Monopoles and dualities in $3d$ $\mathcal{N}=2$ quivers

Seiberg-like dualities in $2+1$d quiver gauge theories with $4$ supercharges are investigated. We consider quivers made of various combinations of classical gauge groups $U(N)$, $Sp(N)$, $SO(N)$ and $SU(N)$. Our main focus is the mapping of the supersymmetric monopole operators across the dual theories. There is a simple general rule that encodes the mapping of the monopoles upon dualising a single node. This rule dictates the mapping of all the monopoles which are not dressed by baryonic operators. We also study more general situations involving baryons and baryon-monopoles, focussing on three examples: $SU-Sp$, $SO-SO$ and $SO-Sp$ quivers.


Introduction and results
Quiver quantum field theories are gauge theories with product gauge group and matter content in rank-2 representations, such as bifundamental and adjoint representations. Since strings can end on two different branes, quivers are ubiquitous in string theory compactifications. For this reason, quiver theories have been studied extensively during the past three decades.
Theories with four supercharges, in 3+1 or less dimensions, enjoy Seiberg dualities [4,5,[17][18][19][20][21][22], relating two different UV theories with a single gauge group which flow to the same IR superconformal field theory. Applying Seiberg duality to a node of a quiver gauge theory, one gets a new quiver theory with the same number of nodes.
We study examples of 3d N = 2 quiver dualities. In each case we work out the map of chiral ring generators (in the algebraic sense) across the duality, i.e. from one theory and a one obtained by dualizing a specific node in the quiver. It is quite easy to map mesonic operators, which also exists in 3+1 dimensions. In 3d gauge theories, however, there are also monopole operators [23,24], that is local disorder operators which under duality can map to standard operators polynomial in the elementary fields .
One of the main theme of this paper is the mapping of the monopole operators under Seiberg duality inside a quiver. In the case of linear quivers with unitary gauge groups such issue was important in a series of recent works [9-12, 14, 15]. The process of applying a duality on a node of a quiver requires to take into account possible contact terms that may become non-trivial BF couplings when the duality is applied inside a quiver [6,25,26] (we will be more explicit about this in the main text).
We investigate various quivers with two gauge groups, studying in each example two different duals. In the original model we take the superpotential to be vanishing. It is possible to turn on superpotential or real mass deformations, even if we do not study such deformed dualities in this paper.
The gauge groups we consider are a combination of classical groups U (N ), Sp(N ), SO(N ) and SU (N ). Although we study quivers with only two nodes, the results allow to find the general rule for the mapping of supersymmetric monopole operators under Seiberg duality. The rule is valid for quivers with an arbitrary number of gauge groups and generalizes the findings of [14,15]. The quivers need not be linear, but let us state the monopole mapping rule in the case of linear quivers, since it is simpler.
We denote monopoles in a linear quiver with the notation M 0,0,•,•,... : a 0 in the i th position means that there is vanishing GNO flux for the i th gauge group, while a • in the i th position means that there is minimal GNO flux for the i th gauge group. For unitary gauge groups U (N ) the •'s can be either all +'s or all −'s.
The monopoles which are chiral ring generators have minimal GNO fluxes for each node, and the non-zero fluxes are turned on in a single connected group of nodes (of arbitrary length), of the form M ...,0,0,•,•,•,0,... . Dualising node i, the rule is as follows: • a monopole with zero flux under i − 1, i, i + 1, stays the same:  From the mapping of these basic monopoles, the mapping of generic dressed monopoles follows.
Let us emphasize that the rule stated above is not enough to deal with baryonic operators and baryon-monopoles, which are present when there is a gauge group defined by the determinant = 1 condition (SU (N ) or SO(N )), or when U (N ) is present together with O, Sp, SU , SO. 2 We study various examples of quivers with baryons and baryonmonopoles, with two gauge groups. In these specific examples we are able to find the full mapping of the chiral ring generators. One interesting observation (Section 3.2) is that in the case of SO − SO quivers a monopole operators with three magnetic fluxes (two non-zero fluxes in one SO node and one non-zero flux in the other SO node) turned on is a chiral ring generator. This is to be contrasted with the SO gauge theory, where the chiral ring generators have at most one magnetic flux turned on.
We will present an application of the results in this note in two companion papers [33? ]. In [33] we study the duals of a theory with a single gauge group and matter in the a rank-2 representation, U (N ) with a single adjoint field and flavors, or Sp(N ) with a single antisymmetric field and flavors. We produce a dual which is a quiver with N gauge nodes, obtained by sequentially deconfining the rank-2 field. The process requires to use many times a Seiberg-like duality inside a quiver, and at each step, in 1 Notice that we do not consider M ...,•,0i,•,... . This operator is not a chiral ring generator, its mapping is obtained from the mapping of the two monopoles M ...,•,0i,0,... and M ...,0,0i,•,... . 2 Theories with alternating Sp/(S)O groups appears naturally in the context of 3d N = 4 theories via their realization as gauge theories on the D3 branes on top of orientifold planes. A large class of such theories is realized in the context of S-duality walls in N = 4 SYM with real gauge groups [27]. Various properties of such 3d theories with 8 supercharges have been widely discussed [28][29][30][31].
Applications to theories with four supercharges are studied in [32].
order to control the superpotential, it is crucial to have the mapping of all the chiral ring operators, including the monopoles.
The paper is organized as follows.
In section 2 we consider quivers with U (N ) or Sp(N ) gauge groups (so there are no baryonic operators, and no monopoles dressed by baryonic operators). In each case we study in detail a two-node quiver with flavors: we produce two different dual theories, obtained by dualizing the left or the right node. We tried to make each subsection readable independently from the other.
In section 3 we study quivers with baryons and baryon-monopoles, including SO and SU gauge groups. We work out three examples, with gauge groups SO(N 1 ) × Sp(N 2 ), SO(N 1 ) × SO(N 2 ) and Sp(N 1 ) × SU (N 2 ). In each example we are able to find the mapping of all the chiral ring generators.
The main tool we employ to perform our analysis is the 3d supersymmetric index, that can be computed as the partition function on S 2 × R [35][36][37][38][39][40][41][42], whose fundamental definitions and properties we review in appendix A.

