N = 1 dualities in 2+1 dimensions

: We consider minimally supersymmetric QCD in 2+1 dimensions, with Chern-Simons and superpotential interactions. We propose an infrared SU ( N ) ↔ U ( k ) duality involving gauge-singlet ﬁelds on one of the two sides. It shares qualitative features both with 3 d bosonization and with 4 d Seiberg duality. We provide a few consistency checks of the proposal, mapping the structure of vacua and performing perturbative computations in the ε -expansion.


Introduction and results
The last couple of years saw the discovery of three-dimensional bosonization: infrared (IR) dualities enjoyed by gauge theories with Chern-Simons interactions , similar in spirit to the particle-vortex duality [23,24] and to dualities with extended N ≥ 2 supersymmetry studied for more than two decades [25][26][27][28][29].
In this paper we want to tackle the case of 3d minimal supersymmetry, namely N = 1 (see [30]), an interesting bridge between N = 0 and N ≥ 2 for many reasons.
With 3d N = 1 supersymmetry all supermultiplets (except for those containing conserved currents) are long: there is no protected sector analogous to the chiral ring.Moreover, the interactions are not protected.This makes the analysis of N = 1 theories similar to the N = 0 case, and the dualities quite powerful: usually for N ≥ 2 dualities, it is only known how to map the protected operators to the dual theory.On the other hand if we have an N = 1 duality and we know the mapping of a supermultiplet, we can, in principle, deform the N = 1 duality to an N = 0 duality.
Let us also mention that there might be experimental realizations of (2+1)-dimensional systems with low amount of supersymmetry.This is due to the phenomenon of emergent supersymmetry [40][41][42][43][44][45][46].Roughly speaking, if the massless matter of some model is supersymmetric, then also the interactions of the infrared fixed point may be supersymmetric: the renormalization group flow may land on the SUSY fixed point.

duality with gauge singlets
In this paper we focus on 3d N = 1 models with unitary gauge groups and fundamental flavors.We are interested in a supersymmetric analog of the non-Abelian bosonization.Bosonization maps critical scalars (i.e. with quartic φ 4 interactions) to regular fermions.In a supersymmetric theory, we expect critical scalars to be paired with critical fermions (i.e. with interactions similar to the Gross-Neveu-Yukawa model), and regular fermions to be paired with regular scalars.A supersymmetric U (N c ) or SU (N c ) gauge theory with a matter multiplet Φ in the fundamental representation does not admit interactions of the form φ 4 : the generic superpotential W = |Φ| 4 leads to interactions of the form φ 6 + φ 2 ψψ, so we have regular matter.In order to have an N = 1 duality mapping critical matter to regular matter, we expect it is necessary to introduce additional fields, the simplest option being gauge-singlet supermultiplets.
In the case of a single flavor, we introduce one gauge-singlet real superfield H and propose the following duality: with 1 flavor P and a gauge-singlet H (1.1) The sign of the superpotential on the left-hand side is important: the physics would be different with the other sign.Notice that a parity transformation changes the sign of both the Chern-Simons terms and the superpotential, while the relative sign remains unchanged.On the right-hand side, one can redefine H → −H and change the overall sign of the superpotential, while the relative sign between the two terms is important.
On the U (k) side we have regular matter, on the SU (N ) side we have critical matter.
The "topological" global symmetry on the U (k) side is mapped to the baryonic symmetry of the SU (N ) gauge theory.Accordingly, monopole operators are mapped to baryonic operators.This is similar to what happens in the non-supersymmetric bosonization dualities.
As for the operators in the sector with vanishing global U (1) charge, our proposal is the that the meson is mapped to the singlet: This is analogous to the mapping in 4d N = 1 Seiberg duality [47] or 3d Aharony dualities [27]. 1  Deforming the two theories with a superpotential term m QQ † ↔ m H, the vacua display an interesting behavior: for m ≤ 0 there is one vacuum, while for m > 0 there are two isolated vacua. 2 Each vacuum is gapped and hosts a certain topological sector.Since the same vacuum structure and TQFTs appear on both sides of the duality, we have a first consistency check of our proposal.
A non-trivial implication of the duality (1.1) and of (1.2) is that (QQ † ) 2 is mapped to H 2 .Since (QQ † ) 2 is in the superpotential, in the IR it must be the case that the dimension ∆ (QQ † ) 2 IR > 2, and so ∆ d 2 θ(QQ † ) 2 IR > 3.In order for the duality to be correct, then it must also be the case that on the SU (N ) side ∆[H 2 ] IR > 2. This is a somewhat surprising statement, since H 2 is a mass term, with ∆[H 2 ] U V = 1.
In order to gain further insight, we performed a perturbative computation in the ungauged model (cubic Wess-Zumino model) with superpotential The fact that H 2 is irrelevant in the IR is a consistency check of our proposed duality, and justifies our choice of superpotential in (1.1) which does not include an H 2 term.We expect this picture and the duality (1.1) to be correct for any N ≥ 1 and k ≥ 1.
1 Notice however that here the operators QQ † and H are not protected by supersymmetry: they are long supermultiplets. 2For N = 1 the structure is a bit different.We study the case N = 1 in detail in Section 2.
There are related but different versions of the duality (1.1).One example3 is This version is very similar to (1.1): the superpotential contains H 3 but not H 2 and upon mass deforming the theories, there are two vacua merging into a single vacuum.Now on the U (k) side we have critical matter, on the SU (N ) side we have regular matter.The duality (1.4) is expected to be valid for k ≥ 1 and N > 1.However, (1.4) cannot be valid for N = 1: in this case the SU (N ) side becomes a free complex superfield, that displays enhanced N = 2 supersymmetry, has a nontrivial moduli space of vacua and a different vacuum structure upon mass deformations (a single vacuum for both signs of the mass).Our proposal in this case is4

