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Monopole operators from the $4-ε$ expansion

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 $4-ε$ expansion by generalizing them to codimension-3 defect operators in $d = 4-ε$ 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 $S^2 \times \mathbb{H}^{2-ε}$ in the presence of magnetic flux through the $S^2$, and calculate this quantity to next-to-leading order in $ε$. Extrapolating the conformal weight to $ε= 1$ 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 $ε$ expansions in their overlapping regime of validity.