S-confinements from deconfinements

We consider four dimensional $\mathcal{N}=1$ gauge theories that are S-confining, that is they are dual to a Wess-Zumino model. S-confining theories with a simple gauge group have been classified. We prove all the S-confining dualities in the list, when the matter fields transform in rank-$1$ and/or rank-$2$ representations. Our only assumptions are the S-confining dualities for $SU(N)$ with $N+1$ flavors and for $Usp(2N)$ with $2N+4$ fundamentals. The strategy consists in a sequence of deconfinements and re-confinements. We pay special attention to the explicit superpotential at each step.


Introduction and summary
The infrared (IR) behavior of four-dimensional gauge theories is a problem of central importance in theoretical physics. When minimal supersymmetry is present, substantial progress can be made [1][2][3]. One possible IR behaviour of a gauge theory is Sconfinement, where S stands for smooth. More precisely this terminology has been introduced in [4] for "smooth confinement without chiral symmetry breaking and with a non-vanishing confining superpotential". The IR behavior of an S-confining gauge theory, by definition, can be captured by a theory with trivial gauge dynamics, that is a Wess-Zumino (WZ) model. The elementary fields of the IR WZ description map with the gauge invariant operators of the ultraviolet (UV) gauge theory, more precisely they are in one-to-one correspondence with the generators of the chiral ring of the UV gauge theory. In addition, we require that this WZ description is valid everywhere on the moduli space including the origin where therefore all global symmetries are unbroken. The two paradigmatic, and simplest, examples of S-confining gauge theories are • SU (N ) with N + 1 flavors, described by a theory of mesons and baryons.
• U sp(2N ) with N + 2 flavors 1 , described by a theory of mesons.
There are also other examples of S-confining gauge theories. [5] classified all the Sconfining gauge theories with a simple gauge group and vanishing tree-level superpotential. These theories were argued to be S-confining by proposing a WZ description and checking that all the 't Hooft anomalies of the UV gauge theory match the 't Hooft anomalies of the WZ description. Our aim in this paper is to prove these S-confining dualities. By proof we mean that we find a sequence of dualities that takes us from the gauge theory to the WZ model. At each step, we write the explicit superpotential. The dualities we use are only the two basic S-confining dualities discussed above, that is SU (N + 1)/U sp(2N ) with N + 2 flavors. Our strategy, especially for the case of U sp(2N ) with antisymmetric and 6 fundamentals, is similar to the strategy of [6][7][8], implemented in 3 dimensions. 2 We are able to do this for all S-confining one node quivers, that is gauge theories with a simple gauge group and matter in rank-1 and/or rank-2 representations. The set of theories includes 3 infinite series: U sp(2N ) with antisymmetric and 3 flavors, SU (N ) with antisymmetric and (4, N ) flavors, SU (N ) with anti-symmetric, conjugate antisymmetric and (3,3) flavors. Moreover there are 4 exceptional cases, with SU (5), SU (6), SU (7) gauge group and 2 or 3 anti-symmetrics plus flavors. 3 In the two sporadic SU (5) cases, we will be confronted to extra subtleties in the computation of the superpotential associated to degenerate operators, see also [12].
The main tool we are going to use is the deconfinement method [13,14] that allows us to trade theories with antisymmetric field for theories involving only fields in the fundamental representation of simple gauge factor. More precisely, we use the deconfinement method of [15][16][17]. This deconfinement method in 4d N = 1 has been used recently [18] to construct Lagrangians for families of N = 2 superconformal field theories (SCFT).
One immediate lesson that can be drawn from these results is that the basic Seiberg dualities, involving only fundamental matter, seem to be strong enough to prove dualities involving more general matter content. A recent results that corroborates this expectation appeared recently in [19,20], where it is shown that 4d mirror symmetry dualities [21,22] (and hence 3d mirror symmetry dualities [23,24]) can be proven using only Intriligator-Pouliot (IP) duality, relating theories with U sp gauge group and fundamental matter. Moreover, in [12] we will use the sequential deconfinement techniques of [25,26] to find a quiver dual of U sp(2N ) with F flavors, and use this deconfined version to prove a knwon self-duality of U sp(2N ) with 4 flavors [27]. This paper is organized as follows. In section 2 we describe the way we deconfine the anti-symmetric fields which appear in the S-confining theories with one node and vanishing superpotential.