←→
Free N = 2 chiral multiplet P . ( In this way one-loop effects balance the superpotential term H2 and allow a non-trivial moduli space of vacua for the U (k) theory.Also the vacuum structure upon mass deformations is the same.Summing up, our proposal is the following: the dual of an N = 1 SU/U gauge theory with regular matter (and hence no singlets) is a U/SU theory with critical matter (so there are singlet fields).The superpotential term for the singlet fields is cubic.The only exception is when the regular matter is actually free: then the qualitative structure is different and there is a quadratic superpotential term for the singlet field on the dual side.

Further directions
As for the non-Abelian dualities, it would be nice to investigate what happens outside their range of validity (especially in the generalized case with arbitrary number of flavors N f > 1 discussed in Section 3.4), and study possible quantum phases as in [16,17].
It would also be interesting to study non-supersymmetric deformations and possibly make contact with the bosonization dualities and/or the non-supersymmetric dualities with bosons and fermions studied in [5,20,21].From the mapping of low-lying operators in terms of N = 1 superfields that we provide, it is possible to read off the mapping of non-supersymmetric deformations.In particular notice that even if the top component of the superfield H 2 is an irrelevant SUSY deformation, the bottom component of H 2 is a relevant non-SUSY deformation.
In this paper we have analyzed the Wess-Zumino model (1.3) at two-loops in the ε-expansion.It would be interesting to increase the precision, either going to higher loops and possibly interpolating to a solvable 2d model [43,44,48], or employing the numerical bootstrap [49][50][51].
The Abelian duality we discuss in Section 2 has a simple Type IIB brane description à la Hanany-Witten, involving a D3-brane stretching between an NS5-brane and the simplest pq-web, or the S-dual configuration.The setup is similar to [31][32][33][34][35]37].It would be nice to find a brane description also for the non-Abelian cases.
Finally, let us mention that the dualities of this paper might be useful to understand walls and boundary conditions of 4d N = 1 SQCD [17,37,52,53].

Organization of the paper
In Section 2 we study in detail the Abelian case N = k = 1.In this case the duality (1.1) relates N = 1 SQED with one flavor and a quartic superpotential on one side, and a cubic Wess-Zumino model on the other side.The vacua form a circle for m < 0, while there are two isolated vacua for m > 0. The duality (1.4) instead becomes the well-known duality between N = 2 U (1) 1 2 SQED with one flavor and a free chiral field.Both cases are parity-invariant, a symmetry which on the SQED side emerges in the IR.We also find two additional U (1) ↔ U (1) dualities relating regular to critical matter.
In Section 3 we discuss the non-Abelian case.We present four different versions of the duality, involving in various ways gauge groups SU (N c ) and U (N c ).The various dual pairs can be related to each other by gauging a U (1) global symmetry.Moreover, we state the natural conjectures for number of flavors greater than one.
In Appendix A we discuss some details of the perturbative computations for the cubic N = 1 Wess-Zumino model.Appendix B provides the topological level-rank dualities in N = 1 notation.
Note added.When this work was under completion, we received [54] which, among other things, discusses 3d SU/U (N ) N = 1 gauge theories with a fundamental flavor.The dualities studied in [54] are similar to ours, but have different superpotential interactions and do not involve gauge-singlet fields.
2 N = 1 SQED with N f = 1 and its Wess-Zumino dual In this Section we discuss in detail the case N = k = 1 of the proposed dualities (1.1).
We use the 3d N = 1 superfields of [30], that in the spin-0 case expand as We use upper-case letters to denote the whole superfield and lower-case letters to denote the bottom component.The Lagrangians have standard kinetic terms, and the interactions are encoded in a real superpotential W = W(A i ).
One important remark is that d 2 θ is parity-odd.Therefore a gauge theory with zero Chern-Simon level is parity invariant if it is possible to assign parity quantum numbers to the matter superfields such that W is parity-odd.

