Constant gauge fields and symmetry breaking on torus

This paper examines symmetry breaking in an SU(2) gauge theory with a massless fermion defined on space- time manifold R1,d-d x T2. Vacuum gauge field configuration is taken to have a constant field strength on the torus. Effective Potential in this background is evaluated and compared with that of pure gauge vacuum configuration. It is found that, in d ≡ 5,7 (mod 8), vacuum with sufficiently large color magnetic flux on the torus could have lower energy density. This result applies to a U(1) theory as well. For SU(2) theory, the symmetry is broken to U(1)