New 3d $\mathcal{N}=2$ Dualities from Quadratic Monopoles

Aspects of three dimensional $\mathcal{N}=2$ gauge theories with monopole superpotentials and their dualities are investigated. The moduli spaces of a number of such theories are studied using Hilbert series. Moreover, we propose new dualities involving quadratic powers for the monopole superpotentials, for unitary, symplectic and orthogonal gauge groups. These dualities are then tested using the three sphere partition function and matching of the Hilbert series. We also provide an argument for the obstruction to the duality for theories with quartic monopole superpotentials.


Introduction
The three dimensional N = 2 dualities is a classic subject [1][2][3][4][5] and remains to be an active field of research for several reasons. This is partly because of its connection with four dimensional dualities, as the Seiberg duality [6] and the S-duality. It is also because of the recent interest on 3d theory with lower supersymmetry, namely N = 1 (see e.g. [7][8][9][10][11]), and on the non supersymmetric case (see e.g. [12][13][14][15][16]). One of the main reasons why the web of 3d N = 2 dualities is rich and interesting is related to the presence of a Coulomb branch, that corresponds to the moduli space of a dynamical real scalar inside the vector multiplet. The Coulomb branch can be parameterized in terms of complex coordinates, obtained by combining the real scalar discussed above with the dual photon. This is possible because the gauge symmetry on the Coulomb branch is broken to the maximal abelian torus. The complex combinations obtained from the real scalars and the dual photons correspond to the insertion of monopole operators in the path integral.
It is possible to modify the superpotential by the contribution of such monopole operators, parameterized in terms of the Coulomb branch coordinates [2,3,5,[17][18][19][20]. These superpotential have been shown to play a crucial role in 3d N = 2 dualities. For example they have been used in the circle reduction of 4d Seiberg duality to define new effective dualities on S 1 [21,22]. Moreover, monopole superpotentials have been used as a tool to study other field theories, such as dualites among SU (2) gauge theories with eight chiral doublets [23], dualities involving pq-webs [24], T-brane theories [25], compactification of Argyres-Douglas theories 1 to three dimensions and their mirror theories [29][30][31][32], mirror theories of 3d N = 2 supersymmetric QCD (SQCD) with zero superpotential [33]. Higher order monopole potentials also have some applications in condensed matter and statistical physics (see e.g. [34]).
Recently, a number of new 3d dualities for U (N c ) gauge theories with monopole superpotentials have been obtained in [35]. In most of these dualities the monopole superpotentials appear with a linear power. They modify the constraint between the global charges, breaking the otherwise generated axial and/or topological symmetry. Their presence modifies the rank of the dual gauge group, with respect to the case without such linear monopole superpotential deformations. Moreover an interesting proposal appeared in [35], regarding a duality between U (N c ) gauge groups and quadratic monopole superpotentials.
In this paper we elaborate on this proposal. The three main points are as follows. First, we obtain new classes of unitary, symplectic and orthogonal/special orthogonal gauge groups with quadratic monopole superpotentials. Secondly, we study the moduli space of a large class of theories with monopole superpotentials using the Hilbert series [36][37][38][39][40]. Thirdly, we check our proposals matching the Hilbert series and showing the consistency of the RG flow to the IR. This last step is done using localization on S 3 [41][42][43][44][45]. The dualities that we mostly discuss in this paper are summarized in table 1.
Sec. 3.3 Table 1: Summary of the dualities The paper is organized as follows. In section 2 we review the known dualities with monopole superpotentials. After this brief review the Hilbert series for theories with monopole superpotentials are computed. For theories that are dual to each other, we also match their Hilbert series. This is by itself a new result and it will be useful in the discussion of the Hilbert series for the cases with quadratic monopole superpotentials. In section 3 we introduce the dualities with quadratic monopole superpotentials. Most of the dualities proposed in this section are new. In section 4 we show how to use the 3d partition function as a consistency check for the new dualities. In section 5 we compute the Hilbert series for the new dualities that we are proposing and match them across the duality. In section 6 we provide an argument for the obstruction to the duality of quartic monopole superpotentials, as well as discuss other interesting aspects that we do not cover in the paper but deserve further investigations.

Dualities with linear monopole superpotentials
In this section, we consider theories with linear monopole superpotentials.

Review
Dualities with linear monopole superpotential have been obtained in [35] by studying the dimensional reduction of the four dimensional Seiberg duality between the U Sp(2N c ) gauge theory with 2N f fundamentals Q, and the U Sp(2(N f − N c − 2)) gauge theory with 2N f fundamentals q, an anti-symmetric meson M = QQ, and superpotential W = M qq [46]. The reduction of this theory on S 1 was studied in [21], and the duality was shown to be preserved if a Kaluza-Klein (KK) monopole superpotential was added to both the phases. The 3d limit considered in [35] consists of a large positive shift of the real masses of N f fundamentals and a large negative shift for the remaining N f fundamentals. Furthermore a similar shift was considered for the real scalar σ in the vector multiplet, in both the electric and the magnetic theory. The two shifts have an opposite sign in order to keep the duality in the IR. This construction led to a 3d duality between the U (N c ) gauge theory with N f fundamental flavors and superpotential W = X + + X − (2.1) and the U (N f − N c − 2) gauge theory with N f fundamental flavors, N 2 f singlets M and superpotential W = M q q + X + + X − (2. 2) The superpotential terms for the monopoles X ± and X ± break the axial and the topological symmetry and constraint the R-charge of the fundamentals. A similar duality was constructed in [35], involving only one monopole superpotential. This duality can be constructed from four dimensions as well and it consists of the U (N c ) gauge theory with N f fundamental flavors and superpotential where S + corresponds to the monopole X + of the electric theory, acting as a singlet in the dual phase. Another duality considered in [35] was obtained from this case by turning on a large real mass for k anti-fundamentals. This flow needs a large shift for the FI as well. The final result consists of a relation between the U (N c )k Dualities with linear monopole superpotential have been constructed for real gauge groups as well. They consists of the circle reduction of four dimensional Seiberg duality for U Sp(2N c ) and O(N c ) gauge groups. These dualities have been constructed in [21] for the symplectic case and in [22] for the orthogonal case. In the second case different constructions were needed, depending on the global properties of the gauge group. The linear monopole superpotential in these cases is associated to the KK monopole, which is constructed algebraically from the affine root of the B n , C n and D n series.

Matching the Hilbert series
In this section, we study the moduli spaces and compute the Hilbert series of a number of gauge theories with linear monopole superpotentials. Let us briefly discuss some general features of such theories. Suppose that V is one of the basic monopole operator of the theory 2 . By putting V in the superpotential, say W = V , the Rcharge of V is fixed to 2. This also results in fixing the R-charge of the chiral fields in the theory. Moreover, as pointed out in [2,19], we can rewrite V as V = e z . The F -term ∂ z W = 0 implies that e z = V = 0. Indeed, the Coulomb branch generated by the basic monopole operator V is lifted [21]. If V is the only basic monopole operator in the theory, then the gauge group is left unbroken by the monopole operator and the residual gauge theory is the same as the original theory. However, if there are other basic monopole operators, there remain Coulomb branches parametrised by those monopole operators that need to be analysed.
Let us now discuss this in several examples below.

The Aharony duality for unitary gauge groups
As a warm-up, let us consider the Aharony duality, proposed in [22]: Theory A: U (N c ) gauge theory with N f flavours and W = 0.
Theory B: U (N f − N c ) gauge theory with N f flavours q and q, N 2 f singlets M , singlets S ± and superpotential W = M qq + S − V + + S + V − , where V ± are the basic monopole operators in theory B.
Note that S ± in theory B are mapped to the basic monopoles operators in theory A.
In what follows, we discuss the moduli space and the Hilbert series of theories A and B. In the following, we analyse the moduli space and compute the Hilbert series of these theories.

