Curious patterns of IR symmetry enhancement

We study several cases of IR enhancements of global symmetry in four dimensions. In particular, we consider a sequence of $Spin(n+4)$ supersymmetric gauge theories ($8\geq n\geq 1$) with $n$ vectors and spinor matter with $32$ components. We show that the subgroup of the flavor symmetry of these theories rotating the matter in the spinor representations in the UV, when proper gauge singlet fields are added, enhances to the commutant of $SU(2)$ in $E_{9-n}$. We discuss several other interesting cases of enhanced symmetries and the interplay between symmetry enhancement and self-duality. We also make some observations about possible interconnections between chiral ring relations and symmetry enhancement. Finally, we conjecture relations of the discussed models to compactifications of certain conformal matter models in six dimensions on tori. The conjecture is based on deriving a relation between five dimensional models with $Spin$ gauge groups and conformal theories in six dimensions. As a by product of our considerations we discover a new instance of a simple self-duality of a theory with an $SU(6)$ gauge group.


Introduction
Quantum field theories exhibit various interesting phenomena at strongly coupled fixed points of renormalization group flows. One of these is a possible enhancement of global symmetry of the UV model in the IR. We know of many examples of such enhancements. However, we lack a general understanding of when such an enhancement could happen and what is the general mechanism behind this phenomenon.
The situation in the case of supersymmetric field theories is somewhat more tractable. In particular we have quantitative tools which allow us to access strongly coupled fixed point physics by studying weakly coupled UV descriptions. Some of the supersymmetric quantities one can discuss do not depend on the RG flow [1] and thus should be consistent with the physics along the flow and in particular with the symmetry properties of the fixed point. Typically the tools take the form of different types of indices, that is partition functions with supersymmetric boundary conditions on manifolds which have non contractible cycles, or limits of these. The former partition functions can be thought of as counting certain types of operators in the theory. As the partition functions are not sensitive to the RG scale, the counting captures also the physics of the strongly coupled fixed point. In particular, the counted objects should form the representations of the different symmetry groups. If the symmetry group is bigger in the IR than in the UV, the representations seen in the computation need to be consistent with the enhancement. Moreover, in some cases one can identify in the computation contributions which can only come from conserved currents. If such a contribution can be found, then it serves as a definite footprint of the enhancement, given that all the assumptions of the computation, such as the identification of the superconformal R symmetry, are true.
In five and three space-time dimensions there are many examples of enhancements of symmetry which were widely studied. A reason for that is that the enhancement often happens because the additional components of the conserved currents come from interesting non perturbative physics, monopoles in the latter case [2] and instantons in the former [3]. Actually, in these cases the enhancement was predicted even without resorting to more modern partition function techniques by discussing when monopole and instanton operators can produce additional components of the conserved currents. However, the additional components of the conserved currents might not arise from such non perturbative effects but from more mundane local vector operators which become conserved at the fixed points. In fact in four dimensions this is the only way the enhancement can happen. In this paper we will discuss symmetry enhancement in four space-time dimensional theories.
Another interesting phenomenon in quantum field theory is that two different UV theories might flow to the same fixed point. That is they are dual to each other, or in statistical physics jargon are in the same universality class. A particularly simple phenomenon of duality is when the two dual models are but slightly different. For example, the models can be given by the same gauge sector differing only by gauge invariant fields. Such dualities are often named self-dualities. Although the two models are similar the map of operators between them can be non obvious. Also, the IR fixed point might be on a conformal manifold of theories, in which case the weakly coupled dual models can flow to different points on that manifold. In other words, the duality group can also act on the space of conformal couplings. In fact one can construct examples of dualities where the UV theory is the same for the two models of the pair but one can still show that there is a non trivial map of operators as well as an action on the conformal couplings which gives identical physics. An example of this is the S duality of N = 4 SYM, in which the UV theory is actually conformal and there is no flow. Another example is that of N = 1 SU (2) SQCD with four flavors and a quartic superpotential preserving an SU (4) × U (1) subgroup of the SU (8) symmetry of the UV theory. This theory flows to a fixed point with a large conformal manifold possessing an SU (4) × U (1) symmetry preserving subspace along which the duality group acts. This is a simple example of Seiberg duality [4]. In fact, SU (2) SQCD with four flavors can be further deformed by the addition of gauge singlet fields to a model preserving only SU (2) × SU (6) × U (1) symmetry group, but which is dual to itself under Seiberg duality and can be argued to flow to a fixed point which is isolated. In this case the duality acts only by reorganizing the states of the model and in particular leads to an enhancement of the symmetry to E 6 × U (1) [5]. The phenomenon of duality and of symmetry enhancement can be thus related in certain examples. We will exploit such a relation here to both discover new enhancements of symmetry starting from dualities and to discover new dualities from observations of enhancements of IR symmetries.
An indirect indication to whether a symmetry of some model can be enhanced is obtained by engineering it as a fixed point of a different flow for which the symmetry is manifest. That is, in some cases, one can claim that the theory of interest flows to the same fixed point as another model which manifests a bigger symmetry group. A simple example is again given by N = 1 SU (2) SQCD but now with five flavors. The symmetry is SU (10), however the dual model is an SU (3) theory with five flavors, where only SU (5) × SU (5) × U (1) symmetry is manifest. Here, the enhancement to SU (10) only happens at the fixed point. In fact, in recent years, starting with [6], we have discovered an amazingly rich set of flows which exhibit similar effects. One can engineer fixed points in four dimensions by starting with six dimensional models, put them on a compact two dimensional surface in the presence of background gauge fields for the global symmetries and flow to an effective theory in four dimensions. The symmetry of the theory in four dimensions is determined by the symmetry of the theory in six dimensions and by the details of the background fields. On the other hand the same models can be argued to appear as fixed points of RG flows of four dimensional gauge theories for which the symmetry is often smaller than the symmetry expected in the IR. To obtain claims of this kind one needs to derive a dictionary between four dimensional constructions and compactifications. Such a dictionary in fact can be made very systematic by realizing that one can understand compactifications on complicated geometries by splitting them into simple geometric blocks and then understanding the compactifications on the blocks and how to combine them in four dimensions. The symmetry of the blocks might be enhanced when they are combined to theories corresponding to compactifications which preserve bigger symmetries. Examples of this type of studies can be found in [7][8][9][10][11] following the works of [6,[12][13][14]. This issue of developing the dictionary between six dimensional and four dimensional models, provides one of the motivations for this paper.
In this work we will study several known self-dualities. Most of these dualities were discussed in [16,17] as generalizations of self-dualities of N = 1 SU (n) SQCD with 2n flavors.
For additional examples see [18] and [19]. In particular we will consider a sequence of self-dual models with Spin(n + 4) gauge groups and matter in the spinor and vector representations. We will take the spinors to have thirty two components, which will mean that the number of spinors depends on the gauge group, and n chiral fields transforming as vectors of Spin(n+4). The construction is bounded by Spin (12) as the spinor of Spin(13) has 64 components. First, by adding certain gauge invariant fields we will construct models which are dual to themselves and that exhibit a rather significant enhancement of symmetry in the IR.
The tool we will use to make such a statement is the supersymmetric index [20][21][22] . When the theory is conformal, the index can be thought of as a sum over different representations of the superconformal group. It is independent of the RG flow [1], and thus has to be consistent with the properties of the theories anywhere along the flow. In particular the supersymmetric index, which is defined as a partition function on S 3 × S 1 along the flow, becomes equivalent to the superconformal index at the fixed point. At the fixed point we can make full use of the superconformal symmetry to extract the information about the states. In particular, it has been shown in [23] that although the index counts states up to cancellations due to recombinations of various short superconformal multiplets to long multiplets, at low orders of the expansion of the index it reliably captures the information about some very important operators. These involve the relevant operators and the marginal operators minus the conserved currents. In particular any negative contribution to the difference between the latter comes from conserved currents. The only assumptions in such computations are that
Here s denotes components in the spinor representation, v vector, a is two index antisymmetric, and f flavors which is the number of fundamentals for U Sp and fundamentals plus anti-fundamentals for SU . The flavor symmetry is denoted by G. The symmetry of the SU (2n) sequence with n > 2 is expected to be SO(12) × U (1) × U (1) 2 .
the model flows to an interacting fixed point without free field, and that we have correctly identified the superconformal R symmetry. As the R symmetry can mix with abelian symmetries, we can use a maximization [24] to determine the superconformal one, which should give the correct result if no accidental U (1) symmetries appear at the fixed point. We will always assume that this is the case, and our results are consistent with the assumption. The main result of the paper will be to show that this sequence of theories has a rather regular enhancement of symmetry. The symmetry rotating the spinor fields will enhance to the commutant of SU (n) × SU (2) in E 8 , and the SU (n) symmetry rotating the vectors enhances to SU (n) × SU (n). This pattern of enhancement suggests an interpretation of the theories as fixed points of compactifications of six dimensional models. In particular, we conjecture that taking the E-string theory, which has E 8 global symmetry, and gauging an SU (n) subgroup with the addition of 2n hypermultiplets is the six dimensional model we need to study. When one takes such a model on a torus with flux breaking the SU (2n) symmetry to SU (n) × SU (n) × U (1) one obtains a theory in four dimensions with E 9−n ×SU (n) × SU (n) × U (1) symmetry. We conjecture that the theories we discuss in four dimensions are farther deformations of these models breaking the SU (2) in E 9−n . We will make this statement very precise for low values of n. We will also discuss a variety of other examples of symmetry enhancements, duality, and compactifications. In Figure 1 we summarize the main results regarding symmetry enhancement. We will support our conjectures by discussing first the reduction of a variety of six dimensional conformal theories on a circle to five dimensions.
We will also discuss a curious property of chiral ring relations which many of the models we encounter exhibit. Namely, consider relevant operators O a in representations r a of a global symmetry G, such that O a O b | r ab ∼ 0 in the chiral ring with r ab a representation appearing in the decomposition of r a ⊗ r b . In all examples we have considered (⊕r ab ) ⊕ Adj(G) is an adjoint representation of some group G. This group is not in general the symmetry group of the model. In some cases we can construct, by making use of gauge singlet fields, from a given theory a model so that G is it's symmetry.
The paper has the following outline. In section two we discuss the sequence of four dimensional Spin(4 + n) gauge theories and summarize the enhancement of symmetries. In section three we will consider five dimensional models with similar gauge groups to the ones discussed in four dimensions and conjecture UV six dimensional completions of such models. In section four we use the five dimensional understanding to discuss the relation of the models to compactifications from six dimensions and use this relations to discover more cases of symmetry enhancements and dualities. In section five we discuss an interconnection between chiral ring relations and symmetry enhancements. Several appendices will contain more technical results.
2 Four dimensional symmetry enhancement in Spin(n + 4) models We consider a sequence of theories with gauge groups ranging from Spin(12) down to Spin (5), with matter content in the spinorial and vector representations. Each such theory has at least one dual theory [16,17] with the same gauge group but with additional gauge singlet fields and with some of the matter in different representations of the global symmetry. We will in what follows construct models, motivated by the existence of such self-dualities, which have enhanced global symmetries in the IR. For each model in this sequence we will add certain gauge singlet fields and relevant superpotentials to form models with an enhanced symmetry. With this choice of gauge singlet fields most of the models have a conformal manifold in the IR which is topologically a point and the duality group will then act as the Weyl symmetry of the enhanced global symmetry of the IR theory. In what follows we present this new sequence of theories along with their enhanced symmetry in the IR. We will give evidence, in fact a proof in most cases under some assumptions, of such statements.
The readers unfamiliar with the supersymmetric index terminology should consult the first part of appendix A for a brief summary of the terminology and the technology. Here we just remind the reader that in the expansion of the superconformal index computed with the superconformal R-symmetry in the canonical fugacities q and p [22] at order qp one obtains the difference between the marginal operators and the conserved currents [23]. In what follows we will quote the result of the computation of this order and deduce the symmetry from this argument. In certain cases we will observe only negative contributions which can come only from the currents, which is a proof following the superconformal representation theory that the symmetry is at least the one corresponding to the currents. The only assumptions are that we have flown to an interacting fixed point without free fields, and that we have identified the R symmetry correctly. In appendix A we give all the details of the computation.
The following models contain various fields which are singlets under the gauge symmetries. We choose these fields in such a manner that the theories will exhibit the enhancement of symmetry clearly. This happens if one can identify unambiguously the conserved currents in the computation of the index, which typically occurs when the conformal manifold is a point. In most of the models in the sequence one can choose the singlets so that the conformal manifold is a point and the enhancement of the symmetry is maximal. In some cases, the conformal manifold is a point with some choice of singlets, but one can find evidence for a larger enhancement, using a different choice of singlets, which leads to a conformal manifold bigger than a point. In such cases we detail both options for the singlet fields.

