Elementary particle vacuum functionals for dark matter and why fermions can have only three families

One of the open questions is how dark matter hierarchy reflects itself in the properties of the elementary particles. The basic questions are how the quantum phase q=ep(2iπ/n) makes itself visible in the solution spectrum of the modified Dirac operator D and how elementary particle vacuum functionals depend on q. Considerable understanding of these questions emerged recently. One can generalize modular invariance to fractional modular invariance for Riemann surfaces possessing Z_n symmetry and perform a similar generalization for theta functions and elementary particle vacuum functionals.

In particular, without any further assumptions n=2 dark fermions have only three families. The existence of space-time correlate for fermionic 2-valuedness suggests that fermions quite generally correspond to even values of n, so that this result would hold quite generally. Elementary bosons (actually exotic particles) would correspond to n=1, and more generally odd values of n, and could have also higher families.

For more details see the chapter Construction of Elementary Particle Vacuum Functionals .