Sequential deconfinement in $3d$ $\mathcal{N}\!=\!2$ gauge theories

We consider $3d$ $\mathcal{N}\!=\!2$ gauge theories with fundamental matter plus a single field in a rank-$2$ representation. Using iteratively a process of"deconfinement"of the rank-$2$ field, we produce a sequence of Seiberg-dual quiver theories. We detail this process in two examples with zero superpotential: $Usp(2N)$ gauge theory with an antisymmetric field and $U(N)$ gauge theory with an adjoint field. The fully deconfined dual quiver has $N$ nodes, and can be interpreted as an Aharony dual of theories with rank-$2$ matter. All chiral ring generators of the original theory are mapped into gauge singlet fields of the fully deconfined quiver dual.


Introduction and results
The fascinating phenomenon of infrared dualities seems ubiquitous in strongly coupled gauge theories living in d ≤ 4 dimensions.In the special subset of supersymmetric theories with 4 supercharges, many examples of such dualities have been discovered, starting from [1].Impressive checks of the dualities are possible: matching of the infrared global symmetry, of the chiral ring and of various supersymmetric partition functions.
In the case of 3d N = 2 gauge theories [2][3][4][5], the simplest and paradigmatic examples are the Aharony dualities [5], which relate a pair of theories with a single gauge group.U sp(2N ) with 2f flavors is dual to U sp(2f − 2N − 2) with 2f flavors, while U (N ) with (F, F ) flavors is dual to U (F − N ) with (F, F ) flavors.All chiral ring generators, both mesons and monopoles, of the electric theory are mapped into gauge singlet fields in the magnetic theory.
In this paper we consider 3d N = 2 gauge theories with matter content consisting of an arbitrary number of fundamental flavors and a single field in a rank-2 representation.A rank-2 field can sometimes be deconfined, as shown in the early days of Seiberg dualities in 4d N = 1 models [6][7][8].In 3d N = 2 the story is similar, with the difference that in 3d monopole operators play a crucial role.Examples studied in 3d N = 2 include [9][10][11].In particular, the approach of Pasquetti-Sacchi is particularly interesting, since it allows to find the dual of U (N ) with one adjoint field and one flavour [10] and the one for U (N ) with one adjoint field and k + 1 flavours [11] starting from free field correlators in 2d Liouville CFT.These results have also been uplifted to four dimensions and are related to the compactification of rank-Q E-string theory on a torus with flux [12], and subsequently lead to the discovery of an analogue of 3d mirror symmetry for 4d N = 1 theories [13].
The main complication in the process is given by the supersymmetric monopole operators [26,27].Monopole operators appear in the superpotential, both linearly and through flipping-type interactions.Moreover it is important at each step to keep track of the mapping of the monopoles across the dualities.Hence we need to control the monopoles in 3d N = 2 quivers, for we which we use the results of [10,11,28].

Results
In this paper we focus on two examples: U sp(2N ) with an antisymmetric and U (N ) with an adjoint.Let us state the final results.
Similarly, for unitary gauge group, we find in Section 3 that U (N ) with an adjoint, F fundamentals, F antifundamentals, W = 0, is dual to the following quiver In the main text we explain the notation and derive these dualities, together with the mapping of the chiral ring generators.
The dualities (1.1) and (1.2) are valid for vanishing superpotential in the electric single-node theory, so it is possible to turn on any superpotential and obtain new duals.Similarly, turning on real masses it is possible to obtain duals of theories with non zero Chern-Simons level.We explore various such deformed dualities in the main text.
One noteworthy feature of the dualities (1.1) and (1.2) is that all chiral ring generators, both mesons and monopoles, of the electric theory are mapped into gauge singlet fields in the magnetic theory.So in this sense they are a natural generalization of Aharony dualities to the case with a single rank-2 matter field.

