Flips, dualities and symmetry enhancements

We present various 4d $\mathcal{N}=1$ theories enjoying IR global symmetry enhancement. The models we consider have the $USp(2n)$ gauge group, 8 fundamental, one antisymmetric chirals and various numbers of gauge singlets. By suitably turning on superpotential deformations involving the singlets which break part of the UV symmetry we flow to SCFTs with $E_6$, $SO(10)$, $SO(9)$, $SO(8)$ and $F_4$ IR global symmetry. We explain these patterns of symmetry enhancement following two arguments due to Razamat, Sela and Zafrir. The first one involves the study of the relations satisfied by marginal operators, while the second one relies on the existence of self-duality frames.


Introduction
The 4d N = 1 SU (2) gauge theory with 8 fundamental chiral fields admits 72 dual frames which are rotated into each other by the action of the coset group W (E 7 )/S 8 [1]. In addition to the original description, there are 35 Seiberg dual frames [2], 35 Csaki-Schmaltz-Skiba-Terning (CSST) dual frames [3] and 1 Intriligator-Pouliot (IP) dual frame [4]. Since the theory has 8 fundamental chirals without any superpotential, it preserves the SU (8) global symmetry. In the Seiberg and CSST dual frames, however, this SU (8) symmetry is broken to SU (4) × SU (4) × U (1) in the UV whereas it is restored at the IR fixed point.
In [1] it was shown that those 72 dualities form an orbit of W (E 7 )/S 8 and also found that this structure persists for higher rank U Sp(2N ) theories provided an extra matter in the traceless antisymmetric representation of the gauge group U Sp(2N ) is added. In the higher rank case 35 frames correspond to the duality discussed in [5] while the other frames correspond to generalizations of Seiberg and IP dualities.
It is natural to wonder whether it is possible to construct theories which actually display E 7 , or other enhanced symmetries, rather than being rotated to a dual frame by the E 7 Weyl action. The first theories with E 7 were constructed in [6]. This E 7 model as well as many other models with enhanced global symmetries can be realised geometrically by compacftifying the 6d N = 1 SCFTs on a Riemann surfaces with fluxes for the global symmetry of the sixdimensional theory [7][8][9][10][11][12][13][14][15][16]. It is indeed expected that the subgroup of the 6d symmetry preserved by the flux will also be the global symmetry of the resulting 4d N = 1 SCFT. It also often turns out that the expected global symmetry is not visible from the UV quiver description of the SCFT but it emerges in the IR.
The 6d perspective allows us to make interesting predictions for models with symmetry enhancements in 4d. These can then be tested with a more direct 4d analysis, like the computation of the superconformal index [17][18][19]. On the other hand it is also possible to develop purely 4d QFT strategies to understand and predict symmetry enhancements.
In this note we follow two main strategies to discuss several models with SU (2) gauge group and 8 chirals and various amounts of singlets displaying E 6 , SO(10), SO (9), SO (8) and F 4 symmetry in the IR 1 . We also construct higher rank versions of these models, showing that N = 1 U Sp(2N ) theories with 8 fundamental and one antisymmetric chirals, with various selections of singlets and superpotentials, display E 6 , SO(10), SO (9), SO (8) and F 4 symmetries in the IR.
The first strategy relies on the relation between the symmetry enhancement and the chiral ring relations of marginal operators, which was discussed in [15] and can be summarised as follows. As observed in [21] conserved currents and marginal operators contribute to the order pq of the superconformal index: I = · · · + (χ ind-mar − χ cur )pq + . . . One can remove those marginal operators by flipping either O (1) or O (2) . Say we flip O (2) by an extra flipping field F in the representation R (2) . Then, we have additional contributions originating from F and its supersymmetric partner ψ F (where ψ F denotes the fermionic partner of the scalar F ): (1.5) which cancel the contribution of O (1) O (2) and leave −χ τ pq. The remaining contribution −χ τ pq, which used to be the chiral ring relation of marginal operators, now joins the current.
Thus, the total current multiplet contribution is enlarged to −(χ current + χ τ )pq and may form the adjoint representation of an enhanced global symmetry. In section 2 we will extend this method and consider rank one models where only part of the marginal operators are removed. This has the effect of breaking the UV manifest symmetry to a subgroup, but we will gain a variety of interesting IR enhancements.
The second strategy was proposed in [22] and relies on the interplay between self-dualities and enhanced symmetries. With "self-duality" we mean that the dual theory has exactly the same gauge group, the same matter content (including gauge singlets) and the same superpotential of the original theory, but the two theories are related by a non-trivial map on the operator spectrum and on the global symmetries. The existence of self-dual frames implies that the theory is invariant in the IR under a larger set of transformations than the manifest UV symmetry group. In favorable situations, these additional transformations can lead to a symmetry enhancement in the IR. More precisely, if G IR is the IR symmetry group, then we should be able to identify as many equivalent frames of the theory as the dimension of the Weyl group of G IR . In the case in which part of this symmetry is enhanced in the IR, that is G U V ⊂ G IR , then we expect the transformations W (G IR )/W (G U V ) to come from non-trivial self-dualities of the theory while the transformations of W (G U V ) are trivial invariances of the UV Lagrangian.
Reversing the argument, whenever we have self-duality frames we might expect an enhanced symmetry. For example for the U Sp(2N ) theories with 8 fundamental and one antisymmetric chiral fields mentioned above, by adding extra singlets, we might find a subset of the 72 dual frames which are actually self-dualities in the sense specified above and provide the missing frames to account for an enhanced IR symmetry. Following this strategy [22] constructed an SU (2) theory with E 6 global symmetry and [23] a U Sp(4N ) model with E 7 × U (1) global symmetry, which for n = 1 is related to the model of [6].
In section 3 we will apply this line of reasoning to the models of section 2, listing the extra self-duality frames accounting for various types of enhancement and checking the superconformal index. We will also discuss various deformations which break some of the manifest UV symmetries leading to further interesting IR enhancement. In 4 we will present the higher rank version of these models involving U Sp(2N ) gauge groups. In the appendix A we quickly revise the action of the Seiberg, CSST and IP dualities on the global symmetries. In appendix B we will discuss the IR behavior of the F E[U Sp (4)] theory, which is part of the family of F E[U Sp(2N )] theories discussed in [14], arguing that it enjoys SO(10) × U (1) 2 global symmetry. Finally in appendix C we study the plethystic logarithm of the index to extract the relation satisfied by marginal operators.

Flips, relations and enhanced global symmetries
In this section we apply the first strategy to look for models with interesting symmetry enhancement patterns. We begin by revisiting the discussion of the E 7 model of [23] from this perspective.
