Monopole operators from the $4-ε$ expansion

TL;DR

This work introduces monopole operators in the $4-\epsilon$ expansion by promoting them to codimension-3 defects in $d=4-\epsilon$ and defining their conformal weight via the free energy density on $S^2\times\mathbb{H}^{2-\epsilon}$ in a magnetic background. It provides a controlled calculation of this conformal weight to next-to-leading order in $\epsilon$, derives the fixed-point gauge coupling, and yields an explicit leading $1/\epsilon$ term plus $O(1)$ corrections for the monopole free energy, while demonstrating consistency with the large-$N$ expansion in overlapping regimes. The results bridge the $4-\epsilon$ and large-$N$ approaches, reveal the large-$q$ behavior, and suggest a route to estimating 3d monopole scaling dimensions without relying on $1/N$ methods, potentially enabling Padé-type resummations for the 3d theory. Overall, the paper lays a robust framework for analyzing nonlocal disorder operators in 3d gauge theories using higher-dimensional conformal perturbation theory and cross-checks against large-$N$ techniques.

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 $4-ε$ 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.

Figures (3)

• Figure 1: The vacuum Feynman diagram $\mathbb{D}^{(q)}$ that gives the logarithmic contribution \ref{['Diag3Main']} to the free energy.
• Figure 2: This plot shows the difference between the large $q$ approximation \ref{['largqFinalFerm']} and the arbitrary $q$ numerics (with the exact $q=0$ part subtracted).
• Figure 3: Large $N$ finite $\epsilon$ functional determinant for $q=1/2,1,3/2,2,5/2$.