Further directions
A similar sequential deconfinement procedure can be worked out for theories involving orthogonal gauge groups and/or rank-2 matter in a symmetric representation.We study such a process in [29].The deconfined quivers alternate a symplectic and an orthogonal group.Moreover, it turns out that the quivers display a saw structure.
3d N = 2 gauge theories with a single gauge group, rank-2 matter Φ, fundamentals and superpotential W = Tr(Φ k+1 ) are known to admit a single node dual of Kutasov-Schwimmer type, that is the dual has a single node, a tower of gauge singlets and a superpotential term W = Tr( Φk+1 ) [30][31][32][33][34][35].Such dualities appear different from the dualities discussed in this paper, which have W = 0 on the l.h.s. and a linear quiver on the r.h.s.It would be interesting to investigate a possible relation between the Kutasov-Schwimmer type dualities and our sequential deconfinement procedure.
Another possible direction, which was one of the main motivation for this study, is to extend these results to 3d theories with N = 1 supersymmetry and rank-2 matter (see [36][37][38][39][40][41][42][43][44][45][46][47] for recent results in 3d N = 1 gauge theories).Very little is known on the dynamics of rank-2 matter for N = 1 theories.We hope that a story similar to the one in the present paper is valid in the 3d N = 1 realm, which might be at midway between the N = 2 and the non-supersymmetric case [48][49][50].In particular, the IR dynamics of non-supersymmetric theories with two real adjoint fields, unveiled in [50], displays an intricate duality chain reminiscent of the N = 2 sequential deconfinement.

A sequence of duals for U sp(2N ) with an antisymmetric
In this section we find dual descriptions of U sp(2N ) = Sp(N ) (Sp(1) = SU (2)) with a field A in the traceless antisymmetric representation of Sp(N ) and 2f complex flavors Q i , W = 0. U sp(2N ) theories have been recently studied in [33,51,52].We consider f ≥ 3.If f = 1, the theory does not have a supersymmetric vacuum.If f = 2, the fully deconfined dual is a Wess-Zumino model, see [33,51].
We find a total of 2N dual theories, that are quivers with a number of nodes ranging from 1 to N , the most natural one being the fully deconfined dual, with N nodes.
In each model we describe the chiral ring, giving the list of the chiral ring generators and their global symmetry quantum numbers.As usual in 3d gauge theories, we need to pay special attention to the monopole operators.
We first consider the case of vanishing Chern-Simons interactions, with this result, it will be easy to turn on a real mass deformation and hence a Chern-Simons term in section 2.7.
We start with theory T 1 , that is Sp(N ) with a traceless antisymmetric field A and 2f complex flavors Q i .We take the superpotential to be vanishing.Using the standard quiver notation for theories with four supercharges, T 1 reads The chiral ring is generated by the (dressed) mesons tr (Q i A l Q j ), l = 0, . . ., N − 1, the powers of the antisymmetric traceless field tr (A j ), j = 2, . . ., N , and the (dressed) monopoles {M A k }, k = 0, 1, . . ., N − 1.In terms of the R-charges of the elementary fields Q i and A, which we denote r F and r A , the R-charge of the basic, undressed, monopole M is

