Monopole operators and symmetry enhancement in ABJM theory revisited

We construct monopole operators for 3d Yang-Mills-matter theories and Chern-Simons-matter theories in canonical formalism. In this framework, monopole operators, although as the disorder operators, could be written in terms of the fundamental fields of the theory, and thus could be treated in the same way as the ordinary operators. We study the properties of the constructed monopole operators. In Chern-Simons-matter theories, monopole operators transform as the local operators with the classical conformal dimension $ 0 $ under the action of the dilation and are also covariantly constant. In supersymmetric Chern-Simons-matter theories like the ABJM model, monopole operators commute with all of the supercharges, and thus are SUSY invariant. ABJM model with level $ k=1,2 $ is expected to have enhanced $SO(8) $ R-symmetry due to the existence of the conserved extra R-symmetry currents $ j^{AB}_{\mu} $ involving monopoles. With the explicit form of the monopole operators given, we prove the current conservation equation $\partial^{\mu} j^{AB}_{\mu} =0$ using the equations of motion. We also compute the extra $ \mathcal{N}=2 $ supercharges, derive the extra $ \mathcal{N}=2 $ SUSY transformation rules, and verify the closure of the $ \mathcal{N}=8 $ supersymmetry.


INTRODUCTION AND SUMMARY
In any 3d gauge theory with the gauge group containing a U (1) factor, there is a current J µ = 1 4π µνλ trF νλ whose conservation is equivalent to the Bianchi identity. The conserved topological charge is called the vortex charge, and the related global symmetry is referred to as the U (1) J symmetry. Monopole operators are local operators creating (annihilating) the charge Q [1, 2]. As disorder operators, monopole operators cannot be written as the polynomials of the elementary fields at the insertion point, but dualities can map them onto the operators of that kind [3][4][5][6][7][8][9].
It is expected that a better understanding of monopole operators can make the 3d dualities more transparent. One such example is the 2d duality relating the massive Thirring model and the sine-Gordon model [10,11]. In the sine-Gordon model, local disorder operators for the creation and annihilation of topological solitons (kinks) are constructed in canonical formalism, while by rewriting the sine-Gordon model in terms of these dual variables, the massive Thirring model is obtained [11]. In this paper, we will give a similar construction for monopole operators in 3d, although the main concern is not duality.
Monopole operators are usually defined by specifying the singularities of the gauge fields at the insertion point and expanding the quantum fields around this singular background [2]. For example, in a U (N ) gauge theory, the gauge field singularity is supposed to have the form for the north and south charts so that there is some magnetic flux on S 2 surrounding it. H must satisfy the quantization condition e 4πiH = I and is an element of the Cartan subalgebra of the form H = diag( m 1 2 , · · · , m N 2 ), m i ∈ Z [27]. For a CFT on S 2 × R, with the specified magnetic flux on S 2 given, one can compute quantum numbers like the R-charges and energies of the states, which, by state-operator correspondence, are mapped onto the vortex-charged operators [2,4,5]. In this way, R-charges and conformal dimensions of the vortex-charged operators can be obtained with the quantum corrections taken into account. Since the method only applies for the weakly coupled theories, some strategies must be used to get the strong-coupling result in ABJM theory [22,23,28].
In this paper, we will return to 't Hooft's original definition of the monopole operator as a singular gauge transformation acting on states [1]. In canonical formalism, such operators can be written in terms of the fundamental variables of the theory, and thus could be treated in the same way as the ordinary operators. This is similar to [11] where the kink operator is constructed in 2d sine-Gordon model. In Yang-Mills-matter theories and the Chern-Simons-matter theories, we will give the explicit canonical expressions for the monopole operator M R (x), which is labeled by the representation R of the gauge group and is realized as a gauge transformation singular at x. The singular gauge transformation M R (x), although labeled by R, does not transform in representation R. In Chern-Simons-matter theories, we will define a dressed monopole operator M R (x) that would behave as a local operator in representation R under the local gauge transformation. In non-Abelian gauge theories, M R (x) is the monopole operator that is used to build the vortex-charged gauge invariants.
We also study the properties of the constructed monopole operators. We prove the contraction relation (5.27), which is required in ABJM theory for R-charges to form the so(8) algebra.
We show that in Chern-Simons-matter theories, under the action of the dilation, monopole operators transform as the local operators with the conformal dimension 0-which, of course, may get quantum corrections from the interactions. We compute the supersymmetry transformation of the monopole operators in ABJM theory. It turns out that monopole operators are invariant under the N = 6 supersymmetry, as well as the N = 8 supersymmetry when k = 1, 2. It seems that in supersymmetric Chern-Simons-matter theories, monopole operators always commute with the supercharges, which is not a surprise, since monopole operators are just singular gauge transformations, while the ordinary gauge transformations commute with all gauge invariant operators. On the other hand, in supersymmetric Yang-Mills-matter theories, monopole operators are not SUSY invariant. With the suitable scalar fields added, one may construct BPS scalar-dressed monopole operators preserving part of the supersymmetry.