Duality in quivers without baryons
In this section we study quivers with gauge groups U (N ) and Sp(N ). With this choice of gauge groups, there are no baryonic operators, and no monopoles dressed by baryonic operators (we consider examples with special gauge groups, SU (N ) or SO(N ), and hence baryon and baryon-monopoles operators, in section 3).
In each case we study in detail a two node quiver with flavors: we produce two different dual theories, obtained by dualizing the left or the right node. We discuss in great detail the global symmetries of the supersymmetric monopole operators which are generators of the chiral ring and we exhibit the map of the full set of chiral generators. This is enough to find the general rule for the mapping of the chiral ring operators in arbitrary quivers, which is the main goal of this paper.

Unitary gauge groups
We assume N 2 ≥ N 1 otherwise the strongly coupled gauge dynamics of the left node breaks supersymmetry (via a runaway potential). The global symmetry group is where U (1) B and U (1) Q are the axial symmetries acting on B,B and Q,Q and U (1) T 1 and U (1) T 2 are the topological symmetries for U (N 1 ) and U (N 2 ) respectively. The R-charge of monopole operators M m, n reads Specialising this formula for the monopoles with minimal GNO flux M ±,0 , M 0,± , M ±,± we get The chiral ring generators include both mesons and monopole operators. The former ones are represented by dressed mesons of the form Tr(Q i (BB) J Q j ) for J = 0, . . . , N 1 and Tr (BB) J with the same range for J. The monopole generating the chiral ring In the following we will discuss the two Aharony dual frames and discuss the map of chiral ring generators.

First dual
Let us apply Aharony duality to the U (N 1 ) node, we get: (2.7) The basic monopole operators have the following R-charges (2.10) The R-charges of fundamental fields can be mapped to the ones of theory T A using that (2.11) and the superpotential term Tr(bφ b b), implying that The mesonic part of the chiral ring is generated by dressed operators Tr(q i φ J bq j ) and Tr(φ J b ), for J = 0, . . . , N 2 . As already discussed in the case of the triality for Sp(N 1 ) × Sp(N 2 ) gauge theory, the chiral ring generators include dressed monopoles. In detail, we have the singlets σ ± B flipping the monopoles M ±,0 B and the monopoles with flux on both gauge nodes M ±,± B . The monopoles M 0,± B are the ones that can be dressed with the adjoint field φ B , and we get the tower {(M 0,± B ) φ J B } with J = 0, . . . , N 2 . Observe that in theory T B there is no dressed monopole of the form {(M ±,± B ) (bb) J } since the F -term equations set to zero the matrix bb on the chiral ring.

Second dual
The formula for the R-charge of a monopole with general fluxes is completely analogous to (2.7), hence we do not repeat it here. We write down the R-charges of monopoles with minimal GNO flux The R-charges of fundamental fields can be mapped to the ones of theory T A using the map of mesons to singlets implied by Aharony duality (2.17) combining this piece of information to the superpotential terms in theory T C we find The chiral ring generators include the singlets M i j and σ ± B , the mesonic-like operators Tr(φ J c ) and Tr(p i φ J cp j ) for J = 0, . . . , N 1 . The monopole operators generating the chiral ring are similar to the ones discussed for T B : we have the ones with flux on the gauge nodes M ±,± C and the ones dressed with the adjoint field φ C :

Operator map
In order to discuss the map of chiral ring generators across the triality it is useful to write down the map of the R-charges of the fundamental fields Using the R-charge map we have just summarised and the representation of the global symmetry group under which the various operators transform we can study how the chiral ring generators map across duality. The complete mapping is the following one In the next section we will show the supersymmetric index for the triality just discussed with a particular emphasis on the chiral ring generators.

Supersymmetric index
In this last section we compute the supersymmetric index for the three dual frames T A , T B and T C for N 1 = 1, N 2 = 2, F = 2 and with the choice of R-charges given by R B = 1/5, R Q = 3/8: where y B and y Q are the fugacities for the two axial symmetries. The charges of the fundamental fields under these symmetries are assigned as follows: The topological symmetries map is straightforward in going from T A to T B , while in going from T A to T C they mix non-trivially in a way that can be summarised in the following fugacity map where the label 1 and 2 refers to the U (N 1 ) and U (N 2 ) for T A and U (N 1 ) and (2.31)

Symplectic gauge groups
Consider a theory with two gauge groups and flavours We assume N 2 ≥ N 1 + 1, otherwise the strongly coupled dynamics of the left node breaks supersymmetry via a runaway superpotential [43]. It is possible to find two dual descriptions of this theory by applying Aharony duality [4] to each gauge node. Similar techniques of studying quiver theories through node by node dualisation using Aharony and one-monopole duality [44] have been applied in order to get the dual of U (N ) with one flavour and adjoint [14] and the generalisation to k + 1 flavours [15]. The goal of our analysis is to study the map of chiral ring generators across duality, with particular focus on the monopole operators, including the dressed ones. For theories with N = 4 supersymmetry this problem has been beautifully addressed in [45] using the Hilbert series [46], while in the case of N = 2 theories the same tool has been applied to vector-like theories [47] and to CS theories [48] to discuss the structure of the moduli space of vacua.
The global symmetry group of the theory is SU (2F ) × U (1) B × U (1) Q , where the label in the Abelian symmetries denote on which chirals it is acting on. In order to study the chiral ring generators it is useful to write down the R-charge of a monopole operator with general magnetic fluxes. Denoting the most general monopole for T A as M m, n , its R-charge is as follows where the various contributions correspond respectively to the bifundamental B, the fundamentals Q and the last two lines are the gauginos for the two gauge groups. In this formula R Q and R B gives the mixing coefficients of the UV R-symmetry with the axial symmetries. The coefficients of R Q and R B gives the charge under the corresponding axial symmetry of the monopole under study. In particular, we may specialise (2.33) to the monopoles with minimal flux 3 : These are the bare monopole operators, however in general there are also dressed monopoles which enters the chiral ring as generators. It is well known that, in the context of theories with Sp(N ) gauge theories with antisymmetric field φ and fundamental chirals, the chiral ring is partly generated by M φ J with J = 0, . . . , N [49]. For theories with more than one gauge node there are new monopoles which enter the chiral ring. In the case of two-node quiver these are monopoles with flux on both nodes M •,• A dressed with powers of the bifundamental chirals (BB) J . Thus, the monopoles generating the chiral ring are for J = 0, . . . , N 1 .