Deconfine and dualize: first step
We now use the confining duality 1 In this duality the chiral ring generators map as tr (q i q j ) ↔ A ij (the monopole M and the singlet γ are zero in the chiral ring).
Starting from theory T 1 , we deconfine the antisymmetric field into a two-node quiver theory.That is we consider theory T 1 : 1 This is a variation of a duality introduced by Aharony in [5]: Wess-Zumino w/ 2N × 2N antisymmetric matrix of chiral fields A, and a singlet σ In this duality the monopole is mapped to σ (M ↔ σ), so if we flip the monopole in the l.h.s. with a gauge singlet γ, on the r.h.s.we obtain a superpotential term σγ, so σ and γ become massive, integrating them out the superpotential becomes zero and we obtain the duality (2.4).
Applying the duality (2.4) to the left node of T 1 , the node Sp(N − 1) confines and one readily obtains T 1 .So T 1 and T 1 are dual.We introduced the gauge singlet field β field so that A in T 1 is traceless.The mapping of the R-charges between T 1 and T 1 is simply In linear quivers made of Sp gauge groups, we denote by M 0,•,0,0,... monopoles with non-zero minimal flux in the nodes with • and zero flux in nodes with o.
In theory T 1 , M •,0 , γ, β are zero in the chiral ring: M •,0 is set to zero by the Fterms of γ. γ and β cannot take a vev because of quantum generated superpotentials, so we expect them to be zero in the chiral ring 2 .The monopoles M 0,• and M •,• are instead non-zero the chiral ring, their R-charges are The basic monopole M in T 1 maps to the 'extended' monopole M •,• in T 1 .We will give the full map of the chiral ring generators in (2.17).As explained in [28], the monopole M •,• in T 1 can be dressed with the square of bifundamental field, that is bb , in same way that M in T 1 can be dressed with the antisymmetric field A.
The next step is to dualize the right node Sp(N ) in T 1 .We use the Aharony duality [5] Sp(N ) w/ 2F flavors, in the quiver T 1 and obtain T 2 : We decomposed the dual mesons into the two fields φ and p.Because of the F-terms of the singlet β, that we integrated out, the antisymmetric field φ is traceless.Notice that the monopole M •,0 in T 1 maps to M •,• in T 2 , (2.9); here we are applying the rules of [28] for the mapping of monopole operators under dualities in quivers made of Sp nodes.
The mapping between the R-charges of theories T 1 and T 2 is (2.10) The R-charges of the monopoles and of the flipping fields for the monopoles are The mapping of the chiral ring generators of the three theories constructed so far, T 1 , T 1 and T 2 is 17) It is possible to check the mapping of the dressed mesons using

Deconfine and dualize: second step
We now repeat the same procedure.First we deconfine the antisymmetric traceless in T 2 (2.9) into a bifundamental b connected to a Sp(N − 2) node, introducing a flipping field γ 2 for the Sp(N −2)-monopole.The superpotential term tr (bφb) becomes tr (b bb b) and we get T 2 : T 2 : . This is agreement with the rules of [28], since dualizing the leftmost node in T 2 (and forgetting that the rank of the leftmost group becomes zero), the rule says that M  Integrating out the massive fields and removing the 's for notational simplicity, we get theory 3: We can express the R-charges of the elementary fields in theory 3 as a function of the R-charges in theory 1 r F and r A : The mapping of the chiral ring generators of the three theories T 1 , T 2 and T 3 is

After k steps
After k steps of deconfining and dualizing, we arrive to a quiver with k + 1 nodes: (2.25) We can express the R-charges of the elementary fields in theory k + 1 as a function of the two independent R-charges in theory 1, r F and r A : The mapping of the chiral ring generators with the starting theory T 1 is

Fully deconfined tail
After N − 1 steps, the leftmost group is Sp(1) so there is no antisymmetric traceless to deconfine, and we just dualize the Sp(1) using Aharony duality.
The final result is that our starting theory T 1 is dual to a fully deconfined quiver with N gauge nodes . . .
(2.29) The R-charges of the elementary fields in the fully deconfined theory are given in terms of the two independent R-charges in theory 1, r F and r A as follows: The mapping of the chiral ring generators in the duality T 1 ↔ T DEC is Notice that all chiral ring generators of T 1 map to gauge singlets in T DEC .This is similar to Aharony duality for Sp gauge group without rank-2 matter fields.Also, in the case of Sp(1) gauge group, our duality T 1 ↔ T DEC reduces to Aharony duality.

Superpotential deformation:
In this section and in section 2.6, we discuss complex deformations of the duality between our original theory with a single node Sp(N ) (2.1) T 1 and the fully deconfined quiver T DEC (2.29).As usual in Seiberg like dualities, a complex deformation on the electric side will induce a Higgsing of the gauge groups on the magnetic side.
In this section we consider a superpotential deformation in T 1 of the form tr Without going too much into the details, the final result is that in (2.29) the N − J gauge groups on the left while the J remaining gauge groups on the right are not Higgsed.The last flavor p 2f −1 , p 2f migrates from the left-most node to node Sp(J(f − 2)).More precisely, since the node Sp((J +1)(f −2)) is Higgsed down to Sp((J +1)(f −3)+J), the bifundamental field b J splits into a new Sp((J + 1)(f − 3) + J) − Sp(J(f − 2)) bifundamental and a flavor for the node Sp(J(f − 2)).The flipping fields for the mesons split into two sets M H , H = 1, . . ., N , and (M ) K , K = 1, . . ., J.

