A tale of exceptional 3d dualities

We consider interesting Seiberg dualities for $Usp$ gauge theories with an antisymmetric, $8$ fundamentals and no superpotential. We reduce to three dimensions and systematically analyze deformations triggered by real and complex masses, reaching a plethora of $\mathcal{N}\!=\!2$ dualities for $U(N)$ and $Usp(2N)$ gauge theories, possibly with monopole superpotentials and Chern-Simons interactions. Special cases of these"exceptional dualities"are: supersymmetry enhancement dualities,"duality appetizers"and many known dualities relating rank-$1$ gauge groups. The $4d$ $\mathcal{N}\!=\!1$ $Usp$ dualities provide a unified perspective on many curious phenomena of $3d$ and $4d$ gauge theories with four supercharges. Finally, we propose a free mirror for $A_{2N}$ Argyres-Douglas, with its related adjoint-$Usp(2N)$ duality, and we construct a mirror for adjoint-$U(N)$, with an arbitrary number flavors and zero superpotential.


Introduction and summary
Soon after the first Seiberg duality [1], [2] found a very interesting class of 4d N = 1 dualities for U sp(2N c ) with an antisymmetric field and 8 fundamentals. The set of dual theories was enlarged in [3,4], and in the case N c = 1 it includes both Seiberg duality [1] and Intriligator-Pouliot duality [5].
It was recently realized that these U sp theories have a higher dimensional ancestor: the 6d (1, 0) superconformal field theory called E-string [6][7][8][9]. They are also related to domain walls of 5d N = 1 U sp(2N c ) with an antisymmetric field.
In this paper we study the reduction of the U sp(2N c ) ↔ U sp(2N c ) 4d dualities to 3d. In 3d we can turn on real masses, which do not exist in 4d, this process leads to 3d N = 2 dualities for U sp(2N c ) or U (N c ) gauge theories with a smaller amount of flavors (and possibly monopole superpotentials). As usual, each 3d N = 2 duality implies an integral identity for its S 3 supersymmetric partition function. Such Z S 3 's were studied independently in the mathematical literature [10,11], where it is also shown that the set of theories is naturally organized in terms of Weyl groups of a sequence of groups E 7 → SO(12) → SU (6) → . . . . Because of their 6d origin with E 8 global symmetry, and of the relations with this sequence of groups, we call the dualities of [2][3][4] and their 3d descendants exceptional dualities.
All these N c = 1 dualities admit a generalization replacing U (1) → U (N c ) with adjoint, and U sp(2) → U sp(2N c ) with antisymmetric, while keeping the same super-potential and the same number of fundamentals, which we describe in this paper. 2 For generic N c , it is possible to flow from the dualities of [2][3][4] to two other interesting classes of dualities: • as pointed out by Aghaei, Amariti and Sekiguchi in [29], which provided us the motivation for this paper, one can reach the duality SU (N c ) with adjoint and 1 flavor ↔ N = 4 U (1) with N c flavors. This duality was found in [30,31], where it was derived deforming an N = 4 mirror symmetry [32,33]. It explains the Lagrangians of [34,35] for 4d N = 2 Argyres-Douglas theory (A 1 , A 2Nc−1 ), so the 4d supersymmetry enhancements have their origines in the dualities [2][3][4].
• the "duality appetizer" for SU (2) 1 gauge group of [38] was generalized in [39] to U sp(2N c ) 2 with antisymmetric and to U (N c ) 1 with adjoint. These are dualities between a gauge theory and a set of free fields. They can be reached from the dualities of [2][3][4], turning on complex masses and "axial" real masses, at the end of the process (no flavors left). See also [22]. Along the way we find new, similar, infrared free gauge theories with a non zero amount of flavors (section 4). It looks like the choice of name, "appetizers", was appropriate.
Moreover, [40] recently proposed that U (1)3 2 with 1 chiral flavor displays supersymmetry enhancement, in section 5 we reach a duality which is consistent with the proposal, and naturally leads to conjecture that U (N c ) ± 3 2 with adjoint and 1 chiral flavor displays N = 2 → N = 4 supersymmetry enhancement for any N c .
We hope this convinces that the 4d N = 1 dualities of [2][3][4] allow for a unified perspective on many curious phenomena of 3d and 4d gauge theories with 4 supercharges. Therefore, we embark in a systematic investigation of the 3d N = 2 dualities descending from [2][3][4].
In section 2 we review the duality web of [2][3][4], and turn on 3d "vector-like" real masses to study self-dualities of U (N c ) and U sp(2N c ) theories without Chern-Simons interactions, leaving somewhat more exotic U (N c ) ↔ U sp(2N c ) and U (N c ) W=0 ↔ U (N c ) W=M + dualities with Chern-Simons interactions to sections 5 and 6.
In section 3 we deform the previous dualities with a complex mass, leading to new "confining" dualities. We work out the full superpotential of the Wess-Zumino duals in each case.
At the end of the paper, we find new dualities which cannot be obtained from the 4d models of [2][3][4], but can be guessed by analogy with the results discussed previously: • in section 7 we argue that the 3d mirror of (A 1 , A 2N ) AD models is simply N free hypermultiplets. This statement, together with the Lagrangians of [34,35], allows us to discover a 3d duality for U sp(2N c ) with adjoint and 2 flavors. We check numerically the equality of the S 3 partition function, but we do not know how to "derive" the duality from a more fundamental one analog to the ones [2][3][4]. We nevertheless expect such an uplift to exist.
• in section 8 we generalize the mirror duality [32] for U (N c ) with adjoint and 2 flavors, proposing a mirror dual for U (N c ) with adjoint and a generic number N f flavors, which is a quiver with N f −1 U (N c ) gauge groups. We use its associated brane setup. This is similar to the N = 2 mirror symmetries discussed in [41], for instance for U (N c ) with N f flavors and no adjoint. These mirror symmetries have the property that the rank of the UV global symmetries is different on the two sides. On the other hand, other complications that arise in [41] (non trivial quantum relations in the chiral ring on the quiver side) are absent here. Since it should be possible to turn on a mass for the adjoint and flow to the duality of [41], we hope this will shed light on those issues.
Note. This work overlaps with [42]. We are grateful to the authors of [42] for informing us of their results.
2 Vector-like real masses: self-dualities without Chern-Simons An interesting class of 4d N = 1 dualities [2][3][4] states the IR equivalence of two N = 1 U sp(2N c ) gauge theories with 8 flavors and an anti-symmetric field. We call these dualities self-dualities, even if to be precise they are duality between two different theories, since on the magnetic side there are also gauge singlet fields. These singlet fields appear in the superpotential multiplying mesonics operators, so they can be called "flipping fields". 3 A more precise name is "self-dualities modulo flips".
As explained in [4,43], each duality implies an identity for the associated superconformal index, i.e. the S 3 × S 1 partition function. Such integral identities were studied independently in the mathematical literature [10], where it is also pointed out that the set of dual theories is made of 72 frames, acted upon by the Weyl group of E 7 . 4 We first describe the 4d U sp ↔ U sp with 8 flavors dualities (there are 3 of them, 2.3, 2.6, 2.8). Reducing them to 3d, one basically obtains the same duality for 3d N = 2 theories. The only difference is that the linear monopole term in the superpotential δW = M U sp(2Nc) is generated on both sides of the duality, so the continuos global symmetry is the same in 3d. Because of the linear monopole superpotential, on both sides of the duality the monopole operators are not in the chiral ring, so the 3d chiral ring is isomorphic to the 4d chiral ring.
