Peculiar index relations, 2D TQFT, and universality of SUSY enhancement

We study certain exactly marginal gaugings involving arbitrary numbers of Argyres-Douglas (AD) theories and show that the resulting Schur indices are related to those of certain Lagrangian theories of class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{S} $$\end{document} via simple transformations. By writing these quantities in the language of 2D topological quantum field theory (TQFT), we easily read off the S-duality action on the flavor symmetries of the AD quivers and also find expressions for the Schur indices of various classes of exotic AD theories appearing in different decoupling limits. The TQFT expressions for these latter theories are related by simple transformations to the corresponding quantities for certain well-known isolated theories with regular punctures (e.g., the Minahan-Nemeschansky E6 theory and various generalizations). We then reinterpret the TQFT expressions for the indices of our AD theories in terms of the topology of the corresponding 3D mirror quivers, and we show that our isolated AD theories generically admit renormalization group (RG) flows to interacting superconformal field theories (SCFTs) with thirty-two (Poincaré plus special) supercharges. Motivated by these examples, we argue that, in a sense we make precise, the existence of RG flows to interacting SCFTs with thirty-two supercharges is generic in a far larger 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} = 2 SCFTs arising from compactifications of the 6D (2, 0) theory on surfaces with irregular singularities.


Introduction
In this paper, we begin by focusing on a particularly simple -yet surprisingly rich -class of strongly interacting 4D N = 2 SCFTs called the D 2 (SU(N )) theories, with N = 2n + 1 an odd integer [1]. These theories are often imagined as arising in type IIB string theory 1 at local Calabi-Yau singularities and are part of a larger class of theories called the D p (G) JHEP01(2020)187 theories, where G is the ADE flavor symmetry of the SCFT. However, using the methods of [3], we will primarily think of these theories as coming from twisted compactifications of the 6D (2, 0) theory on Riemann surfaces with an irregular puncture. 2 While the strongly coupled D 2 (SU(2n + 1)) SCFTs are of Argyres-Douglas (AD) type 3 and therefore lack N = 2 Lagrangians, they behave in various surprising ways like collections of free hypermultiplets: • The role of the D 2 (SU(3)) theory in the S-duality studied in [4][5][6] is reminiscent of the role played by some of the hypermultiplets in the S-duality of N = 2 SU(3) Supersymmetric Quantum Chromodynamics (SQCD) with N f = 6 flavors [7].
• The so-called "Schur" limits of the 4D N = 2 superconformal indices of the D 2 (SU(2n + 1)) theories are related to the Schur indices of free hypermultiplets by a simple rescaling of the superconformal fugacity and a specialization of the flavor fugacities [8,9].
Given these parallels, it is interesting to ask if at least some of these close relations with Lagrangian theories persist upon conformally gauging subgroups of the flavor symmetry of the D 2 (SU(2n + 1)) theories. As we will see below, the answer to this question is a resounding, "yes". In particular, we will show that the Schur indices of an infinite set of theories gotten by gauging various diagonal flavor symmetries of collections of arbitrarily large numbers of D 2 (SU(2n + 1)) theories and hypermultiplets are related to the Schur indices of certain Lagrangian theories of class S [12] by simple transformations. Rephrasing these relations in the language of 2D TQFT allows us to efficiently study the action of Sduality on the flavor symmetries of the D 2 (SU(2n + 1)) quiver gauge theories (see [13][14][15] for recent discussions of other S-duality properties of these theories).
Beyond the action on flavor symmetries, one of the most interesting aspects of N = 2 S-duality is the emergence of exotic isolated theories at cusps in the space of exactly marginal gauge couplings. For example, Argyres and Seiberg found the exotic E 6 Minahan-Nemeschansky theory in SU (3) SQCD with N f = 6 emerging at a dual cusp with a weakly coupled SU(2) ⊂ E 6 gauge group [7]. This construction was then vastly generalized to find new classes of isolated non-Lagrangian N = 2 SCFTs (e.g., see [12,16]).
As we will see, the TQFT relations we find between the AD quivers and their Lagrangian cousins lead to an interesting new expression for the Schur index of the exotic AD analog of the E 6 theory, the so-called "T X " SCFT, arising via the S-duality studied in [4][5][6]. Moreover, we are able to find the Schur indices for infinitely many generalizations of the T X theory arising via various AD generalizations of S-dualities involving only regular punctures. For example, we find indices for AD analogs of the R 0,p theories (with 2 Depending on the realization, the twisted compactification may or may not be accompanied by an extra regular singularity. 3 In other words, they have N = 2 chiral operators (i.e., operators annihilated by the anti-chiral half of N = 2 superspace sometimes called "Coulomb branch" operators) of non-integer scaling dimension.

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AD theory Class S fixture analog Flow to 32 supercharges D 2 (SU(2n + 1)) Y simple , Y full , Y full ; (free) no 2 , Y full , Y full ; (interacting) yes m 3 ; (interacting) yes Table 1. Three important classes of isolated SCFTs we study in this paper are in the leftmost column (note that we assume, without loss of generality, that m 3 ≥ m 2 ≥ m 1 ; these quantities obey further constraints discussed in the main text). The middle column indicates the corresponding regular puncture class S fixture (specified by a triple of Young diagrams) in the sense described in section 3, where Y ( ) k is the Young diagram shown in figure 6. The parenthetical comment in this column indicates whether the class S fixture is interacting or not. The final column indicates if the theory admits an RG flow, of the type described in the main text, to an interacting SCFT with thirty-two (Poincaré plus special) supercharges. The above AD relatives of interacting class S fixtures always admit such flows while relatives of free fixtures do not. All the above theories can be realized as type III in the nomenclature of [3]. In section 6, we vastly generalize these results.
p ∈ Z ≥0,odd ) arising via the S-dualities studied in [16]. We call these theories R 2,AD 0,p SCFTs. In all cases, the AD index expressions we find are related to those of their regular puncture relatives (e.g., see [17]) by simple transformations on the fugacities. We term these types of AD theories "AD fixtures" in reference to the terminology for the corresponding isolated theories arising from three-punctured spheres in class S (e.g., see the terminology in [16]). In this context, one may also think of the D 2 (SU(2n + 1)) theories as AD relatives of free regular puncture fixtures. On the other hand, the R 2,AD 0,n SCFTs (and other theories we construct below) are AD relatives of interacting regular puncture fixtures (see table 1).
However, the TQFT index expressions we find for these isolated exotic theories are rather illuminating in their own right. For example, unlike the usual expressions for regular puncture theories, the AD indices feature products over TQFT wave functions that are not independent. We then interpret this lack of independence in terms of the topology of the corresponding quivers of the 3D mirrors associated with the AD theories [3]. As we will see, the quiver topology of our AD relatives of interacting fixtures is characterized by a loop of non-abelian gauge nodes in the 3D mirror. This loop has interesting physical consequences: it guarantees that one can take these isolated AD theories, compactify them on S 1 , and flow (up to free decoupled matter fields) to interacting theories with thirty-two (Poincaré plus special) superchages (thereby generalizing the examples in [6]). 4 We believe that these latter fixed points uplift to 4D N = 4 theories, but we leave a detailed study of this correspondence to future work. 5 Based on the generic existence of RG flows with enhancement to thirty-two supercharges in the exotic isolated AD theories we study, 6 we ask more generally when such JHEP01(2020)187 flows can occur. As we will see, the existence of these types of flows is in fact generic in the space of 4D N = 2 SCFTs (with known 3D Lagrangian mirrors) obtained by compactifying the 6D (2, 0) theory on a Riemann surface with an irregular singularity (we may or may not add an additional regular singularity). 7 Combined with the results of [19][20][21][22][23][24][25][26][27], our work here and in [6] suggests that AD theories naturally live along RG flows with accidental SUSY. 8 We discuss further implications of these ideas in the conclusions.
The outline of the rest of the paper is as follows. In the next section we give more details regarding the D 2 (SU(N )) theories, the resulting quiver gauge theories, and the index relations between these quivers and certain Lagrangian theories of class S. We then move on to construct the 2D TQFT expressions for our indices and study S-duality using these expressions. We conclude this section by computing indices for various exotic type III AD fixtures that arise via S-duality and relating them to indices of better-known theories consisting purely of regular punctures. In the following section, we analyze the implications of these expressions for the quivers of the corresponding 3D mirrors. We then move on to a discussion of the resulting RG flows with accidental supersymmetry enhancement to thirty-two supercharges and conclude by proving a theorem on the universality of such flows in the class of theories arising from compactification of the (2, 0) theory on surfaces with irregular punctures and known 3D mirrors.
Note that throughout our discussion below, we will use the following shorthand to refer to the D 2 (SU(N )) theories in order to ease notational burden: (1.1) 2 Conformal gauging of aD N ≡ D 2 (SU(N )) theories with N ∈ Z Z Z ≥0,odd In this section we introduce relevant technical aspects of the AD N ≡ D 2 (SU(N )) SCFTs (with N odd) and the quiver theories built by conformally gauging them. In particular, we first construct an intermediate building block, T ( ) n 1 ,n 2 , and then construct the main quiver theories of interest, T ( ) n 1 ,n,n 2 . We then move on to construct Schur indices for these quivers and relate them to Schur indices of certain Lagrangian theories.

More details of the AD quiver building blocks
The AD N theories are a class of isolated strongly coupled 4D N = 2 SCFTs. Their Coulomb branch chiral rings are generated by operators of dimensions [1]. Since N is odd, these theories have N = 2 chiral primaries (i.e., "Coulomb theories as well. The fact that we gauge some symmetries to build these theories means that the 3D mirror interpretation of their indices is more subtle. 7 In this sense, the word "exotic" for our isolated AD theories is inappropriate. Indeed, although flows to thirty-two supercharges of the type we describe are not common among the AD theories often studied in the literature, we will see that this is because such theories are actually rather special. 8 Although note that here and in [6] we imagine that the accidental SUSY enhancement arises along RG flows emanating from the AD theories in the UV. On the other hand, in [19][20][21][22][23][24][25][26][27] the accidental SUSY enhancement mainly arises for flows ending on AD theories in the IR.

