Duality and Confinement in 3d $\mathcal{N}=2$"chiral"$SU(N)$ gauge theories

We study low-energy dynamics of three-dimensional $\mathcal{N}=2$ $SU(N)$"chiral"gauge theories with $F$ fundamental and $\bar{F}$ anti-fundamental matters without a Chern-Simons term. Compared to a naive semi-classical analysis of the Coulomb branch, its quantum structure is highly richer than expected due to so-called"dressed"Coulomb branch (monopole) operators. We propose dualities and confinement phases for the"chiral"$SU(N)$ theories. The theories with $N>F>\bar{F}$ exhibit spontaneous supersymmetry breaking. The very many Coulomb branch operators generally remain exactly massless and are non-trivially mapped under the dualities. Some dualities lead to a novel duality between $SU(N)$ and $USp(2 \tilde{N})$ theories. For the 3d $\mathcal{N}=2$ $SU(2)$ gauge theory with $2F$ doublets, there are generally $F+2$"chiral"and"non-chiral"dual descriptions.


Introduction
Asymptotically-free gauge theories exhibit various phases depending on gauge groups, matter contents, space-time dimensions and so on. It is generally difficult to analytically study the low-energy dynamics because those theories are often strongly-coupled and non-perturbative. Duality is a very powerful tool for investigating the dynamics which is not accessible by a perturbation theory [1]. Duality gives us other descriptions of the theories, which are more tractable than the original theories. Supersymmetry is also very powerful and allows us to exactly obtain the non-perturbative information [2].
Recently, a deeper understanding of the three-dimensional N = 2 supersymmetric gauge theories was gained a lot. In particular, the 3d Seiberg dualities were derived from the corresponding 4d dualities [3,4]. In this derivation, it was important to correctly understand the Coulomb branch (monopole) operators and the non-perturbative superpotential coming from a circle compactification of the 4d theories. The analysis of the Coulomb branch was developed also for theories with various matters and various gauge groups [5][6][7][8][9][10]. For 3d "chiral" theories with unequal numbers of (anti-)fundamental matters, the dualities were discussed in the U(N) cases [11]. For the SU(N) cases, a completely chiral theory with no anti-fundamental matter was discussed in [3]. In [12], various chiral theories were extensively studied for the U(1) and SU(2) 1 cases with and without a Chern-Simons coupling.
However, the quantum structure of the Coulomb branch for the SU(N) "chiral" gauge theory with F fundamental andF (< F ) anti-fundamental matters 2 has been less understood until now. Although the "chiral" SU(N) theories were also studied in [12], its analysis was semi-classical and discussed only a very limited class of the Coulomb branch which corresponds to the breaking SU(N) → SU(N − 2) × U(1) × U (1). Since there are so many other classical Coulomb branches, this analysis [12] would be incomplete and miss some important quantum aspects of the "chiral" theories. For instance, [12] claims that there is no s-confining phase for the 3d "chiral" SU(N) theories since we cannot contract the flavor indices of the meson operator and then a confining superpotential is not available. [12] also claims that the supersymmetry will be broken for N >F without mentioning the allowed region of F . They concluded these two statements only from the semi-classical treatment of the Higgs branch and one particular Coulomb branch. By carefully studying various Coulomb branches, we will find that the first statement is incorrect and the second statement should be refined into N > F >F .
In this paper, we will consider the three-dimensional N = 2 SU(N) gauge theory with (F,F ) (anti-)fundamentals with F >F . We will claim that the very many classical Coulomb branches can survive quantum corrections and remain exactly massless in the "chiral" SU(N) theories. This situation is very different from the vector-like theory with F =F , where only a single Coulomb branch remains flat. Naively, one might consider that many Coulomb branches are not gauge invariant and they cannot be moduli coordinates due to the "chiral-ity". However, we can construct the so-called "dressed" monopole operators by appropriately combining the bare Coulomb branch and matter operators, which are gauge-invariant and quantum-mechanically flat. By taking into account these "dressed" operators, we will present various phases: For F > N +1, we will propose a dual description in terms of the SU(F −N) gauge theory. For F = N + 1, we will find a confinement phase where the low-energy degrees of freedom are governed by a cubic superpotential. For N > F >F , the theory exhibits a spontaneous supersymmetry breaking. In contrast to the "chiral" U(N) duality, the "chiral" SU(N) duality gives rise to an additional problem of matching the baryon operators under the duality. We will demonstrate that the anti-baryonic operator is mapped to one of the dressed Coulomb branch operators. By focusing on the duality with the particular matter contents, we will also propose a novel duality between the chiral SU(N) and USp(2Ñ ) theories.
The rest of the paper is organized as follows. In Section 2, we will introduce a set of the Coulomb branch coordinates which appear in the 3d N = 2 "chiral" SU(N) theory. We divide the argument into SU(2N) and SU(2N + 1) cases. In Section 3, we propose the dualities of the "chiral" SU(N) theory. Since the precise structure of the Coulomb branch depends on F,F and N, we will show dualities case-by-case in the subsequent subsections. The first two cases lead to the novel dualities between SU(N) and USp(2Ñ ) theories. In Section 4, we study the confinement phases in the 3d "chiral" SU(N) theories. By deforming the confinement phases, we can find supersymmetry-breaking phases and quantum-deformed moduli spaces for particular matter contents. In Section 5, we will summarize our findings and discuss the future directions. In Appendix, we will show an example of the various chiral and non-chiral dualities for the 3d SU(2) gauge theory with 2F fundamental matters, especially focusing on the F = 3 case.
2 Coulomb branch in "chiral" SU (N ) theories In this section, we will introduce a "typical" set of the Coulomb branch coordinates for the 3d N = 2 SU(N) gauge theories with chiral matter contents. By carefully studying the gauge invariance of these coordinates and by taking into account induced Chern-Simons couplings for unbroken gauge groups, we can find which coordinates remain exactly massless at a quantum level. Note that in this section we will only consider the "typical" Coulomb branch in the 3d chiral SU(N) theories. Then, there might be additional Coulomb brach operators which could appear in case of particular matter contents and the ranks of the gauge group. In some cases, the Coulomb branch operators which will be introduced below are not minimal-monopole-creating operators and we must take n-th root of these operators in those cases. However, the understanding of these "typical" Coulomb branches will be sufficient and strong enough to understand various confinement phases and dualities.