Once we are in 3d, we can turn on supersymmetric real masses, which do not exist in 4d, and deform the dualities to new 3d dualities. This process generically breaks both the gauge and the global symmetry and was studied at the level of S 3 partition functions Z S 3 in [11]. [11] studied the full set of dual frames at each level, demonstrating the existence of an action of the Weyl groups of a sequence of Lie groups of decreasing rank: Instead of going through the whole set of dual theories at each level, we decided to organize the presentation in a different way. In the remaining of this section we describe many dualities that can be obtained from 2.3, restricting ourselves to "vector-like" real masses: in this way no Chern-Simons terms are generated and for U (N c ) gauge group the number of fundamentals is equal to the number of anti-fundamentals. We study self-dualities for each of the following five 3d N = 2 models: In the scheme above we also indicated the real masses used in the RG flows. Green arrows represent RG flows triggered by "axial" real masses, that will not be discussed in this section. For each of the 5 models we describe both an Intriligator-Pouliot like self-duality (with the maximal number of gauge singlets on the magnetic side) and a Csaki et al. like self-duality, (with a smaller number of gauge singlets on the magnetic side).
We start from an interesting 4d duality found more than 20 years ago in [2], which states the IR equivalence of two N = 1 U sp(2N c ) gauge theories with 8 flavors and an anti-symmetric field. The antisymmetric field is traceless, with N c (2N c − 1) − 1 components, so in the case N c = 1 (i.e. SU (2) gauge theory) it is absent. The precise statement of the duality is: The 8 2 = 28 towers of N c gauge singlet fields µ ij;r (i < j = 1, . . . , 8 r = 0, . . . , N c − 1) appearing on the r.h.s. are Seiberg mesons flipping the gauge invariant "dressed mesons" tr(Q i A r Q j ) 5 of the magnetic theory.
The global symmetries on both sides of the duality are where the U (8) acts on the 8 flavors, the U (1) axial = U (1)q×U (1)a U (1) anomaly acts on both the antisymmetric field and the 8 fundamentals (and in the r.h.s. on the gauge singlets), the U (1) anomaly is related to the NSVZ beta function constraint. 6 For simplicity, throughout the paper, we simply write SU (8) × U (1) (or its analogs) and omit the U (1) R factor.
For the duality 2.3, the chiral ring generators map as Generically, for small r and l, these operators violate the unitarity bound R > 2 3 , so they are not in the chiral ring and should be flipped introducing appropriate gauge singlets β O as prescribed in [30].
In the case of N c = 1, 2.3 reduces to the 4d Intriligator-Pouliot duality for SU (2) gauge theory (in the Intriligator-Pouliot duality the magnetic theory has N f 2 gauge singlets). The SU (2) gauge theories also enjoy Seiberg duality, where the magnetic theory has N f 2 2 gauge singlets (we are counting real flavors here). In analogy, all the U sp(2N c ) theories with antisymmetric and 8 flavors satisfy other dualities, which we now describe.
Let us call 2.3 Intriligator-Pouliot like duality. The Seiberg like duality splits the 8 real flavors into 4 "fundamentals" and 4 "anti-fundamentals", and on the magnetic side only the "mesons" are flipped, so there are 4 2 = 16 towers of singlets m ij;r : U sp(2N c ) w/ antisymmetric a and 8 flavors q i .
The UV global symmetry on the r.h.s. is only SU (4) 2 ×U (1) 2 . The chiral ring generators map as the paper. 6 In the case N c = 1 the rank-2 fields are absent and hence the factor U (1) a is absent. This comment applies to each theory we discuss in this paper, except for section 7.
Combining 2.3 and 2.6, one finds a third duality, which also splits the 8 flavors into 4 "fundamentals" and 4 "anti-fundamentals", and in the magnetic side only the "baryons" are flipped, so there are 28 − 16 = 12 towers of singlets b i<j;r ,b i<j;r (i, j = 1, . . . , 4). Similarly to [18], we call it Csaki et al. like duality: U sp(2N c ) w/ antisymmetric a and 8 flavors q i .
Also in this case the UV global symmetry on the r.h.s. is only SU (4) 2 × U (1) 2 . The chiral ring generators map as Considering all the different ways to split the 8 flavors into 4 +4, it is actually possible to produce a total of 72 different dual frames [4,10,18]: 1 frame with no singlets, 1 Intriligator-Pouliout like frame with 28N c singlets,

The most fundamental duality
Here we show that the Csaki et al. like duality 2.8 can be iterated to obtain the other two dualities 2.3 and 2.6.
Let us consider the same 2.8 duality twice, but in one case relabeling q 3 , q 4 → q 5 , q 6 , in other words on the r.h.s. we have flipping fields All of the above is true also in 4d, now we begin the study of truly 3d models. Starting from the Intriligator-Pouliot like duality 2.3 compactified to 3d, and turning on real masses +1 for q 7 and −1 for q 8 we break the global symmetry to where U (1) a acts on the antisymmetric field a and U (1) q acts on the 6 massless flavors. The presence of two independent U (1) axial symmetries is related to the fact that the linear monopole superpotential disappears on both sides. This implies that there is a Coulomb branch: the N c dressed monopoles M a l (M a l is the monopole dressed with r powers of the antisymmetric field a, l = 0, . . . , N c − 1) are in the chiral ring on the electric side.
The tower of singlets µ 78;r does not couple to dressed mesons tr(Q i Φ r Q j ) anymore, but interactions with monopoles are now allowed, and new superpotential terms preserving the SU (6) × U (1) a × U (1) q symmetry are generated: so on the magnetic side, M A r (the monopole dressed with r powers of the antisymmetric field A) is set to zero in the chiral ring by the F-terms of the singlets. In other words the µ 78;r gauge singlets become flipping fields for the dressed monopoles.
The new IR duality reads U sp(2N c ) w/ antisymmetric a and 6 flavors q i .
The case N c = 1 was discovered in [13].
The Z S 3 integral identity associated with this duality is Theorem 5.6.14 of [11].
We can perform the same procedure of giving real masses +1 for q 7 and −1 for q 8 to the Csaki et al. like duality 2.8 (that has 12 towers of singlets on the r.h.s.). On the r.h.s. the UV global symmetry breaks as SU (4) 2 × U (1) 2 → SU (4) × SU (2) × U (1) 3 . The quarksQ 3 ,Q 4 and the singletsb i3;r ,b i4;r (i = 1, 2) become massive. The tower of singletsb 12;r pairs with the tower of dressed monopoles M A r in a quantum generated superpotential.
The new IR duality reads U sp(2N c ) w/ antisymmetric a and 6 flavors q i .
The Z S 3 integral identity associated with this duality is Theorem 5.6.11 of [11].
There is also a U sp(2N c ) ↔ U (N c ) W=M + +M − dual, which can be obtained turning on real masses in the self-duality for U sp(2N c ) with 8 flavors. We do not discuss this duality here, see section 5. A dual model to U (N ) gauge theory with monopole superpotential M + + M − (without the adjoint field, but with an arbitrary number of flavors N f ) was proposed in [20] (see also [45]). 7 , and it was obtained turning on real masses in the 3d reduction of the Intriligator-Pouliot 4d duality for U sp(2N ) with 2N f flavors.