In ABJM theory, the gauge invariant combination of the monopole operators and the matter fields gives a new set of local operators carrying the vortex charge. In AdS/CFT, part of the KK modes of supergravity on AdS 4 ×S 7 /Z k are mapped onto the vortex-charged operators [12].
Since the monopole operators are SUSY invariant and have the classical conformal dimension 0, these operators could be 1/2 BPS and thus have the protected conformal dimension matching the spectrum on gravity side.
In the pure Abelian Chern-Simons theory and the U (1) k × U (1) −k ABJM theory, monopole operators can be written as the Wilson lines ending at the inserting point and are covariantly constant [12,24,[29][30][31]. However, when the charged matter is included or the gauge groups are non-Abelian, monopole operators will not be Wilson lines anymore [12,26]. There is also a controversy on their covariant constancy. In [24,25], the covariant constancy condition for monopole operators was assumed when deriving the N = 8 supersymmetry transformation rules in ABJM theory. Whereas in [26], it was shown that such a condition would lead to Eq (6.11), a severe constraint on monopole operators and the scalar fields. In fact, even in [24], for the SO(8) invariant trial Bagger-Lambert-Gustavsson (BLG) Lagrangian to be identical to the ABJM Lagrangian, the algebraic identities given by Eq (2.7) should also hold, while the closure of the N = 8 supersymmetry requires a few more similar identities. These identities are all constraint equations relating monopole operators and the matter fields.
In ABJM theory, monopole operators commute with the supercharges, while the anticommutator of supercharges gives either a covariant derivative or a field-dependent gauge variation, so from (9.27), one may conclude that monopole operators are both covariantly constant and invariant under that gauge variation. The latter property could be stated as the algebraic identities for monopole operators and the matter fields. This argument is based on supersymmetry.
In section 8, we will give a proof for the covariant constancy of monopole operators in generic Chern-Simons-matter theories without relying on supersymmetry. We will also show that in Yang-Mills-matter theories, monopole operators are not covariantly constant.
With the explicit form of the monopole operators given, the global symmetry enhancement is revisited. In 3d gauge theories, if the original global symmetry algebra is R, with the U (1) J charge Q added, one may consider the possibility for the enlarged symmetryR = R ⊕ Q ⊕ R off−diag , where R off−diag denotes off-diagonal elements charged under Q. The form of the global symmetry currents is entirely determined by the field content. The current for R off−diag must involve monopole operators and does not always exist. In the k = 1, 2 ABJM theory, R andR are su(4) and so (8) algebras with the currents for R and R off−diag given by (9.1) and (9.3). We compute the current conservation equation ∂ µ j AB µ = 0 with the equations of motion plugged in. It turns out that for ∂ µ j AB µ = 0, monopole operators should satisfy the constraints (9.10). The first is the covariant constancy condition. The rest are the invariance condition under some particular field-dependent gauge variations [stronger than that in (9.27)], which is also the origin of the algebraic identities in [24,26]. We show that the constructed monopole operators satisfy (9.10), and so the global symmetry is enhanced to SO(8).
In fact, in a 3d unitary CFT, the dimension-2 currents are always conserved, so to prove the R-symmetry enhancement, it is enough to show j AB µ has the conformal dimension 2, which is the approach in [23]. In our proof of ∂ µ j AB µ = 0, conformal invariance or supersymmetry is not the necessary condition. For example, consider a truncated ABJM model with no fermionic fields, where the monopole operators satisfy the first two equations in (9.10). Among all SU (4) invariant scalar potentials, only the quadratic mass term and the sextic potential in ABJM model could make ∂ µ j AB µ = 0. In this perspective, the symmetry enhancement comes from the classical symmetry of the Lagrangian together with the properties of the monopole operators.
With the extra R-charges added into the N = 6 superconformal algebra of the ABJM theory, two additional supercharges are generated. We get the extra N = 2 supercharges in (10.17).
For the commutation relation (10.16) to take the presented form, the properties (9.10) are used again. We write down the extra N = 2 SUSY transformation rules, which are similar to those obtained in [24]. We verify the closure of the N = 8 supersymmetry and also discuss the BPS multiplet structure of the theory following the line of [32].
The rest of the paper is organized as follows. In section 2, we define the monopole operator M R (x) as a singular gauge transformation. In section 3, we write down the explicit form of M R (x) in Yang-Mills-matter theories and Chern-Simons-matter theories. In section 4, we construct the dressed monopole operator M R (x) that could transform in representation R. In section 5, we derive some contraction relations for M R (x). In section 6, we compute the classical conformal dimension of M R (x) and M R (x). In section 7, we calculate the supersymmetry transformation of M R (x) and M R (x). In section 8, we compute the derivative and covariant derivative of M R (x) and M R (x). In section 9, we prove the current conservation equation ∂ µ j AB µ = 0 in ABJM theory. In section 10, we study the enhanced N = 8 supersymmetry.
Section 11 is the conclusion and discussion.

