Six-dimensional gauge theory on the chiral square

We construct gauge theories in two extra dimensions compactified on the chiral square, which is a simple compactification that leads to chiral fermions in four dimensions. Stationarity of the action on the boundary specifies the boundary conditions for gauge fields. Any six-dimensional gauge field decomposed in Kaluza-Klein modes includes a tower of heavy spin-1 particles whose longitudinal polarizations are linear combinations of the extra-dimensional components, and a tower of heavy spin-0 particles corresponding to the orthogonal combinations. These linear combinations depend on the Kaluza-Klein numbers, and are independent of the gauge fixing. If the gauge symmetry is broken by the vacuum expectation value of a six-dimensional scalar, at each Kaluza-Klein level three spinless fields in the adjoint representation mix to provide the longitudinal polarization of the spin-1 mode, leaving the orthogonal states as two spin-0 particles. We derive the interactions of the Kaluza-Klein modes for generic gauge theories, laying the groundwork for the Standard Model in two universal extra dimensions, and more generally for future model building and phenomenological studies.