Theory A
The Hilbert series of theory A was studied in detail in [37]. Let us review the computation briefly here. By the Callias index theorem, The monopole flux takes the form (m 1 , 0, . . . , 0, m Nc ) with m 1 ≥ 0 ≥ m Nc . The residual theories are as follows 3 : Adding up the mesonic Hilbert series of these residual theories weighted by the factors associated to bare monopole operators, we obtain the Hilbert series of the total moduli space of theory A as follows: and so where r is the R-charge of the quarks, and the mesonic Hilbert series for U (N c ) with N f flavours is given by [37,47] with y the fugacity for the U (1) axial symmetry, and P is the R-charge of the basic monopole operators: The moduli space is generated by X + , X − and the N f ×N f meson matrix M , subject to the following relations (2.11) The first set of relations implies that the rank of M is at most N c , the second set of relations implies that if X + = 0 or X − = 0 the rank of M is at most N c − 1, and the third set of relations implies that if both X + = 0 and X − = 0 the rank of M is at most N c − 2.
Theory B The rank of the gauge group of each residual theory in theory A put a restriction on the rank of the mesons. Since the meson of theory A is mapped to M in theory B, we look at various possible rank of M here. Subsequently, we follow the argument of [5].
If we give M a vacuum expectation value (VEV) of rank N c , the low energy theory consists of a U (N f − N c ) gauge group with N f − N c flavours. The latter has a dual description as the WZ theory with superpotential 4 Hence the effective superpotential of theory B in this case is (2.12) The F -terms ∂ V ± W B = 0 imply that S ∓ = V ∓ det(q q). On the other hand, the F -terms ∂ S ± W B = 0 imply that V ∓ = 0. Thus, S ± = 0 in the chiral ring. Since S ± are mapped to the basic monopole operators of theory A, this case corresponds to case 1 of theory A, in which the monopole fluxes m 1 = m Nc = 0. Indeed, the Hilbert series of the space generated by an N f × N f matrix M such that rank(M ) ≤ N c is equal to H U Nc,N f . On the other hand, if we give M a VEV of rank N c −1, the low energy theory consists of a U (N f − N c ) gauge group with N f − N c + 1 flavours. The latter has a low energy effective description as the WZ theory with superpotential −2(V + V − det(q q)) 1/2 (see (8.6) and the discussion just above (8.7) of [3]), which is valid away from the origin where V + = V − = 0. The effective superpotential of theory B is therefore (2.13) The F -terms ∂ S ± W B = 0 imply that V ± = 0, i.e. the Coulomb branch of theory B is lifted. The case in which S ± = 0 corresponds to case 1 of theory A. Away from the origin, the F -terms ∂ V ± W B = 0 imply that 14) or equivalently We see that S ± can take arbitrary values. The case in which S + = 0 or S − = 0 corresponds to cases 2 and 3 of theory A in which the monopole fluxes m 1 > 0, m Nc = 0 or m 1 = 0, m Nc < 0. If M is given a VEV of rank N c −2, the low energy theory consists of a U (N f −N c ) gauge group with N f − N c + 2 flavours. The effective superpotential of theory B is The case in which S ± = 0 corresponds to case 1 of theory A. Away from the origin, using the F -terms ∂ V ± W B = 0, we find that . Substituting these back to W B , we obtain the effective superpotential W B = −(V + V − det(q q)) 1/3 . Hence, S ± can take arbitrary values. Again, the case in which S + = 0 or S − = 0 corresponds to cases 2 and 3 of theory A. The case in which both S + and S − are non-zero corresponds to case 4 of theory A, in which m 1 = 0 and m Nc = 0.
If the VEV of M has rank greater than N c , say rank(M ) = N c +p with p > 0, the low energy theory consists of a U (N f − N c ) gauge group with N f − N c − p flavours 5 . We will see that there is no stable supersymmetric vacuum for any p > 0.
• If p > 1, from [3, (8.6)], the effective superpotential is (p−1)(V + V − det(q q)) − 1 p−1 , and for the superpotential Using the equations of motion, we obtain and so we have runaway vacua in this case.
• If p = 1, according to [3], this theory is described by the relation V + V − det(q q) = 1. The effective superpotential of theory B is therefore where λ is a Lagrange multiplier. Using the equations of motion, we can rewrite , and so we also have runaway vacua.

2.2.2
The duality involving W = X + + X − Let us consider the following pair of theories [35]: Theory A: U (N c ) gauge theory with N f flavours and gauge theory with N f flavours q and q, N 2 f singlets M and superpotential In theory A, the flux of the monopole operator takes the following form m = (m 1 , 0, . . . , 0, m Nc ) with m 1 ≥ 0 ≥ m Nc . Here X + denotes the monopole with the flux (1, 0, . . . , 0) and X − denotes the monopole with the flux (0, . . . , 0, −1). Let us write X ± = e z ± . The F -terms with respect to z ± , namely ∂ z ± W A = 0, imply that X ± = e z ± = 0. Hence, all monopole operators vanish in the chiral ring. The Hilbert series of theory A is therefore the mesonic Hilbert series of U (N c ) with N f flavours and zero superpotential: where the R-charge r of the quarks fixed by the superpotential W A : and the expression for H U Nc,N f (t, u, v; r) is given by (2.9). Here there is no fugacity y because the U (1) axial symmetry is completely broken.
We compute the Hilbert series of theory B as follows. We first perform the Aharony dual to theory B and obtain Theory B : U (N c + 2) gauge theory with N f flavours q and q, two singlets X ± and superpotential where V ± are the basic monopoles in theory B . The F -terms ∂ X ± W B = 0 gives This means that the monopole operators have non-zero fluxes ( m 1 , 0, . . . , m Nc+2 ) with m 1 = m Nc+2 = 0. Thus, the gauge group U (N c + 2) is broken to U (N c ) × U (1) 2 . Moreover, writing V ± = e z ± and considering the F -terms ∂ z ± W B = 0, we find that X ± V ∓ = 0. From (2.24), we find that i.e. X ± vanish in the chiral ring. Thus, the residual theory is U (N c ) gauge theory with N f flavours and zero superpotential. The R-charge R of the flavour fields q and q in theory B is given by Indeed, the Hilbert series of theory B, which is dual to theory B , is given by Let us now consider the following pair of theories [35]: Theory A: U (N c ) gauge theory with N f flavours Q and Q and W = X − .
gauge theory with N f flavours q and q, N 2 f singlets M , singlet S + and superpotential W = M qq + X + + X − S + , where X ± are the basic monopoles in theory B, and S + is dual to the monopole X + in theory A.
The first observation is that the monopoles in both theories do not appear in the superpotential in a symmetric way, thus the charge conjugation is broken. Moreover, the U (1) topological symmetry and the U (1) axial symmetry are broken to a diagonal subgroup which we shall henceforth refer to as U (1) T . As a consequence we need to slightly modify the expression for the monopole R-charge.
Theory A Let us consider theory A and define The R-charges of the X ± are given by where α parametrizes the mixing of the R-charge and the U (1) T symmetry. We shall soon show that this parametrisation is consistent with the proposed duality. The superpotential fixes the R-charge of X − to be R[X − ] = 2, and so so the meson has R-charge Now let us turn our attention to the Hilbert series. Writing X + = e z + and differentiating with respect to z + to get the F −term we find X − = 0, so X − vanishes in the chiral ring. However, we still have the Coulomb branch generated by the basic monopole operator X + . Hence, to compute the Hilbert series we can use a similar argument to that of [22]. We have two cases to analyse: where z is the fugacity for the U (1) T symmetry.
• m 1 > 0: with residual theory U (N c − 1) with N f flavours and W = 0. The Hilbert series is given by where z x is the fugacity for the monopole operator X + with flux (1, 0, . . . , 0). The sum gives (2.36) Thus, the Hilbert series of theory A reads Theory B Let us analyze now the theory B. As we did for theory A, let us define For the monopoles, we parametrises by β the mixing of the R-charge and the U (1) T symmetry and obtain (2.40) The superpotential imposes R[ X + ] = 2, from which we get r Using this expression for r we find 42) and also that For the singlets M we find Let us now relate the R-charges of various fields in theory A to those in theory B. In theory B, both X + and X − vanish in the chiral ring. The former can be seen by writing X + = e z + and considering the F -term with respect to z + , whereas the latter follows from ∂ S + W = 0. Moreover, X − in theory A and X + in theory B both have R-charge 2 due to the superpotentials. We thus propose the following duality maps: mesons ←→ M . (2.45) Due to the term M qq in the superpotential, we also have The map between the mesons and M implies that Using (2.47) and (2.48), we find that N f (1−R)−(N c −1)+α = 2, which is consistent with (2.31). This shows that our choice of parametrisation of the mixing between U (1) R and U (1) T is consistent with the proposed duality. In order to study the moduli space of theory B, we consider its Aharony dual, which is given by 6 Let us compare the charge assignment here with that in [22]. In the latter, R[Q] and R[ Q] are chosen to be R = Theory B : U (N c + 1) with N f flavours and singlets X ± and S + with superpotential W B = X + V − + X − V + + X + + X − S + , where V ± are the basic monopole operators in this theory.
Consider first the F −terms obtained by differentiating with respect to the singlets: Plugging these equations into W B we get Observe the following (it will be needed later): V − = −1 implies that the flux m Nc+1 = 0, hence the gauge group breaks to U (N c + 1) → U (N c ) × U (1); moreover V + = −S + implies that we need to consider two cases for the computation of the Hilbert series, namely S + = 0 and S + = 0. We shall see below that these correspond to the cases of m 1 = 0 and m 1 = 0 in the theory A, namely (2.34) and (2.35) respectively. Let us turn our attention to the R-charges. Let q and q be the fundamentals and antifundamentals of theory B . Let us also define ). The superpotential W B fixes the R-charge of X + to be R[ X + ] = 2, and so R[V − ] = 0.
The R-charges of the monopole operators V ± of theory B is given by where γ parametrise the mixing between the U (1) R-symmetry and the U (1) T symmetry. We thus have (2.55) The solution for R B reads and so γ = 1 − α . Now we are ready to compute the Hilbert series. As we said, we have two cases: for S + = 0 we have U (N c ) with N f flavours and W B = 0, so (2.58) When S + = 0 the gauge group breaks to U (N c − 1) with a dressing factor due to the presence of V + ; thus the Hilbert series is Indeed, the Hilbert series of theory B , dual to B, reads Note that this expression matches with H (A) since R B = R B .

