Charges of Monopole Operators in $\widehat{ADE}$ Chern-Simons Quiver Gauge Theories

We compute R-charges of the BPS-monopole operators in $\mathcal{N} = 3$ $\widehat{ADE}$ Chern-Simons quiver gauge theories, along the lines of the work of Benna, Klebanov and Klose in \cite{bkk}. These theories have a weakly coupled UV completion in terms of $\mathcal{N}=3$ supersymmetric Chern-Simons Yang-Mills theories. In the UV limit the monopole operators are well approximated by classical solutions. We construct classical BPS and anti-BPS monopole solutions to these theories which preserve $\frac{1}{3}$ supersymmetry all along the RG flow. We compute the $SU(2)_R$ charges in these backgrounds and show that the smallest possible value of quantised $SU(2)_R$ charge is zero in each quiver theory.


Introduction
Study of AdS 4 /CFT 3 correspondence [2]received a lot of interest after the discovery of a N = 6 superconformal Chern-Simons(CS) matter theory which describes the world volume theory of multiple M2 branes [16](also see [3]- [7]) in the low energy limit. This theory, widely known as ABJM theory, has gauge group U (N ) k × U (N ) −k where k is the CS level. In the matter sector there are four complex scalar fields in (N,N ) representation and their complex conjugate fields in the (N , N ) representation along with their fermionic partners. ABJM theory can be obtained as IR limit of a brane construction [17] [18] which preserves N = 3 supersymmetry. The field content in the brane construction is similar to ABJM theory but the gauge fields and their superpartners become dynamical in the high energy regime. One of the important objects in ABJM theory is the monopole operator which are crucial for the supersymmetry enhancement from N = 6 to N = 8 for special values of the CS levels k = 1, 2.
Monopole operator was first studied in the context of QED and supersymmetric QED in [8] [9] where it is defined as a vortex creating operator with unit vortex charge. The vortex charge is the conserved charge of the current J µ = 1 4π µνρ F νρ , which exists in any three dimensional gauge theory and conserved by virtue of Bianchi identity. Monopole operator can also be thought of as 't Hooft operator which is a topological disorder operator and naturally arises as dual to an "order" operator in a topological quantum field theory [13]. Disorder operators can not be expressed as polynomials of basic field variables in the Lagrangian. Therefore they are defined by specifying the singularities of the classical fields in the theory and performing the path integral with a boundary condition that the fields take those specified configurations at the point of singularity. For example a U (1) Dirac monopole in R 3 is defined by specifying the classical gauge field configuration, whereê ϕ is the unit vector in spherical polar co-ordinate system (r, θ, ϕ), the upper and lower sign is for northern and southern hemisphere respectively, q is the magnetic charge. Observe that the gauge field has a real singularity at r = 0. Therefore the statement that there exists such a monopole operator in a theory implies that we are inserting a singular gauge field at a point in space-time. Insertion of such a monopole operator at a point p amounts to integrating over the gauge fields which have a singularity at x = p such that the magnetic flux through a 2-sphere surrounding x = p is q. In U (N ) gauge theory a monopole operator is obtained by defining a homomorphism, One such mapping takes a U (1) element e iα to diag(e i q 1 α , e i q 2 α , ..., e i q N α ) ∈ U (N ). An algebra element gets mapped to H = diag(q 1 , q 2 , ..., q N ) ∈ u(N ). It is shown by Goddard, Nuyts and Olive in [29] that when H is written as a linear combination of Cartan generators then (q 1 q 2 , ..., q N ) are the weights of the dual of U (N ). The monopole operator will transform in the U (N ) representation with highest weight state labelled by (q 1 , q 2 , ..., q N ) [15].
Monopole operators play crucial role in establishing various non-perturbative dualities in three dimensional quantum field theories [11] [12] and are often useful in condensed matter systems [35][36] [37]. In AdS/CFT studies it has been found that monopole operators with zero conformal dimensions are the ones important for matching the spectra with supergravity. Such monopole operators in ABJM theory which are singlets under the global symmetries were first studied by Benna, Klebanov and Klose(BKK) in [1]. Since ABJM theory is strongly coupled for small values of k, which is the only coupling in the theory, it is difficult to compute the conformal dimension of the monopole operators. To overcome this difficulty BKK introduces a method which goes as follows: • A small coupling g is introduced through Yang-Mills deformation of the action which provides a weakly coupled UV completion of the theory. The UV completion is a N = 3 CS Yang-Mills theory 1 .
• To retain N = 3 supersymmetry one has to add dynamical fields in the adjoint representation.
Thus the R-symmetry group of the theory is SU (2).
• In the UV limit the monopole operators are described by classical BPS(anti-BPS) monopole solutions of the CS-Yang-Mills theory.
• Since the final aim is to compute the spectrum of conformal dimension of monopole operator which is valid at the IR fixed point, they compute a quantity in the UV which is related to the conformal dimension and does not change along the renormalisation group(RG) flow. This quantity is the quantised SU (2) R charge of the monopole operator. Due to the non-abelian nature of the R-symmetry group the quantised charges will take discrete values and will not be changed under the continuous RG flow in the parameter g.
• At the conformal fixed point these charges are related to the conformal dimensions of the BPS monopole operators by state operator correspondence.
Monopole operators in ABJM theory have been extensively studied over many years now. In the light of the important role played by monopole operators in ABJM theory we study monopole operators in a wide class of ADE CS quiver gauge theories. ABJM theory is the low energy world volume description of M2 branes probing a transverse toric hyperKähler manifold which has singularities of the form C 4 /Z k . Therefore it is natural to construct theories which can arise as world volume theories of multiple M2 branes probing other hyperKähler singularities. In [19] a N = 3 superconformal quiver CS matter theory was constructed which is a world volume theory of M2 branes placed at singularities in the transverse eight dimensional hyper-Kähler manifold and are dual by the AdS/CFT conjecture [24][25] [26] to M-theory on AdS 4 times the seven dimensional tri-Sasakian manifold whose cone is the eight dimensional hyper-Kähler manifold. The corresponding quiver of the field theory is a Dynkin diagram of A algebra. It is known that one can always construct a three dimensional N = 3 superconformal CS matter theory whose field content can be summarised by a ADE quiver diagram [31].
In this note we find classical monopole solutions and compute R-charges of the BPS monopole operators in N = 3 CS quiver gauge theories with A n−1 , D n and E 6 quiver diagrams along the lines of the work of BKK. The rest of the paper is organised as follows. In section 2 we construct Keeping in mind that in three dimension the R-symmetry group of N = 3 theory is SO(3) which is isomorphic to SU (2) at the algebra level, the on-shell component fields in the superfield expansion are arranged in R-symmetry representations in BKK as follows: The non-auxiliary scalars in the vector multiplet are arranged as 2 , where, the lower/upper index is the row/column index, i = 1, 2, 3 is so(3) vector index. φ a b(j) forms a 3-dimensional representation of SU (2) algebra. Fermions in the vector multiplet are written in terms of a 2 × 2 matrix, Therefore λ ab (j) transforms in the reducible representation 2 × 2 = 3 + 1 of SU (2) R . The SU (2) R indices are raised and lowered by the SU (2) metric ab with, 12 = 21 = +1 and the following relations hold.
The bifundamental matter fields are written as SU (2) R doublets as follows, The component action and the supersymmetry variations will be written in terms of the above R-symmetry representations.

