Monopole operators in Chern-Simons theories with charged matter
have been studied using the state-operator map in CFTs, as states on

Article funded by SCOAP3

0$ take the form \bea \phi & \ = \ \sum_{j,m} a_{jm} r^{-\beta_j - \frac 12} Y_{qjm} \cr \ol\phi & \ = \ \sum_{j,m} \ol a_{jm} r^{\beta_j-\frac 12} \ol Y_{qjm} \label{SingProfileIntro} \eea with $\beta_j = \frac 12 \sqrt{(2j+1)^2 - q^2}$ and $Y_{qjm}$ are the monopole harmonics with background magnetic charge $q$. The allowed values of the angular momentum are $j = \frac{|q|}{2} + n$, $n \in \bZ_{\ge 0}$. The moduli $|a_{jm}|$ are constrained by Gauss's law, while the phases $e^{i\lambda_{jm}}$, defined by $a_{jm}=|a_{jm}| e^{i\lambda_{jm}}$, are dynamical variables. Removing a ball of radius $\epsilon$ around the insertion point and requiring a boundary term on the $S^2$ boundary that cancels the boundary piece of the bulk field variation, we find that in the $\epsilon \to 0$ limit the monopole insertion must be dressed with a factor $\prod_{j,m} (e^{-i\lambda_{jm}})^{n_{jm}}$, where $n_{jm}$ is a collection of positive integers satisfying $\sum_{jm} n_{jm} =kq$ and in terms of which the $|a_{jm}|$ moduli are fixed (see Equations~\ref{GenSingProfile}),~(\ref{GenGL}),~(\ref{GenDressing}). This dressing term also restores the gauge invariance of the monopole insertion in the Chern-Simons theory. Because the dressing factors $e^{-i\lambda_{jm}}$ have spin $j$, the monopoles transform in non-trivial $\Spin(3)$ representations, which are easily worked out in this simple theory. Our findings mimic to a large extent the construction of monopole states of the theory on $\bR\times S^2$ in Hamiltonian quantization as presented in~\cite{Chester-ml-2017vdh}. We extend our construction to the abelian theory with a charged fermion field in section~\ref{sec:CSftheory}. The only qualitative difference with respect to the scalar field case is that the ``occupation numbers'' $n_{jm}$ of each fermion mode take values $0$ or $1$ only, due to the fermionic statistic. We then carry on to study supersymmetric monopoles in section~\ref{sec:Susy}. The simplest instance arises in supersymmetric $\cN=2$ SQED theory with a single charged chiral multiplet and Chern-Simons level $k$. To define a supersymmetric monopole insertion one starts by requiring a half-BPS monopole singularity~\eqref{BPSmonop}. In this background the singular profiles for the boson and fermion fields are slightly modified (or rather simplified). By studying the BPS conditions we find that $\frac{1}{4}$-BPS monopoles can be constructed if one restricts the singular profiles and dressing factors to certain scalar and fermion modes which obey BPS conditions (see Equation~\eqref{GenProfileSusyFinal}). A schematic summary of the construction of BPS monopoles is given in table~\ref{tab:BPSmonop}. In an infrared CFT, the BPS monopoles belong to short multiplets at threshold $A_1$, or $\ol A_1$, in the language of~\cite{Cordova-ml-2016emh}. We compute the quantum numbers of the BPS monopoles, including their dimension, as a function of the infrared R-charges of the fields. We compare our results with the superconformal index of the abelian SQED theory and find an exact agreement. The analysis in this paper should be understood as performed in a small coupling limit of the theories, which is the large $k$ limit (or the small $g_{YM}$ coupling limit if one uses a Yang-Mills UV regulating term). However most of the qualitative results, such as the monopole operator content and their spin representations, does not change as continuous couplings are turned on, including quadratic and quartic scalar potentials, which might be fine-tuned to reach infrared interacting CFTs.