Here we start from the Intriligator-Pouliot like duality 2.3, that also has matter in a rank-2 representation of the gauge group, but we can follow the same steps of [20]: we split the 8 flavors of the U sp(2N c ) gauge theory in two sets of 4 flavors, and give positive real mass +1 to the first 4 flavors and negative real mass −1 to the second 4 flavors.
The real masses break the continuos global symmetry where the U (1) is a combination of the two 'axial' U (1)'s, U (1) q and U (1) φ , which act on the three set of fields {q,q, φ} with charges {1, 1, 0} and {0, 0, 1} respectively. A non trivial part of the story is that we focus on a vacuum in which both the gauge groups are Higgsed as On the magnetic side the story is similar, but there are also the gauge singlets. The singlets µ ij;r are massless if i = 1, . . . , 4 and j = 5, . . . , 8, so they remain part of the magnetic theory at low energies. The µ ij;r with i < j = 1, . . . , 4 or i < j = 5, . . . , 8 are instead massive, and disappear.
Monopole terms in the superpotential are generated because of the Higgsing of the gauge group, through the Polyakov-Affleck-Harvey-Witten mechanism: The two linear monopole superpotential terms break the U (1) top topological symmetry of U (N c ) and one of the two abelian axial symmetries, so they ensure that the global symmetries are correct. Moreover, monopole superpotential lifts completely the Coulomb branch: no monopole operators are in the chiral ring on either side of the duality.
Putting all these ingredients together, in the IR we get a 3d U (N c ) ↔ U (N c ) duality with 16 towers of gauge singlets µ ij;r : The chiral ring generators map as The N c = 1 case was discussed in [18].
We can perform the same procedure of giving real masses starting from the Csaki et al. like duality 2.8. 8 We give real mass +1 to the flavors 1, 2,1,2 and real mass −1 8 We can also consider the same real mass deformation on all the theories in the duality web of 72 models. The discussion of [18] for the N c = 1 case generalizes in a straightforward way. The 72 models of U sp(2N c ) w/2N f = 8 reduce to 32 theories U sp(2N c ) w/ 6 fundamentals (labelled by elements of the coset W (SO(12))/W (SU (6))) plus 40 theories U (N c ) w/ 4 flavors (labelled by elements of the coset W (SO(12))/W (SU (4) × SU (4))). The E 7 action is broken to a to the flavors 3, 4,3,4. Out of the 12 towers of "baryon"-singlets b i<j;r ,b i<j;r , 4 towers (b 12;r , b 34;r ,b 12;r ,b 34;r ) become massive, the other 8 towers stay massless. Also in this case a monopole superpotential M + + M − is generated. Relabelling the r.h.s. fields, in the IR we get the following U (N c ) ↔ U (N c ) duality with 4 + 4 towers of gauge singlets b ij;r andb ij;r : The chiral ring generators map as 24) The N c = 1 case was discussed in [18]. The U (1) W=M + +M − theory with 4 flavors and 8 flipping fields appears as a boundary condition for N = 2 rank-1 class-S SCFT's [52].
Now we start from 2.21 and turn on real and opposite masses for q 4 andq 4 , following the same strategy of section 8 of [20]. On the r.h.s. the flavors Q 4 andQ 4 , and the singlets µ i4;r and µ 4i;r (for i = 1, 2, 3) becomes massive. The singlets µ ij;r and µ 44;r stay massless. We rename µ 44;r → µ −;r .
The SU (4) 2 × U (1) global symmetry breaks to SU (3) 2 × U (1) × U (1), so the process removes one linear monopole superpotential term (if both monopole terms M + and M − are present, the new U (1) symmetry would be broken). New superpotential terms preserving all the magnetic side symmetries are generated: We denote the monopole operators dressed by k powers of the adjoint field φ as M φ k . For U (N c ) gauge group, all the dressed monopoles can be written as an algebraic combination of the 2N c monopoles M ± φ k , k = 0, . . . , N c − 1 [53].
The duality reduces in the IR to The surviving gauge singlets on the r.h.s. now flip both the mesons and the dressed monopoles M + Φ r , while the tower of dressed monopoles M − Φ r is not in the chiral ring because of the superpotential term M − .
The chiral ring generators map as In the case N c = 1 this duality appeared in [20], as a special case of a duality valid for U (N c ) without adjoint, but with an arbitrary number of flavors N f [20].
Let us mention that the duality 2.26 can also be reached starting from the U sp ↔ U sp duality 2.16, turning on real masses (+, +, +, −, −, −) for the 6 flavors on the electric side. On both sides the gauge groups Higgs down to U (N c ), and the massless magnetic singlets are µ 78;r and µ ij;r with i = 1, 2, 3, j = 1, 2, 3.
We can also start from the duality 2.23, and we get a duality different from 2.26. We turn on real and opposite masses for q 4 andq 4 , so that b 14;r , b 24;r , b 41;r , b 42;r become massive, while b 13;r , b 23;r ,b 31;r ,b 32;r stay massless and keep flipping the "baryons". This time no quantum superpotential is generated, and the duality reads On the r.h.s. UV global symmetry is only SU (2) 2 × U (1) 4 . The chiral ring generators map as As far as we know, duality 2.28 is new also in the case N c = 1.
Let us emphasize that the above mapping and the mapping 2.24 resemble the mapping of mirror symmetries [32]: in mirror symmetries monopoles map to mesons, while mesons map to either singlets, mesons or monopoles. We will indeed see in section 2.5.1 that 2.28 can be deformed to a mirror symmetry like duality, with a brane setup such that the gauge theory duality corresponds to S-duality in the brane setup.
We start from 2.26, in order to remove the last monopole superpotential, we turn on real and opposite masses for q 3 andq 3 .
The continuos global symmetry breaks as As before, the superpotential terms linear in the monopoles disappear on both sides, while a new tower of monopole-flipping superpotential terms is generated on the r.h.s. On the r.h.s. of 2.26, the flavors Q 3 andQ 3 , and the singlets µ i3;r and µ 3i;r (for i = 1, 2) becomes massive. The singlets µ ij;r , µ 33;r , µ −;r (i, j = 1, 2) stay massless.
Renaming µ 33;r → µ −;r , at low energy we get a Aharony like duality: If N c = 1 the duality reduces to the Aharony duality for U (1) with 2 flavors [13]. The chiral ring generators map as Let us display the charges of the relevant fields under the global symmetry SU (2) The Abelian global symmetries U (1) q and U (1) φ mix with the R-symmetry, accordingly r q and r φ are the two independent variables appearing in Z-extremization, which can be used to determine the superconformal R-charges.

A mirror like duality and its brane setup
We can also find a different dual for U (N c ) with adjoint and (2, 2) flavors, starting from 2.23, giving real mass +1 to q 2 and real mass −1 toq 2 . Q 2 ,Q 2 and the singlets b 23;r ,b 32;r become massive, the other 2 flavors and the singlets b 13;r ,b 31;r stay massless. Renaming q 3 ,q 3 → q 2 ,q 2 and {Q 1 ,Q 1 , Q 3 ,Q 3 , b 13;r ,b 31;r } → {Q L ,Q R , Q R ,Q L , µ L;r , µ R;r } (so that the singlets flip the diagonal instead of the off-diagonal mesons), the low energy duality reads: If N c = 1 the duality reduces to the N = 2 mirror symmetry duality for U (1) with 2 flavors of [12].
