  Three-dimensional quantum electrodynamics with $N$ charged fermions contains
monopole operators that have been studied perturbatively at large $N$. Here, we
initiate the study of these monopole operators in the $4-\epsilon$ expansion by
generalizing them to codimension-3 defect operators in $d = 4-\epsilon$
spacetime dimensions. Assuming the infrared dynamics is described by an
interacting CFT, we define the "conformal weight" of these operators in terms
of the free energy density on $S^2 \times \mathbb{H}^{2-\epsilon}$ in the
presence of magnetic flux through the $S^2$, and calculate this quantity to
next-to-leading order in $\epsilon$. Extrapolating the conformal weight to
$\epsilon = 1$ gives an estimate of the scaling dimension of the monopole
operators in $d=3$ that does not rely on the $1/N$ expansion. We also perform
the computation of the conformal weight in the large $N$ expansion for any $d$
and find agreement between the large $N$ and the small $\epsilon$ expansions in
their overlapping regime of validity.
