  It has been conjectured that fermions minimally coupled to a Chern-Simons
gauge field define a conformal field theory (CFT) that is level-rank dual to
Chern-Simons gauged Wilson-Fisher Bosons. The CFTs in question admit relevant
deformations parametrized by a real mass. When the mass deformation is
positive, the duality of the two deformed theories has previously been checked
in detail in the large $N$ limit by comparing explicit all orders results on
both sides of the duality. In this paper we perform a similar check for the
case of negative mass deformations. In this case the bosonic field condenses
triggering the Higgs mechanism. The effective excitations in this phase are
massive $W$ bosons. By summing all leading large $N$ graphs involving these $W$
bosons we find an all orders (in the 't Hooft coupling) result for the thermal
free energy of the bosonic theory in the condensed phase. Our final answer
perfectly matches the previously obtained fermionic free energy under the
conjectured duality map.