SU (2N ) with (F,F )
We start with the Coulomb branch of the 3d N = 2 SU(2N) gauge theories with (F,F ) (anti-)fundamentals. The matter contents and their representations are summarized in Table 1 below. Notice that, due to a parity anomaly, F ±F should be even since we do not introduce a tree-level Chern-Simons term to the gauge interaction.
The Coulomb branch (or Coulomb moduli space) is a flat direction of the adjoint scalar fields which come from the SU(2N) vector multiplet. Typically, we will consider the following Coulomb branch which breaks the gauge group as where a = 1, · · · , N − 1. For a = 0, the breaking pattern is slightly modified to There are N types of the Coulomb branches which are related with U(1) 1 . The corresponding operators are denoted as Y bare a . For the theory with (F,F ) (anti-)fundamentals, along these directions, the effective Chern-Simons terms are introduced. The (anti-)fundamental fields are massive and integrated out, which results in an Chern-Simons theory. If we have an SU(M) k CS theory with 0 ≤ |k| < M, there is no SUSY vacuum [12]. Therefore, we require N − a ≤ F −F 2 , which means a must satisfy Notice that the effective Chern-Simons term is not generated for U(1) 1 since the low-energy U(1) 1 theory is vector-like. That is why these Coulomb branches can be flat directions.
These Coulomb branch operators are generally not gauge invariant and charged under the U(1) 2 since the theory is "chiral". Therefore, we have to multiply it by the chiral superfields (·, ·, ) 0, N−a a ∈Q which belong to an SU(2a) anti-fundamental representation. Then, we have to construct the baryonic operators from them in order to make an SU(2a) singlet, which requires 2a ≤F for a = 0. (2.10) By collecting all the conditions for a, we find (2.12) The charges of these operators can be computed from the mixed Chern-Simons terms as in Table 2 below.
Finally, we must be careful of minimal monopole operators. Here we constructed the naive monopole operators and there might be more smaller operators whose magnetic charges are smaller than here. Since the SU(N − a) × SU(N − a) gauge symmetry is unbroken in addition to U(1) 1 , the minimal U(1) 1 magnetic charge could be more smaller. In some cases, the corresponding (minimal) magnetic charge becomes 1 N −a and the minimal bare operators are the (N − a)-th root of the naive ones. However, for those minimal operators, we cannot make them gauge-invariant by usingQ 2a in many cases. If this is possible, we have to construct more minimal operators from the naive ones by taking a root. Thus, our analysis below will become case-by-case depending on the matter contents and the rank of the gauge group.
Along the Coulomb branch, the matter fields are massive and integrated out. This introduces the Chern-Simons terms for the two SU(N − a) groups. The resulting low-energy theory is an Chern-Simons theory. Since this theory must have a supersymmetric vacuum, it requires [12] N − a ≤ F −F 2 . (2.18) Furthermore, since we need to construct a baryonic operatorQ 2a+1 in order to dress the bare monopoles, there is a further constraint, 2a + 1 ≤F . By collecting all the constraints for a, we find The Coulomb branch Y dressed a exists only for this range of a. In this subsection, we focused on the general structure of the Coulomb branch. When we will study a specific example in the next section, there are more-smaller monopole operators available in some cases. For instance, we can consider a monopole operator which has a smaller magnetic charge and is still not gauge invariant. In many cases, we cannot construct a gauge invariant composite from Y minimal bare a andQ 2a+1 . However, in some cases (special choices of N, F,F and a), we can do it. Because the correct choice of the bare monopoles depend on the matter contents, we have to consider the dualities case-by-case.