Warm-up: the N = 2 duality
It is instructive to start with the basic N = 2 duality between the Chern-Simons gauge theory U (1) 1/2 with 1 flavor Q and a free chiral field P .Written in N = 1 notation this is Here Ψ is a real scalar superfield, that completes the N = 1 vector multiplet to the N = 2 vector multiplet.Note that one cannot integrate out the field Ψ, since the coefficient in front of Ψ 2 is not parametrically large with respect to the scale of the gauge coupling: it is fixed in terms of the CS level.
The free complex superfield P has a U (1) symmetry that rotates it.This corresponds to the magnetic (or topological) U (1) M symmetry on the left-hand side (LHS).The right-hand side (RHS) is manifestly time-reversal invariant; instead on the LHS time reversal is an emergent symmetry in the infrared.
The operators, collected into supefields, are mapped according to Here M is the gauge-invariant supersymmetric monopole.We used that the bottom component ψ of Ψ(θ) = ψ + θλ Ψ + θ 2 F Ψ is in the same N = 2 supermultiplet as the magnetic U (1) M current, and has dimension ∆[ψ] = 1.From the superpotential we can instead find the top component of the superfield Ψ: where q is the bottom component of Q(θ).Thus a particular linear combination of qq † and ψ is a supersymmetry descendant of ψ, and has ∆[qq † − ψ] = 2. Another linear combination-simply ψ-has ∆[ψ] = 1.We can now take the top component (2.4) and use it as the bottom component of a new superfield.This allows us to write the map of superfields where D α is the superderivative.This could be used to infer new relations between their components, and so on.Notice also that the duality implies the quantum relation in the gauge theory.
SUSY deformations.The free complex superfield P has a single N = 1 relevant deformation compatible with the U (1) symmetry: a superpotential mass term δW = mP P † with m ∈ R. In N = 2 notation, this is a "real mass".The quartic superpotential deformation δW = 1 4 αP P † P P † , that gives a sextic potential V = α 2 |p| 6 , is marginally irrelevant.
Therefore, one interesting consequence of the duality is that, on the LHS, the superpotential deformations Ψ 2 and QQ † are irrelevant (more precisely, marginally irrelevant since ∆ IR = 2) in the infrared CFT, even though they are clearly relevant in the ultraviolet where they have ∆ U V = 1.In other words, in the UV there are 3 relevant global-symmetry-invariant deformations: Ψ, Ψ 2 , QQ † .In the IR only one of them (Ψ) is relevant.
The relevant N = 1 deformation of the IR CFT, invariant under the global U (1) symmetry, is thus δW = m Ψ ←→ δW = m P P † . (2.7) As we said, this is actually the N = 2 preserving "real mass" deformation, and it breaks parity invariance.The free theory of P has, obviously, a single gapped vacuum both for m > 0 and m < 0. Let us quickly analyze the phases on the SQED side.For m > 0 there is a vacuum where ψ gets a positive VEV and Q gets positive mass.Integrating it out, the CS level is shifted to U (1) 1 and the quadratic superpotential term is shifted to −Ψ 2 by a one-loop effect.The F-term equation for the effective superpotential W eff = mΨ − Ψ 2 is consistently solved by ψ = m/2, q = 0 and the vacuum is gapped.For m < 0 there is a vacuum where |q| 2 = |m| and ψ = 0: the Higgs mechanism takes place and the vacuum is gapped.Therefore the phases match.
For m < 0 we could have considered the possibility that ψ gets a negative VEV and Q gets negative mass.Integrating it out, the CS level is shifted to U (1) 0 and the quadratic superpotential term is shifted to zero by a one-loop effect.The F-term equation for the effective superpotential W eff = mΨ does not allow for a VEV of ψ, leading to a contradiction.Notice also that since P is a free field, P P † is a positive operator: if the duality is correct, Ψ should not be able to get a negative VEV.