Brane interpretation The duality 2.34 admits a brane interpretation in Type IIB superstringà la Hanany-Witten [33]: 36) The N S5 branes stretch along 012345, the N c D3 branes stretch along 0126 and the D5 branes along 012347. The N c D3 branes give rise to a U (N c ) gauge theory with a massless adjoint φ on the l.h.s. and Φ on the r.h.s. (more precisely φ and Φ have N 2 c degrees of freedom so there is also a gauge singlet, which however decouples).
On the l.h.s. the central D5 's provide the 2 flavors q i ,q i to the U (N c ) gauge theory (D3 − D5 strings), which do not couple to φ.
On the r.h.s. the D5 's are outside the N S branes, each D5 still provides a flavor, but on the r.h.s. there are N c + N c complex degrees of freedom associated the vertical motion of the external D3 segments. Such d.o.f. are represented by 2 towers of N c gauge singlets b L;r , b R;r , flipping the dressed mesons tr(Q L Φ rQ L ), tr(Q R Φ rQ R ). The two brane setups in 2.36 are related by Type IIB S-duality, so we can interpret the duality 2.34 as a mirror symmetry, with N = 2 supersymmetry. This suggests a natural generalization to a mirror duality for N = 2 adjoint-U (N c ) with N f flavors and W = 0. We will discuss this mirror symmetry in section 8.
It would be interesting to discover a brane interpretation of the dualities discussed in the rest of this section. Dualities like 2.21, 2.26 and 2.31 involve monopoles in the superpotentials, so the relevant setups might be similar to the ones constructed in [46][47][48][49].
3 Turning on a complex mass: cubic Wess-Zumino duals In this section we start from the dualities of section 2 and turn on a complex mass for the electric theories (i.e. the side with no gauge singlet fields). We consider giving mass to a minimal number of flavors, that is 2 fundamentals for U sp(2N c ) and 1 fundamental and 1 anti-fundamentals for U (N c ).
The dualities we start from have the following property: turning on a complex mass to a minimal number of flavors on the electric side of an Intriligator-Pouliot like duality, the magnetic gauge group is completely broken.
This happens because on the magnetic side a superpotential term linear in a singlet field is turned on, and in order to satisfy the F-terms, the 2-index field (antisymmetric for U sp(2N c ) and adjoint for U (N c )) takes a maximal nilpotent vev. So we end up with a duality between a U sp(2N c ) or U (N c ) gauge theory and a theory with no gauge interactions (for this reason such dualities are usually called confining). The gauge theories are: A non trivial superpotential is generated for the low energy Wess-Zumino model, which we find as the most general interaction preserving all the symmetries. In each case the superpotential is cubic. This implies, in particular, that all the Wess-Zumino models are relevant deformations of a free theory and flow to a non-trivial SCFT in the IR.
For each model, the duality for the case N c = 1 already appeared in the literature.
We first discuss at length the case of U (N c ) with 3 flavors, W = M + + M − , in section 3.1. The other cases are very similar and we skip many details. Let us also mention that each theory in 3.1 (except for the top one, U sp(2N c ) with 6 flavors) can be obtained either by a complex mass deformation of the corresponding theory in 2.2, or by a real mass deformation of a "higher" theory in 3.1 (using the real masses indicated). This is consistent with the superpotential being always cubic. The vev of Φ is a maximal N c × N c Jordan block. Similar vev's were discussed in [50,51]. The vev's 3.3 are such that tr(Q 4 Φ r Q 4 ) = δ r,Nc−1 , while the other dressed mesons tr(Q i Φ r Q j ) are zero.
We now want to determine the precise superpotential of the Wess-Zumino model. If N c = 1 the fields µ 44;r are absent, the duality was studied in [19] and the Wess-Zumino superpotential was found to be a determinant: 1975. This implies that R[tr(φ 2 )] < 1 2 , so tr(φ 2 ) is a free field in the infrared. We are then led to apply the prescription of [30]: introduce N c − 1 gauge singlets β i that "flip" the fields tr(φ j ). On the dual side, the N c − 1 singlets µ 44;r become massive and disappear from the Wess-Zumino model.
In [30,31] (which studied a 4d duality that reduced to 3d becomes equivalent to the one we discuss in section 3.3), the β j fields cannot take a vev for quantum reasons: if such a field takes a vev, an ADS-like superpotential is generated, and there is no vacuum. This supports the conjecture that the β j 's are not in the quantum chiral ring. (Logically there is the possibility that the β j 's are nilpotent elements of the chiral ring, so they do not give rise to flat directions but are non-trivial chiral ring operators. In the analogous cases discussed in [30,31,54] this possibility was discarded showing that the β j 's are descendants under an emergent IR additional supersymmetry.) We can also argue that the N c − 1 fields µ 44;r are free in the IR on the dual side: on the Wess-Zumino model (as we will see in the next paragraph), the 9N c µ ij;r fields enter cubically the superpotential. In order to preserve all the SU (3) 2 × U (1) axial symmetries, the µ 44;r fields should enter the superpotential via terms of higher degree, that (in the absence of gauge interactions) are however always irrelevant, so the µ 44;r 's would decouple from the rest of the SCFT.
The most general superpotential. We then need to generalize 3.4 and find the superpotential for N c > 1, as a function of the µ ij;r . Cubic couplings similar to a 3 × 3 determinant, of the form So the superpotential W = M + + M − imposes the relation Using the map 2.27, we deduce the trial R-charges of the surviving fields µ ij;r on the magnetic side: so for the cubic terms : From these it is possible to find all the interactions with total R-charge 2 and U (1) axial charge 0: only the cubic terms µ iĩ;r µ jj;s µ kk;t with r + s + t = N c − 1 are present in the Wess-Zumino superpotential.
The Z S 3 integral identity associated to 3.10 is the first formula of Theorem 5.6.7 of [11]. So the magnetic gauge group is broken completely, the low energy theory is a Wess-Zumino with model with massless fields µ ij;r , µ −;r (i, j = 1, 2, r = 0, . . . , N c − 1). As before, the N c − 1 fields µ 33;r map to tr(φ j ), do not enter the superpotential and are free fields, so we move them on the l.h.s. adding the β j fields. The case N c = 1 duality was found in [20,21] and states the IR equivalence between U (1) with 2 flavors, W = M + and a cubic Wess-Zumino with N c = 1 : The analog duality for U (N c ) with no adjoint and N f = N c + 1 flavors, found in [20], plays a crucial role in the "sequential confinement" of 3d quiver tails [31], see also [41,54]. We can find the quantum generated superpotential preserving the global symmetry SU (2) 2 × U (1) 2 following very similar steps to the ones in section 3.1. The result is that the duality can be written as where i, j = 1, 2 and r, s, t = 0, . . . , N c − 1. The Z S 3 integral identity associated to 3.14 is the second formula of Theorem 5.6.7 of [11].
