Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories

It has been conjectured that Chern-Simons (CS) gauged `regular' bosons in the fundamental representation are `level-rank' dual to CS gauged critical fermions also in the fundamental representation. Generic relevant deformations of these conformal field theories lead to one of two distinct massive phases. In previous work, the large $N$ thermal free energy for the bosonic theory in the unHiggsed phase has been demonstrated to match the corresponding fermionic results under duality. In this note we evaluate the large $N$ thermal free energy of the bosonic theory in the Higgsed phase and demonstrate that our results, again, perfectly match the predictions of duality. Our computation is performed in a unitary gauge by integrating out the physical excitations of the theory - i.e. W bosons - at all orders in the 't Hooft coupling. Our results allow us to construct an exact quantum effective potential for ${\bar \phi} \phi$, the lightest gauge invariant scalar operator in the theory. In the zero temperature limit this exact Landau-Ginzburg potential is non-analytic at ${\bar \phi \phi}=0$. The extrema of this effective potential at positive ${\bar \phi}\phi$ solve the gap equations in the Higgsed phase while the extrema at negative ${\bar \phi} \phi$ solve the gap equations in the unHiggsed phase. Our effective potential is bounded from below only for a certain range of $x_6$ (the parameter that governs sextic interactions of $\phi$). This observation suggests that the regular boson theory has a stable vacuum only when $x_6$ lies in this range.