Symmetry enhancement and duality walls in 5d gauge theories

Gauge theories in four dimensions can exhibit interesting low energy phenomena, such as infrared enhancements of global symmetry. We explore a class of 4d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 gauge theories arising from a construction that is motivated by duality walls in 5d gauge theories. Their quiver descriptions bear a resemblance to 4d theories obtained by compactifying 6d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (1, 0) superconformal field theories on a torus with fluxes, but with lower number of flavours and different number of gauge singlets and superpotentials. One of the main features of these theories is that they exhibit a flavour symmetry enhancement, and with supersymmetry enhancement for certain models, in the infrared. Properties of the superconformal fixed points of such theories are investigated in detail.


Introduction
Enhancement of global symmetry in the infrared is one of the most fascinating phenomena in quantum field theory. This can occur when certain operators become conserved currents at the fixed point in the infrared (IR), and make the global symmetry in the IR larger than that in the ultraviolet (UV). One of the reasons that makes the symmetry enhancement intriguing is due to the lack of a general principle and mechanism to explain such a phenomenon, especially in four spacetime dimensions. Nevertheless, supersymmetry allows JHEP06(2020)159 one to study the enhancement of symmetry in a more tractable fashion. This is due to the presence of quantities that do not depend on the renormalisation group flow [1], such as the supersymmetric index in four dimensions [2][3][4], that enable us to easily extract information about the conserved currents at the strongly coupled fixed point by a calculation in the weakly coupled regime.
In this paper we focus on a class of 4d N = 1 supersymmetric gauge theories arising from a construction that is motivated by duality walls in 5d N = 1 gauge theories [5]. Their quiver descriptions are very similar to those studied in [6][7][8][9][10], but with lower number of flavours and different number of gauge singlets and superpotentials. One of the main features of such gauge theories is that they exhibit a flavour symmetry enhancement, as well as supersymmetry enhancement for some models, in the IR. Those with supersymmetry enhancement can be regarded as the complements to the models considered in [11][12][13][14][15]. 1 In the following, we describe the construction of the aforementioned 4d N = 1 gauge theories in detail.
Duality walls in 5d N = 1 gauge theories. Four dimensional theories associated with duality walls in 5d N = 1 gauge theories were proposed and studied in [5]. For definiteness, let us consider 5d N = 1 SU(N ) gauge theory with 2N flavours of fundamental hypermultiplets, and Chern-Simons level zero. For N = 2 this 5d theory has a UV completion as a 5d N = 1 SCFT with an enhanced flavour symmetry E 5 ∼ = SO(10) [16], whereas for N ≥ 3 the UV completion is a 5d N = 1 SCFT with an enhanced symmetry SU(2N ) × SU(2) 2 [17,18]. The 4d N = 1 theory in question is a Wess-Zumino model that can be represented by the following quiver diagram [5, figure 12]: Through out the paper, a white node labelled by n denotes the group SU(n). We denote each factor of the gauge symmetry by a circular node and the flavour symmetry by a rectangular node. The superpotential is taken to be of interesting gauge theories by simply gluing the building blocks together. For example, one can concatenate two duality walls in this 5d theory, and the corresponding 4d theory has the following quiver description [5, figure 13]: The E-string theory on Riemann surfaces with fluxes. Theory (1.1) can be modified in order to describe 4d theories associated with a duality wall in other 5d theories. An interesting modification was studied in [6] in the context of the compactification of 6d rankone E-string theory on Riemann surfaces with fluxes. In that reference, the case of N = 2 is investigated and the top SU(2N ) = SU(4) node is replace by SU (8) The corresponding 5d N = 1 theory is the SU(2) gauge theory with 8 flavours, whose UV completion is the 6d rank-one E-string theory [19][20][21]. The flavour symmetry of theory (1.4) is SU(2) 2 × SU(8) × U(1) F × U(1), where SU(8) × U(1) F is a subgroup of the E 8 symmetry of the E-string theory. Theory (1.4) can be interpreted as coming from the compactification of the rank-one E-string theory on a two punctured sphere (i.e. a tube) with a particular choice of 6d flux that breaks the E 8 symmetry to E 7 × U(1) F . Note that each puncture brings about an SU(2) symmetry and breaks E 7 × U(1) F to SU(8) × U(1) F . From the 5d perspective, the U(1) F symmetry implies the presence of a duality domain wall such that the mass parameter for U(1) F flips its sign as we go from one side of the wall to the other. As discussed in [6, section 3], one way to see the E 7 × U(1) F symmetry is to glue the two punctures together (i.e. close the tube) to form a torus. The corresponding 4d theory can be obtained by taking two copies of (1.4) and 'self-gluing' by identifying their SU (8) nodes and commonly gauging each SU(2) from each copy of (1.4). As a result, one obtains The index of this theory was computed in [6, (3.3)], where it can be written in terms of characters of E 7 × U(1) representations. In fact, a plethora of 4d SCFTs with interesting IR properties, including enhancement of flavour symmetry, can be obtained by compactifying various 6d theories on a torus or a more general Riemann surface, see e.g. [7][8][9][10][22][23][24][25][26].
Modifying the theories. An interesting question that could be asked is whether it is possible to glue together the basic building block (1.1) in a similar fashion as described above in order to obtain a theory analogous to (1.5); for example, for N = 2, we have We emphasize that the crucial difference between (1.7) and (1.5) is that the 5d gauge theory associated with the former has a UV completion in 5d, whereas that associated with the latter has a UV completion in 6d. Therefore, (1.5) has a natural interpretation as coming from the compactification of the 6d theory on a torus, which can be obtained by closing the tube, whereas (1.7) does not. In fact, the superpotential and the condition for the nonanomalous R-symmetry fixes the R-charges of (U, D, L, R, F U , F D ) to be (0, 0, 1, 1, 2, 2). At this stage, we should further introduce the flipping field F U D together with superpotential F U D (U D) that flips the operator U D, which falls below the unitarity bound. This leads to the conformal anomalies (a, c) = 3 16 , 1 8 , which implies that the theory flows to the theory of a free vector multiplet. This implies that such a simple and naive modification of (1.5) to (1.7) does not lead to an interesting interacting SCFT.
