 3-vertices for throats since SU(3) symmetry would exclude more complex reaction vertices such as n-boson vertices corresponding the gluing of n wormhole contact lines along their 3-dimensional ends. The couplings of singlet representation for bosons would have same coupling to all fermion families so that the basic experimental constraint would be satisfied.

3. Both fermions and bosons cannot correspond to octet and singlet of SU(3). In this case reaction vertices should correspond algebraically to the multiplication of matrix elements e_ij: e_ij e_kl = δ_jk e_il allowing for instance F(g₁,g₂) +Fbar(g₂,g₃)→ B(g₁,g₃) . Neither the fusion of entire wormhole contacts along their ends nor the splitting of wormhole throats before the fusion of partonic 2-surfaces allows this kind of vertices so that BFF vertex is the only possible one. Also the construction of QFT limit starting from bosonic emergence led to the formulation of perturbation theory in terms of Dirac action allowing only BFF vertex as fundamental vertex.

4. Weak electric-magnetic duality brings in an additional complication. SU(3) symmetry poses also now strong constraints and it would seem that the reactions must involve copies of basic BFF vertices for the pairs of ends of weak strings. The string ends with the same Kähler magnetic charge should meet at the vertex and give rise to BFF vertices. For instance, FFbarB annihilation vertex would in this manner give rise to the analog of stringy diagram in which strings join along ends since two string ends disappear in the process.

If one accepts this picture the remaining question is why the number of genera is just three. Could this relate to the fact that g≤ 2 Riemann surfaces are always hyper-elliptic (have global Z₂ conformal symmetry) unlike g>2 surfaces? Why the complete bosonic de-localization of the light families should be restricted inside the hyper-elliptic sector? Does the Z₂ conformal symmetry make these states light and make possible delocalization and dynamical SU(3) symmetry? Could it be that for g>2 elementary particle vacuum functionals vanish for hyper-elliptic surfaces? If this the case and if the time evolution for partonic 2-surfaces changing g commutes with Z₂ symmetry then the vacuum functionals localized to g≤ 2 surfaces do not disperse to g>2 sectors.

These and many other questions are discussed in the chapters of p-Adic length scale hypothesis and dark matter hierarchy, in particular in the chapter Elementary Particle Vacuum Functionals.

By the way, I have performed and updating of several books about TGD in order to achieve a more coherent representation. I have also added three new chapters to the book Topological Geometrodynamics: an Overview discussing TGD from particle physics perspective (see this, this, and this).

Also the chapters of p-Adic length scale hypothesis and dark matter hierarchy are heavily updated.