Let's consider the U Sp(4) gauge theory with 8 fundamental chirals Q i , i = 1, . . . , 8 one traceless antisymmetric chiral X and a singlet x 2 with W = x 2 X 2 . Its superconformal index is given by 2 I = 1 + 28 t − 1 2 (pq) ∆ Q + 28 t 1 2 (pq) 1−∆ Q + t −2 (pq) 1−∆ A + (336 + 70) t −1 (pq) 2∆ Q + 28 t − 1 2 (p + q)(pq) ∆ Q + (378 + 336 − 63 − 1) pq + . . . (2.1) where ∆ Q is the R-charge of the 8 fundamental chirals and ∆ A is that of the antisymmetric chiral. ∆ Q and ∆ A satisfy a relation 4∆ Q + ∆ A = 2 which comes from requiring the existence of a non-anomalous R-symmetry. The first three terms are the contributions of the chiral ring generators 3 m 0,ij ≡ Tr g (Q i Q j ) , m 1,ij ≡ Tr g (Q i A Q j ) , x 2 (2.2) respectively, where the first two are in the antisymmetric representation of the SU (8) flavor symmetry while the last one is a singlet. In addition, one can see the current multiplet contribution −(63 + 1) pq, which reflects the SU (8) × U (1) t global symmetry. In this theory, the marginal operators are given by satisfying the relation which can be explained as follows [23]. Consider the object Q a i Q b j Q c k Q d l X ef , where both the U Sp(4) gauge indices a, b, c, d, e, f and the SU (8) flavor indices i, j, k, l are not contracted. We want to show that if we antisymmetrize all the flavor indices, there is no way of contracting the gauge indices to make a gauge invariant object. This is because Q a , where all the flavor indices are antisymmetrized, transforms in the fourth antisymmetric power of the fundamental representation of U Sp(4) (since the Qs are bosons), which is just a singlet. Hence, there is no way of multiplying this by the antisymmetric A ef and contracting the gauge indices so to make a non-vanishing gauge invariant object. Thus, those in (2.3) subject to the relation (2.4) are 28 × 28 − 70 = 378 + 336 independent marginal operators.
One can remove the marginal operators by flipping either m 0 or m 1 , where the two choices are merely related by a duality. Once the marginal operators are removed, the 70 relations among them now join the current. Thus, the total number of conserved currents is 63+1+70 = 133+1, which form the adjoint representation of E 7 × U (1). The model where m 0 is flipped by M 0 is exactly the model with E 7 × U (1) global symmetry found in [23].
One may wonder whether we can analogously construct an SU (2) model with E 7 symmetry. If we look at the index of the SU (2) theory with 8 chirals and W = 0: we note two things: first, the global symmetry is SU (8) without additional U (1) because there is no antisymmetric matter for the SU (2) theory; second, the only chiral ring generators are the mesons m 0,ij = Tr g (Q i Q j ), which transform in the antisymmetric representation of such SU (8) flavor symmetry. The marginal operators in this theory are given by m 0,ij m 0,kl (2.6) subject to the relation One way to see where these relations originate is along the lines of the argument we used for the U Sp(4) gauge theory. If we consider the combination Q a where all the SU (2) gauge indices are not contracted while the SU (8) flavor indices are antisymmetrized, it should transform in the fourth antisymmetric power of the fundamental representation of SU (2), which doesn't exist. Hence, we can't construct a gauge invariant object out of it. Thus, those in (2.6) subject to the relation (2.7) are 28·29 2 − 70 = 336 independent marginal operators.
One may attempt to remove the marginal operators by flipping m 0 such that the 70 relations are translated into 70 conserved currents as above. However, even though we introduce new flipping fields, say M 0,ij , which flip m 0,ij such that the original marginal operators are removed, M 0,ij provide new marginal operators M 0,ij M 0,kl subject to the same number of relations M 0,[ij M 0,kl] = 0. Thus, there is no change in the contributions of the relation and the current multiplet. This is because the assumption O (1) = O (2) in the above argument fails to hold. Indeed, a similar situation happens for higher odd ranks; there are operators of R-charge 1 whose squares give marginal operators which cannot be removed by the flipping of the operators of R-charge 1. Therefore, one cannot obtain an E 7 model for odd-rank theories, at least in this way.
Since in SU (2) models we cannot achieve the symmetry enhancement by removing completely the marginal operators we can try to introduce flips of operators which break the SU (8) global symmetry into subgroups and only partially remove the marginal operators. We will see that while those partial flips reduce the manifest global symmetry in the UV, they eventually lead to intriguing patterns of symmetry enhancements in the IR.
Since the partial flips break the UV global symmetry, they can be organised along the line of the symmetry breaking pattern of the SU (8) global symmetry. The maximal subgroups of SU (8) are: The R-charges of the chiral fields in the original theory preserving SU (8) are determined by the anomaly condition and do not change along the RG-flow because there is no U (1) that can be mixed with U (1) R . On the other hand, if the UV symmetry is broken as in (2.8) by introducing flipping fields we do have a U (1) symmetry. If this U (1) is not mixed with U (1) R , the R-charges of the operators do not change and we cannot remove the marginal operators for the same reason we explained before for the E 7 case. Thus, nothing interesting happens in this case. On the other hand if the U (1) is mixed with U (1) R , the contributions charged under U (1) at order pq before the flip won't appear at order pq anymore after the flip. Thus, only the U (1)-neutral contributions will remain at order pq and, among these, those with negative sign that used to correspond to relations before the flip are of our interest since they may now combine to the flavor current. Thus, when we will decompose operators, relations and currents using the branching rules for the (2.8) cases, we will only look at the U (1)-neutral contributions. As we are going to discuss below only the symmetry breaking to SU (6) × SU (2) × U (1) has neutral sectors suitable for our discussion and leads to E 6 × U (1) IR global symmetry.
We can then further break the SU (6) × SU (2) × U (1) symmetry. We may consider either the breaking of SU (6) or SU (2). The breaking of the latter into U (1), however, doesn't have U (1)-neutral relations, thus, we only need to consider the breaking of SU (6) into: In this case only the SU (4) × SU (2) × U (1) breaking has U (1)-neutral sectors and as we will see leads to SO(10) × U (1) 2 IR global symmetry. Lastly, we consider the breaking of SU (4), which includes where only SU (2) × SU (2) × U (1) has U (1)-neutral relations and as we will see leads to We begin by decomposing SU (8) representations in terms of SU (6) × SU (2) × U (1) v (we use the branching rules of [24]): • Conserved currents: • Relations: For each contribution above, the U (1)-neutral sectors are as follows.
Our analysis actually holds regardless of the gauge rank. The only change is that there is another U (1) a in the UV global symmetry acting on the traceless antisymmetric matter. Hence, we expect higher rank models exhibiting E 6 × U (1) v × U (1) a obtained by partial flips of operators constituting the marginal operators. We will discuss this in section 4.
In section 3.1 we will check this enhancement using the superconformal index. The higher rank version of this model is discussed in section 4.

Flips, self-dualities and symmetry enhancements
In the previous section we observed that by considering three different partial flips breaking the SU (8) UV global symmetry of the SU (2) theory with 8 fundamental chirals to we expect to find models exhibiting enhanced IR global symmetries respectively. In section 3.1 we will check that these models indeed enjoy the expected symmetry looking at the superconformal index expansion and in addition, following the second strategy discussed in the introduction, we list all the duality frames accounting for the enhanced global symmetry. In section 3.2 we will consider various deformations, which in particular lead to models with SO(9) and F 4 symmetries.
We begin by introducing all the fields and their charges. In this section we will work with conventions in which only the SU (2) 4 × U (1) t × U (1) u × U (1) v subgroup of the full UV symmetry is explicitly manifest. This is done by splitting the 8 fundamental chiral fields into four doublets Q 1 , . . . , Q 4 , one for each SU (2) flavor symmetry. The singlets we introduced in the previous section to break the SU (8) symmetry will also split accordingly. All models will have bifundamental singlets D 1 , D 2 , D 3 coupled as: 3) where Tr 1,2,3,4 denote the traces over the SU (2) 1,2,3,4 flavor indices. For some models we will also consider singlets b i with i = 1, · · · , 4 contributing b i Tr g Tr i (Q i Q i ) to the superpotential.