Spin(5)
We start with a Spin(5) g gauge theory with an octet of spinors and a single vector field. The group is isomorphic to U Sp(4) and from that perspective we have an octet of fundamental chiral fields and a single field in the two index antisymmetric traceless representation. This particular case was discussed in much detail in [25]. The matter content is given in the table below.
Here U (1)r is the superconformal R symmetry obtained by a maximization [24] and the superpotential is given by, We will in this section denote by M gauge singlet chiral fields which are added to the models so as to make them be completely self dual under the dualities of [16], while fields denoted by Φ are added to have all operators above the unitarity bound. In this model, considered in [25], the contribution of the conserved currents to the index at order pq and having zero charge under U (1) a is given by suggesting that the symmetry SU (8) × U (1) enhances to E 7 × U (1) in the IR.
Here 133 E 7 decomposes as 63 SU (8) + 70 SU (8) . Assuming that the theory flows to an interacting SCFT and that we have identified the superconformal R symmetry correctly this is a proof that the symmetry enhances to this group. Note that as discussed in [25] the model has marginal operators charged under U (1) a ; in particular, on the conformal manifold the theory is equivalent to the E 7 surprise model of [15] which is a Spin(4) theory with eight spinors in the representation ( 1 2 , 0) and eight in the (0, 1 2 ). On the conformal manifold the field V acquires a vacuum expectation value breaking Spin (5) to Spin (4).
We note that this model has a generalization. The group Spin(5) is the same as U Sp(4) and if one takes U Sp(4n) models with an antisymmetric field, which generalizes the vector, and eight fundamentals, which generalize the spinor, one also obtains enhancement to E 7 × U (1) as was discussed in [25]. We will not discuss this generalization here.

Spin(6)
Let us now consider a Spin(6) g gauge theory. The matter content is given in the table below.
The superpotential is given by, and the superconformal R charge byr = r − 0.142q a − 0.057q b . In this model, analyzed in [5] and in appendix A, the contribution to the index at order pq is given by suggesting that the symmetry SU (2) × SU (4) 2 × U (1) 2 enhances to SU (2) × SO (12) × U (1) 2 in the IR. Moreover, the dimension of the conformal manifold vanishes if there are no accidental U (1)s, evidence for which we do not see. We note that also this sequence has a generalization. Note that Spin(6) is the same as SU (4). Taking SU (2n) with two antisymmetric, which generalize the vector, four fundamentals and four anti fundamentals, which generalize the spinor, we can construct theories which have SU (4) × SU (4) symmetry enhanced to SO(12) given appropriate gauge singlet fields are coupled. This sequence is somewhat outside our main thread of discussion and we will consider it in appendix C in full detail. In particular we will discuss a new duality related to symmetry enhancement in this model.

Spin(7)
We consider next a Spin(7) g model. The matter content is given in the table below.
The superpotential is given by and the superconformal R charge byr = r + 0.039q a . The contribution to the index at order pq is given by, in the IR. Note that we observe the adjoint representation of SU (3) v twice in the index at order qp with the negative sign. If the basic assumptions are correct this indicates that the SU (3) symmetry we see in the UV is a diagonal combination of two SU (3) symmetries in the IR. The SU (4) s ∼ Spin(6) symmetry is enhanced to SU (6) with the vector representation being the (anti)fundamental of SU (6). Note that in this case the rank of the IR symmetry is bigger than the rank of the UV symmetry. In appendix A we discuss how the conserved currents of the enhanced symmetries are constructed from the operators of the UV theory. This is not a clean statement as there are many recombinations of short multiplets into long ones. We note that also here the dimension of the conformal manifold vanishes as there are no terms with positive sign appearing at order qp.

Spin(8)
We consider a Spin(8) g gauge theory with matter in all the three 8 dimensional representations. We split the discussion into two. In the first part, we consider a model in which the conformal manifold is a point, and in the second part a closely related model with an IR global symmetry that corresponds to the commutant structure mentioned above. The matter content of the first model is given in the table below.
as before, but in addition to that the index is now consistent with a combination of the two U (1) symmetries enhancing to SU (2). Explicitly, if we define we get that U (1) e enhances to SU (2) e . Writing the index in terms of SU (2) e and SU (4) ⊃ SU (2) s × SU (2) c , we obtain (2.11) In this case the evidence from the index does not give a proof for the enhancement but rather an indication that the supersymmetric spectrum of states is consistent with such a claim. This is because we only see representations of the symmetry appearing but not the conserved currents. If the conjecture is correct the theory contains marginal operator in the adjoint representation of SU (2) which is canceled in the index with the conserved currents.
We should also note that in this model we do not observe any evidence for the enhancement of SU (4) v to SU (4) 2 . Particularly, we do not see the additional conserved currents. However, there is no direct contradiction either, as the index can be written in characters of that symmetry and the additional conserved currents can cancel against a marginal operator in the adjoint representation of SU (4) v . One reason why we may expect this enhancement is that we can start from the previous case with M 0 , where we observe the conserved currents, add a free chiral field and use it to flip M 0 . This is a relevant deformation compared to the fixed point and so should initiate a flow that is expected to lead to the model without M 0 . Furthermore, we do not expect this to break the enhanced symmetry as the only other operator with the same R-charge is C 2 V 2 and there is no way to combine them to a representation such that the flipping breaks the enhanced symmetry.

