Sequential deconfinement and self-dualities in $4d$ $\mathcal{N}\!=\!1$ gauge theories

We apply the technique of sequential deconfinement to the four dimensional $\mathcal{N}\!=\!1$ $Usp(2N)$ gauge theory with an antisymmetric field and $2F$ fundamentals. The fully deconfined frame is a length-$N$ quiver. We use this deconfined frame to prove the known self-duality of $Usp(2N)$ with an antisymmetric field and $8$ fundamentals. Along the way we encounter a subtlety: in certain quivers with degenerate holomorphic operators, a naive application of Seiberg duality rules leads to an incorrect superpotential or chiral ring. We also consider the reduction to $3d$ $\mathcal{N}\!=\!2$ theories, recovering known fully deconfined duals of $Usp(2N)$ and $U(N)$ gauge theories, and obtaining new ones.


Introduction and summary
Dualities are among the most powerful tool to analyze quantum field theories at strong coupling. In this paper we are interested in four dimensional N = 1 gauge theories with rank-2 matter. Various recent works derived dualities involving gauge theories with rank-2 matter using only basic dualities involving gauge theories with fundamental matter, like Seiberg [1], Intriligator-Pouliot [2] and Aharony dualities [3]. modulo flips 3 of U sp(2N ) with antisymmetric and 8 fundamentals, proposed long time ago in [29]. By self-dual modulo flips we mean that the electric and magnetic theory share the same gauge structure, but differ by gauge singlets fields of flip type. Selfdualities modulo flips have been discussed in [29][30][31][32], the simplest case is SU (2) with 8 doublets. Interestingly, given a self-duality modulo flips, it is possible to move the singlets across the duality and construct exactly self-dual theories with enhanced infra-red global symmetry, see for instance [32][33][34][35][36]. In our case of U sp(2N ) with antisymmetric and 8 fundamentals, [35] discussed various exactly self-dual theories and discussed the associated symmetry enhancements. Sometimes these symmetry enhancements can be understood compactifying a 6d (1, 0) SCFT on a Riemann surface. For U sp(2N ) with antisymmetric and 8 fundamentals, [36] related the self-duality and specific symmetry enhancements to a compactification of the rank-N E-string 6d SCFT on a 2-sphere.
We encounter one subtlety (which is also present in the 3d cases but was overlooked in [9]) during the process that we call degenerate holomorphic operator ambiguity. As the name suggests, this phenomenon appears when we reach a frame that contains more than one gauge invariant holomorphic operator with the same global symmetry quantum numbers (including U (1) R ), but only one combination is a chiral protected operator. If such an operator is flipped by a gauge singlet, only one specific combination appears in the superpotential. In the examples we encounter in this paper, it happens that if we follow the rules of Seiberg duality (as is usually done) we end up with the incorrect result. In some cases we can determine which is the precise combination of operators appearing in the chiral ring (equivalently, the combination that can be flipped) by going in a dual frame and using classical F-terms relations there. Hence, in the original theory with degenerate holomorphic operator ambiguity, the ambiguity is resolved by quantum relations. In the case of F = 4, the precise superpotential is crucial in the proof of the self-duality modulo flips, so this case provides a good consistency check of our procedure.