The basic N = 1 duality
From the N = 2 duality and exploiting the operator map (2.3), we can obtain a genuine N = 1 duality.We "flip" the real N = 1 superfield Ψ ↔ P P † on both sides, i.e. we introduce a parity-odd real superfield H and we couple it through the interactions with large coefficient.Notice that this deformation is relevant on both sides of the N = 2 duality.On the LHS both Ψ and H become massive and can be integrated out.This generates quartic superpotential interactions for the flavor Q, and the coefficient will be renormalized to a critical value.On the RHS we obtain an interacting Wess-Zumino (WZ) model.No symmetry prevents a superpotential term H 3 to be generated by quantum effects.In this way we obtain a new genuinely N = 1 duality (2.9) Here Q and P are complex superfields, while H is a real superfield.In the WZ model there is also an extra gravitational coupling −CS g .On the LHS we tuned the term QQ † to zero.The coefficient − 1 4 was chosen for later convenience, but its sign is physical and important.The WZ model on the RHS is manifestly parity invariant (H is parity-odd).For this reason the terms H 2 and P P † are not generated.We tuned the term H to zero.Since we can redefine H → −H, only the relative sign of the couplings in front of HP P † and H 3 has physical meaning.Performing loop computations in the ε-expansion, we will confirm that the term H 3 is generated, and we will see that HP P † and H 3 have opposite sign at the RG fixed point.
The basic operator map is Similarly to above, supersymmetry imposes some relations.We focus on the operators neutral under the U (1) global symmetry.The top components of the fundamental superfields on the RHS are (2.11) Thus the operator pp † − h 2 is a supersymmetry descendant of h.Another linear combination of pp † and h 2 , which we formally denote as pp † + h 2 , is instead a superconformal primary.(Moreover, a linear combination of hpp † and λ P λ P is a descendant of pp † .)On the LHS of the duality we have a descendant of qq † , while another combination, which we formally denote as λ Q λ Q + (qq † ) 2 , is a superconformal primary (the precise coefficients in front of the two terms depend on the computation scheme used).We thus have one more operator mapping: which can be upgraded to a map between supermultiplets: (2.14) SUSY deformations.From the last operator mapping, one crucial feature of our proposed scenario follows.On the SQED side, we know that the primary operator Let us stress that it is essential, for the duality to work, to make sure that there are no relevant deformations corresponding to the superpotential terms P P † and H 2 .The WZ model on the RHS is time-reversal invariant in the UV, therefore those deformations will not be generated in any case.The theory on the LHS, instead, develops timereversal invariance only in the IR.If the deformations P P † or H 2 were relevant, they would be activated in the IR and the SQED theory on the LHS could not hit the time-reversal invariant fixed point (one would need extra tuning, which however is not available in the SQED theory in the UV). (2.16) Notice in particular that the couplings in front of HP P † and H 3 have opposite sign.The two-loop scaling dimensions of the elementary fields are Pade 0.644 .
(2.17There are two quadratic singlets under the O(2) global symmetry.One is g 2 P P † +g 3 H 2 : this is a supersymmetry descendant of H therefore its scaling dimensions is ∆[H] + 1 at the IR fixed point.
The other singlet operator is the superconformal primary O s .Its precise form in our computation scheme and at two-loops is (2.19) We see that ∆[O s ] > 2, as required by our proposed duality.
On the other hand, ∆[P ], ∆[P 2 ] < 2 so the two superpotential monopole deformations M + M † and M 2 + (M 2 ) † (which break the U (1) topological symmetry completely and to Z 2 , respectively) are relevant deformations in the IR CFT. 5