Spin(9)
Next consider a Spin(9) g model. The matter content is given in the table below.
The superpotential is given by, 12) and the superconformal R charge byr = r + 0.0376q a . The contribution to the index at order pq is given by

Spin(10)
The next model in our list is a Spin(10) g gauge theory. The matter content is given in the table below.
The superpotential is given by 14) and the superconformal R charge byr = r − 0.076q a . The contribution to the index at order pq is given by corresponding to the enhancement of symmetry from SU (2) × SU (6) × U (1) to SU (3) × SU (6) 2 × U (1). We note that the results are also consistent with the marginals only being in 175 SU (6)v and the symmetry rotating the fields in the vector representation not enhancing to two copies. We conjecture that the symmetry does enhance because the model is part of a sequence with such a property but we do not have a proof of this statement. The symmetry SU (2) s does enhance to SU (3) with the adjoint of the former being the (anti)fundamental of SU (3). Here the results of the index are a proof of the statement modulo the usual assumptions.

Spin(11)
We consider a Spin(11) g model with seven chiral fields in the vector representation and a single spinor. The matter content is given in the table below.
The superpotential is given by 19) and the superconformal R charge byr = r + 0.030q a . The contribution to the index at order pq is given by suggesting that the symmetry SU (7) × U (1) enhances to SU (7) 2 × U (1) 2 in the IR. The symmetry rotating the vectors enhances to two copies in the IR. Moreover, the dimension of the conformal manifold vanishes.

Spin(12)
The last model is a Spin(12) g gauge theory. The matter content is given in the table below.
The superpotential is given by 21) and the superconformal R charge byr = r − 0.110q a . The contribution to the index at order pq is given by suggesting that the symmetry SU (8)×U (1) enhances to SU (8) 2 ×U (1) in the IR. Moreover, the dimension of the conformal manifold vanishes.

Summary of the symmetry enhancements
We summarize the various symmetry enhancements that occur in the models described above in the table below.

Gauge
UV symmetry IR symmetry UV rank IR rank In the case of Spin(8) we write two options for the enhancement which are determined by the singlets. We note that the sequence of groups, is precisely the commutant of SU (n) × SU (2) in E 8 . The commutant of SU (n) in E 8 is known as the E 9−n algebra. We will discuss a possible way to understand this enhancement of symmetry by flows of six dimensional theories on a compact space.

From four dimensions to five and six
We would like to try to find a more fundamental reason as to why such an enhancement should occur. One way to do this is to realize the 4d theories as a compactification of a 6d model, preserving the enhanced symmetry, which is a subgroup of the symmetry of the 6d model. For example, it was argued that the E 7 enhancement in the [15] E 7 surprise model can be understood as resulting from a compactification of the rank 1 E-string theory on a torus with fluxes breaking E 8 to E 7 × U (1) [9]. Specifically, the E 7 surprise is a mass deformation of a 4d theory that is the above mentioned E-string compactification. The E 7 surprise can also be regarded as the Spin(4) case in the Spin sequence discussed in the previous section. Thus, it is plausible that there is a generalization of the 6d construction also to this case, and one may hope that this can shed light on the observed enhancements. To try and make a concrete proposal, we first recall some aspects of the E-string compactifications, see [9]. The important observation here is that the 4d theory contains an SU (2) gauge theory with 8 doublet chiral fields, which in turn is related to the fact that the E-string theory when compactified on a circle has a gauge theory description as a 5d N = 1 SU (2) gauge theory with 8 doublet hypermultiplet. The reasons regarding this connection were elucidated in [9], and further developed in [10,11], and we refer the reader to these papers for the details.
This suggests that, as a starting point in proposing a 6d interpretation, we should consider the 5d N = 1 versions of the 4d Spin gauge theories we introduced in the previous section. These have the same matter content as their 4d version, but with the 4d N = 1 matter replaced by 5d N = 1 matter, so that 4d chiral fields become 5d hypermultiplets and so forth.
We are then lead to consider the 5d N = 1 Spin(n + 4) gauge theories with n hypermultiplets in the vector representation and spinor matter with 32 component whose exact splitting into hypermultiplets depends on n. The structure of this family is such that they sit on a Higgs branch flow line generated by a vev to the vector hypermultiplets. The maximal value of n here is 9, despite the fact that the spinors of Spin(13) are 64 dimensional, as these are pseudo-real allowing half-hypers. Furthermore for Spin (12), whose spinors are also pseudoreal, we have two possibilities: either a single spinor hyper of a chosen chirality or two spinor half-hypers of opposite chirality 1 . The Spin(12) parity invariant case is the one generated via Higgsing of the Spin(13) case, but it is the other case that appears to be important for the 4d story. Nevertheless, we shall include these cases as those naturally fit in the 5d to 6d story.
Next, we inquire as to the UV behavior of these theories. Specifically, we remind the reader that 5d N = 1 gauge theories may be realized by a mass deformation of a 5d or 6d SCFT, where in the latter case the deformation is given by circle compactification with possible flavor holonomies. This, however, is only true if the matter content is sufficiently small. For the theories we consider, there is evidence that these can be UV completed to 6d SCFTs on a circle. This is by itself a promising indication in our search for a 6d explanation.
We next briefly review the evidence for the 6d UV completion. First, as previously mentioned, these theories sit on a line of Higgs branch flows generated by a vev to the vector hypermultiplets. The end point of the line leads to theories that are known to have a 6d lift notably the cases of n = 0, which lifts to two decoupled rank 1 E-string theories [26], n = 1, which lifts to the rank 2 E-string theory [26] and n = 2, which lifts to the E 7 × Spin(7) conformal matter SCFT [27]. Thus, it is plausible that the other cases also lift to 6d SCFTs. This is further supported by analyzing the symmetry enhancement pattern in these theo-ries. Five dimensional gauge theories have non-perturbative instanton excitations that can in some instances provide additional conserved currents that lead to symmetry enhancement at the UV fixed point. It is possible in some cases to determine what these additional currents are using field theory methods, providing a glimpse to the UV global symmetry. In theories where the UV fixed point is a 5d SCFT the additional currents always combine with the IR global symmetry to form a finite group. However, in theories where the UV fixed point is a 6d SCFT the additional currents combine with the IR global symmetry to form an affine Lie group. Furthermore, the type of affine group is the 6d global symmetry, possibly twisted by an outer automorphism. For the theories we consider here, such an analysis for the 1-instanton case was preformed in [28], and it was found that the enhancement spectrum is indeed inconsistent with a finite group, but consistent with an affine one. For convenience, and as this will play a role later, we have summarized the resulting symmetries in table 3.
Finally, there are various criteria that have been conjectured for the existence of a 5d or a 6d SCFT fixed points [29], and they can also be used. Indeed, when applied to these cases, they too suggest that these 5d gauge theories posses a 6d SCFT UV completion.