In general we will denote by T i with i = 1, 2, 3 the theory where the interactions involving respectively b 1 or b 1 , b 2 or b 1 , b 2 , b 3 are turned on. We will also denote by T i the theory where also the interaction involving b 4 is turned on. We use a different notation in this case because this latter interaction is not involved in the symmetry enhancement process but we might need to turn it on to avoid having decoupled fields.
For convenience we will also work in a different basis for the U (1) 3 symmetry. Specifically, we will use a parametrization of the abelian symmetries that we will denote by U (1) the previous section by the following redefinition of the charges: 4) or equivalently at the level of the fugacities in the index The charges of the fields with this new parametrization are as in table 1, where we also give a possible choice of UV trial R-symmetry U (1) R 0 .

Self-dualities and Enhancements
We start discussing two models enjoying the SO(8) × U (1) 3 enhancement. In the first model T 0 we introduce only the singlets D 1 , D 2 , D 3 interacting with W 0 . The label for the theory stands for the fact that we don't introduce any of the b i fields in this case. The matter content of the theory is summarized in the quiver diagram of figure 1.
Combining the information on the U (1) 3 charges of table 1 with the redefinition (3.4) we can see that the operators flipped by D 1 , D 2 , D 3 in W 0 are precisely the operators (2, 2, 1, 1) 0,2,2 , (1, 2, 2, 1) −1,−1,2 , (2, 1, 2, 1) 1,−1,2 of section 2.3. Hence, we expect in this case that the manifest SU (2) 4 × U (1) 3 UV global symmetry gets enhanced in the IR to SO(8) × U (1) 3 . This can be checked computing the superconformal index of the theory T 0 . We first perform a-maximization [25] to find the values of the mixing coefficients of the R-symmetry with U (1) corresponding to the superconformal R-symmetry, which we approximate to The superconformal index then reads 4 (3.8) In the expression of the index, each number is the character of an SO(8) representation and, in particular, the term −(28 + 3)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of SO(8)×U (1) 3 . In this case we also have 72−48 = 24 marginal operators. Notice that the fact R 1 = R 2 = R 3 implies, among the three U (1) symmetries, only U (1) v mixes with the R-symmetry. Thus, although we have used approximate R-charges for the expansion of the index, the terms independent of fugacity v are exact; the terms of order pq, corresponding to 24 marginal operators and 31 conserved currents, are such examples. Now according to the argument in [22], since the size of the Weyl group of SO(8) is given by Figure 1. Quiver diagram for the T 0 model. and W (SU (2) 4 ) = 2! 2! 2! 2! is manifest, we expect 12 = 4! 2 3 /(2! 2! 2! 2!) self-dual frames including the original theory itself. To find the self-dual frames we proceed as follows. We specialize the SU (8) fugacities vector u defined in (A.2) according to the breaking of SU where y i is the SU (2) i fugactity. This corresponds to choosing one particular representative in the oribit of the UV SU (2) 4 Weyl group. Now we implement the Seiberg, CSST and IP dualities. As explained in appendix A, these dualities transform the fugacities vector respectively as in eqs. (A.8), (A.12), (A.5). Inspecting the transformed vectors we can identify the self-dual frames. Those will correspond to frames where we have a collection of charged chirals with the same R-charge and U (1) t,u,v charges as in the original frame. We also checked that the self-dual frames have the same collection of singlets. In the end we have found that the 12 self-dual frames are realized by 1 original , Another model exhibiting SO(8) × U (1) 3 enhancement is T 1 , obtained including the singlet b 1 with the superpotential 5 5 Given the symmetry of the quiver in figure 2 this is equivalent to introducing the singlet b2.
where W 0 is given by The matter content of the theory is now summarized in the quiver diagram of figure 2. Notice that the new b 1 field is flipping the meson Tr g (Q 1 Q 1 ) corresponding to the operator (1, 1, 1, 1) 2,2,2 in the notation of section 2.3.
Performing a-maximization we find the following approximate values of the mixing coefficients: . (3.14) The index computed with this R-symmetry is then given by Again the index organises into characters of SO(8) and at order pq we can see the contribution of the SO(8) × U (1) 3 current highlighted in blue.
We can explain the enhancement of theory T 1 in terms of self-dualities exactly in the same way as for theory T 0 . Indeed, the operator Tr g (Q 1 Q 1 ) is trivially mapped to itself under all the self-dualities (3.11). Hence, these are also self-dualities of theory T 1 and the same counting we did for T 0 explains the SO(8) enhancement for T 1 .
The singlet b 4 is also a spectator from the point of view of the self-dualities (3.11). This means that theories T 0 and T 1 , where we also turn on b 4 Tr g Tr 4 (Q 4 Q 4 ) in the superpotential, will still exhibit the SO(8) × U (1) 3 enhancement.

SO(10) × U (1) 2 model
Now we consider a model with the IR SO(10) × U (1) 2 symmetry. We denote this model by T 2 as, in addition to the usual D 1 , D 2 , D 3 singlets, we also introduce the singlets b 1 , b 2 . The superpotential is where W 1 is given by The matter content of the theory is summerized in the quiver diagram of figure 3, but the full manifest UV global symmetry is actually In particular D 3 and b 1 , b 2 form the antisymmetric representation of SU (4), which flip the mesonic operators corresponding to the operator (6, 1, 1) 2,2 of section 2.2. Indeed, as it can be seen combining the data contained in table 1 and the map of the charges (3.4), their charges are compatible with those expected from the branching rule (2.27). The singlets D 1 , D 2 also recombine to form the bifundamental representation between SU (4) × SU (2) 3 , which flips corresponding to the operator (4, 2, 1) −1,2 .
Performing a-maximization we find the following approximate values of the mixing coefficients: Notice that equal R 1 and R 2 imply U (1) t doesn't mix with the R-symmetry, which is consistent with the fact that U (1) t is part of the nonabelian symmetry SU (4). The index computed with this R-symmetry is then 6 Each number is the character of an SO(10) representation and, in particular, the term −(45+ 2)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of SO(10) × U (1) 2 . Let's now discuss the self-duality frames responsible for the enhancement. The size of the Weyl group of SO(10) is given by where W (SU (4) × SU (2) 2 ) = 4! 2! 2! is manifest. Thus, 20 = 5! 2 4 /(4! 2! 2!) self-dual frames including the original theory itself are expected. Also in this case to find the self-dual frames we specialize the SU (8) fugacities vector u defined in (A.2) according to the breaking of where w i are the SU (4) fugacities subjected to the constraint 4 i=1 w i = 1. In this case we collect a subset of the transformations (A.8), (A.12), (A.5) corresponding to Seiberg, CSST and IP dualities respectively for which we get a transformed vector associated to a collection of chirals with the same R-charge and U (1) u,v charges, as well as the same set of singlets. In the end we have found that the 20 self-dual frames are realized by

dualities.