In sections 3, 4 and 5 we consider the infinite series. In order to have a simple close formula for the superpotential at generic N (for these infinite series, [5] only gave the superpotential for very small values of N ), we flip some operators in the gauge theory. The analysis of the superpotential at all intermediate steps of our procedure is quite involved, so we relegate many details in Appendix A.
In section 6 we discuss the four sporadic S-confining models.
In appendix B we give a unitary presentation of the S-confining dualities discussed in this paper, flipping the operators that violate the unitarity bounds.
Note added: while completing this work, we became aware of [28], which has some overlap with the results of Section 3. We thank the authors of [28] for coordinating the submission.

Elementary S-confining theories and deconfinement method
In this section we present the elementary S-confining theories that will be the building blocks to obtain the more complicated cases of the next sections.
The first one is the original Seiberg S-confining theory 4 [1], SU (N ) SQCD with F = N + 1 flavors. In quivers notation it reads SU building block The red circle means a SU gauge node and the square node is for the SU flavor symmetry. An oriented arrow between two nodes means a bi-fundamental field (transforming in the fundamental/antifundamental representation for the ingoing/outgoing arrow). We have also set the dynamical scale Λ that should appear in the superpotential in the r.h.s to 1 since it won't play any role in our work. The second building block is the IP S-confining theory [30], U sp(2N ) with 2N + 4 fundamental fields.

2N
2N + 4 The blue circle means a U sp(2N ) gauge group and the half-circle with arrows represents an antisymmetric field. The examples that we are going to discuss in the next sections are gauge theories with rank-2 matter, more precisely antisymmetric fields. Therefore if we want to use only the building block confining theories, we need a way to get a situation in which only fundamental fields are present. This is possible and it is called deconfinement. The price we have to pay is an additional gauge group. We trade the rank-2 field by a confining gauge group. This method was introduced by Berkooz [13] and further developed in [14]. The original deconfinement method of [13] was for SU (2N ) with an antisymmetric field and it reads The justification is straightforward, we start on the r.h.s, we notice that the U sp(2N −4) is coupled to 2N fields, so we apply the U sp building block (2.2) and we get the l.h.s. We would prefer a situation where the superpotential is zero on the side with the antisymmetric field. A way has been found by Luty, Schmaltz and Terning [14]. It applies for any group G and it says On the r.h.s, N is the dimension of the fundamental representation of G (the antisymmetric field has indices X ij with i, j = 1, . . . , N ), K is the smallest integer such that N + K − 4 is even and β is an antisymmetric field of the flavor group. Few remarks • Depending on the group G, there is additional matter, that contributes to cancel the gauge anomaly. However G is a spectator in this deconfinement.
• Planar Triangle corresponds to the term lbc with the obvious contraction of indices.
We put the mapping into quotation mark because it does not correspond to gauge invariant operators.
• If K > 1 we have a fictitious SU (K) global symmetry. It is fictitious because the only fields that transform under this symmetry are not present in the low-energy theory.
