Chiral Compactification on a Square

We study quantum field theory in six dimensions with two of them compactifiedon a square. A simple boundary condition is the identification of two pairs ofadjacent sides of the square such that the values of a field at two identifiedpoints differ by an arbitrary phase. This allows a chiral fermion content forthe four-dimensional theory obtained after integrating over the square. We findthat nontrivial solutions for the field equations exist only when the phase isa multiple of \pi/2, so that this compactification turns out to be equivalentto a T^2/Z_4 orbifold associated with toroidal boundary conditions that areeither periodic or anti-periodic. The equality of the Lagrangian densities atthe identified points in conjunction with six-dimensional Lorentz invarianceleads to an exact Z_8\times Z_2 symmetry, where the Z_2 parity ensures thestability of the lightest Kaluza-Klein particle.Comment: 28 pages, latex. References added. Clarifying remarks included in section 2. Minor corrections made in section