Turning on a complex mass in the duality 2.31 breaks the SU (2) 2 × U (1) 3 global symmetry to U (1) 3 . On the r.h.s. the gauge symmetry U (N c ) breaks completely, so the low energy theory is a Wess-Zumino model with massless fields µ 11;r , µ +;r , µ −;r (r = 0, . . . , N c − 1) and µ 22;r (r = 0, . . . , N c − 2). The N c − 1 fields µ 22;r do not enter the superpotential and are free fields, so we move them on the l.h.s. adding the β j fields. We rename the fields q 1 ,q 1 , µ 11;r → q,q, µ r . The low energy duality can be phrased as U (N c ) w/ adjoint φ and 1 flavor q,q W = Nc j=2 β j tr(φ j ) ⇐⇒ Wess-Zumino model w/ N c fields µ r ∼ tr(qφ Nc−1−r q) and 2N c fields µ ±;r ∼ M ± φ r W = Nc−1 r,s,t=0 µ r µ +;s µ −;t δ r+s+t,Nc−1 (3.16) The case N c = 1 is the well known duality between U (1) with 1 flavors and the XY Z model [12]. Notice that for any N c there is a Z 3 symmetry on the r.h.s., which emerges in the IR on the gauge theory side. The Z S 3 identity associated to 3.16 is Theorem 5.6.8 of [11], as was pointed out in [29]. One new thing is that here we spell out the explicit form of the superpotential, which contains Nc(Nc+1) 2 cubic terms. An equivalent version of 3.16 (with all singlets µ r except one on the l.h.s., and with the topological symmetry gauged) was derived in [30,31] starting from the N = 4 mirror symmetry for SU (N c ) 2N c flavors hyper (for which the equality of the round S 3 partition function can be proven as a function of the N = 4 mass and Fayet-Iliopoulos parameters [55]), and using the sequential confinement technique.
Interestingly, [56] recently derived the Z S 3 identity starting from 5d N = 1 gauge theories, through S-duality for pq-webs and topological string / instantons partition functions. Similar ideas have been used in [19,57], and might be useful to derive new 3d N = 2 dualities from 5d.
Similarly to the duality 2.34, also 3.16 admits a brane interpretation in Type IIB superstringà la Hanany-Witten analogous to 2.36: This is accomplished taking the vev for the antisymmetric field A to be 0 1 1 0 ⊗ Jordan Nc×Nc , where Jordan Nc×Nc is a maximal Jordan block. The vev breaks completely the magnetic gauge group, the 12N c fields µ 7j;r and µ 8j;r (j = 1, . . . , 6, r = 0, . . . , N c −1) get a mass and disappear from the spectrum. The fields µ 78;r decouple, so we add the β j fields and W = β j tr(a j ). 10 10 It is interesting to perform A-maximization [58,59] for the electric gauge theory in 4d, U sp(2N c ) with antisymmetric a and 6 fundamentals q i , with W 4d = Nc j=2 β j tr(a j ). The result for the superconformal R-charges, for all N c , is very simple So all the electric mesons tr(q i a r q j ) have R = 2 3 and are free fields in the IR. Since the tr(q i a r q j ) are precisely at the unitarity bound we can leave the µ ij;r on the r.h.s. of the duality. Accordingly, in 4d the cubic Wess-Zumino interactions are marginally irrelevant. We see another example of how unitarity issues are mapped under duality to asymptotic freedom issues.
In the case N c = 1, found by Intriligator and Pouliot [5], the Wess-Zumino superpotential is the SU (6)-invariant Pfaffian of the 6 × 6 mesonic matrix: The generalization to N c > 1 follows the same steps of the previous cases. The IR duality reads U sp(2N c ) w/ antisymmetric a and 6 flavors q i W = M + Nc j=2 β j tr(a j ) ⇐⇒ Wess-Zumino model w/ 15N c chiral fields µ ij;l ∼ tr(q i a l q j ) W = 1 2 3 3! rst ijklmn ε ijklmn µ ij;r µ kl;s µ mn;t δ r+s+t,Nc−1 (3.21) The associated Z S 3 integral identity is formula 5.3.7 of [11].

Axial + complex masses: "Duality appetizers"
In this section we start from the confining dualities of section 3 and turn on real masses for the flavors such that the Wess-Zumino models become free. 11 In this section we Notice that since R[a] = 0, the Z S 3 ×S 1 of the electric theory "factorizes" into a product of the Z S 3 ×S 1 of N c SU (2) gauge theories with 6 doublets. This is analogous to the SU (N c ) ↔ U (1) Nc−1 Abelianization in 3d [31]. 11 It would be easy to flip enough fields in the Wess-Zumino models and obtain dualities with a set of free fields (for instance in the U (1) with 1 flavor q,q ↔ XY Z duality, we can flip the meson qq ↔ X allow for non-vector-like (axial) real masses, so in particular Chern-Simons terms are generated. If zero flavors are left, we recover two classes of dualities that were discovered in [39] generalizing the findings of [38] for adjoint-SU (2): 12 The free fields on the r.h.s. are simply the Casimir invariant combinations tr(a j ) and tr(φ j ) (recall that both a and φ are traceless, tr(a) = tr(φ) = 0, so j = 2, . . . , N c ). There are no monopoles in the chiral ring because of the Chern-Simons interaction. Let us mention that, in section 7, inspired by a 4d susy enhancement duality, we discover new dualities for U sp(2N c )1 2 with an adjoint field (instead of an antisymmetric) and 2 fundamentals q, p, from which, turning on a real mass for q, we get: It is easy to see that the dualities 4.2 and 4.1 can be obtained from the dualities 3.22 and 3.16 (respectively), turning on same sign real masses for all the flavors.
It is however interesting to spell out all such dualities obtained from the confining dualities of section 3. We need to turn on axial-like real masses which do not involve the 2-index matter, so we need at least two abelian factors in the global symmetry: we cannot turn on an axial real mass for in U sp(2N c ) with 6 flavors and W = M, or in U (N c ) with (3, 3) flavors and W = M + + M − (the global symmetry is only SU (6) × U (1), or SU (3) 2 × U (1)). So in the rest of this section we turn on axial real masses for the three dualities 3.14, 3.16 and 3.22, producing more examples of infrared free gauge theories.
In all the dualities of this section, the gauge theory is not time-reversal invariant in the UV (because of the Chern-Simons interactions), but the dual side is free so it is obviously time-reversal invariant. The duality then implies that time-reversal and obtain the duality between N = 4 U (1) with 1 flavors and two free chirals Y, Z), but this not what we are interested in here. Here we want a duality between a gauge theory with no flipping fields on the l.h.s., and a set of free fields on the r.h.s. 12 To be precise, at low energies the gauge theory is not completely free, there is also a topological quantum field theory in the infrared. Describing the topological field theory in each case goes beyond the scope of this paper.
invariance in these gauge theories is emergent. In other words the statements hold also reversing the sign of the Chern-Simons term.
In this section we present the dualities leaving all the free fields on the r.h.s., as in [38] and [39].
The case N c = 1 gives rise to the most basic N = 2 duality: U (1)1 2 with 1 charged chiral dual to a free chiral field, found in [17].