The justification of (2.4) goes as follows. We start once again from the r.h.s and notice that U sp(N + K − 4) is connected to N + K so we can use our U sp building block (2.2). The result is Then sincec and β are massive we can integrate them out which put to zero p and α. Now if we look at the Pfaffian term, we see columns of zero and so it vanishes. Therefore we obtain the l.h.s of (2.4). For G = SU (2N )/U sp(2N )/SO(2N ) K is an even integer greater or equal to 2, so using this method there is a fictitious global symmetry (this SU (2N ) deconfinement appears in Terning [31]). For our purposes this is not enough, we need a deconfinement without this additional fake symmetry group. When the theories have at least one matter fields in the fundamental representation, we will use the following deconfinement 6) This type of 4d N = 1 deconfinement reduces to the 3d N = 2 deconfinement used in [26] and also appeared in [17]. The mapping of the chiral ring generators is The trick was to split the F fundamental fields into F − 1 and 1 and use this extra 1 to deconfine without introducing any extra "fake" global symmetry. The proof of (2.6) is similar to the previous ones. We start on the r.h.s by confining the U sp(2N − 2) gauge group. The initial superpotential terms become mass terms of the unwanted mesons which therefore are set to 0. This kills the would-be Pfaffian term because, as before, we get a vanishing column.
The U sp(2N ) case is similar 2N 2F The cross, ×, on the r.h.s quiver stands for the singlet β 1 . This singlet is called a flipper. In this situation we should add this flipper because we want the antisymmetric A 0 on the l.h.s to be traceless 5 .
In this case the mapping is Let's write more explicitly the indices of the operators.
To summarize with these two ways of deconfining. The advantages are • W = 0 • No additional matter fields in the deconfined frame that introduces fictitious global symmetry The disadvantage is the apparent breaking of the global symmetry For G = SU (2N + 1) with F fundamentals we have K = 1 and the deconfinement reads The mapping of the chiral ring generators is the following: This form of deconfinement appears first in Pouliot [32]. In the following sections, we will apply the two S-confinements (2.1), (2.2) and the deconfinements (2.6) (2.8) to prove all S-confining dualities for single node quivers. Sconfining theories with a single gauge group were classified by [5]. By "proof" we mean using only the basic building blocks (2.1) and (2.2). The strategy is to first deconfine the rank-2 matter, then confine one by one all the gauge groups.
3 U sp(2N ) with + 6 series Let us start from U sp(2N ) gauge theory with one antisymmetric and 6 fundamental fields. This theory has a continuous global symmetry SU (6) × U (1) (on top of the U (1) R symmetry). We also turn on a superpotential 2N 6 where α i are gauge singlets. The F-term equations of these flippers set the tr (A i ) to 0 on the chiral ring. In addition, on the quantum chiral ring these flippers α i are not generators. We can understand it as follows: Start with the theory with W = 0. After a-maximization [33], we discover that the operators tr (A i ) violate the unitary bound, see Appendix B. Therefore, we expect that in the IR the theory breaks into a free and an interacting part. If we want to focus on the interacting part, the procedure is to flip these operators [34] and the flippers are not generators of the quantum chiral ring. The conclusion is that the quantum chiral ring generators of (3.1) are the dressed mesons: tr (q A a q), a = 0, . . . , N − 1 and these are the operators that we have to map. We turned on this superpotential because it will be much easier to keep track of the superpotential when doing the sequence of confinement/deconfinement. Now we start by using the trick of splitting the 6 fundamentals into 5 + 1 as in [6][7][8] T 0 : The first step is the use of the deconfinement (2.8).
T 0 : Let us remark that the h 1 field has been integrated out using the equation of motion (E.O.M) of the massive α N . Now we see that the U sp(2N ) is coupled to 2N −2+5+1 = 2N + 4 fundamental fields. Therefore we can apply the basic S-confining result (2.2). We get The E.O.M of β 1 and d 1 set tr (A 1 ) ≡ (A 1 ) α 1 α 2 J 2N −2 α 1 α 2 = 0 and p 1 = 0. Therefore the Pfaffian term becomes Let us focus on the first term: We claim that we can drop this term. Indeed the part ε 2N −2 A N −1 1 can be written, by linear algebra, as the product of all traces of A i (which are all set to 0 on the chiral ring by the F-term equations of α i ). Then using the chiral ring stability argument 6 [34] on ε 2N −2 ε 5 A N −1 1 M 2 1 O 1 we conclude that we can remove this term from the full superpotential in (3.4). More generally, the chiral ring stability allows us to drop terms of the form: Therefore from now on we will discard these terms. This is the reason why we turn on the superpotential in (3.2), to avoid the proliferation of this kind of term.