A-type quiver
Before writing down the superspace action for quiver gauge theories let us first set up the notation to express the above field content via a quiver diagram. Figure 1 is a Dynkin diagram of affine A-algebra and can be used to represent the N = 3 field content discussed above. Each of the circles(nodes) are associated with a gauge group factor U (N (j) ), a CS level k (j) and contains a 2 The subscript (j) was not present in BKK and will be explained shortly gauge multiplet (V (j) , Φ (j) ). The subscript in parentheses (j) is used to label the nodes and the edges in the quiver which runs from 1 to n. The (j)-th edge is the one that joins the (j)-th node to the (j + 1)-th node and we should take (n + 1)-th edge to be (1)-st edge. The arrows represent the bi-fundamental hypermultiplets Z (j) and W (j) , in the representation (N (j) , N (j+1) ) and (N (j) , N (j+1) ) respectively. The gauge group of the theory is U ( The CS levels satisfy n (j)=1ñ (j) k (j) = 0 [31], whereñ (j) is the co-mark of the (j)-th node which is 1 for all (j) in this case, making n (j)=1 k (j) = 0. The above described field content can be used to write down a N = 3 superconformal CS matter theory. Such a theory was constructed by Jafferis and Tomasiello in [19]. The superspace action of Yang-Mills deformed CS theory consists of five parts, The first three parts involve only vector multiplet fields and the last two parts involve the hypermultiplet fields together with their minimal couplings to the vector multiplet fields. The CS term, The Yang-Mills term, g is a coupling of mass dimension 1 2 which is responsible for the RG flow. U α(j) = 1 is the super field strength.
The kinetic terms of the adjoint scalar and fermionic fields arise from, After introducing the dimensionful coupling g the theory is not conformal any more. At the IR fixed point g → ∞ which sets S YM and S adj to zero thus making the gauge fields and the adjoint fields non-dynamical. We can then integrate out the Φ (j) 's and recover N = 3 superconformal theory.
For the bifundamental matter fields we have minimally coupled action, The last part of the action is a superpotential term, where, We can write the gauge transformations of the fields under which the action is invariant as, where, Λ (j) ∈ U (N (j) ) is a chiral superfield.
We do Grassmann integration and integrate out the auxiliary fields to get the component action.
Since the R-symmetry group SU (2) is preserved all along the RG flow it is desirable to write the component action as follows where the SU (2) R symmetry is manifest.
The component action on R 1,2 with signature (− + +), written in terms of the above SU (2) R multiplets is, σ i are the usual Pauli matrices and the (σ i ) a b = σ T i are the transpose of the Pauli matrices. The various gauge covariant derivatives above are, It can be checked that when the above action is unpacked following the notation introduced in 2 reproduces the expressions given in appendix B.1. The supersymmetry transformation parameter ε in a N = 3 theory is in the 3 of SU (2) R : ε ab = ε i (σ i ) ab . The supersymmetry transformations of the non-auxiliary component fields that leave the action (2.14)+(2.15) invariant are, Figure 2 is a Dynkin diagram of D n algebra. The main differences in D quiver diagram from the A-type quiver are the external nodes (j) = 1, 2, 3, 4 which have only one edge attached to them and the nodes labelled by (5) and (n + 1) which have three edges attached to them. The hypermultiplet corresponding to an external edge say (1) is (Z (1) , W (1) ), which are in the gauge representations (N (1) , N (5) ) and (N (1) , N (5) ) respectively. The representations of the hypermultiplets for other external edges can be written similarly. Rest of the quiver, i.e from (j) = 6, ..., n is similar to the previous case. The CS levels satisfy (j)ñ (j) k (j) = 0.ñ (j) = 1 for (j) = 1, 2, 3, 4 andñ (j) = 2 for (j) = 5, 6, ..., n + 1 which implies,