\footnote{The Chern-Simons level is not a continuous coupling, but it is likely that the spectrum of monopoles that we describe is correct at any value of $k$. This is certainly the case for BPS monopoles.} Because they can acquire large anomalous dimensions, it is difficult to assess the fate of non-supersymmetric monopole operators along RG-flows in strongly coupled theories (small $k$). \sloppy{Although we study only simple abelian theories, the construction presented in this note should generalize to other abelian theories and non-abelian theories without \hbox{major~modifications. }} ]]>

\frac 12 \,. \label{beta} \ee We can thus solve the eom $(ii)$ with the scalar profiles \be \phi = \frac{a}{r^{\beta_j + \frac 12}} Y_{qjm} \,, \ee or \be \phi = a r^{\beta_j-\frac 12} Y_{qjm} \,, \ee with $a\in \bC$. However, if we adopt the standard reality condition $\ol \phi = \phi^\ast$, we find that no such profile can solve Gauss's law~\eqref{GaussLaw}. Actually, with $\ol\phi = \phi^\ast$ the left-hand-side of~\eqref{GaussLaw} is imaginary, whereas we need it real. Instead, if we think of $\phi$ and $\ol\phi$ as independent fields of charge $1$ and $-1$ respectively, with the equations of motions $D^2\phi = 0$, $D^2\ol\phi = 0$, we have the solutions \be \phi = \frac{a}{r^{\beta_j+\frac 12}} Y_{qjm} \,, \quad \ol\phi = \bar a r^{\beta_j-\frac 12} \ol Y_{qjm} \,, \ee with \be \int_{S^2} \star(\ol\phi D\phi - \phi D\ol\phi) = -2\beta_j a \ol a \,. \ee Thus Gauss's law is satisfied with \be a \ol a = \frac{kq}{2\beta_j} \,. \ee While we do satisfy Gauss's law, we remark that the solutions do not satisfy the local equations $(i)$ in~\eqref{CSbEOM}. Trying to impose a solution to $(i)$ seems a too strong requirement and we find that imposing only Gauss's law, which is the integrated equation, will fit our purposes.\footnote{See also section 3.1.4 in~\cite{Chester-ml-2017vdh} for a comment and a possible explanation on this point.} We conclude that, assuming $kq>0$, one should impose the profiles at the origin \bea \phi &= \frac{e^{i\lambda}}{r^{\beta_j+\frac 12}}\sqrt{\frac{kq}{2\beta_j}} Y_{qjm} + \text{sub} \,, \cr \ol\phi &= e^{-i\lambda}r^{\beta_j-\frac 12} \sqrt{\frac{kq}{2\beta_j}} \ol Y_{qjm} + \text{sub} \,, \label{SingProfile} \eea where the phase $e^{i\lambda}$, $\lambda \sim \lambda + 2\pi$, is a fluctuating field and ``sub'' denotes subleading terms in small $r$. The angle $\lambda$ cannot be chosen as a fixed background because it transforms under gauge transformations: \be A \to A + \dd\Lambda \,, \quad \lambda \to \lambda + \Lambda(0) \,, \label{lambdatransfo} \ee where $\Lambda(0)$ is the evaluation at the origin of the gauge parameter. If $kq <0$, one has to exchange the roles of $\phi$ and $\ol\phi$, i.e.\ changing $\beta_j \to -\beta_j$ in the profiles~\eqref{SingProfile}. The surprising feature of the profiles~\eqref{SingProfile} is that the leading behavior of $\ol\phi$ is related to that of $\phi$ by an unusual reality condition $\ol\phi = r^{2\beta_j}\phi^\ast$. In general one can think of $\ol\phi$ and $\phi$ as independent complex fields and define a half-dimensional slice in field space to integrate on in the path integral. The choice of slice should make the action positive definite. The standard choice is $\ol\phi =\phi^\ast$, but there could be other choices. We will not study how to choose a slice, or define proper reality conditions, compatible with~\eqref{SingProfile} and simply assume that it can be done. We notice that if we perform a conformal map to the cylinder $\bR\times S^2$ and