{\,,}\label{eq:J4free}\end{aligned}$$ where parentheses around indices denote an averaging over all permutations of the indices. When discussing the large $N$ limit we shall call such scalar bilinears “single-trace” operators. Since they saturate the unitarity bound, these primaries are also conserved currents, $\partial_\mu J^{\mu}_{\;\;\mu_1\dots\mu_{s-1}} = 0$, and therefore the free theory has an infinite number of conserved currents. In addition there is a scalar singlet operator $J_0 = \frac{1}{\sqrt{N}} \phi^i \phi^i$, also a primary, with dimension $\Delta=1$. In the large $N$ limit, all operators in the theory are products of these basic “single-trace” operators, or descendants of such products. Note that adding the Chern-Simons sector does not add any additional non-trivial local operators.

Let us now turn on the Chern-Simons coupling $\lambda$. As we showed in section \[allorders\], the theory is still conformally-invariant at infinite $N$. The currents of the free theory, as written above, are not gauge-invariant, but they can be made gauge-invariant by promoting derivatives to covariant derivatives and projecting onto the symmetric traceless part. The promoted currents, which will also be denoted $J_s$, are the “single-trace” primary operators of the new theory. At finite $N$ they are generally not conserved, and they also mix with “multi-trace operators”; however, as 