Also in this case the singlet b 4 is a spectator from the point of view of the self-dualities (3.24), so the theory The last model we consider is the one exhibiting the E 6 × U (1) symmetry enhancement of [22]. 7 Following the same nomenclature of the previous cases, we call this theory T 3 as on top of the singlets D 1 , D 2 , D 3 we also introduce all the singlets b 1 , b 2 , b 3 , b 4 with superpotential where W 2 is given by In this case we have to include also the singlet b 4 since otherwise the operator Tr g Tr 4 (b 4 Q 4 Q 4 ) would be free in the IR. The matter content of the theory is summerized in the quiver diagram of figure 4, but the full manifest UV global symmetry is actually SU (6)×SU (2) 4 ×U (1), since SU (2) 1 , SU (2) 2 , SU (2) 3 and two U (1) out of U (1) 3 , specifically U (1) t and U (1) u , recombine into SU (6). In particular the singlets D 1 , D 2 , D 3 and b 1 , b 2 , b 3 form the antisymmetric representation of SU (6), which flips the mesonic operators which indicates U (1) t and U (1) u do not mix with the R-symmetry because they are part of the nonabelian symmetry SU (6). The index computed with this R-symmetry is then 9 Each number is the character of an E 6 representation and, in particular, the term −(78+1)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of In [22] this enhancement was explained by studying the self-dualities of the model, similarly to what we did for the previous cases. The size of the Weyl group of E 6 is given by where |W (SU (6) × SU (2))| = 6! × 2! is manifest. The remaining 36 = 51840/(6! 2!) should be realized as self-dualities. In order to determine which of the 72 frames correspond to self-dualities, we specialize the SU (8) fugacities vector u defined in (A.2) according to the breaking of SU where w i are the SU (6) fugactities subjected to the constraint 6 i=1 w i = 1. In this case we collect a subset of the transformations (A.8), (A.12), (A.5) corresponding to Seiberg, CSST and IP dualities respectively for which we get a transformed vector associated to a collection of chirals with the same R-charge and U (1) v charges, as well as the same set of singlets. In the end we have found that the 36 self-dual frames are realized by Performing a-maximization in the theory without the singlet b4 we would find a value of the mixing coefficients for which the operator Trg(Q4Q4) is below the unitarity bound, meaning that it becomes a decoupled free field in the IR. 9 We choose to parametrize the U (1) symmetry that doesn't participate in the enhancement with U (1)v. dualities.

Symmetry breaking deformations
So far we considered singlets b i coupling to SU (2) flavor singlet mesons Tr g Tr i (Q i Q i ) which ensures that the manifest symmetry includes SU (N ) groups. On the other hand we can introduce extra deformations of the form b i Tr g Tr j (Q j Q j ) with i = j.
In the E 6 × U (1) modelT 3 all the SU (2) flavor singlet mesons are already flipped and thus trivial in the chiral ring, so this model cannot be deformed in this way. We will thus focus on the deformations of the SO(10) × U (1) 2 model T 2 and the SO(8) × U (1) 3 model T 1 (the theory T 0 can't be deformed in this way as it doesn't contain any of the b i singlets).
We begin with the T 2 model. Looking at W 2 in eq. (3.16) we see that Tr g Tr 1 (Q 1 Q 1 ) and Tr g Tr 2 (Q 2 Q 2 ) are trivial in the chiral ring while Tr g Tr 3 (Q 3 Q 3 ) is not. Thus, we can obtain a new theory T 2 by introducing a stable 10 deformation term 11 where W 2 is given by Here i can be either 1 or 2. For definiteness, we will take i = 2 and we sahll denote the deformed theory by T 2 . This deformation breaks the manifest UV symmetry SU (4)×SU (2) This can be seen as follows. The deformation (3.33) breaks one combination of U (1) Specifically, at the level of fugacities it imposes the constraint which means that a combination of U (1) x 2 ,x 3 or equivalently of U (1) t,u is broken. We decide to parametrize the surviving combination of these two U (1), which we shall denote by U (1)ũ, as At the level of the charges this means One can then check for example that the fields Q 2 and Q 3 have the same R-charge and U (1)ũ ,v charges and can thus be organized into the fundamental representation of U Sp (4). Similarly, the singlets D 1 and b 2 can be organized into the traceless antisymmetric representation of U Sp (4). Finally, the singlets D 2 and D 3 can be collected into the bifundamental representation of SU (2) 1 × U Sp (4).
Similarly to what we did in the previous subsection, one can check that out of the 20 selfdualities (3.24) of the original T 2 theory only 12 map ∆W 2 to itself and remain self-dualities of the deformed theories. These correspond exactly to the self-dualities (3.11) of the SO(8) models, which were 1 original , Therefore, the expected size of the Weyl group of the enhanced symmetry is now Thus, we expect that the SO(10) × U (1) 2 model T 2 deformed by (3.33) has the SO(9) × U (1) 2 IR symmetry with the UV symmetry SU (2) 1 × U Sp(4) × SU (2) 4 recombining into SO(9) in the IR. This is confirmed by the superconformal index expansion. Performing a-maximization we find the following approximate values of the mixing coefficients: The index computed with this R-symmetry is then 12 (3.42) 12 We choose to parametrize the U (1) symmetries, which don't participate in the enhancement, with U (1)ũ,v.
Each number is the character of an SO (9) representation and, in particular, the term −(36 + 2)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of SO(9) × U (1) 2 . Thus, the deformation (3.33) added to the SO(10) × U (1) 2 model T 2 leads to a new theory exhibiting the IR symmetry enhancement We can further deform the T 2 theory. From W 2 in eq. (3.33) we see that one linear combination of Tr g Tr 2 (Q 2 Q 2 ) and Tr g Tr 3 (Q 3 Q 3 ) is flipped by b 2 , but there is another linearly independent combination, say Tr g Tr 3 (Q 3 Q 3 ), which is non-trivial in the chiral ring. Thus, we can obtain a new theory T 2 by introducing another stable deformation term 13 where W 2 is given in (3.33). This deformation breaks the manifest UV symmetry SU (2) This can be seen as follows. The deformation (3.44) breaks one combination of the two U (1) symmetries. Specifically, at the level of fugacities it imposes the constraint which means that the U (1)ũ symmetry is broken. Hence, the surviving abelian symmetry is U (1) v , which we recall was defined in (3.4)-(3.5). One can then check for example that the fields Q 2,3 , which we already collected into the fundamental representation of U Sp (4), have also the same R-charge and U (1) v charge of Q 1 and we can thus organize them into the fundamental representation of U Sp (6). Similarly, the singlets D 1 , D 2 , D 3 and b 1 , b 2 can be organized into the traceless antisymmetric representation of U Sp (6).
The manifest Weyl group is W (U Sp (6)×SU (2)), whose size is 3!×2 3 ×2!. In addition, all the 12 self-dualities of the SO(9)×U (1) 2 model still remain self-dualities after the deformation (3.44). The expected size of the Weyl group of the enhanced symmetry is then This is confirmed by the superconformal index expansion. Performing a-maximization we find the following approximate value of the mixing coefficient: The index computed with this R-symmetry is then 15 19 + (324 + 1)v −4 (pq) 17 19 −(52 + 1)pq + · · · . (3.49) Each number is the character of an F 4 representation and, in particular, the term −(52+1)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of F 4 × U (1). Thus, the deformation (3.44) added to the SO(9) × U (1) 2 model T 2 leads to a new theory exhibiting the IR symmetry enhancement Note that T 2 can be also obtained from T 3 by integrating out the trace part of the antisymmetric representation of flavor SU (6) constituted by (D 1 , D 2 , D 3 , b 1 , b 2 , b 3 ), which breaks SU (6) into U Sp (6). This can be done by introducing an additional singlet c with a superpotential ∆W = c (b 1 +b 2 +b 3 ), which makes both c and the trace part b 1 +b 2 +b 3 massive. Once those massive fields are integrated out, the resulting theory is exactly T 2 with the superpotential (3.44). The enhanced IR symmetry E 6 × U (1) is then partially broken to F 4 × U (1).

SO(8) → SO(9) → SO(8) deformations
Now we consider the T 1 model with the SO(8) × U (1) 3 symmetry. While the SO(8) × U (1) 3 symmetry can be realized either with or without b 1 , we stick to the model with b 1 because we need at least one b i field to deform the theory.