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n + AD 2n+ AD 2n+3 Figure 1. The quiver diagram of two conformally gauged AD N SCFTs. The left box stands for an AD 2n+ theory, the right box stands for an AD 2n+3 theory, and the middle circle stands for an SU(n + ) vector multiplet diagonally gauging the two AD theories. Here n is an integer, and is an odd integer. This is the simplest example of a conformally gauged AD building block for the more complicated class of quivers we will focus on (see figure 3).
branch" operators) of non-integer dimension and are therefore of AD type. 9 The conformal anomalies of AD N are given by a AD N = 7 96 (N 2 − 1) and c AD N = 1 12 (N 2 − 1). Most importantly for us in what follows, the flavor symmetry of AD N is SU(N ), and the corresponding flavor central charge is given by where a fundamental hypermultiplet of SU(N ) contributes as k SU(N ) = 2. From the isolated AD N theories, we can construct an intermediate building block for the theories we are interested in as follows. Consider AD 2n+ and AD 2n+3 for a positive integer n and an odd positive integer (so that 2n + and 2n + 3 are odd). These theories have SU(2n + ) and SU(2n + 3 ) flavor symmetries respectively. We can couple an SU(n + ) vector multiplet to these SCFTs by gauging a diagonal SU(n + ) flavor symmetry. The flavor central charge (2.1) implies that this gauging is exactly marginal. The resulting theory is an N = 2 SCFT described by the quiver diagram in figure 1 and has U(n) × U(n + 2 ) flavor symmetry.
Given this flavor symmetry, we can further gauge an SU(n + 2 ) ⊂ U(n + 2 ) subgroup. This gauging is exactly marginal when the SU(n + 2 ) vector multiplet is coupled to an additional AD 2n+5 theory in such a way that the residual flavor symmetry of the AD 2n+5 sector is U(n + 3 ). The resulting theory now has U(n) × U(1) × U(n + 3 ) flavor symmetry.
By continuing this procedure, we obtain a series of conformal linear quiver theories whose matter sector is comprised of various AD N theories. The quiver diagram for these theories is shown in figure 2, where the gauge group is SU(n 1 + ) × SU(n 1 + 2 ) × · · · × SU(n 2 − ) for a positive odd integer, , and two integers, n 1 and n 2 , such that (n 2 −n 1 )/ is a positive integer. We denote this theory by T ( ) n 1 , n 2 , and it has U(n 1 )×U(1) n 2 −n 1 −2 ×U(n 2 ) flavor symmetry. 10 From the quiver diagram, we see that the flavor central charge of the SU(n 1 ) and SU(n 2 ) subgroups are 2n 1 + and 2n 2 − , respectively.

2.2
The main quiver theories of interest: the T ( ) n 1 , n, n 2 SCFTs Now we come to the main quiver theories of interest that are built from the above SCFTs and also from fundamental hypermultiplets. To be more explicit, let us take T ( ) In particular, AD3 is identical to the H2 Argyres-Douglas theory [2] and is sometimes also called the (A1, D4) theory [28]. 10 Note that, when we write T ( ) n 1 , n 2 , we always have n1 < n2 so that (n2 − n1)/ is a positive integer. Figure 2. The quiver diagram of the T ( ) n1, n2 building block for the larger quiver we will consider in figure 3 and focus on in the next section. Here is a positive odd integer, and n 1 and n 2 are two integers such that (n 2 − n 1 )/ is a positive integer. The gauge group of the quiver is SU(n 1 + ) × SU(n 1 + 2 ) × · · · × SU(n 2 − ). The flavor symmetry is U(n 1 ) × U(1) and fundamental hypermultiplets of SU(n). 11 By the discussion in the previous subsection, if we gauge a diagonal SU(n) flavor subgroup of these theories, the beta function vanishes:

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where T ( ) n 1 ,n and T ( ) n 2 ,n both contribute 2n − , the fundamental hypermultiplets contribute 2 , and the SU(n) vector multiplet contributes −4n. The resulting theory is an N = 2 SCFT described by the quiver diagram in figure 3 and has U(n 1 ) × U(n 2 ) × U( ) × U(1) 2n−n 1 −n 2 −2 flavor symmetry. We denote this theory by T ( ) n 1 , n, n 2 , where the middle n in the subscript stands for the largest rank of the simple components of the gauge group.

Schur index
In this subsection, we construct the Schur indices of the T ( ) n 1 ,n,n 2 SCFTs from the various building blocks described previously. As we will see, these quantities turn out to be closely related to the indices of certain Lagrangian theories of class S.
To understand these statements, first recall that the Schur index of a general N = 2 SCFT, T , is defined as [17,29] 12 where H is the Hilbert space of local operators of T , E is the scaling dimension, R is the Cartan generator of SU(2) R normalized so that the fundamental representation has 11 Note that n, n1, n2, and are positive integers such that is odd, and (n − n1)/ and (n − n2)/ are positive integers. 12 The Schur index is a particular limit of a more general superconformal index [30,31].

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eigenvalues ± 1 2 , G F is the flavor symmetry of the theory, and f i is the i th Cartan generator of G F (i.e., the i th flavor charge).
In the case of AD N , the Schur index was conjectured to be [8] (see also the mathematical results in [32,33] is the fugacity for the SU(N ) flavor symmetry, χ SU(N ) adj (x) is the character of the adjoint representation, and P.E. is the "plethystic exponential". This latter quantity is defined as Let us focus on the case N = 2n 1 + for a positive integer n 1 and an odd positive integer , since these theories enter the quivers we are interested in. In order to make contact with the index of the T ( ) n 1 ,n 2 SCFT, it is useful to consider the splitting of the SU(2n 1 + ) fugacity, x, into those for the SU(n 1 ) × SU(n 1 + ) × U(1) ⊂ SU(2n 1 + ) subgroup. In particular, x splits into y = (y 1 , · · · , y n 1 ), z = (z 1 , · · · , z n 1 + ), and a such that n 1 i=1 y i = n 1 + i=1 z i = 1. 14 In terms of these variables, the Schur index (2.4) for N = 2n 1 + is 15 (2.9) 13 The N = 3 case is also discussed in [34][35][36][37], and the formula in (2.4) agrees with the formula found in these references. 14 The precise relation between x and (y, z, a) is given by , yi = xi/a for i = 1, · · · , n1 , zi = xia for i = n1 + 1, · · · , 2n1 + .

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with "fund" and "afund" being fundamental and anti-fundamental representations, respectively. Note that the last factor of (2.8) is identical to the Schur index of a bifundamental hypermultiplet of SU(n 1 ) × SU(n 1 + ) with q replaced by q 2 . This expression will be important in our discussions below.
2.3.1 The index of the T ( ) n 1 ,n 2 building block Let us now evaluate the Schur indices of the T ( ) n 1 ,n 2 quiver building blocks we will eventually use to construct the Schur indices of the quivers of ultimate interest. Since the T ( ) n 1 ,n 2 SCFTs are obtained by conformally gauging AD N theories, their indices are evaluated as integrals of products of the indices associated with each sector of the quivers.
To describe this gauging, let z 0 , z n 2 −n 1 , and a ≡ a 1 , · · · , a n 2 −n 1 be fugacities for SU(n 1 ), SU(n 2 ), and U(1) n 2 −n 1 subgroups of the T ( ) n 1 ,n 2 flavor symmetry, respectively. Then the quiver diagram in figure 2 implies that where the integral is taken over SU(n 1 + ) × SU(n 1 + 2 ) × · · · × SU(n 2 − ), dµ i is the Haar measure on SU(n 1 + i ), z i for 1 ≤ i ≤ n 2 −n 1 − 1 is the SU(n 1 + i ) fugacity associated with dµ i , and is the index contribution from an SU(N ) vector multiplet. Note that, up to adjoint-valued pre-factors (whose role we will clarify below) and a q → q 2 fugacity rescaling, the Schur indices of the AD 2n 1 + SCFTs in (2.8) are just the indices of bifundamental hypermultiplets. As a result, the indices of the T ( ) n 1 ,n 2 SCFTs will also have a close connection with those of Lagrangian theories. Indeed, using the identities (A.2) and (2.8), one can rewrite (2.10) as where  is the Schur index of the Lagrangian theory described by the quiver in figure 4. Note that this quiver has the same gauge group as in figure 2, but its matter sector is composed purely of fundamental and bifundamental hypermultiplets. 16 The expression (2.12) shows that the Schur index of T ( ) n 1 ,n 2 has a close connection with that of L ( ) n 1 ,n 2 (we need only multiply by adjoint-valued prefactors and rescale q → q 2 ).
Let us briefly comment on the plethystic exponential pre-factor in front of I L ( ) n 1 ,n 2 on the r.h.s. of (2.12). This term is inherited from the AD theories at the ends of the quiver and is independent of the abelian flavor fugacities, a. On the other hand, this pre-factor does depend on the fugacities, x and y, for the non-abelian flavor subgroup. The role of this dependence can be understood by noting that I T ( ) (2.14) where 1 ≤ i ≤ n 2 −n 1 , and a = b 1 , · · · , b n 2 −n 1 −i , c 1 , · · · , c i . There is a similar recursive relation for I L ( ) Let us now assemble our previous results and compute the Schur indices of the quivers we will ultimately be interested in for our discussion below -the T ( ) n 1 ,n,n 2 SCFTs. To begin, we let (x 1 , a 1 ), (x 2 , b 1 ), and (y, c) denote the fugacities for the flavor U(n 1 ), U(n 2 ), and U( ) subgroups, respectively. We also let a 2 , · · · , a n−n 1 and b 2 , · · · , b n−n 2 represent the fugacities for the residual U(1) 2n−n 1 −n 2 −2 flavor subgroup. From its quiver description 16 The L ( ) n 1 ,n 2 theory has the same flavor symmetry as T ( ) n 1 ,n 2 unless there is an accidental enhancement. Therefore, its Schur index is a function of the same set of fugacities as I T ( ) where a ≡ a 1 , · · · , a n−n 1 and b ≡ b 1 , · · · , b n−n 2 .
As in the case of T ( ) n 1 ,n 2 , this Schur index is also related to the index of a quiver gauge theory with a Lagrangian description. Indeed, using (A.2), (A.4) and (2.12), one can rewrite (2.15) as 18 is the Schur index of a Lagrangian theory described by the quiver diagram in figure 5. We call this quiver gauge theory L ( ) n 1 ,n,n 2 . Note that the flavor symmetry of L ( ) n 1 ,n,n 2 is U(n 1 ) × U(n 2 ) × U(2 ) × U(1) 2n−n 1 −n 2 −2 .
In (2.17), (x 1 , a 1 ) and (x 2 , b 1 ) are fugacities for the U(n 1 ) and U(n 2 ) flavor subgroups respectively, while (y 1 , y 2 , c 1 , c 2 ) are fugacities for the U(2 ) flavor subgroup. Note that the flavor symmetry of L ( ) n 1 ,n,n 2 is not the same as the flavor symmetry of T ( ) n 1 ,n,n 2 . Indeed, the rank of the flavor symmetry of L ( ) n 1 ,n,n 2 is larger than that of T ( ) n 1 ,n,n 2 by . Therefore, in the relation (2.16), 2 fugacities for the U(2 ) flavor subgroup of L ( ) n 1 ,n,n 2 are restricted to fugacities (y, c). Finally, note that the P.E. factor depending on x 1 and x 2 plays the same role as in the case of T ( ) n 1 ,n 2 .