First dual
Applying Aharony duality to the Sp(N 1 ) node in T A to get theory T B we get a dual of the quiver where φ b is a traceful antisymmetric field generated by the Aharony duality. The The dimensions of monopoles with minimal fluxes are given by The global symmetry group is SU (2F ) × U (1) × U (1). Observe that, even though we have three sets of chiral fields b, φ b and Q we only have two independent abelian global symmetries due to the superpotential constraint Tr(bφ b b). The R-charge map with respect to theory T A can be obtained by looking at the mesonic operators. The operator Tr(QQ) in T A maps to Tr(QQ) in T B . Moreover, the singlet Tr(BB) in T A maps to Tr(φ b ), which gives The chiral ring generators include mesonic operators B is removed from the chiral ring by the superpotential term σ B M •,0 B , leaving instead σ B as a generator. There are also dressed monopoles as discussed at length in the previous sections. In this particular case we have B dressed with bifundamental chiral b since the superpotential set to zero the matrix bb on the chiral ring.

Second dual
Similarly to what has been done in the previous section, one may apply Aharony duality to the Sp(N 2 ) node in T A to get the following dual frame In analogy to the case of theory T B there are only two abelian symmetries because of the superpotential terms, giving is SU (2F ) × U (1) × U (1). The formula for the Rcharge of a monopole with general flux is completely analogous to (2.39), so we do not repeat it here. Instead, we can write the R-charges of the various minimal monopole operators: It is possible to get the R-charge map with respect to theory T A by looking at mesonic operators. The mesons Tr(Q i Q j ) map to M ij so we get R M = 2R Q that further gives, from the superpotential term Tr(qM q), R q = 1 − R Q . In a similar fashion, the operator Tr(BB) maps to φ c and gives R φc = 2R B , that is related, via the superpotential coupling Tr(bφ c b), to R b , implying:

Operator map
In the previous sections we discussed two dual Aharony-dual frames of the two node quiver theory T A . At this point we are ready to study the map of chiral ring operators among the three theories T A , T B and T C . It is then useful to summarise the R-charge map between T B , T A and T C , T A : Using these maps and the transformation properties of the various operators under the global symmetry it is possible to get the following chiral ring map It is interesting to observe how the various (dressed) monopole operators map across the two dual frames: apart from the usual monopole-singlet map coming from the application of Aharony duality, e.g.
From the point of view of dressed monopoles this implies that "long" monopoles dressed with the bifundamental in theory T A map to single-node flux monopoles dressed with antisymmetric fields, as can be seen from the last line of (2.51).
Before closing this section, we need to comment the operator map in the following sense. In theory T A , Tr (BB) J gives algebraically independent operators on the chiral ring for J = 1, . . . , N 1 . The same comment holds for For this reason, the map involving these operators for N 1 < J ≤ N 2 (including monopoles dressed with them) has to be understood such that the operators in T A and T C are composite. Their composite-ness, while evident in theory T A , is due to some quantum relation in T B and T C .

Supersymmetric index
In this last section we compute the supersymmetric index for the three dual frames T A , T B and T C for N 1 = 1, N 2 = 3, F = 3 and with the choice of R-charges given by R B = 2/5, R Q = 2/7: where y B and y Q are the fugacities for the two abelian symmetries. The charges of the fundamental fields under these symmetries are assigned as follows: In (2.52) we highlighted in blue the various chiral ring generators already discussed in the previous sections. For definiteness, let us take the generators in the frame T A and identify them with the various terms we obtain in the supersymmetric index (2.60)