Turning on a positive real mass for q in 4.5, on the l.h.s. no monopole is left in the chiral ring, and on the r.h.s. the free monopoles get a mass. 4.5 flows to 4.1: The associated Z S 3 integral identities are in Theorem 5.6.8 of [11]. The chain of dualities in this subsection was already discussed in [22]. Since we have a brane setup for 4.4, we can easily produce brane setups for the 4.5 and 4.6. First we move the D5 on top of a N S5 so that they form a pq-web with 4 semi-infinite 5-branes. Adding the first real mass corresponds to deform the pq-web to the minimal pq-web with 3 semi-infinite 5 branes of type (1, 0), (0, 1), (1,1). Adding the second real mass corresponds to deforming the minimal pq-web with 3 semi-infinite 5 branes into a single 5-brane of type (1,1). It is indeed known that N c D3's streching between a (1, 0) and (1, k) 5-brane give rise to a U (N c ) k gauge theory. We then turn on a real mass for q 3 , q 2 and q 1 , flowing to U sp(2N c ) 1 w/ antisymmetric a and 2 flavors q i , W = 0 ⇐⇒ N c free mesons ∼ tr(q 1 a l q 2 ) and N c −1 free fields ∼ tr(a r ), The associated Z S 3 integral identities are in Theorem 5.6.6 of [11].

Axial masses
In this section we describe a descending chain of dualities relating U (N c ) and U sp(2N c ) gauge theories. Each line in the following scheme represents a duality 1) Except for the first two flows, from the U (N c ) perspective, a single axial mass is turned on at each step (it is easy to read from the flavors which disappear). We use red and blue for RG flows triggered by vector-like real masses, and green color for RG flows triggered by axial real masses.
We choose to put all the gauge singlets on the U sp side, but of course they can be moved on the U (N c ) side, providing instances of U (N c ) theories with enhanced global symmetries.
Let us mention that if we turn on a complex mass in the dualities of the scheme 5.1, we flow to a confining duality only for the cases with no Chern-Simons, otherwise we flow to a "duality appetizer" of section 4.
The set of U sp theories on the right of 5.1 satisfies self-dualities of form On the electric side there are 4 positive and 4 negative masses, so the same arguments leading to 2.21 apply, and we are left with U (N c ) with an adjoint φ and (4, 4) fundamentals q i ,q i . However, on the magnetic side the dual vacuum is the origin of the moduli space, that does not break the U sp(2N c ) gauge group. The flavors Q 1,2,3 ,Q 1,2,3 and the 6 towers of gauge singlets b i4;r andb i4;r (i = 1, 2, 3) are massless, while the flavors Q 4 ,Q 4 and the singlets b ij;r andb ij;r (i < j = 1, 2, 3) become massive.
In the IR we get the following U (N c ) ↔ U sp(2N c ) duality with 3 + 3 towers of gauge singlets b ij;r andb ij;r , flipping the "baryons" of the 6 = 3+3 split: The global symmetry is SU (4) 2 × U (1). On the r.h.s. the UV global symmetry is SU (3) 2 × U (1) 3 . The Z S 3 integral identity associated with this duality is Theorem 5.6.15 of [11]. The chiral ring generators map as Notice the mapping meson → monopole. Throughout this section, on the U sp side, the singlets flip all the "baryons" of the split.
In 5.3 we can only turn on vector-like real masses. Turning on +m for q 1 and −m forq 4 on the l.h.s, one of the monopole superpotentials gets lifted. On the r.h.s. only Q 1 gets a mass, the monopole disappears from the chiral ring, and the 2 towers of singletsb 24;r ,b 34;r become massive. At the end we flow to The Z S 3 integral identity associated with this duality is Theorem 5.6.16 of [11]. The chiral ring generators map as . . , N c (5.6) At this point the global symmetry is SU (3) 2 × U (1) 2 , so we have a choice: we can turn on another vector-like real mass, or an axial mass. We explore the second option in section 6. Turning on a vector-like real mass (0, 0, +1; −1, 0, 0) in 5.5 we flow to a duality studied (for generic N c ) in [22]: The Z S 3 integral identity associated with this duality is 2nd formula in Theorem 5.6.17 of [11]. In 5.7 (and in all the dualities that follow in this section), U (N c ) mesons map to U sp(2N c ) "mesons", U (N c ) monopoles map to singlets, that flip the U sp(2N c ) "baryons". The case N c = 1 of 5.7 (composed with Aharony duality of U (1) with 2 flavors or with the Intriligator-Pouliot like 5.2), is equivalent to a duality appeared in [23]. 13 13 We need to set F =F = N = 2, k = 1 in the duality of [23], that is Moving the singlets on the l.h.s. we get a U (N c ) gauge theory with IR enhanced global symmetry SU (2) 2 × U (1) 3 → SU (4) × U (1) 2 . Such symmetry enhancement can also be proven composing the Aharony like duality 2.31 with the mirror like duality 2.34: the arguments are the same as in [24], simply replacing U (1) → U (N c ) and the gauge singlets with a tower of N c gauge singlets. Turning on masses (0, 0; 0, 1) in 5.7 we flow to U (N c )1 Also all the next dualities in this section, for N c = 1, are special cases of 5.9.
5.1 One chiral flavor and supersymmetry enhancement [40] recently proposed that the N = 2 U (1)3 2 theory with 1 chiral flavor (with full UV global symmetry U (1) top × U (1) R ) displays supersymmetry enhancement to N = 4 supersymmetry. Highly non trivial evidence was given in support of the statement [40].
The U (1)3 2 theory is the case N c = 1 of the l.h.s. of the duality 5.13. 14 Notice that N = 2 SU (2) k theories with precisely one complex doublet q α have UV global symmetry SU (2) F : in N = 1 language, the superpotential (including the real adjoint N = 1 superfield) is Integrating out the massive SU (2)-adjoint S αβ , one gets a quartic superpotential, and only one structure can appear The last expression is the only quartic SU (2) F -invariant written doubling the components of q α to q A,I and imposing a reality condition q A,I = ε AB ε IJ q * B,J , where A, B are gauge indices, and I, J are SU (2) F indices. So 5.13 "explains" part of the IR symmetry enhancement to SO(4) R . What should happen is that only if k = ± 5 2 also the U (1) R-symmetry of SU (2) k with one doublet enhances the IR U (1) R → SU (2), and the full IR global symmetry is SO(4) R , which is the R-symmetry of N = 4 supersymmetry.
Another curious facts about the theories U (N c )3 In this section we start from duality 5.5, and turn on an axial mass for q 1 . On l.h.s. a Chern-Simons term is generated, and the monopoles M − exit the chiral ring. On the r.h.s. only one fundamental and the singletsb 12;r become massive, so we flow to: The chiral ring generators map as Combining 6.1 and 5.7 (after moving the singlets on the U (N c ) side using the chiral ring maps), we get a U ↔ U duality: The chiral ring generators map as As can be seen from the above mapping, a curious phenomenon is at work here: the l.h.s. UV global symmetry includes an SU The case N c = 1 appeared recently in [25], which pointed out that it explains the symmetry enhancement SU (2) × U (1) top → SU (3) discovered in [26,27] and studied via an N = 1 Wess-Zumino model in [24,28]. 7 The mirror of A 2N Argyres-Douglas and adjoint-U sp dualities As pointed out in [29], starting from the 4d U sp ↔ U sp dualities, one can flow to the duality 3.16. Upon gauging the U (1) symmetry, one gets the equivalent duality for the SU (N c ) theory: where the N c dressed mesons tr(qφ r q) map to singlets µ r , the 2 dressed baryons map to monopoles and the N c (N c − 1) dressed monopoles of SU (N c ) map to mesons µ +;s µ −;t . See [31] for details about the map. It is important that for the dualities 7.1 and 3.16 the UV global symmetry is U (1) 3 on both sides. Accordingly, the Z S 3 's match as a function of 3 fugacities.
