Topology of Electroweak QDMs

Quantum dynamical manifolds (QDMs) are solutions of the quantum dynamical manifold equations (QDMEs) describing mass-spacetimes having specified internal color, gauge, and flavor symmetry. The electron momentum-space manifold ($k$-space representation of the color Lie algebra $su(2)$ QDM) is topologically orientable, being topologically equivalent to an $S^{2}$-sphere, and the photon $k$-space manifolds are not orientable being equivalent to a Klein bottle, $K^{2}$. A \textquotedblleft new\textquotedblright\ kind of particle having non vanishing mass-parameter is found. As this parameter vanishes it represents the Dirac neutrino. Because of the dimension of the color algebra is three, when including many-body spacetime effects, there are \textit{exactly} three leptons and lepton neutrinos. By examining the topology of the new neutrino solutions in $k$-space, an argument for the existence of only left-handed neutrinos is found. These neutrino manifolds are topologically equivalent to the 2D projective space, $\mathbb{R}$P$^{2}$. Tentative vector boson $(W^{+},Z^{0}(% \bar{Z}^{0}),W^{-})$ solutions to 3D $su(2)$ representation color algebra symmetric, 3D $SU(2)$ representation flavor group symmetric QDMEs contain the $T^{2}$ torus manifold. Together the electrons ($S^{2}$), neutrinos ($% \mathbb{R}$P$^{2}: S^{2}\#\mathbb{R}$P$^{2})$, photons ($ % K^{2}: S^{2}\#\mathbb{R}$P$^{2}\#\mathbb{R}$P$^{2})$ and the vector bosons ($T^{2}$) form a topological semigroup ($S^{2}$, $T^{2}$% , $\mathbb{R}$P$^{2}$, $\#$) under the topological connected sum ($% \#$). Thus $k$-space representations of electroweak particles can be joined describing interacting electroweak manifolds.