Future directions
One natural question is wether there is a relation between our sequentially deconfined dual of U sp(2N ) with an antisymmetric and 2F fundamentals and Kutasov-Schwimmer type dualities [38][39][40], which for Usp were proposed in [41]. Namely one can turn on a superpotential term tr(A j ) on the electric side, such term maps to a singlet on the magnetic, so a Higgsing process is induced. The study of this Higgsing might shed light on the dualities of [38][39][40][41]. We expect the degenerate holomorphic operator ambiguity encountered in this paper to play an important role. There are quite a few self-dualities modulo flips proposed in the literature [29][30][31][32], such as SU (2N ) with antisymmetric, conjugate antisymmetric and (4, 4) fundamentals, or SU (6) with 2 antisymmetrics and (2, 6) fundamentals, or SU (8) with 2 antisymmetrics and (0, 8) fundamentals. A natural question is if such self-dualities can be proven using only the basic Seiberg and Intriligator-Pouliot dualities, as done in this work for U sp(2N ) with antisymmetric and 8 fundamentals. Notice that many self-dual gauge theories have been constructed simply 'adding one flavor' to an S-confining gauge theory [29,31], so the fact that the S-confining dualities can be proven [15] is encouraging.
Related to the above point, many S-confinements and many self-dualities have been proposed for 4d N = 1 Spin(N ) theories with spinors and vectors [29,31,32,34]. It would be very interesting to find a way to deconfine spinorial matter and try to prove such proposals.
Limits to gauge theories with orthogonal and/or symplectic gauge groups. Sconfinements for 3d N = 2 theories with SO/Usp gauge groups and adjoint matter where recently proposed in [42], and [43] pointed out a relation to 4d N = 1 Sconfinements for Usp gauge group and antisymmetric matter. It might be interesting to deform the 4d N = 1 sequential deconfinements as in [43]. See [28] for the sequential deconfinement of 3d N = 2 rank-2 matter with SO/Usp gauge groups.
It would also be interesting to study degenerate holomorphic operator ambiguity in other examples, possibly involving different kind of gauge groups.

Structure of the paper
This paper is organized as follows.
In section 2 we recall our notation and the three basic dualities which will be iteratively used in the rest of the paper.
In section 3 we discuss in detail a simple but non-trivial example, namely we sequentially deconfine U sp (6) with an antisymmetric and 8 fundamentals. This example is the simplest one where the issue of the degenerate holomorphic operator ambiguity appears. We construct the fully deconfined dual, then we reconfine the quiver tail in order to prove the self-duality of the theory.
In section 4 we present the general sequential deconfinement of U sp(2N ) with an antisymmetric and 2F fundamentals.
In section 5 we set 2F = 8, which allows to sequentially reconfine the quiver tail, and prove the self-duality modulo flips of the theory.
In section 6 we show how to reduce our 4d N = 1 U sp(2N ) story to 3d N = 2, re-obtaining the results of for U (N ) and U sp(2N ) found in [9]. Along the way we also derive new sequentially deconfined duals, namely for U (N ) with adjoint and (F, F ) fundamentals with monopole superpotentials.

Basic S-confining and dualities moves
In this section we present the basic ingredients that we use in this paper to obtain the more complicated dualities involving rank-2 matter field. The first basic move is the deconfinement of an antisymmetric field with a U sp gauge group as in [15,16], which is a modification of the original Berkooz deconfinement [44]. This form was used in [15,16] to tackle Usp(2N) with 6 fundamentals, which is S-confining.

2N
2F On the r.h.s quiver in (2.1), we didn't include the flipper β 1 on the drawing. Sometimes, as in [15] this flipper is represented on the quiver by a cross, ×, on the bifundamental field b 1 . Moreover, the trace "tr" is taken using the U sp invariant antisymmetric matrix. Throughout the paper, a trace of an operator in the antisymmetric of a U sp group is taken using the U sp invariant antisymmetric matrix.
In order to get this deconfinement we have to use another basic move. It is the S-confining result of Intriligator-Pouliot (IP) [2] that involves U sp(2N ) gauge group and 2N + 4 fields in the fundamental representation. IP S-confining duality: The last move concerns the same U sp theory but with 2F ≥ 2N + 4 fundamentals now.
It is the IP duality [2]. In quiver notation it reads IP duality: 3 Case study: U sp(6) with + 8 In this section we study the U sp(6) gauge theory with matter in the antisymmetric representation and 8 fundamentals, that is 4 flavors. It is the simplest example that contains all the ingredients that we want to exhibit. In the next sections, we will show the general case. It is known that this theory is self-dual modulo flips [45]. We will prove the self-duality using the basic moves of the previous section (2.1), (2.2) and (2.3).

