Monopole Operators in $U(1)$ Chern-Simons-Matter Theories

TL;DR

This work analyzes monopole operators in U(1) Chern-Simons-matter theories at IR fixed points by evaluating the S^2×S^1 partition function in flux sectors with large N and k, keeping κ=k/N fixed. The authors develop a systematic large-N expansion F_q = N F_q^{(0)} + F_q^{(1)} + ..., extracting leading scaling dimensions Δ_q^{(0)} and entropies S_q^{(0)}, and they provide a microstate interpretation via a Landau-level dressing picture that explains degeneracies and their lifting at subleading order. They apply this framework across non-supersymmetric QED$_3$, scalar QED$_3$, and supersymmetric SQED$_3$ theories with ${ m N}=1,2$, finding near-degeneracies at leading order and proposing a universal pattern for energy splittings ∝ ℓ^2/N for spins ℓ up to ∼√N; in the ${ m N}=4$, k=0 bootstrap they obtain evidence that these degeneracies persist to small N. The work also shows that in ${ m N}=2$ SQED$_3$ the lowest monopole is generically non-BPS, revealing richer IR spectra and providing tests for dualities and bootstrap approaches in Abelian gauge theories.

Abstract

We study monopole operators at the infrared fixed points of $U(1)$ Chern-Simons-matter theories (QED$_3$, scalar QED$_3$, ${\cal N} =1$ SQED$_3$, and ${\cal N} = 2$ SQED$_3$) with $N$ matter flavors and Chern-Simons level $k$. We work in the limit where both $N$ and $k$ are taken to be large with $κ= k/N$ fixed. In this limit, we extract information about the low-lying spectrum of monopole operators from evaluating the $S^2 \times S^1$ partition function in the sector where the $S^2$ is threaded by magnetic flux $4 πq$. At leading order in $N$, we find a large number of monopole operators with equal scaling dimensions and a wide range of spins and flavor symmetry irreducible representations. In two simple cases, we deduce how the degeneracy in the scaling dimensions is broken by the $1/N$ corrections. For QED$_3$ at $κ=0$, we provide conformal bootstrap evidence that this near-degeneracy is in fact maintained to small values of $N$. For ${\cal N} = 2$ SQED$_3$, we find that the lowest dimension monopole operator is generically non-BPS.

Figures (9)

• Figure 1: The Landau level $\tilde{j}$ and the filling fraction $\xi_{\tilde{j}}$ for $q=1/2$ as a function of $\kappa=k/N$.
• Figure 2: The leading order coefficients $\Delta_q^{(0)}$ and $S_q^{(0)}$ of the scaling dimension and entropy of the lowest-dimension monopole operators in QED$_3$, as a function of $\kappa\equiv k/N$. The scaling dimension coefficient is plotted for $1/2 \leq q \leq 2$, while the entropy coefficient $S_q^{(0)}$ is plotted only for $q=1/2$ in order to avoid clutter.
• Figure 3: Top: The leading order coefficients $\Delta_q^{(0)}$ and $S_q^{(0)}$ of the scaling dimension and entropy of the lowest-dimension monopole operators $1/2\leq q \leq 2$ in scalar QED$_3$, as a function of $\kappa\equiv k/N$. The exact value $\Delta_{1/2}^{(0)}=1$ at $|\kappa|=1$ is explained in \ref{['Lucasmagic']}. Bottom: The saddle point value of the Lagrange multiplier $\mu$ as a function of $\kappa$. Note that $\mu(\kappa)$ is not exactly linear.
• Figure 4: The leading order in $1/N$ monopole scaling dimension $\Delta_{1/2}=N\Delta_{1/2}^{(0)}+O(N^0)$ for $1/2 \leq q \leq 2$ as a function of $\kappa\equiv k/N$.
• Figure 5: Top: Leading order scaling dimension $\Delta_{1/2}^{(0)}$ of lowest lying $q=1/2$ monopole operator as a function of $\kappa\equiv k/N$ and $\delta n\equiv\frac{N^+-N^-}{N^++N^-}$. Thick dashed lines separate different phases described in the table on the right: the transitions I to IX, II to X, V to XI and VI to XII are smooth in the value of $\sigma$ and $D$, with $D$ changing sign, while all other transitions are discontinuous. For convenience, $\sigma$ and $D$ are shown in Appendix \ref{['SQEDN2SigmaAndD']}, in Figure \ref{['fig:univ-bounds-sigmaAndD']}, for several values of $\delta n$ and $\kappa$. Dashed lines with large dashing separate the regions in which the bare monopole is dressed with fermionic ($-1/2<\kappa<1/2$) or scalar modes. Along the (anti-)diagonal line the lowest lying monopole is (anti-)BPS and is shown in (blue) green. Center: Horizontal bisection of the density plot along the line $\delta n=0$ (left) and $\delta n =1/2$ (right) showing the scaling dimension $\Delta_{1/2}^{(0)}$ as a function of $\kappa$. The (blue) green line shows the (anti-)BPS monopole operator dimension \ref{['localBPSb']} and the orange curves show the dimension of the lowest monopole operator. Bottom: Vertical bisection of the density plot for $\kappa = 0$ (left) and $1/4$ (right) showing the scaling dimension $\Delta_{1/2}^{(0)}$ as a function of $\delta n$. The color coding is the same as in the center.