Monopole operators from the $4-\epsilon$ expansion

TL;DR

This paper studies monopole operators in 3D quantum electrodynamics using the $4-psilon$ expansion, defining their conformal weight via free energy on curved space and comparing results with large N expansion.

Contribution

It generalizes monopole operators to $d=4-psilon$ dimensions and computes their conformal weights to estimate their scaling dimensions in 3D.

Findings

Conformal weight computed to next-to-leading order in psilon

Agreement found between psilon and large N expansions

Provides estimates of monopole scaling dimensions in 3D

Abstract

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.