Relevant deformations and vacua
Having established that there is only one deformation that preserves N = 1 supersymmetry and the U (1) Z C 2 global symmetry (where we now proceed to study the different phases that one obtains when turning on such a deformation, in the two dual theories respectively.We find that for m > 0 there are two isolated gapped vacua (corresponding to the broken IR time-reversal symmetry), while for m < 0 there is an S 1 Goldstone boson with a free fermion. 6s we will show below, it is convenient to keep track of counteterms for background fields, in particular for a gauge field B that couples to the U (1) global symmetry and for the metric.We use here the following notation for the various component fields.In the SQED theory the fields are a complex supermultiplet Q = (q, ψ, F Q ), the gauge field a and the gaugino λ.In the WZ model there is a real supermultiplet H = (h, η, F H ) and a complex supermultiplet P = (p, χ, F P ).We recall that when integrating out a Majorana fermion with negative mass we generate the gravitational coupling −CS g , and that U (1) 1 is equivalent to −2CS g .SQED side.We study the theory with superpotential We also couple to the topological symmetry via 1 2π adB.The F-term is This gives the following potential and fermionic interactions: It is useful to write fermionic interactions in a real notation. 7Defining Q = Q 1 + iQ 2 in terms of real superfields Q a , we find (2.25) Depending on the sign of m we find the following vacuum structure: • m > 0. There are two vacua.
One vacuum is at q = 0 where Q has mass m.Integrating it out, we get N = 1 U (1) 1 CS.Since the gaugino is a free fermion with negative mass, it can be integrated out and we get U (1) 1 CS.This is a trivial gapped vacuum.If we keep into account the background counterterms, we have − 1 4π BdB − 3CS g .The other vacuum is at |q| 2 = m.The gauge symmetry is Higgsed, both ψ and λ are massive, and we are left with a trivial gapped vacuum.More precisely, 7 Recall that Majorana fermions satisfy ψ c a = Cγ T 0 ψ * a = ψ a .For Majorana fermions, ψχ = χψ.
consider q 1 = √ m and q 2 = 0. We see that the radial superfield δQ 1 has mass −2m, the angular scalar q 2 participates in the Higgs mechanism, the angular fermion ψ 2 and λ mix.The mass matrix is , therefore one fermion has positive mass and one negative.We thus have −2CS g .
• m < 0. There is a vacuum at q = 0 where Q has mass −|m|.Integrating it out, we get N = 1 U (1) 0 SYM, i.e. a free massless fermion λ and an S 1 free compact boson.We also have −2CS g .
Wess-Zumino side.We study the deformed superpotential Recall that we also have an extra gravitational coupling −CS g .The superfield P has charge 1 under the background gauge field B. The F-terms are They give potential and fermionic interactions (2.28) Using the real notation P = P 1 + iP 2 , the fermionic interactions are with a = 1, 2. Depending on the sign of m we find the following vacua:8 • m > 0. There are two vacua at p = 0 and h = ∓ √ m, where P has mass ∓2 √ m.Also H is massive, with mass ±2 √ m around its VEV.Integrating them out we are left with two trivial gapped vacua.
Notice that the VEV for H breaks time-reversal symmetry, and the two vacua are related by that symmetry.Taking into account background counterterms, the vacuum with upper sign has − 1 4π BdB − 3CS g , the vacuum with lower sign has −2CS g .
• m < 0. There are vacua at |p| 2 = |m| and h = 0.The global U (1) symmetry that rotates P is spontaneously broken: we get a massless fermion and an S 1 free compact boson.
Using the real notation, we see that H mixes with the radial part of P around its VEV, giving two modes of opposite mass ±2 |m|.When we integrate them out we are left with −2CS g .
We see that the various phases perfectly match, including the counterterms for background fields.

Other Abelian dualities
We can produce other Abelian dualities employing a gauging procedure.On both sides of the duality we add a CS term at level = 0, 1 or −1 for the background gauge field B, we couple it to a new background field C and then make B dynamical (we rename it b).In order to preserve N = 1 supersymmetry, we should also introduce a real fermion γ suitably coupled to the theory.
On the SQED side we have the Lagrangian terms Let us analyze the three cases in turn.For = 0, the gauge field b can be integrated out setting a = C. Integrating out γ sets λ = 0. We are left with an N = 2 chiral multiplet Q with charge 1 under C and a background counterterm 1 4π CdC (we neglect here gravitational couplings).In this theory the quartic superpotential interaction is marginally irrelevant and can be dropped.
For = 1, the gauge field b can be integrated out, together with a massive fermionic eigenmode.We are left with CdC . (2.31) The new theory has gauge group U (1) −1/2 .For = −1, the gauge field b can be integrated out, leaving a bare CS level 2, together with a massive fermionic eigenmode.We are left with CdC . (2.32) The new theory has gauge group U (1) 3/2 .On the WZ side we have the Lagrangian terms This is an Abelian gauge theory coupled to P and H.
We thus find three new dualities.
• The case = 0 leads to Free complex field Q with 1 flavor P and a singlet H (2.34) On the LHS we dropped the quartic superpotential since it is marginally irrelevant.On the RHS, instead, we have used that in this particular case the superpotential deformation H 2 is relevant while H 3 becomes irrelevant.Indeed, the RHS is the N = 2 SQED theory with one flavor.In this theory we know that H 2 is turned on at the fixed point, and we know that H 3 is irrelevant there.The gauging procedure allows then to go back from the N = 1 duality to the N = 2 duality.
• In the case = 1 we find (2.35) The theory on the left is not the time-reversal of (2.9) because the CS level has opposite sign, but not the superpotential term.The duality suggests that the theory on the LHS has an N = 1 fixed point, besides the N = 2 fixed point.
Even though both the N = 2 (with W = HP P † − 1 2 H 2 ) and the N = 1 fixed point (with W = HP P † − 1 3 H 3 ) are reachable from the same free UV model, let us emphasize that there are no N = 1 RG flows going from one IR fixed point to the other one.
The phases of the two theories can be analyzed as before.For m > 0 there is a gapped vacuum and an S 1 worth of vacua (parametrized by a Goldstone boson and with a free fermion).For m < 0 there is a unique gapped vacuum.
• In the case = −1 we find (2.36) The phases are as follows.For m > 0 there is a trivial gapped vacuum and a gapped vacuum with a topological theory U (1) 2 ∼ = U (1) −2 .For m < 0 there is a trivial gapped vacuum.