D-type quiver
Such quiver gauge theory is dual to M-theory on AdS 4 ×Y , where, Y is the base of the hyperKähler [32]. The superspace action for S CS , S YM and S adj remains same except now the limit of summation runs form (j) = 1 to n + 1. The rest of the action is as follows: The minimally coupled action in the matter sector, (2.19) and with, which gives the following component action after doing the Grassmann integral and eliminating the auxiliary fields.
The interaction part of the action, The kinetic part of the action, where the covariant derivatives of the bi-fundamental fields are, Rest of the covariant derivatives are same as A case. The supersymmetry transformations of the vector multiplet fields are, Note: The reason that the transformations of λ ab (n+1) and λ ab (5) are different in spite of their symmetry in the quiver diagram is because of our convention of labelling the X (j) 's. There is no arrow coming out of the n + 1-th node like X (5) was coming out of the 5-th node. Therefore the terms g 2 iX a (5) X † c(5) ε cb and ig 2 2 ( 2 (j)=1 −(XX † ) (5) )ε ab are not present in δλ ab (n+1) . One other way to check this is, the term g 2 i (j)=3,4,n X a (j) X † c(j) ε cb is a matrix of size 3 × 3 where the L.H.S δλ ab (n+1) , is an (n + 1) × (n + 1) matrix.
The supersymmetry transformations of the hypermultiplet fields are, The E 6 quiver is similar to D n quiver diagram after adding some extra nodes and mapping 7-th node of the former to n + 1-th node of the latter. Therefore all the notions can be generalised easily from the D n case. The comarks in this case are,ñ (j) = 1 f or (j) = 1, 3, 5,ñ (j) = 2 f or (j) = 2, 4, 6 andñ (j) = 3 f or (j) = 7 which implies that the CS levels satisfy, The first three parts of the action (2.6) remains same as except the summation limit changes for 1 to 7. We write down the last two parts of the action, with, We write down the interaction part of the action since the kinetic part is straight forward from the previous cases.
The supersymmetry variations are,

Action and supersymmetry variations on R × S 2
Our main goal is to compute the scaling dimensions of the BPS monopole operators in the theory which is related to its charges by operator state correspondence in a radially quantized system.
Following the method of BKK, to obtain the action on R × S 2 from the action on R 1,2 one has to carry out the following steps. First we do a Wick rotation to go from R 1,2 to R 3 in the following way, The next step is to do a co-ordinate transformation from (x 1 , x 2 , x 3 ) to the polar co-ordinates (r, θ, ϕ). Under this co-ordinate transformation all the terms in the action except the measure is invariant since they are written in a co-ordinate independent form. Then we translate the action to R × S 2 by introducing a radial variable τ by the relation, where τ ∈ R, runs from −∞ to +∞, is the Euclidean time. Now one has to perform a Weyl rescaling of the metric by a factor of e −2τ so that the theory lands on R × S 2 described by (τ, θ, ϕ) co-ordinate system. The metric after Weyl rescaling is, g mn = diag(1, 1, sin 2 θ). m, n = τ, θ, ϕ are used to denote the space-time indices on R × S 2 . One has to also rescale the R 3 fields to obtain R × S 2 fields in the following way, where, X is any generic field. The coupling g is also rescaled by g is the Yang-Mills coupling on R × S 2 (for the component fields we don't use any tilde).
The vector multiplet fermions remains auxiliary in the IR and drop out form the theory. But since they were dynamical in the UV and can effect the charges of the monopole operators by quantum fluctuation one has to rescale them before doing the Weyl rescaling in the following way,

