Absence of confinement in the absence of vortices

We consider the Wilson loop expectation in SU(2) lattice gauge theory in the presence of constraints. The constraints eliminate from the functional measure gauge field configurations whose physical interpretation is that of thick center vortices linking with the loop. We give a simple proof that, for dimension $d\ge 3$, the so constrained Wilson loop follows perimeter law, i.e. non-confining behavior, at weak coupling (low temperature). Thus the presence of vortex configurations is a necessary condition for confinement.