The final result is that
. . .
Complex mass deformation Let us start from f > 2 and turn on a complex mass for 2 flavors, δW = tr (Q 2f −1 Q 2f ).This is the special case J = 0 of the discussion above.The flavor p and the gauge singlets M are absent for J = 0, and all the gauge groups in T DEC get partially Higgsed.The final result is precisely (2.29) with f → f −1.
We thus get a consistency check of the duality T 1 ↔ T DEC .
2.6 N = 4-like deformation: We now consider adding f cubic terms to T 1 , obtaining Sp(N ) with 2f chiral flavors and W = f j=1 tr (Q 2j−1 AQ 2j ).Using the results just obtained in (2.33), on the dual side, all the flavors migrate to the right-most node Sp(f − 2), and out the tower of singlets (M J ) ij , only (M 1 ) ij survive.The tail of gauge groups Notice that the right-most gauge group Sp(f − 2) is not Higgsed.
The final result is that . . .
This result is strictly speaking valid for f > 2N .If f ≤ 2N the dual quiver becomes shorter and some of the flipping fields γ and σ decouple.This is due to the fact the theory Sp(N ) with W = f j=1 tr (Q 2j−1 AQ 2j ) if f ≤ 2N becomes "bad" in the Gaiotto-Witten sense, so some Coulomb branch operators (that is tr (A h ) and {M A k }) become free and decouple.