MONOPOLE OPERATOR AS A SINGULAR GAUGE TRANSFORMATION
Monopole operators were first introduced by 't Hooft to define an alternative criterion for confinement [1]. The basic idea was to define an operator that creates or annihilates topological charges. It is known that solitons are pure gauge configurations singular at their locations, so the soliton operators can be constructed as the gauge transformations singular at the insertion points. The generic relation between solitons and soliton operators is given in Appendix A.
Monopole operators in [1] can also have the nontrivial winding n, which excludes the existence of quarks. We will only consider the situation with n = 0 so that the action of the monopole operators on fields in (anti)fundamental representation is also well defined.
Let us start with a brief introduction on group theory. For a group G with the rank r, {t M | M = 1, 2, · · · , dim G} are generators for the Lie algebra of G in fundamental representation, among which, {H A | A = 1, 2, · · · , r} are generators of the Cartan subalgebra.
R is also in one-to-one correspondence with in Cartan subalgebra, where H = (H 1 , H 2 , · · · , H r ). exp{4πiH m } = I. When G = U (N ), for an irreducible representation R labeled by m = (m 1 , · · · , m N ), the corresponding H m is Now consider a 3d gauge theory with the canonical coordinate (A i , Φ) and the conjugate A i is the gauge field in adjoint representation. Φ is the matter field in adjoint or (anti)fundamental representation. The complete orthogonal basis of the Hilbert space can be selected as the In canonical formalism, it is more appropriate to call the monopole operator the vortex operator since it creates the vortex in 2d space. A vortex at the position x carrying one unit of the vortex charge Q can be descried by the gauge configuration 1 2 a i (x, y) with b(x, y) = ij ∂ y i a j (x, y) = 4πδ 2 (x − y) , where by ∂ y i we mean the derivative with respect to y. a i can be solved as for the arbitrary scalar σ(y). a i is not a pure gauge, even though one can still construct some everywhere except for a singularity at x.
The monopole operator M R (x) labeled by R is defined via its action on |A i , Φ . When Φ is in adjoint representation, When Φ is in fundamental representation, Ω m is single valued and e −4πiH m = I amounts to selecting the winding number n = 0 in [1].
Away from x, is satisfied, so M R (x) is a local gauge transformation everywhere except for a singularity at x.

MONOPOLE OPERATORS IN CANONICAL FORMALISM
It is straightforward to write down the operator expression of M R in canonical formalism.
We will consider two typical situations: M R in 3d Yang-Mills theory coupling with the matter and M R in 3d Chern-Simons theory coupling with the matter. Although in these cases, the actions of M R on canonical variables are identical, the explicit forms of M R are different due to the distinct kinetic terms for gauge fields.
In 3d Yang-Mills-matter theory, the Gauss constraint is where ρ is the charge density of the matter fields. Local gauge transformation operator U (α) with the transformation parameter α is given by where α(y) is a Lie-algebra valued function well defined everywhere. As a singular gauge transformation with the parameter H m ω, M R (x) could be written as In a 3d Chern-Simons-matter theory with the level k, the canonical commutation relation for the gauge field is [33] [ 4) and the Gauss constraint is The monopole operator M R (x) is given by (3.6) In (2.4), a i is determined up to a local gauge transformation, an ordinary local gauge transformation. In particular, in U (1) pure Chern-Simons theory, One can always select the suitable σ so that a i is nonvanishing only at a Dirac string y(s) with 0 ≤ s < ∞, y(0) = x, y(∞) = ∞. In this case, is a Wilson line starting from x and extending to infinity, as expected [29][30][31].
Similar to the local gauge transformation, the action of M R on canonical fields is and for Φ in fundamental representation or adjoint representation. However, for the field strength we have 14) The extra δ-term indicates M R (x) is a gauge transformation singular at x. When G = U (N ), for Q given by (1.1), from (3.14), where R 1 and R 2 are two irreducible representations labeled by H m 1 and H m 2 , respectively.
The Gauss constraint in ABJM implies trF ij = trF ij [12]. For the topological charge with trH m 1 = trH m 2 assumed.