TQFT expressions for the Schur indices and S-duality
In this section we begin by focusing on the T ( ) n 1 ,n,n 2 SCFTs and studying the resulting Sdualities via the connection with L ( ) n 1 ,n,n 2 discussed in the previous section. In particular, this connection leads us to simple TQFT expressions for the Schur indices of the T ( ) n 1 ,n,n 2 SCFTs and makes it straightfoward to read off the action of S-duality on the corresponding abelian flavor symmetries. 19 Moreover, as we will see, the TQFT approach gives rise to 18 In the case of ni = 1, the factor χ SU(n i ) adj is replaced with 0. In the case of ni = 0, it is replaced by −1. 19 The generalization of this discussion to T ( ) n 1 ,n 2 is straightforward but involves extra decoupled hypermultiplets. interesting new expressions for indices of certain exotic AD building blocks that appear at certain cusps in the conformal manifolds of the T ( ) n 1 ,n,n 2 theories. One useful aspect of the Lagrangian quiver theory, L ( ) n 1 ,n,n 2 , is that it can be obtained by compactifying the 6D (2,0) A n−1 theory on a sphere with 2n−n 1 −n 2 simple punctures and two additional regular punctures associated with Y ( ) figure 5) [12]. This fact implies that the superconformal index of L ( ) n 1 ,n,n 2 can be computed via a TQFT on the sphere [38]. In this context, its Schur index, I L ( ) n 1 ,n,n 2 , is written as a correlation function of q-deformed Yang-Mills (q-YM) theory [17]. Moreover, since the compactificaiton of the 6D (2, 0) theory involves only regular punctures, the TQFT expression for I L ( ) n 1 ,n,n 2 is particularly simple.

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On the other hand, AD theories arise from the compactifications of the (2, 0) theory with one irregular puncture and, depending on the case, at most one additional regular puncture. The resulting TQFT index expressions tend to be considerably more elaborate [34,39].
However, the simple TQFT expression for I L ( ) n 1 ,n,n 2 and the relation (2.16) imply that the Schur index of the non-Lagrangian quiver theory T ( ) n 1 ,n,n 2 also has a simple TQFT expression. Indeed, applying the transformation in (2.16) to the q-YM expression for I L ( ) n 1 ,n,n 2 , we obtain

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where y * ≡ (y −1 1 , · · · , y −1 ) and The parameters e i and f i are functions of a, b, c, and q satisfying is the n-box Young diagram with k columns of height one and columns of height n−k (see figure 6). We use the short-hand notation Y simple ≡ Y where w is an SU(n) fugacity such that figure 6), and χ U( ) adj (y) = χ SU( ) adj (y)+1. 21 Note here that the wave function factors in (3.1) are not directly given by (3.6) but involve the rescaling q → q 2 .
Note also that the expression (3.1) for the Schur index of T ( ) n 1 ,n,n 2 is invariant under the permutations of e 1 , · · · , e n−n 1 , f 1 , · · · , f n−n 2 . It turns out that such permutations 20 Note that not all ei and fj are independent. Indeed, we see that there is one constraint on them: (3.5) 21 For Y simple and Y full , this expression reduces to Figure 6. The left picture shows the Young diagram Y ( ) k with n boxes. Here k and are nonnegative integers such that n−k is a positive integer. There are k columns of height one and columns of height n−k . We also use the shorthand notation Y simple ≡ Y in the main text. The right picture shows how the SU(n) fugacity w in (3.6) is related to the SU(k) × SU( ) × U(1) fugacities (x, y, f ) (in the particular case of n = 8, k = 2, and = 3), where w 1 , · · · , w n are assigned to the boxes. are realized by reparameterizing a i , b j and c. Indeed, e i ←→ e i+1 is realized by with the other fugacities kept fixed. Similarly, f i ←→ f i+1 is realized by a transformation of b i and b i+1 . Finally, e n−n 1 ←→ f n−n 2 is realized by a n−n 1 → a n−n 1 c → a n−n 1 with the other fugacities kept fixed. Note that all these transformations keep e 0 and f 0 invariant, and therefore preserve the wave functions f This discussion shows that the Schur index of T ( ) n 1 ,n,n 2 is invariant under the action of S 2n−n 1 −n 2 . As discussed below, this invariance can be regarded as a natural generalization of an S 2n symmetry of the index of T (1) 1,n,1 = (A 2n−1 , A 2n−1 ), which was identified in [39] as the action of the S-duality group (see also [41]). It is therefore natural to interpret the above S 2n−n 1 −n 2 invariance as a consequence of the S-duality invariance of T ( ) n 1 ,n,n 2 .
In the next section, we carefully study two special cases, T (n) 0,n,0 and T (1) 1,n,1 , and show that, from various S-dual descriptions of these theories, one can read off the Schur indices of various infinite series of exotic type III AD theories that decouple at cusps in the space of gauge couplings.

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set of examples gives rise to theories that generalize the T X theory discussed in [6] and are AD analogs of the R 0,n theories studied in [16]. Some of the theories in the second set of examples are AD analogs of other regular puncture fixtures (although, we will see there are some interesting subtleties in this analysis).