Dualities
Here, we propose a new duality for the 3d N = 2 SU(N) gauge theory with F fundamental matters Q andF anti-fundamental mattersQ. The following discussion assumes F >F without loss of generality. The proposed dual description is a 3d N = 2 SU(F − N) gauge theories with F fundamental (dual) matters q,F anti-fundamental (dual) mattersq and a meson singlet M with a following superpotential W = Mqq. (3.1) In order to illustrate how the duality works, we restrict ourself to the case where the anti-baryonic operators cannot be constructed on the electric side. Furthermore, in order to explain a generic structure of the duality, let us focus on the 3d N = 2 SU(2N) with (2F, 2F ) (anti-)fundamentals. One can easily generalize our analysis to more generic cases. The proposed dual description is a 3d N = 2 SU(2F −2N) with (2F, 2F ) (anti-)fundamentals and a singlet M with a superpotential (3.1). The matter contents and their quantum numbers are summarized in Table 5.
The charge assignment of the fields in the dual description can be fixed from the superpotential and the matching of the baryon operator. The mesonic branch labeled by M :=QQ is mapped to a singlet M. The baryonic operator B := Q 2N is mapped to the baryon b := q 2F −2N constructed from the dual quarks q. WhenF ≥ F − N, we can construct the anti-baryonic operator on the magnetic side. One might consider that there is no counterpart of the anti-baryon on the electric side. However, as we will see in the next subsection, the anti-baryon corresponds to one of the Coulomb branch operators on the electric side.
The electric and magnetic theories are both UV-complete because the duality does not include a monopole superpotential. In addition, the proposed duality is very similar to the conventional 4d Seiberg duality [1]. Therefore, this 3d duality can straightforwardly pass several simple tests. By applying the duality twice, we can go back to the electric theory. The complex mass deformation on the electric side is mapped to the Higgsing of the dual gauge group and the duality is correctly preserved with reduction of F andF . The parity anomaly matching is also satisfied. The generic real mass deformation is now complicated and let us focus on one particular simple case. By weakly gauging the global U(1) symmetries, all the electric matters can obtain the positive real masses and the electric theory flows to the SU(2N) F +F SUSY CS theory. This flow is only possible for F +F ≥ 2N otherwise the supersymmetry is broken and we have to find a more complicated flow. On the magnetic side, the fundamental matters obtain positive real masses while the anti-fundamental matters obtain negative masses. The theory flows to SU(2F −2N) F −F SUSY CS theory. This flow is possible for F −F ≥ 2F −2N. Therefore, only forF = 2N −F , these flows are simultaneously realized. In this case, the electric side becomes the SU(2N) 2N CS theory while the magnetic side becomes the SU(2F − 2N) 2F −2N CS theory. The both sides have the same Witten index [12,13] which is one. The most non-trivial part of this duality is the matching of the moduli operators and hence we focus on it in what follows.
Since the correct Coulomb branch operators will change case-by-case depending on N, F,F and a, we here only give a general rule of matching the Coulomb branch operators on both sides. The more precise matching (including the anti-baryon) will be shown in the following subsections. As we explained in the previous section, the Coulomb branch of the "chiral" theory is labeled by Y a . The charges of these operator on both sides are summarized in Table 6 below. Notice that the dressed operators are generally charged under the non-abelian flavor symmetry, but we omitted it for simplicity.
Let us consider the matching of the dressed monopole operators. From Table 6 above, the identification becomes The index a e on the electric side runs over while the magnetic a m runs over Then, in what follows, we will consider the specific cases of the duality for the SU(N) "chiral" theories.