MONOPOLE OPERATORS IN REPRESENTATION R
The monopole operator M R (x) discussed in section 2 and 3, although labeled by R, does not transform in representation R under the gauge transformation. In the following, we will compute the gauge transformation of M R (x) explicitly.
First, in 3d electromagnetic theory coupling with the matter, which is invariant under the U (1) gauge transformation.
In 3d U (1) Chern-Simons theory coupling with the matter, Under the U (1) local gauge transformation, M (x) transforms as an operator at x carrying the U (1) charge k.
In 3d non-Abelian gauge theory with the group G, suppose G is the group composed by the local gauge transformation operator U , ∀ U ∈ G, where u ∈ G is the transformation matrix for U . Concretely, for U (α) given by (3.2), the related u(y) = e −iα(y) . Actions of U on M R (x) in (3.3) and (3.6) are given by and respectively. Obviously, M R (x) does not transform in representation R.
It is desirable to construct the monopole operator that would transform as a local operator in representation R under the action of U . Suppose {|α |α = 1, 2, · · · , dim R} are bases for the representation R, among which |1 is the highest weight state. The group element u in and where R is an arbitrary irreducible representation. ∀ V ∈ G, It seems that M α β (R,R ) (x) with the fixed β transforms as a local operator in representation R, while M +α β (R,R ) (x) with the fixed α transforms as a local operator in representationR. However, in most cases, M α β (R,R ) (x) and M +α β (R,R ) (x) constructed in (4.9) and (4.10) are actually 0. Since which requires from the gauge transformation rule (4.5), On the other hand, for M R in Chern-Simons-matter theory, from the gauge transformation rule In Yang-Mills-matter theory, (4.15) requires D γ β R [v] = δ γ β , ∀ v ∈ G H , which is possible only when R is the identity representation I, in which case, So the only non-zero M that can be constructed is M (I,R) (x) which is gauge invariant.
In Chern-Simons-matter theory, M R transforms as (4.16) under the action of G H . When from (4.16), the corresponding V i will make So, in addition to the condition (4.14) for V satisfying (4.12), M R and M α β (R,R ) should also have the opposite U (1) charges, which is possible only when R is the representation labeled by kH m and |β is the highest weight state in representation R.

For example, when
R must be the km 1 -symmetric representation with |β the highest weight state When R must be an irreducible representation with the Young tableau containing k boxes in the first two rows. |β is the highest weight state To conclude, in Chern-Simons-matter theory with the level k, the monopole operator in representation R labeled by H m can be constructed as where |1 is the highest weight state in presentation R and M R/k (x) is the monopole operator labeled by H m/k . Since m i /k must be integers, for k > 1, not all of R can be realized.
For example, if R is the fundamental representation, k must be 1; if R is the 2-symmetric representation, k can only be 1 or 2. The integration measure DU is normalized with We also add a divided factor √ N R . Suppose G S ⊂ G is the stationary group of the integrand, N R is given by By stationary group, we mean ∀ V ∈ G S , which is equivalent to the condition The associated v could also compose a stationary group G s , under which |1 is invariant up to Similarly, the monopole operator in representationR is will be gauge invariant. In this way, the vortex-charged gauge invariant operators can be constructed.
Recall that in (2.4), a i can differ by a local gauge transformation. With a i → a i + ∂ i σ, up to a phase. The different a i will also give rise to the different M R . Such ambiguity can be eliminated by Gauss constraint, which is imposed via a projection to the physical Hilbert space H phy . ∀ |ψ ∈ H phy , ∀ U ∈ G, U |ψ = |ψ . From (4.28), (4.36) and (4.29), the action of the physical operator M α in H phy remain the same. We may select an arbitrary a i to construct M R , and the obtained physical operators in H phy are identical. In fact, when computing the matrix element in H phy , (4.38) can be equivalently interpreted in path integral language.
From (A.7), the action of M R/k (x) produces a vortex-charged configuration, which is then combined with O 1R (x) to compensate the U (1) charges. The integration over gauge equivalent configurations gives the matrix element in physical Hilbert space.
In Chern-Simons-matter theory with the gauge group (3.16), and the corresponding monopole operator in representation (R 1 ,R 2 ) can be constructed as where M α R 1 (x) and Mα R 2 (x) are monopole operators for the first and second U (N ) groups. Under the local gauge transformation VV , The topological charge Q + is gauge invariant, so from (3.18) and (4.40), . In ABJM theory, monopole operators with R 1 = R 2 are used to construct gauge invariant chiral operators. The generic monopole operators with R 1 = R 2 should also exist and are required to make the large N spectrum of the protected operators agree with that of the gravity states [22].