S-duality of the T
where y and c are fugacities for SU(n) ⊂ U(n) and U(1) ⊂ U(n) subgroups of the flavor U(n) symmetry, respectively. 22 Note that, unlike in the case of regular puncture theories, the two full puncture wave functions are not independent of each other since they have conjugate fugacities (the same statement applies for the simple puncture wave functions). We will discuss some implications of this fact in the context of the isolated theories that emerge from cusps in the T (n) 0,n,0 gauge coupling space. 23 Let us now discuss the different S-duality frames of the T (n) 0,n,0 SCFTs and the exotic fixtures that appear at certain cusps in the gauge coupling constant space. In order to proceed, it is useful to first review the corresponding story for the L (n) 0,n,0 theories. To that end, recall that the L (n) 0,n,0 theory has another S-dual description in terms of the quiver diagram on the left of figure 7, where the SU(2) gauge group is coupled to a fundamental hypermultiplet and an isolated SCFT / fixture called R 0,n [16]. The flavor symmetry of R 0,n is generically SU(2) × SU(2n), which is enhanced to E 6 in the case n = 3. 24 The gauge coupling, τ , of the dual description is related to the coupling, τ , of the original description by τ = 1 1−τ . In terms of the punctured sphere on the right of figure 5, this description corresponds to the pants decomposition shown in figure 8. This dual description implies that the Schur index of L (n) 0,n,0 can also be expressed as where (w i , c i ) are U(n) fugacities as in (2.17), z = (z, z −1 ) is an SU (2) n−1 . We also note that e0 = f0 = 1 in this case. 23 In fact, some of these implications apply to the gauged theories as well. However, the 3D mirror analysis is more complicated in this case. 24 Since its Coulomb branch operators are all of integral dimension, the R0,n theory is not an AD theory. JHEP01(2020)187 0,n,0 (left) and T (n) 0,n,0 (right). In the left quiver, an SU(2) gauge group is coupled to a fundamental hypermultiplet and an isolated SCFT called R 0,n . In the right quiver, an SU(2) gauge group is coupled to AD 3 (playing the role of the hypermultiplet) and an exotic fixture we call R 2,AD 0,n (this latter theory is a type III theory in the nomenclature of [3]).  of R 0,n given by [17] where only an SU(2) × U(1) × SU(n) 2 subgroup of the flavor symmetry is manifest. As we discuss in appendix B, the T (n) 0,n,0 theory has a similar S-dual description, which is described by the quiver shown on the right of figure 7. The gauge group is again SU(2), which is now coupled to an AD 3 theory (acting as an AD generalization of hypermultiplets) and a type III AD theory in the language of [3]. This type III AD theory is labeled by three Young diagrams Y 1 = Y 2 = [n − 1, n − 1, 2] and Y 3 = [2, · · · , 2, 1, 1] with 2n boxes and generically has SU(2) × SU(n) flavor symmetry (the n = 3 case has SU(3) × SU(2) × SU(2) flavor symmetry). We denote this type III theory by R 2,AD 0,n since it can be regarded as an AD counterpart of the R 0,n fixture. This quiver description implies that the Schur index of T (n) 0,n,0 can also be expressed as where I R 2,AD 0,n is the Schur index of the R 2,AD 0,n theory. Note that previously this index was obtained only for the special case n = 3 [5], while here we describe it for all odd n ≥ 3. 25 25 The identification of the flavor U(1) fugacity in IAD 3 (q; z, c n ) can be understood as follows. From the quiver description in figure 5, we see that T This equation is solved by Indeed, there exists an inversion formula [42] that extracts the integrand of (4.5), which implies that (4.6) is the unique solution to (4.5). Combining (4.6) and (4.3), we obtain the following TQFT expression for the Schur index of R 2,AD 0,n . (4.7) Note that, even though the flavor is set to q, one can show that (4.7) only has integer and half-integer powers of q as it should. Moreover, one can check that for n > 3 the index does not have an O q 1 2 term and so the theory does not have free hypermultiplets. In appendix C we find another proof of this fact by bounding monopole operator dimensions in the 3D mirror. 26 For n = 3, one can perform a stronger consistency check of the above result. Indeed, the R 2,AD 0,3 theory was carefully studied in [5], where it was shown that R 2,AD 0,3 splits into an exotic AD theory called T X and a decoupled half-hypermultiplet in the fundamental representation of the flavor SU(2). 27 The Schur index of the T X SCFT is then where the first two factors comprise the Schur index of the free matter fields. One can check, order by order in q, that (4.8) with (4.7) substituted in is identical to the following expression for the index of T X obtained in [5]: flavor SU(2) × SU(n) symmetry (this operator corresponds to a mirror monopole of scaling dimension (n − 2)/2). Let us denote it by O a with a = 1, 2 being the SU(2) index. Let us also denote by O a ± two flavor SU(3) moment maps in the doublet of SU(2) ⊂ SU(3), where the subscript stands for the charge under U(1) ⊂ SU(3). Then we see that ab O a ± O b can be identified as the baryonic Higgs branch operators mentioned above. This discussion implies that the flavor U(1) fugacity in IAD 3 is c n . 26 Due to the non-trivial quiver topology of the 3D mirror that will be discussed further in the next section, this computation is non-trivial and does not follow directly from the results in [43]. 27 In [5], the R 2,AD 0,3 theory is denoted as is the character of a representation R of su(N ) −N , and R λ and R λ,λ are the highest weight representations of su(2) −2 and su(3) −3 corresponding to the Dynkin labels (−2 − λ, λ) and (−3 − 2λ, λ, λ), respectively. Let us further analyze the two equivalent expressions in (4.8), with (4.7) substituted in, and (4.9). Note that these two expressions have very different origins. Indeed, the expression in (4.9) is written in terms of affine Kac-Moody representations 28 while (4.7) is closely related to the correlator of a TQFT on a sphere with three regular punctures. Moreover, (4.9) takes the form of a sum over a full set of SU(2) representations (with the SU(3) representations restricted in terms of the SU(2) data), while (4.7) takes the form of a sum over a full set of SU (3) representations (here the SU(2) data is fixed in terms of the larger SU(3) data). In spite of these differences, the two formulas both take the form of a product of group theoretical factors of SU(2) × SU(3). Indeed, the second SU (3) wave function in (4.7) is dependent on the first SU(3) wave function since their fugacities are complex conjugates of each other (therefore, in some sense, both expressions involve restrictions on SU(3) data). In section 5, we will reinterpret this dependence of the wave functions in terms of the topology of the corresponding 3D mirrors of the R 2,AD 0,n SCFTs.

S-duality of T
(1) Next let us consider the T 1,n,1 theories for positive integer n ≥ 2. Taking = n 1 = n 2 = 1, the TQFT expression (3.1) reduces to where e i and f i are determined by (3.3) and (3.4). Note that this index is invariant under the S 2n that permutes e 0 , · · · , e n−1 and f 0 , · · · , f n−1 . These permutations are realized by transforming the flavor fugacities as in (3.10) and (3.11), but now for i = 0, · · · , n − 1. In particular, the permutation symmetry is "accidentally" enhanced in this case from S 2(n−1) to S 2n . This S 2n invariance can be interpreted as reflecting the S-duality invariance of the theories. Indeed, it has been argued in [4,45] that the T 1,n,1 theories are identical to the so-called (A 2n−1 , A 2n−1 ) SCFTs [28], whose S-duality group acts on the flavor fugacities through S 2n [39]. Our formula (4.10) clarifies how this S 2n acts on the (2n − 1) flavor fugacities, ( a, c, b), of T 1,n,1 . As in the case of T (n) 0,n,0 , other S-dual descriptions of our theories lead us to expressions for the Schur indices of a series of exotic type III AD fixtures. Indeed, by applying the technique developed in [45], we see that the T We focus on the case in which 2 ≤ m i < n for all i = 1, 2, 3 (we will discuss relaxing JHEP01(2020)187 the condition that m i < n below). Then this dual description is characterized by the quiver diagram shown on the right of figure 9. The quiver has three tails corresponding to three T 1,n,1 , we see that its Schur index can also be written as where s i ≡ (s i,1 , · · · , s i,m i −1 ), t j are some functions of a, c and b, and the last factor is the Schur index of T 2,AD (m 1 ,m 2 ,m 3 ) . This latter index has not been worked out in the literature before.

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with t 3 ≡ 1 t 1 t 2 (as in the case of the R 2,AD 0,n SCFTs, this fugacity dependence will have consequences for the corresponding 3D mirrors to be discussed in the next section). While we don't have a full proof that this is the only expression consistent with (4.11) and (4.12), we see that it gives a physically meaningful result, since there are only integer and halfinteger powers of q (which is necessary for the quantity to be a Schur index of an N = 2 SCFT), and it has the expected S 3 symmetry acting on the z i and t i .
Finally, let us note that the expression in (4.12) assumes that m i < n (at least for the corresponding 4D regular puncture theory to make sense). Indeed, for m i ≥ n, we would end up with a Young diagram with m i columns of height one and one column of non-positive height, n − m i ≤ 0. 30 On the other hand, the expression in (4.11) may in principle make sense for m i ≥ n. It would be interesting to understand if we can analytically continue the expression in (4.12) to the regime of m i ≥ n and understand the corresponding regular puncture theory, T (m 1 ,m 2 ,m 3 ) , as a non-unitary 4D theory (perhaps generalizing the discussion in [46][47][48][49][50]).