The next step is to deconfine again the antisymmetric field. We get The chiral ring stability argument says the following [34]: Start with a theory with superpotential For each i, consider the modified theory T i where the term W i is removed. Then, check if the operator W i is in the chiral ring of T i . If it is not, drop W i from the full superpotential W. 7 Even if we don't want to use the chiral ring stability criterion, these terms will disappear once we reach the final frame. Indeed every power of the antisymmetric field will at some point be mapped to a singlet h a field. This singlet will enter in the superpotential as a mass term with one flipper α. Now we confine the U sp(2N − 2). It is similar to the previous step between the frames T 0 and T 1 . There is once again a confining superpotential given by a Pfaffian term. What is interesting and non-trivial is the mapping of the existing superpotential terms in (3.7).
Now since we reach U sp(2) SU (2) the traceless antisymmetric field simply does not exist anymore. Therefore there is nothing to deconfine and we can directly apply the building block once more. We reach the final "Deconfined" frame In the last equality, we have repackage the superpotential in a manifestly SU (6) invariant way. We recover the result of [5] and the superpotential for a generic N proposed in [10]. The mapping of the chiral ring generators between the original frame and the final one is the following In the repackaged form, the mapping becomes: tr (q A a q) ⇐⇒ µ a+1 , a = 0, . . . , N − 1.
The first series is SU (2N + 1) gauge theory with fields in the antisymmetric and conjugate antisymmetric representation and (3, 3) fundamentals, antifundamentals. The continuous global symmetry is The chiral ring generators are We deconfine the two antisymmetric fields with the help of (2.10).
Now we confine the SU node with (2.1).
Let's give more details. The first 7 terms in the superpotential (4.3) come from the determinant of the meson matrix Φ The "XH N M 0 +6 Planar Triangles" terms come from the cubic interaction "meson × baryon × meson" when the SU group confines. We also rescale the fields such that the coefficient in front of each term is +1.
The next step is to confine the left U sp(2N − 2) using (2.2).
Let's explain how to get the superpotential in (4.5). The first 7 terms in (4.3) are mapped to the first 10 terms in (4.5). The additional 3 terms come from Then X andỸ of (4.3) get massive from the planar triangles terms. Indeed we obtain the following mass terms: P 1 X and P 2Ỹ where P 1 is the meson [K 1 Y ] and P 2 the meson [K 2 Y ] in the frame T 2 . Therefore after integrating out massive fields, we are left with only two "triangles terms": The remaining terms come from the Pfaffian confining superpotential (2.2) As we said P 1 and P 2 are massive fields and their expression are obtained by the E.O.M of X andỸ . We get (after rescaling fields) Putting all together, we get the superpotential written in (4.5). The next step is to confine the U sp(2N − 2) node using the result (3.11). In order to apply it and get a superpotential in a close form, we need to flip the tower of traces of the antisymmetric field B. Using the mapping (4.7), it amounts to flip in the frame T 1 the tower of (AÃ) j . To simplify even further we also flip the two singlets ε 2N +1 (A N −1 Q 3 ) and ε 2N +1 (Ã N −1Q3 ). Therefore we kill terms with T N , B 3 andB 3 .
The summary of this flipping procedure is the following duality We can now use our confining result of Section 3. We rewrite (3.11) after splitting the 6 fundamentals into two groups of 3, getting Apply this result to our frame T 3,f lip in (4.8) we get the final WZ T 4,f lip : As a consistency check, for N = 2 we recover the superpotential given in [5] Section 3.1.4. The mapping of the chiral ring in the flipping case is

Even rank: M = 2N
Let us now study the even rank case of the previous gauge theory.
The chiral ring generators are The next step is to deconfine the antisymmetric and the conjugate antisymmetric fields. To do so we use a variant of the deconfinement method (2.6), where we don't have to split the flavor symmetry. By doing so we don't have to split the chiral ring generators and it is then easier.