a Wick rotation to Lorentzian signature, the reality condition that we observe in~\eqref{SingProfile} becomes usual complex conjugation. We will also not study the precise form of the subleading term ``sub''. When introducing a diverging profile at the origin, a common procedure is to cut a small ball $B_{\epsilon}$ of radius $\epsilon>0$ and allow for a boundary term on $S^2_\epsilon = \p B_\epsilon$. Such a boundary term is fixed, in principle, by requiring a well-defined variation principle, namely the cancellation of boundary terms coming from the field variation of the action. The variation of the scalar action produces the boundary term \bea \delta S |_{\rm bdy} &= -\int_{S^2_\epsilon} \omega_2 \epsilon^2 (\delta\ol\phi \, \p_r\phi + \p_r\ol\phi \, \delta \phi) \cr &= - \int_{S^2_\epsilon} \omega_2 i kq |Y_{qjm}|^2 \delta\lambda + O(\epsilon^\alpha) \cr &= - i kq \delta\lambda + O(\epsilon^\alpha) \,,\phantom{\frac{1}{1}} \eea with $\alpha>0$. To cancel this term (in the limit $\epsilon \to 0$) we should thus add the boundary~term\footnote{In our conventions the integrand of the path integral is $e^{-S - S_{\rm bdy}}$.} \be S_{\rm bdy} = i kq\int_{S^2_\epsilon} \omega_2 |Y_{qjm}|^2 \lambda = ikq\lambda \,. \label{SbdyWithMonop} \ee Therefore we find that we should insert $e^{-ikq\lambda}$ at the origin to complete the operator insertion.\footnote{Describing this insertion as a local operator insertion in terms of $\phi$ and $\ol\phi$ is not convenient.} This result agrees nicely with the analysis of the gauge invariance of the monopole operator. Let us explain this point. Under a field variation $A \to A +\delta A$ the Chern-Simons action changes by a bulk plus a boundary term \be \delta S_{\rm CS} = \frac{ik}{2\pi} \int \delta A\wedge F + \frac{ik}{4\pi}\int_{S^2_{\epsilon}} \delta A \wedge A \,. \ee In order for the boundary term $\delta A \wedge A$ to cancel we can fix one component of the gauge field to zero, say $A_{\theta}=0$, on the boundary~\cite{Moore-ml-1989yh}. This is compatible with the presence of the Dirac monopole singularity. The variation $\delta S_{\rm CS}$ reduces only to the bulk term \be \delta S_{\rm CS} = \frac{ik}{2\pi} \int \delta A\wedge F \,. \ee Specializing to a gauge transformation $\delta A = d\Lambda$, we find \be \delta_\Lambda S_{\rm CS} = \frac{ik}{2\pi} \int \dd\Lambda \wedge F = -\frac{ik}{2\pi} \int_{S^2_\epsilon} \Lambda F \,. \ee We see that $\delta_\Lambda S_{\rm CS}$ is a boundary term which is non-vanishing in the presence of a non-zero magnetic flux~\eqref{MonopSingbis}. In the limit $\epsilon \to 0$, we have \be \delta_\Lambda S_{\rm CS} = - i k q \Lambda(0) \,. \label{CSgaugetransfo} \ee Note that the transformation is compatible with $\Lambda$ being $2\pi$-periodic, since $kq \in \bZ$. This gauge variation is nicely canceled by the gauge variation of the dressing factor $e^{-ikq\lambda}$, \be \delta_\Lambda (e^{-ikq\lambda} e^{-S_{\rm CS}}) = 0 \,. \ee The insertion of $e^{-ikq\lambda} = (e^{-i\lambda})^{kq}$ can be thought of as a dressing with $kq$ modes of $\ol\phi$ at the origin. To summarize, with $kq >0$, a monopole insertion at the origin $M_q(0)$ is defined in the path integral formulation by requiring the Dirac monopole singularity~\eqref{MonopSingbis}, the scalar profiles~\eqref{SingProfile} and the insertion of $e^{-ikq\lambda}$, with $\lambda$ the phase defined in~\eqref{SingProfile}. For $kq <0$, the scalar profiles are exchanged ($\beta_j \to -\beta_j$) and the dressing $e^{-ikq\lambda} =(e^{i\lambda})^{-kq}$ can be thought of as a dressing with $-kq$ modes of $\phi$ at the origin. \paragraph{Generalization and spin of monopoles.