Dualities in quivers with baryons and baryon-monopoles
In this section we study quivers with more general gauge groups, including SO and SU gauge groups. We work out three examples, with gauge groups SO( . In each example we are able to find the mapping of all the chiral ring generators. Unlike the cases discussed in section 2, for the examples in this section we do not have a general rule for the mapping of the monopoles which can be extended to quivers with an arbitrary number of gauge groups. We leave this issue to future work.
Let us consider the following theory: with N 1 + F even. In order to avoid runaway superpotential breaking supersymmetry to be generated or confining dynamics, we assume 2N 2 ≥ N 1 and The continuous global symmetries of the model are acts on the bi-fundamental fields and U (1) F acts on the flavor fields. Moreover, two discrete factors, the charge-conjugation Z C 2 and the magnetic symmetry Z M 2 , are associated to the orthogonal node. The charge-conjugation acts on flavor and bifundamental chirals as an orthogonal reflection transformation and also possesses a non-trivial action on monopoles; in particular, as reviewed in section B, two kind of monopoles can be defined, whose fluxes are denoted by ±, i.e. their charge under C. The magnetic symmetry M instead, is related to center of the orthogonal group and acts trivially on bi-fundamental and chiral fields while it charges −1 all the monopoles with minimal flux.
Let us study the chiral ring generators of this model. Because of the presence of an SO(N 1 ) gauge group, we have both mesonic and baryonic operators. The first kind consists of traces involving both the bi-fundamental field B and the flavors Q i : Observe that, because the orthogonal invariant form is symmetric and the symplectic one is antisymmetric, it is not possible to construct a meson operator that is quadratic in the bi-fundamental fields. The unique baryon operator is: where · denotes the contraction of the Sp(N 2 ) indices through the symplectic form and r = N 1 /2 is the rank of the orthogonal group. The term colored in red is only present when N 1 is odd, in which case we need an extra flavor in order to obtain a gauge invariant operator: this implies that when N 1 is odd, such baryon transforms non-trivially under SU (F ). 4 As reviewed in appendix B, also the monopoles operators can be of non-baryonic and of baryonic type. The non-baryonic monopoles are charge-conjugation even and consists of M +,0 A , M 0,• A and M +,• A . M +,• A , with minimal fluxes turned for both the nodes, can be dressed with powers of bi-fundamental fields, (M +,• A ) (BB) j , j = 0, . . . , N 1 /2 . The baryonic monopoles, instead, are chargeconjugation odd and always have non-trivial fluxes turned on for the orthogonal group. They need to be dressed with N 1 − 2 chiral fields in the fundamental representation of SO(N 1 ); in the case at hand we have two possibilities: As before, the Sp(N 2 ) indices of the dressing factors are contracted with the symplectic form and possibly with an extra chiral field Q i if N 1 is odd; in this latter case, both the baryon monopoles transform in the fundamental representation of SU (F ). The R-charges of the various monopoles can be computed using the general formula: where the red terms must be taken into account whenever N 1 is odd. Specializing the previous formula to the case of the monopole with minimal fluxes, we obtain the following result: , it is straightforward to obtain the baryon-monopole R-charges taking into account the contribution of the dressings.

First dual
Using the ASRW duality reviewed in appendix B, one can dualize the orthogonal node in (3.1), obtaining the following dual frame: where S is a symmetric (adjoint) field. As before, the chiral ring contains both nonbaryonic and baryonic operators. Among the possible traces, all the ones containing the symmetric product bb are flipped while the traces involving the adjoint field are now part of the chiral ring. The unique baryon is instead ε(bÑ 1 Q i ) whereÑ 1 = 2N 2 − N 1 + 2. As usual, the operators Tr(BB) 2 and TrQ i Q j in T A map to Tr S 2 and TrQ i Q j respectively in T B . From this map we immediately read the constraints on the R-charges of b, S and Q i in order for the duality to hold: In particular, the R-charges of the charge-conjugation even monopoles are: (3.11)

Second dual
This time we apply Aharony duality to the symplectic node in T A : where A transforms in the rank-two antisymmetric (adjoint) representation of SO(N 1 ) and M transforms in the rank-two antisymmetric representation of the SU (F ) flavor group. Because Tr(BB) 2 and Q i Q j in T A map to Tr A 2 and M ij respectively in T C , the R-charges of the chiral fields in this third frame read: Because of the constraints imposed by the superpotential, a unique baryon can be built using A (and p whenever N 1 is odd) as follows: where we remind r = N 1 /2 to be the rank of the orthogonal group. As in T A , one can wonder whether more general baryons, built substituting one or more adjoint fields with some antisymmetric combination p i p j in (3.14), are chiral ring generators. As we will see, the mapping between T A and T C suggests that such baryonic-like objects are actually composite operators that can be expressed as products of the baryon (3.14) and the meson M ij due to quantum relations. Similarly, the baryonic-monopoles generators in such frame are (M −,0 C ) A n−1 p i and (M −,• C ) A n−1 p i The R-charges of the fundamental charge-conjugation even monopoles are:

Operator map
In this section we want to present the mapping between the chiral ring generators in the three dual frames. The proposal is supported by the agreement between the global charges of the various operators and further checked using the supersymmetric index. The map is: As in the ARSW duality, the baryon monopole (−, 0) is mapped to the baryon (and viceversa) when the orthogonal group is dualized. Also observe that the charge-conjugation even monopoles still behave in agreement with the prescription presented in the introduction. The charge conjugation-odd monopoles behave in a way similar to the the general rule under Sp-duality: M −,0 extends while M −,• shortens.