Then we confine the SU node with (2.1).
Once again some terms in the superpotential in (4.14) come from the determinant of the (2N + 1) Then we confine the left U sp(2N ) node with (2.2) to get We notice that the fields n andX get a mass. To obtain (4.16) from (4.14) we have to compute the Pfaffan of the following (2N + 4) × (2N + 4) antisymmetric matrix All in all we get (4.16). The next step is to confine the U sp(2N − 2) node using the result (3.11). Therefore we have to flip the whole tower of tr (B k ). Notice that the last one tr (B N ) is already flipped by α N . We also flipB 0 and B 0 because we do not want the flipper α N to appear elsewhere in the superpotential. By doing so we kill 3 terms in the superpotential of (4.16) : Using the mapping (4. 19) we see that in the original frame it amounts to flip the tower of (AÃ) k , k = 1, . . . , N − 1 and also ε 2N A N , ε 2NÃ N .
The summary of this flipping procedure is the following duality The final step is to confine the U sp(2N ) using the result from (4.9). The WZ is The fieldsH 1 and H N have been set to 0 by the E.O.M of Y andñ. It kills N terms in the sum. As a consistency check, for N = 2 we recover the superpotential given in [5] Sec.3.1.5. The mapping of the chiral ring in the flipping case is The gauge theory is SU (2N +1) with one antisymmetric field, 2N +1 antifundamentals and 4 fundamentals, with continuous global symmetry SU (2N + 1)Q × SU (4) Q × U (1) 2 .
T 1 : The chiral ring generators are We deconfine the antisymmetric field using (2.10). We get We then confine the SU gauge node using (2.1). The field l becomes massive. Then the determinant of the meson matrix is easily computed. After rescaling of the fields to put a +1 in front of each term we obtain We then S-confines the U sp(2N − 2) gauge node using (2.2). The result of the integration of the massive field Y , the computation of the Pfaffian superpotential and the rescaling of the fields is the following WZ model We recover the result of Section 3.1.3 of [5]. The mapping of the chiral ring generators across the different frames is given in (5.5).

Even rank: M = 2N
Now we study the even rank version of the previous theory. The continuous global We deconfine the antisymmetric using (2.6) which implies the breaking of the SU (4) Q into SU (3) q × U (1) F . The chiral ring generators in the split form are The next step is once again to confine the SU gauge group using (2.1).
The last step is to confine the U sp(2N − 2). We get the following WZ theory The final superpotential can be repackaged in a manifest SU (4) invariant way as 2 } is an antisymmetric of SU (4) Q . We recover the result of Section 3.1.2 of [5]. The mapping of the chiral ring generators across the frames is given in (5.11).
Four 'sporadic' cases: SU (5), SU (6), SU (7) In this section we show how the strategy of deconfining antisymmetric fields allow proving the remaining four S-confining cases in the list of [4,5]. In the two SU (5) cases, there are extra subtelties when computing the superpotential because of the presence of degenerate operators, see also [12]. The gauge theory is SU (7) with two antisymmetric fields and 6 antifundamentals, with continuous global symmetry SU (2) A × SU (6)Q × U (1).
We will deconfine each antisymmetric field using (2.10). Notice that this step breaks the global SU (2) A symmetry into a U (1) A . In (6.1), the subscript of the two antisymmetric fields, A ±1 correspond to the U (1) A charge (corresponding to the weight of the representation of SU (2) A ). So let's give the chiral ring in this split form Notice that B +3 , B −3 and V 0 are gauge singlets but are zero in the chiral ring, this is a quantum effect that can be seen dualizing the U sp nodes.