Real masses and Chern-Simons terms
Starting from the dualities discussed above, it is easy to turn a Chern-Simon interaction at level k: we simply start from the theory with 2f + 2k flavors and turn a positive real mass for 2k flavors.We obtain Sp(N ) k with 2f flavors and W = 0. Now f and k are either integers or half-integers, but f + k is always an integer.
The real mass is in the supermultiplet of the U (2f + 2k) global symmetry current, so in the fully deconfined dual (2.29) the real mass is mapped to a real mass for some of the flavors p (the bifundamental fields b i are not charged under the U (2f + 2k) global symmetry) and some of the gauge singlets M .In the fully deconfined dual (2.29) (with f → f + k), 2k flavors p's get a negative real mass, which induces a negative Chern-Simons level −k for the leftmost node, while the Chern-Simons levels of all the other nodes do not get any shift.
If k = 0, the monopoles {M A J } are not in the chiral ring of Sp(N ) k , accordingly the singlet fields σ i disappear from the deconfined dual of Sp(N ) k .
Summing up, the fully deconfined dual of Sp(N ) k with antisymmetric and 2f flavors, W = 0, is 4 The relation among R-charges of the elementary fields is the same as before: The mapping of the chiral ring generators is 4 If f + |k| = 2, k = 0, ±1, ±2, all the ranks in the quiver tails vanish.In the case of the fully deconfined theory, the full gauge group is trivial.This means that the deconfined theory is replaced by a Wess-Zumino, with a non trivial superpotential constructed out of the gauge singlet fields γ i , σ i , M i , as in [33,51].
, and this can be sequentially confined.We start from the right-most node, which is a Sp(1) 0 with 2 • 2 flavors, so it confines.Moreover, the antisymmetric for the Sp(2) 0 node is removed.We then dualize the Sp(2) 0 with 3•2 flavors, which also confines.After N −1 confining steps, we end up with Sp(N ) −k , with an antisymmetric plus 6 − 2|k| flavors, and some flipping fields.The same Sp(N ) +k ↔ Sp(N ) −k duality can be achieved in a different way, see [33] and eq.5.2 in [51].
The main difference with respect to the case of 2 is that we deconfine the adjoint using the "one monopole duality" of [24], which introduce superpotential terms in the quiver which are linear in the monopoles.Such terms break the topological symmetries and give rise to some complications, for instance the R-charge of the monopoles M ...,+,... is not equal to the R-charge of the monopoles M ...,−,... .(In linear quivers made of U gauge groups, we denote by M 0,0,±,±,... monopoles with non-zero minimal flux in the nodes with ± and zero flux in nodes with 0).In detail, the presence of a linear monopole superpotential leads to a modification of the usual R-charge monopole formula; in fact, every time we have a superpotential term W = M ...,−,... we need to ensure the marginality of such monopole.The main idea is to start with a simple ansatz for the additional corrections to the standard monopole R-charge formula, and fix the additional terms using the marginality of the monopoles contained in the superpotential and the operator map across duality to completely fix the coefficients of such terms.Physically, the added terms corresponds to mixed contact terms between R-symmetry and gauge symmetry, that may be computed, for instance, using localisation techniques.However, this goes beyond the aim of the present work.
Let us now explain a bit more in detail the procedure we are going to use.As we said, monopole operators in the superpotential are not symmetric under charge conjugation.Thus, the modification of the usual R-charge formula should distinguish the different signs of the fluxes, so, given a general linear quiver with N gauge nodes, we start from the ansatz: where (standard) refers to the usual R-charge contributions from matter fields and gauginos, for instance for the following quiver with matter in the (bi-)fundamental and adjoint i − m (2) The parameters α i are the ones that will be fixed imposing the marginality of the monopoles in the superpotential and the use of the duality map.The use of the duality map can be considered as a weakness of such an effective procedure: given a general quiver theory with an arbitrary combinations of linear monopole superpotential we are not able to provide an expression for the monopole R-charge; moreover, in this way we may only find the parameters α i only in terms of the mixing parameters of the the starting theory.Nonetheless, as we will concretely see later, the procedure we employ works perfectly in order to study the deconfinement of a traceless U (N ) adjoint field.
A first check of the validity of the procedure is that the result for the parameter fixed via the operator map does not depend on which operator we map.Another strong test comes from the computation of the supersymmetric index, where the presence of such contact terms is crucial, since it enters the sum over the gauge magnetic fluxes.
We start from the case of vanishing Chern-Simons interactions, with this result, it will be easy to turn on a real mass deformation and hence a Chern-Simons term in section 3.5, where we discuss the duals U (N ) k with adjoint and flavors.
We start with theory T 1 , that is U (N ) with a traceless antisymmetric field Φ and F flavors Q i , Qi .We take the superpotential to be vanishing, W = 0. Using the standard quiver notation for theories with four supercharges, T 1 reads Throughout most of this section, the square node denotes a SU (F ) × SU (F ) global symmetry.

The global symmetry is SU
The chiral ring is generated by the (dressed) mesons tr ( Qi Φ l Q j ), l = 0, . . ., N − 1, the powers of the antisymmetric traceless field tr (Φ j ), j = 2, . . ., N , and the (dressed) monopoles {M Φ k }, k = 0, 1, . . ., N − 1.In terms of the R-charges of the elementary fields Q i and Φ, which we denote r F and r Φ , the R-charge of the basic, undressed, monopole