Sequential deconfinement
The first step is the deconfinement of the antisymmetric with (2.1). We get the following two frames T 0 and T 0 T 0 : The global symmetry of the original theory is SU (8) × U (1) A × U (1) R . When we split the 8 chirals into 7 + 1, we split the SU (8) into SU (7) × U (1) P . So in the splitted form, the R-charges of the fields should be a function of two variables (for the two global U (1)'s that can mix with the U (1) R ). We choose to write the R-charges in terms of R Q and R A . Then the R-charges of the other fields are determined by the U (1) R ABJ anomaly 5 and by the requirement that any superpotential term should have R-charge equal to 2. We have written the R-charges of the fields next to them. Then we dualize the U sp(6) node with (2.3). The fields d 1 and β 1 get a mass. After integrating them out and rescaling the fields to put a +1 in front of each term in the superpotential, we get 5 This is the same as requiring the vanishing of the NSVZ β function.
In T 1 the antisymmetric field, Φ, is traceless because the trace component has been killed by the equation of motion (E.O.M) of the flipper β 1 . The mapping of the chiral ring generators after this first step is the following It can be checked that the mapping (3.2) is consistent with the R-charges of the operators. Now we iterate the procedure.
We now deconfine the traceless antisymmetric field Φ.
Now we dualize the U sp(4) node. There will not be any antisymmetric field for the other gauge groups because they are U sp(2) 6 . The fields q 1 , d 2 and β 2 get a mass. Moreover, tr (c 1 c 2 c 2 c 1 ) becomes a mass term therefore there will be no link between the two U sp(2) gauge groups. After integrating these massive fields out and a rescaling we get T 2 : At this step, we face a feature that we call the degenerate holomorphic operator ambiguity. It arises when we ask what is the operator flipped by the singlet H 2 .

Degenerate holomorphic operator ambiguity
If we apply the rules of Seiberg duality locally in the quiver, as is usually done, using the mapping (2.3), we would conclude that it is O 1 = V 2 C 2 P 3 , so the superpotential should contain H 2 O 1 . However, for the quiver at hand, there is another candidate, O 2 = V 1 R 1 . Both O 1 and O 2 are gauge singlets, singlets under the SU (7) flavor symmetry and have R-charge: R(V 1 R 1 ) = R(V 2 C 2 P 3 ) = 2 − 2R(A) (which implies that the two operators have the same charges under the flavor U (1) s). Therefore they are degenerate operators. 7 . So it might be that the precise operator flipped by H 2 is not exactly O 1 , but some linear combination of O 1 and O 2 .
We claim that the correct answer is that the operator flipped by Our argument in favor of this statement comes from dualizing some nodes in the quiver, as we will explain soon. So in a sense the fact the the correct operator is not the naive one is due to quantum relations, which become classical relation after Seiberg duality. 7 Let us emphasize that there is no ambiguity with H 1 because there is only one operator which is singlet under the gauge symmetry, the SU (7) flavor symmetry and has R-charge equal to 2 − 3R(A). This operator is Our strategy to decide the correct operator is to use dualities in order to go to a frame where F-term equations can answer the question. In this case, we apply IP S-confining duality (2.2) on the left U sp(2) gauge node of theory T 2 with the singlet H 2 removed. We consider the same theory with the flipping removed, the question becomes which linear combination of the two degenerate holomorphic operators O 1 and O 2 is non zero in the chiral ring. Since the answer, as we will see, is that O 2 is non zero in the chiral ring, the quantum relation in the unflipped theory is O 1 = 0.
The U sp(2) we dualize is coupled to 6 fundamentals and so it confines, producing a traceless antisymmetric field B (the trace part is killed by the flipper α 1 ). We get 4 2 7 The Pfaffian term gives: Pfaff The fields X and V 2 are massive. The E.O.M of X gives: In addition, the F-term equation for the singlet H 1 gives: P C 2 P 3 = 0. Combining these two informations, we can resolve the ambiguity about the operators O 1 and O 2 . Indeed in this frame these operators become The symbol in the last of equality in (3.6) means an equivalence in the chiral ring. Therefore we conclude that the non-zero operator in the chiral ring is O 2 = V 1 R 1 and so it should be this one that enters in the superpotential with H 2 .