Non-Abelian dualities
We propose that the Abelian N = 1 dualities of the previous Section generalize to non-Abelian dualities.The four families of non-Abelian dualities are related by the gauging procedure described in Section 2.5.

SU/U duality
The first family, that we call SU/U and generalizes (2.34), is as follows.Theory A is while Theory B is with 1 flavor P and a real singlet We consider this duality in the range N ≥ 2, k ≥ 1 (outside this range, it might still be possible to make sense of the duality along the lines of [16,17]).The global symmetry is where the second factor is charge conjugation.In Theory B we can perform a field redefinition H → H + const.to remove a possible term P † P from the superpotential; we always assume we have removed such a term.Next, the absence of a superpotential term H 2 is justified as follows.We start from the Wess-Zumino model with k complex and 1 real superfield, W = HP † P − 1 3 H 3 .In Appendix A we show that for any k > 0, the singlet quadratic operator of the form Gauging the global U (k) symmetry with a large enough Chern-Simons level will not spoil the relation ∆[O s ] > 2. So, for large enough Chern-Simons level, the correct superpotential does not contain the term H 2 .Finally, we have performed a tuning to zero of the mass term Q † Q in Theory A and of the linear term H in Theory B. Those represent the only N = 1 relevant deformation, identified on the two sides of the duality: We study the resulting phases below.The case N = 1 is special, because Theory A is a free N = 2 chiral multiplet Q.In this case, the duality suggests that in Theory B-because of the particularly small CS level-the superpotential term H 2 is relevant and present in the theory.We propose that Theory B is with 1 flavor P and a real singlet H while Theory A is as in (3.1) with N = 1 and k ≥ 1.For k = 1 this is the gauge theory with one chiral field discussed in Section 2.1 and in (2.34).For k > 1 the theories are N = 1.
Relevant deformations and vacua.Let us compare the behavior of the two theories under the relevant deformation.
In Theory A (for N ≥ 2) we take the deformed superpotential The F-term is F Q = q m − |q| 2 .One finds the following vacuum structure: • For m > 0 there are two vacua.
One vacuum is at q = 0 where Q has mass m.Integrating it out we get the topological theory We have indicated both the N = 1 and the standard N = 0 notation (see [56] and Appendix B).
The other vacuum is at |q| 2 = m, where the gauge symmetry is broken and the radial mode is massive.The breaking SU (N ) → SU (N − 1) eats 2N − 1 real bosonic modes, while the real radial mode is massive, therefore all modes of q are massive.Taking q = √ m , 0, . . ., 0 and analyzing the quadratic fermionic action, one finds that the modes ψ 1 and λ 11 acquire a mass, which does not affect the CS level of the unbroken group.The modes ψ a and λ 1a (a = 1) give two modes of opposite mass, therefore the bare CS level of the unbroken gauge group is shifted by −1 (such a bare level was k + N ) while the effective CS level is not shifted.Therefore we are left with • For m < 0 there is a single vacuum at q = 0 where Q has mass −|m|.Integrating it out we get For k = 1 this is a trivial gapped vacuum.
In Theory B (for N ≥ 2) we take the deformed superpotential The F-terms are F H = m + |p| 2 − h 2 and F P = 2hp.One finds the following vacuum structure: • For m > 0 there are two vacua at p = 0 and h = ∓ √ m, where P has mass ∓2 √ m.Also H is massive, with mass ±2 √ m around its VEV.Integrating them out, in the vacuum with upper sign we get while in the vacuum with lower sign we get • For m < 0 there is a vacuum at |p| 2 = |m| and h = 0, where the gauge symmetry is broken and the radial mode of p is massive.The singlet H mixes with the radial part of P around its VEV, giving two modes of opposite masses ±2 |m|.
Since χ a and λ 1a (a = 1) give two modes of opposite mass, we are left with 9 The shift in the level k is only apparent: if we write U (n) k,k+mn we see that m is not shifted.
In all cases we find a perfect match between the two descriptions.
For N = 1, Theory A is a free N = 2 chiral multiplet.Under both positive and negative mass deformation, it gives a trivial gapped vacuum.The analysis of deformations of Theory B in (3.4) requires to keep into account the one-loop effects, as we did in Section 2.1.For m > 0, h gets negative VEV, P acquires negative mass and integrating it out one gets a shift δW = 1 2 H 2 .This leads to U (k) −1 CS theory, which has a trivial gapped vacuum.For m < 0, we only find the Higgsed vacuum |p| 2 = |m| leading to U (k − 1) −1 CS theory with a trivial gapped vacuum.The classical vacuum where h gets a positive VEV is lifted quantum mechanically, because of the one-loop Let us mention that, if we consider a theory as in (3.1) but with opposite sign of the superpotential, i.e.W = 1 4 Q † QQ † Q, then its vacuum structure is reproduced by a theory as in (3.2) but with superpotential W = HP † P + α 2 H 2 (with large positive α) i.e. with quadratic rather than cubic term in H. Then H could be integrated out leading to a theory with no singlets and superpotential W = − 1 2α P † P P † P .