The Giveon-Kutasov duality
In this subsection, we consider the following thories [48]: Theory A: U (N c ) k gauge theory with N f flavours and W = 0.
Theory B: U (N f + |k| − N c ) −k gauge theory with N f flavours q and q, N 2 f singlets M , and superpotential W = M qq.
For simplicity, let us assume that k > 0 and that N f + |k| − N c ≥ 0. As pointed out in [39, sec. 3.3], the Coulomb branch is completely lifted and the full Hilbert series of theory A is the mesonic Hilbert series of U (N c ) with N f flavours: (2.62) This Hilbert series corresponds to the space generated by the mesons, which can be regarded as an N f × N f matrix, subject to the condition that the rank is at most N c . In theory B, the Coulomb branch is also lifted. The moduli space is generated by N f × N f matrix M . We shall argue that there is a quantum condition on the rank of M : rank(M ) ≤ N c (this is the classical condition of the meson in theory A). This can be seen as follows: If we give a VEV to M of rank greater than N c , say N c + p with p > 0, then the lower energy theory is where V ± is the basic monopole operators in this low energy effective theory. Since p > 0 and |k| > 0, we have runaway vacua. We just matched the moduli space of theory B with that of theory A. Thus, the Hilbert series of theory B is also given by (2.62).

The Benini-Closset-Cremonesi (BCC) [p, 0]a duality
We consider the following theories [20]: In theory A, the U (1) gauge charges of the basic monopole operators X ± are Hence, the Coulomb branch that is generated by X − is lifted. In theory B, the Fterm ∂ S W = 0 implies that X + vanishes in the chiral ring. We propose the following duality map: Let us discuss about the R-charges of various fields. The R-charge of the basic monopole operators in theory A is where r is the R-charge of the quarks in theory A. Since X + is mapped to S under the duality, we have Since the mesons of theory A are mapped to M in theory B, we have and so the monopole V + of theory B has R-charge where the blue term comes from the mixed gauge-R symmetry CS terms; see [49, (4.4)]. Indeed, this is compatible with the superpotential term S X + in theory B.

Theory A
Since the Coulomb branch that is generated by X − is lifted in theory A, the monopole flux thus takes the form (m 1 , 0, . . . , 0) with m 1 ≥ 0.
• If m 1 = 0, the residual theory is U (N c ) gauge theory with N f fundamentals and N a antifundamentals, whose mesonic Hilbert series is [0 N f −Nc , n Nc , . . . , n 1 ; n 1 , . . . , n Nc , The mesonic chiral ring is generated by the N a N f meson matrix M a a = Q a i Q i a of rank at most N c .
• If m 1 = 0, the residual theory is U (N c − 1) gauge theory with N f fundamentals and N a antifundamentals. The mesons in this theory is to be dressed with the monopole operators generated by X + . The Hilbert series in this case is Here z is the fugacity for the topological symmetry, y is the fugacity for the axial symmetry, and k gA is the mixed Chern-Simons level between the central gauge U (1) and the axial U (1) A symmetry, which is quantized to ensure that the exponent of y is an integer.
The full Hilbert series of theory A is therefore The chiral ring of the theory is generated by the N a × N f meson matrix M , of rank at most N c , and by the bare monopole operators X + subject to the extra relation that the rank of X + M is at most N c − 1, i.e. X + minor Nc (M ) = 0.
Theory B Now let us consider theory B. The U (1) gauge charges of the basic monopole oper- Hence X − is not in the chiral ring and the Coulomb branch parametrised by X − is lifted. There are quantum conditions that give bounds on the rank of M . These correspond to the classical conditions for the rank of the mesons in theory A. We can derive such quantum conditions below.
Since the F -terms with respect to S implies that X + vanishes in the chiral ring. The moduli space of theory B is generated by the M . Therefore, after imposing such quantum conditions on M , we conclude that the Hilbert series of theory B is equal to that of theory A.
Let us turn on a VEV of M with rank N c +p. The low energy theory is described by U (N f − N c ) −k gauge theory with N a − N c − p fundamentals and N f − N c − p antifundamentals. We can use the topological and axial symmetries to constraint the form of the effective superpotential. We claim that the only possible consistent combination that can appear in the effective superpotential is for some power P , which can be worked out from the R-charges of S and X + as follows. Since the singlets M are mapped to the mesons in theory A, for p < 0, the gauge group U (N c ) of theory A is broken to U (|p|). On the other hand, for p ≥ 0, U (N c ) is completely broken. Since S is mapped to the monopole operator X + , the R-charges of S is given as follows: where the R-charges of the magnetic quarks are 1 − r. The R-charges of X + can be computed as usual Hence, the power P is 1 for p ≤ −1 and 2 1−p for p ≥ 0: For p ≥ 1, we have runaway vacua. This agrees with the analysis of the theory A, which says that the mesons have rank at most N c . For p ≤ 0, we have a positive power of S X + in the superpotential. If X + = 0, we have S = 0 and this corresponds to the case of m 1 = 0 in theory A. If X + = 0, then the value of S is arbitrary; when S = 0, this corresponds to the case of m 1 = 0. This is in agreement with theory A.

The duality involving W = X − , chiral flavours and Chern-Simons terms
Let us consider the following theories [35]: In theory A the U (1) gauge charge of the monopoles are given by Thus, the Coulomb branch generated by X + is lifted due to the non-zero Chern-Simons level. However, by putting X − in the superpotential, we can write X − = e Z − and the F -term with respect to Z − implies that X − vanishes in the chiral ring. Thus, the Coulomb branch generated by X − is also lifted. The residual theory is the full U (N c ) gauge theory with (N f , N a ) fund/antifund flavours, whose mesonic Hilbert series is given by where observe that there is no fugacity for the axial symmetry since it is broken by the presence of X − in the superpotential. The R-charge R of the quarks and antiquarks are fixed by the monopole superpotential: The generators of the mesonic chiral ring are the mesons, which are matrix N f × N a of rank at most N c . In theory B the Coulomb branch is lifted for the same reason of theory A (i.e. due to both the non-zero Chern-Simons level and the monopole superpotential). The moduli space is generated by the singlets M , which can be viewed as an N f × N a matrices. We shall argue that there is a quantum condition on the rank of M : rank(M ) ≤ N c (this is the classical condition of the meson in theory A). Let us give a VEV to M with rank N c + p with p > 0. The low energy effective theory is a fund/antifund. At this step, we can use the BCC duality to obtain the dual theory. Since the dual gauge group is the unitary group of rank we see that for p > 1, supersymmetry is broken. For p = 1, the dual theory is a WZ model with singlets S and X + with superpotential W = S X + . In which case the effective superpotential for theory B is where the first term of W B comes from the first term of the original superpotential of theory B. The equations of motion imply that both S and X + are massive and we do not have a supersymmetric vacuum.

