Monopoles in 2+1-dimensional conformal field theories with global U(1) symmetry

TL;DR

This work studies monopole insertions ${\cal M}_q$ in 2+1D CFTs with a global $U(1)$ symmetry by coupling the theory to a background gauge field and employing the state-operator correspondence to extract scaling dimensions $\Delta_q$. The authors develop a systematic large-$N_b$ expansion for the Wilson-Fisher fixed point of $N_b$ complex scalars, deriving the leading term $\mathcal{F}_q^{\infty}$ and the next-to-leading correction $\delta\mathcal{F}_q$ from fluctuations of the Lagrange multiplier, encoded in a kernel $D^q_\ell(\omega)$. They compute $\mathcal{F}_q^{\infty}$ by saddle-point analysis and provide numerical evaluations of $\delta\mathcal{F}_q$ using a careful regularization strategy, yielding concrete predictions for $\Delta_q$ up to $q=5$ (e.g., $\Delta_1 = 0.125\,N_b - 0.057 + O(1/N_b)$). The results offer new critical exponents for the Wilson-Fisher CFT and connect with known large-$N$ results from the CP$^{N_b-1}$ model, with potential checks via Monte Carlo or series expansions.

Abstract

In 2+1-dimensional conformal field theories with a global U(1) symmetry, monopoles can be introduced through a background gauge field that couples to the U(1) conserved current. We use the state-operator correspondence to calculate scaling dimensions of such monopoles. We obtain the next-to-leading term in the 1/N_b expansion of the Wilson-Fisher fixed point in the theory of N_b complex bosons.

Figures (1)

• Figure 1: The coefficient $\delta {\cal F}_1$ evaluated numerically from \ref{['FqFinalReg']} using the relativistic cutoff \ref{['Relativistic']} as a function of the inverse cutoff scale $1/L$. The solid line is a quadratic fit from which we extract the value $\delta {\cal F}_1 \approx -0.057$ as we take $L \to \infty$.