A-type quiver
Carrying out all the steps described above we convert the action on R 1,2 to R × S 2 . The kinetic part of the action, The interaction part is, The supersymmetry variations obtained for the action on R 1,2 can also be converted to R × S 2 by following same steps as stated above. The supersymmetry variation for the action on R × S 2 with the rescaled variation parameterε ab = e − τ 2 ε ab are as follows,

D-type quiver
The kinetic part remains similar as the previous case. We write down the interaction part of the action on R × S 2 , Supersymmetry variations on R × S 2 of the vector multiplet fields, Variations of the hypermultiplet fields, TheÊ 6 action on R × S 2 is calculated following the same steps. We do not write the expression of the action here since it is straightforward from the D-type quiver. Let us write down the supersymmetry variations which are needed to find the monopole solutions later.
Variations of the hypermultiplet fermions, In this section we compute the classical bosonic BPS and anti-BPS monopole solution to the theories under consideration. To find a monopole solution we start with a Dirac monopole solution where, H = diag(q 1 , q 2 , ...q N ) and q i ∈ Z are the magnetic charges. The upper sign is for the northern hemisphere and lower sign for the southern one. The the dual of the field strength of the above gauge field, is a constant. If there exists a BPS monopole solution with such a gauge potential (3.1), then corresponding fermionic variation δλ ab (j) , δξ a (j) , δξ † a(j) should be equal to zero for a non trivial supersymmetry variation parameter. As we can see form (2.38) that δλ ab (j) contains terms of orderg andg −1 . The goal of finding solutions all along the flow means we have to set them separately to zero. In our following calculation we are also assuming that the background fields (A, φ) are in the Cartan subalgebra of the gauge group factors, which makes all commutator vanish.
δλ ab (j) at order 1 g : Equating orderg −1 terms of (2.38) to zero, Recalling that δλ ab (j) is in the reducible representation 2 × 2 = 1 + 3 of SU (2) R we isolate the 1 part i.e. the SU (2) R trace, by computing δλ ab (j) ba . The first term is zero since ε a a = 0. Therefore only the trace of the second term contributes to the SU (2) R trace of δλ ab (j) at order 1 g . Since, mnk F mn is a constant, we can have a simple situation where φ's are also constant. Then the second term of the above equation is zero. The third term in (3.3), after using the the Killing spinor equation on gives, i.e, the supersymmetry variation parameter is orthogonal to the background scalar. We can solve the above by choosing, Then we isolate the 3 part by computing δλ ab (j) (σ i ) ba and we get, easily solvable in two ways: In the first case, the preserved supersymmetry is ε 1 + iε 2 and is the BPS solution and the second case, the preserved supersymmetry is ε 1 − iε 2 and is the anti-BPS solution. In both cases, since one of the three supersymmetry parameters are preserved by the solution we have 1 3 -BPS solutions. Now let us examine the orderg terms in δλ ab (j) .
δλ ab (j) at orderg: We obtain from (2.38) We analyse the above as before by considering the 1 and the 3 parts separately. The SU (2) R trace of the first term is κ (j) ε i φ i(j) which vanishes on using (3.5) and the third term is traceless. The second term gives By isolating the 3 part of (3.10) one obtains the following equation, by which we see that the hypermultiplet scalars are already constrained by the CS levels and the magnetic charges in addition to other constraints that are yet to come from analysing the hypermultiplet fermions variation equations.
Moving on to the hypermultiplet fermion variations, δξ a (j) : We obtain from (2.39) The last two terms cancel 3 because in the special case we are working in, all the φ (j) 's are equal.
Using Killing-spinor equation we get, Expanding the above, and Demanding δξ a (j) = 0, fixes the functional dependence of X a (j) to, X 1 (j) ∼ exp(−ητ /2) and X 2 (j) ∼ exp(ητ /2). Then we can make all δξ a (j) 's zero either by the functional dependence or by the condition derived above ε 1 − iηε 2 = 0. The solution chosen for the X's also satisfy their classical equation of motion. The only other equations of motion that remain to be satisfied by the so-far obtained background are for the gauge fields: To summarise, we have obtained the background gauge and adjoint scalar fields, with ε 1 +iη ε 2 the preserved supersymmetry. And we have several constraints on the hypermultiplet scalar fields: (i) equation (3.11), (ii) equation (3.12) (iii) the τ dependence that follow from the hypermultiplet fermion variations and (iv) the equation (3.15). We will analyse these constraints and find solutions first for the three node quiver and with the experience gained thus, we can then generalize for a generic n-node quiver.

