A remark on higher dimension induced domain wall defects in our world

There has been recent interest in new types of topological defects arising in models with compact extra dimensions. We discuss in this context the old statement that if only SU(N) gauge fields and adjoint matter live in the bulk, and the coupling is weak, then the theory possesses a spontaneously broken global Z(N) symmetry, with associated domain wall defects in four dimensions. We discuss the behaviour of this symmetry at high temperatures. We argue that the symmetry gets restored, so that cosmological domain wall production could be used to constrain such models.