This, on the other hand, suggests that the superpotential we turned on in (1.7) is too restrictive. We may further modify the theory by dropping the term LDR and the flipping field F U and consider instead the following theory As it will be discussed in section (2.1), this theory turns out to flow to a decoupled free chiral multiplet, which is identified with the operator U D, together with a 4d N = 2 JHEP06(2020)159 SCFT, described by the 4d N = 2 SU(2) gauge theory with four flavours of fundamental hypermultiplets. The latter has an SO(8) flavour symmetry. We see that not only the flavour symmetry gets enhanced from SU(4) × U(1) to SO (8), but supersymmetry also gets enhanced from N = 1 to N = 2. This naturally leads to a question whether we can obtain more 4d N = 1 gauge theories with interesting IR properties by modifying the quivers in a similar way as described above. The main objective of this paper is to construct and study a number of such theories. Our approach is as follows. We start with 4d N = 1 gauge theories arising from compactification of 6d SCFTs on a torus with fluxes, discussed in [6][7][8][9][10]. The theories are then modified by (1) reducing the number of flavours if this is allowed by gauge anomaly cancellation, (2) dropping some superpotential terms, and (3) adding or dropping flipping fields. As a result, we find several theories that flow to SCFTs with enhanced flavour symmetry, and possibly with enhanced supersymmetry in some cases. Note that as a result of step (1), it is tempting to regard the resulting theory as being obtained by gluing together certain basic building blocks that are associated with duality walls of some 5d gauge theory whose UV completion is in 5d [5], instead of 6d. However, while these theories are inspired by theories related to 5d domain wall theories, in this paper we do not explicitly study the theories living on the 5d domain walls. The theories studied in this paper were mostly chosen by the existence of interesting IR dynamics, and may or may not have an higher dimensional interpretation. We reserve a more in-depth study of such an interpretation to future work.
Organization of the paper. The paper is organized as follows. In section 2, we propose a 4d N = 1 gauge theory that flows to the 4d N = 2 SU(N + 1) gauge theory with 2N + 2 flavours of fundamental hypermultiplets and a decoupled free chiral multiplet. In section 3, a 4d N = 1 gauge theory that flows to the (A 1 , D 4 ) Argyres-Douglas SCFT is investigated. This theory turns out to be Seiberg dual to the theory proposed in [13]. In section 4, we consider modifications of quivers from the minimal (D 5 , D 5 ) conformal matter on a torus with fluxes. In particular, we discuss a 4d N = 1 gauge theory that flows to the 4d N = 2 SO(4) gauge theory with 2 flavours of hypermultiplets in the vector representation. In section 5, we study a 4d N = 1 quiver gauge theory containing an SCFT known as E[USp(2N )], which was first proposed in [10] and is reviewed in appendix A, as a component. We discuss the enhancement of the flavour symmetry in the IR. In section 6, we study a quiver theory with the USp(4) × SU(3) gauge group that is a modification of the (D 5 , D 5 ) conformal matter on a torus with fluxes [7,9]. For the model that we propose, it is found that the flavour symmetry gets enhanced in the IR. We also discuss a subtlety regarding the accidental symmetry of this model. We then conclude the paper in section 7. The basic notion of the supersymmetric index of 4d N = 1 SCFTs is summarized in appendix B.
2 Flowing to the 4d N = 2 SU(N + 1) with 2N + 2 flavours In this section, we consider a 4d N = 1 quiver gauge theory that flows to the N = 2 SU(N + 1) gauge theory with 2N + 2 flavours of fundamental hypermultiplets. We start by exploring the case of N = 1 and then move on to the case of general N . Let us consider the following theory: where F is the flipping field for the gauge invariant quantitiy DD ≡ αβ α β (D) α α (D) β β , with α, β = 1, 2 the indices for the left gauge group, and α , β = 1, 2 the indices for the right gauge group. This is a modification of the rank-one E-string theory on a torus with a flux that breaks E 8 to E 7 × U(1) [6, figure 3]. In comparison with that reference, we lower the number of flavours from 8 to 4, drop the flipping field for U U , and drop the superpotential term LDR.
The superpotential and the condition for the non-anomalous symmetry imply that this theory has one non-anomalous U(1) flavour symmetry, whose fugacity is denoted by d. The superconformal R-charges of the chiral fields can be determined using a-maximisation [27]. We summarize these charges in the following diagram where the powers of the fugacity t denote the exact superconformal R-charges. Observe that the gauge invariant quantity U D has R-charge 2 3 and is therefore a free field, which decouples. Subtracting the conformal anomalies of a free chiral multiplet, (a, c) free chiral = ( This turns out to be the conformal anomalies of 4d N = 2 SU(2) gauge theory with 4 flavours. In particular, this suggests that supersymmetry gets enhanced in the IR. Let us compute the index of 2.1, whose details are collected in appendix (B.1). After factoring out the contribution from the free chiral multiplet (which can be achieved, for JHEP06(2020)159 example, by flipping U D) we obtain where u = (u 1 , u 2 , u 3 ) denotes the SU(4) fugacities corresponding to the square node in quiver (2.1). This can be compared with the index of the N = 2 SU(2) gauge theory with 4 flavours, whose SO(8) flavour symmetry is decomposed into a subgroup SU(4) × U(1) b : The blue terms correspond to the moment map operators transforming under the adjoint representation of SO (8), written in terms of representations of SU(4) × U(1) b ; these operators are mapped to the gauge invariant combinations LDR in (2.1). The term d 4 t 4 3 corresponds to the Coulomb branch operator; this is mapped to U 2 in (2.1). Here the SU(2)×U(1) R-symmetry of the N = 2 theory is decomposed into a subgroup U(1) R ×U(1) d symmetry, where U(1) R is the R-symmetry of the N = 1 theory and U(1) d commutes with U(1) R . The fugacity b corresponds to the baryonic symmetry of the N = 2 theory. This is not manifest in the description (2.3) of the N = 1 theory but is emergent in the IR. This is the reason why we cannot refine the index (2.6), which was computed using (2.3), with respect to the fugacity b.
Finally, we note that it is possible to understand and motivate this result as follows. First, we note from figure (2.3) that the field D has zero charges under all global symmetries and so there is no impediment to it acquiring a vev. Therefore, under the usual way of thought in quantum field theory, we expect this field to acquire a vev dynamically during the RG flow. The effect of this vev should be to identify the two SU(2) gauge groups, leading to only a single SU(2) gauge group, the diagonal one. The additional vector multiplets are Higgsed together with most of the components of the bifundamental D. The bifundamental U , becomes a field in the adjoint representation of the remaining SU(2) and a singlet chiral field. The superpotential LU R then couples the adjoint field with the fields L and R. Overall, we end up precisely with the N = 2 SU(2) gauge theory with 4 flavours, plus a single free chiral field that can be identified with the gauge invariant given by U 2 .