Example of a three node case
In the previous section we saw how to choose the fields in the vector multiplet to make the fermionic variations(both vector and hypermultiplet) zero. In this section we will consider a simple example of a three node case and see how to completely fix the hypermultiplet scalar fields to obtain full Figure 4: A quiver diagram with three gauge groups monopole solution. For that we need to solve, where, l = 1, 2, 3.
(3.17) when expanded in terms of components gives the following set of equations, Similarly, For a simple case, there can be three possibilities which solve the aove six equations.
Now expanding φ 3 , in terms of X and X † from (3.17) we get the following equations, We need to solve the above three equations for the three cases written above. Since the Chern-Simons levels satisfy, are solved by, The above equations are satisfied for any diagonal matrix A with entries ≥ 0 assuming, κ (1) , κ (3) > 0 and κ (2) < 0.
anti-BPS η = −1: Similarly the anti-BPS case is solved by, where, where A α and q α denote the α-th entry of the matrix A and H respectively. Similarly the rest of the cases can be solved.

n node case
Now we can move on to more general case with gauge group U (N ) × U (N ) × ... × U (N ) and apply similar steps as above to obtain the full monopole solution in this theory. From (3.18) and (3.19) we can see that, Because all Z (j) 's and W (j) 's are diagonal, if we take d α to be the α-th entry of the matrix Z (j) W (j) , then Also from (3.20) we can write, Using the above equation again and again we get, where, Using (3.24) and substituting in (3.25) Above is a quadratic equation in u (j) with a solution, These seem to exist for any value of κ (j) . These form a moduli space of solutions classified by . The above is a general solution.
Let us examine the case if we choose d α = 0. Then from (3.24) either v (1) or u (1) is zero. Also from (3.25) Now, we have a constraint that κ (1) + κ (2) + ... + κ (n) = 0. Therefore some of the κ (j) 's are positive and the rest are negative. In other words there will be some positive K (j) 's , let us call that set [K + ] and rest of them will be negative which are denoted by [K − ]. Now let us solve case by case as we did in the three node example. For, Since u (1) is positive, positivity of the LHS implies solutions exist for K (j) > 0. For negative K (j) 's There is no restriction on u (1) for [K − ]. But for [K + ] we get, There is no restriction on v (1) Similarly, Now, let us look at the solutions. For the nodes where v (j) = 0, from (3.29) For, Now this K (j) will either belong to For the nodes where u (j) = 0, we have, For, There is no solution for [K u − ]. But for [K u + ] we get, The above four cases exhaust all the possibilities. To summarize,

BPS solutions
Therefore we can conclude that we have obtained full monopole solution in the CS Yang-Mills A quiver theory with N = 3 supersymmetry and with gauge group U (N ) × U (N ) × ...U (N ).

D-type quiver
We consider the gauge group of the theory to be U (2N ) n−3 × U (N ) 4 [30][31] [32] since these are the ones whose gravity dual is AdS 4 × M 7 vacua of M-theory, where M 7 is a tri-Sasakian manifold.
Also this class of theories have been considered in the context of ADE matrix models since they have a nice large N limit. Since the ranks of all gauge groups are not equal like the previous case, we need to choose two ansatz for the gauge fields as following, The logic here is same as the A quiver case, i.e we need to make all the supersymmetry variations of the theory zero for such gauge ansatz with a non trivial variation parameter. Since the order g −1 terms in δλ ab (j) is same for (j) = 1 to n + 1. The analysis is same as before and we get analogues of (3.5)(3.8), Contracting with σ ab to isolate the 3 part we get, Similarly, δλ ab (5) |g = 0 gives, δλ ab (j) |g = 0, for (j) = 6, ...n gives, At this stage we have obtained solutions for A (j) and φ i(j) explicitly. But to call it a monopole solution we need to check if these solutions make all δξ a (j) 's and its complex complex conjugates zero. Let us verify it, Using Killing spinor equation (3.4) above, we get, for a = 1 In the anti-BPS case (ε 1 + iε 2 ) = 0 hence the above supersymmetry variation vanishes. But in the BPS case the following has to be zero, To cancel the first two terms we fix the functional dependence of X 2 (j) ∼ e ητ /2 . To cancel the last two terms for the simplest case, one has to fix the diagonal entries of H (1) and H (2) to be all equal, say q (of course there can be non-trivial choices) (3.46) By taking the above ansatz one can check that all the supersymmetry variations δξ a (j) and their complex conjugates δξ † a(j) will vanish and we take the functional dependence of the hypermultiplet scalars to be X 1 (j) ∼ e −ητ /2 , X † 1(j) ∼ e ητ /2 , X † 2(j) ∼ e −ητ /2 . Now we can generalise same steps used in the A case to solve for the hypermultiplet scalars. For (j) = 1, ..., 4: First two equations imply that Z (j) W (j) = 0. Therefore we have the following two cases.
For a positive semi-definite solution we get more constrains on the CS levels associated to the external nodes of the quiver, i.e ηκ (j) ≤ 0.
For (j) = 5: Using the solutions from the previous the last summed over terms vanish and the remaining can be solved exactly as before.
For (j) = 6, ..., n: This analysis is similar to the A case.
While fixing the coefficients of X and X † 'one has to keep in mind that they should satisfy the equations of motions of the gauge fields, , (j) = 6, ..., n (3.50)