The Aharony duality for symplectic gauge groups
We consider the following duality, which was proposed in [5]: Theory A: U Sp(2N c ) with 2N f fundamental chirals Q i and superpotential W = 0.
singlets M and singlet Y (that are dual to the mesons and the monopole of theory A) and superpotential W = M qq + Y S, where Y is the fundamental monopole of theory B.
Let us fist study R-charges of various fields. In theory A, let the R-charge of the fundamentals be R Thus, by the superpotential it follows that the R-charge of the singlet S is . Indeed, this is consistent with the expectation that the monopole operator Y in theory A is mapped to the singlet S in theory B under the duality.
Theory A In the following we discuss the Hilbert series of theory A. This has been analysed in detail in Section 6 of [37]. It has two contributions, depending on the value of the magnetic charge m in the magnetic flux (m, 0 Nc−1 ): • m = 0: the residual theory is U Sp(2N c ) with 2N f fundamentals, whose mesonic Hilbert series reads [50]: (2.83) • m > 0 : the residual theory is U Sp(2(N c − 1)) with 2N f fundamentals; since now the magnetic flux is non vanishing the Hilbert series contains a dressing factor taking into account of the monopole Y m with flux (m, 0 Nc−1 ) such that m ≥ 1: where y is the fugacity for the U (1) axial symmetry.
Thus the total Hilbert series of the theory is the given by adding the two contributions H I(A) and H II(A) : The Hilbert series tells us that the moduli space is generated by the antisymmetric 2N c × 2N c meson matrix M and by the fundamental monopole operator Y , subject to the condition: Note that the first equality implies that the rank of M is at most 2N c . The second equality implies that for Y = 0, the rank of M is at most 2(N c − 1).

Theory B
Let us now analyse the moduli space of theory B. Since the mesons are mapped to the singlets M , we give a VEV to M of rank 2(N c + p) and study at the moduli space for various values of p. The low energy effective theory is a U Sp(2(N f − N c − 1)) gauge theory with 2(N f − N c − p) massless quarks 8 . Below we analyse the possible cases of p. We shall see that, for p > 0, there is no stable supersymmetric vacua, whereas for p ≤ 0, the moduli space agrees with that of theory A.
• p < 0: The effective description away from the origin of the moduli space is given by the superpotential The equations of motion are the following • p = 0: the effective superpotential of the form: The equations of motion for S and Y are Recalling that the singlet S is mapped to the basic monopole operator Y of theory A, this case correspond to the magnetic flux m = 0 in theory A, and hence the corresponding Hilbert series is H I(A) .
• p = 1: In this case instantons in the low energy theory generate the constraint Y Pf(qq) = 1, which can be put in the superpotential through a Lagrange multiplier λ: Using the solution for ∂ λ W = 0, we can rewrite the superpotential as follows so we have runway vacua.
• p > 1: The effective superpotential is given by The equation of motion for the basic monopole reads Substituting into the superpotential we get and, again, we have runway vacua.

The duality involving symplectic gauge groups and W = Y
We consider the following duality [21]: In theory A, we write Y = e z and the F -terms ∂ z W = 0 implies that Y = 0 in the chiral ring. The Hilbert series thus get the mesonic contribution: where R is the R-charge of the quarks. This means that the moduli space is generated by the mesons M , with the constraint rank(M ) ≤ 2N c . The R-charge of the meson is fixed the presence of the monopole in the superpotential as usual, since R[Y ] = 2 and is also given in terms of the R-charge of the quarks: from which we get In theory B, the R-charge r of the fundamentals is fixed by the monopole superpotential term Y , whose R-charge is 2: (2.101) Therefore, the R-charge of the singlets M is in agreement with (2.100).
In order to analyse the moduli space of theory B, let us now perform the Aharony duality. We get a U Sp(2(N c + 1)) gauge theory with 2N f fundamentals, singlet S and superpotential W = S Y + S, where Y are the basic monopoles of this theory. The F −terms ∂ S W = 0 gives a non-zero VEV to the monopole Y = −1. Since the vacuum expectation value of Y is non-zero, the gauge group U Sp(2(N c + 1)) is broken to U Sp(2N c ), with an additional U (1) factor which decouples in the IR. If we substitute back Y = −1 to the superpotential, we end up with W = 0. Thus, the residual theory is a U Sp(2N c ) gauge theory with N f flavours, whose mesonic Hilbert series is given by H U Sp 2Nc,2N f (t, y, x; R). This is indeed in agreement with the Hilbert series of theory A.

The BCC duality for orthogonal gauge groups
Let us consider the following duality [20,51]: The R-charge of the monopole operator in theories with orthogonal gauge group reads The Coulomb branch is parametrized by the fundamental monopole operator Y . The magnetic flux (m, 0 n−1 ) (where n is given by N c = 2n for the even case and N c = 2n + 1 for the odd case) of Y m gives the following two contributions to the Hilbert series • m = 0: the residual theory is O(N c ) gauge theory with N f flavours and, in this case, the Hilbert series is is the mesonic Hilbert series of the aforementioned residual theory. It can be obtained from that of the SO(N c ) gauge theory with N f flavours [50, (2.29)] by projecting out the baryons and reads: where y is the fugacity for the axial symmetry and x denotes the fugacities associated with the flavour symmetry SU (N f ).
• m > 0: the residual theory is O(N c − 2) gauge theory 9 with N f flavours and, in this case, the Hilbert series is the mesonic contains a dressing factor taking into account of the monopole operators Y m : Hence the Hilbert series of the theory reads The Hilbert series tells us that the moduli space of theory A is generated by the symmetric N f × N f meson matrix M and by the fundamental monopole Y , subject to the following relations The first relation implies that the mesons have at most rank N c , while the second implies that, for Y = 0, M has at most rank N c − 2.
Since the mesons are mapped to the singlets M we give a VEV to M of rank N c + p and we study the moduli space for various p. The low energy effective theory is 10 . In the following analysis of the moduli space we shall see that for p ≥ 0 there is no stable supersymmetric vacua, while for p < 0 the moduli space agrees with the one in theory A. Let us analyze the various cases we get depending on p: • p < 0: the F -terms we get from the effective superpotential W = S Y + 110) The first of these equations implies that the Coulomb branch is completely lifted, and, since the effective superpotential is valid away from the origin of the moduli space the singlet S can take any arbitrary VEV. We can have two cases: for S = 0 we recover the case of magnetic flux m = 0 in electric theory, corresponding with the Hilbert series H I(A) , while S = 0 corresponds m > 0, where the Hilbert series is equal to H II(A) .
• p = 0: the R-charges of Y and det(qq) implies that the combination Y 2 det(qq) has zero R-charge, meaning that there is a constraint Y 2 det(qq) = 1 generated. Thus, we have the effective superpotential where λ is the Lagrange multiplier. Using the equation of motion, we have which gives runway vacua.
• p ≥ 1: the effective superpotential reads (2.113) 10 The R−charges of Y and det(qq) are the given by: where R[q] = r. The combination Y 2 det(qq) thus has R-charge −2p.
The equations of motion for Y reads Substituting into the superpotential we finally obtain which again gives runway vacua.

The ARSW duality for special orthogonal gauge groups
In this section we consider the following duality: This duality was proposed by Aharony, Razamat, Seiberg and Willet (ARSW) in [22]. A crucial difference between this duality and the duality involving gauge group O(N c ), discussed in the previous section, is the presence of the baryons and the baryon monopoles in the former.
Theory A Let us start with theory A. The magnetic flux of the monopole operator Y takes the form (m, 0, . . . , 0) in the Dynkin label notation. For m = 0, the residual theory is SO(N c ) with N f flavours (for N f ≥ N c ), in which case the Hilbert series is given by [50, (2.29)]: where R is the R-charge for the quarks and y is the fugacity for the axial symmetry. We emphasise that H SO Nc,N f (t, y, x; R) counts the operators generated by the mesons and the baryons, subject to algebraic relations among themselves.
For m = 0, the gauge group is broken to S(O(N c − 2) × O(2)); this includes a Z 2 group corresponding to transformations with determinant (−1) both in SO(N c − 2) and in SO (2). As discussed in [22], the gauge invariant monopole operator Y is charge conjugation even in SO (2), and it will be denoted by W + . There is also a charge conjugation odd monopole operator W − in SO (2). In order to obtain an invariant quantity under the Z 2 part of the gauge group, one can form a baryon monopole β = Q Nc−2 W − , where the product Q Nc−2 is invariant under the SO(N c − 2) × SO(2) residual gauge symmetry. The R-charges and the U (1) A charges of the monopole operators are as follows: The Hilbert series for m = 0 can be obtained as follows: where the factor in the bracket is the dressing factor coming from the monopole operator Y m = W m + with m ≥ 1. The total Hilbert series of theory A can be obtained in a similar way as in (2.107):   in agreement with [22, (2.22)]. We emphasise that, in the Hilbert series (2.122), the factor corresponding to the baryon monopole β emerges only after summing the second term with the first term.