General N
An interesting generalization of (2.1) is to consider the following model: This model can also be thought of as a modification of a 4d theory descending from the compactification of a 6d (1, 0) SCFT, similarly to the previous model. Here the 4d theory in question is the one in [7, figure 7], which comes from a compactification of the 6d (1, 0) SCFT known as the (D N +3 , D N +3 ) conformal matter [28]. Like in the previous case, the 4d theory in [7] is based on 5d domain walls between different 5d gauge theory descriptions of the 6d SCFT on the circle. In line with our general approach here, the modification in (2.8) then corresponds to changing the 5d matter content by the removal of fundamental fields such that the 5d gauge theory now has a 5d SCFT as its UV completion. 2 Nevertheless, this does not guarantee that the theory in figure (2.8) has an interesting higher dimensional origin as it may not be a domain wall theory associated with the modified 5d gauge theory and its associated 5d SCFT.
In the same way as (2.3), this theory has one non-anomalous U(1) flavour symmetry, whose fugacity is denoted by d. The U(1) d charges and superconformal R-charges of each chiral field are depicted in the following diagram: This turns out to be precisely the conformal anomalies for 4d N = 2 SU(N + 1) gauge theory with 2N + 2 flavours. We compute the index of (2.8) for N = 2 and obtain [0,0,1,0,0] (u) + d 6 t 2 + . . . .
The (DN+3, DN+3) conformal matter on the circle for N > 1 has several different 5d gauge theory description, leading to multiple interesting domain wall theories. However, not all cases support a generalization to a smaller number of flavors, while others are more intricate making the calculation we wish to perform involved for generic N . We shall return to consider cases based on other domain walls between 5d gauge theory descriptions for the (D5, D5) conformal matter theory in section 4.

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This can be compared with the index for the N = 2 SU(3) gauge theory with 6 flavours: [0,0,1,0,0] (u) + d 6 t 2 + . . . . (2.14) where b is the fugacity for the baryonic symmetry U(1) b of the N = 2 theory. This symmetry is not manifest in the description (2.10) of the N = 1 theory, but is emergent in the IR. Similarly to the N = 1 case, the U(1) d symmetry is the commutant of the N = 1 R-symmetry in the N = 2 SU(2) × U(1) R-symmetry. The blue terms correspond to the moment map operators in the adjoint representation of SU(6) × U(1) b ; these are mapped to the gauge invariant combinations LDR in (2.8). The term d 4 t 4 3 denotes the Coulomb branch operator tr(φ 2 ), where φ is the complex scalar in the N = 2 vector multiplet; this operator is mapped to U 2 D in (2.8). The marginal operators are represented by the positive terms at order t 2 , and they are as follows. The brown terms correspond to the baryons and antibaryons in the N = 2 theory; they are mapped to L 3 and R 3 in (2.8). The term d 6 t 2 corresponds to the Coulomb branch operator tr(φ 3 ) of the N = 2 theory; it is mapped to the operator U 3 in (2.8). The negative terms at order t 2 confirm that the non-R global symmetry of the theory is indeed SU(6) × U(1) b . 3 Like in the N = 1 case, we can understand and motivate this result as the field D has zero charges under all global symmetries and so there is no impediment to it acquiring a vev. Therefore, we again expect such a vev to be dynamically generated, leading to the identification of the two SU(N + 1) groups and the collapse of the quiver to a single SU(N + 1) gauge theory. Following what happens to the matter content, we again see that we just get the N = 2 SU(N + 1) gauge theory with 2N + 2 fundamental flavours, plus a single free chiral field.

Flowing to the (A 1 , D 4 ) Argyres-Douglas theory
Let us now consider the following theory: and turn on the superpotential: This is again the modification of the rank-one E-string theory on a torus with a flux that breaks E 8 to SO(14) × U(1) [6, figure 12].
Here the superpotential term (U D)(LD) 2

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breaks the SU(2) flavour symmetry associated with the top square node in quiver (3.1) to SO (2). This follows as the indices are contracted in the following manner. The U D part is contracted so as to form a singlet under the left gauge SU (2), but under the right gauge SU(2) it transforms as a triplet plus a singlet. The gauge singlet component of U D, denoted by tr(U D), is massive together with the flipping field F 1 and hence do not play any role in the IR dynamics of the theory. The indices under the left gauge SU(2) are contracted so that LD forms an effective bifundamental between the right gauge SU (2) and the SU(2) flavour symmetry associated with the top square node. It is then contracted with itself so as to be in the triplet under both SU(2) groups. We then contract the triplet indices of U D and (LD) 2 so that we get an invariant under the right SU(2) gauge group. The resulting operator is then in the triplet of the flavour SU(2) associated with the top square node, and so breaks it to SO(2). The superpotential and the condition for non-anomalous R-symmetry imply that there is one non-anomalous U(1) flavour symmetry, whose fugacity is denoted by d.
The theory (3.1), with superpotential (3.2), therefore has a manifest flavour symmetry SO(2) × SU(2) × U(1) d . The U(1) d charges and superconformal R-charges of each chiral field are depicted in the following diagram: where the powers of the fugacity t denote the exact superconformal R-charges. The conformal anomalies are This turns out to be those of the (A 1 , D 4 ) or H 2 Argyres-Douglas theory. In order to see the relation between (3.1) and the (A 1 , D 4 ) theory, it is more convenient to apply Seiberg duality [29] to the lower left SU(2) gauge node. We shall discuss this dual model in detail in the next subsection. There, we shall also argue that the SO(2) × SU(2) symmetry gets enhanced to the SU(3) flavour symmetry of the (A 1 , D 4 ) theory, whereas the U(1) d symmetry plays a role as the commutant of the N = 1 R-symmetry in the N = 2 SU(2) × U(1) R-symmetry of the (A 1 , D 4 ) theory.

Seiberg dual of theory (3.1)
Let us apply the Seiberg duality [29] (see also the Intriligator-Pouliot duality [30]) to the lower left SU(2) gauge node in (3.1), which has six fundamental chiral fields (3 flavours) transforming under it. As a result, we obtain a Wess-Zumino model with 15 singlets, which can be written as M IJ (with I, J = 1, . . . , 6), transforming under the rank two antisymmetric representation of the SU(6) acting on the six fundamental chirals, with the cubic superpotential Pf(M).