} As such this operator does not transform nicely under $\Spin(3)$ rotations. If we label $\lambda = \lambda_{jm}$ the phase appearing in the profile~\eqref{SingProfile}, we observe that $e^{i\lambda_{jm}}$ (or $e^{-i\lambda_{jm}}$) transforms as a component of the spin $j$ representation. The dressing operators $e^{-ikq\lambda_{jm}} = (e^{-i\lambda_{jm}})^{kq}$ however do not transform in a representation of the rotation group, or rather transform into operators that we have not yet discussed . To find the missing operators, we need to generalize the monopole insertions. The generalization goes as follows. Assuming $kq >0$, we require, in addition to the monopole flux singularity~\eqref{MonopSingbis}, the scalar field profile at the origin \bea \phi &= \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j \frac{a_{jm}}{r^{\beta_j+\frac 12}} Y_{qjm} + {\rm sub} \,, \cr \ol\phi &= \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j \ol a_{jm} r^{\beta_j-\frac 12} \ol Y_{qjm} + {\rm sub} \,, \label{GeneralSingProfile} \eea with $\ol a_{jm} = (a_{jm})^\ast$, and with only a finite number of non-zero $a_{jm} \in \bC$. Gauss's law~\eqref{GaussLaw} imposes the constraint \be \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j 2\beta_j |a_{jm}|^2 = kq \,. \label{GenGLconstr} \ee The boundary contribution in the variation of the action is now canceled by adding the boundary term \bea S_{\rm bdy} &= \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j \int_{S^2_\epsilon} \omega_2 |Y_{qjm}|^2 i (2\beta_j) |a_{jm}|^2 \lambda_{jm} \cr & = \label{GenSbdy} \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j i (2\beta_j) |a_{jm}|^2 \lambda_{jm} \,. \eea Now recall that the phases $\lambda_{jm}$ are $2\pi$ periodic, so, for the boundary term $\exp(-S_{\rm bdy})$ to make sense, we must impose the quantization conditions $2\beta_j |a_{jm}|^2 := n_{jm} \in \bN$ for all $j,m$. To satisfy Gauss's law~\eqref{GenGLconstr} one must then choose a collection of non-negative integers $n_{jm}$, such that $\sum_{j,m} n_{jm} = kq$. The scalar profiles become \bea \phi &= \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j \frac{e^{i\lambda_{jm}}}{r^{\beta_j+\frac 12}} \sqrt{\frac{n_{jm}}{2\beta_j}} Y_{qjm} + {\rm sub} \,, \cr \ol\phi &= \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j e^{-i\lambda_{jm}} r^{\beta_j-\frac 12} \sqrt{\frac{n_{jm}}{2\beta_j}} \ol Y_{qjm} + {\rm sub} \,, \label{GenSingProfile} \eea with \be {\text{\emph{Gauss's law}}}: \qquad \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j n_{jm} = kq \,, \quad (n_{jm} \in \bZ_{\ge 0} ) \label{GenGL} \ee and the monopole insertion is completed by the dressing operator \be {\text{\emph{Dressing term}}}: \qquad \prod_{j\ge \frac{|q|}{2}} \prod_{m=-j}^j \exp(-i n_{jm}\lambda_{jm}) \,. \label{GenDressing} \ee Once again the dressing operator restores the gauge invariance of the monopole insertion, since the phases $\lambda_{jm}$ all transform as $\lambda_{jm} \to \lambda_{jm} + \Lambda(0)$ under a gauge transformation. This defines the insertion at the origin of a monopole operator $M_{q {\bf n}}$ with ${\bf n} = (n_{jm})$ with $n_{jm} \ge 0$ and $|{\bf n}| := \sum_{jm} n_{jm} = kq$. As is sometimes done in the literature, one can think heuristically of the dressing as the insertion of $kq$ factors, where each factor is thought of as a $\p^n\ol\phi$ insertion at the origin, with $\ol\phi$ an operator of spin $\frac{|q|}{2}$ and $n = j-\frac{|q|}{2}$. This is however more of a book-keeping device rather than a correct statement. For $kq<0$ we need to invert the roles of $\phi$ and $\ol\phi$ by exchanging the