Theory B
The gauge invariant combination qq in theory B vanishes due to the F -term ∂ M W = 0. The singlets M are subject to the quantum relation rank(M ) ≤ N c , which can be derived in the same way as in the previous subsection for orthogonal gauge groups. Also, the singlet Y in theory B is mapped to the monopole operator Y in theory A. As discussed in [22, (2.28)], the baryon monopole operators and the baryons in theory B are mapped to the baryons and the baryon monopole operators in theory A, and the former satisfy the same set of relations as those in theory A. As a consequence, the Hilbert series of theory B is equal to that of theory A.

The duality involving orthogonal gauge groups and W = Y
In this section we consider the following duality [22]: As usual, the R-charge R of the quarks in theory A is fixed by the superpotential, which gives Writing Y = e z , we see that Y = 0 in the chiral ring due to the F -term ∂ z W = 0, so the Hilbert series of theory A is given by the mesonic Hilbert series of O(N c ) gauge theory with N f flavours where H O Nc,N f (t, y, x; R) is given by (2.105). Let us now analyse theory B. It is convenient to consider its BCC dual, which is an O(N c +2) theory with N f fundamentals and a singlet Y and with a superpotential W = Y Y + Y , where Y is now the basic monopole operator of this new theory. Taking the equations of motion of the singlet we see that the VEV of the monopole is fixed, Y = −1, hence the original gauge group O(N c + 2) is broken to O(N c ) with an additional O(2) factor that decouples in the IR. Moreover, putting Y = −1 into the superpotential we end up with W = 0. Therefore, the residual theory is O(N c ) with N f flavours. The R-charge of the quarks in this theory is also equal to R; this is determined by the superpotential, which implies N f (1 − R) − (N c + 2 − 2) = 0. Thus, the Hilbert series is H O Nc,N f (t, y, x; R), in complete agreement with theory A. There is also a similar duality for the special orthogonal gauge groups [22]: In theory A , Y = 0 in the chiral ring and so the Hilbert series of theory A is where H SO Nc,N f (t, y, x; R) is given by (2.116). The moduli space of theory B can be conveniently studied by applying the ARSW duality and obtain SO(N c + 2) with N f flavours, a singlet Y and W = Y Y + Y , where Y is the basic monopole operator in this theory. The F -term ∂ Y W = 0 gives Y = −1, and so the gauge group SO(N c +2) is broken to SO(N c ). The residual theory is therefore SO(N c ) gauge theory with N f flavours and zero superpotential. The R-charge of the quarks in this theory is also equal to R. Thus the Hilbert series of theory B is also H SO Nc,N f (t, y, x; R).

Dualities with quadratic monopole superpotentials
In this section, we study theories containing quadratic monopole superpotentials and their dualties. We start our discussion with the duality involving models with superpotential W = (X + ) 2 + (X − ) 2 , which was first proposed in [35]. We then proceed to new dualities, including that involving W = (X − ) 2 and chiral flavours and Chern-Simons levels, as well as those with symplectic and orthogonal gauge groups. Below we state explicitly the dualities, global symmetries, R-charges of various chiral fields, and some of their important features. In section 4, we present the three sphere partition functions of the theories study presented here and show that they match across the duality. As a further test of these new dualties, it will also be shown that by giving appropriate real masses to certain chiral multiplets, we can flow from the proposed dualities to the known ones. Subsequently, in section 5, we study the moduli spaces, compute the Hilbert series and show that they also match between the dual theories. This duality was proposed in [35]. Due to the monopole superpotential, we expect the U (1) topological symmetry and the U (1) axial symmetry to be broken. In theory A, the Coulomb branch is complete lifted due to the F -terms ∂ X ± W = 0, which implies that X + = X − = 0. The same phenomenon happens also in theory B. The R-charge R of the quarks and antiquarks in theory A is fixed by the monopole superpotential: The mesons in theory A is mapped to the singlets M in theory B. We shall discuss the rank condition of M and other details in 5.1.1.

W = (X − ) 2
Theory A: U (N c ) gauge theory with N f flavours Q and Q and W = (X − ) 2 .
Theory B: U (N f − N c ) gauge theory with N f flavours q and q, N 2 f singlets M , singlet S + and superpotential W = M qq + ( X + ) 2 + S + X − , where X ± are the basic monopoles in theory B, and S + is dual to the monopole X + in theory A.
In these theories, the topological symmetry and the axial symmetry is broken to a diagonal subgroup, which we denotes by U (1) T symmetry. In theory A, the part of the Coulomb branch that is generated by X − is lifted due to the F -terms ∂ X − W = 0. However, the part that is generated by X + still remains. The R-charges of X − is fixed to be equal to 1, whereas that of X + depends on how we parametrise the mixing between the R-symmetry and the U (1) T symmetry. We shall postpone the detailed discussion until section 5.1.2. We propose the following operator maps between theories A and B: mesons ←→ M .

(3.4)
In section 4.1, we show that this duality can be obtained by flowing from the duality W = (X + ) 2 +(X − ) 2 discussed in the previous subsection. In this way, we also match the three sphere partition functions of the two theories in (4.16). Furthermore, we demonstrate that, by appropriate shifts of real masses, one obtains the Aharony duality. These constitute non-trivial tests of the proposed duality.
In theory A, The monopole operator X − vanishes in the chiral ring due to the F -term ∂ X − W = 0, and the Coulomb branch generated by X + is lifted due to the non-zero CS level. Thus, the moduli space is generated by the meson matrix with the rank at most N c . The R-charge R of the fundamentals and antifundamentals is fixed by the monopole superpotential:

Models with symplectic gauge groups
Theory A: U Sp(2N c ) with 2N f fundamentals and superpotential W = Y 2 .
In theory A the Coulomb branch is lifted because of the F -terms ∂ Y W = 0. Again, the moduli space is generated by the meson matrix with the rank at most 2N c . The R−charge R of the fundamentals is fixed by the monopole superpotential: A similar analysis can be carried out for theory B. More details will be provided in section 5.2. The matching of the partition functions for the two theories will be discussed in section 4.2.

Models with orthogonal gauge groups
Theory A: O(N c ) with N f chirals in the vector representation and superpotential W = Y 2 .
The analysis is very similar to the case of the symplectic gauge groups. In theory A, the Coulomb branch is lifted because of the F -terms ∂ Y W = 0. Again, the moduli space is generated by the meson matrix with the rank at most N c . The R−charge R of the quarks is fixed by the monopole superpotential: A similar analysis can be carried out for theory B. More details will be provided in section 5.3. The matching of the partition functions for the two theories will be discussed in section 4.3. Finally, it is worth mentioning that there is also a similar duality for the special orthogonal gauge groups:

The three sphere partition function
In this section we provide some analytic checks of the dualities proposed above. We study the consistency of the real mass flows connecting the dualities with quadratic monopole superpotentials in the UV to Aharony duality in the IR. In each case we conjecture in the UV an identity between the squashed three sphere partition functions obtained by localization. Such identities are sketchily of the form where Z ele and Z mag refer to the electric and to the magnetic partition functions. The parameters µ are in general complex combinations of real masses and R-charges. The presence of the monopole superpotentials break some combinations of the topological and of the axial global symmetries and it reflects in some constraints on the µ parameters. We refer to the these constraints as balancing conditions. Then we simulate, on the partition functions, the real mass flows that lead to other IR dualities. Such real mass flows correspond to infinite and real shifts on some of the parameters µ. In general we arrive, in the IR, to identities between infinite quantitites. If we can drop the divergent terms in these final identities we interpret the identities between the finite parts as the ones between the IR partition functions obtained after the real mass flows. When the expected IR duality corresponds to Aharony duality we read the final identity between the electric and magnetic partition functions and compare with the ones that already appeared in the literature. If these agree the whole procedure has furnished a consistency check of (4.1) and consequently of the conjectured UV duality. We apply the procedure described above to the dualities with quadratic monopole superpotential in presence of unitary, symplectic and orthogonal gauge groups. As a general remark we observe that in each case we need to perform a dual Higgs flow [52] in order to recover Aharony duality in the IR. Such dual higgsing is necessary to reconstruct the correct scaling of the divergent term and to reconstruct the correct matter and gauge content of the dual theories.