Fully deconfined frame
We now go back to our deconfining procedure, and dualize the right U sp(2) node in T 2 using (2.3). We reach a frame that we call "fully deconfined" as in [9].
T Dec : The antisymmetric field a 2 is traceless, as all the antisymmetric field of U sp that will appear in this paper. Once again there is the question of the operator flipped by h 2 because v 2 r 2 has the same quantum numbers as v 1 r 1 . Using the same procedure of confining from the left, we would obtain that the operator v 2 r 2 is 0 on the chiral ring. Therefore we claim that the correct final superpotential is the one with this switching procedure and not the one that we would have got using naive iteration of IP dualities.
The final mapping of the chiral ring generators is Combining the two mappings (3.2) and (3.9), we get the mapping between T 0 and T Dec

Self-duality
We already said that this theory is self-dual [45]. Let us see now how we can use our T Dec frame to prove the self-duality. The strategy is to reconfine the quiver tail. We notice that the left U sp(2) has 6 fundamentals, so we start confining from the left. The effect of this confinement is to kill the antisymmetric field a 2 . In addition, the fields α 1 , v 2 and h 2 get a mass and we produce a Pfaffian superpotential as in (3.5). We get R 1 : Then we can confine the U sp(4) node. We will reach the self-dual frame of the original theory. Indeed, we produce a traceless antisymmetric field, B, for the U sp(6) and the fields h 1 , α 2 and v 3 get a mass. The final quiver reads We can repackage the final result into a manifestly SU (8) invariant way Where we define During this sequential reconfinement, from T Dec and R f inal , the mapping is Comparing with the mapping (3.10), we read the mapping for the self-duality This is precisely the mapping given in [45].
Notice that in the reconfinement the precise operator flipped by the singlet H 2 was crucial to obtain the duality with the correct amount of gauge singlets.
In the next section we generalize our discussion to arbitrary N and F .
In this section we study the general case. The U sp(2N ) gauge theory with a traceless antisymmetric field A and 2F complex chiral fields (the number of fundamentals should be even to avoid the global anomaly). We proceed as in [9] to derive a chain of 2N dual frames consisting of quiver theories, with number of gauge nodes ranging from 1 to N . Let us start with the first T 0 quiver

Deconfine and dualize: first step
We start by using the deconfinement (2.1), we obtain Then we dualize the U sp(2N ) node with (2.3). This step is the same as in (3.1).
The mapping of the chiral ring generators after this first step is the following Now we iterate the procedure. We deconfine the traceless antisymmetric field, Φ and then we dualize. Let us write explicitly another step and then it will be enough to obtain the general story.

Second step
After the deconfinement we get Now we dualize the U sp(2N − 2) node. The fields q 1 , d 2 and β 2 get a mass. In addition tr (c 1 c 2 c 2 c 1 ) becomes a mass term therefore there will be no link between the U sp(2F − 6) and U sp(2N − 4) gauge group. After the integration of these massive fields and a rescaling we get We recall that the antisymmetric field A 1 is traceles, as well as Φ.
The mapping after this second step is given by 8

After k steps
After the iteration of k steps, we get the following quiver The last term in the superpotential should be taken with a grain of salt. Indeed as explained in the appendix A, when k is great enough some operators become degenerate and then the superpotential should be modified. This is the degenerate holomorphic operator ambiguity that we described in (3.4). Since it is a k-dependent statement, we decided to be cavalier when writing this term in (4.8) and write the modified version in T N −1 .
The mapping of the chiral ring generators is the following

After N-1 steps
The last term in the superpotential is the one after switching, see Appendix A. Since the last node is U sp (2), there is no antisymmetric and we can directly use (2.3).

Final step and fully deconfined frame
. . .
We have obtained the fully "deconfined" frame. The mapping is Collecting all the mappings, we write the mapping from the T 0 frame to T Dec One interesting property of the "deconfined" frame is that all chiral ring generators are elementary gauge singlets.