Supersymmetric index
Let us compute a the supersymmetric index of the three dual theories in the particular case of N 1 = 3, N 2 = 2 and F = 7. We fix the R-charges to be R B = 2 5 and R Q = 1 2 ; moreover, the charges under the U (1) global symmetries are: while the charges of mesons and rank-two field are constrained by the form of the superpotential. We denote with y B and y Q the fugacities of U (1) B and U (1) Q respectively, while we denote with x the R-charge fugacity and with z i the SU (F ) ones. With the mentioned choices, the supersymmetric index reads: We highlighted in blue the generators of the chiral ring. Let us explicitly write down the generators we read from the index: Let us consider a theory involving orthogonal gauge groups only: (3.20) We assume N 2 ≥ N 1 and N 1 + F ≥ N 2 in order to avoid the generation of a runaway superpotential or confining dynamics. In the following it will be useful to define the rank of the two groups,  It is also possible to build three kind of baryons: The set of the monopole chiral-ring generators is more intricate in this case and it can be arranged in terms of the charges under the C i symmetries. To be more precise, we will consider the bare charge under charge-conjugation: for instance, a monopole such as M 0,+ , which at the bare level is even with respect to both charge conjugation symmetries, at the quantum level can transform non-trivially under C 1 , because of the dressing of bi-fundamental fields. The even monopoles (at the bare level as just explained) are M +,0 , M 0,+ and M +,+ , where the latter can be dressed with powers of BB. Using the standard formulas introduced in the previous sections, the R-charges of the monopoles can be easily computed: The two dressings are possible only if F ≥ N 1 −2 and F ≥ N 2 −N 1 +2 respectively, because full antisymmetrizations of indices are needed. In the dressing of the second operator in (3.24) the Levi-Civita tensor of the full SO(N 2 ) node appears, making this monopole someway exotic: it behaves like a baryon with respect to the second node. For this reason, we dub this operator dibaryon monopole. In the same way, we can construct: Observe that M −,+ cannot be dressed using the Levi-Civita tensor of the second node. Indeed, when it gets a VEV, the non-trivial + flux causes the breaking of the SO(N 2 ) gauge group down to S(O(N 2 − 2) × O(2)) and the presence of the ε (2) symbol would make the operator not invariant with respect to the residual gauge group.
• In a similar fashion, we can consider monopoles which are, at the bare level, charge-conjugation odd with respect to C 2 only. These can be built appropriately dressing M 0,− A and M +,− A . In the former case we can build: The two baryon monopoles only exists if F ≥ N 2 −2 and N 1 ≤ N 2 −2 respectively. The latter, in particular, is a dibaryon monopole in the notation introduced previously. In a similar manner, starting from M +,− A we can build As before, we cannot use ε (1) in the dressing of M (+,−) because it would make the dressed monopole transforming under the residual gauge group.
• Finally, the last monopole to consider, M −,− A , can be dressed in a unique way: (3.28) This monopole always exists because F ≥ N 2 − N 1 by assumption.
As observed in [42], when studying the reduction of orthogonal gauge theories with a single node from four to three dimensions, a second type of baryonic operators play a relevant role. Such operators would correspond, in three dimensions, to non-minimal monopoles, i.e. monopoles with magnetic fluxes m 1 = −m 2 = 1 for two different Cartan generators of the gauge group turned on. In SO(N ) SQCD with F flavors, such monopoles are not actually chiral and do not map in a simple way across the duality. However, using the supersymmetric index, one can observe that monopoles with three magnetic fluxes turned on, that we will denote as M ÷,+ and M +,÷ , must be part of the chiral ring. 5 The "symbol" "÷" is used to remember which node has two fluxes turned on; when such a monopole gets a VEV, the gauge group factor SO( (3.29) Their R-charges are: (3.30)

First dual
Using the ARSW duality on the left orthogonal node of T A we obtain the following quiver theory: where S B is a traceful symmetric field. As usual, using a Seiberg-like duality Tr(BB) maps to Tr (S B ) and Q i Q j maps to Q i Q j , implying the following constraints on the R-charges: The chiral ring generators consist of mesons, baryons and (baryonic) monopoles, built in the same fashion we presented for T A . Because of the constraints imposed by the superpotential, traces involving the bi-fundamental fields are set to zero and the chiral ring actually contains The theory again contains three baryons: The baryonic monopoles (minimal and non-minimal) are defined as in the dual frame T A performing the substitution {B → b, N 1 →Ñ 1 }. For this reason, we will not write down them again.

Second dual
The last dual frame is obtained applying the ARSW duality to the SO(N 2 ) node: The usual map of the ARSW duality implies Tr(BB) → TrS C and Q i Q j → M ij fixing the chiral field R-charges in the third frame: Given the constraints imposed by the superpotential, the mesonic operators are Tr(p i S J C p j ) and Tr(S J C ), while the conjugation-even (dressed) monopoles are M +,+ and (M +,0 ) S J C ; the monopole M 0,+ is flipped by σ C instead. Again, as in the other frames, there are three different baryons: Finally we have to define the baryon monopoles in this duality frame. This time, they need some particular attention because there is no simple rules to relate them to the ones in T A . There are two monopoles of charge (−, +) with respect to (C 1 , C 2 ) that are dressed using the chiral field p only: while there are two baryon monopoles of charge (+, −) that are dressed usingq: Finally, there are three monopoles constructed usingb andq appropriately contracted with Levi-Civita symbols:

Operator map
The operator map across the triality is quite intricate: Let us observe that the presence of the non-minimal monopole is crucial in order to have a consistent map among the three frames. It is interesting to observe that, in order the previous map to be consistent, the discrete charge-conjugation symmetries must be also identified as follows across the duality: (3.44)

Supersymmetric index
In this section we present the supersymmetric index of the three dual theories with the particular choice case of N 1 = 3, N 2 = 4 and F = 3. We fix the R-charges to be R B = 1 2 and R Q = 1 3 ; moreover, the charges under the U (1) global symmetries are fixed as in (3.17). As before, we denote with y B and y Q the fugacities of U (1) B and U (1) Q respectively, while we denote with x the R-charge fugacity and with z i the SU (F ) ones.The supersymmetric index reads: We highlighted in blue the generators of the chiral ring and correspond to: Let us observe that, for this choice of ranks of the gauge groups, not all the possible baryon and baryon monopoles are present. In particular, there are no non-minimal monopoles for T A with the current choice.