The unitary case
We start analyzing the RG flow from the W = (X + ) 2 + (X − ) 2 duality to Aharony duality. As an intermediate step we obtain the duality with W = (X − ) 2 . Starting from this last duality we discuss the case with a CS term as well.

Flowing from
We start considering the duality between 3d N = 2 U (N c ) with N f flavors and W = (X + ) 2 + (X − ) 2 to W = (X − ) 2 and U ( N c = N f − N c ) with N f dual flavors, the meson M and W = M q q + ( X + ) 2 + ( X − ) 2 .
We conjecture the identity relating the UV partition functions to be of the form 3) In this relation the parameters µ i and ν i are complex combinations of real masses and R-charges of the N f fundamentals and anti-fundamentals respectively. They can be explicitly expressed as follows where i m i = i m i = 0. The real parameter m A is the axial mass and the Rcharge ∆ coincides in this case for the fundamentals and for the anti-fundamentals. The parameter ω is purely imaginary, and it is explicitly given by 2ω , where b is the real squashing parameter of the ellipsoid S 3 b . The ellipsoid is defined by the relation The one loop determinants Γ h are hyperbolic Gamma functions 11 (4.6) 11 The notations adopted by this paper is related to that adopted by [35] as follows: Moreover, the notations of real masses in this paper are related to [35] as follows: The dimension of the Weyl group is denoted by |W |. In (4.2) the real mass parameters are constrained by the balancing condition that corresponds to the constraint enforced by the monopole superpotential on the global symmetries 12 . We now consider this duality for N f + 1 fundamentals and anti-fundamentals. The gauge group of the dual model is U (N f − N c + 1). We study the real mass flow to the W = (X − ) 2 duality by shifting µ N f +1 and ν N f +1 as and consider the limit s → ∞. The flow on the dual side must be supported by the Higgs flow x N f −Nc+1 → y + s. The balancing condition becomes At large s we can integrate out the massive flavors by using the formula This formula corresponds to the generation of half-integer CS levels, for the gauge and for the flavors symmetries, when integrating out heavy fermions with large real mass. By using (4.10) the partition function of the electric theory becomes (4.11) where the exponential factor in the integrand represents the contribution of a generalized FI term in the classical action. The dual partition function is 12 Let us explain briefly the origin of this balancing condition. The superpotential W = (X + ) 2 + (X − ) 2 constraints the R-charges of X ± to be 1, i.e. R[X ± ] = 1 = N f (1 − ∆) − (N c − 1), which implies that ∆ = 1 − Nc N f . This is in agreement with the matching of the coefficients of ω in the left and the right hand sides of the balancing condition, namely 2N f ∆ = 2(N f − N c ).
The first integral in (4.12) can be simplified, because it corresponds to SQED with one flavor, and it is mirror dual to a triple of singlets. This duality corresponds to the integral identity In our case it corresponds to Then, substituting this integral in the magnetic partition function and using the identities we arrive at the relation The first term on the RHS corresponds to the singlet S. It can be re-written, by using the balancing condition, as At this point of the discussion we can shift η → η + 2ω, so that now − 1 2 η can be interpreted as a real FI parameter of the theory, as it appears in the exponential functions in the first and the second lines of (4.16). This shift modifies the balancing condition and the final relation properly. We can then read the charge of the singlet in (4.17) and relate it to the one of the electric monopole X + as follows. The term η 2 implies that S carries the topological charge +1, in the same way that X + does. Moreover, after the shift, the R-charge of the monopole operator X + in the theory A and the singlet S in theory B can be determined by mixing the topological symmetry and the R-symmetry as follows: where the LHS are the complexified axial masses of X + and S and the RHS are the complexified real masses for the topological symmetry (complexified FI parameters) for X + and S. Thus, R[X + ] = R[S], as expected. The final relation (4.16) is therefore compatible with the duality between U (N c ) with N f fundamental flavors and W = (X − ) 2 and U (N f − N c ) with N f dual fundamental flavors and W = M q q +( X + ) 2 +S X − , where the singlet S corresponds to X + in the electric side. The term S X − , which enters the superpotential, has R-charge 2; this can be seen as follows. From (4.17), the R-charge of S is given by where each term comes from the coefficients of ω in and ω(N f − N c ), respectively. Moreover, the R-charge of X is given by 20) and so R[S] + R[ X] = 2, as expected.

Flowing from W = (X − ) 2 to the Aharony duality
We can study a further flow, from this duality to the Aharony duality. This flow is engineered by considering the W = (X − ) 2 duality with with N f + 1 flavors and shift the masses as We also shift the vector multiplet by x i → x i + s, i =, . . . , N c . Furthermore, in the dual theory we need to consider the shift x i → x i − s, for i = 1, . . . , N f − N c . There is also a dual Higgsing corresponding to the shift x N f −Nc+1 → y + sN f . The balancing condition (4.9) becomes (4.23) By computing the large s limit we arrive at the identity The integral in the last line can be computed explicitly and it corresponds to The first term in (4.25) simplifies with the term Γ h (η 2 ) in the first line of (4.24), due to the identity Γ h (2ω − η 2 )Γ h (η 2 ) = 1. The second term is equivalent to Γ h (ω) = 1 because of the balancing condition (4.23). The last term in (4.25), which is simplified to Γ h (η 2 − ω) upon using the balancing condition (4.23), can be identified with the monopoles of the electric theory. The antimonopole is still identified with Γ h (η − ω) in (4.24). Indeed by using the balancing condition (4.23) we can see that they are equivalent to where η 2 − η ≡ ζ is the effective FI that can be read from the partition function. We have obtained the identity This is the correct expression for the matching of the electric and the magnetic partition function in the Aharony duality. Summarizing we started from the conjectured identity (4.2) between the partition functions of the W = (X + ) 2 + (X − ) 2 duality. Then we have obtained the identity between the partition functions of the W = (X − ) 2 duality. Eventually we have obtained the known identity corresponding to the matching between the electric and the magnetic Aharony dual phases. This corroborates the validity of the dualities with quadratic monopole superpotentials.

The W = (X − ) 2 duality with the Chern-Simons term
We conclude this section by studying the RG flow from the W = (X − ) 2 duality to the case with CS term. The masses, the FI and the scalar σ in the electric theory are shifted as follows While in the magnetic theory we read the masses from the duality map and provide the opposite shift on σ i → σ i + ks, i = 1, . . . , N c . We arrive to a duality between a U (N c )k 2 theory with N f − k fundamentals and N f antifundamentals and superpo- theory with N f − k fundamentals and N f antifundamentals, a meson M with N f (N f − k) components and superpotential We perform the infinite shift on the identity (4.16). There is a divergent phase that cancels between the electric and the magnetic side of the identity. Observe that the linear divergent term in the phase cancels because of the relation The final identity is where the phase corresponding to the contributions of the contact terms [20,53,54] is and the effective electric and magnetic FI terms are where in the last equalities we made use of the balancing condition (4.9) and of the relation (4.30).

The symplectic case
In this section we perform a similar analysis for the duality between a 3d N = 2 U Sp(2N c ) theory with 2N f fundamentals and superpotential W = Y 2 and an U Sp(2(N f − N c − 1)) theory with 2N f fundamentals, an anti-symmetric meson and superpotential W = M qq+ Y 2 . Observe that a similar duality with a linear monopole superpotential has already been studied in the literature [21] and it corresponds to the duality obtained by circle reduction of 4d U Sp(2N c ) Seiberg duality [46]. Assuming the validity of this duality the identity between the partition functions on S 3 b is with the balancing condition As a check we show that (4.33) becomes the identity between the partition functions of the electric and of the magnetic phases of Aharony duality. We trigger the real mass flow by performing the shifts where we impose s to be large and positive. In the dual theory we need to perform an higgsing as well, σ N f −Nc → y + s. The balancing condition becomes One can show that the divergent term coincide and arrive to the identity This is a duality between a U Sp(2N c ) theory and a U Sp(2(N f − N c − 1)) × U (1) theory. In order to arrive at a more conventional duality we can reformulate this identity by integrating over the U (1) factor. The last integral in (4.40) is equivalent to the product The second term in (4.38) is exactly equal to 1 while the last term in (4.38) simplifies against Γ h (2α) in (4.40). We are left with the term Γ h (2α − ω). By applying the balancing condition this is equivalent to that is the contribution of the electric monopole acting as a singlet in the dual phase. The final identity is (4.40) and it represents again the matching of the partition functions of Aharony duality for U Sp(2N c ) gauge theories.