Wave function relations and topology of 3D mirrors
In this section, we interpret the TQFT formulas (4.7) and (4.14) for the Schur indices of the R 2,AD 0,n and T 2,AD (m 1 ,m 2 ,m 3 ) SCFTs in terms of the corresponding 3D mirrors given in figure 12  and figure 14 respectively. In the following subsection, we argue that this discussion implies the existence of RG flows with accidental SUSY enhancement to thirty-two (Poincaré plus special) supercharges.
We begin by discussing the TQFT formula for the R 2,AD 0,n index, which we reproduce below for ease of reference (2) adj (z) I R 0,n (q 2 ; z, q, y, y * ) . (5.1) Using the expression for I R 0,n (4.3) we then have .
Let us pay special attention to the transformation on the flavor fugacities when we go from the TQFT expression for R 0,n to that for R 2,AD 0,n . At the level of flavor symmetries, recall that R 0,n has a G R 0,n = SU(2) × SU(2n) flavor symmetry (which is enhanced to E 6 for n = 3) [16]. On the other hand, R 2,AD 0,n has flavor symmetry G R 2,AD 0,n = SU(2) × SU(n) (which is enhanced to SU(2) 2 × SU(3) for n = 3). 30 If mi > n for some i ∈ {1, 2, 3}, the decomposition of the punctured sphere shown in figure 10 leads to a different S-dual description of L (1) 1,n,1 from the one described by the left quiver of figure 9. In particular, the central three-punctured sphere corresponds to a different fixture from T (m 1 ,m 2 ,m 3 ) . It would be interesting to find an AD analog of this class S fixture. JHEP01(2020)187 Figure 11. The 3D mirror of the S 1 reduction of the R 0,n SCFT. Nodes labeled by "N " represent U(N ) gauge nodes and lines between nodes denote bifundamental hypermultiplets (an overall decoupled U(1) is removed). The R 0,n index has a TQFT expression with three independent wave functions corresponding to the three quiver tails in the above diagram of the 3D mirror. The two quiver tails circled in red generate monopoles which are responsible for the SU(n) 2 ⊂ SU(2n)×SU(2) flavor symmetry of the theory (the SU(2) ⊂ SU(2n) × SU(2) factor comes from the third tail, and the balanced central node is responsible for the U(1) × SU(n) 2 → SU(2n) ⊂ SU(2n) × SU(2) enhancement). When we perform the transformation that takes us from the Schur index of R 0,n to that of R 2,AD 0,n , the two SU(n) tails fuse to form a single SU(n) line of nodes as in figure 12.
The TQFT expression in (4.3) makes manifest a U(2)×SU(n) 2 ⊂ G R 0,n flavor subgroup via the wave function with U(2) symmetry, f Y (1) 2 R (q, z, r), and the two wave functions with SU(N ) symmetry, f Y full R (q, w 1 ) and f Y full R (q, w * 2 ). These wave functions, and the flavor symmetries they describe, are related to punctures in the A n−1 (2, 0) theory. The punctures appear in the 3D mirrors of the S 1 compactifications of our 4D theories via the presence of certain quiver tails radiating off a central SU(n) node as in figure 11 [16,51]. In particular, the two tails with gauge groups U(n − 1) × · · · × U(1) correspond to punctures described by the f Y full R wave functions, while the tail with gauge group U(2) × U(1) corresponds to the puncture described by f Y (1) 2 R . Indeed, by the linear quiver rules given in [43], the dimension one monopole operators with fluxes supported on, say, one of the U(n − 1) × · · · × U(1) tails give rise to multiplets containing the additional symmetry currents that enhance the corresponding U(1) n−1 topological symmetry to SU(n). 31 This statement follows from the fact that the corresponding line of nodes is "balanced", i.e., each U(n c ) node has n f = 2n c flavors. A similar phenomenon occurs in the other U(n − 1) × · · · × U(1) tail and the U(2) × U(1) tail, thereby giving rise to the U(2) × SU(n) 2 ⊂ G R 0,n non-abelian symmetry (the U(1) × SU(n) 2 → SU(2n) enhancement occurs because of monopole operators with flux through the central U(n) node).
Given this discussion and the relations between (4.3) and (5.2), let us give an explanation for the form of the quiver tails for the 3D mirror of R 2,AD 0,n shown in figure 12. First, note that the two independent SU(n) TQFT R 0,n wave functions in (4.3) are no longer 31 Recall that any 3D U(nc) gauge group has a corresponding topological symmetry current, jµ = µνρF νρ , where F νρ is the field strength corresponding to the trace part of U(nc). Note that this is a global flavor symmetry acting on the Coulomb branch. In the direct reduction (i.e., the mirror of the mirror quivers we are discussing), the topological symmetry (along with any additional enhanced symmetry via monopole operators) acts on the Higgs branch and descends from the usual 4D flavor symmetry. JHEP01(2020)187 Figure 12. The 3D mirror of the S 1 reduction of the R 2,AD 0,n theory. Nodes and lines are defined as in figure 11. The two quiver tails corresponding to the TQFT wave functions with conjugate fugacities are fused together to give one linear set of nodes generating an SU(n) symmetry. The corresponding nodes are inside the red oval. The remaining quiver tail gives an SU(2) symmetry and is symmetrically fused with the SU(n) nodes via an unbalanced SU(2) node. As a result, the quiver contains a closed non-abelian loop, and the theory can flow to an interacting N = 8 SCFT via the procedure described in the text. independent in (5.2). Indeed, we must set w 1 = w 2 = y (in addition to taking q → q 2 ) and so there is just one independent set of SU(n) fugacities. Since the two SU(n) wave functions are no longer independent, it is natural that in going from figure 11 to figure 12 we should fuse the two previously independent quiver tails into a single tail giving rise to a single SU(n) symmetry. 32 Indeed, note that the line of nodes in the red oval in figure 12 have a bifundamental connecting the two previously independent tails and consist of n − 1 total balanced nodes. By the rules of [43], this line of nodes gives rise to the SU(n) ⊂ G R 2,AD 0,n symmetry.
Since the two previously independent SU(n) wave functions are now related by complex conjugation of fugacities, non-chirality demands that their corresponding line of nodes connects to the quiver tail corresponding to the SU(2) wave function in a symmetric fashion. Indeed, the loop of nodes appearing in figure 12 is precisely such a symmetric connection. The shortening of the remaining tail reduces the U(2) global symmetry factor to SU(2) and also ensures that the line of nodes generating the SU(n) symmetry are indeed balanced. 32 In fact, since the wave functions have conjugate fugacities, it is tempting to write f y), whereR is the SU(n) representation conjugate to R. We may then write the product of SU(n) wave functions in (5.2) as where IV is the Schur index of the SU(n) vector multiplet. The appearance of a single wave function suggests that the SU(n) symmetry should be associated with a single line of nodes in the 3D mirror. Moreover, the additional inverse factor of I − 1 2 V reminds us that this symmetry was associated with two punctures in the original regular puncture theory. At the level of the mirror quiver, this factor reflects the fact that the ranks of the gauge groups in the red oval increase by two between successive nodes in the tails.  Figure 13. The 3D mirror of the S 1 reduction of the T (m1,m2,m3) theory. Nodes and lines are defined as in figure 11. The three quiver tails correspond to TQFT wave functions carrying U(m i ) global symmetry.

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Note that this loop topology of the R 2,AD 0,n quiver will be important in arguing for flows to theories with thirty-two (Poincaré plus special) supercharges in the next section.
Next let us discuss the case of T 2,AD (m 1 ,m 2 ,m 3 ) . For ease of reference, we again write the TQFT expression for the Schur indices of these theories originally appearing in (4.14) wave function corresponds to an independent U(m i ) × · · · × U(1) quiver tail of balanced nodes which, by the rules of [43] gives rise to monopole operators leading to the U(1) m i → U(m i ) flavor enhancement (again, this statement holds assuming generic m i such that m 1 +m 2 +m 3 = 2n and m i < n).
On the other hand, the T 2,AD (m 1 ,m 2 ,m 3 ) theory no longer has independent wave functions carrying U(m i ) flavor symmetry since the t i fugacities in (5.4) are constrained so that t 3 = 1 t 1 t 2 . Indeed, only the SU(m i ) parts of the wave functions are still independent. We can then give an argument in favor of the 3D mirror of the S 1 reduction of T 2,AD n shown in figure 14. The point is that the three balanced U(m i − 1) × · · · U(1) tails correspond to the independent SU(m i ) parts of the TQFT wave functions, while the loop of three U(m i ) JHEP01(2020)187 Figure 14. The 3D mirror of the S 1 reduction of the T 2,AD (m1,m2,m3) SCFT. Nodes and lines are defined as in figure 11. The three U(m i ) × · · · × U(1) quiver tails generate independent SU(m i ) fugacities and correspond to the independent SU(m i ) parts of the three TQFT wave functions. On the other hand, the U(1) parts of the three TQFT wave functions are symmetrically dependent. This dependence is reflected in the quiver by the loop of three U(m i ) nodes. nodes appears because of the constraint on the U(1) parts of the TQFTs wave functions. Again, this difference in the topology of the AD mirror relative to the T (m 1 ,m 2 ,m 3 ) mirror gives rise to the RG flows to theories with thirty-two supercharges that will be discussed further in the next section.

Flows to thirty-two supercharges
As alluded to in the previous section and also in the introduction, one important characteristic of the isolated AD fixtures we are discussing is that, unlike the regular puncture theories they are related to, the AD theories have RG flows (triggered by vevs and genuinely relevant deformations) with accidental SUSY enhancement to interacting theories with thirty-two (Poincaré plus special) supercharges (thereby generalizing the examples in [6]). In the next section, we will argue that such flows are in fact generic in the landscape of AD theories with known 3D mirrors. Note that these flows will proceed via reduction to 3D and via flowing onto the moduli spaces of the resulting theories. We briefly discuss the possibility of uplifting this discussion to 4D at the end of this subsection while postponing a more detailed analysis for future work.
To first understand why the RG flows to interacting theories with thirty-two supercharges occur in the R 2,AD 0,n and T 2,AD (m 1 ,m 2 ,m 3 ) theories discussed above, it is sufficient to compactify these theories on S 1 and consider the corresponding 3D mirrors. Let us start with the mirror in figure 12. Flowing to generic points on the Coulomb branch of the two lines of nodes with gauge groups U(2) × U(4) × · · · × U(n − 3) and also onto points of the Coulomb branch of U(n − 1) × U(n − 1) with symmetry breaking pattern JHEP01(2020)187 , we obtain the quiver in figure 15, where we have dropped decoupled U(1) factors. This is the mirror of the lowest rank theory studied in [6], which we know from that reference flows to N = 8 via mass terms in the direct reduction. However, it will be useful for our more general discussion below to analyze a purely moduli space flow to N = 8 in the mirror theory itself. 33 To that end, consider turning on Higgs branch vevs where the Q i , Q i pairs correspond to the three edges in the loop of figure 15 so that we break U(2) 3 → U(2) diag leaving the quiver in figure 16 after dropping decoupled fields. 34 In terms of the squark fields, we may imagine turning on vevs for i = 1, 2, 3. This latter theory flows directly to N = 8 in the IR. Therefore, we see that through a combination of Coulomb and Higgs branch flows in the mirror theory, we flow to an interacting 3D N = 8 SCFT. Now consider the T 2,AD (m 1 ,m 2 ,m 3 ) SCFTs. Since the T 2,AD (m 1 ,m 2 ,m 3 ) mirror clearly flows to the R 2,AD 0,n mirror via excursions on the Coulomb branch, we can appeal to the above discussion and conclude that there are flows from T 2,AD (m 1 ,m 2 ,m 3 ) to theories with thirty-two supercharges. On the other hand, the T 2,AD (m 1 ,m 2 ,m 3 ) theories also admit other flows to a richer set of N = 8 theories, which we now describe.
Without loss of generailty, we will assume that m 3 ≥ m 2 ≥ m 1 . Now, let us flow to points on the Coulomb branches of the two quiver tails of length m i with i = 3, 2 such JHEP01(2020)187  that the corresponding gauge groups break as U( where the ellipses on the r.h.s. of the breaking contain only abelian gauge groups). Simultaneously, we flow to a generic point on a Coulomb sub-branch in the third tail specified by SU(m 1 − 1) × U(m 1 − 2) × · · · × U(1) and obtain the theory in figure 17. We can then turn on vevs as in (6.1) where the Q i , Q i pairs are now bifundamentals of U(m 1 ) × U(m 1 ). 35 This procedure produces the interacting N = 8 theory described in figure 18. Note that in all RG flows described in this subsection, we flow on both the Coulomb and Higgs branches of the 3D mirror. Therefore, by mirror symmetry, in order to reach the N = 8 fixed points, we flow on both the Higgs and Coulomb branches of the direct S 1 reductions of our 4D SCFTs. It would be interesting to understand if these flows uplift to 4D flows along the Higgs and Coulomb branches of our 4D theories (i.e., if the corresponding 4D RG flows commute with the S 1 reduction as in [6]).
If the flows do uplift, then it would also be interesting to understand if the 3D N = 8 fixed points map to N = 4 theories in 4D. In principle, if the flows are well behaved enough, then the detailed properties of these possible N = 4 fixed points -e.g., if they are of Super-Yang Mills (SYM) type or not -can be studied.

