Compactification of gauge theories and the gauge invariance of massive modes

We study the gauge invariance of the massive modes in the compactification of gauge theories from D = 5 to D = 4. We deal with Abelian gauge theories of rank one and two, and with non-Abelian ones of rank one. We show that Stückelberg fields naturally appear in the compactification mechanism, contrarily to what usually occurs in literature where they are introduced by hand, as a trick, to render gauge invariance for massive theories. We also show that in the non-Abelian case they appear in a very different way when compared with their usual implementation in the non-Abelian Proca model