We can "move" the singlets µ r on the l.h.s. (i.e. flip them with fields α r , recalling that tr(qφ r q) ↔ µ r ), then the global symmetry enhances. Starting from 7.1 and moving N c − 1 singlets, we get the duality SU (N c ) w/ adjoint φ and 1 flavor q,q . W = Nc−2 r=0 α r tr(qφ r q)+ + Nc j=2 β j tr(φ j ) ⇐⇒ U (1) w/ N c flavors µ ±;r . W = µ Nc−1 Nc s=0 µ +;s µ −;Nc−s N = 4 susy (7.2) This is precisely the duality studied in [30,31], and obtained using "sequential confinement" from a N = 4 mirror symmetry for U (N c ) with 2N c flavors. The r.h.s. of 7.1, however, displays N = 4 supersymmetry and non-R global symmetry U (1) top × SU (N c ). This symmetry enhancement is obvious on the r.h.s., but the Cartan generators of the new symmetries only emerge in the IR on the l.h.s., making it impossible to match the S 3 partition functions as functions of all the UV fugacities visible on the r.h.s. So in a sense the presentation of the duality 7.1 is more natural than the presentation 7.2.
U sp(2N c ) with adjoint and 2 flavors, W = tr(pφp): A 2Nc AD models. We now study the analogous case of A 2Nc Argyres-Douglas.
The 4d A 2Nc Argyres-Douglas theory, as far as we know, does not have a proposed 3d mirror. 15 It is however known that 4d Higgs Branch is just a point. Since Higgs Branches are not expected to change compactifying theories with 8 supercharges, we assume that the Higgs Branch is trivial also in 3d. This means that the Coulomb Branch of the mirror must be trivial. Assuming that the 3d mirror is an N = 4 theory with an N = 4 Lagrangian, the only possibility is that the 3d mirror has no gauge interactions, so it must be free. Since the 4d Coulomb Branch has rank-N c , we reach the conclusion the 3d mirror of A 2Nc AD is N c free hypermultiplets.
Maruyoshi and Song [34,35] proposed that a certain 4d U sp(2N c ) with adjoint and φ and 2 flavors q, p flows in the IR to A 2Nc AD: U sp(2N c ) with adjoint φ and 2 flavors p, q, W 4d = tr(pφp) + Nc r=1 α r tr(qφ 2r−1 q)+ + Nc−1 j=0 β j tr(φ 2j+2 ) =⇒ (A 1 , A 2Nc ) Argyres-Douglas theory (7.3) Using the prescription of [30,31], we added the N c singlets β j to deal with the (apparent) unitarity violations for the operators tr(φ 2j ). The UV global symmetry in 4d is just U (1), so one performs A-maximization in one variable, the result is: The operators α r and β r have R-charges The interpretation of these relations is that the N c holomorphic operators tr(pq) and β r (r = 2, . . . , N c ) are not in the chiral ring, but sit inside the N = 2 supersymmetric Coulomb Branch multiplet: they are descendant (under the emerging N = 2 supersymmetry) of the α r 's (r = 1, . . . , N c ). See [30,31] for the analog discussion in the A 2N −1 case. It is not clear to us what happens to β 0 in this case (it is easy to see that β 0 cannot take a vev, so it is likely not in the chiral ring). Now we flow to 3d. No monopole superpotential is generated, as for the SU (N c ) case in [30,31]. Combining with the 3d mirror proposed above, we are then led to conjecture the following duality U sp(2N c ) with adjoint φ and 2 flavors p, q, W = tr(pφp) + Nc r=1 α r tr(qφ 2r−1 q)+ + Nc j=1 β j tr(φ 2j ) ⇐⇒ 2N c free chiral fields ∼ U sp(2N c )-monopoles (7.7) The free fields on the r.h.s. are the monopoles, indeed, for a U sp(2N c ) with adjoint, we can construct 3N c N = 4 Coulomb Branch operators [53]: the Casimirs of the adjoint tr(φ 2j ), j = 1, . . . , N c , and 2N c BPS dressed monopoles M φ j , j = 0, 1, . . . , 2N c −1. The UV global symmetry on the l.h.s. is U (1) 2 , both U (1) factors mix with R-symmetry.
We checked numerically the duality 7.7. Performing Z-extremization, we find Recalling that (the last contribution comes from the gauginos), we see that the superconformal Rcharge of all the dressed monopoles is 1 2 , so they are free fields. All the N c α r fields in 3d have R-charge 1, so they must be a quadratic combination of the monopoles. These are quantum relations that are non-trivial to see in the gauge theory. It is easy to see that the global symmetries allow for the quantum relations, for instance if N c = 1, R[α 1 ] = 2 − 2r q − r φ and R[M] = 1 − r q − 3 2 r φ , so a chiral ring relation is consistent with the global symmetries. It would be nice to uplift the duality above to a 3d confining duality and possibly to a 3d or 4d duality for U sp with adjoint, flavors and cubic "dressed meson" superpotential. We leave these interesting issues to future work. 8 A "mirror" dual of U (N c ) with adjoint and N f flavors, W = 0 In this section we consider a generalization of the duality discussed in section 3.3. We look for a dual of N = 2 adjoint-U (N c ) with N f flavors, with arbitrary N f . Let us read off the duality from the generalized brane setup. On the l.h.s. we stretch N c D3's between two NS5's, and add N f flavor D5's branes. As in figure 3.17, the N S5 branes stretch along 012345, the D3 branes along 0126 and the D5 branes along 012347. The central D5 on the l.h.s. provides the flavor q 1 ,q 1 to the U (N c ) gauge theory.
The theory on the l.h.s. is adjoint U (N c ) with N f flavors in the fundamental, N f flavors in the anti-fundamental (corresponding to strings stretching from the D3 to the D5' branes). The superpotential vanishes. Notice that the adjoint field is an SU (N c )-adjoint plus a decoupled singlet.
The theory on r.h.s. is not so simple to read off. The theory is a quiver U (N c ) N f −1 , with N f −2 bifundamentals b J ,b J . Each node has an adjoint field Φ J . These fields enter the superpotential in an N = 4 fashion, i.e. with cubic terms of the form tr(b J Φ JbJ − b J Φ J−1bJ ). At this point is each adjoint field Φ J contains also a U (N c )-singlet field, which enters the superpotential.
The external D5' branes provide a single flavor, q L ,q L and q R ,q R , for each of the two external gauge nodes. This flavor enters the superpotential in a non trivial way. There are also two towers of chiral gauge-singlet fields α L,r , α R,r , r = 0, . . . , N c − 1 (these complex α modes are associated to the motion of the leftmost and rightmost D3 branes along the 34 directions). The gauge-singlets couple to the "dressed mesons", providing an N = 2 part in the superpotential: Out of the N f − 1 singlets in the adjoint fields Φ, only N f − 2 linear combinations (which we call η J ) actually couple with the rest of theory. The decoupled one match with the decoupled singlet on the U (N c ) side. So from now on all the adjoint fields are traceless.