Deconfine and dualize with the one-monopole duality
In order to deconfine the adjoint field, we use a variation of the confining one monopole duality of [24], which reads In this duality qq ↔ Φ and M + ↔ s.
The mapping M + ↔ s is in agreement with the R-charge computation.On the l.h.s. the topological symmetry is broken by the superpotential term, so the R-charge of the monopoles mixes with the topological symmetry: Imposing R[M − ] = 2 we get δ = −(N − 1)r q and thus We will need the following variation of (3.6): we start from (3.6), flip the monopole M + in the l.h.s. with a gauge singlet γ, on the r.h.s. a superpotential term sγ arises, s and γ become massive, integrating them out the superpotential becomes zero and we obtain the following deconfining duality: In this duality the chiral ring generators are only the quadratic mesons, which map as tr (q i qj ) ↔ M i j .
Starting from theory T 1 , we use (3.9) to deconfine the adjoint field into a two-node quiver theory, that is we consider theory T 1 : In T 1 the monopoles M 0,± , M +,+ and M −− are non trivial elements of the chiral ring, their R-charges read r Φ , we see that these monopoles map into T 1 as follows From the mapping we learn the following rule: deconfining and adjoint with the one monopole duality (3.9), that has M − in W, the monopole M + extends to M +,+ , while the monopole M − becomes M 0,− .This rule will be useful to fully deconfine the theory.We will give the full map of the chiral ring generators in (3.18).The next step is to dualize the right node U (N ) in T 1 using Aharony duality [5] U (N ) w/ (F, F ) flavors, in the quiver T 1 and obtain T 2 : +tr ( bφb) + tr (bqp) + tr ( bq p) + M tr (q q) (3.14) We decomposed the Seiberg dual mesons into the fields φ, M and p.Because of the F-terms of the singlet β, that we integrated out, the antisymmetric field φ is traceless.Notice that the monopoles M ±,0 in T 1 became M ± (1,1) in T 2 , here we are applying the rules of [11,28] for the mapping of monopole operators under dualities in quivers.
The mapping between the R-charges of theories T 1 and T 2 is dictated by the mapping of the mesonic operators and is The R-charges of the monopoles and of the flipping fields for the monopoles are where, the procedure to find α 1 , α 2 explained in 3, gives In T 1 , some monopole operators can be dressed using the meson made by bifundamental fields b, b, as discussed in [11,28].In T 2 , some monopole operators can be dressed with the adjoint φ.
The mapping of the chiral ring generators of the three theories constructed T 1 , T 1 and T 2 is Notice that {M + Φ J } monopoles map to singlets σ + N −J , in the same way of the monopoles of Sp(N ) with an antisymmetric, (2.31).On the other hand {M − Φ J } monopoles map to σ − J+1 .This is due to the fact that every time we deconfine the rank-2 field, the positive charge monopoles of U and the monopoles of Sp extend (M +,... becomes M +,+,... , M •,... becomes M •,•,... ), while the negative charge monopoles of U do not extend (M −,... becomes M 0,−,... ).
The general formula for the monopole R-charge in T DEC reads where Observe that the superpotential for T DEC contains N − 1 linear monopoles, and their marginality fixes N − 1 of the α i parameters; the remaining one has to be fixed using the duality map.
Let us finally comment that, as for Aharony duality for U gauge group without rank-2 matter fields, all the chiral ring generators of T 1 map to gauge singlets in T DEC .

Superpotential deformation
In this section and in section 3.4, we discuss complex deformations of the duality between our original theory with a single node U (N ) T 1 (3.4) and the fully deconfined quiver T DEC (3.20).As usual in Seiberg-like dualities, a complex deformation on the electric side will induce a Higgsing of the gauge groups on the magnetic side.
In this section we consider a superpotential deformation in T 1 of the form tr ( QF Φ J Q F ).This meson, according to (3.22), is mapped in T DEC to (M J+1 ) F,F , the flipping field for the meson tr (p F . . .bJ+1 b J+1 . . .p F ).We take J < N .Turning on a linear superpotential term in M J means that this F × F matrix of long mesons must take a non-zero vacuum expectation value of minimal non zero rank.This is achieved by giving a vev to the bifundamentals b J+1 , . . ., b N −1 and to the flavors p, such that tr (p F bN−1 . . .bJ+1 b J+1 . . .b N −1 p F ) = 1, while all other mesons are zero.
The final result is that Turning on a complex mass for a single flavor is a special case J = 0 of the discussion above.The flavors p and the gauge singlets M are absent for J = 0, and all the gauge groups in T DEC get partially Higgsed.The final result is precisely (3.20) with F → F − 1.
3.4 Deformation to the N = 4 theory: W = F j=1 tr ( Qj ΦQ j ) We now add f cubic terms to T 1 , obtaining U (N ) with F flavor hypers and W = j=1,...,F tr ( Qj ΦQ j ), that is the N = 4 theory U (N ) with F flavors and flavor sym-metry SU (F ) × U (1) top . 5sing the results obtained in Section 3.3, on the magnetic side all the flavors migrate to the right-most node U (F − 1), and out of the tower of singlets (M J ) ij , only (M 1 ) ij survive.The tail of gauge groups The right-most group U (F − 1) is not Higgsed.The final result is that is dual to ,0 i−1 + M tr (pp)+ + N −1 i=1 (tr(Φ i bi b i ) + tr(Φ i bi−1 b i−1 ) + φ i tr( bi b i )) (3.28)This result is strictly speaking valid for F > N .If F ≤ N the dual quiver becomes shorter and some of the flipping fields γ and σ decouple.This is due to the fact the theory U (N ) with W = j=1,...,F tr( Qj ΦQ j ) if F ≤ N becomes "bad" in the Gaiotto-Witten sense, so some Coluomb branch operators (that is tr (Φ h ) and {M Φ k }) become free and decouple.

