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 S²× ℍ ^{2 − ϵ} in the presence of magnetic flux through the S², 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.