We can now S-confine the left U sp(4) with (2.2). After integrating out massive fields, computing the Pfaffian superpotential and rescale the fields to put a +1 in front of each terms we get Finally, we repeat the same operation with the last gauge group and get the WZ model with quartic superpotential The final superpotential can be repackaged in a manifest SU (2) A invariant way as where is a 2-symmetric tensor of SU (2) A (also called the triplet). We recover the result of Section 3.1.10 of [5]. The mapping of the chiral ring generators is The second sporadic case is the SU (6) gauge theory with two antisymmetric fields, 5 antifundamentals and 1 fundamental, with continuous global symmetry SU (2) A × SU (5)Q × U (1) 2 . T 1 : The chiral ring generators are We will deconfine A 1 , using (2.6) and then our result in Section 5.2. This step breaks the SU (2) A symmetry into U (1) A . In (6.8), the subscript of the two antisymmetric fields, A ±1 correspond to the U (1) A charge. So let's give the chiral ring in this split form Now we want to use the result of the section (5.2), specialized in the case N=3. Before using the superpotential (5.10), we should split the 6 antifundamentals into 5 + 1. More precisely,H, the antisymmetric of the global SU (6)Q there, split intoH −1 , an antisymmetric of SU (5)Q and D +1 a fundamental. Similarly, M , the fundamental of SU (6) there, split into a fundamental, N + 1 2 and a singlet (under the global SU (5)Q symmetry) (bc) + 5 2 . We should also split B 2 , the antisymmetric of SU (4) Q into a traceless antisymmetric tensor of U sp(4) B −1 and a singlet s −1 . Finally, we rename the three singlets B 4 , B 0 andB there as s +1 , s −3 and β +2 . After this splitting, the use of (5.10) give 4 5 s +3 + singlets (s +1 , s −1 , s −3 & β +2 ) (6.11) Now we want to confine the U sp(4) gauge group with the antisymmetric and 6 flavors. Unfortunately, we cannot immediately use our result about U sp(2N ) of Section 3 for the following reason. The flipper β +2 appears in (6.11) in three different terms but we know the final superpotential only when we flip the whole tower of the traces of the antisymmetric as in (3.2). In this case since the rank is small, it's easy to apply our strategy of the Section 3 starting with W = 0 to get The mapping is We now use this result into (6.11). We see that the singlets β +2 and T −2 become massive. After integrating them out and rescaling fields, we get the final result 13) The final superpotential can be repackaged in a manifest SU (2) A invariant way as where We recover the result of Section 3.1.9 of [5]. The final mapping of the chiral ring generators is The third case is the SU (5) gauge theory with 2 antisymmetric, 4 antifundamental and 2 fundamental fields with continuous global symmetry SU The chiral ring generators are We now deconfine the two antisymmetric using (2.10) and so we break SU (2) A into U (1) A . In (6.16), the subscript of the two antisymmetric fields, A ±1 correspond to the U (1) A charge (corresponding to the weight of the representation of SU (2) A ). After the splitting, the chiral ring generators become Once again, the subscripts in (6.17) correspond to the U (1) A charges of the fields. The next step is to confine the SU (5) gauge group with (2.1). Notice that l − 5 2 and l + 5 2 will become massive. After integrating them out, computing the det(mesons) of degree 6 and rescaling the fields we obtain M 0 (6.18) Then we use (2.2) for the left U sp (2). The fields B +2 and V 0 get a mass. We have to integrate them out and compute the Pfaffian superpotential. Let us write more explicitely the Pfaffian term because it will be useful later. The mesons involving in the Pfaffian are ]. We didn't give a name to the last two mesons because they are massive and will be integrate out by the E.O.M of V 0 and B +2 . The Pfaffian is then given by Where we rescaled the fields. Therefore the Pfaffian gives the following contribution The theory after this U sp(2) lef t confinement is (6.21) The last step is to confine the other U sp (2). The field B −2 gets a mass. As in the last step, let us write the Pfaffian term. The mesons involve are: ]. The last one is massive and will be integrate out with the E.O.M of B −2 . The Pfaffian is Where we rescaled the fields. Therefore the Pfaffian gives the following contribution We get the following WZ model The final superpotential can be repackaged in a manifest SU (2) A invariant way as We recover the result of Section 3.1.8 of [5].