E 6 quiver
Solving for the hypermultuplet scalars forÊ 6 is straight forward from the D case. The theory we analyse has gauge group U (2N ) 3  Equating orderg −1 of δλ ab (j) one chooses the background scalars to be, δλ ab (j) |g = 0 for (j) = 1, 3, 5 δλ ab (j) |g = 0 for (j) = 2, 4, 6 To make the variations of the hypermultiplet fermions vanish, lets say for (j) = 2, 4, 6, one has to make the following zero, for a = 1 Equating this to zero we get the functional dependence of the X's same as before and, The Dirac operator, / D = γ τ ∂ τ + / D S , where / D S is the operator on S 2 , contains a U (1) monopole solution with charge q. The associated conserved current is, which has the following conserved charge, Our goal is to find the normal ordering constant of (4.2). To solve for ψ from (4.1) we use the machinery of monopole spinor harmonics [20].

C is an integration constant
Substituting this back in equation of motion we get, Similarly one obtains, where, P, Q, R, S are integration constants and fixed by using normalisation of ψ and canonical anti commutation relations of the operators, E j = j + 1 2 is the energy of the corresponding state. Quoting the final solution from BKK, (Note: In the expression of ψ † we reversed sign of τ since we are working with Euclidean time.) The wave-functions for the, BPS case (η = +1), anti-BPS case (η = −1), Now, the U (1) R charge can be computed by using point splitting regularisation [9] where β > 0. In the end we take the limit β → 0. Now substituting (4.9) above and using properties of monopole spinor harmonics and canonical anti-commutation relations we get, whose normal ordered piece at β = 0 is, with a normal ordering constant, where in the last sum the zero modes with j = |q|−1 2 is also included. Observing that ε u ε † j u ε = 1 for every positive energy state(∼ e −E j τ ) and ε v ε † j v ε = 1 for every negative energy state(∼ e E j τ ), we can write, The above quantity would be zero when we have a symmetric spectrum with respect to E = 0. But after turning on the scalar fields one finds that the energy corresponding to the zero mode is, − |q| 2 for BPS states and |q| 2 for anti-BPS states, i.e the zero mode energy spectrum is not symmetric 4 But for non zero modes both positive and negative energy states are present for a fixed value of j.
Therefore the normal ordering constant is non zero only in the case of zero modes, factor |q| arises because of the sum over zero modes, which has multiplicity |q|. Bosonic fields do not contribute to the U (1) R charge because their spectrum is symmetric [9].

Application to A-type quiver
Now we can apply the above method to N = 3 Yang-Mills deformed CS ADE theories. We first compute the SU (2) R current and then extract the U (1) R part from that. Under infinitesimal SU (2) transformation fundamental and anti-fundamental SU (2) R indices transform as, where X represents a generic field. Hence the fields in the vector multiplet transform as, The fields in the hypermultiplet transform as, The energy spectrum plot can be found in BKK.
The conserved SU (2) R current, Now the expression of charge in (4.2) has become, The equations of motion of the fermions in the far UV limitg → 0, where, q mr ≡ q m − q r . comparing this with (4.1) one sees that now the effective monopole charge is q m − q r . Using the result (4.16) form the previous section and adding contributions from hyper and vector multiplet fermions, we get, Therefore U (1) R charges of the monopoles of N = 3 A-type quiver gauge theory is zero. This result is as anticipated as in BKK who obtained, Q mon where N f is the number of hypermultiplets between two nodes. Therefore for N f = 2 which is ABJM we get Q mon R = 0. Our result indeed matches with ABJM(i.e n = 2) as it should.

D-type quiver
This case is almost similar to the previous case except for the external edges. The U (1) R component of the Noether current is, The equation of motions in the far UV limit of the hypermultiplet fermions, (j) = 5, .., n : / equations of motions of the vector multiplet fermions, It is straight forward to generalise (4.24) for the internal edges, i.e (j) = 5, ..., n except now we have q m = q r . This implies that the fermions associated to the internal edges do not interact with the monopole hence do not contribute to the U (1) R charge. For the external edges, where the hatted gauge indices imply that ζ (1) is an N × 2N matrix and so on. Therefore the hyperinos associated with the external edges also do not contribute to the charge. This happens because of our choice of the gauge ansatz in (3.46). Therefore we find that the U (1) R charges of the monopole operators in the D case do not receive any quantum corrections. This readily implies that U (1) R charges of the monopole operators in the E 6 case also do not receive any quantum corrections.