The duality with the full superpotential on the quiver side is stated as: The case N c = 1 In this case the duality 8.4 becomes the well known N = 2 abelian mirror symmetry (with no adjoint fields): The global symmetry on the l.h.s. is l.h.s. : where U (1) q is an axial symmetry under which q i andq i have charge +1, and U (1) top is the magnetic (or topological) symmetry. As for the UV global symmetry on the r.h.s., the are 2N f charged fields p i ,p i r.h.s. : Where U (1) b is the non gauged combination of vector-like symmetries (q L , b i , q R have charge +1 andq L ,b i ,q R have charge −1) and U (1) axial;i gives charge +1 to b i andb i . The rank of the UV global symmetries is 2N f on both sides. In the IR the r.h.s. global symmetry U (1) . It is well known that the S 3 partition functions match exactly as a function of 2N f parameters.
The case N f = 2 In this case 8.4 becomes the duality between two U (N c ) theories 2.34. Notice that the duality is different from the duality 2.31: 8.4 is a generalization of the Abelian mirror symmetry and involve 4 towers of gauge singlet flipping fields, while 2.31 is a generalization of the Aharony duality for U (1) with 2 flavors, and involves 6 towers of gauge singlet flipping fields. The global symmetry on the l.h.s. is l.h.s. : As for the UV global symmetry on the r.h.s., the are 4 fields p i ,p i plus 1 adjoint field, there are no independent relations coming from the superpotential, se we have r.h.s. : So the rank of the UV global symmetry is 5 on both sides. Let us recall the map of the chiral ring generators: tr(q 1 φ r q 1 ), tr(q 1 φ r q 2 ) tr(q 1 φ r q 2 ), tr(q 2 φ r q 2 ) The general case: N f > 2 and N c > 1 The general case involves some subtleties. Let us start from the UV global symmetry. The global symmetry on the l.h.s. is l.h.s. : SU (N f ) 2 × U (1) q × U (1) φ × U (1) top (8.12) As for the r.h.s., the are 2N f fields p i ,p i plus N f − 1 adjoint fields, subject to N f − 1 (if N f > 2 and N c > 1) independent relations coming from the superpotential terms J tr(p J Φ JpJ + p J+1 Φ JpJ+1 ), so the UV global symmetry is r.h.s. : We see that there is a mismatch between the rank of the UV global symmetry: 2N f + 1 vs N f + 3, on the quiver side the rank is smaller. This is similar to what happens for the mirror of U (N c ) with N f > 2 flavors and no adjoint, studied in [41]. We propose that the chiral ring generators map in analogy to 8.8 and 8.11: • the first and last diagonal dressed mesons map to gauge singlets α L,R;j • "central" diagonal mesons map to gauge singlets η j • "central" dressed diagonal mesons map to traceless linear combinations of tr(Φ r i ) • off-diagonal dressed mesons map to dressed monopoles of the form M 0,..,0,1,..,1,0,..,0 • dressed monopoles map to dressed 'long mesons' • the Casimirs of the adjoint map to a symmetric linear combination of tr(Φ r i ) More in details, (again we only display the case N f = 3), the map is tr(q 1 φ r q 1 ), tr(q 1 φ r q 2 ), tr(q 1 φ r q 3 ) tr(q 1 φ r q 2 ), tr(q 2 φ r q 2 ), tr(q 2 φ r q 3 ) tr(q 3 φ r q 2 ), tr(q 3 φ r q 2 ), tr(q 3 φ r q 3 ) Where the . . . stand for η 1 if r = 0, for tr(Φ r 1 )−tr(Φ r 2 ) if r > 0. Let us emphasize one difference with respect to the case without adjoint field studied in [41]. Without the adjoint field, on the l.h.s. the are no dressed monopoles and no dressed mesons. But the dual is still a quiver similar to the one we are studying here, locally N = 4 with adjoints. So on the dual quiver side, the chiral ring operators come in towers of dressed operators (mesons and monopoles). In the case without adjoint there must be quantum relations on the quiver side, 16 which are highly nontrivial to see without knowing the duality. We see that adding matter, hence in a sense "uplifting" the duality can help with some of the issues.
It would be nice to uplift the duality 8.4 to a duality where the ranks of the UV global symmetry is the same. One option might be to look for a dual of the r.h.s. quiver where the cubic "N = 4" superpotentials are absent.

Complex deformations
To finish, we check the picture proposed above turning on superpotential terms.
Mass deformations. A complex mass for the "last" meson tr(q N f q N f ) ↔ α R;Nc−1 induces on the quiver side a maximal nilpotent vev for Φ N f −1 . Also q L andq L take a vev. So the rightmost gauge group in the quiver is Higgsed U (N c ) → U (0). The quiver becomes one node shorter. This is consistent with the duality.
A complex mass for a "central" meson tr(q i+1 q i+1 ) ↔ η i induces on the quiver side a vev for the operator tr(b ibi ). This implies that b i ,b i take a vev, which Higgses the two gauge groups adjacent to b i to the diagonal subgroup: U (N c ) × U (N c ) → U (N c ). Again the quiver loses one node, consistently with the duality.
Cubic deformations. A more complicated operation is turning on cubic superpotentials, in the form of a meson dressed by one factor of φ. On the l.h.s. we flow to U (N c ) with adjoint and N f flavors and a cubic superpotential. No field disappears.
In the brane setup 8.1 this operation, on the l.h.s. rotates a D5 into a D5 brane, so on the r.h.s. it rotates an N S5 into a N S5 brane. (N S5/N S5 branes stretch along 345/589, D5/D5 branes stretch along 789/347).
Let us consider turning on the "last" once-dressed meson tr(q N f φq N f ) ⇐⇒ α R;Nc−2 (8.15) On the quiver side of the duality, the F-terms of α R;Nc−2 induce the following vev: q R = (1, 0, . . . , 0) Φ Nc×Nc = Jordan Nc−1×Nc−1 0 0 0 q R = −(0, . . . , 0, 1, 0) (8.16) while all other fields stay at zero vev. This vev breaks the rightmost gauge group U (N c ) → U (1). This is consistent with the r.h.s. brane setup, since the rightmost N S 16 We are grateful to Simone Giacomelli for discussions about this. brane now is a N S5 , so, because of the S-rule, only one D3 brane can stretch between it and the right D5 brane, the other N c − 1 D3's now attach to the rightmost N S5, and the rightmost gauge group is only a U (1).
Let us also consider turning on a "central" once-dressed meson, which, according to our proposal, map as On the quiver side, the two adjoint fields Φ i−1 and Φ i become massive, integrating them out creates a quartic superpotential coupling involving the bifundamental stretching between the two gauge groups that lose the adjoints δW = tr(b i−1bi−1 b i−1bi−1 ). Again, this result is confirmed on the brane picture: now on the r.h.s. we are rotating a middle N S5 into an N S5 , this makes the two adjacent adjoints massive, since the D3's attached to the N S5 cannot slide up and down anymore.