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However, in the quiver theory we do not have the SU(6) as part of it is gauged by the right SU(2) gauge group, and so we should split these 15 singlets into representations of the SU(2) gauge group and its commutant. Specifically, this gives 4 mesons M U = LU and 4 mesons M D = LD, both transforming in the bifundamental of the SU(2) gauge group and the top flavour node, 1 baryon L 2 , 1 baryon U 2 , and 1 baryon D 2 , which are singlets, and the 4 fields U D. The latter can be split into the trace part tr(U D) and the traceless part X; in other words, tr(X) = 0 and The field X then is a chiral field in the adjoint of the gauge SU(2), while tr(U D) becomes a singlet chiral field. From the superpotential (3.2), all of the baryons and the trace tr(U D) are flipped, so they are set to zero in the chiral ring. We then obtain the following dual theory The superpotential (3.2) of the original theory contains the term U LR → M U R. This implies that the fields R and M U acquire a mass and can be integrated out. We are thus left with the following theory The superpotential of this theory can be determined by putting all of the possible gauge and flavour invariants that map to the combinations of the fields in (3.3) with R-charge 2 and U(1) d charge 0: This theory was in fact studied in section 3.2 of [13] and section 2.1 of [15]. The term 4 This superpotential term breaks the SU(2) flavour symmetry corresponding to the left square node to SO(2) ∼ = U(1).

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This, together with the SU(2) corresponding to the right square node, gets enhanced to SU(3) in the IR. There is also a non-anomalous U(1) symmetry, which can be identified with U(1) d of the original theory. The superconformal R-charges and U(1) d charges of the chiral fields are summarised as follows: The conformal anomalies are which are equal to those of the (A 1 , D 4 ) Argyres-Douglas theory, whose index was computed in (5.12) in [13]. Using the notation of (3.9), this can be written as 5 where u is the SO(2) ∼ = U(1) fugacity corresponding to the left square node in (3.9), and x is the SU(2) fugacity corresponding to the right square node in (3.9). The brown terms correspond to the decomposition of the adjoint representation of SU (3)  The quivers for 4d theories arising from the compactification of the 6d minimal (D 5 , D 5 ) conformal matter on a torus with various fluxes were presented in figures 29, 30 and 31 of [8]. The idea of constructing such theories was to start from a suitable building block theory corresponding to a sphere with two punctures (i.e. a cylinder) associated with appropriate 6d flux. Such a flux can be viewed as introducing domain walls in certain 5d gauge theories, whose UV completion is the 6d conformal matter. Every building block contains an SU(4) × SU(4) flavour symmetry, which are subgroups of the 6d SO (20) global symmetry group that were preserved by the fluxes. To form a torus with a given flux, the two punctures of an appropriate cylinder are then glued together.
In this section, we consider a variation of the above 4d theories. Similarly to the preceding sections, we modify the building block such that the flavour symmetry is SU(2) × SU(2), instead of SU(4) × SU(4) as mentioned above. We then glue such building blocks together. The resulting theories have the same structure as those in figures 29, 30 and 31 of [8] but with SU(4) flavour symmetry nodes replaced by SU (2). The flipping fields and superpotential are then introduced such that the gauge theory has interesting IR properties. 5 The notation in (5.12) in [13] can be translated to our notation as follows: t = t There are two non-anomalous U(1) symmetries whose fugacities are denoted by d 1 and d 2 .
Each chiral field in the quiver carries the global charges as indicated in the diagram below: where the powers of the fugacity t denote the approximate superconformal R-charges. 6 Since the chiral fields Q DU and Q DD transform under the bifundamental representation of the same gauge groups, it is expected that there is an SU(2) global symmetry acting on them. Indeed, as we shall discuss in more detail below, a combination of U(1) d 1 and U(1) d 2 becomes a generator of such SU(2), and so we have an SU (2) We can Seiberg dualise the top left and bottom right nodes of (4.1), in a similar way to that described in section 3.1. As a result, we obtain the following quiver where the chiral fields of this theory are mapped to the combinations in (4.1) as follows: where we remark that the traces of Q LL Q LR and Q RL Q RR are flipped by F 2L and F 2R according to (4.2), and so X L and X R transform under the adjoint representation of each SU(2) gauge group. Each chiral field in the dual theory carries the global charges as indicated in the diagram below: (4.7) The superpotential of the dual theory can be determined by gauge and flavour invariant combinations in the above quiver that have R-charge 2: The conformal anomalies of (4.5) are indeed equal to (4.4), as it should be. In fact, the SU(2) global symmetry in (4.1) and (4.5) can be made manifest by setting where w is the SU(2) fugacity. This SU (2) is just the one rotating the two SU(2) × SU(2) bifundamentls in (4.5), or the diagonal ones in (4.1), and is visible already in the UV theories. This model, then, does not actually manifest any symmetry enhancement in the IR, and we present it here mostly for completeness. The index can be written as (w) t 2 + . . . .

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The terms at order t 2 indicate that the theory has the flavour symmetry SU (2) From quiver (4.5), one may expect to consider instead the superpotential Note that the last two cubic terms break the SU(2) u and SU(2) v flavour symmetries to SO(2) u and SO(2) w respectively. This is actually the 4d N = 2 theory with an SU(2) × SU(2) gauge group, one bifundamental hypermultiplet, and one flavour of fundamental hypermultiplet for each gauge group. However, since each SU(2) gauge group has three flavour of fundamental hypermultplets charged under it, this theory flows to a theory of two free vector multiplets (after flipping the operators tr(X 2 L ) and tr(X 2 R ), which fall below the unitarity bound). The latter can be seen from the conformal anomalies: (a, c) = (3/8, 1/4) = 2(3/4, 1/2).