Looking at W 1 in eq. (3.12) we can see that both Tr g Tr 2 (Q 2 Q 2 ) and Tr g Tr 3 (Q 3 Q 3 ) are non-trivial in the chiral ring. Thus, we can obtain a new theory T 1 by introducing a stable deformation term 14 where W 1 is given by Here i can be either 2 or 3. For definiteness, we will take i = 2 and we shall denote the deformed theory by T 1 . This deformation modifies the manifest UV symmetry SU This can be seen as follows. The deformation (3.51) breaks one combination of the three U (1) symmetries. Specifically, at the level of fugacities it imposes the constraint which means that the symmetry U (1) t is broken. Hence, the surviving abelian symmetries are U (1) u,v , which we recall were defined in (3.4)-(3.5). One can then check for example that the fields Q 1 and Q 2 have the same R-charge and U (1) u,v charges and can thus be organized into the fundamental representation of U Sp (4). Similarly, the singlets D 3 and b 1 can be organized into the traceless antisymmetric representation of U Sp(4). Finally, the singlets D 1 and D 2 can be collected into the bifundamental representation of U Sp(4) × SU (2) 3 . The Weyl group W (U Sp(4) × SU (2) 2 ) of the manifest symmetry is of size 2! × 2 2 × 2! × 2!. Similarly to what we did in the previous subsection, one can check that all of the 12 self-dualities (3.11) of the original T 1 theory are still self-dualities of the deformed theory. Therefore, the expected size of the Weyl group of the enhanced symmetry is now (3.55) Thus, we expect that the SO(8) × U (1) 3 model T 1 deformed by (3.51) has IR symmetry SO(9) × U (1) 2 . This is confirmed by the superconformal index expansion. Performing a-maximization we find the following approximate values of the mixing coefficients: (3.56) The index computed with this R-symmetry is then 15 (3.57) Each number is the character of an SO(9) representation and, in particular, the term −(36 + 2)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of SO(9) × U (1) 2 . Thus, the deformation (3.51) added to the SO(8) × U (1) 3 model T 1 leads to a new theory exhibiting the IR symmetry enhancement This T 1 model can be also obtained from T 2 by integrating out the trace part of the antisymmetric presentation of flavor SU (4) constituted by (D 1 , D 2 , b 1 , b 2 ), which breaks SU (4) into U Sp (4). This also partially breaks the enhanced IR symmetry SO(10) × U (1) 2 into SO(9) × U (1) 2 .
We can further deform the T 1 theory. From W 1 in eq. (3.51) we see that the operator Tr g Tr 3 (Q 3 Q 3 ) is still non-trivial in the chiral ring. Thus, we can obtain a new theory T 1 by introducing another stable deformation term 16 where W 1 is given in (3.51). This deformation breaks the manifest UV symmetry U Sp(4) × Indeed, the deformation (3.59) breaks one combination of the two U (1) symmetries. Specifically, at the level of fugacities it imposes the constraint which means that the U (1) u symmetry is broken. Hence, the surviving abelian symmetry is U (1) v , which we recall was defined in (3.4)-(3.5). The superpotential now doesn't preserve the U Sp(4) symmetry since the singlet b 1 , which in T 1 formed a U Sp(4) together with D 3 , now appears (without D 3 ) in the deformation (3.59). The manifest Weyl group is W (SU (2) 4 ), whose size is (2!) 2 . In addition, all the 12 self-dualities of the SO(9) × U (1) 2 model T 1 still remain self-dualities after the deformation (3.59). The expected size of the Weyl group of the enhanced symmetry is then Thus, we expect the SO(9) × U (1) 2 model T 1 deformed by (3.59) to have the IR symmetry SO(8) × U (1). Indeed, we have confirmed it by looking at the superconformal index. Performing amaximization we find the following approximate value of the mixing coefficient: (3.63) 16 Using the superconformal R-charge of theory T 2 we can check that this deformation has R-charge R[b1 Trg Tr3(Q3Q3)] 1.95598 < 2 so it is a relevant deformation.  9 10 −(28 + 1)pq + · · · (3.64)

The index computed with this R-symmetry is then
Each number is the character of an SO(8) representation and, in particular, the term −(28 + 1)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of SO(8) × U (1). Thus, the deformation (3.59) added to the SO(9) × U (1) 2 model T 1 leads to a new theory exhibiting the IR symmetry enhancement

Higher rank theories
In section 2 we have argued that enhanced symmetries are expected for U Sp(2N ) theories once we flip a set of operators breaking the SU (8) UV global symmetry to particular subgroups. Since the argument holds regardless of the gauge rank, we have an infinite family of theories for a given enhanced symmetry in the IR. 17 For higher rank theories there can be multiple operators in a given representation of the global symmetry and we need to flip all of them to realise the enhanced symmetry.
In this section, we show how this works explicitly in some examples. We consider 4d N = 1 U Sp(2N ) theories with one antisymmetric chiral A and 8 fundamental chirals, and extra gauge singlets D i 1,2,3 and a subset of b i 1,2,3,4 for i = 0, . . . , N − 1, whose global charges are shown in table 2. The D i 1,2,3 singlets couple to the fundamental chirals via the following superpotential terms: while the b i 1,2,3,4 singlets, if present couple through In such higher rank cases we can also construct gauge invariant operators of the form Tr g A i . These turn out to fall below the unitarity bound in all the examples we are going to consider, so we will also need additional singlets a i that flip them through the superpotential terms a i Tr g A i .  Similarly to the rank one case, we will denote by T N i with i = 1, 2, 3 the theory where the interactions involving respectively b 1 or b 1 , b 2 or b 1 , b 2 , b 3 are turned on and by T N i the theory where also the interaction involving b 4 is turned on, where now the upper index denotes the rank of the gauge group.

E 6 × U (1) 2 model
In section 2.1 we have shown that to realize the E 6 symmetry we have to flip the operators either in the representation (6, 2) −2 or in the representation (15, 1) 2 of SU (6) × SU (2) × U (1) v ⊂ SU (8). For definiteness we may take the latter, which are then given by Furthermore, we also flip because those with low powers of A, violate the unitarity bound. Thus, we introduce the following flipping fields in total: with the superpotential given in (4.1)-(4.2)-(4.3). In the following, we will denote this theory by T N 3 , where the lower index represents the number of towers b-singlets included, while the upper index is the rank of the gauge group.