Universality of flows to interacting SCFTs with thirty-two supercharges
In this section, we briefly state and prove a theorem governing how universally we may expect the existence of RG flows to interacting theories with thirty-two (Poincaré plus special) supercharges. This discussion is motivated by our TQFT formulae for the Schur indices of the R 2,AD 0,n and T 2,AD (m 1 ,m 2 ,m 3 ) theories and our reinterpretation of these formulae as leading to closed loops of non-abelian nodes in the corresponding 3D mirrors. Indeed, we 35 In analogy with the previous case, we turn on vevs Qi = Qi = v1m 1 ×m 1 = 0 for all i = 1, 2, 3.

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saw that the existence of such closed loops generically led to RG flows ending on interacting SCFTs with thirty-two supercharges.
Combined with the infinite class of examples in [6], it is then tempting to wonder whether such flows are generic in the class of (untwisted) type III theories (and therefore, perhaps, in the space of N = 2 theories coming from compactifications of the (2, 0) theory on surfaces with untwisted punctures). In fact, it is straightforward to show this is the case, if we assume the classification of such theories given in [13,14]. In this classification, the space of type III theories is specified by N ≥ 2 Young diagrams (the theories discussed above have N = 3). The N = 2 theories cannot flow to theories with thirty-two supercharges (we do not consider turning on additional gauge couplings in the UV), and so we focus on the more generic theories with N ≥ 3. 36 The Young diagrams in question take the form [13,14] 1 , · · · , a 1,n 1 , a 2,1 · · · , a 2,n 2 · · · a p,np ] , (6.3) where the column heights h i and a i,b are non-decreasing (from left to right) positive integers satisfying The above Young diagrams correspond to the degeneracy of the eigenvalues of the singular terms in the Higgs field one obtains in the Hitchin system describing the type III compactification [3] (although note that in our conventions Y 1 corresponds to the most singular piece). At the level of the 3D mirror, the quiver consists of a core with gauge group 5) and N − 2 bifundamentals between each node. 37 The final Young diagram, Y N , describes the quiver tails. For example, if the column of height h b is broken up into [a b,1 , · · · , a b,n b ], we attach a tail to U(h b ) with gauge group (6.6) and bifundamentals between each corresponding node (and also a single bifundamental between the U(h b −a b,1 ) and U(h b ) node). One repeats this procedure for all b ∈ {1, · · · , p}. Given this setup and assumptions, we can prove the following theorem on the universality of non-perturbative flows from sixteen to thirty-two supercharges: Theorem. If the quantities h 3 and n 1 in (6.3) satisfy h 3 , n 1 > 1, the corresponding type III SCFT flows, up to free decoupled factors, to an interacting theory with thirty-two (Poincaré plus special) supercharges upon compactification to 3D, flowing to certain points on the moduli space of the theory, and, for N > 3, turning on mass terms in the 3D mirror. 38 36 Interestingly if one adds a regular singularity one finds, among the N = 2 theories, 3D mirrors equivalent to the star-shaped quivers found in the case of some theories with regular punctures (and no irregular punctures). 37 In the case of the R 2,AD 0,n and T 2,AD (m 1 ,m 2 ,m 3 ) theories, the cores are the triangular loops in figure 12 and figure 14 respectively. 38 The same caveats described at the end of the previous section apply in lifting these flows to 4D.

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Proof. We would like to reduce the 3D mirror to the diagram in figure 17 with m 1 = h 3 > 1.
To accomplish this task, we can first move along the Coulomb branch to reduce our theory to a diagram similar to the one in figure 17, but containing N − 2 bifundamentals between each node. To get to this diagram, first go to generic points on the Coulomb branches of the subset of the core nodes (see (6.5)) characterized by U(h 4 ) × · · · × U(h p ) ⊂ G and to generic points on the Coulomb branches of all their tails (if any exist). Next, we go to generic points on the Coulomb branches of the tails of the U(h 2 ) × U(h 3 ) nodes to remove them as well. Then, we go to generic points on the Coulomb branch of the U(h 1 −a 1,1 −1)× · · · × U(h 1 − a 1,1 − · · · − a 1,n 1 ) part of the U(h 1 ) quiver tail. This procedure leaves us (up to decoupled U(1) factors, which we drop) with a U(h 1 ) × U(h 2 ) × U(h 3 ) group of core nodes connected by N −2 bifundamentals between each node and a U(1) node connected to U(h 1 ) via a fundamental. To proceed, we now go to a point on the U(h 1 )×U(h 2 ) Coulomb branch that breaks the gauge symmetry as U( Up to decoupled U(1)'s, we have a diagram equivalent to that in figure 17 with m 1 = h 3 except for the fact that there are N − 2 bifundamentals between each non-abelian node. We may add mass terms to remove N −3 of the bifundamentals between each node to end up with a diagram identical to the one in figure 17. Combined with the Higgs branch flow described below figure 17, we flow to an interacting N = 8 theory. Therefore, if we are willing to go on the Coulomb and Higgs branches of the 3D mirror and, at the same time, add mass terms for some of the bifundamentals between the remaining non-abelian nodes, we flow to a theory with thirty-two supercharges. 39

Conclusions
In this paper we found various new relations between theories with non-integer scaling dimension N = 2 chiral operators (i.e., AD theories) and those with purely integer dimensional N = 2 chiral operators (the regular puncture class S theories). The latter theories have TQFT index expressions that are typically simpler (and more uniformly presented) than those of the former. The additional complication in the TQFT expressions for the case of AD theories (e.g., see [39,52]) is related to the fact that the corresponding singularities in the compactification from 6D to 4D generally contain more data. However, we saw that we can, in some sense, encode this additional data by taking TQFT data for regular puncture theories (which only have integer dimension N = 2 chiral operators) and demanding interdependence of the different TQFT wave functions through intricate fugacity relations. This fugacity interdependence has important physical consequences: a large class of AD theories flow to interacting IR SCFTs with thirty-two (Poincaré plus special) supercharges via flows of the type discussed in section 6. Using these index relations, we also found expressions for the Schur indices of various classes of exotic type III AD theories. Clearly, there is a lot more to be said. We conclude with some open problems (and potential solutions): 39 Note that adding a regular singularity to the above set of theories does not change the above proof: we can decouple the additional nodes associated with this singularity via flowing to generic points on the corresponding Coulomb branches.

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• It would be interesting to understand if the RG flows we discussed above can be lifted to 4D (for some flows, we know this is the case; e.g., see [6]). If so, then it would be particularly intriguing to try to compute the indices of some of the resulting IR theories and see if they are N = 4 theories or not. If they are N = 4 theories, then it would be interesting to understand if they are Lagrangian (SYM theories) or not.
• One way to address the above point would be to try to construct better-behaved RG flows in the class described in section 6. This might involve better understanding the role that monopole operators can play in the corresponding mirror RG flows. Alternatively, this might involve a better understanding of non-abelian mirror symmetry.
• Another approach to the problem in the first bullet point might be as follows. The authors of [53] find N = 1 Lagrangians for certain class S regular puncture theories by considering excursions along N = 1 conformal manifolds that include these N = 2 SCFTs as special points. In their discussion, the authors find N = 1 Lagrangians on certain conformal manifolds containing N = 2 SCFTs that have both dimension three Higgs branch and dimension three Coulomb branch operators. Some of the theories discussed in the present article satisfy this condition. Moreover, given the similarity of the Schur indices of our theories to those in the regular puncture class S case, it would be interesting to see if one can find N = 1 Lagrangians for some of the R 2,AD 0,n and T 2,AD (m 1 ,m 2 ,m 3 ) theories in this manner. Having an N = 1 Lagrangian or, at the very least, an N = 1 conformal manifold might in turn make it easier to study flows to N = 4.
• The ubiquity of RG flows to interacting theories with thirty-two supercharges emanating from compactifications of the 6D (2, 0) theory on Riemann surfaces with irregular punctures strongly suggests the existence of another way of understanding these theories via D3 branes probing type IIB/F-theory backgrounds far beyond what has been explored in the literature.
• It would be interesting to understand the most general class of N = 2 SCFTs with non-integer dimensional N = 2 chiral operators (i.e., Coulomb branch operators) that are involved in RG flows with SUSY enhancement either as UV or IR end points.
• We had to rescale fugacities as q → q 2 in order to find a match between the indices of the AD theories and those of the regular puncture theories. In the process, we had to consider going from the A n−1 to the A 2n−1 6D (2, 0) parent theories. It would be interesting to understand why this is the case and also to see if more general q → q m rescalings are meaningful.
• It would also be interesting to generalize the peculiar index relations we found to other physical quantities. A natural candidate for such a quantity is the Macdonald limit of the superconformal index [29]. Unfortunately, the Macdonald index is not known in closed form for the AD N theories with N > 3. In the case of the AD 3 theory, however, the index is known in closed form [37]. Moreover, the S-duality discussion in [37] is consistent with the one appearing here. In appendix A we comment JHEP01(2020)187 further on generalizing other results we presented in this paper to the Macdonald limit. In particular, we argue that the important identities (A.2) and (A.4) are easily generalized to the Macdonald index. On the other hand, even in the case of the AD 3 theory, the crucial identity (2.8) is somewhat more subtle.
• Finally, we saw that there is a close relation between regular puncture class S fixtures and our AD fixtures. It would be interesting to understand if to each class S fixture there exists an AD counterpart and, if so, how many such counterparts exist. In addition, we saw that in our class of theories, the AD fixtures with interacting regular puncture relatives admitted RG flows to interacting thirty-two supercharge theories.
On the other hand, AD fixtures with free class S relatives did not admit such flows (even though the corresponding AD theories are strongly interacting). It would be interesting to understand if this story is completely general in the space of theories of class S.