Theory
Symmetry Table 1. The minimal symmetry consistent with the perturbative plus 1-instanton contribution for the 5d Spin gauge theories considered here. The spectrum of the expected additional current is such that it can only be accommodated by an affine group, at least for some factors of the global symmetry. This is interpreted as the theory lifting to a 6d SCFT in the infinite coupling limit, whose symmetry is the finite Lie group associated with the affine case. The superscript here denotes whether the affine group is the twisted (2) or the untwisted version (1). This lifts to whether the compactification of the 6d SCFT involves a twist or not. We also use the notation of U (1) (2) for a U (1) group projected out by charge conjugation.
The remainder of this section will be devoted to conjecturing the 6d SCFT fixed points associated with these theories, and then subjecting these conjectures to various consistency checks. This is an interesting problem by itself, and can be tackled independently of the 4d story. Furthermore, the answer will assist us in formulating a conjectural explanation for the substantial symmetry enhancement we observed in the previous section. Next, we shall state our conjecture and lay out the supporting evidence in its favor.
We conjecture that this family of theories uplifts to a family of 6d SCFTs that can be engineered as follows. Consider taking the rank 1 E-string theory and gauging an SU (n) subgroup of E 8 , while also adding 2n hypermultiplets in the fundamental representation of SU (n). With this combination, the SU (n) gauge anomaly can be canceled by adding a tensor multiplet as is usual in 6d low-energy gauge theory descriptions of 6d SCFTs. It is thus plausible that the infinite coupling limit would correspond to the origin of the tensor branch of some 6d SCFT. This is further supported as this SCFT has string theory realizations, both in brane constructions [30,31] and in F-theory compactifications [32].
Then our claim is that the 5d SCFTs in this family, parametrized as Spin(n + 4), lift to this theory where the compactification is done with a twist in the charge conjugation symmetry of the 6d SCFT 2 . We next present some evidence for our claim. First we note that the flow pattern is correctly reproduced. The largest group in 5d is Spin (13) and this maps to the group theory fact that the largest SU subgroup inside E 8 is SU (9). There is a Higgs branch flow in 6d that maps an SCFT of this type with some value of n to ones with lower values of n. This is seen in the low-energy description as Higgsing down the SU (n) group using the flavor hypermultiplets.
Another strong piece of evidence is that the global symmetry matches, where in cases where the symmetry is complex we indeed see the twisted version in 5d. The global symmetry of the 6d SCFT is SU (2n) × E 9−n 3 . Here E 9−n is the commutant of SU (n) inside E 8 , and it is probably recognizable for those familiar with 5d SCFTs as the global symmetry of the 5d SCFT whose IR description is SU (2) + (8 − n)F [3]. Such people will also know that taking n to be 8 there are in fact two possibilities the so called E 1 = SU (2) andẼ 1 = U (1) theories, and that onlyẼ 1 can flow to E 0 . Indeed this also appears in our case though the flow direction is now reversed.
In 6d this comes about as there are two inequivalent ways of embedding SU (8) inside E 8 , where one preserves an SU (2) while the other preserves a U (1). This can be seen by using the Spin(16) subgroup of E 8 under which the adjoint decomposes as 248 → 120 + 128. We can now embed SU (8) using the U (1) × SU (8) subgroup of Spin (16), but there are two distinct ways of performing this embedding, that differ by how each spinor decomposes. As only one spinor appears inside the adjoint of E 8 , this leads to two inequivalent embeddings of SU (8) inside E 8 . What is even more interesting is that this matches the behavior in 5d where for Spin (12) we have two options depending on the chiralities of the spinors. These two options match the two possible embeddings as is apparent from the symmetry enhancement pattern in this theories.
However, we have seen that the Spin(13) theory can flow to only one of them. This leads us to suspect that only theẼ 1 embedding can be reached from the n = 9 case. This comes about as embedding the SU (8) inside E 8 using its SU (9) maximal subgroup automatically selects this embedding, as can be confirmed using the branching rules.
Besides the global symmetry and flow pattern, we can also compare the dimension of the Coulomb branch. When we compactify a 6d theory with tensor and vector multiplets, we expect the Coulomb branch of the 5d theory to be spanned by the tensor multiplets and the vectors on the circle. Its dimension should then be the dimension of the 6d tensor branch plus the total rank of the 6d gauge groups. In our case we compactify with a charge conjugation twist which projects out some of these directions. Particularly, the Coulomb branch of an SU (n) group is spanned by the operators T r(φ k ), for k = 2, 3, ..., n. Under the action of charge conjugation the operators with k even are even while those with k odd are odd. Thus, we see that in the compactification, the Coulomb branch directions associated with k odd operators are projected out. Therefore, we conclude that compactifying the 6d SCFT in this way we expect a n 2 + 2 dimensional Coulomb branch. This exactly matches the Coulomb branch dimension of the Spin(n + 4) 5d gauge theory.
Some lower dimensional cases have already been analyzed, particularly the cases of n = 1, 2. In the case of n = 1 the 5d gauge theory is Spin(5) + 1V + 8S = U Sp(4) + 1AS + 8F which is known to lift to the 6d rank 2 E-string [26], which is indeed the 6d SCFT we find in this case. For n = 2 the 5d gauge theory is Spin(6) + 2V + 8S = SU (4) 0 + 2AS + 8F which is known to lift to the 6d E 7 × Spin(7) conformal matter [27], which is indeed the 6d SCFT we find in this case. In both cases charge conjugation acts trivially so the compactification can be done without the twist.
Finally we can also consider other Higgs branch flows. The 5d theory possesses also ones that are associated with giving vevs to the spinor hypermultiplets. These lead to theories outside the family we considered so far. In 6d we can Higgs the tensor in the rank 1 E-string theory, replacing it with 29 hypers. This process is known to break E 8 down to SU (2) × E 7 , where the SU (2) is locked to the SU (2) R-symmetry. Of course in order to access the Higgs branch the commutant of SU (n) in E 8 must be at least SU (2). Under the unbroken E 7 , the 29 hypers transform as 1 + 1 2 56. In our case the E 7 will be partially gauged by the SU (n) and these will provide additional matter for the SU (n). As we shall now see the remaining flow diagram is also consistent.
First it should be noted that in both 5d and 6d the Higgs branch at infinite coupling usually differs from the one visible at finite coupling, see for instance [34][35][36][37]. Thus, although in principle the Higgs branches at infinite coupling should agree this does not guarantee that the ones at finite coupling, which are the ones we will be mostly relying on, agree.
With that in mind consider first the Spin(13) SCFT. In this case the minimal Higgs branch flow Higgses SU (9) down to SU (8), and maps to giving a vev to a vector on the 5d side. One cannot Higgs down the rank 1 E-string theory here because there is no ungauged SU (2) in E 8 . Alternatively ,on the 5d side, it is known that one cannot give a vev just to a half-hyper in the spinor so any Higgs branch flow must involve at least a vev to one vector.
Going down to Spin(12), we have two theories. Again, one cannot give a vev just to the spinors when we have a half-hyper spinor for each chirality, but such a vev is possible when we have a full hyper in the spinor representation. In the latter case the theory flows to an SU (6)+16F gauge theory, which is known to lift to the 6d SCFT associated with the 6d gauge theory U Sp(8) + 16F [38,39]. On the 6d side we can only Higgs down the rank 1 E-string theory when the commutant of SU (8) in E 8 is SU (2). This is precisely the case mapped to the full hyper spinor. In that case the 6d gauge theory flows to an SU (8) + 1AS + 16F gauge theory which can be further Higgsed to U Sp(8) + 16F . As this theory is real, the charge conjugation twist does not effect it and we find the 6d and 5d flow patterns consistent.
The flow patterns for Spin (11) and Spin(10) are similar, both having a Higgs branch flow generated by spinor vevs which leads to an SU (5) + 14F gauge theory. This theory is known to lift to the 6d SCFT associated with the 6d gauge theory U Sp(6) + 14F [38,39]. This maps in the 6d side to Higgsing down the rank 1 E-string theory to get SU (n) + 1AS + (n + 8)F and then further Higgsing down to U Sp(6) + 14F in both cases. One curious thing is that for n = 6 one can also Higgs down the rank 1 E-string theory to get SU (6) + 1 2 20 + 15F instead, though we do not see any good candidate for this flow on the 5d side. It is possible that this flow appears only non-perturbatively there.
The Spin (9) and Spin(8) spinor vevs are again very similar, both leading to Spin(7) + 1V +6S which we can further Higgs to SU (4)+12F . In 6d this flow gives SU (n)+1AS +(n+ 8)F which can be further Higgsed to U Sp(4) + 12F , which reduces in 5d to the SU (4) + 12F gauge theory [38,39]. So we see again that the flow appears consistent. Interestingly in both cases we now pass through an intermediate theory. Comparing symmetries and Coulomb branch dimensions we come to the conjecture that the Spin(7) + 1V + 6S should lift to the twisted compactification of the 6d SCFT associated with SU (4) + 1AS + 12F .
Finally, for Spin (7), a vev to a spinor hyper leads to a G 2 + 6F gauge theory which can be further Higgsed to SU (3) + 10F . In the 6d side this flow maps to Higgsing down the rank 1 E-string which leads to SU (3) + 12F and can be further Higgsed to SU (2) + 10F . The latter, again, is known to be the 6d lift of the 5d SU (3) + 12F gauge theory [38,39]. Interestingly, it has been suggested in [40] that 5d G 2 + 6F gauge theory lifts to a twisted compactification of SU (3) + 12F , and our results are consistent with that.