U/SU duality
The second family, that we call U/SU and generalizes (2.9), is with 1 P and H W = HP † P − 1 3 H 3 . (3.7) We consider this duality in the range N, k ≥ 1.The case k = 1 is special because the RHS becomes the WZ model that we have studied in Section 2.
Relevant deformations and vacua.Theory A (on the LHS) has the following vacuum structure: • For m > 0 there are two vacua.In the vacuum at q = 0 the field Q has mass m.
Integrating it out gives the topological theory U (N ) k .
In the Higgsed vacuum at |q| 2 = m we are left with U (N − 1) k .
Notice that for k = 1 both gapped vacua are trivial, and for N = 1 the second gapped vacuum is trivial.
• For m < 0, in the vacuum at q = 0 the field Q has mass −|m|.Integrating it out gives N = 1 U (N ) k+ N 2 −1,k−1 CS theory.For k > 1 this is the topological U (N ) k−1 CS theory, while for k = 1 this is an S 1 free scalar together with a free fermion.
Theory B (on the RHS) has the following vacuum structure: • For m > 0 there are two vacua at h = ∓ √ m where P has mass ∓2 √ m and H is massive as well.Integrating them out, in the vacuum with upper sign we get SU (k) −N while in the vacuum with lower sign we get SU (k) −N +1 .
• For m < 0 there is a Higgsed vacuum at |p| 2 = |m|, leading to SU (k − 1) −N for k > 1.When k = 1, the global symmetry is broken and we get an S 1 Goldstone boson with a free fermion instead.
The two descriptions match.

U/U duality
The third and fourth families, that we call U/U and generalize (2.35) and (2.36), are We consider these dualities in the range N, k ≥ 1.
Relevant deformations and vacua.Theory A (on the LHS) has the following vacuum structure: • For m > 0 there are two vacua.In the vacuum at q = 0 the field Q has mass m.
Integrating it out gives the topological theory U (N ) k,k±N .
There are some special cases: when the second level is 0 we get an S 1 free scalar and a free fermion.
• For m < 0, in the vacuum at q = 0 the field Q has mass −|m|.Integrating it out gives Theory B (on the RHS) has the following vacuum structure: • For m > 0 there are two vacua at h = ∓ √ m where P has mass ∓2 √ m and H is massive as well.Integrating them out, in the vacuum with upper sign we get U (k) −N,−N ±k while in the vacuum with lower sign we get U (k) −N +1,−N +1∓k .
The two descriptions match.

