Monopole Taxonomy in Three-Dimensional Conformal Field Theories

TL;DR

This work computes the spectrum and quantum numbers of monopole operators in non-supersymmetric 3D gauge theories at large $N_f$ by mapping monopole backgrounds to states on $S^2\times\mathbb{R}$ via the state-operator correspondence. The authors derive stability criteria for GNO backgrounds, compute the leading and subleading contributions to the monopole operator dimensions from fermion, ghost, and gauge-field determinants, and reveal that stable monopoles can occur in multiple backgrounds per topological class. They determine the $SU(N_f)$ flavor representations of bare monopole operators in $U(N_c)$ QCD with fundamental fermions (rectangular Young diagrams with $N_f/2$ rows and $2\sum_a|q_a|$ columns) and provide detailed results for the QED case $(N_c=1)$ as well as various non-Abelian groups. The findings shed light on confinement and chiral symmetry breaking in 3D gauge theories and have implications for algebraic spin liquids and related dualities. Overall, the paper furnishes a systematic large-$N_f$ framework to classify monopole operators and their quantum numbers across a broad class of gauge theories.

Abstract

We study monopole operators at the infrared fixed points of Abelian and non-Abelian gauge theories with N_f fermion flavors in three dimensions. At large N_f, independent monopole operators can be defined via the state-operator correspondence only for stable monopole backgrounds. In Abelian theories, every monopole background is stable. In the non-Abelian case, we find that many (but not all) backgrounds are stable in each topological class. We calculate the infrared scaling dimensions of the corresponding operators through next-to-leading order in 1/N_f. In the case of U(N_c) QCD with N_f fundamental fermions (and in particular in the QED case, N_c =1), we find that the monopole operators transform as non-trivial irreducible representations of the SU(N_f) flavor symmetry group.

Figures (19)

• Figure 1: We plot the terms in the infinite sum over $j$\ref{['relnice']} that give the matrix element $\left[{\bf K}^J_{q q'}(\Omega)\right]_{UU}$ for $q=-1,\, q'=1/2,\, \Omega=1,$ and $J=35/2$. We show the stage of the calculation where all the finite sums (over $\delta q,\, l',\, l$, and $j'$) in \ref{['relnice']} have been done and only the infinite sum over $j$ remains. The dots represent the actual terms in the sum, while the solid line is the asymptotic expansion of the summand to ${\cal O}(1/j^{18})$ that we determined analytically. We perform the infinite sum by zeta-function regularization of the asymptotic form for $j>j_c$, where $j_c$ is the value below which we use the numerical values of the terms in the sum. We check the numerical precision by changing $j_c$ and we reach our goal of $10^{-12}$ precision by choosing $j_c\approx 40$. This precision is needed to get the free energy with $10^{-3}$ precision.
• Figure 2: The eigenvalues of $\textbf{K}_{qq'}^J(\Omega)$ for some example $q,\, q'$ and low $J$ values as a function of $\Omega$. Zero eigenvalues corresponding to pure gauge modes are omitted. Note that the eigenvalues are monotonic in $J$ and $\Omega$, hence it suffices to examine the $\Omega=0$ behavior of the lowest $J$ mode for stability. Also note that in both examples $\left|Q\right|\geq1$ and the two lowest lying $J$ modes have one non-zero eigenvalue, while higher $J$ modes come with two eigenvalues. (The smaller number of eigenvalues corresponds to the reduced size of the matrix $\textbf{K}_{qq'}^J(\Omega)$.)
• Figure 3: We plot the ratio of the non-zero eigenvalues $\lambda_\text{gauge}^J(\Omega)$ of the gauge kernel divided by their asymptotic behavior $\lambda_\text{asymp}^J(\Omega)$. We chose $q=-1,\, q'=1/2$ for this example. Because $|Q|=3/2$ the $J=1/2,\, 3/2$ modes contribute one eigenvalue, while for higher $J$ eigenvalues come in pairs. We used the same colors to plot the pair of eigenvalues for these higher $J$ modes. Because the ghosts give a contribution proportional to $\lambda_\text{asymp}^J(\Omega)$ this plot shows that the low energy modes are the most important in determining the free energy.
• Figure 4: We plot the subleading term in the free energy, $\delta F(q,q')$ for $q=-1,\, q'=1/2$ as a function of the cutoff $\Lambda$. We extrapolate to $1/\Lambda\to 0$ by fitting the data points by a second order polynomial. Our results are reliable to $10^{-3}$ precision.
• Figure 5: The lowest eigenvalue $\lambda = {\bf K}^{\left\lvert Q_{ab} \right\rvert - 1}_{q_a q_b}(0)$ of the $a_\mu^{ab}$ component of the gauge field fluctuations around the GNO monopole background \ref{['MonopoleGeneral']}. We have marked explicitly the plane $z = 0$. The region where this eigenvalue dips below zero corresponds to an instability of $a_\mu^{ab}$. If this eigenvalue is positive, then the action for $a_\mu^{ab}$ is positive-definite.