Symmetry enhancement and compactification from 6d
In this section we shall suggest a connection between the observed 4d symmetry enhancement phenomena to compactifications of six dimensional theories on a torus. Previously, we have found that the 4d Spin(n + 4) models have symmetry enhancement consisting, typically, of a U (1) × SU (n) × SU (n) part and a remaining symmetry which can be described as the This suggests the interpretation that the 4d Spin(n + 4) models can be obtained from a deformation of a compactification of the 6d SCFT introduced in the previous section, probably on a torus with fluxes in the global symmetry of the SCFT. We remind the reader that the 6d SCFTs in question can be constructed as the UV completion of an SU (n) gauge theory with 2n fundamental hypermultiplets connected to a rank 1 E-string theory via gauging an SU (n) subgroup of E 8 . The symmetry of the six dimensional theory is then E 9−n × SU (2n). When compactifying down to four dimensions on a surface with flux for some subgroup of the global symmetry, the symmetry can be further broken. We conjecture that a relevant deformation of such a compactification can lead to the sequence we have discussed. The combined effect of the relevant deformation and the flux is to break SU (2n) to SU (n) 2 × U (1) and E 9−n to the commutant of SU (2) in E 9−n .
These expectations also hang on some recent understandings regarding the relation between theories in 6d, 5d and 4d. Particularly, compactifying a 6d SCFT on a torus with flux, we can first reduce to 5d where in lucky cases the 6d SCFT has an effective IR free gauge theory description. Reducing on another circle to 4d leads to a 4d gauge theory possessing similar matter content as the 5d gauge theory. An example of this is the (2, 0) theory. Reducing the type A k−1 (2, 0) model on a circle one obtains the maximally supersymmetric SU (n) gauge theory, and further reduction to four dimensions leads to N = 4 SU (n) gauge theory. Other examples involve compactifications of ADE conformal matter on the torus [8][9][10][11].
Thus, from this view point, it would not be surprising if the 4d theories are related to compactifications of these 6d SCFTs which can be described by Spin gauge theories with vectors and spinors in five dimensions. In section 3 we have discussed how the six dimensional models above, when compactified to five dimensions with a twist, are described by such effective IR free gauge theories. One issue that needs to be addressed is the twisting. The 5d theories are related to the 6d ones under a twisted compactification. However, in the 4d theories we observe the untwisted version of the global symmetry. These two observations are not contradictory for the following reason. We can build the torus from two tubes both containing a twist. Then, on one hand as the tubes contain a twist we expect them to have a Spin description, but, on the other hand, the full surface is constructed from two such tubes, and as the twist is Z 2 valued, the two twists should cancel out. In this way we suspect there should be a Spin construction of an untwisted compactification.
We will now make the conjecture precise for the case of n = 1 and n = 2. We begin with the case of Spin (5). Here the 6d SCFT is the rank 2 E-string theory 4 . The direct compactification of this model to four dimensions is not known, but a mass deformation of it breaking the SU (2) global symmetry is known [9]. We can consider such a theory corresponding to flux 1 2 breaking E 8 → U (1) × E 7 . Due to the fractional flux the E 7 is broken to SO(8) so the global symmetry of the 4d theory is U (1) × SO (8). See figure 2 with the four dimensional theory. We can now preform an additional deformation by giving a vev to the flip field. This breaks both the U (1) and the R-symmetry which is replaced by a new Rsymmetry under which the symmetric and lower antisymmetric have R-charge 1. Therefore, a mass term for these two fields is consistent with all the symmetries and we expect it to be generated, eliminating these fields from the resulting theory.
We now see that the charged matter content of the resulting theory is the same as the one in the Spin(5) theory we studied though the two differ by the flipping field spectrum. As some operators hit the unitary bound in this theory we expect them to decouple. We do Figure 2. The 4d theory conjectured to be the result of the compactification and mass deformation of the rank 2 E-string. Here the gauge group is U Sp(4) and S and AS denote the symmetric, that is the adjoint for U Sp groups, and antisymmetric representations, respectively. Here there is a superpotential coupling the symmetric chiral to the quark bilinear and the middle antisymmetric to the symmetric bilinear. The theory has an SO(8) global symmetry represented by the square as well as a U (1) whose charges we denoted using the fugacity t. The black X on the symmetric chiral stands for an additional chiral field which couples linearly to the symmetric bilinear. In fact, due to the charges of the fields, the flip field also couples linearly to the lower antisymmetric bilinear, and so efficiently flips some combination of the two symmetric bilinears. There is also an anomaly free R symmetry under which the symmetric and lower antisymmetric have R charge 0, the quarks R charge 1 and the middle antisymmetric and flip fields have R charge 2.
note that due to the superpotential we started with, the theory we get is at a point on the conformal manifold of the Spin(5) theory where E 7 × U (1) has been broken to SO (8).
We thus see that the Spin(5) theory can be connected to a compactification of the rank 2 E-string theory up to decoupling of free fields and flipping of various operators. This still leaves a bit to be desired as the starting point does not posses all of the 6d symmetry apparently visible in the end point.
Another case where we understand the details is the case of Spin (6). The six dimensional theory is the Spin(7) × E 7 conformal matter, which is the same as an SU (2) gauging of the rank 1 E-string theory with additional four fundamental hypermultiplets. The E 7 is the commutant of SU (2) in E 8 and Spin (7) is the symmetry rotating the four hypermultiplets. Figure 3 shows the 4d theory resulting from the compactification of the Spin(7)×E 7 conformal matter on a torus with flux 1 in Spin (7) preserving U (1) × SU (2) × SU (2) [41]. Next consider separating the 8 flavors to a pair of 4. This splits the SU (8) to U (1) × SU (4) × SU (4). As the SU (8) enhances to E 7 , the U (1) should enhance to SU (2), and the SU (4) × SU (4) should enhance to SO(12) which together form the maximal subgroup of E 7 .
We next consider giving a vev to a baryon. This breaks the U (1) and breaks one of the SU (4) gauge symmetries which is then identified with one of the SU (4) global symmetries. This is expected to preserve a U (1) × SU (2) × SO(12) global symmetry, which is expected to be the global symmetry of the resulting theory. The resulting theory is shown in figure 4. We indeed see that it is the theory we proposed up to a different choice of flipping fields.  (7) preserving U (1) × SU (2) × SU (2). Here there is a cubic superpotential along the triangle and a quartic one coupling the two antisymmetrics via the bifundamentals. There are also two singlet chiral fields flipping the baryons made from the bifundamentals. The visible global symmetry here is There is also a U (1) R R-symmetry under which the bifundamentals have R-charge 0, the flippers have R-charge 2, and the rest have R-charge 1.

More symmetry enhancement and duality
The way we obtained the Spin(6) theory with enhanced symmetry can be generalized in various ways. Instead of starting with the Spin(7) × E 7 conformal matter in six dimensions we can start from closely related models which have the symmetry E 7 ×SU (2)×SU (2)×SU (2). These are obtained by taking a collection of M5-branes near the end of the world M9-brane and on a C/Z 2 singularity. This type of models also depends on the choice of action of the Z 2 orbifold group on the E 8 symmetry of the M9-brane, where there are three distinct choices [32]. Here we concentrate on the choice preserving the E 7 × SU (2) subgroup of E 8 .
The resulting 6d SCFTs can be described on the tensor branch as a linear SU (2) quiver  (2). There is also a U (1) R R-symmetry under which the bifundamentals have R-charge 0, the flippers have R-charge 2, and the rest have R-charge 1.
gauge theory, connected by bifundamental hypermultiplets, with two fundamental hypermultiplets for both of the edge SU (2) groups. One of the edge SU (2) groups is further connected to a rank 1 E-string theory via gauging. We shall denote the number of SU (2) groups in the quiver by n − 1. Then the case of n = 1 is the rank 1 E-string theory, the case of n = 2 is the Spin(7) × E 7 conformal matter and the higher n are the generalizations. For generic n the theory has an E 7 global symmetry coming from the commutant of SU (2) gauge in E 8 as well as an SU (2) 3 factor coming from the various fundamental and bifundamental rotation groups 5 . The three SU (2) groups are not symmetric, and we shall refer to one of them, roughly associated with rotations of the fundamental flavors at the edge of the quiver not connected to the rank 1 E-string, as SU (2) E . When reduced on a circle to 5d, this class of theories has an affective 5d description as an SU (2n) gauge theory with two antisymmetric hypermultiplets, 8 fundamental hypermultiplets and Chern-Simons level 0. This can be used to conjecture the 4d theories resulting from torus compactifications with fluxes of this class of 6d SCFTs [41]. Specifically, the gauge theory description upon compactifying the models on a torus with flux breaking SU (2) E to U (1) is shown in figure 5. This is very similar to what we had above (see Figure 3) except the gauge groups are now SU (2n). In particular it is then natural to split the eight flavors into 8 − 2n and 2n and give vacuum expectation value to the baryons generalizing what we did before. This procedure will Higgs one of the two gauge groups and we will be left with an SU (2n) gauge theory with 2n fundamentals, 8 − 2n antifundamentals, additional fields in the antisymmetric representation and singlet fields. Note that this procedure makes sense only for n = 1, 2, 3, 4. The SU (8) symmetry we break by the procedure enhances to E 7 and thus the symmetry we expect is the commutant of the U (1) in E 7 associated with the split, which is then in order of increasing n: E 6 , SO(12), E 6 , and E 7 . The first case was discussed in [5], the second is the case we discussed here, and finally the fourth case was discussed in [16] in the context of self duality. The third case to the best of our knowledge was not considered till now.
We will next consider each case in detail discussing the theories and the enhancement of symmetry. In addition, we present a new self-duality associated with the n = 3 case. The models obtained from six dimensions have a particular set of gauge singlet fields. In the models below we alter this set in such a way so that the enhanced symmetry will remain but the conformal manifold will reduce to a point and thus the symmetry of the theory will appear in the index computation through the contribution of the conserved currents.
We have an SU (2) gauge theory with matter content as follows, Here U (1)r is the superconformal R charge and the superpotential is given by This model was analyzed in [5], where it was shown that the index at order pq is given by and that the UV symmetry SU (6) × SU (2) × U (1) enhances to E 6 × U (1) in the IR.
The model we obtain is the SU (4) (or Spin (6)) gauge theory considered in section 2 in table 2, which we reproduce here for convenience, The superpotential is given by and the superconformal R charge byr = r − 0.142q a − 0.057q b . As discussed in section 2 and derived in the appendix, the index at order pq is given by We have an SU (6) gauge theory with matter content as follows, The superpotential is given by and the superconformal R charge byr = r − 0.0175q a − 0.0481q b . The index at order pq equals (see the appendix for more details) and the UV symmetry SU (2) 2 × SU (6) × U (1) 2 enhances to SU (2) × E 6 × U (1) 2 in the IR.
This model was discussed in [16]. We have an SU (8) gauge theory with matter content as follows, The superpotential is given by and the superconformal R charge byr = r + 0.0693q a . The index at order pq equals (see the appendix for more details) and the UV symmetry SU (8) × SU (2) × U (1) enhances to E 7 × SU (2) × U (1) in the IR.