SU (2N ) with (2N + 2, 2F )
As a first example of our duality, let us start with the case where the dual gauge group becomes SU (2). This case is simplest since the bare monopole operator of the magnetic SU(2) theory is gauge invariant without "dressing". The electric theory is a 3d N = 2 SU(2N) gauge theory with (2N + 2, 2F ) (anti-)fundamental matters. The magnetic description is a 3d N = 2 SU(2) gauge theory with 2N + 2 + 2F fundamentals (q,q) and a singlet M. The magnetic theory contains the superpotential W = Mqq. Table 7 shows the quantum numbers of the matter contents.
The bare monopole operators Y bare a are allowed for a =F ,F − 1. These two operators are not gauge invariant and require "dressing" byQ 2a . For a =F − 1, the gauge invariant combination can be constructed fromQ 2a and the minimal bare monopole instead of the naive one Y barē F −1 . Then, the dressed Coulomb branch will be described by this smaller monopole. For a =F , the minimal monopole cannot be made gauge invariant bỹ Q 2a and we must use Y bare a=F −1 for constructing the gauge invariant operators. The dressed coordinates become In Table 8, the U(1) 2 charges of the Coulomb branch operators are summarized. We can see The operator mapping is manifest from Table 7. Notice that the SU(2) magnetic theory allows only one Coulomb branch operator [14] and an additional Coulomb branch comes from the anti-baryonic operatorB :=q 2 . The identification becomes (3.8) ForF = N, the anti-baryon operatorb :=Q 2N is available while the Coulomb branch operator YF =N is not defined. The anti-baryonb is identified with the SU(2) monopole operatorỸ SU (2) in this case. In this way, the duality works even forF = N.
Since the dual gauge group SU(2) can be regarded as a member of the USp(2n) series, one can apply the Aharony duality [15] to the magnetic side. The Aharony dual is given by a 3d N = 2 USp(2(N +F − 1)) gauge theory. The matter contents are summarized in Table 9.
Since the USp(2(N +F − 1)) theory has only the fundamental matters, the corresponding Coulomb branch is simple and one-dimensional, which is denoted by Y U Sp . Table 9: The superpotential takes Since the mesonic fields M and N are massive, they can be integrated out. The F-term conditions lead to N = 0 and M ∼ bb. The matching of the flat directions with the SU(2N) theory is manifest from Table 7 and Table 9. The unnecessary flat directions are all lifted by the above superpotential.
3.2 SU (2N + 1) with (2N + 3, 2F + 1) For completeness, we study the SU(2N + 1) duality whose dual is again given by an SU(2) gauge theory. The electric theory is a 3d N = 2 SU(2N + 1) gauge theory with 2N + 3 fundamentals and 2F + 1 anti-fundamentals without superpotential. We assumeF ≤ N in what follows. The Coulomb branch operators Y bare a are available only for a =F andF − 1. These are not gauge invariant and need "dressing". For a =F − 1, the more minimal operator (Y bare a=F −1 ) 1/(N −F +1) can be dressed byQ 2a+1 while the U(1) 2 charge of (Y bare a=F ) 1/(N −F ) cannot be dressed byQ 2a+1 . Therefore, we need to introduce the following Coulomb branch operators where the flavor index ofQ 2a+1 is anti-symmetric as it should be. The matter contents and their charges are summarized in Table 10 below.
On the dual side, the gauge group is SU(2) and there are 2N + 2F + 4 fundamental matters (no difference between 2 and2). The dual theory has a superpotential W = Mqq. (3.13) We need not introduce any dressed operators and there is a single Coulomb branch operator Y SU (2) [14]. The operator matching again shows that one of the Coulomb branch operators is transformed into the anti-baryonic operatorB constructed from the dual quarks q.

USp(2(N +F )) dual
Since the dual gauge group is an SU(2), we can obtain additional dual description by adopting the Aharony duality [15]. The USp(2(N +F )) dual theory contains 2N + 2F + 4 fundamental matters and singlets M, N, B,B and Y U Sp . Table 11 shows their quantum numbers.
The superpotential takes  In this case, the Coulomb branch is highly simplified and only Y bare a=0 can survive because we cannot anti-symmetrize the anti-fundamental matters. Table 12 shows the matter contents and their quantum numbers. The Coulomb brach Y bare a=0 is gauge invariant and does not need "dressing".
The dual description is a 3d N = 2 SU(2F − 2N + 1) gauge theory with (2F + 1, 1) (anti-)fundamental (dual) matters and a singlet M with the superpotential The charges of the dual fields can be fixed from the superpotential and from the matching of the baryon operator as Table 12. The magnetic Coulomb branch is also allowed only for Y a=0 . Since the dual gauge group is SU(2F − 2N + 1),Ỹ a=0 need "dressing" in contrast to the electric side. The matching of the moduli fields are transparent from Table 12.

