Properties of families of spinors in $d=(5+1)$ with zweibein of an   almost $S^2$ and two kinds of spin connection fields, allowing massless and   massive solutions in $d=(3+1)$
  We studied properties of spinors in a toy model in $d=(5+1)$, when ${\cal M}^{(5+1)}$ breaks to an infinite disc with a zweibein which makes a disc curved on an almost $S^2$ and with a spin connection field which allows on such a sphere only one massless spinor state, as a step towards realistic Kaluza-Klein theories in non compact spaces. Previously we allowed on $S^2$ two kinds of the spin connection fields, those which are gauge fields of spins in and those which are the gauge fields of the family quantum numbers, both as required for this toy model by the spin-charge-family theory. This time we study, by taking into account families of spinors interacting with several spin connection fields, properties of massless and massive solutions of equations of motion, with the discrete symmetries ($\mathbb{C}_{{ \cal N}}$, ${\cal P}_{{\cal N}}$, ${\cal T}_{{ \cal N}}$) included. We also allow nonzero vacuum expectation values of the spin connection fields and study the masses. 
D. Lukman, N.S. Mankoc Borstnik
hep-th