SU (6) self-duality
As discussed above, the symmetry enhancements in the n = 1, 2, 4 theories are related to self-dualities, and we expect the same to be true in the case of the n = 3 model. We find a self-duality as follows. We begin by decomposing the SU (6) flavor symmetry to SU (2) 3 × SU (4) × U (1) c and write the first duality frame as the matter content of Table (4) without the gauge singlet fields and with no superpotential, Then, the second duality frame is given by the following matter content, along with the superpotential Note that the representations of the matter content of the two dual theories under the gauge groups are different. The 't Hooft anomalies of the two models match and the indices agree.

Ring relations and hidden symmetries
So far we have discussed an interplay between symmetry enhancement and duality. There is another interesting field theoretical effect related to enhancement of symmetry. Supersymmetric theories in general have chiral operators forming an algebraic ring. This ring is specified by generators and relations, see [43] for a nice exposition. In this section we explore some interconnections between the ring relations and enhancements of symmetry. In particular we will show that a theory with chiral ring relations of a certain type entails a theory with an enhanced symmetry. We will also make an experimental observation that the representations under the global symmetry of the chiral ring relations of a certain type seem to always complement the adjoint representation of the symmetry to the adjoint representation of a bigger group, which is not in general a symmetry group of the model. In more detail, we will discuss chiral ring relations in theories with at least four supercharges in four dimensions appearing at R charge ≤ 2. By studying many examples we observe that the representations of the relations under the global symmetry group G extends the adjoint representation of G to an adjoint representation of a bigger group G. We will give examples related to the Spin(n + 4) sequence of models we discussed here and also discuss some different cases.

The basic setup for the claims
We examine a model, which we will denote T O , with chiral relevant operators O i such that O i O j are marginal, where i and j can be either equal or different. We assume that the model has at least N = 1 supersymmetry, an R symmetry U (1) r , and a global non R symmetry G under which the operators O i are in the representations R i . Now, these operators can satisfy chiral ring relations, where r ij are some representations of G which appear in the decomposition of the products R i × R j . In general, the set of representations r ij can be either non trivial or empty. We will show next that ij r ij + Adj(G) (5.2) forms in many examples the adjoint representation of some group that we denote by G. In addition, in the cases where i = j and O i and O j are oppositely charged under all the abelian factors of G, this observation can be proven, under certain assumptions, and the group G turns out to serve as the global symmetry of another model related to T O by the addition of some gauge singlets and a superpotential. Note that the sum ij r ij might be in the adjoint representation of some group by itself or it can extend Adj(G) to the adjoint representation of a bigger group for which G is a non trivial subgroup.
We first discuss the case mentioned above for which we have a proved argument, and then continue with nontrivial examples that illustrate the observations here.

Mapping relations into conserved currents
We begin by considering models in which all the marginal operators are of the form O where F i are gauge singlet fields we add to the model. This superpotential sets O (2) i to zero in the chiral ring and as a result there are no marginal operators in T F . Note that since the operators F i are charged under the abelian symmetries, the superconformal symmetry of T F is not the same as of T O ; nonetheless, we assume that all the operators are above the unitarity bound.
We assume that in T O , Next, in order to find the IR symmetry of T F , we analyze the supersymmetric index of this model at order pq and deduce the spectrum of marginal operators and the conserved currents. The various contributions are as follows. Each operator O i ), respectively. In addition, we have the contribution of the UV conserved currents −χ(Adj(G)), and so in total we obtain We see, as stated above, that in the model T F the dimension of the conformal manifold vanishes and that there is an enhancement of the symmetry from G in the UV to a larger group G in the IR (if r i are not empty). In particular, we see that in the model T O the sum of the representations of the relations extends the adjoint representation of the global symmetry G to the adjoint representation of some other, larger group.
We can now apply the same argument for theories that contain in addition to the marginal operators which are of the form O We finally comment that the argument is also applicable to theories in which O We now turn to analyze examples that illustrate our claim.

SU (2) with eight fundamentals
We consider an SU (2) gauge theory with eight chiral fields Q i in the fundamental representation, that is SU (2) N f = 4 SQCD. In the infrared, this theory flows to a superconformal field theory in which the gauge invariant composite operator Q i Q j (where the gauge indices are suppressed and contracted with ε αβ ) has an R-charge which equals to 1 and all the marginal operators are of the form (Q i Q j )(Q l Q n ). Moreover, these marginal operators satisfy a relation and therefore this theory serves as a nontrivial example and we denote the operator None of these operators satisfy a relation since N c > 2 and so all the representations in (5.8) are present. Therefore, r is the empty set and we have G = G. Note that in the case N c = 2 the second representation in (5.8) (which is (6, 6) 0 ) is "missing" since it corresponds to the following product of (gauge singlet) operators, (Q i ) 2 (Qj) 2 , in contrast to the fact that the gauge indices are contracted between Q i andQj in each O ij .
In this case, discussed in detail in [15], two copies of the SU (2) gauge theory with four flavors are taken such that the chiral fields Q (1) i of the first copy transform in the fundamental representation 8 of its global symmetry group SU (8) 1 , while those of the second copy Q (2)i are in the anti-fundamental representation 8 of its group SU (8) 2 . This product theory is then deformed by the exactly marginal operator where j , and analogously for O (2) . The superpotential identified the two SU (8) symmetries, that is preserves a diagonal combination of the two symmetries. The flavor symmetry of the deformed theory is SU (8) d , the diagonal part of SU (8) 1 × SU (8) 2 . The exactly marginal operator in (5.9) that preserves this SU (8) d is unique and therefore λ parametrizes a line of fixed points with SU (8) d flavor symmetry. As was shown in [15], there is a special point on this line of fixed points at which the flavor symmetry enhances to E 7 . (In fact, it enhances to E 7 × U (1) as discussed in [25].) We would like to apply our general argument to two kinds of points on this line, one is the special point with the enhanced E 7 symmetry and the other is a generic point where the symmetry is SU (8)  , where 63 out of the product 28 × 28 = 1 + 63 + 720 combined with the off-diagonal currents of SU (8) 1 × SU (8) 2 that were broken by the deformation (5.9). We therefore have in the chiral ring. Note that while the relations r 1 and r 2 are kinematical, r 3 is dynamical. Now, since r 1 + r 2 + r 3 does not form adjoint representation of some group, we expect it to extend the adjoint representation of the flavor symmetry SU (8) d to the adjoint representation of some other, bigger group. We get where G = E 7 × E 7 .
Next, we consider the special point on the line of fixed points where the flavor symmetry SU We see that in this case the relations form an adjoint representation by themselves. The sum of the representations of the relations and the adjoint of the symmetry group E 7 yields of course the same result as in (5.12).

Spin(n + 4) and conformal matter sequence
Most of the theories in the sequences considered in sections 2 and 4 are examples of T F . For each such theory, one can easily find the related model T O in which there are marginal operators and relations. These relations, in turn, extend the adjoint representation of the symmetry group of T O to the adjoint of the larger group that serves as the IR symmetry in the model T F . We now turn to analyze two T O models related to two theories from the Spin(n + 4) sequence, such that these models do not satisfy all the assumptions of the proof (in particular, some of the relations involve operators that are charged under U (1) factors of the symmetry group) we gave but we still observe that the relations combine with the conserved currents to form the adjoint representation of a bigger group.