Reconfinement and self-duality for U sp(2N ) with + 8
It was proposed in [45] that in the special case of F = 4 the theory studied in the previous section, U sp(2N ) with antisymmetric and 2F fundamentals, is self-dual modulo flips. In this section we use our T Dec frame (4.11) to prove this result. Let us rewrite it, specifying F = 4: We see that the U sp(2) gauge group is coupled to 4+1+1 chiral fields in the fundamental representation and so it confines (2.2). This step is similar as in (3.11). The confinement will give a mass to the traceless antisymmetric field a 2 as well as v 2 , α 1 and h N −1 . After integrating them out and computing the Pfaffian superpotential (See discussion below (3.5)) we get 3) The mapping of the chiral ring generators is it will stay like that i = 1, . . . , N − 2 (5.4) We can now iterate. Indeed the U sp(4) group is coupled to 6 + 1 + 1 fundamentals and so it also confines We can iterate. After k confinement, we get The mapping between two successive reconfinement is For k = N − 2, there won't be any antisymmetric field left. We get Now we can do the last confinement with the U sp(2N − 2) group. It will produce the traceless antisymmetric field, B for U sp(2N ) (the trace part is killed by the flipper α N −1 ). We get . . .
The mapping for this last reconfinement is given by . . .
We can repackage the last frame into a manifestly SU (8) invariant way to obtain the final frame . . . Now combining the mappings, we see that the reconfinement procedure gives Now if we compare the original frame T 0 and the last frame after reconfinement R f inal we see the self-duality and we obtain the following mapping Which is precisely the mapping proposed in [45].