profiles in~\eqref{GenSingProfile} (this is implemented by $\beta_j \to -\beta_j$, $n_{jm} \to - n_{jm}$) and dressing the insertion with $-kq$ factors $e^{i\lambda_{jm}}$. Heuristically we dress the monopole with modes of $\phi$ instead of $\ol\phi$. Gauss's law becomes in this case \be \sum_{j\ge \frac{|q|}{2}}\sum_{m=-j}^j n_{jm} = -kq > 0 \,. \ee We can now reconsider the question of the transformation under $\SU(2)= \Spin(3)$ rotations. From the definition of the profiles~\eqref{GenSingProfile} we understand that the operators $e^{-i\lambda_{jm}}$ for $|m| \le j$ form a spin $j$ representation, which we denote ${\bf j}$. We deduce that the set of operators $M_{q{\bf n}}$ with fixed $n_j := \sum_{m=-j}^j n_{jm}$ (satisfying $\sum_j n_j = kq$), transform in the tensor product representation $\bigotimes_{j\ge \frac{|q|}{2}} [\mathbf{j}^{\otimes n_{j}}]_{\rm sym}$, where $[\ldots ]_{\rm sym}$ takes the symmetric product of the factors in the bracket. \be \left\lbrace M_{q{\bf n}} \, \bigg| \, \sum_{m=-j}^j n_{jm} = n_j \right\rbrace \quad \longrightarrow \quad \SU(2) \ \text{rep} \quad \bigotimes\limits_{j\ge \frac{|q|}{2}} [\mathbf{j}^{\otimes n_{j}}]_{\rm sym} \,. \ee This is a reducible representation.\footnote{It contain the symmetric traceless representations and the traces.} In the minimal case $k=q=1$, the occupation numbers ${\bf n}$ of the monopoles $M_{q{\bf n}}$ have a single non-zero entry $n_{jm}=1$. In this case the $2j+1$ monopoles with $n_j=1$ form a spin $j$ representation, the smallest spin being $j=\frac q2 = \frac 12$. These monopole operators, with these spin quantum numbers, exist in the theory deformed by a scalar mass term and scalar quartic potential, since continuous deformations do not affect the $\SU(2)$ representations in which the monopole transform. We can think about the monopole operators in the theory with critical quadratic and quartic interactions, which flows to an infrared CFT\@. Ideally one would like to know the conformal dimension of these monopoles in the CFT\@. Computing the dimension of the monopole operators (or any unprotected operator) is a notoriously hard problem in a strongly coupled field theory. In the limit of large number of charged fields $N_f \gg 1$ the infrared theory is effectively weakly coupled and the monopole dimension can be computed pertubatively in $1/N_f$~\cite{Borokhov-ml-2002ib,Pufu-ml-2013vpa} (see also a $d=4-\epsilon$ approach in~\cite{Chester-ml-2015wao}). In the Chern-Simons theory, 't Hooft-like limits were considered (with large $k,N_f$ or large $k,N_c$ and fixed ratio). The monopole dimension is then extracted from the leading contribution to the free energy of the theory on $S^1_\beta\times S^2$ in the small temperature limit $\beta \to \infty$~\cite{Radicevic-ml-2015yla,Chester-ml-2017vdh,Dyer-ml-2013fja}, sometimes relying on numerical evaluations. These results are not directly applicable to the theory of a $\U(1)$ gauge group with a single~flavor.\footnote{In the large $k$ limit, the saddle point analysis of~\cite{Chester-ml-2017vdh} should be valid, even with a single charged scalar, which corresponds to taking large $\kappa$. The results presented for the bosonic theory are numerical. We were not able to isolate a result which applies to our situation.} ]]>

