Grassmannian duality and the particle spectrum

Schemes based on anticommuting scalar coordinates, corresponding to properties, lead to generations of particles naturally. The application of Grassmannian duality cuts down the number of states substantially and is vital for constructing sensible Lagrangians anyhow. We apply duality to all of the subgroups within the classification group $\mathrm{\text{SU}}(3)\times \mathrm{\text{SU}}{(2)}_L\times \mathrm{\text{SU}}{(2)}_R$, which encompasses the standard model gauge group, and thereby determine the full state inventory; this includes the definite prediction of quarks with charge $-4/3$ and other exotic states. Assuming universal gravitational coupling to the gauge fields and parity even property curvature, we also obtain $4{sin}^2{𝜃}_w=1-2\alpha /3{\alpha }_s$ which is not far from the experimental value around the $M_Z$ mass.