Once those operators are flipped, the manifest UV symmetry is given by where, similarly to what we discussed in section 3.1.3 for the rank one case, SU (6) is formed by SU (2) 1 × SU (2) 2 × SU (2) 3 and the two combinations of U (1) x 1 ,x 2 ,x 3 corresponding to the U (1) t,u symmetries we defined in (3.4)-(3.5). According to the argument in section 2.1, this is supposed to be enhanced in the IR to This enhancement can be checked by computing the superconformal index of the theory for low values of N , for example N = 2. In this case, performing a-maximization we find the following approximate values of the mixing coefficients of the R-symmetry with U (1) The superconformal index of the theory computed with these R-charges then reads  Each number is the character of an E 6 representation and, in particular, the term −(78+2)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of It is worth mentioning that the existence of the N = 2 copies of the gauge singlets listed in (4.6) is crucial to realize the E 6 symmetry. Let us look at the second term 27 v −2 a − 1 2 (pq) 69 176 . 27 is decomposed into the representations of SU (6) × SU (2) 4 as follows:
Once those flipping fields are taken into account, the UV symmetry is broken to where, similarly to what we discussed in section 3.1.2 for the rank one case, SU (4) is formed by SU (2) 1 × SU (2) 2 and the combination of the abelian symmetries 2(U (1) According to the argument in section 2.2, this is supposed to be enhanced in the IR to This enhancement of the global symmetry can be checked using the superconformal index for low values of the rank of the gauge group. For instance, for N = 2 we find the following approximate values of the mixing coefficients from a-maximization: The superconformal index of the theory computed with these R-charges then reads 19 15 22 + a −2 (pq) 11 15 + · · · −(45 + 3)pq + · · · . (4.25) Each number is the character of an SO(10) representation and, in particular, the term −(45 + 3)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of SO(10) Analogously to the SU (2) case, one can deform the theory by introducing extra superpotential terms of the form b i m Tr g Tr n (A i Q n Q n ) with m = n. The first term we introduce is 20 and we label the theory obtained from this deformation as T 2 N . Similarly to what happened in the rank one case of section 3.2.1, this deformation breaks the global symmetry of theory where again we are parametrizing the two surviving combinations of U (1) x 1 ,x 2 ,x 3 with U (1)ũ ,v which are defined in (3.37)-(3.38). Note, for example, that D i 1 and b i 2 construct the traceless antisymmetric representation of U Sp (4).
Given the approximate mixing coefficients of U (1) Each number is the character of an SO(9) representation and, in particular, the term −(36 + 3)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of We should comment that the index can be also written in terms of SO(8) characters because any SO(9) representation can be decomposed into SO (8) representations. In that case, the term (9 − 36 − 3) pq would be written as −28 SO(8) − 2 pq, where 28 SO(8) is the character of the adjoint representation of SO (8). Indeed, if there is no marginal operator that can be constructed, the conserved current must be in the representation 28 SO(8) and the enhanced non-abelian symmetry is SO(8) rather than SO (9). Thus, we have to show that the marginal operators in the representation 9 exist in order to claim that the enhanced symmetry includes SO(9).
Let us list the operators contributing to order pq except some boson-fermion pairs trivially canceled. We first define the following single trace operators organized into representations of the manifest symmetry SU (2) 1 × U Sp(4) × SU (2) 4 , where we omit the charges under the abelian symmetries: (1, 4, 2) ,     Now let us look at the bosonic operators M 0 Π 1 , M 1 Π 0 , L 0 P 1 and L 1 P 0 , which satisfy the relations (4.32) as well as the F-term conditions realized by parts of the fermionic operators σ 0 Π 1 , σ 1 Π 0 , τ 0 P 0 and τ 1 P 1 . The remaining independent one is only (2, 1, 2), which cannot be lifted because there is no fermionic operator to be paired up. Similarly, M 0 Σ 0 and M 1 Σ 1 can be paired up with σ 0 Σ 0 and σ 1 Σ 1 respectively, leaving Note that those cancelations reflect the F -term conditions from the superpotential. The remaining operators in (4.34) are in the representation 2×(1, 5, 1). One combination of them can become a long multiplet being paired up with the traceless antisymmetric part of µ. On the other hand, the other combination still remains short and combines with the remaining (2, 1, 2) of M 0 Π 1 , M 1 Π 0 , L 0 P 1 , L 1 P 0 into 9 of SO (9). Therefore, we have found 9 marginal operators consisting of (2, 1, 2) from M 0 Π 1 , M 1 Π 0 , L 0 P 1 , L 1 P 0 and (1, 5, 1) from (4.34). Moreover, the remaining fermionic operators constitute the supersymmetric partners of the conserved current in the adjoint representation of SO (9) The second deformation we introduce is 22 and we label the theory obtained from this deformation as T 2 N . Similarly to what happened in the rank one case of section 3.2.1, due to this deformation U (1)ũ is broken whereas SU (2) 1 × U Sp(4) gets enhanced to U Sp (6). Thus, the entire UV global symmetry is now given by where once again U (1) v is defined as in (3.4)-(3.5). Note, for example, that D i 1 , D i 2 , D i 3 and b i 1 , b i 2 are organized into the traceless antisymmetric representation of U Sp (6). Given the approximate mixing coefficients of U (1) with the R-symmetry of the theory for rank N = 2 we have the superconformal index for N = 2 as follows: Each number is the character of an F 4 representation and, in particular, the term −(52+2)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of Again we need to check if the marginal operators in the representation 26 really exist. We first define the following single trace operators organized into representations of the manifest symmetry U Sp(6) × SU (2) 4 : where we have used the same names of the operators as those in the previous SO(9) case. The operators contributing to order pq are now given by    In appendix C we check this looking at the plethystic logarithm of the superconformal index. Comparing (4.40) and (4.41), we find that there are exactly 26 bosonic operators in the representation (6, 2) + (14, 1) that cannot be paired up with any of the fermionic operators in (4.41). Therefore, the theory has the marginal operators in the representation (6, 2)+(14, 1), which is 26 of F 4 . Moreover, the remaining fermionic operators constitute the supersymmetric partners of the conserved current in the adjoint representation of

SO(8) × U (1) 4 models and SO(8) → SO(9) → SO(8) deformations
The last two examples are the models exhibiting the SO(8) × U (1) 4 global symmetry. As we discussed in section 2.3, in order to obtain this model we need to flip, for example, the operators in (2, 2, 1, 1) 0,2,2 + (1, 2, 2, 1) −1,−1,2 + (2, 1, 2, 1) 1,−1,2 + (1, 1, 1, 1) 2,2,2 , which are representations of SU (2) 4 × U (1) t × U (1) u × U (1) t ⊂ SU (8). The corresponding operators are for i = 0, . . . , N − 1 respectively. In addition, we need to flip because those with low i violate the unitarity bound. Thus, we introduce the following flipping fields: with the superpotential given in (4.1)-(4.2)-(4.3). We denote this theory by T N 1 . Once those flipping fields are taken into account, the UV symmetry is broken to which is supposed to be enhanced in the IR to where U (1) t,u,v are again defined as in (3.4)- (3.5). This enhancement of the global symmetry can be checked using the superconformal index for low values of the rank of the gauge group. For instance, for N = 2 we find the following approximate values of the mixing coefficients from a-maximization: . (4.48) The superconformal index of the theory computed with these R-charges then reads 23 2772 + a −2 (pq) 8 11 + + · · · + (8 c u −3 t + 8 c u 3 t −1 −28 − 4)pq + · · · . (4.49) Each number is the character of an SO(8) representation and, in particular, the term −(28 + 4)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation Analogously to the SU (2) case, one can deform the theory by introducing extra superpotential terms of the form b i 1 Tr g Tr l (A i Q l Q l ). The first term we introduce is 24 and we label the theory obtained from this deformation as T 1 N . Similarly to what happened in the rank one case of section 3.2.2, this deformation breaks U (1) t and makes SU (2) 1 × SU (2) 2 recombine into U Sp(4). Therefore, the manifest symmetry is now given by where as usual U (1) u,v are defined in (3.4)-(3.5). Note, for example, that D i 1 and b i 1 construct the traceless antisymmetric representation of U Sp (4).
Given the approximate mixing coefficients of U (1) x 1 × U (1) x 2 × U (1) x 3 × U (1) a with the R-symmetry of the theory for rank N = 2 , (4.52) the superconformal index for N = 2 is given by 25 Each number is the character of an SO(9) representation and, in particular, the term −(36 + 3)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation of Note that there exist the marginal operators in the representation 9, which can be explicitly constructed in a similar way to the SO(9) × U (1) 3 model in the previous subsection.