A model with an enhanced SU(2) symmetry
We consider the following modification of figure 30 of [8]: with the superpotential (4.13) There are three non-anomalous U(1) symmetries, whose fugacities are denoted by q 1 , q 2 and x. Each chiral field in the quiver carries the global charges as indicated in the diagram below: JHEP06 (2020)159 where the powers of the fugacity t denote the approximate superconformal R-charges 7 that are used in the computation of the index below. We claim that U(1) x gets enhanced to SU (2) x in the IR. The evidence for this is as follows. First of all, the 't Hooft anomalies involving odd powers of U(1) x vanish, as it should be in order for U(1) x to get enhanced to SU (2) x . Moreover, one can compute the index of this theory: (v) t 2 + . . . , (4.16) where u and v are the fugacities associated with the SU(2) flavour symmetry of the left and right square nodes in the quiver. We see that the index can be written in characters of U(1) q 1 × U(1) q 2 × SU(2) u × SU(2) v × SU (2) x , at least to the evaluated order. Note that we do not see the negative term −χ SU(2) [2] (x) at order t 2 . However, this can be accounted for by a cancellation with certain marginal operators.  (2)x can be written as follows: (4.17) Notice that these combinations do not carry fugacities q1 and q2, as required. with the superpotential This theory has three non-anomalous U(1) symmetries, whose fugacities are denoted by d 1 , d 2 and d 3 . Each chiral field in the quiver carries the global charges as indicated in the diagram below: 1 where the powers of the fugacity t denote the exact superconformal R-charges. The conformal anomalies of this theory are (a, c) = 19 12 , 5 3 . (4.21) It is interesting to point out that these are coincident with those of the 4d N = 2 SO(4) gauge theory with 2 flavour of hypermultiplets in the vector representation, or equivalently the SU(2) × SU(2) gauge theory with 2 bifundamental hypermultiplets. We will shortly JHEP06(2020)159 describe the connection between (4.18) and this N = 2 theory. The index of (4.18) is (v) where u and v are fugacities for the SU(2) u and SU(2) v flavour symmetries denoted by the square nodes on the left and right of quiver (4.18).
In order to make a connection with the aforementioned N = 2 theory, we remark that both flipping fields F LL and F RR have R-charge 2, and they can be turned on in the superpotential (4.19), again this is assuming that there are no accidental U(1) symmetries and we can trust the results of the a-maximisation procedure. Under the U(1) p = U(1) d 1 + U(1) d 2 symmetry (so that the fugacity p 2 = d 1 d 2 ), they carry charges p 2 and p −2 respectively. Therefore there is a Kähler quotient implying that this combination is exactly marginal. Thus, adding F LL + F RR in the superpotential (4.19) amounts to moving along a one dimensional subspace of the conformal manifold. In this subspace, Q LL and Q RR acquire a vacuum expectation value (vev). This can be seen as follows. We have the superpotential terms F LL Q 2 LL + F RR Q 2 RR + F LL + F RR , and the F -terms with respect to F LL and F RR force Q 2 LL and Q 2 RR to acquire a vev. In other words, moving along this subspace breaks the U(1) p symmetry, and without this symmetry there is nothing that prevents Q LL and Q RR from acquiring a vev. In either way, the vevs cause (4.18) to collapse to the N = 2 quiver with two SU(2) gauge groups and two bifundamental hypermultiplets.
The index of theory (4.18) with the superpotential deformation F LL + F RR in (4.19) can be obtained from (4.22) by setting (In this parametrisation d 1 d 2 = 1, and so the U(1) p symmetry defined above is broken.) As a result, we obtain (w) + 2 t 2 + . . . .

(4.24)
This is precisely equal to the index of the 4d N = 2 SU(2) × SU(2) gauge theory with two bifundamental hypermultiplets, whose flavour symmetry is USp (4). Observe that the U(1) x and U(1) w symmetries of the deformed N = 1 theory get enhanced to SU(2) x and SU(2) w . Indeed SU(2) x × SU(2) w is the subgroup of USp(4) that is preserved everywhere on the conformal manifold, as can be seen from the negative terms at order t 2 of the index (4.24).

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The blue terms at order t 4 3 correspond to the USp(4) moment map operators, and the term 2q 2 t 4 3 corresponds to the Coulomb branch operators of the two SU(2) gauge groups in the N = 2 theory. The SU(2) × U(1) R-symmetry of the N = 2 theory can indeed be decomposed into U(1) R ×U(1) q , where U(1) R is the N = 1 R-symmetry and U(1) q commutes with U(1) R . Note that the SU(2) u and SU(2) v flavour symmetries completely decouple along the conformal manifold, as can be seen from the index (4.24). A way to see this is to use the fact that the only non-vanishing 't Hooft anomaly involving them is with U(1) p , so once the latter is broken there is no obstruction for them to disappear in the low-energy theory.

Quiver with the E[USp(2N )] theory as a building block
Let us now consider a 4d N = 1 theory whose quiver description contains the E[USp(2N )] theory as a component. The E[USp(2N )] theory is a 4d N = 1 SCFT with USp(2N ) × USp(2N ) × U(1) × U(1) flavour symmetry [10,31]; see also appendix A for a review. It admits a quiver description (A.1), where only the symmetry USp(2N ) × SU(2) N × U(1) × U(1) is manifest. One may use one or many copies of E[USp(2N )] as a building block to construct several interesting 4d SCFTs by commonly gauging the USp(2N ) symmetries, including those that are not manifest in the quiver (A.1), and couple them to matter fields. 9 In [10], a number of such quivers were studied in the context of compactification of the 6d rank N E-string theory on a torus with fluxes.
In this paper, the general strategy is as described in the preceding sections, namely we modify such quivers by lowering number of flavours (say to N f < 8). The resulting quivers are expected to correspond to theories on the domain wall of the 5d N = 1 USp(2N ) gauge theory with an antisymmetric hypermultiplet and N f < 8 flavours of fundamental hypermultiplets. We also modify the superpotential and flipping fields so that the theory has interesting IR properties. In the following, we focus on the theory that is a higher rank USp(2N ) generalisation of (2.1). This theory turns out to have an enhanced flavour symmetry in the IR.

A higher rank USp(2N ) generalization of (2.1)
Let us consider the following model:  9 We remark that such a construction is in the same spirit of that of the 3d S-fold SCFTs in the sense that two U(N ) or SU(N ) symmetries of the T (U(N )) or T (SU(N )) theory [32] are commonly gauged and possibly coupled to matter fields [33][34][35] (see also [36][37][38]). Note that in the Lagrangian description of the T (U(N )) or T (SU(N )) theory only one U(N ) or SU(N ) symmetry is manifest, whereas the other is emergent in the IR.