SU (2) R charges of the monopole operators
In this section we compute the quantised SU (2) R charges of the monopole operators which is the main goal of this note. In the previous sections we have shown that the Yang-Mills deformations to the superconformal CS theories under ADE classification preserve N = 3 supersymmetry. Also we have explicitly solved for the hypermultiplet scalars, which implies that monopole solution exists for more than two gauge groups. Therefore it is legitimate to use BKK method, in ADE quiver theories as well to compute the SU (2) R charges. We report in this note that, the smallest possible representation of the SU (2) R charge is zero. This result was anticipated in [1] for the A-type quiver and here it is verified by explicit calculations.
Let us first briefly describe the method used in BKK which is collective co-ordinate quantisation method to obtain the SU (2) R charge of the BPS background. The first step in obtaining the charges is to generalise the BPS background in (3.16)(and their analogs for DE) to arbitrary SU (2) R orientation, where, n i is the unit vector on two sphere SU (2) R /U (1) R . It can be checked that these are the bosonic zero modes by computing the equation of motion for φ (j) . In the previous section φ i(j) was chosen in the 3 direction with n i = δ i3 . Therefore the conserved quantity with that background was U (1) R charge. Now, to specify the the collective co-ordinates one makes the unit vector n i time dependent, n i (τ ) is the collective co-ordinate of the BPS background. Now the global SU (2) R symmetry can act on this background whose action is to rotate n i (τ ) on the two sphere. Since the collective co-ordinate is interacting with the fermions of the theory the motion is not free. The effect of these interactions are obtained by calculating the effective action of the collective co-ordinate, i.e by integrating out the fermions. From the effective Lagrangian we compute the conserved quantity by Noether's procedure. The conserved quantity will be the angular momentum since SU (2) R acts as rotational symmetry. After computing the angular momentum we compute its quantized values which are the SU (2) R charges of the BPS-monopole operators in ADE theories.

Quantum effective action of the collective co-ordinate
In this section we present the details for obtaining the effective action of the collective co-ordinates.
Following BKK we first consider a simple model with one fermion in an abelian gauge theory. Then we will generalise the results of the former to Yang-Mills deformed quiver theories.
Let us consider a fermion ψ a (τ ) in the fundamental representation of SU (2), on R × S 2 with the action, where the operator, The orthogonality property of monopole harmonics ensures that modes with different (jm) values and zero modes do not couple to each other. Therefore, the effective action can be computed easily for each (jm) separately.
The effective action for this system, To evaluate this functional determinant first we write the general form of effective action using derivative expansion in n i (τ ), Now we will expand both (5.5) and (5.6) and compare to find A i ( n). We are keeping terms upto first order derivative in n i , because the higher orders will be suppressed(in the far UV limit g → 0) by a term proportional to 1 g 2ṅ 2 i which comes from the bosonic kinetic term in the action. Now, to expand the effective action we write, where,n i is a constant, satisfyingn 2 = 1 andñ i (τ ) is a small fluctuation. Expanding (5.6) around The above says that to determine A i we will have to look at the terms with two powers ofñ i with one derivative. Such term is denoted by Γ (2,1) ( n). We present the final result, details of which can be found in BKK.
where, q 2n i :=m i , q 2ñ i :=m i , ω is the energy. The contribution to the effective action from non-zero modes cancels because of the fact that ∆ + = −∆ − . comparing the above with, (5.8) we get, The factor |q| occurs because of the sum over zero modes which has multiplicity m = 2j + 1 = |q|.

Application to A-type quiver
Now we are ready to generalise the previous result in our case. The relevant part of the action in (2.36) and (2.37) for computing effective action is, Now, to take care of the gauge indices we apply the same trick as (4.24) and rewriting the action with gauge indices, To write the vector-multiplet fermion action in the above form so that we can use the result from the abelian case, we have defined the following similarly as BKK, The above action is similar to (5.3) except for the vector multiplet where the sign of interaction term is changed. Now we can treat each matrix elements of the fields as abelian fields and use the result from previous section to compute the effective action. The only modification we have to make is to put a negative sign in the final result for vector multiplet fermions. We obtain, which means the induced monopole charge, in the SU (2) R moduli space, due to fermionic interaction is zero. This happens because of the field configuration of A-quiver. Quoting the result of BKK who obtained, which is zero for ABJM where N f = 2 and hence our result is consistent.
The effective action adding all contributions from bosons and fermions, where, M = n 2g 2 tr(H 2 ) = n 2g 2 e −2τ tr H 2 (5.16) The above action can be thought of as a free particle of mass M moving on a unit sphere due to the presence of Lagrange multiplier(last term) in (5.15). The conserved angular momentum, L = iM n ×˙ n (5.17) whose quantized values are l = 0, 1, 2.... Now, to find the conformal dimension of the monopole operators we solve the Schrodinger equation and read off the scaling dimension from the wave function by using state operator correspondence at IR. It can be checked that to get the correct behaviour of the wave function one has to include second order correction(Γ 2,2 ( n)) in the effective action. By doing so we recover the correct exponential behaviour of the wave function and read off the conformal dimension of the monopole operator by using state operator correspondence. This also verifies that the lowest energy state l = 0 is BPS in our case and hence the lowest possible value of the conformal dimension is zero.