The second deformation we introduce is 26 and we label the theory obtained from this deformation as T 1 N . Similarly to what happened in the rank one case of section 3.2.2, this deformation breaks U (1) u as well as U Sp(4) into SU (2) 1 × SU (2) 2 . Hence, the entire UV global symmetry is now given by where as usual U (1) v is defined in (3.4)-(3.5). Given the approximate mixing coefficients of U (1) 1 × U (1) 2 × U (1) 3 × U (1) a with the R-symmetry of the theory of rank N = 2 we have the superconformal index for N = 2 as follows: (4.57) 25 We choose to parametrize the U (1) symmetries, which don't participate in the enhancement, with U (1)u,v,a. 26 Using the superconformal R-charge of theory T 1 N =2 we can check that this deformation has R-charge R[b i 1 Trg Tr3 A i Q3Q3 ] 1.96449 < 2 for i = 0, 1, so it is a relevant deformation.
The remaining fermionic operators constitute the supersymmetric partners of the conserved current in the adjoint representation of SO (8) We conclude considering the other model exhibiting the SO(8) × U (1) 4 IR global symmetry. Indeed, similarly to what happens in the rank one case of section 3.1.1, the operators Tr g Tr 1 A i Q 1 Q 1 in the representation (1, 1, 1, 1) 2,2,2 of SU (2) 4 × U (1) t × U (1) u × U (1) t ⊂ SU (8) are spectators from the point of view of the SO(8) enhancement. Hence, we expect that theory T N 0 including the same singlets (4.45) that define theory T N 1 except the b i 1 fields will have the same enhancement. More precisely, the manifest UV symmetry is supposed to be enhanced in the IR to where U (1) t,u,v are again defined as in (3.4)- (3.5). This enhancement of the global symmetry can be checked using the superconformal index for low values of the rank of the gauge group. For instance, for N = 2 we find the following approximate values of the mixing coefficients from a-maximization: The superconformal index of the theory computed with these R-charges then reads 27 (4.65) Each number is the character of an SO (8) representation and, in particular, the term −(28 + 4)pq highlighted in blue reflects the current multiplet, which is in the adjoint representation where This theory enjoys three different types of dualities [1], which are generalizations of the Intriligator-Pouliot [4], Seiberg [2] and Csaki-Schmaltz-Skiba-Terning [3] dualities for N = 1. The generalization of Intriligator-Pouliot duality first appeared in [5]. The dual theory is still a U Sp(2N ) gauge theory with one antisymmetric chiralÂ and 8 fundamental chirals q a , but in addition we have 28N gauge singlet chiral fields M ab;i with a < b = 1, . . . , 8 and i = 1, . . . , N interacting with the superpotential The action of the duality on the global symmetries can be easily expressed in terms of the fugacities we introduced in (A. Accordingly, we have the following operator map: The generalization of Seiberg duality breaks the manifest SU (8) v symmetry to the subgroup SU (4) 2 × U (1) in the dual frame. Indeed, the dual theory is again a U Sp(2N ) gauge theory with one antisymmetricÂ and 8 fundamental chirals, but now the fundamentals are naturally divided into two groups of four that we denote by q a and p b with a, b = 1, . . . , 4. This is because we also have additional 16N gauge singlets M ab;i with a, b = 1, . . . , 4 and i = 1, . . . , N interacting with the superpotential The action of the duality on the global symmetries can again be expressed in terms of the fugacities we introduced in (A.2). In order to do so, we have to make a choice on how to break SU (8) v to the subgroup SU (4) 2 × U (1), which is equivalent to choosing how to split the 8 chirals Q a of the original theory into two groups of four. The most intuitive option is to split Q 1,2,3,4 from Q 5, 6,7,8 . With this choice, we have where we defined u 4 + = 4 a=1 u a and u 4 − = 8 a=5 u a . Accordingly, we have the following operator map 28 Clearly, this is not the unique choice for splitting 8 chirals into two groups of four. In total we have 1 2 8 4 = 35 different possibilities that will give rise to inequivalent dual frames. Finally, the generalization of the Csaki-Schmaltz-Skiba-Terning duality also breaks the manifest SU (8) v symmetry to the subgroup SU (4) 2 ×U (1) in the dual frame. Indeed, the dual theory is once more a U Sp(2N ) gauge theory with one antisymmetricÂ and 8 fundamental 28 Here and in the following we use that the two-index antisymmetric representation of SU (4) is real to freely lower its indices. For example explicitly (A.9) chirals, where the fundamentals are naturally divided into two groups of four that we denote by q a and p b with a, b = 1, . . . , 4. This time this is due to the presence of additional 12N gauge singlets µ ab i , ν ab i with a, b = 1, . . . , 4 and i = 1, . . . , N interacting with the superpotential Also in this case in order to express the action of the duality on the global symmetries in terms of the fugacities of (A.2) we have decide how to split the 8 chirals Q a of the original frame into two groups of four. Using the most natural decomposition into Q 1,2,3,4 and Q 5,6,7,8 we have the transformation where recall that we defined u 4 + = 4 a=1 u a and u 4 − = 8 a=5 u a . Accordingly, we have the following operator map:  [14]. 29 The F E[U Sp(2N )] theories where shown to enjoy U Sp(2N )×U Sp(2N )× U (1) 2 global symmetry with one of the U Sp(2N ) factor emerging in the IR from SU (2) n . As we will see, for n = 2 we have a further enhancement with U Sp(4) × U Sp(4) recombining into SO (10). F E[U Sp (4)] is an N = 1 SU (2) gauge theory with 8 fundamental chiral fields and 15 gauge singlets, so it belongs to the class of theories studied in the main text. Following similar conventions to [30], we split the fundamental chirals in three groups that we denote by D (1) α , 29 See also [26,29] for discussions of the more general F E[U Sp(2N )] theory and applications in models with global symmetry enhancements.   Table 3. Transformation rules under the global symmetries and the trial R-symmetry U (1) R0 of all the chiral fields of the theory.
V α and q a and the gauge singlets in other four groups that we denote by b, D αa , A and O H,[ab] , with α = 1, 2 and a = 1, · · · , 4 and with O H such that Tr 4 O H = 0. The superpotential is W = A Tr g Tr x (q q) + Tr g Tr x (O H q q) + Tr g Tr x Tr y 2 V q D (2) + b Tr g Tr y 1 D (1) D (1) , where Tr x and Tr y i are the traces over the U Sp(4) x and SU (2) y i flavor indices respectively. The matter content is summarized in the quiver diagram of Figure 5.
The non-anomalous manifest global symmetry is In [14] it was argued that the symmetries 2 n=1 SU (2) y i of the saw are enhanced to a second U Sp(4) y symmetry in the IR.
In order to detect the enhancement we can compute the superconformal index of F E[U Sp (4)]. This can be done using the assignment of charges for the chiral fields summarized in Table 3.