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where we have used the notation as in appendix A. Here two copies of E[USp(2N )] are glued together by commonly gauging USp(2N ) symmetries from each copy, so that we have a pair of USp(2N ) gauge groups, denoted by blue circular nodes in the quiver. The fields H, C, Π coming from the upper (resp. lower) copy of E[USp(2N )] are labeled by the subscripts U (resp. D), standing for up (resp. down). In the above we introduce the flipping fields F D and F U D , as well as the chiral fields Φ L and Φ R in the traceless antisymmetric representation of the left and right node respectively. The superpotential is taken to be: 10 Notice that the F -terms with respect to Φ L and Φ R have the effect of making a combination of H U , H D and a combination of C U , C D massive, thus leaving only one massless operator in the antisymmetric of the left gauge node and one in the antisymmetric of the right gauge node. We denote the surviving operators by A L and A R and we represent them in the quiver as arcs on the two nodes. The resulting quiver is therefore The superpotential and the condition for the existence of a non-anomalous R-symmetry imply that this theory has two non-anomalous U(1) flavour symmetries, whose fugacities we denote by d and τ . The UV R-charges of the chiral fields L, R, F D and of the operators A L , A R , Π U and Π D are where R d and R τ are the mixing coefficients of the R-symmetry with the abelian global symmetries U(1) d and U(1) τ . To relate these notations to those adopted in appendix A, we remark that the U (1)

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The values of R d and R τ that correspond to the superconformal R-charge can be determined via a-maximisation. For generic N we find For N = 1 we recover exactly the results of section 2.1, with the opererator U D being flipped. In this case, the theory flows to the 4d N = 2 SU(2) gauge theory with four flavours. From now on we will focus on the case N = 2.

The case of N = 2
We have In order to compute the index, we approximate R d = 4 5 . Using (5.5), we summarise the charges of each chiral field as follows: where the powers of t denote the approximate R-charges. Using the charge assignment as in (5.9), we find that the index is SU (4) [0,1,0] (u) + 2 t  we claim that the SU(4) flavour symmetry in the description (5.9) gets enhanced to SO (8) in the IR. Note that the aforementioned U(1), which is a commutant of SU(4) in SO (8), is not manifest in the description (5.9); it is in fact emergent in the IR and combines with JHEP06(2020)159 SU(4) to become SO (8). Moreover, we claim that the U(1) τ gets enhanced to SU(2) τ . Indeed, the above index can be rewritten as (8) [0,1,0,0] (x) + 1) t 11 5 + . . . .

(5.12)
Let us now discuss the symmetry enhancement in further detail. We first consider the enhancement of SU(4) to SO (8). Note that such enhancement also occurs in the N = 1 case, as discussed in section 2.1, where the theory flows to 4d N = 2 SU(2) gauge theory with four flavours, whose flavour symmetry is SO (8). First of all, we notice that the index rearranges into characters of SO (8). For example, at order t 6 5 , we have the terms d −1 χ SO (8) [0,1,0,0] (x) + 1 , which come from the following operators in the following representations of SU(4) × U(1) d : Regarding the enhancement from U(1) τ to SU(2) τ , we again notice that the index rearranges into characters of SU(2) τ . In particular, at order t Such marginal operators and their fugacities are as follows: FD.

(5.15)
Notice that these gauge invariant combinations are neutral under U(1) d , as required.

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where the superscript (U ) is there to emphasize that these are operators coming from the upper E[USp(2N )] theory. Note also that the 't Hooft anomalies involving odd powers of U(1) τ vanish. This is indeed a necessary condition for the enhancement to SU(2) τ . Finally, we again note that we do not observe the conserved currents for this SU(2) τ in the t 2 order in the index. This again might be explained by a cancellation with some marginal operators. For instance, there are the marginal operators, with their index contributions: These could cancel the contribution of the conserved current in the adjoint representation of SU(2) τ .

General N
Let us briefly comment on the case of a general value of N .
We claim that the U(1) τ gets enhanced to SU(2) τ in the IR. The reasons are as follows. Notice that the vanishing of the 't Hooft anomalies with odd powers of U(1) τ holds for any N , and so the necessary condition for such enhancement is satisfied. Moreover, from (5.5) and (5.6), we have R[Π D ] = 0 and R τ = 1 for any N , and so we will have the same set of marginal operators (5.18) in the triplet of SU(2) τ for general N . Finally, E[USp(2N )] enjoys a self-duality (see appendix A) that acts on the τ fugacity of the index as τ → pq/τ = t 2 /τ , which implies that τ will appear in the index of our model with characters of SU(2) τ . All these facts suggest that the enhancement of U(1) τ to SU(2) τ may also occur for higher N .
Regarding the enhancement of SU(4) to SO(8), we do not have crystal clear evidence for it taking place for N ≥ 3. This is partly because it is very cumbersome to compute the index for E[USp(2N )] for N ≥ 3 as a power series in t to a satisfactory order. Nevertheless, one can still see some signals of the SO (8)

A model with an enhanced SU(9) symmetry
In this section, we consider a quiver theory with a USp(4) × SU(3) gauge group that is a variation of figure 4(b) of [7] and figure 6 of [9], associated with the (D 5 , D 5 ) conformal matter on a torus with flux 1 2 . The modification is such that the gauge anomalies are JHEP06(2020)159 cancelled. In particular, we study the following model: where the blue circular node with the label 4 denotes the USp(4) gauge group, and the white circular node with the label 3 denotes the SU(3) gauge group. Let us first focus on the zero superpotential case: The condition for the non-anomalous R-symmetry implies that the R-charges of the chiral fields can be written as a-maximisation fixes (x, y, z) to be  Observe that the gauge invariant combination LL has R-charge 0.392, falling below the unitarity bound. We therefore introduce the flipping field F L and add the superpotential term F L (LL).

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Adding the superpotential term U LR. Let us deform the theory by turning on the relevant deformation U LR, whose R-charge is 101 72 ≈ 1.403, in the superpotential so that The flavour symmetry that is manifest in the quiver is SU (6) We will see below that there is, in fact, an accidental symmetry. This renders the Rcharges (6.10) obtained using a-maximisation unreliable. 12 To understand this point, it is more transparent to consider the Intriligator-Pouliot dual [30] of (6.5).