SU (2N ) with (2F, 2)
The next simple example is a 3d N = 2 SU(2N) gauge theory with (2F, 2) (anti-)fundamental matters. The dual description is given by a 3d N = 2 SU(2F − 2N) with (2F, 2) (anti-)fundamentals and a singlet M. The dual theory includes the superpotential The matter contents and their quantum numbers are summarized in Table 13 below.
On the electric side, the Coulomb branch is allowed for Y bare a (a = 0, 1). Since the U(1) 2 is absent for a = 0, Y bare a=0 is gauge invariant. For a = 1, Y bare a=1 is not gauge invariant and needs "dressing". The indices of the dressing factorQ 2 is anti-symmetrized and the dressed operator Y dressed   The matter contents and their representations are summarized in Table 14.
The Coulomb branch operators are allowed only for a = 0 on both sides. The electric Coulomb branch operator Y bare 0 is not gauge invariant and must be dressed byQ. On the other hand, the magnetic Coulomb branchỸ bare 0 is gauge invariant and can be used for a moduli coordinate. From Table 14, we can see the exact matching these two operator under the duality.

SU (2N + 1) with (2F + 2, 2)
The final example is a 3d N = 2 SU(2N + 1) gauge theory with 2F + 2 fundamental and 2 anti-fundamental matters. The proposed dual description is given by a 3d N = 2 SU(2F − 2N + 1) gauge theory with 2F + 2 fundamental and 2 anti-fundamental (dual) matters in addition to a singlet M. The dual theory has a cubic interaction The matter contents and their quantum numbers are summarized in Table 15. Since we consider the region with F > N, anti-baryon operators are not available on both sides.
The electric Coulomb branch is allowed only for Y bare a=0 , which corresponds to the breaking

Confinement and SUSY breaking
In this section, we will consider the confinement phases of the 3d N = 2 "chiral" SU(N) gauge theory with (F,F ) (anti-)fundamentals. (We again assume F >F .) The appearance of the confinement phase can be regarded as a special limit of our duality proposed in the previous section. We will claim that (s-)confinement phases are more ubiquitous than expected from a naive semi-classical analysis of the moduli space of vacua since the dressed monopole operators would not be flavor singlets and then a confining superpotential is available by properly contracting the flavor indices of the meson, baryon and monopole operators. In earlier works [12], these phases were overlooked due to lack of understanding of the quantum Coulomb branches. In [14], it was discussed that the 3d N = 2 SU(N) gauge theory with (F,F ) = (N, N) vector-like matters exhibits s-confinement. Then one can naively guess that the 3d N = 2 SU(N) gauge theory with (F,F ) = (N + 1, N − 1) also shows a similar confinement since this theory differs very little from the former theory and since the total number of the (anti-)fundamental matters does not change. We here claim that this guess is indeed correct.
Let us consider the Coulomb branch labeled by Y = Y 1 · · · Y N −1 , where Y i corresponds to the fundamental monopoles. The expectation value for Y breaks the gauge group to SU(N − 2) × U(1) 1 × U(1) 2 and Y is related to a U(1) 1 subgroup. Due to the "chiral" nature of the theory, the effective Chern-Simons level is induced for k along the RG flow. Therefore, Y is no longer gauge-invariant and charged under the U(1) 2 symmetry. The effective CS term is calculated as and then the U(1) 2 charge of Y is −2 in the current case. In order to construct the gauge invariant Coulomb branch, we can appropriately combine Y andQ into Y d := YQ N −2 . As a result, the Coulomb branch is not a flavor singlet. The quantum numbers of the moduli fields are summarized in Table 16.  Table 16, we see that the low-energy effective degrees of freedom are described by three singlets M, B and Y d and that the effective superpotential becomes One can check that the parity anomaly matching is satisfied by this confining description. The another check of this phase is that we can derive the same IR description by using the duality discussed in a previous section. Since the dual gauge group is absent, the dual (anti-)quarks q andq are gauge invariant, which are identified with B and Y d respectively. The dual superpotential takes the same form as (4.2). This superpotential can be also derived from the 4d s-confinement phase. Let us consider the 4d N = 1 SU(N) gauge theory with N + 1 flavors [2], which is s-confining and described by the superpotential By putting the theory on a circle and introducing the real masses for the SU(2) subgroup of the second SU(N + 1) flavor symmetry, the electric theory flows to the "chiral" SU(N) theory in Table 16. The effects of the twisted instanton are turned off in this deformation [3]. On the confined side, the real masses are introduced to the meson and the anti-baryons. By identifying the anti-baryon as the dressed Coulomb branch and integrating out the massive components, the superpotential is correctly reduced to (4.2).
In what follows, we will deform this confining phase by introducing the complex masses and flow to the 3d N = 2 SU(N) gauge theory with (F,F ) = (N − a, N − a − 2), where a ≥ 0.
By introducing a complex mass to one pair of (anti-)fundamental matters, we can flow to a 3d N = 2 SU(N) gauge theory with (F,F ) = (N, N − 2). The quantum numbers of the moduli coordinates are summarized in Table 17. In this case, the flavor indices ofQ N −2 inside the dressed monopole Y d is completely anti-symmetrized and Y d is now a flavor singlet.  while M has no constraint and will become a free field. This phase can be also derived from the previous subsection via a complex mass deformation ∆W = mM N +1,N −1 .
In this case, there is no Coulomb branch operator since we cannot construct the baryonic op-eratorQ N −2 . By introducing the complex masses for the previous case, one finds no solution for the F-flatness condition of M. Therefore, the supersymmetry would be spontaneously broken for a 3d N = 2 SU(N) gauge theory with (F,F ) = (N − a, N − a − 2), a ≥ 1. (2N ) with (F,F ) = (2N + 1, 2F + 1)

