a more detailed weak-weak coupling duality. We will consider both the $O(N)$ case with a real scalar field in the fundamental representation, and the $U(N)$ case with a complex scalar. These theories were previously studied perturbatively in [@Chen:1992ee; @Avdeev:1992jt].

We begin in section \[On\] by introducing our action and our methods of regularizing and renormalizing it. In section \[confSym\] we study whether the theory at small $\lambda$ is still conformally invariant. The Chern-Simons level is quantized and does not run [@Deser:1981wh; @Deser:1982vy]. One problem that may arise whenever we have interactions is that relevant operators of the form $\phi^2$ and $(\phi^2)^2$ (where $\phi$ is the scalar field) may be generated, even if they are tuned to zero at some scale. However, in our renormalization scheme these couplings do not run away if they are initially set to zero. A more serious problem is that this theory has a classically marginal $\lambda_6 (\phi^2)^3 / N^2$ coupling, which could start running once we turn on $\lambda$. However, we provide an argument (and check explicitly at two-loop order) that at infinite $N$ the beta function for this coupling vanishes. Therefore, there is a two-dimensional family of large $N$ conformal field theories, parameterized by 