With this parametrization, we get from a-maximization the following values for the mixing coefficients of U (1) c and U (1) t with U (1) R 0 : In computing the index we approximate these mixing coefficients with R c We can also see at order pq the contribution of SO(10) conserved current. Indeed we find −45 − 2, which is the character of the adjoint representation of SO(10) This enhancement of the global symmetry is quite peculiar, as U Sp(4) 2 is a maximal subgroup of SO(10) but its rank is lower than the one of SO(10). This curious behaviour can be explained as follows. It is useful to work in a basis of fields that makes manifest the SU (2) 2 subgroup of U Sp (4), similarly to what we did in section 3. Specifically, we split the four fundamental chirals q into two groups of two that we denote by Q 1 and Q 2 . In these new conventions, the first two terms in the superpotential (B.1) read where we wrote explicitly all the traces Tr x , Tr y 1 , Tr y 2 over the U Sp(4), SU (2) y 1 , SU (2) y 2 flavor indices and we are omitting Tr g for the contraction of color indices. From this expression, it is clear that this superpotential violates the chiral ring stability criterion [27]. Suppose that we deform the theory by removing the first term A Q 2 1 . The equation of motion of the field O H,12 set the operator Q 2 1 to zero, meaning that the operator A Q 2 1 that we removed vanishes in the chiral ring of the deformed theory. Hence, this term is unstable in the superpotential (B.5) and should be dropped. Notice that this operation doesn't really modify the theory, since it can be realized with a trivial linear field redefinition It is easy to check that with such a modification we recover nothing but the T 2 theory of section 3. Hence, after the stabilization the manifest UV symmetry is actually the SU (4)×SU (2) 2 × U (1) 3 symmetry of T 2 , which we know is enhanced in the IR to SO(10) × U (1) 2 .
Furthermore, it has been discussed that the compactification of F E[U Sp(2N )] on a circle and its real mass deformations lead to many interesting 3d theories exhibiting similar properties of F E[U Sp(2N )] [14,31]. This is certainly true for F E[U Sp (4)]. For example, the 30 Remember from the discussion in section 3.1.2 that this field is a spectator from the point of view of the SO(10) enhancement. direct reduction of F E[U Sp (4)] gives rise to the 3d theory with the same matter contents and a monopole superpotential, which exhibits the same enhancement of the global symmetry into SO(10) × U (1) 2 . In addition, one can also take a subsequent real mass deformation, as explained in [14], such that each U Sp(2n) factor is broken to U (n). With a certain traceless condition imposed, the resulting theory is F M [SU (2)] proposed in [31]. While the manifest UV symmetry of F M [SU (2)] is SU (2)×U (1) 3 , we have checked that its superconformal index exhibits the characters of SO(6) × U (1) 2 . Thus, we expect F M [SU (2)] enjoys the symmetry enhancement: Moreover, this SO(6) enhancement is closely related to an example discussed in [32]. Indeed, as explained in [31], one can further deform F M [SU (2)] to obtain the theory called F T [SU (2)] [33], which has one less U (1). F T [SU (2)] is basically the same theory as the model in [32] exhibiting SO(6) but with an unstable superpotential. After the stabilization of the superpotential as above, we obtain a U (1) gauge theory with two flavors (Q α ,Q β ), four gauge singlets η αβ for α, β = 1, 2 and the superpotential which is exactly the model in [32] showing the enhancement of the global symmetry:

C The plethystic logarithm and the representations of the relations
In section 4, we have seen that higher rank theories mostly have marginal operators satisfying some relations. The correct identification of such relations is important to argue the existence of the independent marginal operators. In this appendix, we explain how to read the relations of the marginal operators by examining the superconformal index.
We introduce the plethystic logarithm [34] PL which is the inverse function of the plethystic exponential The coefficient µ(k) is the Möbius function defined by k has repeated prime factors, 1 , k = 1, (−1) n , k is a product of n distinct primes.
If we take the plethystic log of the superconformal index, it will give the generating function of the single trace operators, either bosonic and fermionic, as well as the relations among them, which reflect the interaction of the theory. For example, N free chiral multiplets have the PL index where a i is the fugacity for each U (1) rotating each chiral multiplet. There will be extra terms if the theory is interacting. In addition, the power of pq will be varied by the shift a i → a i (pq) i where i is determined by the interaction. In fact, one should remember that there are not only the relations of bosonic and fermionic operators but also those of relations themselves. Here, however, we focus on the relations of bosonic operators, which then give negative contributions to the PL index. One should note that such relations of bosonic operators appear in the PL index in two different ways: one is realized by the negative contribution of a fermionic single trace operator while the other is the negative contribution corresponding to the absence of a bosonic operator in the original index. For example, let us consider an F-term condition coming from a superpotential where S is a gauge singlet and O is some gauge invariant bosonic operator. The F-term condition demands that This relation for the bosonic operator is realized in the index as the contribution of the fermionic operator Ψ S , which cancels the contribution of O such that it vanishes in the index. On the other hand, the other type of relation does not come up with a fermionic operator. In order to understand this, let us consider the example of the SU (2) gauge theory with 8 fundamental chirals and no superpotential that we reviewed in section 2. Recall that in this theory the chiral ring generators are the mesons m 0,ij = Tr g (Q i Q j ) which transform in the antisymmetric representation of the SU (8)  The term 28(pq) 1 2 represents the chiral ring generators m 0,ij , while the term −133pq corresponds to the sum of 70 relations m 0,[ij m 0,kl] = 0 and of the fermionic superpartners of the conserved current in the adjoint representation of SU (8).
In order to distinguish the two types of relations, we can introduce a fictitious fugacity F i in the numerator of the 1-loop determinant of each matter multiplet. Then the contributions involving any matter fermion will come up with extra factor i F n i i with some power n i . Such contributions can be either the independent fermionic operators or the fermionic operators corresponding to relations. On the other hand, we said that there are also the relations that are not realized by fermionic operators. The contributions of those relations do not include any factor of F i .
If we go back to our example of the SU (2) gauge theory with 8 fundamental chirals and turn on the same fictitious fugacity F for all the fermions contained in the chirals, we obtain the following PL of the index PL [I] = 28(pq) The positive term correspons indeed to the contribution of Tr g λ 2 , which recombines with one of the 64 fermionic operators into a long multiplet in the true index that we get in the limit F → 1. The remaining 63 fermionic operators correspond to the superparteners of the SU (8) flavor current, while the 70 relations do not carry any power of F , meaning that they do not come from fermionic operators as we anticipated before. Now let us apply this strategy to some of the models we considered in section 4. Consider for example the rank-2 T 2 N =2 model in section 4.2. This has the manifest symmetry SU (2) 1 × U Sp(4) × SU (2) 4 × U (1) 3 , (C. 12) which is enhanced to SO(9) × U (1) 3 (C. 13) in the IR. In order to prove the symmetry enhancement to SO(9) × U (1) 3 , it was important to argue the existence of the marginal operators in the representation 9 of SO(9). Those independent marginal operators can be found by constructing the candidate marginal operators and their relations, which are either realized by fermionic operators or not. In particular, we have claimed in (4.32) that the relations not realized by fermionic operators are in the representation (2, 1, 2) + (2, 5, 2) (C.14) of SU (2) 1 ×U Sp(4)×SU (2) 4 . Here we explain how to read this representation of the relations from the PL index. Turning on a fugacity F i = F for each matter fermion, the plethystic log of the superconformal index is given by where we have turned off the fugacities for SU (2) 1 × U Sp(4) × SU (2) 4 for simplicity. If they are turned on, all the numeric coefficients are written as the characters of SU (2) 1 × U Sp(4) × SU (2) 4 , which are enhanced to those of SO(9) in the F → 1 limit. For the first several terms, one can read bosonic single trace operators as well as their relations, especially the Fterm conditions, realized by fermionic operators. For example, the first term (1 − F ) a 2 r 18/11 indicates the operator Tr g A 2 (C. 16) and its F-term relation Tr g A 2 = 0 (C. 17) due to the superpotential term a 2 Tr g A 2 .