THE CONTRACTION RELATIONS FOR MONOPOLE OPERATORS
In this section, we will prove some contraction relations for monopole operators M α R (x) and M αR (x) that will be used in section 10.
Consider the Chern-Simons-matter theory with the gauge group U (N ) and the level k = 1.
If R is the fundamental representation N, the related H m is The corresponding monopole operator is is the monopole operator in antifundamental representationN. M + a (x) = M a (x). Under the local gauge transformation, For the given U , one can always find some W with As a result, (5.6) = 0 only when U ∈ G S , u ∈ G s = U (1) × U (N − 1), and then M a and M a in representations N andN do not exist when k > 1.
When k = 1, the 2-symmetric representation N 2 sym is labeled by The corresponding monopole operator is . Under the local gauge transformation, The contraction of M ab (x) and M bc (x) is (5.14) For the given U , one can always find some W with We should at least have where we have used we can also get M ab (x) in 2-antisymmetric representation N 2 asym and M ab (x) in conjugate representationN 2 asym . In Chern-Simons-matter theory with the gauge group The monopole operators in (N 2 sym ,N 2 sym ) and (N 2 sym , N 2 sym ) representations are which is the contraction relation needed in section 10.
When k = 1, except for M ab ab (x) and Mâb ab (x), there are 6 additional monopole operators carrying the vortex charge ±4 : (5.28) in (N 2 asym ,N 2 asym ) and (N 2 asym , N 2 asym ) representations, (5.29) in (N 2 sym ,N 2 asym ) and (N 2 sym , N 2 asym ) representations and (5.30) in (N 2 asym ,N 2 sym ) and (N 2 asym , N 2 sym ) representations. As will be discussed in section 10, in ABJM theory, such operators can be used to construct the superconformal current multiplets. In (d+1)-dimensional conformal field theory, for a field φ with the conformal dimension ∆, the conjugate momentum π should have the conformal dimension ∆ = d − ∆. The action of the dilatation operator D is where i, j = 1, 2. The monopole operator M R (x) is given by (3.6).
where we have used Therefore, The classical conformal dimension of M R (x) is 0. (6.8) applies for M R (x) with x 0 = 0; however, can be recovered. from which, (6.8) and then (6.9) are obtained again.
As for the scaling transformation of M R (x), from (4.28), M R 's constructed from the gauge equivalent a i will all have the conformal dimension 0 under the action of D. We will return to this point in section 9.

SUPERSYMMETRY TRANSFORMATION OF MONOPOLE OPERATORS
In this section, we will study the supersymmetry transformation of monopole operators in As a result, for where Γ's are 32 × 32 gamma matrices in 10d spacetime, we have for M R (x) to be a genuine gauge transformation is ij ∂ i a j (x, y) = 4πδ 2 (x − y) = 0, so when computing the supersymmetry variation of M R (x), we only need to consider terms producing ij ∂ i a j . The supersymmetry variation of Π i is where "· · · " are terms with no derivatives, so is obtained again.
It is possible to construct the scalar-dressed monopole operator preserving part of the supersymmetry. Since for the operator exp{−4πtr[H m Φ I (x)]}M R (x), the supersymmetry variation is Next, consider the U (N ) k × U (N ) −k ABJM theory with N = 6 supersymmetry. Following the convention in [34], the field content consists of complex scalars X a Aâ , spinors Ψ Aâ a , and their adjoints X Aâ a , Ψâ Aa , transforming as (N,N ) and (N , N ) representations of the gauge group. A = 1, · · · , 4, a,â = 1, · · · , N . The U (N ) gauge fields are Hermitian matrices A a b andÂâ b . The covariant derivatives are with similar formulas for the spinors. µ = 0, 1, 2. The action of the ABJM theory is [34] S = S kin + S SC + S F + S S (7.10) with the kinetic term the Chern-Simons term the fermionic interaction 13) and the scalar potential Under the N = 6 supersymmetry transformation [34], where The 6 supercharges are given by [34]

