Monopole Operators in $U(1)$ Chern-Simons-Matter Theories

We study monopole operators at the infrared fixed points of $U(1)$ Chern-Simons-matter theories (QED$_3$, scalar QED$_3$, ${\cal N} =1$ SQED$_3$, and ${\cal N} = 2$ SQED$_3$) with $N$ matter flavors and Chern-Simons level $k$. We work in the limit where both $N$ and $k$ are taken to be large with $\kappa = k/N$ fixed. In this limit, we extract information about the low-lying spectrum of monopole operators from evaluating the $S^2 \times S^1$ partition function in the sector where the $S^2$ is threaded by magnetic flux $4 \pi q$. At leading order in $N$, we find a large number of monopole operators with equal scaling dimensions and a wide range of spins and flavor symmetry irreducible representations. In two simple cases, we deduce how the degeneracy in the scaling dimensions is broken by the $1/N$ corrections. For QED$_3$ at $\kappa=0$, we provide conformal bootstrap evidence that this near-degeneracy is in fact maintained to small values of $N$. For ${\cal N} = 2$ SQED$_3$, we find that the lowest dimension monopole operator is generically non-BPS.