SU (4) model
We look at an SU (4) gauge theory with 2 antisymmetric tensors, 4 flavors and a gauge singlet field. Explicitly, the matter content is given as follows, The superpotential is given by W = ΦX 2 , (5.14) the superconformal R-charge byr = r+0.12845q a and all the operators are above the unitarity bound. We begin as usual by identifying the relevant operators that constitute marginal ones. They are listed in the following table, The where G = SO(12) × SO(5) × U (1).

Quiver example
We now turn to examine a model with more than one gauge group. This theory, analyzed in [8], is given by the following matter content, The superpotential is given by all the operators are above the unitarity bound and the superconformal R-charge isr = r + 0.027q b . This model is Klebanov-Witten [44] with gauge singlets added. In order to identify the relations in this case, we consider the superconformal index. As was discussed in [8], in the IR the UV symmetry SU (2)×U (1) a ×U (1) b is expected to enhance to U Sp(4) × U (1) b . The reason is that this model is engineered as a compactification on a torus with flux of class S 2 with N = 2, that is two M5 branes probing Z 2 singularity. The flux is such that the SO(7) symmetry of the six dimensional model is broken to SO(5) × U (1) and that is the reason we expect this symmetry. In 4d it appears to be further extended to U Sp(4) × U (1) b , which might be accidental from the 6d viewpoint. Note that the fact that the symmetry of the fixed point here is enhanced is new for this example as compared to the other examples we have studied. Then the index is computed in [8] and given by It is consistent with an enhancement of the symmetry to U Sp(4) × U (1) and from now on we will assume that this is the symmetry of the conformal theory. Denoting the representations under the UV global symmetry G by the relevant operators and their representations are as follows, As can be seen in the expression for the index, the operators Φ 1 , Φ 2 and Q 2 form the 5 −2 of the IR symmetry U Sp(4) U (1) b , and Q 1 Q and Q 2 Q form the 4 0 . At order pq (with a vanishing U (1) b charge), we have the following contribution: and if we take into account the conserved currents 10 U Sp(4) +1 of the IR symmetry (and recall that the contribution to the index at order pq comes from the marginal operators minus the conserved currents), we find that the contribution of the marginal operators is 2 10 U Sp(4) + 1 .

(5.33)
On the other hand, from the product of relevant operators Q 1 Q 2 (Φ 1 + Φ 2 + Q 2 ) we would expect to have a marginal operator in the representation 5 U Sp(4) , which is not there. We deduce that there is a relation r = 5 U Sp (4) . Adding this relation to the IR currents, we observe that We obtain that G = SU (4) × U (1). Here the vector of so(5) ∼ usp(4) is embedded in so(6) ∼ su(4) as 6 so(6) = 5 so(5) + 1 so (5) .
Note that we can consider a relevant deformation of the theory turning on a superpotential with the operators charged as b −4 qp. This is a relevant deformation and it is a singlet of U Sp(4) obtained from the symmetric square of b −2 5 U Sp(4) (qp) Note that one of the singlets should be absent as a marginal operator since it is the superpotential deformation we turned on. That singlet is the one in the decomposition of the symmetric square of 5 U Sp (4) , and it recombines with the broken current of the abelian symmetry U (1) b . Thus, we have a (dynamical) relation in the singlet representation. We can now look at the marginal operators as they appear in the index and deduce that these are in the representations 2 1 U Sp(4) + 2 10 U Sp(4) + 14 U Sp(4) + 16 U Sp (4) . (5.37) We assume that there is no enhancement of symmetry in the fixed point and thus the current is in the representation 10 U Sp (4) . From here we see that also the representation 5 U Sp (4) and twice the 4 U Sp(4) are missing. We deduce that the relations are, Combining these with the adjoint of U Sp(4), we get the adjoint of SU (5). The embedding is first so(5) ∼ usp(4) in so(6) ∼ su(4) as before and then su(4) in su(5) such that 5 su(5) = 4 su(4) + 1 su (4) . We conclude that G = SU (5).

A Finding the IR symmetries of the 4d Spin(n + 4) sequence
In this appendix we present the calculation of the superconformal index at order pq corresponding to the sequence of Spin(n + 4) gauge theories described in section 2. We then use it to extract the global symmetry in the IR of the models under consideration, as discussed below. After reviewing the definition of the index, we begin with a general description of the method and continue with its application to the Spin(n + 4) sequence of theories.

The superconformal index
The index of a 4d superconformal field theory is defined as the Witten index of the theory in radial quantization. Denoting by Q one of the Poincaré supercharges 6 , the index is defined as the following weighted trace over the states of the theory quantized on S 3 × R [20][21][22]45]: Here j 1 and j 2 are the Cartan generators of the SU (2) 1 × SU (2) 2 isometry group of S 3 , r is the U (1) r charge, and u a and e a are the fugacities and charges of the global symmetries, respectively. Moreover, δ is defined as the following anti-commutator: Note that even though the chemical potential β appears in the definition (A.1), the index is in fact independent of it since the states with δ > 0 come in boson/fermion pairs and therefore cancel. Only the states with δ = 0, corresponding to short multiplets of the superconformal algebra, contribute. The most salient feature of the index, which we exploit repeatedly, is its invariance under the RG flow. By calculating the index of an asymptotically free theory at the UV fixed point, we can extract information about the operator spectrum and the global symmetries of the theory at the IR fixed point. Below, we concentrate on calculating the coefficient that appears in the expansion of the index at order pq, where p and q are the superconformal fugacities appearing in (A.1). From this coefficient we will be able to identify the enhanced symmetry of the theory in the IR, as explained below.

The method
To evaluate the index of a gauge theory we need to list all the operators one can build from the fields and then project on gauge invariants. This is often done by writing a matrix integral, see for example [22,46]. The integrand corresponds to the generating function of all the operators and the integral projects on gauge invariant states. The theories we consider have a relatively high rank and a lot of matter making the evaluation of such integrals time consuming. Instead, we will just use representation theory to deduce the needed terms in the index computation. We begin every calculation with listing the operators that contribute at order pq to the index. In order to do that, we first note that for a chiral multiplet χ only the scalar component χ and the anti-spinorψ χ contribute to the index [45]: The scalar contributes (pq) with r being the R charge, along with contributions resulting from their charges and representations under the global symmetries. Note that if some U (1) factors of the global symmetry mix with the R symmetry (such that the corresponding mixing coefficients are not some simple rational numbers), then the operators that we build out of the scalars and anti-fermions should have vanishing charges under these U (1)s so that their contribution to the index will indeed be at order pq. In addition, the only contribution from the vector multiplet that concerns us comes from the gaugino λ, and since it is not charged under the (non R) global symmetry and its R charge equals to 1, the corresponding contribution to the index at order pq will always be 1 and will result from the operator λλ.
Next, after listing the operators that contribute at order pq to the index, we turn to find their representations under the global symmetry. It is important to note that in this list of operators only the forms of the operators are written (i.e. the contraction of gauge indices is not specified) and so it often happens that more than one (gauge invariant) operator corresponds to each form. We then write a table with all the operators that one can build corresponding to each form. After summing the characters of the representations of all these operators with the correct signs, we finally obtain the index at order pq.
At this stage, in order to extract the IR symmetry of the model we use the following property of the index [23]: The contribution at order pq comes from the marginal operators minus the conserved currents. In all the theories considered below except for the case of Spin(10) (and in one of the models in the Spin(8) case), we get only negative contributions at order pq that correspond to the IR conserved currents and to the absence of marginal operators (so that the dimensions of the conformal manifolds vanish). In the cases of Spin (10) and Spin (8), we also find a positive contribution at order pq and so we first need to identify the marginal operators before obtaining the conserved currents.
It is important to note that in all the theories analyzed below, the superpotential is a relevant deformation. It is checked in each case by first considering the model without the superpotential and then coupling gauge singlet fields through new superpotential terms to the operators that violate the unitarity bound (i.e. we flip these operators). This way, we obtain a model in which all the operators satisfy the unitarity bound and the corresponding superconformal R charge is the correct one. The gauge singlet fields that do not appear in the new superpotential are of course free (and have IR R charge 2/3). Then, it is easy to verify that the original superpotential terms (that are not in the new superpotential) are relevant operators in this model. We show it below explicitly for the Spin(6) theory.
We now turn to the application of this procedure to the Spin(n + 4) sequence, starting from n = 2 (the n = 1 case was discussed in [25]). In the simple cases (up to n = 4) the results were confirmed by a calculation of the index in Mathematica.