Before moving on, we should comment on the red term in (6.24):H −1 n +1 S 0 M 0 . Indeed, if we combine the superpotential in (6.21) with the Pfaffian term (6.23) we get the superpotential in (6.24) without this red term. So why did we add it and where does it come from? First, we remark that without this term it would not be possible to repackage the superpotential in a manifestly SU (2) A invariant way as in (6.25). In addition, this term is invariant under all the global symmetries including the U (1) 2 × U (1) A . There is another argument that suggest the presence of this term. Suppose that after the frame T 2 , we decide to confine in the reverse order meaning that we first confine the U sp(2) right and then the U sp(2) lef t . With this order, the termH −1 n +1 S 0 M 0 is present in the frame T 4 and it would be the termH +1 n −1 S 0 M 0 missing. The last observation is that it would have been possible to add an extra term that respect all the global symmetries in all the previous frame (T 1 to T 3 ) and which lead to the red term in T 4 . However, we believe that this extra term is forbidden in the frames T 1 , T 2 , T 3 because of chiral ring stability. Therefore, chiral ring stability will force us to wait up to the last frame before adding this extra term allowed by the global symmetries.
This subtle point seems to come from the fact that we have degenerate operators in ] and S 0 M 0 . It suggests that in presence of degenerate operators, applying a duality locally (to a particular node inside a quiver) would miss some informations 11 .
One prescription that lead to the correct final superpotential is the following: When going from T 3 to T 4 , during the computation of the Pfaffian superpotential (6.23), we should not use f 0 but the combination f 0 +S 0 M 0 . One can understand this prescription in the following way: In the frames T 1 to T 2 there is a Z 2 symmetry, corresponding to the Weyl reflection inside SU (2) A , which maps a field with U (1) A charge x to the field with charge −x. When we go to the frame T 3 , this symmetry is not explicit anymore. Imposing the restoration of this symmetry in T 4 is enough to give the correct superpotential. This is the role of this prescription.
The mapping of the chiral ring generators is We stress that the last line in the mapping (6.26) is ambiguous in the intermediate frames. Indeed, in the frames T 2 and T 3 there are multiple holomorphic operators with the same quantum numbers under all the global symmetries which should map to a single chiral ring generator.
The chiral ring generators are Now we deconfine the three antisymmetric, breaking SU (3) A to U (1) 2 A . Contrary to the others "sporadic" cases, the subscripts here do not correspond to the U (1) A charges. After the splitting the chiral ring generators are given by The next step is to confine the U sp(2) up , B 2 , B 4 and B 7 get a mass. The result is In the next two steps, we confine U sp(2) lef t and U sp(2) right . It is really similar to the previous step. We obtain T 4 : O transforms in the adjoint, S as a symmetric 2-tensor andH in the fundamental of SU (3) A . We recover the result of Section 3.1.7 of [5]. The mapping of the chiral ring generators is

Conclusions and outlook
In this paper we have shown that all 4d N = 1 S-confining gauge theories with a single gauge group, vanishing tree-level superpotential and rank-1 and/or rank-2 matter can be obtained from the basic Seiberg (2.1) and Intriligator-Pouliot (2.2) S-confining dualities [1,30]. We have also obtained the confining superpotential in a closed form for all theories. We did this using new versions of the deconfinement technique of [13].
Our result participates to the project of reducing the number of apparently independent dualities started in [13]. It would be interesting to know how far we can go with this technique and in particular if dualities with matter only in the fundamental representation are strong enough to reproduce all the more complicated dualities.
There are many other directions to explore. An obvious one is trying to go beyond the classification of [5] by considering more than one node quivers and/or non-vanishing tree-level superpotential [35]. There are also S-confining theories involving non-quivers type of matter as rank-3 and Spin gauge theories with chirals in the spinor representation [5]. It would be really interesting if we can also obtain these theories from simpler dualities.
