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 compactification of some of dimensions of space [E. Witten, Nucl. Phys. B 186 (1981) 412; E. Witten, Princeton Technical Rep. PRINT-83-1056, October 1983]. We assume a M(1+3)× 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.