Acknowledgments
It is a pleasure to acknowledge P. Argyres, T. Creutzig, S. Giacomelli, Z. Laczko, and M. Martone for interesting discussions and correspondence. We also thank Z. Laczko for collaboration on closely related topics. M.B. thanks the Centro de Ciencias de Benasque Pedro Pascual and organizers for a wonderful working environment during the excellent workshop "Gauge theories, supergravity, and superstrings". T.N. is grateful to the organizers of the conferences "Infinite dimensional algebras, geometry and integrable systems" at RIMS and "Vertex algebras, factorization algebras and applications" at Kavli IPMU where he had many useful discussions. M.B.'s research is partially supported by the Royal Society under the grant, "New Constraints and Phenomena in Quantum Field Theory". T.N.'s research is partially supported by JSPS Grant-in-Aid for Early-Career Scientists 18K13547.

A Useful identities
In this appendix, we derive useful identities for index contributions from a vector multiplet and a bifundamental hypermultiplet. We also comment on their generalizations to the Macdonald index. The index contribution from an SU(n) vector multiplet is given by . Amusingly, the above identities have a natural generalization to the Madonald index [29]. To see this, first recall that the Macdonald index is defined by where H, E, R and f i are the same as in (2.3), j 1 is the "left spin" associated with su(2) L ⊂ so (4) which can be regarded as Macdonald generalizations of (A.2) and (A.4). From these results, it is natural to expect that the replacement q → q 2 for the Schur index is generalized to (q, t) → (q 2 , tq) for the Macdonald index. However, a similar generalization of (2.8) is more subtle. For example, let us consider the case of AD 3 = (A 1 , D 4 ), whose Macdonald index is [37] where x is the fugacity for the flavor SU(3) symmetry. It is possible to apply the inverse function of the plethystic exponential to obtain the function i AD 3 (q, t; x) such that JHEP01(2020)187 I AD 3 (q, t; x) = P.E. i AD 3 (q, t; x) . However, the resulting i AD 3 (q, t; x) does not have a simple form as in the Schur index case (2.4). Indeed, i AD 3 (q, t; x) contains various terms with positive and negative signs. Moreover, i AD 3 (q, t; x) contains characters of many different irreducible representations of SU (3). These are in contrast to the Schur index case (2.4). This discussion suggests that generalizing (2.8) to the Macdonald index involves a deformation whose interpretation is currently more mysterious.
B III 2×[n−1,n−1,2],[2,··· ,2,1,1] theory In this appendix, we argue that theory described by the right quiver in figure 7 is equivalent to T (n) 0,n,0 . To that end, first note that the former theory is equivalent to the type III AD theory associated with three Young diagrams, Y 1 = Y 2 = [n − 1, n − 1, 1, 1] and Y 3 = [2, · · · , 2, 1, 1], in the language of [3]. Indeed, the prescription of [45] suggests that this type III theory has a weak coupling description corresponding to the splitting of 2n boxes in Y 1 = [n − 1, n − 1, 1, 1] into the two groups, [1,1] and [n − 1, n − 1]. 40 From the 3d mirror analysis, we see that the sector corresponding to [1,1] is D 2 (SU(3)) = AD 3 , the one corresponding to [n − 1, n − 1] is R 2,AD 0,n , and an SU(2) vector multiplet is coupled to them. 41 Therefore, all we have to show here is that this type III AD theory is equivalent to T (n) 0,n,0 . To see the equivalence of the above-mentioned type III theory and T (n) 0,n,0 , let us consider a weak coupling description of the type III theory corresponding to the splitting of 2n boxes in Y 1 into [n − 1, 1] and [n − 1, 1]. From the prescription of [45] and the spectrum of N = 2 chiral operators, we see that the sector corresponding to each [n − 1, 1] is the type IV AD theory (in the language of [3]) associated with an irregular puncture labeled by three Young diagrams Y 1 = Y 2 = [n − 1, 1] and Y 3 = [2, · · · , 2, 1], and a full (and therefore regular) puncture. 42 We also see that an SU(n) vector multiplet is coupled to these type IV AD theories as well as an extra fundamental hypermultiplet. Therefore, this weak coupling description corresponds to the quiver diagram in figure 19.
Hence, all we need to show is the equivalence of T (n) 0,n,0 and the theory described by the quiver in figure 19. Note that, for this purpose, it is sufficient to show that the type IV theory involved in the quiver is equivalent to the T ( ) 0,n = AD n with n−1 2 extra fundamental hypermultiplets of SU(n). 43 In the rest of this appendix, we show that the 40 Here, the idea of [45] is that there exists an S-dual frame for each splitting of boxes in Y1 into two groups. 41 Recall that R 2,AD 0,n is the type III AD theory associated with Y1 = Y2 = [n − 1, n − 1, 2] and Y3 = [2, · · · , 2, 1, 1]. 42 A type IV theory is obtained by compactifying the 6d (2,0) An−1 theory on sphere with an irregular puncture and a regular puncture. These punctures are characterized by the singularity of an sl(n)-valued meromorphic (1, 0)-form, ϕ, around them. Suppose that a regular puncture is at z = 0. Then ϕ behaves near z = 0 as ϕ ∼ M z + · · · dz with M ∈ sl(n), up to conjugation. When the regular puncture is a full puncture, the eigenvalues of M are all different. When an irregular puncture associated with Y1, Y2 and Y3 are at z = 0, ϕ behaves as ϕ ∼ M 1 z 3 + M 2 z 2 + M 3 z + · · · dz up to conjugation, where M1, M2, M3 ∈ sl(n) and the eigenvalues of Mi are such that the ordered list of the numbers of equal eigenvalues is identical to Yi. 43 Recall here that n is odd, and therefore n−1 1 Figure 19. Another weak coupling description of the type III AD theory associated with the Young diagrams Y 1 = Y 2 = [n − 1, n − 1, 2] and Y 3 = [2, · · · , 2, 1, 1]. The left and right boxes each stand for one copy of the type IV AD theory described in the main text, while the top box stands for a fundamental hypermultiplet. We argue that this quiver theory is identical to T (n) 0,n,0 .
Seiberg-Witten (SW) curves of these two theories are indeed identical, which strongly suggests the equivalence of these two theories.

B.1 Curve of type IV theory
Let us first write down the SW curve of the above-mentioned type IV theory. Since the theory is obtained by compactifying the 6d (2,0) A n−1 theory on a sphere with one irregular puncture and a regular puncture, its SW curve is where xdz is the SW 1-form and ϕ = ϕ z dz is a meromorphic (1, 0)-form valued in sl(n).
We take a holomorphic coordinate, z, on the sphere so that the irregular puncture is at z = ∞ and the full puncture is at z = 0. The Young diagrams characterizing the irregular puncture, Y 1 = Y 2 = [n − 1, 1] and Y 3 = [2, · · · , 2, 1], imply that ϕ behaves near z = ∞ as where, up to conjugations, T 1 = diag(a, · · · , a, −(n − 1)a), where M = diag(M 1 , · · · , M n ) such that n i=1 M i = 0. By a change of coordinates that preserves the SW 1-form, the first two matrices can be mapped to T 1 = diag(0, · · · , 0, −1) and T 2 = diag(0, · · · , 0, − b). Here, m i and M i are identified as mass parameters, and b is identified as a relevant coupling of the type IV theory.
While the masses and couplings of the 4D theory are encoded in the singular terms described above, the vacuum expectation values (vevs) of Coulomb branch operators are encoded in less singular terms. To write down the most general expression for the curve including these vevs, let us consider the first correction, U/z 2 , to the terms in the bracket of (B.2), where we parameterize U as The parameters u i and v i are not fixed by the boundary conditions, but they are partially restricted so that det(x − ϕ z ) has only integer powers of x and z. Let us now turn to the SW curve of the AD n theory with n−1 2 extra fundamental hypermultiplets of SU(n). Our strategy is the same as in appendix B of [4], i.e., we start with the curve of AD n , weakly gauge its SU(n) flavor symmetry, introduce n−1 2 extra fundamental hypermultiplets of SU(n), and then turn off the SU(n) gauge coupling. The SW curve of AD n = D 2 (SU(n)) is [1] where M i are the mass parameters associated with the SU(n) flavor symmetry and therefore subject to N i=1 M i = 0, U 0 is the relevant coupling of dimension 1 2 , and U i for i ≥ 1 are the vevs of Coulomb branch operators. The SW 1-form is given by λ = w dt t . When we weakly gauge the SU(n) flavor symmetry, the curve becomes where Λ is the corresponding dynamical scale, and M i is identified with the vevs of the Coulomb branch operators arising from the SU(n) vector multiplet. When we introduce n−1 2 extra fundamental hypermultiplets of SU(n), the curve becomes

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In terms of z ≡ t/ n−1 2 i=1 (w − m i ) and x ≡ w/z, the curve is 11) and the 1-form is λ = xdz up to exact terms. We finally turn off the SU(n) gauge coupling by setting Λ = 0. We then see that the resulting curve is precisely identical to the curve in (B.6), where U 0 is identified as b and U i for i ≥ 1 are identified as u i . This strongly suggests that the type IV theory discussed in the previous sub-section is identical to the AD n theory with n−1 2 extra decoupled hyper multiplets of SU(n). The last identification then implies the equivalence of T (n) 0,n,0 and the theory described by the quiver in figure 19.