DERIVATIVE AND COVARIANT DERIVATIVE OF MONOPOLE OPERATORS
In this section, we will compute the derivative and covariant derivative of the monopole operators M R (x) and M R (x) in 3d Yang-Mills-matter theory and Chern-Simons-matter theory.
We find that in Chern-Simons-matter theory, M R (x) and M R (x) are covariantly constant.
The monopole operator M R (x) at the point x is characterized by the function H m a i (x, y).

Under the infinitesimal translation
In 3d Yang-Mills-matter theory, M R (x) is given by We can also calculate the covariant derivative of M R (x).
The covariant derivative of M R (x) is Compared with (8.2), the right-hand side of (8.6) only contains local operators.
Of course, the derivative could be directly obtained from the action of the momentum operator P i . For example, in a 3d U (N ) Yang-Mills-matter theory with the matter part composed of a scalar φ and a spinor ψ, both in adjoint representation of U (N ), where Π φ and Π ψ are conjugate momenta of φ and ψ.
which is consistent with (8.2).
The gauge invariant completion of P i is . For the gauge field A j , From (8.5), (8.9) and (8.10), (8.6) is recovered.
In 3d Chern-Simons-matter theory with the level k, the monopole operator M R (x) is Still, consider a Chern-Simons-matter theory with the gauge group U (N ) and the field content (A i , φ, ψ). The momentum operator is The gauge invariant momentum operator is In particular, in pure CS theory, P i = 0, and moreover, with A 0 = 0, H = 0 [33]. When . (8.18) in contrast to (8.11). P i is gauge invariant and does not contain F , so Next, let us compute the derivative and the covariant derivative of the monopole operator ] also has the explicit x-dependence, so we should check whether ∂ i M R (x) can still be realized by the action of P i .

The action of the translation operator exp{iP
is the gauge transformation with the related transformation matrix u (x) = u(x−ξ), DU = DU . So is still valid.
The covariant derivative operator P i is gauge invariant, [U, P i ] = 0, can be written explicitly. Consider the gauge invariant operator O αR (x)M α R (x) with O αR (x) in representationR constructed from the matter fields.
When acting on the gauge invariant operators, P i reduces to P i , so For the 3d Chern-Simons-matter theory with the gauge group We should also consider the derivative and the covariant derivative along the time direction.
and a single current j µ = 1 4π µνλ (trF νλ + trF νλ ) (9.2) whose conservation follows from the equations of motion and the Bianchi identity. Using the Gauss constraint, (9.2) can also be written as the global U (1) charge current.
When k = 1, 2, with the help of the monopole operator, 12 extra currents [28,35] and their adjoints j µAB = j AB+ µ can be constructed, which, if are conserved, will offer the off-diagonal charges, making the original SU (4) × U (1) J symmetry enhanced to SO (8).
Let us consider the current conservation equation for j AB µ . In Appendix B, with the equations of motion plugged in, ∂ µ j AB µ could be written as W C D is a U (N ) × U (N ) gauge variation whose actions on X B and M are given by and respectively. V CD and U CB are U (N ) × U (N ) gauge variations whose actions on M are given by 14) The supersymmetry transformations of X C X D , X D X C and M are So For example, Up to now, (9.10) is verified.
In ABJM theory, from the SUSY invariance of the monopole operator, we can arrive at a conclusion similar to (9.10). In superspace formulation of the gauge theory, commutators of the covariant derivatives in superspace yield the field strengths, on which the constraints can be imposed. Supersymmetric gauge theories are entirely characterized by these constraints [36]. For ABJM theory in N = 6 superspace parametrized by coordinates (x µ , θ I ), covariant derivatives should satisfy the constraints [26] {D I ,D J } = 2iδ IJ γ µ D µ + iF IJ , (9.24) where D I is the covariant derivative along θ I , D µ is the covariant derivative in 3d spacetime, and F IJ = −F JI is the IJ component of the field strength in superspace whose lowest θ-expansion (9.25) Σ IJ C C = 0, so effectively, The operator realization of D I andD J is Q I andQ J . One may compute the anticommutator of supercharges directly, Usually, the current conservation equation may impose some constraints on the classical Lagrangian. For example, ∂ µ j A µB = 0 requires that the Lagrangian must be SU (4) invariant. To study the constraints imposed by ∂ µ j AB µ = 0, consider a truncated ABJM model with no spinor fields for simplicity, where the monopole operators satisfy D µ M = 0 and W C D M = 0. Among all of the SU (4) invariant scalar potentials, only the mass term m 2 tr(X A X A ) and the sextic potential in (7.14) could make j AB µ conserved. The requirement of symmetry enhancement almost fixes the interaction potential.
From the conserved currents (9.1) and (9.3), 15 + 6 + 6 = 27 conserved R-symmetry charges and are obtained, which, together with (3.17), compose the generators of the SO(8) group. The commutation relations of the R-symmetry charges obey the so(8) algebra. In particular, requires M ab ab Mĉb cb = δ a c δĉ a , which is proved in (5.27).