Reduction to 3d N = 2 sequential deconfinement
It is possible to reduce 4d N = 1 theories on a circle, obtaining 3d N = 2 theories. Generically, this steps introduces a superpotential term linear in the basic monopole operator (exceptions are, for instance, theories with 8 supercharges). Once in 3d, it is possible to turn on real mass deformations, that do not exist in 4d. Starting from a 4d U sp(2N ) gauge theory, 3d real masses allow to flow to U sp(2N ) or U (N ) gauge groups with or without various types of monopole superpotentials. This process has been discussed in detail in the case without rank-2 matter [10], and for the case of 2F = 8 [7,8]. A brane interpretation has been found in [46]. Examples of 4d N = 1 simplectic quivers reduced and deformed to 3d N = 2 or 3d N = 4 unitary quivers have been discussed in [5,37], mostly from the superconformal index perspective.
On the electric side, the story is as follows: Where the rank-2 field is a traceless antisymmetric for U sp(2N ) and a traceless adjoint for U (N ). The monopoles M, M ± are the monopoles with minimal GNO charges. See [10] for more details. We could also turn on different real masses, possibly leading to non-zero Chern-Simons terms as in [7][8][9], but we refrain to do this in the present paper.
In the remaining of this section we perform the reduction and deformation of the fully deconfined dual, recovering the results found in [9] working in 3d.
Reduction to the deconfined dual of U sp(2N ) with antisymmetric and 2F + 2 fundamentals, W = M We put the 4d duality on a circle. On the electric side we get 3d N = 2 U sp(2N ) with antisymmetric and (2F + 1) Q + 1 P fundamentals, W = M, with global symmetry SU (2F + 2) × U (1). On the magnetic side we obtain the same quiver as in 4d, (4.11), with F → F + 1. The difference is that the superpontential now includes N additional terms, linear in the monopole operators with GNO charges for a single gauge group, 2) For convenience we reproduce also the mapping of the chiral ring generators: This is the same mapping as in 4d, at this level there are no monopoles in the chiral ring, due to the presence of linear monopole terms in the superpotential.
Flow to the deconfined dual of U sp(2N ) with antisymmetric and 2F fundamentals, W = 0 We now discuss what happens on the fully deconfined quiver 6.2 upon turning on real masses. We first turn on a real mass of the form (0 2F , +, −) (that is we are moving to the left in the diagram 6.1), on the electric side Q 2F +1 and P become massive. Notice that the rank of the global symmetry decrease by one unit. Accordingly the mesons tr (Q I A i Q 2F +1 ) and tr (Q I A i P ) have non-zero real mass, for I = 1, . . . , 2F . Notice that tr (Q 2F +1 A i P ) has zero total real mass. The electric theory becomes 3d N = 2 U sp(2N ) with antisymmetric and 2F fundamentals, W = 0, with global symmetry It follows from the mapping 6.3 that the singlets (l i ) I and (m i ) I,2F +1 become massive for I = 1, . . . , 2F , while (l i ) 2F +1 remain massless. The l j 's and (m i ) I,2F +1 's become massive imply that also the elementary gauge variant fields v i , r i and (q N ) 2F +1 become massive. In other words the saw structure in the fully deconfined quiver disappears, and we are left with  [47]). These interactions are generated dynamically, one way to understand them is that such interactions are allowed by all global symmetries, and if they are not generated the gauge singlets would be free fields, which cannot be correct. Equation (6.4) agrees with the results of section 2.4 of [9] (modulo renaming h i → γ i and (l i ) 2F +1 → σ i ), obtained by sequentually decofining in 3d, using the deconfining The only difference is the precise extended monopole flipped by h i , the subtlety related to the degenerate holomorphic operators which can in principle be flipped by h i was not appreciated in [9]. One can check that with the superpotential above, setting F = 3, the tail reconfines appropriately and it is possible to derive the self-duality modulo flips of 3d N = 2 U sp(2N ) with antisymmetric and 6 fundamentals, W = 0, at each step one h i singlet is eaten, while the extended monopoles flipped by (l i ) 2F +1 'shorten' according to the rules of [48].
Flow to the deconfined dual of U (N ) with adjoint and F + 1 flavors, W = M + + M − We now start from the 3d duality U sp(2N ) with 2F + 2, W = M ↔ 6.2, and turn on a real mass of the form (+ F +1 , − F +1 ) (that is we are moving to the right in the diagram 6.1). This type of real mass induces a Higgsing of the form U sp(2N i ) → U (N i ) on both sides of the duality, the antisymmetric fields are replaced by adjoints and a pair of fundamentals is replaced by a fundamental plus an antifundamental. The Higgsing is induced by a vev of 'Coulomb branch type', that is, on both sides of the duality, we are going to a specific sublocus of the moduli space of vacua (inside the so called N = 2 Coulomb branch) where there is the maximum amount of massless fields. Moreover, the monopole superpotentials M's are replaced by (M + + M − )'s. See [5,10,37] for more details.
On the electric side we flow to U (N ) with adjoint and (F Q + 1 P , (F + 1)Q) flavors and W = M + + M − , with global symmetry SU (F + 1) × SU (F + 1) × U (1) 10 . The rank of the global symmetry decreases by one unit, as it should.
On the magnetic side, we end up with a fully deconfined quiver . . .
Where we did not draw the h i singlets 11 . The global symmetry of (6.5) is The mapping is 12 On the magnetic side, using 6.6, the singlets (l j ) I and (M i ) J F +1 become massive. This in turn implies thatq F +1 gets a mass, together with the v i 's and the r i 's. Hence the saw disappears and we end up with . . .