0$ around the origin and study the limit $\epsilon \to 0$. Varying the fermion action at finite $\epsilon$ produces a boundary term \bea \delta S_{\rm fermion} |_{\rm bdy} =\;& \int_{S^2_\epsilon} \omega_2 \epsilon^2 \ol \psi\gamma^r \psi \, \delta\lambda \\[2mm] \xrightarrow{\epsilon \to 0} \;&-i kq \delta\lambda \,, \eea where $S^2_\epsilon = \p B_{\epsilon}$ and we have used the fact that the fermion background satisfies Gauss's law~\eqref{GLfermion} to reach the final result in the limit $\epsilon \to 0$. To cancel this boundary term (in the limit $\epsilon \to 0$) we add to the operator insertion the dressing term \be e^{-i kq \lambda} \,. \ee As in the scalar theory, the dressing factor has gauge charge $-kq$, compensating for the gauge transformation of the Chern-Simons term and restoring the full gauge invariance of the monopole operator insertion. There is however an issue with the dressing factor. The phase $e^{-i\lambda}$, as defined by the fermion profile~\eqref{FermionProfile}, is a Grassmann-odd field. Therefore it vanishes when raised to a power two or bigger. Thus the dressing term $(e^{-i\lambda})^{kq}$ vanishes, except for $kq=1$. To be able to define monopole operators with higher values of $kq$ we need more fermion modes. \paragraph{Generalization.} The monopole insertion can be generalized. We assume $kq >0$. Since this is analogous to the scalar field case, we only go through the main lines, skipping details. We can require a singular profile of the fermion field \bea & \psi = \sum_{j \ge \frac{|q|}{2} -\frac 12}\sum_{m=-j}^j e^{i\lambda_{jm}} r^{-\beta_j-1} \sqrt{\frac{n_{jm}}{q(j+\frac 12)}}\bigg[ \frac{q}{2} T_{qjm} + \bigg( j + \frac 12 + \beta_j\bigg) S_{qjm} \bigg] + \text{sub} \,, \cr & \ol\psi = \sum_{j \ge \frac{|q|}{2} -\frac 12}\sum_{m=-j}^j i e^{-i\lambda_{jm}} r^{\beta_j-1} \sqrt{\frac{n_{jm}}{q(j+\frac 12)}} \bigg[ \frac{q}{2} \ol T_{qjm} - \bigg( j + \frac 12 - \beta_j\bigg) \ol S_{qjm} \bigg] + \text{sub} \,, \label{GenFermionProfile} \eea where it is understood that $T_{qjm} = 0$ if $j=|\frac q2| -\frac 12$, and $n_{jm}$ are positive and satisfy $\sum_{j \ge \frac{|q|}{2} -\frac 12}\sum_{m=-j}^j n_{jm} = kq$. The required dressing term is then \be {\text{\emph{Dressing term}}}: \quad \prod_{j\ge \frac{|q|}{2} -\frac 12} \prod_{m=-j}^j \exp(-i n_{jm}\lambda_{jm}) \,, \ee The periodicity of the phases $\lambda_{jm} = \lambda_{jm} + 2\pi$ implies $n_{jm} \in \bZ_{\ge 0}$. Because $\exp(-i\lambda_{jm})$ are Grassmann-odd fields, there should at most one power of each such factor in the dressing operator. This means that $n_{jm} \in \{0,1\}$. This defines the insertion of the monopole operator $\ti M_{q {\bf n}}$ with ${\bf n} = (n_{jm})$, satisfying \be {\text{\emph{Gauss's law}}}: \quad \sum_{j \ge \frac{|q|}{2} -\frac 12}\sum_{m=-j}^j n_{jm} = kq \,, \quad n_{jm} \in \{0,1\} \,. \ee \paragraph{Spin of the monopoles.} From the definition of the fermion profile we observe that the phases $e^{i\lambda_{jm}}$ transform in the spin $j$ representation. It follows that the set of monopole operators $\ti M_{q {\bf n}}$ with fixed $n_j := \sum_{m=-j}^j n_{jm}$ transform in the $\SU(2)$ representation $\bigotimes_{j \ge \frac{|q|}{2} -\frac 12} [{\bf j}^{\otimes n_j}]_{\rm anti-sym}$, where $[\ldots ]_{\rm anti-sym}$ takes the anti-symmetric product of the factors in the bracket. In the minimal case where $k=q=1$, the bare monopole has gauge charge one and a single occupation number $n_{jm}$ is non-zero for each dressed monopole. The $2j+1$ monopoles $\ti M_{q {\bf n}}$ with $n_j=1$ transform in the spin $j$ representation, and the minimal spin is $j=\frac q2 - \frac 12 = 0$ corresponding to a scalar operator. For $kq<0$, we must exchange the roles of $\psi$ and $\ol\psi$, by exchanging the profiles in~\eqref{GenFermionProfile} and dress the monopole singularity with $-kq$ factors of $e^{i\lambda_{jm}}$. Heuristically we dress the monopole with modes of $\psi$, instead of $\ol\psi$. Gauss's law becomes in this case \be \sum_{j \ge \frac{|q|}{2} -\frac 12}\sum_{m=-j}^j n_{jm} = -kq \,, \quad n_{jm} \in \{0,1\} \,. \ee ]]>

