can you please rephrase this section: "As in other phase transitions, the deconfined vacuum at weak coupling could join con- tinuously to the confined vacuum at strong coupling, or there could be a jump between them at some critical value of the coupling. We begin by analyzing the continuous case, by which we mean that for every value of ΛL there is an AdS-invariant vacuum state, that interpolates between the deconfined state at weak coupling and the confined state at strong coupling. There are two scenarios in which we can imagine this happening, but both are somewhat problematic. The first continuous scenario involves the Higgs mechanism in AdS4 . As we explained above, at weak coupling the mass of the single gluon states – corresponding to the conserved current on the boundary – cannot vary continuously from E = 2/L. This is because the gauge bosons sit in a short multiplet which is protected as long as the gauge symmetry survives. This, of course, is a familiar story in flat space too. In both flat space and in AdS, gauge bosons can only smoothly achieve a mass through the Higgs mechanism, eating a massless scalar field which breathes life into the longitudinal gauge mode. In order to invoke such a mechanism to explain the confinement transition, we need that as g2 increases, a charged scalar state must become massless (with energy E = 3/L) and condense, providing a mass to the gluons. At weak coupling there is no scalar state with E < 4/L, so clearly this can only happen once the coupling becomes of order 