A comment about topological explanation of family replication phenomenon

The topological explanation of family replication phenomenon of fermions in terms of the genus g defined as the number of handles added to sphere to obtain the quantum number carrying partonic 2-surface distinguishes TGD from GUTs and string models. The orbit of the partonic 2-surface defines 3-D light-like orbit identified as wormhole throat at which the induced metric changes its signature. The original model of elementary particle involved only single boundary component replaced later by a wormhole throat. The generalization to the recent situation in which elementary particles correspond to wormhole flux tubes of length of order weak length scales with pairs of wormhole throats at its ends is straight-forward.

The basic objection against the proposal is that it predicts infinite number of particle families unless the g< 3 topologies are preferred for some reason. Conformal and modular symmetries are basic symmetries of the theory and global conformal symmetries provide an excellent candidate for the sought for reason why.

1. For g<3 the 2-surfaces are always hyper-elliptic which means that they have have always Z₂ as global conformal symmetries. For g>2 these symmetries are absent in the generic case. Moreover, the modular invariant elementary particle vacuum functionals vanish for hyper-elliptic surfaces for g>2. This leaves several options to consider. The basic idea is however that ground states are usually highly symmetric and that elementary particles correspond to ground states.

2. The simplest guess is that g>2 surfaces correspond to very massive states decaying rapidly to states with smaller genus. Due to the the conformal symmetry g<3 surfaces would be analogous to ground states and would have small masses.

3. The possibility to have partonic 2-surfaces of macroscopic and even astrophysical size identifiable as seats of anyonic macroscopic quantum phases (see this) suggests an alternative interpretation consistent with global conformal symmetries. For partonic 2-surfaces of macroscopic size it seems natural to consider handles as particles glued to a much larger partonic 2-surface by topological sum operation (topological condensation).

   All orientble manifolds can be obtained by topological sum operation from what can be called prime manifolds. In 2-D orientable case prime manifolds are sphere and torus representing in well-defined sense 0 and 1 so that topological sum corresponds to addition of positive integers arithmetically. This would suggest that only sphere and torus appear as single particle states. Particle interpretation however requires that also g=0 and g=2 surfaces topologically condensed to a larger anyonic 2-surface have similar interpretation, at least if they have small enough size. What kind of argument could justify this kind of interpretation?

4. An argument based on symmetries suggests itself. The reduction of degrees of freedom is the generic signature of bound state. Bound state property implies also the reduction of approximate single particle symmetries to an exact overall symmetry. Rotational symmetries of hydrogen atom represent a good example of this. For free many particle states each particle transforms according to a representation of rotation group having total angular momentum defined as sum of its spin and angular momentum. For bound states rotational degrees of freedom are strongly correlated and only overall rotations of the state define rotational symmetries.

   In this spirit one could interpret sphere as vacuum, torus as single handle state, and torus with handle as a bound state of 2 handles in conformal degrees of freedom meaning that the Z₂ symmetries of vacuum and handles are frozen in topological condensation (topological sum) to single overall Z₂. If this interpretation is correct, g>2 2-surfaces would always have a decomposition to many-particle states consisting of spheres, tori and tori with single handle glued to a larger sphere by topological sum. Each of these topologically condensed composites would possess Z₂ as approximate single particle symmetry.

For more details see the chapter Elementary Particle Vacuum Functionals of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".