0$, since Gauss's law is in this case $\sum_{j,m} n_{jm} = k'_{(-)} q$. We will use here $k':=k'_{(-)}$ and assume $k'>0$ and $q>0$ for simplicity. Generically the monopoles $\ol\cM_{q\bf n}$ do \emph{not} preserve any supersymmetry, but some of them preserve $\ol Q_1$ or $\ol Q_2$. A simple class of monopoles preserving $\ol Q_2$ has $n_{jj} = k'q$ for a chosen $j$, and $n_{jm}= 0$ for all other $j,m$ pairs. Their counterpart monopoles preserving $\ol Q_1$ have $n_{j,-j} = k'q$ and $n_{jm}=0$ for all other $j,m$ pairs. They both belong to the same irreducible spin representation $[{\bf j}^{\otimes k'q}]'_{\rm sym}$ of $\SU(2)$, where $[\ldots ]'_{\rm sym}$ denotes the symmetric traceless product of the representations. This is simply the spin $k'qj$ representation. Let us denote $J=k'qj$ and $V^{(j)}_{q m}$ , with $|m| \le J$, the monopoles in this representation. The $\ol Q_2$ invariant monopole is $V^{(j)}_{qJ}$ and the $\ol Q_1$ invariant monopole is $V^{(j)}_{q,-J}$. The other monopoles $V^{(j)}_{qm}$ are built out of the monopoles $\cM_{q\bf n}$ with $n_{j} := \sum_m n_{jm} = k'q$ and $n_{j'}=0$ for $j '\neq j$. The dimension of the monopoles $V^{(j)}_{qm}$ can be computed using the BPS properties of the BPS operators. It is a sum of two contributions \be \Delta(V^{(j)}_{qm}) = \Delta_{\rm bare} + \Delta_{\rm dressing} \,. \ee The contribution from the dressing factor is the sum of the dimension of each individual factor $e^{-i\lambda_{jm}}$. This is easily extracted from the definition of the $(j,m)$ mode: $\ol\phi \sim e^{-i\lambda_{jm}} r^{j} \ol Y_{qjm}$. In an $\cN=2$ SCFT, the dimension of the anti-chiral field $\ol\phi$ is related to its R-charge\footnote{Hopefully there will be no confusion between the R-charge $r$ and the radial coordinate $r$. Both notations are fairly standard.} $-r$: $\Delta(\ol\phi)= - R(\ol\phi) = r$. It follows that the dimension of the dressing mode is \be \Delta(e^{-i\lambda_{jm}}) = j + r \,. \ee The dressing has $k'q$ modes of $\lambda_{jm}$, leading to \be \Delta_{\rm dressing} = k'q \big(j + r \big) = J + k'qr\,. \ee The contribution $\Delta_{\rm bare}$ from the bare anti-chiral monopole is also related to its R-charge by the BPS condition and can be computed as a sum of zero point energy of all oscillators in the theory on the cylinder~\cite{Borokhov-ml-2002cg}. It is given by~\cite{Imamura-ml-2011su,Benini-ml-2011cma} \be \Delta_{\rm bare} = \frac{1-r}{2}q \,, \ee and the total dimension is \be \Delta(V_{qm}) = J + \bigg( k' - \frac 12 \bigg)qr +\frac{q}{2} \,. \ee Similarly the $\U(1)$ R-charge of the BPS monopoles are computed as the sum of the R-charge of the bare monopole $R_{\rm bare} = -\frac{1-r}{2}q$ and the R-charge of the dressing $R_{\rm dressing} = k'q R(\ol\phi) = -kqr$, \be R(V^{(j)}_{qm}) = - \bigg( k' - \frac 12 \bigg) qr -\frac{q}{2} \,. \ee The exact values of the R-charge and dimension of such BPS operators at the infrared fixed point depend on the $\U(1)$ R-charge at this fixed point. This may not coincide with the UV R-charge, but rather is a combination of $\U(1)_{R,{\rm UV}}$ with the $\U(1)$ global symmetries of the IR fixed point. The parameter $r$ refers to the charge of $\phi$ under this infrared R-symmetry. This can be determined by extremizing the $S^3$ partition function of the $\cN=2$ theory under consideration~\cite{Jafferis-ml-2010un}. We recover that the BPS monopoles obey the BPS condition $\Delta = J_3 - R$ for $V^{(j)}_{q, J}$ and $\Delta = -J_3 - R$ for $V^{(j)}_{q, -J}$. This follows