Application to D-type quiver
It is easy to see now that in D case everything from the previous section follows for the internal edges. For the external edges also one can generalise the results as follows.
The relevant part of the action for charge computation is, The last summation is same as A case. Rest of the interaction terms for the hypermultiplets after substituting the monopole solution for the gauge fields, Therefore the external hypermultiplet fermions do not contribute to the charge. By doing same manipulations as above it is easy to see that the hypermultiplets associated to the internal edges also do not contribute to the charge. This can be anticipated from the A case if we set q m = q r .
Similarly the action for the vector multiplet fermions after substituting the BPS background is, Now the commutator in above expression, which implies that they do not contribute to the SU (2) R charge. Therefore, The same logic applies in the E 6 case as well and we get same result as above.

Discussion
To summarise our results, • In this note we have constructed actions and supersymmetry variations of three dimensional N = 3 Yang-Mills deformed CS quiver gauge theories with A n−1 , D n , E 6 quiver diagrams.
These theories flow to a conformal fixed point in the IR via RG flow.
• We have obtained 1 3 BPS and anti-BPS monopole solutions in the A n−1 quiver case with equal ranks of all gauge groups. • We find that the lowest possible value of quantised SU (2) R charge for each quiver theory is zero. In the A case, it is observed by calculating the path integral for the adjoint fermions that they give negative contribution to the R-charge and therefore cancelling the positive contribution from hypermultiplet fermions and making net quantum correction zero. The contribution from the adjoint fermions wouldn't have been captured in the IR theory, since thy are not dynamical in the IR. In the DE case we find SU (2) R charges to be zero for the same reasons explained in the previous point.
Our result is similar to ABJM theory which has monopole operator of zero conformal dimension.
In fact these monopole operators are the ones needed to match the spectrum of ABJM theory with dual gravity theory.
Some of the interesting questions that maybe worth exploring are as follows: • It is well known that in ABJM theory there is a supersymmetry enhancement from N = 6 to N = 8 for k = 1, 2. In this phenomena monopole operators played an important role. It will be interesting to check if there is any supersymmetry enhancement in ADE theories as well.
• To the best of our knowledge the brane construction for exceptional quivers is not well understood [33]. One can try to see if the results derived here are useful for brane engineering [34].
• In interacting CFT's, monopole operators are usually studied via state operator correspondence, where the monopole operators become states on R×S 2 (considering three dimensional theories), which provides quantized flux through S 2 due to a monopole kept at the centre of S 2 [8]. Then one quantizes the theory in the monopole background and find several quantities like scaling dimensions, superconformal index. A recent study on monopole operators in CS matter theories in [23] proposes a prescription to describe the monopole operators as local operators directly on R 3 . In a N = 2 abelian SQED with single charged chiral multiplet and a CS term with level k, it can be done by giving a singular profile to the bosonic and fermionic fields in the theory along with the singular gauge field at the insertion point, keeping in mind that they should be consistent with the equations of motion and Gauss law constraints. They find 1 4 -BPS monopoles on R 3 and compute their dimensions. As suggested in [23] that this method is applicable to continuous deformation of ABJM theory as it won't affect the discrete global charges of the monopole operators in the theory. Therefore it would be nice to apply this method in ADE theories as a consistency check.
Acknowledgements I am grateful to my advisor Chethan N. Gowdigere for suggesting me the problem. I would like to thank Bobby Ezhuthachan and Chethan N. Gowdigere for collaboration at the initial stage, many helpful discussions and reading the manuscript. I would like to thank Ashoke Sen and Palash Dubey for helpful discussions. I would like to thank RKMVERI and HRI for the warm hospitality where some part of the work was done.

B Component action computation
We use dimensionally reduced N = 2 multiplet of four dimension which are again written in terms of N = 1 superfields. The component expansion of the N = 2 superfields are given as follows. In the gauge multiplet we have a vector superfield, and an adjoint chiral superfield, In the hypermultiplet we have two bifundamental chiral superfields, 4π . The Yang-Mills part, The component expression of S adj , The matter action, The component action for S pot , Auxiliary fields are eliminated by using their equation of motions, After eliminating the auxiliary fields,
We present the component action for only the L pot and L mat rest of the part is straight forward from the previous section.

C Checking supersymmetry variation of D n
We present some steps of checking the supersymmetry variation of D n action. We vary all the fields in (2.21) and (2.22) simultaneously and substitute (2.23) and (2.24). The terms which combine and cancel are labelled by same alphabet. We do not write tr in front of every term, which implies the trace of gauge indices.

C.2 Variation of the interaction part
We now explicitly show the term by term variation of the interaction part of the action, substituting transformation equations in each terms we get the following terms, For (j) = 5: For a simple solution, there can be the following possibilities which solve the above equations. For (j) = 5: Solving the above,  (5) . For a positive semi-definite solution, κ (1) ≥ 0, κ (2) ≤ 0, κ (3) ≥ 0, κ (4) ≤ 0, κ (5) ≤ 0. The anti-BPS case can be similarly solved by reversing the signs of the CS levels.