Mapping the orthogonal baryonic operators in general quivers
Using the operator maps (3.16) and (3.43) we can guess how baryons and baryon monopoles are mapped across the duality in more general quivers. In the case of orthosymplectic quivers, we propose precise mapping rules based on the two-node experience. We propose that, applying an ARSW duality to the i th node: • if the node is symplectic, the general rule presented in the introduction still applies. For instance: (3.47) and so on, where σ i = ±1.
• If the node is special-orthogonal and the monopole is of baryonic type with respect to C i , a monopole (M ...,ω i−1 ,− i ,ω i+1 ,... ) with at least one between ω i+1 and ω i−1 non vanishing, the operator is mapped to another baryon monopole with the same string of fluxes.
Observe that in the case the monopole is baryonic, we do not indicate how the dressing transforms across the duality, such task can be easily addressed matching the quantum numbers between the frames. The knowledge of the mapped string of fluxes is the crucial guideline in order to do so.
When the quiver is purely special-orthogonal, instead, the situation is much more intricate and we do not really have a general prescription but we conjecture some general guideline to follow and that should simplify the task of matching the chiral ring operators. The higher intricacy is mostly due to the fact that we do not know in general which strings of fluxes are admitted, in particular for the monopoles of nonminimal or dybarion type. Let us consider a monopole of the form M ... ω i−1 ω i ω i+1 ... where ω j = 0, 1, −1 denotes the bare charges of the monopole with respect to the charge conjugation symmetry of the j-th node; the operators will be assumed to be not dibaryons or non-minimal unless explicitly stated. Let us assume to apply an ARSW duality to the i th node. We conjecture that: • if σ i = +, 0 the monopole keeps on transforming as explained in the introduction for quivers without baryons. For instance, the monopole M ··· 0,+,ω i+1 ··· maps to a monopole of the form M ··· 0,0,ω i+1 ,··· .
It would be interesting in the future to further investigate the precise content of the chiral ring in longer orthogonal quivers in order to make the previous conjectured prescription more precise.

Sp
In the spirit of studying dualities in two-node quivers, let us consider a theory with Sp(N 1 ) × SU (2N 2 ) gauge group: where the SU (2N 2 ) gauge node has the same number of fundamental and anti-fundamental chirals, hence it is non-chiral. The global symmetry is where the three U (1)'s act on the three sets of chirals B, Q 1 and Q 2 as indicated by the corresponding label.
The R-charge of monopole operators M m, n , m being the magnetic charges for the Sp(N 1 ) node and n the ones for SU (2N In particular, the monopoles with minimal GNO fluxes have R-charges The monopole with flux both on the Sp(N 1 ) and on SU (N 1 ) deserves particular attention and will be discussed at length in the following. The chiral ring of this theory has generators including mesonic and baryonic operators and dressed monopoles. We can construct a tower of baryonic operators with an even number of bifundamental fields B and the appropriate number of Q 1 in order to use the SU (2N 2 ) epsilon tensor where the i indices are the ones for the SU (2N 2 ) gauge group, the a are for the Sp(N 1 ) and are contracted using the symplectic form Ω. For simplicity we suppressed the indices of the flavour symmetry. This operator transforms in the rank-(2N 2 − 2k) antisymmetric representation of the SU (2F ) global symmetry, the antisymmetry being induced by the contraction with the epsilon tensor. Furthermore, we may construct a mesonic-like operator using B and Q 2 as follows

First dual
Let us apply Aharony duality to Sp(N 1 ). We obtain the following dual theory where A is an antisymmetric field. Standard Aharony duality would imply the map Tr(BB) ↔ A; however, being the flavour symmetry of the Sp(N 1 ) gauged in this case, A is not a gauge invariant operator, so we have to look at Tr( (3.56) For the minimal GNO flux monopoles we find This theory has interesting dressed monopoles generating the chiral ring that needs to be discussed. The interesting part of the discussion comes from the presence of the antisymmetric field A for the SU (2N 2 ) group.
Dressed monopoles in a SU (2N ) theory with an antisymmetric field The SU (2N ) theory with an antisymmetric field, and in particular its dressed monopoles, has been discussed in [50]. We will closely follow this reference to review the construction of the dressed monopoles. For the moment we generalize the set-up of [50] by taking into account an SU (2N ) gauge theory with N f fundamentals Q,N f antifundamentals, N A antisymmetric andN A conjugate antisymmetric. As already observed, a monopole with minimal GNO flux for SU (2N ) breaks the gauge group to SU (2N − 2) × U (1) 1 × U (1) 2 , where 1 and 2 attached to the U (1)'s are just labels. It turns out that there is a mixed CS term between these two U (1)'s: This mixed CS term induces a gauge charge under U (1) 2 for the bare monopole [51] given by The crucial point is now to use matter fields in the residual gauge theory that may be used to cancel this gauge charge. The residual antisymmetric field asym 0,−2 is a good candidate to do this job. However we need to take N − 1 copies of it, so the U (1) 2 charge is cancelled and there is no U (1) 1 charge brought by the dressing. In the end, the gauge invariant monopole has the form 64) where A N −1 is contracted using the epsilon tensor of the residual SU (2N − 2) This construction may be generalised to get a tower of dressed monopoles. The fundamental field in the residual theory can be used in a way similar to the antisymmetric. Since the U (1) 2 charge is −1 and not −2 as the antisymmetric, every time we "remove" one A from {M A N −1 } we put two fundamentals Q's. In the end, the tower of dressed monopoles that we get has the form We can now go back to our quiver gauge theory. The presence of a tower of dressed monopoles reflects the fact that the chiral ring of T B includes the monopoles Furthermore, the chiral ring includes the singlet σ B and the monopole with flux on both gauge nodes M ·,· B . The mesonic part of the chiral ring generators is given by the mesons Tr(Q 1 Q 2 ) and Tr(AQ 2 Q 2 ). One may also construct baryonic operators. First, we have εQ 2N 2 2 , but we can also form baryons using the antisymmetric A and the fundamental chirals Q 1 . In detail, we get a tower of baryons: where again we suppressed the flavor indices of Q 1 . Again, due to the anti-symmetrization, the baryonic operators that we denote schematically as εA k Q 2N 2 −2k 1 transform in the rank-(2N 2 − k) antisymmetric representation of the SU (2F ).
Let us also comment on the monopole with both fluxes on the Sp(N 2 − N 1 − 1) and on the SU (2N 2 ). In this case also the symplectic gauge node is broken down as (2). Hence, if we want to compute the U (1) 2 charge of the monopole M ·,· we need to take into account that the contribution from the bifundamental fields b is now reduced to 2(N 2 − N 1 − 2). In detail, applying (3.60) So the monopole M ·,· is gauge invariant, and is a generator of the chiral ring