from the fact that the bare monopole and the dressing factors all obey the corresponding BPS condition. Another way to find the BPS monopoles is to consider only those dressed with modes $\ol\phi_{jm},\psi_{jm}$ (or $\phi_{jm},\ol\psi_{jm}$) which obey such BPS~conditions. The fact that these monopoles obey the BPS condition means that they define non-trivial elements of the $\ol Q_1$ or $\ol Q_2$ cohomologies and therefore contribute to the superconformal index defined with $\ol Q_1$ or with $\ol Q_2$. The monopoles $V^{(j)}_{qm}$ are only the simplest BPS monopoles. These considerations apply in general to the BPS monopoles dressed with both bosonic and fermionic modes, as described in the previous subsection. We leave as a exercise to work out the quantum numbers in this general case. \paragraph{Superconformal multiplet.} In the infrared SCFT the $\ol Q_{\alpha}$-BPS monopoles belong to short superconformal mulitplets called $A_1$ in the classification of~\cite{Cordova-ml-2016emh} (or $\chi_S$ in~\cite{Minwalla-ml-2011ma}). These are the only short superconformal multiplets in 3d $\cN=2$ SCFTs which accommodate for non-zero spin. They are $\frac 14$ BPS operators in the sense that they are non-trivial in $\ol Q_1$ cohomology or in $\ol Q_2$ cohomology, but not in both. Moreover they are not the bottom component of the $A_1$ multiplet, but rather descendants. Indeed the bottom component $C$ in this multiplet satisfies $\Delta = J - R + 1$, where $J$ is the $\SU(2)$ spin and $R$ is the $\U(1)$ R-charge. In components we can write it $C_{\alpha_1\cdots\alpha_{2j}}$, for the operator with spin $j$. It satisfies the shortening condition $\ol Q_\beta C^{\beta}{}_{\alpha_1\cdots \alpha_{2j-1}} = 0$.\footnote{If $j=0$, $C$ is rather the bottom component of an $A_2$ multiplet with shortening condition $\ol Q_\alpha \ol Q^\alpha C=0$.} The descendants $D_{\alpha_1,\cdots, \alpha_{2j+1}} = \ol Q_{(\alpha_1} C_{\alpha_2\cdots\alpha_{2j+1})}$ obeys $\Delta = J - R$. The BPS monopoles are identified with the components $D_{11\cdots 1}$ and $D_{22\cdots 2}$, which are non-trivial in $\ol Q_2$ and $\ol Q_1$ cohomology respectively. They are the only operators in the full multiplet contributing to the superconformal index defined with $\ol Q_2$ or with $\ol Q_1$ (see~\cite{Minwalla-ml-2011ma} for details). There is a mirror discussion for $Q_\alpha$-BPS monopoles defined with $u=+1$ backgrounds and $\phi,\ol\psi$ dressing modes. Note in particular that this is different from supersymmetric monopoles in Maxwell (or Yang-Mills) theory without Chern-Simons term, which are not dressed with charged matter field. There, the monopoles are chiral ($\frac 12$-BPS) operators with no spin. They are the bottom components of $B_1$ (or $\ol B_1$) superconformal multiplets~\cite{Cordova-ml-2016emh} and are non-trivial in both $\ol Q_\alpha$ (or both $Q_\alpha$) cohomologies. ]]>

0$, and $V^{-}_{\alpha_1\cdots \alpha_{kq}} = v_{q,q} A_{\alpha_1}\cdots A_{\alpha_{|kq|}}$ for $kq <0$, with zero spin. This was described in~\cite{Cremonesi-ml-2016nbo}. We find that, despite having Chern-Simons terms, the monopoles in ABJM theory are half-BPS chiral operators. In particular they have no spin. This phenomenon carries on to other quiver Chern-Simons theories of a similar nature and in particular in Chern-Simons theories with extended $\cN \ge 3$ supersymmetry, as studied in~\cite{Assel-ml-2017eun}. In these theories the Chern-Simons levels and matter content are constrained in such a way as to allow for such half-BPS monopoles~\cite{Gaiotto-ml-2008sd,Hosomichi-ml-2008jd} (in addition to $\frac 14$-BPS monopoles). ]]>