The orthogonal case
In this section we conclude the analysis by studying the duality between a 3d N = 2 a O(N c ) theory with N f vectors and superpotential W = Y 2 and a O(N f − N c + 2) theory with N f dual vectors, an symmetric meson and superpotential W = M qq+ Y 2 .
Observe that a similar duality with a linear monopole superpotential has already been studied in the literature [22] and it corresponds to the duality obtained by circle reduction of 4d O(N c ) Seiberg duality [55]. We will not specify the global properties in this discussion, meaning that we analyze the duality between O(N c ) + theories in the language of [22]. It should be nevertheless interesting to study the other cases, because in this case the different global properties are expected to have observable effects on the models and consequently on the dualities. We leave this problem for future analysis.
At the level of the partition function wee need to distinguish the even and odd N c case. In the first case, if we consider an O(2N c ) theory with 2N f fundamentals the duality corresponds to the identity with the balancing condition If the rank is odd, corresponding to a O(2N c + 1) theory, the identity is with the balancing condition In the following we will show how to obtain these identities by using a standard trick [20,56]. This correspond to derive them by deforming the ones for the U Sp(2N c ) theories.
Let us end this subsection by briefly commenting on the divergence of the partition function discussed on page 43 of [22] for the case of W = Y . For the case of W = Y 2 , we have a different balancing condition from the linear monopole superpotential case. As a result, we do not have a divergence term that arises from Γ h (0). In fact, we have checked the aforementioned identities numerically for various parameters µ i and found that the results are finite on both sides of the equality.

Even orthogonal case
Consider U Sp(2N c ) with 2N f + 4 fundamentals and assign the masses as follows By using the duplication formula [56] Γ The partition function of the electric theory becomes (4.48) and the balancing condition is We can do the same in the dual theory obtaining such that the contribution of the meson becomes representing the fact that the meson is symmetric in this case. We have obtained an identity between an O(2N c ) theory with 2N f fundamentals and an O(2N f − 2N c + 2) theory with 2N f fundamentals and a symmetric meson. The electric and the magnetic monopole superpotentials, compatible with the balancing condition are quadratic.

Odd orthogonal case
In this case we consider 2N f + 2 fundamentals and fix The electric partition function becomes with the balancing condition The dual partition function becomes with the extra piece By distributing properly the terms on the RHS and LHS we arrive at the expected relation between the orthogonal theories with odd rank and quadratic superpotential

Flowing to Aharony in the orthogonal case
We can engineer the flow from the W = Y 2 duality to Aharony duality in the orthogonal case as well. This is a further check of the proposed duality. In the electric case we consider 2(N f + 1) fundamentals and assign the large real masses as µ 2N f +1 = η 2 + s and µ 2N f +2 = η 2 − s. In the magnetic theory we must consider an higgsing of the dual gauge group as well, giving raise to an O(N f − N c + 2) × U (1) theory. The O(N f − N c + 2) theory has N f fundamentals and superpotential W = M qq, where the meson M is a symmetric tensor with dimension N f (2N f + 1). There is an extra massless singlet H in this theory, coming from the original meson with dimension (N f + 1) (2N f + 3). This interact with the two chiral multiplets, having opposite charge under the U (1) sector. We refer to them as p and p, such that the superpotential of this sector is W U (1) = Hp p. This U (1) sector has an FI term constrained by the choice of real masses. If we dualize this sector this is dual to three, the meson X = p p, and the monopoles Y and Z. The fields H and X are massive because of the superpotential. The duality map fixes Y to have vanishing real mass and R-charge equal to 1. This corresponds to have a massive field. The other field Z has the same quantum numbers of the electric monopole and it is compatible with the dual superpotential of the Aharony dual model. This structure can be reproduced on the partition function by distinguishing the even and odd cases as before.

Matching the Hilbert series
In this section, we compute the Hilbert series for theories with quadratic monopole superpotentials. The analysis is similar to the linear case. If V is one of the basic monopole operators, the superpotential W = V 2 fixes the R-charge of V to be 1 and fixes the R-charge of the chiral fields. The Coulomb branch generated by V is also completely lifted due to the F -terms ∂ V W = 0. Below we demonstrate the computation in various examples.

Models with unitary gauge groups
In this section, we consider the duality presented in section 3.1.1.
Let us first analyse theory A. The F -terms ∂ X ± W A = 0 give X + = X − = 0 in the chiral ring. The residual theory is the whole U (N c ) with N f flavours, with the R-charge Q and Q is given by (3.3). The Hilbert series of theory A is thus the mesonic Hilbert series of U (N c ) with N f flavours where the expression for H U Nc,N f is given in (2.9) and the R-charge R of the quarks and antiquarks are given by (3.3). There is no fugacity y because the axial symmetry is broken by the monopole superpotential.
Let us now analyse theory B. We first perform the Aharony dual to theory B and obtain Theory B : U (N c ) gauge theory with N f flavours q and q, two singlets X ± and superpotential where V ± are the basic monopoles in theory B . The F -terms ∂ X ± W B = 0 gives Substituting this into W B , we find that W B = −( X + ) 2 − ( X − ) 2 , and hence X ± = 0 in the chiral ring. Thus, V ± also vanish in the chiral ring. The analysis of the residual theories is therefore similar to that of theory A. Note that the R-charge R of q and q is equal to that of theory A: Indeed, the Hilbert series of theory B, which is dual to theory B , is given by Let us now discuss about the duality presented in section 3.1.2.
Theory A Let us consider theory A. We mix the U (1) T symmetry with the R-symmetry such that the R−charges of the X ± are given by and α parametrizes the mixing of the UV R charge and the U (1) T charge of the monopole operators. Imposing the marginality of the superpotential we get R[ Also, R[X + ] = 1 − α .
where z is the fugacity for the U (1) T symmetry.
• m 1 > 0 = m N : with residual theory U (N c − 1) with N f flavours. The Hilbert series is given by where we assign the fugacity z x for the U (1) T symmetry of X + , and Thus, the Hilbert series of theory A reads Theory B Let us analyze now the theory B. It is convenient analyse, instead, the Aharony dual of theory B, which is given by Theory B : U (N c ) with N f flavours and singlets X ± and S + with superpotential where V ± are the basic monopole operators of theory B . Consider first the F −terms obtained by differentiating with respect to the singlets: Plugging these equations into W B we get The F -term ∂ X + W B = 0 implies that X + = 0 in the chiral ring. The condition V − = −2 X + implies that V − also vanishes in the chiral ring, like X + does. Moreover V + = −S + implies that we need to consider two cases for the computation of the Hilbert series, namely S + = 0 and S + = 0. We shall see below that these correspond to the cases of m 1 = 0 and m 1 = 0 in the theory A.
Let us turn our attention to the R−charges. Let us define ). The superpotential W B fixes the R-charge of X + to be R[ X + ] = 1, and so from the superpotential term X + V − we have R[V − ] = 1. The R-charges of V ± are given by where we use β to parametrise to mixing between U (1) R and U (1) T . Hence, the solution for R B reads Hence, the R-charge of the mesons in this theory is 2R B = 2R, which perfectly match with the R-charge of the singlets M in theory B. This also suggests that our choice of parametrisation of the mixing between U (1) R and U (1) T is consistent with the proposed duality. Moreover, we have Now we are ready to compute the Hilbert series. As we said, we have two cases: • For S + = 0, we have V + = 0 and so the residual theory is U (N c ) with N f flavours. The Hilbert series is is • For S + = 0, we have V + = 0 and so the residual theory is U (N c − 1) with N f flavours. The Hilbert series is given by

chiral flavours and CS terms
We now consider the duality involving chiral flavours and CS terms presented in section 3.1.3. The analysis of theory A is very similar to section 2.2.6. The monopole operator X − vanishes in the chiral ring due to the F -term ∂ X − W = 0, and the Coulomb branch generated by X + is lifted due to the non-zero CS level. The Hilbert series of theory A is given by (2.80) with the R-charge R given by (3.5).
By the similar argument to section 2.2.6, the moduli space of theory B is generated by M subject to the quantum condition that rank(M ) ≤ N c . In order to prove this, we turn on a VEV of M with rank N c + p with p > 0 and obtain the how energy fund/antifund. We then use the BCC duality to obtain the dual theory. Since the dual gauge group is the unitary group of rank N f − (N f − N c ) = −p, we see that for p > 0, supersymmetry is broken and there is no supersymmetric vacuum.