Second dual
In order to dualize the SU node we use the ARSW duality.
ARSW duality SU ↔ U The duality that we need in order to dualize the SU (2N 2 ) gauge node has been discussed in [41] as the reduction of the original SU (N ) Seiberg duality [17]. Before going into the details of the second dual frame of theory T A it is worth to review the original duality.
The electric theory is an SU (N ) gauge theory with F flavours Q,Q and superpotential Let us discuss the global symmetry of this theory. The non-abelian part is SU (F ) × SU (F ), while the abelian part in principle comprises the topological symmetry U (1) T apart from six independent U (1)'s acting on the chirals M, q,q, Y, b,b. Observe that being the gauge group U and not SU one combination of these U (1)'s is gauged; moreover the superpotential breaks four extra combinations, leaving only two abelian symmetries. Without going into the detail of the precise combination of the abelian symmetries, we just show the mapping of the gauge invariant operators that will be useful in the upcoming analysis where Q N = i 1 ...i N Q i 1 · · · Q i N , and similarly forQ N ; for simplicity we are suppressing flavour indices. Observe that the linear monopole superpotential in the magnetic theory just removes from the chiral ring the monopoles for the U (F − N ) gauge theory. Having discussed the duality involving an SU (N ) gauge theory we now come back to theory T A and analyze the third dual frame: Let us turn to the discussion on the chiral ring generators. We can start from the singlets flipping the mesons q 1 q 2 and cq 2 , namely M and N . However, N is charged under the Sp(N 1 ) gauge node, hence we need to take the gauge invariant combination N 2 where the indices of Sp(N 1 ) are contracted using the symplectic tensor. Then, we have the singlet Y that flips bb. The chiral ring also includes baryonic generators. The obvious one we can construct is using q 2 and b: q 2F +2N 1 −2N 2 2b , where as explained in the previous section the presence ofb is necessary to balance the U (1) gauge charge. This operator transforms in the rank-(2N 2 ) antisymmetric representation of SU (2F + 2N 1 ) global symmetry. There is another baryon we may construct, this time using three chirals: q 1 , c, b contracted as q 2F −2N 2 +2k 1 c 2N 1 −2k b and transforms in the rank-(2N 2 − 2k) antisymmetric of SU (2F ). The monopoles with flux only on the U (2F + 2N 1 − 2N 2 ) enters linearly in the superpotential and are removed from the chiral ring. The monopole operators of the form M ·,0 C needs a detailed discussion. Usually, if we take a Sp(N ) gauge theory with 2F fundamental chirals, the basic monopole operator is charged under the axial symmetry (in particular it has charge −2F .) In theory T C this axial symmetry is gauged because of the U (2F + 2N 1 − 2N 2 ) gauge node, hence the basic Sp(N 1 ) monopole operator is not gauge invariant. To construct a gauge invariant operator out of the Sp(N 1 ) monopole it is possible to dress it using the fields with the fields q 1 and c, which have the correct U (1) gauge charge to cancel the one of the monopole. The reason is the following: the axial charge of M ·,0 C , which is −(2F + 2N 1 − 2N 2 ), has opposite sign with respect to the U (1) charge of c, that we take to be +1. Moreover, we see from the quiver T C that q 1 has the same U (1) charge as c. Thus, the U (1) gauge charge of the bare M ·,0 C is cancelled if we dress with a total of 2F +2N 1 −2N 2 fields, being either c or q 1 . Recall that we need to make the dressed monopole gauge invariant also under Sp(N 1 ) and SU (2F + 2N 1 − 2N 2 ) ⊂ U (2F + 2N 1 − 2N 2 ): this is achieved by taking an even number of c fields and contracting everything with the epsilon tensor of U (2F + 2N 1 − 2N 2 ). The tower of dressed monopoles constructed in this way is as follows c 2k is a schematic expression denoting the baryonic operator

Operator map
Finally we are ready to discuss the map of chiral ring generators across the three dual frames of the SU − Sp gauge theory we studied. As we have done in the other sections, it is useful to have at hand the R-charge map that allows to go from T A to T B and T C : The operator map is as follows Observe the last two lines of the table which involves dressed monopoles. The Sp(N 1 ) monopole M ·,0 A , mapping to the usual Aharony singlet σ B in T B , maps to a dressed monopole in theory T C : {M ·,0 C b}. The last line of the table is similar to the mapping of the monopole with flux on two nodes in the case of Sp − Sp gauge theory, the difference is that now we have non trivial dressing making the monopoles transforming in the rank-2k antisymmetric representation of the SU (2F ) global symmetry group.

Supersymmetric index
We compute the supersymmetric index for the triality at hand in the case of N 1 = 1, N 2 = 2, F = 2 with the choice of R-charges given by R B = 1/5, R Q 1 = 3/8, R Q 2 = 4/7. For theory T B we did not manage to perform the index computation due to machine limitations, so we limited ourselves to the explicit computation for T A and T C theories.