C Monopole dimension bounds
In this appendix, we argue that the dimensions of monopole operators in the 3D mirror SCFTs associated with the R 2,AD 0,n theories, ∆(O i ), satisfy the following bounds ∆ ≥ 1 2 , n = 3 1 , n > 3 (n odd) .

(C.1)
This result is in agreement with our 4D index analysis in the main text. Indeed, we argued that the R 2,AD 0,n SCFT only has a decoupled free field sector for n = 3. Note that the linear quiver discussion in [43] does not directly apply here since, as discussed around figure 12, the mirror quiver contains a closed loop of nodes. Indeed, the fact that the n = 3 case has free hypermultiplets even though it is "good" by the naive application of the criteria of [43] motivates us to examine the case for general n more carefully.
While the bound for n = 3 follows from the mirror symmetry discussion in [5,6] (and also the analysis in [4]), we will prove the result in this case and also for all n > 3 directly via an analytic monopole analysis in the mirror. To that end, the quantity we wish to bound is where i A , j A ∈ {1, · · · , 2A}, i B , j B ∈ {1, · · · , 2B}, and a (A) ∈ Z 2A , b (b) ∈ Z 2B , c ∈ Z 2 label the magnetic flux through each gauge node in the quiver (note that we have dropped JHEP01(2020)187 Figure 20. We reproduce the quiver from figure 12 but rotated and with labels a (A) ∈ Z 2A , b (B) ∈ Z 2B , c ∈ Z 2 denoting magnetic charges through the corresponding gauge nodes (the nodes to the left of the central U(2) node have fluxes labeled by "a", while those to the right have fluxes labeled by "b").
subscripts denoting the particular entry in the flux vector) -see figure 20. Note that the negative contributions in (C.2) arise from the gauge nodes while the positive contributions arise from the (bi)fundamentals. The main strategy in proving (C.1) is repeated use of the triangle inequality to cancel four positive matter contributions to ∆ against single gauge contributions (we perform the cancelation between lines and the nodes that they end on). We will start from the leftmost U(2) node in figure 20 and then inductively argue that we can cancel all the negative contributions from all the nodes in the left tail up to and including negative contributions from the U(n − 3) node that neighbors the left U(n − 1) node. By Z 2 symmetry, the corresponding negative contributions from the U(2) to U(n − 3) nodes from the right tail will also be cancelled by corresponding matter contributions. We then move on to consider the core of the quiver and prove (C.1).
Before continuing, let us note that we may always use Weyl transformations at each gauge node to arrange that for all α, β ∈ 1, 2, · · · , n−1

2
. This maneuver has the effect of removing absolute values from gauge node contributions in (C.2). We may then write the contributions from the U(2A) node as We can cancel the negative contributions from U(2) against four hypermultiplet contributions by using the triangle inequality twice This procedure leaves a surplus of four matter contributions we can use to cancel contributions from the adjoining U(4) node. Moreover, since we have not used matter contributions involving a 1,4 , we can use this surplus to cancel one of the most negative terms from U(4) (i.e., one proportional to a Let us now discuss the U(4) node and adjoining matter contributions more carefully. Since this computation contains contributions from matter fields to the left and right of the gauge node, we can use this discussion to build a base case for an inductive proof of the positivity of contributions to ∆ from the left quiver tail. To that end, consider the contributions We may use the surplus contributions in the second term above to cancel one of the contributions from the U(4) gauge node so that 3 )) + Let us now use twelve of the twenty-four U(4) × U(6) hypermultiplets to cancel the remaining three negative U(4) contributions. To see how this cancelation is done, it is useful to visualize the hypermultiplet contributions via a 4 × 6 matrix with a "1" indicating an unused matter contribution and a "0" indicating a used matter contribution. We start with Our strategy is to leave as surplus the first and last columns while using the remainder of the first and last rows (eight terms in all) to cancel the two U(4) contributions proportional to a We cancel the remaining negative contribution from U(4) by using the middle four entries of L 4,6 so that Indeed, we see that This procedure leaves a surplus of 12 hypermultiplets we can use to cancel negative contributions from U(6). Now that we have shown how the negative U(2) × U(4) contributions in the left quiver tail are cancelled, we can move on to the induction hypothesis in our proof. We assume that all the negative contributions in U(1) × · · · × U(2A) have been canceled. In particular, the Let us understand these statements in more detail. In particular, we should first focus on the L 2(A−1),2A generalization of (C.12) we get after finishing the cancelation of terms in U(2(A − 1)). This matrix has its first column filled with 1's. The next column has all 1's except in the first and last row which are 0. For 2 ≤ p ≤ A, the p th column consists of zeros in positions i such that 1 ≤ i ≤ p − 1 and 2A − p ≤ i ≤ 2(A − 1) with 1's everywhere else. This discussion specifies half the matrix. The remaining half is set by demanding that L 2(A−1),2A is symmetric under reflections through a line running between columns A and A + 1, i.e. 1 0 0 · · · · · · 0 0 1 1 1 0 · · · · · · 0 1 1 . . . 1 1 0 · · · · · · 0 1 1 1 0 0 · · · · · · 0 0 1 (C.14) By using the 2A(A − 1) hypermultiplet contributions corresponding to the 1's in (C.14), we assume we cancel A(A−1) 2 of the negative U(2A) contributions via repeated applications of the triangle innequality.
Next we move to L 2A,2(A+1) . This is a 2A × 2(A + 1) matrix full of 1's. Now, as in the U(4) case, we leave the first column alone. In the p th column, with 2 ≤ p ≤ A + 1, we set to zero all rows i such that 1 ≤ i ≤ p − 1 and 2(A + 1) − p ≤ i ≤ 2A. This procedure again specifies the left half of the matrix. The right half is fixed by requiring the matrix to be symmetric under reflection through a line running between columns A + 1 and A + 2, i.e.
Given these assumptions, we now show that we can cancel the negative contributions in U(2(A + 1)) and complete our inductive proof. The negative contributions in this case take the form ∆ ⊃ − To cancel the remaining negative terms, we must use the U(2(A + 1)) × U(2(A + 2)) bifundamental contributions captured by L 2(A+1),2(A+2) . In particular, this latter matrix JHEP01(2020)187 has 1's in all 2(A + 1) × 2(A + 2) entries. Let us use entries 2 through 2A + 3 of the first and last rows to cancel the −(A + 1)(a where we have repeatedly used the triangle inequality. Proceeding in a similar fashion with rows p and 2(A + 1) − p + 1 (but now using entries p + 1 through 2(A + 2) − p of each row), we find that each contribution is bounded from below by (A + 2 − p)(a (A+1) p − a (A+1) 2(A+1)+1−p ). Therefore, we have succeeded in cancelling all the negative contributions of U(2(A + 1)). Note that, after canceling the U(2(A + 1)) contributions, we have 2(A + 1)(A + 2) contributions from bifundamentals of U(2(A + 1)) × U(2(A + 2)) left over as surplus. By Z 2 symmetry, we have now proven that all the non-core nodes of the quiver have their negative contributions to ∆ canceled, and we are left over with (n−3)(n−1) 2 bifundamental contributions of U(n − 3) × U(n − 1) in both gauge tails of figure 20. In particular, we have shown where the first line contains the only remaining negative contributions (i.e., those from the core U(n − 1) × U(n − 1) × U(2) nodes of the quiver), the first two sums in the second line are restricted to lie in the sets S a,b that run over the surplus U(n − 3) × U(n − 1) nodes in the left and right tails respectively (the "a" and "b" subscripts distinguish these tails), and the final line contains bifundamentals from the core of the quiver. This discussion is summarized in figure 21.
C.2 Analyzing the quiver core and proving (C.1) To complete our proof, we now proceed to the quiver core in figure 21. In particular, let us begin by canceling some of the negative contributions to ∆ from the left U(n − 1) node ∆ ⊃ −

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As a result, we have proven that ∆ ≥ 1 2 (|c 1 | + |c 2 |) + 1 2 (C.30) While our choice of cancelation below (C.22) has the effect of making this inequality less manifestly Z 2 symmetric (the contributions of the "a" side of the quiver have already been taken into account in the above bound), this choice does not affect our conclusions.
To prove (C.1), we need only consider a few simple cases. For c 1 = c 2 = 0, we know that all monopole operators have ∆ ≥ 1 by [43] since the quiver effectively reduces to a linear quiver and all nodes are "good". Moreover, if |c i | ≥ 2 for either i = 1 or i = 2, then clearly ∆ ≥ 1. Similar statements hold if |c 1 | = |c 2 | = 1. Therefore, we need only consider the case where, without loss of generality, |c 1 | = 1 and c 2 = 0. We then have (C.31) For n = 3, this bound reduces to (C.1) since the second and third terms are trivial. For n > 3, if we choose any of the b ( n−1 2 ) j = 0, then ∆ ≥ 1 due to contributions from the second term in (C.31). However, if we set all b ( n−1 2 ) j = 0, then the third term leads to ∆ ≥ 1. Therefore, we have proven (C.1).
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