N = 8 SUPERSYMMETRY IN ABJM THEORY
The dynamical fields of the ABJM theory consist of X a Aâ , Ψâ Aa in4 representation and the adjoints X Aâ a , Ψ Aâ a in 4 representation of the SU (4) R-symmetry. Although there are 8 scalars (X A , X A ) and 8 spinors (Ψ A , Ψ A ), they cannot be transformed into each other by the SO (8) rotations due to the distinct gauge group representations. With the nondynamical Chern-Simons gauge fields A a µb andÂâ µb added, the monopole operator M R (x) can be obtained. In particular, when k = 1, 2, using M ab ab (x) and Mâb ab (x), one may construct the dressed fields [12,24]X The SO(8) symmetry group has 28 = 15 + 6 · 2 + 1 real parameters. The first 15 are ω B A 's satisfying (ω A B ) * + ω B A = 0, ω A A = 0. ω B A R A B generates the SU (4) transformation, under which and for adjoints. The 6 · 2 real parameters are ω AB 's satisfying R AB ω AB and R AB ω AB generate the transformations and for adjoints. For (10.6) and (10.7) to be consistent with (10.1) and (10.2), is again needed.
Aside from the monopole dressed matter fields, one may also introduce the monopole dressed gauge fieldsÃ c µa andÃĉ µâ with With the R-charges R AB and R AB added into the N = 6 superconformal algebra, two additional supercharges will be generated. For convenience, we will use instead of R AB and R AB . The commutator of Q I and R J gives two extra supercharges Q and the adjoint Q + : where of (9.10), the right-hand side of (10.17) can take several different but equivalent forms.
For example, using U CD M = 0, we have (10.19) so effectively, is in4 representation of SU (4). In [26], the constraint X BX A X C − X CX A X B ∼4 was derived in Eq (6.11) as a consequence of D µ M = 0. We have seen that such a condition can indeed be satisfied.
Q, Q + , and Q I altogether comprise 8 supercharges for the N = 8 supersymmetry. Under the action ofQε, K is a gauge variation with 1 + 1 parameters 2(X C X C −X CX C ) and 2(X C X C −X CX C ), When acting on the gauge invariant operators, (10.26) becomes 30) completing the N = 8 superalgebra.
In [32], a systematic classification of the unitary superconformal multiplets in d ≥ 3 spacetime dimensions is given. In the following, specified to ABJM model, we will give the explicit operator content for the 1/3 BPS stress tensor multiplet, 1/2 BPS extra SUSY-current multiplet, 1/6 BPS higher-spin current multiplet, free hypermultiplet of the 3d N = 6 SCFT and the 1/2 BPS stress tensor multiplet of the 3d N = 8 SCFT. The operators are characterized by [2j] (R) ∆ , where j is the half-integer su(2) spin, ∆ is the conformal dimension, and R is the Dynkin labels of the R-symmetry representation [32].
Besides, 1/6 BPS higher-spin current multiplet with chiral primary operators in [0] (1,0,0) 1 representation is also allowed in 3d N = 6 SCFT [32]. The whole multiplet is given by and fermionic currents (0,2,0) 2 in (10.39). Similarly to the Appendix B, we may find that j AB µ X and j µ AB Ψ are not conserved due to the fermionic interaction (7.13). This is expected, since otherwise, we will get the conserved higher-spin current which can only exist in free theory [37]. So the multiplet (10.39) is unprotected and would not bring the additional symmetry.
Finally, 3d N = 6 SCFT can also have the free hypermultiplets exchanged by complex conjugation [32]: [0] and In U (N ) SYM theories, the decoupled free sector is the trace of the fundamental fields, but here the monopole operators must be used.