SU
We can generalize the confinement phase observed in the previous subsection. The confinement phases appear by taking a special case of the dualities in Section 3. Let us start with the SU(2N) cases. The theory is a 3d N = 2 SU(2N) gauge theory with 2N + 1 fundamentals and 2F + 1 anti-fundamentals, where we assumeF < N.
In this case, almost all the Coulomb branches are lifted and only Y a=F , corresponding to the breaking can survive and remain exactly massless. The bare monopole Y barē F is not gauge invariant and should be dressed byQ 2F . Now, we have to be careful about the choice of the correct monopole operator since the U(1) 2 charge of the minimal monopole is −(2N − 2F ) and this can be dressed byQ 2a=2F . Therefore, the Coulomb branch will be described by the minimal dressed monopole instead of the naive one. (4.7) The matter contents and the quantum numbers of the moduli coordinates are summarized in Table 18. From a dual theory point of view, M :=QQ is regarded as a singlet M. The baryon operator B := Q 2N is identified with a dual quark q which is now a gauge-singlet.
The Coulomb branch operator Y minimal dressed F corresponds to the dual anti-quarkq.
The theory is dual to a non-gauge theory with three chiral superfields M, B and Y minimal dressed F with a cubic superpotential This s-confining phase generalizes the result of the previous subsection. By introducing a complex mass, it is shown that a 3d N = 2 SU(2N) gauge theory with (2N, 2F ) has a single quantum constraint between the baryon and monopole operators BY minimal dressed F = 1 and there is a free meson M, which is the same as (4.4). By further intoducing the complex masses to more flavors, we find that a 3d N = 2 SU(2N) gauge theory with (2N − a, 2F − a) (anti-)fundamentals spontaneously breaks the supersymmetry.

SU (2N + 1) with (2N + 2, 2F )
Finally, we consider the confinement phase for the SU(2N + 1) chiral gauge theory. This case also comes from the SU(2N + 1) duality by taking the special limit where the dual gauge group becomes null. The electric theory is a 3d N = 2 SU(2N + 1) gauge theory with 2N + 2 fundamental and 2F anti-fundamental matters. Since we are interested in a "chiral" theory, we requireF ≤ N without loss of generality and the following argument is applicable only to this region.
The Coulomb branch Y a=F −1 , corresponding to the breaking can remain flat after including the quantum corrections. Since Y bare a=F −1 is not gauge invariant, we have to dress the bare monopole operator byQ 2a+1 =Q 2F −1 . In this particular matter contents, we can dress the more smaller monopole by usingQ 2a+1 . The minimal monopole corresponds to  Table 19 below.
The low-energy dynamics is described by M, B and Y minimal dressed F −1 with a cubic superpotential (4.12) One can easily check that the parity anomaly matching is satisfied by this dual description. The another check is that one can flow to the SU(2N) s-confinement phase by giving an expectation value to M .
By introducing the complex masses to this s-confinement phase, we find that a 3d N = 2 SU(2N +1) gauge theory with (2N +1, 2F −1) (anti-)fundamentals has a quantum-deformed moduli space of the baryon and monopole operators BY minimal dressed F −1 = 1 with a single free field M. It is also found that the supersymmetry is spontaneously broken for an SU(2N + 1) gauge theory with (2N + 1 − a, 2F − 1 − a) (anti-)fundamentals, where a ≥ 1. By combining the results of the SU(2N) and SU(2N + 1) cases, we conclude that the 3d N = 2 SU(N) gauge theory with (F,F ) (anti-)fundamentals exhibits a spontaneous supersymmetry breaking for N > F >F .

