Monopoles, duality, and large charge in Chern-Simons QED$_3$
We compute the scaling dimensions of charge $q$ monopole operators in QED$_3$ with $N$ two-component complex fermions and Chern-Simons level $k$, to subleading order in the limit where $k,N$ are large and $k/N$ is fixed. We use this and previous results for scalar QED$_3$ (sQED$_3$) to check the duality between QED$_3$ with $N=1,k=-3/2$ and sQED$_3$ with $N=1,k=2$, whose $U(1)$ monopole symmetry is enhanced to $SO(3)$. We find that the lowest charge monopole value for QED$_3$ is close to 2 as expected for the emergent $SO(3)$ current, the second lowest charge monopole matches a prediction from a fuzzy sphere calculation, while higher $q$ monopoles match the corresponding sQED$_3$ values with relative error of just one percent. We also find similar evidence for dualities between QED$_3$ with $N=1,k=-(2m+1)/2$ and one scalar coupled to two $U(1)$ gauge fields, which has an effective description as sQED$_3$ with $N=1,k=\frac{m+1}{m}$ and $q_s=mq_f$ for $m>1$. Finally, by fitting many values of $q$ we show that the $q^0$ term at large $q$ for QED$_3$ takes the same nonzero value that appears in the universal EFT for parity preserving $U(1)$ theories, even though parity is broken for $k\neq0$. For sQED$_3$, a similar fit gives zero $q^0$ term for $k\neq0$, unlike the universal nonzero value previously observed for $k=0$.