CONCLUSION AND DISCUSSION
The main result of the paper is composed by two parts. First, based on the original definition in [1], we constructed the monopole operators and computed the contraction relations, classical conformal dimensions, supersymmetry transformations and the covariant derivatives. Second, with the concrete form of the monopole operators given, we studied their role in the global symmetry enhancement of the ABJM theory and proved several assumptions that were made on them to achieve the symmetry enhancement.
In Chern-Simons-matter theories with the nondynamical gauge fields, monopole operators commute with the supercharges, the covariant derivative operators, and some particular fielddependent gauge variation operators. The ordinary gauge transformations also commute with these gauge invariant operators, while the monopole operators are just the singular gauge transformations whose singularity does not have the manifestation here. As a result, except for changing the matter field's gauge representation, monopole operators do not have the side effect when combined with the matter, which makes them the suitable ingredients in the symmetry enhancement.
The conserved charge is the instanton number, and the corresponding global symmetry is the U (1) I symmetry. As the higher-dimensional analogues of the 3d monopole operators, instanton operators are local disorder operators creating instanton number on a S 4 surrounding their insertion point and could be defined by specifying the field configurations carrying the nonvanishing instanton number on S 4 [38][39][40]. Global symmetry enhancement may occur at the UV fixed point of the 5d gauge theories [41]. The original global symmetry algebra R ⊕ Q I of the theory can be enhanced toR = R ⊕ Q I ⊕ R off−diag if the currents for R off−diag can be built from the instanton operators and are conserved [42][43][44][45]. In parallel with the discussion for monopole operators, one may write down the instanton operators, construct the R off−diag currents, and investigate their conservation.

ACKNOWLEDGMENTS
The work is supported in part by NSFC under Grant No. 11605049.

Appendix A: Solitons and soliton operators
In a D-dimensional gauge theory with the gauge group G, the finite-action gauge field configuration should satisfy at the spatial boundary ∂R D−1 ∼ = S D−2 ∞ , i = 1, · · · , D − 1. So, the finite-action configurations provide maps from S D−2 ∞ to G, which is labeled by the homotopy group Π D−2 (G). Two maps are in the same homotopy class if they can be continuously deformed into each other. The surface integral counts how many times the group wraps itself around S D−2 ∞ and is simply 0 when Π D−2 (G) = 0.
When D = 3, G = U (N ), 2d gauge fields are classified by the homotopy group Π 1 (U (N )) ∼ = Z. The integer q ∈ Z is just the vortex charge (1.1) If |A i is in a homotopy class labeled by q, then Vortex operators are "large" gauge transformations that could make the states in different homotopy classes transform into each other. Consider a gauge transformation U with U A i U −1 = [Q, U ] = −kU . U is a vortex operator (monopole operator) carrying −k vortex charges.
U |A i is in a homotopy class labeled by q − k. The ordinary gauge transformation should have k = 0. When k = 0, g must be singular in at least one point, which is the location of the vortex operator. Such g is gauge equivalent to Ω m (ω) in section 2.
When D = 5, G = U (N ) or SU (N ) and N ≥ 2, 4d gauge fields are classified by the homotopy group Π 3 (G) ∼ = Z. The integer q ∈ Z is the instanton number (11.1) For a gauge transformation U with U A i U −1 = g −1 A i g + ig −1 ∂ i g, if 1 24π 2 ijkl S 3 ∞ d 3 S i (g −1 ∂ j g)(g −1 ∂ k g)(g −1 ∂ l g) = k , (A.10) then U Q I U −1 = Q I + 1 24π 2 ijkl S 3 ∞ d 3 S i (g −1 ∂ j g)(g −1 ∂ k g)(g −1 ∂ l g) = Q I + k . Direct calculation gives From the action (7.10)-(7.14), we have We may introduce the gauge variation W C D , the action of which on X Bâ R B 3 could be written as We have Next, consider iMΨ C R ABC 1 , whose explicit form is Construct a gauge variation V CD , the actions of which on X Bâ a andΨ B are given by respectively. iMΨ C R ABC 1 could be written as On the other hand, where (B.20) On the other hand, (B.21) We get 2 + iMΨ C R ABC The conservation condition for j AB [1] G. 't Hooft, "On the phase transition towards permanent quark confinement," Nucl. Phys. B 138, 1 (1978).