Real masses and Chern-Simons terms
It is immediate to start from the duality between (3.19) and the fully deconfined tail (3.20) and derive the corresponding duality in the presence of a non-trivial Chern-Simons level.There are various possibilities to generate a CS level, and our aim in this section is only to give one example and not to treat the most general case, as for instance it has been done in [14] in the case without adjoint matter.The example we focus is as follows.We start from (3.19) with F + k flavours and give a real mass to k of the fundamental chirals Q.The result on the electric side is The effect of having a CS term is to remove some of the monopoles from the chiral ring.
In general, the fundamental monopole operators M ± acquire a gauge charge under the U (1) part of the gauge group given by thus, the monopoles negatively charged under the topological symmetry are removed from the chiral ring since gauge variant.The dual is:6 If F + |k| = 1, all the ranks in the quiver tails vanish.In the case of the fully deconfined theory, the full gauge group is trivial.This means that the deconfined theory is replaced by a Wess-Zumino, with a non trivial superpotential constructed out of the gauge singlet fields γ i , σ i , M i , as in [33,51].
If F + |k| = 2, the quiver tail is U (N ) −k − U (N − 1) − U (N − 2) − . . .− U (1), and this tail can be sequentially confined.We start from the right-most node, which is a U (1) 0 with (2, 2) flavors, so it confines.Moreover, the adjoint for the U (2) 0 node is removed.We then dualize the U (2) 0 with (3, 3) flavors, which also confines.After N − 1 confining steps, we end up with U (N ) −k , with an adjoint plus (2 − |k|, 2) flavors, and possibly some flipping fields.The same U (N ) +k ↔ U (N ) −k duality can be achieved in a different way, for instance turning on some real masses in duality 2.34 of [51].
T CS DEC : (3.31)where observe that, similarly to the electric theory, all the monopoles under flux − under the node with CS level are not gauge invariant and disappear from the superpotential, correspondingly, all the σ − i are removed from the chiral ring (recall that for vanishing CS level these singlets map to the tower of negatively charged dressed monopoles in the electric theory).

21 )
We put the Seiberg flipping terms in the last row.Notice that the monopoles change as follows: M •,0,0 → M •,•,0 and M 0,0,• → M 0,•,• .The quartic term tr (b 1 b 2 b 2 b 1 ) became a quadratic term for the Sp(N −2)-Sp(k −2) bifundamentals N 13 , which are thus massive and can be integrated out, generating a new quartic term tr (b 1 b 2 b 2 b 1 ).Â1 is an antisymmetric for Sp(f − 2), which we split into an antisymmetric traceless A 1 and a trace part a 1 .Same for φ, antisymmetric traceless for Sp(N − 2).
This meson, according to (2.31), is mapped in T DEC to (M J+1 ) 2f −1,f , the flipping field for the meson tr (p 2f −1 . . .b J+1 b J+1 . . .p 2f ).We take J < N .Turning on a linear superpotential term in M J means that this 2f × 2f matrix of long mesons must take a non-zero vacuum expectation value of minimal non zero rank.This is achieved by giving a vev to the bifundamentals b J , b J+1 , . . ., b N −1 and to the flavors p, such that tr (p 2f −1 b N −1 . . .b J+1 b J+1 . . .b N −1 p 2f ) = 1, while all other mesons are zero.