Summary and Discussion
In this paper, we investigated the quantum structure of the Coulomb moduli space of vacua in the 3d N = 2 SU(N) "chiral" gauge theories with F fundamental andF anti-fundamental matters. In case of a vector-like theory with F =F , almost all the flat directions of the classical Coulomb branches are lifted and only a single direction (Y bare N −1 in our notation) remains exactly massless. On the other hand, in the "chiral" SU(N) theories, the very many Coulomb branch directions could generally remain flat, which were described by Y bare a (a = 0, · · · , N − 1). These bare operators are not gauge invariant due to the mixed Chern-Simons term. Then, we constructed the correct gauge-invariant operators by dressing them with the anti-baryonic operatorsQ. Due to this dressing procedure, the Coulomb branch operators are no longer flavor singlets. The previous research [12] was missing this possibility. By using the "dressed" Coulomb branch operators, we proposed the duality for the 3d N = 2 SU(N) gauge theory with (F,F ) (anti-)fundamentals. The dual description is given by a 3d N = 2 SU(F −N) gauge theory with (F,F ) (anti-)fundamental matters and a meson M with a superpotential W = Mqq. Naively, one might think that the baryonic operators do not match on both sides of the duality. However, the dressed monopole operators remedy this and the duality works well. The total number of the (anti-)baryonic and Coulomb branch operators is identical on both sides and one can find the correct matching between the Coulomb branch and anti-baryonic operators. By focusing on the duality where the dual gauge group is SU(2) and by applying the Aharony duality [15], we found a novel duality between the SU(N) and USp(2Ñ ) theories. For the 3d N = 2 SU(2) gauge theory with 2F fundamentals, we can construct the various "chiral" dualities.
We also found the novel confinement phases of the "chiral" SU(N) gauge theory. The confinement phases appear for F = N + 1 and F >F , where the dual gauge group becomes null. The confined degrees of freedom are the meson, the baryon and the dressed Coulomb branch operator. The low-energy dynamics is governed by the cubic interaction of these three fields whose flavor indices are naturally contracted. By introducing the complex masses to these confining phases, we found that the 3d N = 2 SU(N) gauge theory with (N, N − a) (anti-)fundamentals have the free meson M and the quantum deformed moduli space, where a must be even. By further introducing the complex masses, the 3d N = 2 SU(N) gauge theory with (F,F ) (anti-)fundamentals exhibits a spontaneous supersymmetry breaking when N > F >F and F +F is even.
In this paper, we introduced the important class of the Coulomb branch operators Y bare a which typically appear in the "chiral" theories. However, we do not exhaust all the Coulomb branch operators. Generally, there could be other operators. For example, we did not consider the possibility where the dressed monopoles are constructed from the bare monopoles, anti-quarks and gauginos. Typically, they will take the following form where b and c are chosen as the above combination becomes gauge invariant. These operators were studied in case of the U(N) theory [16]. One might consider that the matching of these additional operators ruins the proposed duality. However, the matching of the proposed Coulomb branch operators would be highly strong and non-trivial enough to test the validity of the duality. It is worth studying when these additional operators appear or disappear on both sides of the duality. This would be an additional check of our duality. It would be very important to test our duality and confinement phases from the superconformal index computation [17][18][19][20]. We have partially done this in Appendix only for the SU(2) case. The additional Coulomb branch operators (5.1) which could appear for more higher rank gauge groups can be studied from the index. Probably, in some cases, the lowest Coulomb branch operator which has a minimal conformal weight could be different from the operators which we defined in this paper. In such cases, (5.1) will give the lowest contribution. It is also worth understanding our duality and confinement phases from the four-dimensional Seiberg duality point of view by following the same path as [3]. This could be achieved by introducing the real masses only for the anti-fundamental matters and flowing to the 3d limit. The duality of the "chiral" SU(N) Chern-Simons matter theories was discussed in [21]. The U(N) CS duality was studied in [11,22]. It is curious to find the RG flow from our duality to those Chern-Simons dualities. theory with six fundamental matters. The low-energy dynamics of this theory was studied in [14,23]. In [24] (see also [25,26]), the duality of the SU(2) with six doublets was proposed.
Here, we will consider the several "chiral" dual descriptions. One of them coincides with the duality in [24].
The Higgs branch is described by a meson composite M QQ := QQ while the Coulomb branch is parametrized by a single coordinate Y . When Y obtains a non-zero vacuum expectation value, the gauge group is broken as SU(2) → U(1). The matter contents and their quantum numbers are summarized in Table 20. The theory exhibits a manifest global SU(6) symmetry.
We can regard this theory as the 3d N = 2 SU(2) gauge theory with F fundamentals and 6 − F anti-fundamentals although the explicit SU(6) flavor symmetry is invisible. The Coulomb branch is completely the same as the previous one while the Higgs branch operator is decomposed into the meson, baryon and anti-baryon operators in Table 21. In the following subsections, we can construct various dual descriptions for each F and some of them are known in the literature [3,15,24].
First, we consider the dual of the description in Table 20 (or F = 6 in Table 21). Since the SU(2) group is a member of symplectic groups, we can use the Aharony duality [15] and obtain the USp(2Ñ) dual description. The dual gauge group is again SU(2) in this case.
The dual theory includes the meson and the Coulomb branch operator as elementary fields. Therefore, all the moduli coordinates are introduced as elementary fields on the dual side.  Table 22 shows the matter contents and their quantum numbers of the USp(2) dual theory.
The dual theory has a tree-level superpotential which is consistent with all the symmetries in Table 22. The equations of motion of the superpotential lift the magnetic meson and the dual Coulomb branch operators.
A.2 SU (2) "chiral" dual Next, we consider the dual of the description in Table 21 for F = 4. The low-energy dynamics of the "chiral" SU(N) theories and the dual description is given in this paper. The dual gauge group again becomes SU(2) and this case coincides with [24]. The dual theory contains four fundamentals, two anti-fundamentals and a meson singlet M. The dual theory has a tree-level superpotential The quantum numbers of the chiral superfields are summarized in Table 23. The matching of the moduli operators are as follows.
As we explained in Section 3, the role of the Coulomb branch and the anti-baryonic operator is exchanged. In this dual, only the meson is introduced as an elementary field on the magnetic side. Therefore, the duality has no problem of the UV-completion.