Spin(6)
We start by describing again the model discussed in section 2. The matter content is given in table 2, which we reproduce here for convenience, The superpotential is given by and the superconformal R charge byr = r − 0.142q a − 0.057q b . In order to check that (A.3) is a relevant deformation, we begin by considering the model without a superpotential. When doing this, one finds that the operator V 2 violates the unitarity bound; therefore, it should be flipped and we couple the field Φ to this operator by adding the superpotential W = ΦV 2 . 7 Now all the operators satisfy the unitarity bound and the superconformal R charge in this model is given byr = r + 0.128q a . Ther charges of the operators M 0 CV 2 S and M 1 V C 2 are 1.92 and 5/3, respectively, and since they are smaller than 2, we find that these operators are relevant.

Spin(7)
The matter content is given in the following table, The superpotential is given by and the superconformal R charge byr = r + 0.039q a . We see that U (1) a mixes with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq should have a vanishing charge under U (1) a . These operators are of the forms Next, denoting the irreducible representations (and the corresponding characters) of the nonabelian groups in the theory by we find the following representations of the gauge singlets corresponding to the operators in (A.8),
Next, in the second part, we examine a model which is similar to the previous one but without the "flipper" field M 0 ; that is, we remove this field and its corresponding term in the superpotential (A.11). We obtain the matter content given in the following table, The superpotential is now given by andr written in the table is the superconformal R charge. Since the enhancement that we want to identify is less trivial in this case (in particular, there is a nonvanishing positive contribution at order pq coming from marginal operators), we calculate the full index up to order pq instead of focusing only at this order. The analytical calculation at orders (pq) < 1 is done in the same way as the one performed at order pq, 8 and in addition to that the index was calculated in Mathematica (which yielded the same result). We get We see that this index is consistent with the U (1) corresponding to the fugacity a 2 b 2 enhancing to SU (2) somewhere on the conformal manifold. Defining 16) we get that the representations appearing in the index are consistent with U (1) e enhancing to SU (2) e . Moreover, we get that SU (2) s × SU (2) c enhances to SU (4) as before. Rewriting the index using the representations of SU (2) e and SU (4), we obtain As expected, we see that in this case there is a nonvanishing contribution at order pq from marginal operators. Noticing that we have such operators which are of the forms S 2 V 2 C 2 V 2 , S 2 V 2 2 and C 2 V 2 2 , i.e. (denoting by subscripts the charges under U (1) a and U (1) b ) Operator Representations

Spin(9)
The matter content is given in the following table, The superpotential is given by .22) and the superconformal R charge byr = r + 0.0376q a . We see that U (1) a mixes with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq should have a vanishing charge under U (1) a . These operators are of the forms Next, denoting the irreducible representations (and the corresponding characters) of the nonabelian groups in the theory by Spin (10) The matter content is given in the following table, The superpotential is given by 26) and the superconformal R charge byr = r − 0.076q a . We see that U (1) a mixes with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq should have a vanishing charge under U (1) a . These operators are of the forms Next, denoting the irreducible representations (and the corresponding characters) of the nonabelian groups in the theory by  1, 1, 490) , (1, 1, 840 ) , 3 (1, 1, 896) , 3 (1, 1, 1050 ) , 5 (1, 1, 1134 ) , (1, 3, 35) , 2 (1, 3, 189) , 3 (1, 3, 280) , Summing (with signs) the above characters, we obtain We see that in this case there is a nonvanishing contribution from marginal operators. Noticing that we have such an operator which is of the form S 2 V 3 2 , i.e. 10 Operator Representations (1,1,175) and that the contribution to the index at order pq comes from marginal operators (with a positive sign) and conserved currents (with a negative sign), we can write this sum as

Spin(11)
The matter content is given in the following table, The superpotential is given by 33) and the superconformal R charge byr = r + 0.030q a . We see that U (1) a mixes with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq should have a vanishing charge under U (1) a . These operators are of the forms Next, denoting the irreducible representations (and the corresponding characters) of the nonabelian groups in the theory by we find the following representations of the gauge singlets corresponding to the operators in (A.34), Summing (with signs) the above characters, we obtain which corresponds to the enhancement of the flavor symmetry SU (7) × U (1) to SU (7) 2 × U (1) 2 in the IR. Moreover, the dimension of the conformal manifold vanishes.

Spin(12)
The matter content is given in the following table, The superpotential is given by and the superconformal R charge byr = r − 0.110q a . We see that U (1) a mixes with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq should have a vanishing charge under U (1) a . These operators are of the forms Next, denoting the irreducible representations (and the corresponding characters) of the nonabelian groups in the theory by we find the following representations of the gauge singlets corresponding to the operators in (A.38), We present the calculation of the superconformal index at order pq corresponding to the SU (6) and SU (8) gauge theories of section 4 using the same method as in the previous section. Notice that, as before, the superpotential is a relevant deformation.

SU (6)
The matter content is given in the following table, The superpotential is given by and the superconformal R charge byr = r − 0.0175q a − 0.0481q b . We see that U (1) b and U (1) a mix with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq should have vanishing charges under U (1) b and U (1) a . These operators are of the forms Next, denoting the irreducible representations (and the corresponding characters) of the nonabelian groups in the theory by

SU (8)
The matter content is given in the following table, The superpotential is given by W = M A 5Q2 + ΦA 4 (B.5) and the superconformal R charge byr = r + 0.0693q a . We see that U (1) a mixes with U (1) r in the expression for the IR R symmetry, and so the operators that contribute to the index at order pq C Self-dualities and symmetry enhancement in SU (2N ) gauge theories In [5] two models with SU (2) and SU (4) gauge groups were considered, and the relation between their self-duality and symmetry enhancement properties was discussed. In particular, it was shown explicitly in the SU (4) model how the enhanced symmetry corresponds to a multitude of self-dualities, and it was further used to find a new self-duality. It was also mentioned that these two models are the first two entries in a sequence of SU (2N ) gauge theories having such self-dual descriptions. In this section we extend the results obtained for the SU (4) group to this sequence of SU (2N ) (N > 2) theories, and find for each N a new self-dual frame in addition to the ones considered in [16,49]. Furthermore, we use these self-dualities to construct for each N a model with symmetry enhancement. We begin with presenting the dualities and continue with the description of the related model with symmetry enhancement.

Dualities
We consider the gauge group SU (2N ) and the following matter content: In order to define the various duality frames, we decompose SU (4) L → SU (2) L × SU (2) L × U (1) L , SU (4) R → SU (2) R × SU (2) R × U (1) R and correspondingly define Q → (Q − , Q + ) and Q → Q − , Q + . Moreover, we omit the U (1) r column for clarity. In this notation, the matter content is given by: The first dual frame is obtained by introducing gauge singlet mesons M ±± k , k = 0, . . . , N − 1 which couple to gauge composite operators through the following superpotential, The matter content is given by: The second dual frame is obtained by introducing gauge singlet baryons G ±± , G ±± , H ±± m , H ±± m , m = 0, . . . , N − 2 which couple to gauge composite operators through the following superpotential: The matter content of the gauge charged objects is given by: The matter content of the gauge singlet objects is given by: 12 The matter content of the gauge charged objects is given by: The matter content of the gauge singlet objects is given by: Overall, in the first duality frame we take the matter to be in the conjugate representation of the nonabelian symmetry (SU (4) L × SU (4) R ), in the second we change the signs of the charges under the baryonic symmetry, and in the third we interchange U (1) L and U (1) R (obtained from the decomposition of SU (4) L × SU (4) R ). The 't Hooft anomalies of all the dual frames match.

Enhancement of symmetry
We now turn to use the self-dualities described above in order to construct a model which is mapped into itself under these dualities; as a result, this model will have a symmetry enhancement in the IR. To do that, we flip as usual half of the composite operators involved in the dualities where N is even; however, in the case where N is odd, there is a composite operator the square of which is marginal, and we flip half of the remaining operators (obtained after ignoring the one that squares to a marginal operator).
This construction is analogous to the one used for the SU (4) theory in [5] (or alternatively for the Spin(6) theory in section 2) and the nonabelian part of the UV symmetry SU (4) L × SU (4) R , viewed as SO(6) L × SO(6) R , is expected to enhance to SO(12) in the IR. This was explicitly verified in the case N = 3.
Turning to the details of the model, the matter content is as follows, In this table, m = 0, . . . , N − 2 as before and l = n(N ), . . . , N − 1, where n(N ) = N 2 for N even and n(N ) = N +1 2 for N odd. The superpotential is given by 13 As mentioned above, the UV symmetry SU (4) L ×SU (4) R ×U (1) s ×U (1) b ×U (1) a enhances to SO(12) × U (1) s × U (1) b × U (1) a in the IR. The dualities act as the Weyl symmetry of the enhanced group.