Intriligator-Pouliot dual of theory (6.5)
We apply the Intriligator-Pouliot duality [30] to the USp(4) gauge group of (6.5). Recall that, under this duality, the USp(4) SQCD with 8 fundamentals is a Wess-Zumino model with 28 chiral multiplets, represented by an 8×8 antisymmetric matrix M , with the quartic superpotential W = Pf M . The dual of model (6.5) can be written as where M X denotes the components of M dual to the bilinear X in (6.5). The combination U D (with the USp(4) gauge indices contracted) can be decomposed into a rank-two symmetric field S and a rank-two antisymmetric field A under SU (3). Note that the latter can also be regarded as a chiral field * A in the antifundamental representation of SU(3). of the fields in (6.5) with R-charge 2 and U(1) x , U(1) y charge 0: We remark that the terms in the first four lines come from the superpotential Pf M of the Seiberg duality. Indeed, the index contractions in these terms are compatible with that in Pf M ∝ I 1 J 1 ··· ,I 4 J 4 M I 1 J 1 · · · M I 4 J 4 . The last term M LL F L comes from F L (LL) in (6.9) of the original theory.
Let us now consider the dual of the theory with superpotential (6.9). In the latter, the superpotential term U LR = M LU R implies that the fields R and M LU acquire a mass and so can be integrated out. 13 The resulting quiver is then Indeed, we see that dualising the USp(4) gauge group in the original theory brings about quartic superpotential terms, which correspond to irrelevant operators with respect to the UV fixed point. These indeed lead to an accidental symmetry. The superpotential and the condition for non-anomalous R-symmetry give (6.16) 13 Note that the matter content in (6.12) renders the SU(3) gauge group to be IR free, and so, without the superpotential (6.9), the theory has trivial dynamics in the IR. This implies that the results in (6.8) are inaccurate, as fields are expected to become free in the IR implying accidental symmetries not taken into account in the calculation leading to (6.8).

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If we were to proceed with a-maximisation, we would obtain ( With these values of the R-charges, we would obtain the conformal anomalies (a, c) = (2.167, 2.573) , (6.18) in agreement with (6.11). However, due to the accidental symmetry, the R-charges presented in (6.17) are unreliable. 14 An enhanced SU(9) flavour symmetry. We claim that the theory (6.14) with superpotential (6.15) flows to a superconformal field theory with a global symmetry SU(9) × SU(2) × U(1) 2 , where the U(1) 3 × SU(6) flavour symmetry manifest as rectangular nodes 15 in the quiver (6.14) gets enhanced to SU(9) in the IR. Let us explain this as follows. Let us consider (6.14), without the singlets F L and M LL , and with zero superpotential. We can combine M DD , M U U , * A and Q, which transform in the antifundamental representation of the SU(3) gauge group, into a the chiral field F in the following theory: 14 Another piece of evidence that something goes wrong is the supersymmetric index. Computing the index of theory (6.14) with the R-charges (6.17) and expanding it as a power series in t = (pq) 1 2 , we obtain negative terms at the power lower than t 2 . This is in contradiction with the superconformal symmetry. 15 In fact, one of such U(1) symmetries is broken by quartic superpotential terms. However, since the latter are irrelevant, we gain this factor of U(1) back in the IR.

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The conformal anomalies are To compute the index of (6.19), we choose the values of the R-charges of the chiral fields to be close to the superconformal ones. For convenience, we take (α, β) in (6.20) to be (− 3 100 , − 7 100 ). We also denote the fugacities of U(1) α and U(1) β as α and β. With these values and notations, we obtain the index to be (v)t 37 15 Q 2 S 2 +α 16 9 β We see that the only relevant operators are F Q , S 3 , SF F and F F F . Let us now deform the fixed point of (6.19) by adding the singlets M LL and F L and turning on superpotential (6.15). Note that M DD , M U U , * A and Q are parts of the field F . From the index (6.24), the terms in the second line obviously correspond to irrelevant operators. Since M LL is a singlet that is added to the fixed point of (6.19), we have R[M LL ] = 2 3 , and so each term in the first line of (6.15) corresponds to an irrelevant operator; for example, R[M LL ] + R[S 3 ] ≈ 2 3 + 179 100 > 2. The last term in (6.15) gives mass to the singlet M LL via the vacuum expectation value of F L . In summary, adding the singlets and turning on the deformation (6.15) makes the theory flow back to the original fixed point of (6.19).
In conclusion, theory (6.5) with superpotential (6.9) and the dual theory (6.14) with superpotential (6.15) flow to the same fixed point as that of theory (6.19). As a result, the flavour symmetry of each of these theories is SU(9) × SU(2) × U(1) 2 . We emphasise again that, for theories (6.5) and (6.14), the SU(9) global symmetry is not visible in the UV but is emergent in the IR.

Conclusion and perspectives
A number of 4d N = 1 gauge theories with interesting IR properties, such as flavour symmetry and supersymmetry enhancement, are proposed and studied. The main approach that is used to construct such theories is to start with 4d N = 1 gauge theories obtained by the compactification of 6d SCFTs on a torus with fluxes. We then modify such theories by reducing the number of flavours as well as dropping or adding superpotential terms and flipping fields. Although such a procedure leads to a number of interesting theories, supersymmetry or flavour symmetry enhancement is not guaranteed in the IR limit. It would be nice to have a systematic method to produce such models.