A.3 SU (2) third dual
We can construct the third SU(2) dual description [24] where the (anti-)baryonic operators and the Coulomb branch are introduced as elementary fields. Although this is not a "chiral" duality studied in this paper, we will show this duality for completeness.
The matter contents and their quantum numbers are summarized in Table 24. The third dual description includes the tree-level superpotential which is consistent with all the symmetries in Table 24. The magnetic meson is mapped to the electric meson operator.

A.4 SU (4) "chiral" dual
Let us consider the chiral dual description of Table 21 with F = 6. This case was studied in [3]. Since the SU(2) gauge theory with six fundamentals can be regarded as the SU (2) gauge theory with six fundamentals and no anti-fundamental (F = 6 in Table 21), we can construct the "chiral" SU(4) dual in addition to the USp(2) dual. The dual theory has no tree-level superpotential. The magnetic baryon operator is identified with the electric meson. The Coulomb branch operator Y is mapped to the magnetic Coulomb branch operator Y SU (2)×SU (2) whose expectation value leads to the breaking SU(4) → SU(2) × SU(2) × U(1). The quantum numbers of the magnetic matter contents are summarized in Table 26.

A.5 SU (3) "chiral" dual
By regarding the electric theory as the SU (2) gauge theory with five fundamentals and one anti-fundamental (F = 5 in Table 21), we can also construct the SU(3) dual description. The dual side becomes the 3d N = 2 theory with five fundamentals, one anti-fundamental and a gauge singlet M. The quantum numbers of these fields are summarized in Table 26.
The dual side includes a tree-level superpotential   A.6 U (1) "vector-like" dual Finally, we show the magnetic U(1) dual description which was found in [3]. The SU(2) gauge theory with six fundamentals can be regarded as the SU(2) gauge theory with three fundamentals and three anti-fundamentals (F = 3 in Table 21). By following [3], we can construct the U(F − N) dual description. The magnetic dual is the 3d N = 2 U(1) gauge theory with four electrons q,b and four positronsq, b in addition to the gauge singlets M and Y . The quantum numbers of the dual fields are summarized in Table 27. The theory includes the tree-level superpotential which is consistent with all the symmetries in Table 27. One can easily find the operator matching from Table 27. As we explicitly explained the various chiral dualities for the SU(2) with six doublets, one can easily generalize this argument to the SU(2) gauge theory with 2F doublets. By regarding the theory as the SU(2) gauge theory with 2F fundamentals and no anti-fundamental, the USp dual and the "chiral" SU(2F − 2) dual are available. For the SU(2) gauge theory with 2F − a fundamentals and a anti-fundamentals, the "chiral" SU(2F − a − 2) dual theory, which we proposed in this paper, can be constructed. For a = F , we can also have the "vector-like" dual [3]. In these dual descriptions, the full global symmetry is not manifest and only visible in the far-infrared limit.