Notice that half of the linear monopole superpotential disappeared and that the massless gauge singlets (l i ) F and h i now flip monopole operators instead of mesons constructed with the saw.
In the special case F = 3, we can use the confining duality for U (N ) with (N + 1, N + 1) flavors and W = M − + hM + [10] to reconfine the tail in 6.7, deriving the self- 12 We have checked that the mapping is consistent by computing the R-charges of the operators (as a function of two variables coressponding to the two U (1) symmetries) on both sides. duality modulo flips of 3d N = 2 U (N ) with adjoint and (3, 3) fundamentals, W = M + [7].
Flow to the deconfined dual of U (N ) with adjoint and F − 1 flavors, W = 0 We now turn on real masses (0 F −1 , +; 0 F −1 , −) in the previous duality.
On the electric side one monopole superpotential is lifted and we flow to U (N ) with adjoint and F − 1, F − 1 flavors and W = 0, with global symmetry SU (F − 1) × SU (F − 1) × U (1) 3 . Again, the rank of the global symmetry decreases by one unit.
On the magnetic side, 8) The result 6.8 agrees with section 3.2 of [9] (modulo renaming h i → γ i , (l i ) 2F +1 → σ + i , (M i ) F,F → σ − i and F → F + 1). Also here, the difference is the precise extended monopole flipped by h i , the subtlety related to the degenerate holomorphic operators which can in principle be flipped by h i was not appreciated in [9]. In the special case F = 3, we can use the confining duality for U (N ) with (N + 1, N + 1) flavors and W = M − + hM + to reconfine the tail in 6.8 and derive the self-duality modulo flips of 3d N = 2 U (N ) with adjoint and (2, 2) fundamentals, W = 0, discussed in [7]. 13 13 At each reconfining step, one linear monopole term disappears, one h i is eaten, while the (M i ) J I 's (I, J = 1, 2) and the two towers (l i ) 2F +1 and (M i ) F,F survive. The self-duality reads A R-charges and degenerate holomorphic operator ambiguity in generic frame T k We start by writing the R-charges of the fields in the generic frame T k (4.8). Table 1. R-charges in the frame T k with k = 1, . . . , N − 1, i = 1, . . . , k and j = 1, . . . , k − 1.
Now we want to find the degenerate operators that can couple to H i , as we did in (3.4) in the case of U sp (6). In order to so, we look at the R-charges of the operator V i B i . . . B k−1 C k P k+1 which is the natural candidate to be coupled to H i . We find The potential degenerate operators should be a singlet under the non-abelian global symmetry and should have the same R-charges (A.1). We can build the degenerate operators from the fields V a , B b and R c . Indeed, we start from V m , then we put some B b and end with R n (n ≥ m) 14 . The form of these operators is then: V m B m . . . B n−1 R n . The number of fields B is n − m. The R-charge is If we compare with (A.1), we find the following condition with 1 ≤ m ≤ n ≤ k − 1, i = 1, . . . , k and k = 1, . . . , N − 1.
We can already make two remarks: • For i = 1, the constraint becomes n − m = N − 2 but the maximal value of n − m is k − 2 and k satisfies k ≤ N − 1. We conclude that there is never a solution for i = 1. Therefore there is never a degenerate operator associated to H 1 .
• If k = 1 then n and m don't exist. Conclusion, in order to get degenerate operators we should have N ≥ 3 which means that degenerate operators will pop up in frames with at least 3 gauge groups (which correspond to k = 2).
We can ask the more precise question: What is the first frame, T k min , when some operators degenerate?
In order to answer that we have to try to maximize the l.h.s of (A.3) and minimize the r.h.s. Therefore it is enough to look at n = k min − 1, m = 1 and i = k min to determine k min (it could also have degenerate operators in T k min not associated to n = k min − 1, m = 1 and i = k min ). We obtain  14 We cannot end with V j+1 because the F-term equation of R j sets the combination B j V j+1 to 0.
Conclusion, the degenerate operators (with respect to V i B i . . . B N −2 C N −1 F N ) that potentially coupled to H i in T N −1 are: Finally, we can study the final frame T Dec . The R-charges are the following Table 2. R-charges in T Dec with i = 1, . . . , N and j = 1, . . . , N − 1. Now the question is obvious, which operator is the correct one? In the special case of F = 4, we could use the same argument that we used in Section 3. It goes as follows. When we reach the frame T k min some operators become degenerate. In order to decide the correct operator, we start confining from the left (it is possible because in the case of F = 4 the gauge group becomes U sp (2)). Then at some point we will discover that using the F-term equation for H 1 (which is never associated to a degenerate operator as we saw) we can select the correct operator associated to some H a (as in (3.6)). Then, the procedure is iterative meaning that we should re-use our previous results for H a and do more and more reconfinement to select all the correct operators associated to the other H b . All in all, we end up in the frame T N −1 with the following superpotential term which becomes in the final frame T Dec Unfortunately, in the case of F > 4 the previous argument fails because we cannot reconfine from the left. In this case we can deconfine the antisymmetric field A 1 but we didn't manage to find constraints and remove the degeneracy. Therefore, in this case the superpotential that we wrote in (4.10) and (4.11) are ambiguous. We wrote them with the results (A.5), (A.6) obtain in the case F = 4 but it is logically possible that they are wrong for F > 4.