An example of Kaluza-Klein-like theory with boundary conditions, which   lead to massless and mass protected spinors chirally coupled to gauge fields
  The genuine Kaluza-Klein-like theories (with no fields in addition to gravity) have difficulties with the existence of massless spinors after the ompactification of some of dimensions of space\cite{witten}. We assume a $M^{(1+3)} \times$ a flat finite disk in $(1+5)$-dimensional space, with the boundary allowing spinors of only one handedness. Massless spinors then chirally couple to the corresponding background gauge gravitational field, which solves equations of motion for a free field, linear in the Riemann curvature. 
N. Mankoc Borstnik, H. B. Nielsen