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Another interesting direction is to further study models similar to (5.3), namely those containing E[USp(2N )] as a component, as well as its compactification on a circle to a 3d N = 2 gauge theory with an appropriate monopole superpotential turned on. As we pointed out in footnote 9, the construction of (5.3) is in the same spirit of that of the 3d S-fold SCFTs [33][34][35][36][37][38], which possess 3d N = 3 or N = 4 supersymmetry. The dimensional reduction of E[USp (2N )], as showed in [10], has indeed a limit to the T [SU(N )] theory used in the S-fold construction. 16 Hence, the compactification of the 4d N = 1 theories containing the E[USp(2N )] building blocks on a circle would naturally give rise to the 3d N = 2 analog of the aforementioned 3d S-fold SCFTs. 17 Recently there have been a proposal regarding a class of N = 2 S-fold solutions in Type IIB supergravity of the form AdS 4 × S 1 × S 5 involving S-duality twists of hyperbolic type along S 1 [44]. It would be interesting to see if there is any connection between such a 3d N = 2 analog in the large N limit to this type of supergravity solutions.
where each blue node labelled by an even number m denotes the group USp(m). Here D (n) stand for the chiral multiplets represented by diagonal lines, V (n) stand for the chiral multiplets represented by vertical lines, and A (n) are the chiral multiplets in the ranktwo antisymmetric representation [0, 1, 0, . . . , 0] + [0, . . . , 0] = n(2n − 1) of USp(2n). The superpotential is taken to be where we omitted contractions of indices, which are always performed using the antisymmetric tensor J = I n ⊗ i σ 2 .
The manifest non-anomalous global symmetry of (A.1) is 19 This symmetry gets enhanced in the IR to which is the non R-global symmetry of the E[USp(2N )] theory. The enhancement was argued in [10] by showing that the gauge invariant operators rearrange into representations of the enhanced USp(2N ) y symmetry (e.g. using the supersymmetric index) and by means of a duality, called of mirror-type in [31], which allows us to find a dual frame where USp(2N ) y is manifest while USp(2N ) x is emergent in the IR.  19 It is worth noting that the U(1)τ in this paper was referred to as U(1)t in the original reference [10].
The reason that we change the notation t to τ in this paper is to avoid the confusion with the fugacity t in the index.
We schematically summarise the charges under the abelian symmetries of the chiral fields and a possible choice for the trial R-charge below: 2 4 . . .
The transformation rules of these operators under the enhanced global symmetry (A.2) are listed below: is self-dual with a non-trivial map of the gauge invariant operators. More precisely, the duality interchanges the USp(2N ) x and USp(2N ) y symmetries and redefines the U(1) τ symmetry and the trial R-symmetry, while it leaves U(1) c unchanged. Denoting with R τ the mixing coefficient of U(1) τ with U(1) R 0 , the action of the duality on these symmetries can be encoded in The operators are accordingly mapped as where the superscript ∨ labels the operators in the dual frame.
In the main text, we use the superconformal field theory E[USp(2N )] as a building block to construct a more complicated model by gauging the USp(2N ) x and USp(2N ) y symmetries and coupling them to some additional matter fields. For this purpose, it is useful to represent E[USp(2N )] by a diagram where we explicitly show both its USp(2N ) global symmetries: where the left and right nodes correspond to USp(2N ) x and USp(2N ) y respectively. We display explicitly the operator Π, H and C. We emphasise that these are composites of chiral fields in the quiver description (A.1). The other operators, such as M n and b n , which transform trivially under each USp(2N ) symmetry are omitted from the diagram. One important ingredient to analyse various models in the main text is the contribution of the E[USp(2N )] block to the tr U(1) R anomaly of each USp(2N ) gauge node. When the JHEP06(2020)159 node that we are gauging corresponds to the manifest USp(2N ) x symmetry, its contribution to tr U(1) R is where the first term is the contribution of Q (N −1,N ) , while the second term is the contribution of D (N ) . On the other hand, for the USp(2N ) y symmetry, it is not convenient to use the quiver description (A.1) of E[USp(2N )], since USp(2N ) y is not manifest in that description but is emergent in the IR. Nevertheless, we can take advantage of the self-duality of E[USp (2N )]. Specifically, we can compute the contribution to the U(1) R anomaly of the gauged USp(2N ) y node using its dual frame where this symmetry is manifest. Using (A.11) we find that such a contribution is Another important result that we used in the main text is that the contribution to the U(1)

B Supersymmetric index
In this appendix we briefly summarise basic notion of the supersymmetric index for 4d N = 1 SCFTs [2][3][4]; see also [45] for a more comprehensive review. We follow closely the exposition of the latter reference. The index of a 4d N = 1 SCFT is a refined Witten index of the theory quantized on where Q is one of the Poincaré supercharges; Q † = S is the conjugate conformal supercharge; M i are Q-closed conserved charges, and µ i are their chemical potentials. All the states contributing to the index with non-vanishing weight have δ = 0; this renders the index independent of β. where α = ± andα =± are respectively the SU(2) 1 and SU(2) 2 indices of the isometry group Spin(4) = SU(2) 1 × SU(2) 2 of S 3 . For definiteness, let us choose Q = Q−. With this particular choice, it is common to define the index as a function of the following fugacities I (p, q) = Tr(−1) F p j 1 +j 2 + 1 2 r q j 2 −j 1 + 1 2 r . (B.3) where p and q are fugacities associated with the supersymmetry preserving squashing of the S 3 [4]. Indeed, even if the dimension of the bosonic part of the 4d N = 1 superconformal algebra is four, the number of independent fugacities that we can turn on in the index is two because of the constraints δ = 0 and [M i , Q] = 0. A possible choice for the combinations of the bosonic generators that satisfy these requirements is ±j 1 + j 2 + r 2 , where j 1 and j 2 are the Cartan generators of SU(2) 1 and SU(2) 2 , and r is the generator of the U(1) r R-symmetry. In the main text, we write t = (pq) The index counts gauge invariant operators that can be constructed from modes of the fields. The latter are usually referred to as 'letters' in the literature. The single-letter index for a vector multiplet and a chiral multiplet χ(R) transforming in the R representation of the gauge×flavour group is where {z i }, with i = 1, . . . , N c and Nc i=1 z i = 1, are the fugacities parameterising the Cartan subalgebra of SU(N c ). We will also use the shorthand notation Γ e uz ±n = Γ e (uz n ) Γ e uz −n . (B.8) On the other hand, the contribution of the vector multiplet in the adjoint representation of SU(N c ), together with the SU(N c ) Haar measure, is where the dots denote that it will be used in addition to the full matter multiplets transforming in representations of the gauge group. The integration contour is taken over the maximal torus of the gauge group and κ is the index of U(1) free vector multiplet defined as κ = (ty; ty)(ty −1 ; ty −1 ), (B.10) with (a; b) = ∞ n=0 (1 − ab n ) the q-Pochhammer symbol. A similar discussion for the USp(2N c ) gauge group can be found in appendix B of [10].
At the superconformal fixed point, we employ the superconformal symmetry to extract the information about the states. Although the index counts states up to cancellations due to recombinations of various short superconformal multiplets to long multiplets, it has been shown in [46] that at low orders of the expansion in t the index reliably contains information about certain important operators. In particular, at order t 2 = pq, one obtains the difference between the marginal operators and the conserved currents. We extensively utilise the result of the computation at this order in the main text.

B.1 Example: index of theory (2.1)
Let us now discuss how to obtain the index of theory (2.1) with the charge assignment given in (2.3). A technical problem here is that the chiral field D carries zero R-charge, which causes the problem in obtaining the power series in t of the index (2.6). One way to circumvent this problem is to assign an extra U(1) symmetry, which we shall refer to as a 'fake' symmetry and is denoted by U(1) f , so that it mixes with the R-symmetry in such a way that the chiral fields originally carrying zero R-charge now have positive R-charge.
As an example, we can assign the U(1) f charge of D to be 2 such that the new R-charge