Models with symplectic gauge groups
We now consider the duality involving symplectic gauge groups presented in section 3.2.
In theory A the Coulomb branch is completely lifted by the F -terms ∂ Y W = 0. The Hilbert series of the theory is thus the mesonic Hilbert series where the expression for H U Sp 2Nc,2N f is given by (2.83) and the R−charge R of the fundamentals is given by (3.6). There is no fugacity y because the U (1) axial symmetry is broken.
Let us consider now theory B. It is convenient to study its Aharony duality to theory, we end up with a U Sp(2N c ) gauge theory with a singlet Y and superpotential where Y is now the basic monopole of this theory. The F −terms of the singlet give Substituting back in W we obtain a superpotential and so the F -term ∂ Y W = 0 gives Y = 0. It follows from (5.31) that Y = 0, and so the Coulomb branch is lifted. Note that the fundamentals also have R-charge R; this is fixed by the R-charge of the monopole operator Y : The Hilbert series is thus given by H U Sp 2Nc,2N f (t, y, x; R), in agreement with theory A.

Models with orthogonal gauge groups
We now consider the duality involving orthogonal gauge groups presented in section 3.3. Let us first analyse theory A. The Coulomb branch is completely lifted due to the F -term ∂ Y W = 0, so the Hilbert series of the theory is the mesonic Hilbert series where the expression for H O Nc,N f is given by (2.105) and R is the R-charge of the quarks given by (3.7).
Let us now turn to theory B. As before, it is convenient to consider the Aharony dual of this theory B, which is an O(N c ) gauge theory with a singlet Y and superpotential Substituting back in W we obtain a superpotential W = − Y 2 . The F -term ∂ Y W = 0 implies that Y = 0, and so Y = 0 by (5.36). Hence, the Coulomb branch is lifted. Note that the quarks in this theory also have R-charge R; this is fixed by the R-charge of the monopole operator Y : Thus, the Hilbert series is given by the mesonic one H O Nc,N f (t, y, x; R), reproducing exactly the one of theory A.
For the duality involving the special orthogonal gauge groups, the analysis is very similar. For theory A , the Hilbert series is where H SO Nc,N f is given by (2.116). The moduli space of theory B can be analysed by using the ARSW duality in the same way as the above.

Further developments
Let us conclude this paper by discussing an obstruction to the duality for theories with quartic powers of the monopole superpotential. We then proceed to discuss open problems and other research directions that we leave for future work.

The quartic monopole superpotential
Let us consider the following pairs of theories: The R-charge R A of the quarks and antiquarks in theory A is fixed by the monopole superpotential: A similar computation show that the R-charge R B of the quarks and antiquarks in theory B is . This is equal to the R-charge of the mesons in theory A, where latter is 2R A = 2N f −2Nc+1 N f . Thus, theories A and B have a chance to be dual to each other. However a further analysis shows that this is not the case.
In theory A, the F -terms ∂ X ± W A = 0 imply that (X ± ) 3 = 0. This admits a solution X ± = 0; in which case, the residual theory is a U (N c ) gauge theory with N f flavours. The meson matrix thus has a maximum rank of N c .
On the other hand, in theory B, if we give a VEV of rank N c to the singlet M , which is dual to the meson in theory A, the low energy effective field theory is U (N f − N c + 1) gauge theory with N f − N c flavours. According to [3], this theory is described by the monopole operators X ± and the mesons q q satifying X + X − det(q q) = 1. 13 The effective superpotential of theory B is therefore W B = ( X + ) 4 + ( X − ) 4 + λ( X + X − det(q q) − 1) , (6.5) where λ is a Lagrange multiplier. Using the equations of motion, we find that the effective superpotential becomes W B ∼ (det(q q)) −2 . Thus, we have runaway vacua. This does not match with theory A, where we have the mesonic branch of U (N c ) gauge theory with N f flavours as the moduli space of vacua. Thus, the duality fails.
Another argument is to apply the Aharony duality to theory B and obtain a U (N c − 1) gauge theory with N f flavours, N 2 f singlets M , two singlets X ± , and superpotential W = X + V − + X − V + + ( X + ) 4 + ( X − ) 4 , where V ± are the basic monopole operators in this theory. The F -terms ∂ X ± W = 0 admit the solutions V ± = 0, in which case the meson matrix attains its maximum rank, which is N c − 1. This is not compatible with theory A, in which the maximum rank of the meson matrix is N c . Thus, the duality fails.
This argument can be used to show that the following pairs of theories cannot be dual to each other: The R-charge of the mesons in theory A is equal to that of M in theory B; they are equal to . The maximum rank of the meson matrix in theory A is N c . Applying the BCC duality to theory B, we obtain the dual gauge group being O(N c − 1). The maximum rank of the meson matrix in the latter is N c − 1, which is not compatible with that in theory A.
Let us finally consider the following pairs of theories: Theory A: U (N c ) gauge theory with N f flavours, and superpotential W A = (X − ) 4 .
Theory B: U (N f − N c + x) gauge theory with N f flavours q, q, N 2 f singlets M , a singlet S + and superpotential W B = M q q + S + X − + ( X + ) 4 .
Our aim is to show that these theories cannot be dual for any integer x. As before, the maximum rank of the meson matrix of theory A is N c . Let us consider the Aharony dual of theory B, which is a U (N c − x) gauge theory with N f flavours, singlets S + , X + , X − , N 2 f singlets M and superpotential W B = X + V − + X − V + + S + X − + ( X + ) 4 . The F -terms ∂ X ± W B = 0 admit the solution V ± = 0, in which case the meson matrix attains its maximum rank N c − x. For this to be compatible with theory A, we must have x = 0. Now let us consider the R-charges. Since R[ X + ] = 1 2 due to the superpotential term ( X + ) 4 , we have R[V − ] = 3 2 due to superpotential term X + V − . The R-charges of X − of theory A and V − of theory B are given by where R A and R B are the R-charges of quarks and antiquarks in theories A and B , respectively. 14 Hence the R-charge 2R A of the mesons in theory A is not equal to the R-charge 2R B of the mesons in theory B . Thus, theory A is not dual to theory B and hence not to theory B.

Future directions
Let us discuss some other interesting lines of research that we leave for future analysis. An important aspects that we did not discuss in the paper is related to estimation of the conformal window. This is indeed possibile that some singlets hit the bound of unitarity when maximizing the free energy. This bounds corresponds to the failure of the inequality ∆ Singlet > ∆ F ree = 1 2 . In such cases we are in presence of accidental symmetries that need to be cured by applying the procedure of [57] (see also [58] for a 3d version of this procedure). This can modify the dualities and it should be interesting to have a complete understanding of the conformal window along the lines of [59]. A related analysis consists of finding UV complete models that flow to the ones with quadratic power monopole superpotentials in the IR, of the type discussed here. Similar discussions appeared in [35] and it would be interesting to adapt such analysis to our cases.
Another aspects that we did not discuss is related to the global aspects of the dualities with orthogonal gauge groups in presence of quadratic power monopole superpotentials. In such cases one should follow the discussion of [22] and distinguish O(N ) ± and Spin(N ) cases. Here we restricted to the duality between O(N ) + groups. Furthermore we did not discuss possible dualities between SU (N ) gauge groups and quadratic power monopole superpotentals. The existence of a duality for the U Sp(2) = SU (2) case looks a good starting point for the existence of such duality, but we have not been able to provide a general behaviour for such a case.
Another interesting problem consists of the brane interpretation of the quadratic monopole superpotential. Naively one can think to this superpotential as arising as in the linear case [60], i.e. by placing a 4D theory on a circle, T-dualizing, possibly moving some D-brane (D3 and or D5) along the circle and then adding a D1 brane between two stacks of D3 branes separated along the compact direction. These D1 branes represent the monopole superpotential, KK monopole in the case of real gauge groups [61] and linear monopoles for unitary groups [62]. A similar construction may be engineered for the quadratic monopoles, where the higher power can be for 14 If we choose the parameterisations such that 1 2 , in order to equate the R-charges of the mesons 2R A = 2R B , we see that α − β must depends on N c and N f . This is inconsistent with the duality between theories A and B .
example engineered by multiple stacks of D1 branes. It should be interesting to check this or similar constructions.