Generalization to N f > 1
Our proposed dualities admit a natural generalization to the case with more than one flavor.Even if a detailed analysis of this case is beyond the scope of the present paper, let us state the conjecture and make a few comments.There are four families of dualities: the SU/U duality SU (N ) and the two U/U dualities U (N ) On both sides, the flavors are in the (complex) fundamental representation, while the N f singlets are real.The global symmetry is where the second factor is charge conjugation.The singlets H ij transform in the adjoint plus singlet representation of the global SU (N f ) symmetry factor, and the superpotentials are more complicated than in the N f = 1 case.On the LHS there are two possible terms (that we have schematically indicated by |Q| 4 ): We do not know the precise form of the superpotential nor the structure of the infrared fixed points of the two theories.The dualities are expected to hold when N c ≥ N f on both sides, so we need N ≥ N f and k ≥ N f .Outside this ranges there might be interesting quantum phases.We leave these issues to future work.superfields.One way around this obstacle is the following [44].This strategy should work for any Wess-Zumino model whose interacting Lagrangian is The fermions λ are two-component Majorana fermions.We replace each λ with a tower of 4p fields λ j (where p is an arbitrary integer) and modify the fermionic part of the previous Lagrangian as follows: We do not change the number of scalars nor the quartic scalar interactions.At this point we combine the 4p λ's into p complex Dirac four-component fermions.We obtain a Gross-Neveu-Yukawa model with global symmetry SU (p) that exists for any d ≤ 4. At d = 3 the global symmetry enhances to SO(4p).For this extended model we can use the existing results for loop computations present in the literature.They are the beta-functions of the quartic and Yukawa couplings, and also the scaling dimensions of the fields φ I , λ I and φ I φ J .At some point in the computation we can set p = 1/4 and infer the results for our 3d real fermions, hence for our cubic Wess-Zumino model of interest.This strategy was implemented in [44].
Our interest in this paper lies in the following cubic Wess-Zumino model, 10 with two independent couplings: The global symmetries are O(K) and parity, which forbid other cubic terms Φ 3 i>0 and Φ i>0 Φ 2 0 to be generated.Under parity the Φ i 's are even and Φ 0 is odd.Notice also that only the relative sign between g 2 and g 3 is physical; we will find that g 2 and g 3 have opposite sign at the RG fixed point.
We compute the beta-functions for Yukawa couplings (and hence for our SUSY couplings) using eqn.(7.2) of [58].In terms of the two SUSY couplings g 2 , g 3 , the 10 A similar model, but with N = 2 supersymmetry, was studied in [57].It was found that the coupling g 3 flows to 0 at the IR fixed point, leading to accidental IR symmetries.In our case with N = 1 supersymmetry we find that both g 2 and g 3 are non-vanishing in the IR fixed point.Even starting from g 3 = 0 in the UV, the term Φ 3 0 would be generated in the IR.
beta-functions and scaling dimensions at one-loop are (A.7) We compute the scaling dimension of the fundamental fields in the Wess-Zumino model using eqn.(A.3) of [44].At one-loop they are: From eqns.(A.4) and (A.5) of [44] we can obtain the mixing matrix for the operators quadratic in the fundamental fields.The (K+1)(K+2) 2 quadratic operators transform in the vector, symmetric traceless (O s.t. ) and two singlets representations of SO(K).
The operators Φ 0 Φ i transform as a vector of SO(K), so they do not mix with the other quadratic operators.At one-loop their scaling dimension is given by ∆[Φ 0 Φ i ] = 2 − ε + g 2 3 + (K + 10)g 2 2 + 4g 2 g 3 2(4π) 2  .(A.9) Notice that, upon using β g 2 = 0, the relation ∆[Φ 0 Φ i ] = ∆[Φ i ] + 1 is satisfied.This is consistent with the fact that Φ 0 Φ i is a SUSY descendant of Φ i at the fixed point.We write down the mixing matrix for the K + 1 operators Φ 2 0 , Φ 2 1 , Φ 2 2 , . .., at oneloop: From its eigenvalues we can read off the scaling dimensions of the singlet operator O s Φ 2 0 + Φ 2 i (at one-loop), the singlet g 2 Φ 2 0 + g 3 Φ 2 i (which is a SUSY descendant of Φ 0 ), and the symmetric traceless operator O s.t. .We report the one-loop results in Table A.
Let us make a few comments about the different values of K.

2. 3 = g 2 2 HP P † + g 3 6 H 3 . ( 2 . 15 )
IR irrelevance of the"mass term" O s = |P | 2 + H 2 in the ε-expansion We consider the cubic Wess-Zumino model with N = 1 supersymmetry and superpotential W Working in the 4 − expansion, we have computed (see Appendix A) the two-loop beta-functions for the model.The numerical values of the couplings at the physically sensible fixed point are
in the IR an irrelevant superpotential deformation, since it already appears in the Lagrangian.So, if the duality is correct, it must be the case that on the RHS the operator O s = P P † + H 2 satisifes ∆[O s ] ≥ 2 and is an irrelevant superpotential deformation-even though ∆[O s ] U V = 1 in the free UV theory.This feature is very similar to what happens in the N = 2 duality studied above.In order to test this proposal, we compute ∆[O s ] in the ε-expansion in the next subsection.