Conclusion and open questions
In this note we discussed Seiberg-like dualities for two-node quiver theories with various gauge groups, paying particular attention to the mapping of monopole operators across the duality. We may identify two different class of theories: one class without baryonic operators, such as quivers involving two unitary or symplectic groups, and one class with baryons and baryon-monopoles, such as various combination of special orthogonal, special unitary and symplectic groups. Even though we explicitly studied only twonode quivers, for the first class we find a simple rule to map all chiral ring generators (mesons and monopoles) in a quiver of arbitrary length. We were not able to find such a general rule for the second class of theories. In this case there are subtle issues, especially for orthogonal groups, that do not appear for unitary or symplectic quivers that makes finding such a general map a hard task. Nonetheless, the chiral-ring in the presence of baryons and baryon-monopoles is much richer and involves interesting operators that are not present when discussing single node quivers. A very useful tool in our analysis is the superconformal index, whose explicit expression (as a function of the fugacities for the abelian global symmetries, turning on fugacities for non-abelian symmetries is computationally hard and not necessary for our purposes), allows us to check that our proposed list of chiral ring generators is complete.
The analysis performed here has its own interest as a step towards the understanding of the complete operator map across duality in general 3d N = 2 quivers. Nonetheless, our first motivation was to use the techniques developed here to study the "deconfinement" of rank-2 matter, for instance in the case of an adjoint U (N ) chiral or for the rank-2 anti-symmetric representation for Sp(N ). Aspects of the latter theory have been discussed in [49,52]. These results will be presented in [33? ].
Finally, let us mention some open questions.
There are some two-node quivers for which we have not been able to understand the complete chiral-ring map, such as for Sp(N 1 ) × U (N 2 ) and SU (N 1 ) × U (N 2 ), where the matter fields are such that the full theory is non-chiral. It would be interesting to fully understand these cases. The last comment immediately raises the question of analizing theories with chiral matter content. The difficulty in such cases comes already from the intricate structure of the dualities needed for chiral theories. Some results on dualities for chiral-matter are found in [22,[53][54][55][56]. Furthermore, a more systematic analysis of alternating orto-symplectic quivers is desirable, in the spirit of theories with eight supercharges. All the developments we mentioned up now involve theories with four supercharges. However, another interesting line of research is related to theories with minimal amount of supersymmetry, N = 1 in 2+1 dimensions. Some previous works on the dynamics of such theories can be found in [57][58][59][60][61][62][63][64][65][66][67][68]. To the best of our knowledge there is no analysis for the IR dynamics of N = 1 quivers and it would be worth studying it.
R being the R-charge.
It is not so easy to employ the definition (A.1) to perform an explicit computation of the index; here the localization techniques come at rescue. Indeed, the index can be computed as the partition function on S 2 × S 1 given by the following expression where the integral is taken over the Cartan torus of the gauge group whose fugacities are z, |W m | is the dimension of the Weyl group that is left unbroken by the monopole background specified by the GNO magnetic fluxes m. Localization implies that the only non trivial contribution to (A.3) from non-exact term in the classical action and from 1-loop terms. The various terms Z cl , Z vec , Z mat have the following expressions • Z cl : The classical terms includes only CS couplings and, more generally, BF terms.
Take a gauge group whose rank is rkG. Denoting the fugacity for the topological with ω and the associated flux as n, and given a level k CS term we have • Z mat : The contribution of an N = 2 chiral multiplet with R-charge r transforming in the representations R and R F under the gauge and flavour group, whose weights we denote as ρ, ρ F , is

B Monopoles and dualities for orthogonal gauge groups
In this section we will review the current knowledge about the monopole operators in theories involving orthogonal gauge groups and the related Seiberg-like dualities proposed in [21,42].
Another type of operator is relevant for us, having non-trivial magnetic fluxes with respect to two different Abelian factors in U (1) r . Semi-classically, it can be written as: where the two lined up bullets denote the fact that two different fluxes are turned on. Such monopole breaks the SO(N ) gauge group down to S(O(N − 4) × O(4)), 8 and it is not gauge invariant unless dressed with a conjugation-odd operator in SO (N − 4). This can be done defining the gauge invariant operator εM ÷ Q N −4 , 9 existing only for F ≥ N − 4. In theories with only one gauge group factor, M ÷ is not really a chiral operator; however, it plays a crucial role in dualities between orthogonal quiver theories discussed in this paper.
In the theory under consideration, the last operator that deserve to be mentioned is the usual baryon: In the main text, different baryon-like operators can appear; in that case, we will denote them by an ε followed by the fields contracted with the Levi-Civita symbol: for instance, the baryon in (B.6) could be also denoted by εQ N . Once we have understood which kind of operators can be part of the chiral ring in 3d SQCD with orthogonal groups, we can easily discuss the Seiberg-like duality proposed by Aharony, Razamat, Seiberg and Willett (ARSW) in [42]. The theory dual of T A , SO(N ) SQCD with F flavors Q, W T A = 0, is T B , SO(F − N + 2) gauge theory with F flavors q, F (F + 1)/2 singlets M ij transforming in the symmetric representation of SU (F ) and superpotential: W T B = σM + + M ij Tr(q i q j ) . (B.7) The map of the chiral ring generators is the following: Observe that baryons and baryon-monopoles are mapped to each other. 8 Actually, the SO(4) factors is further broken to U (2). Different gaugings of charge conjugation and the magnetic Z M 2 discrete symmetry leads to different gauge groups, enjoying the same algebra as SO(N ) but differing in their global properties; in particular, the spectrum of chiral operators will be different.
• The gauge group O(N ) + is obtained gauging Z C 2 , i.e. the orthogonal reflection. Such O(N ) group is the most common in literature: the gauging of charge conjugation makes the baryon and the baryon-monopole not gauge invariant and they are not part of the chiral ring anymore.  All such theories enjoy Seiberg-Like duality similar to the ARSW duality [21,42]. O(N ) + SQCD with F flavors is dual to O(F − N + 2) + SQCD with F flavors, N (N + 1)/2 M ij singlets duals of the meson TrQ i Q j , the singlet σ dual of M + and the usual superpotential W = σM + + Tr(q i M ij q j ); an analogous duality holds for Pin(N ) SQCD. Finally, O(N ) − SQCD is dual to Spin(F − N + 2) SQCD (with singlets and appropriate superpotential): further details about the chiral ring mapping can be found in [42].