It would also be worth exploring beyond S-confining theories. For example, more general IR dualities involving rank-2 matter [36][37][38][39] as well as self-dual theories [27,40]. In a separate paper [12] we will report some progress on the last point.

Case
Notice that 2k In T k (p) max : Once again 2k This finishes the second step. The third and last step is to combine all the previous ingredients to write the superpotential in a generic frame. So let's fix k (1 ≤ k ≤ N −1) we want the superpotential in T k . As we said it contains terms with O p with 1 ≤ p ≤ k. Also, depending on p, there are two kinds of terms depending on whether k ≥ k and for p < p : k (p) max < k which is what we wanted. To recap, for 1 ≤ p ≤ p − 1 we have terms A and for p ≤ p ≤ k we have terms B. So the superpotential takes the following form We have included the Heavyside step function: to treat also the case p ≤ 0. ε is a small positive quantity ( 1) to avoid double counting in the case p = 0. Now let's look in turn at terms A and B.
Terms B: By definition they satisfy p ≤ k ≤ k (p) max ∀p ∈ B ≡ ∀p ≥ p . In this case it is easy and Θ (p) is given by (A.1).
Terms A: By definition we have k > k (p) max ∀p ∈ A ≡ ∀p < p . It is more complicated in this case. The first thing to notice is that N + p − 2 (= 2k) is necessarily even. Equivalently we can say that p has the same parity has N . For p = p − 1 : k max . In addition N + (p − 1) − 2 is odd therefore we are in the situation of (A.3) and (A.6). Conclusion Θ (p −1) is given by Then we go on.
and t = 1, . . . , p − 2. In addition N + (p − 1 − t) − 2 has the opposite parity of t. therefore to continue and use our (A.5) and (A.7) we should separate between the odd and even value of t. That is, we rewrite the sum A as Where the " " means t min , t min + 2, t min + 4, . . . It's not complicated to get the expression for t max odd and t max even . They are given by t max odd = 2 p 2 − 3 and t max even = 2 p 2 − 2. Now Θ (p −1−t odd ) in T k is given by (A.2) and (A.5). To use these formulas we need to do the following translation: k Similarly, Θ (p −1−teven) in T k is found using (A.4) and (A.7). Once again to use the formula we need to impose : k ! = k (p −1−teven) max + 2 + t = k − teven 2 + 1 + t ⇒ t = teven 2 − 1.
Where we recall that " " means l min , l min + 2, l min + 4, . . . and similar for m. We can do even better, 1 + 2 can combine together to give Less obviously, 1 + 2 can be packaged together. To do so we notice • M i , M j and O p that enter in the sum satisfy i + j − p = N • In addition, the above i and j satisfy N + p − k ≤ i ≤ j ≤ k • All terms satisfying the above 3 criteria are present in the sums Therefore 1 + 2 can be written So the last version of the superpotential is

B Unitary presentation of the S-confining dualities
The S-confining dualities discussed in this paper are not written in a unitary fashion. This is not by itself a problem, since the gauge theories can be part of larger gauge theory, where no unitary violation is present. Nevertheless, for completeness, we give a presentation of the confining dualities flipping all the operators that violate unitarity.
In the Wess-Zumino there can only be cubic superpotential terms (higher order terms automatically imply that at least one field has R < 2 3 ). So all surviving operators have R charge precisely 2 3 and the WZ superpotential is cubic. Obviously, flipping some chiral ring generator O, removes the operator O from the chiral ring. If R[O] > 2/3, the flipper field appears in the chiral ring of the new theory, but notice that if R[O] < 2/3 the flipper field is not a chiral ring generator of the new theory. This is due to quantum effects: if the flipper takes a vev, the theory develops a quantum generated superpotential that lifts the supersymmetric vacua at the origin [34].
The unitary dualities look somewhat trivial sometimes, having very few fields left in the chiral ring. Let us start from the very simple examples of SU (N ) and U sp(2N ) with only fundamentals. The case of SU (2) = U sp(2) has cubic dual Wess-Zumino and it is unitary.