Conformal Manifolds and 3d Mirrors of Argyres-Douglas theories

Argyres-Douglas theories constitute an important class of superconformal field theories in $4$d. The main focus of this paper is on two infinite families of such theories, known as $D^b_p(\mathrm{SO}(2N))$ and $(A_m, D_n)$. We analyze in depth their conformal manifolds. In doing so we encounter several theories of class $\mathcal{S}$ of twisted $A_{\text{odd}}$, twisted $A_{\text{even}}$ and twisted $D$ types associated with a sphere with one twisted irregular puncture and one twisted regular puncture. These models include $D_p(G)$ theories, with $G$ non-simply-laced algebras. A number of new properties of such theories are discussed in detail, along with new SCFTs that arise from partially closing the twisted regular puncture. Moreover, we systematically present the $3$d mirror theories, also known as the magnetic quivers, for the $D^b_p(\mathrm{SO}(2N))$ theories, with $p \geq b$, and the $(A_m, D_n)$ theories, with arbitrary $m$ and $n$. We also discuss the $3$d reduction and mirror theories of certain $D^b_p(\mathrm{SO}(2N))$ theories, with $p<b$, where the former arises from gauging topological symmetries of some $T^\sigma_\rho[\mathrm{SO}(2M)]$ theories that are not manifest in the Lagrangian description of the latter.


Introduction
Superconformal field theories (SCFTs) in four spacetime dimensions and with eight supercharges attracted much interest over the past decades, as the conformal symmetry and the large amount of supersymmetry often make it possible to achieve exact results even in the strongly coupled regime.
One interesting class of such SCFTs are those of Argyres-Douglas (AD) type. The defining property of an AD SCFT is the presence of at least one Coulomb branch (CB) operator in the spectrum with fractional (non-integer) conformal dimension. The first examples of AD theories were found soon after the discovery of the Seiberg-Witten solutions [1,2]. It was then realized that at singular points in the Coulomb branch of gauge theories, mutually non-local BPS dyons become massless [3]. The low-energy dynamics of the system is thus captured by an intrinsically interacting non-Lagrangian theory. This initial set of AD theories was tremendously enlarged over the years (see, for example, [4][5][6][7][8][9][10][11]). It was found that many AD theories can be realized inside the class S construction [12,13], and also many admit a geometric engineering description [7,8] as the IIB superstring compactified on a non-compact singular Calabi-Yau (CY) threefold. Some AD theories admit both descriptions.
Despite being interacting and non-Lagrangian, many AD theories are not isolated. They can admit exactly marginal operators in the spectrum, and therefore possess a conformal manifold. At weakly coupling cusps in the conformal manifold, the AD theory splits as a sector of vector multiplets gauging other matter sectors, which are themselves possibly non-Lagrangian SCFTs.
In this work, we consider AD SCFTs that can be realized either starting from the 6d N = (2, 0) theory of D-type compactified on a sphere with an irregular and a regular (possibly trivial) puncture or from the IIB geometrical engineering. Our focus is double: we systematically study the structure of the conformal manifold of this class of models, as well as derive their 3d mirror theories [14] upon reduction on a circle. The latter class of theories turns out to be 3d N = 4 gauge theories with the property that their Coulomb branch is identical to the Higgs branch (HB) of the original 4d theory, and that their HB is identical to the CB of the 3d reduction of the original 4d theory on a circle. Such mirror theories can be regarded as magnetic quivers, in the notation of [15][16][17][18][19].
By using the geometric engineering picture, we uncover a complete and systematic pattern for the structure of the conformal manifold. We find that the (A n , D m ) and D b p (SO(2N )) AD theories that admit marginal couplings can be described as an SO or USp gauging of matter sectors which can generically be realized with a twisted irregular (both A and D types can appear) plus a twisted regular puncture [20]. In one class of cases, however, one of the matter sectors does not admit this type of class S description, and corresponds to theories of type VII or X in the notation of [21]. We provide several checks of this proposal, by matching the IIB geometries, the CB spectrum, and the conformal central charges a and c. To provide these checks, we first discuss what is the contribution to the CB spectrum of a partially closed twisted regular puncture in the presence of an irregular puncture. We also discuss how the contribution to the central charges of a regular (possibly twisted) puncture changes in the presence of an irregular puncture, compared to the contribution it would have had in the setup of regular punctures only.
The 3d mirror theories that we find are quivers of orthosymplectic gauge nodes, as well as an overall Z 2 quotient and a free sector. The presence of the free sector is explained by the fact that, at a generic point of the HB of the original 4d theory, the lowenergy effective theory involves a sector of strongly coupled AD theories each of which does not have a HB. 1 In the same way as in [22], we refer to such a sector as the non-Higgsable SCFTs. These theories dimensionally reduce to free twisted hypermultiplets [22][23][24], which then correspond to the free hypermultiplets sector under the mirror map.
Contrary to D b p (SU(N )) theories, whose reduction to 3d always leads to a Lagrangian theory [22,23], we do not have such a description for the class of theories studied in the present paper. As a result, the quiver description of each mirror theory in this paper is obtained by guesswork, based on information about the CB dimension, the value of 24(c − a) and the number of mass parameters of the original 4d theory in question. Such conjectural quivers are then subject to several stringent tests, including matching of the HB and CB Hilbert series with those of the known theories in various special cases, as well as the Maruyoshi-Song (MS) flow [25] of the D b p (SO(2N )) theory. The latter deserves a more detailed discussion here, since it puts highly non-trivial constraints on the theory, as demonstrated in [22] for the D b p (SU(N )) theory. Roughly speaking, this is a renormalization group (RG) flow from the D p (SO(2N )) theory to the (A p−1 , D N ) theory [26], triggered by vacuum expectation values (VEVs) of certain flipping fields. This flow can be realized at the level of the 3d mirror theory, in particular, it is possible to determine how the flipping fields turn into each component of the mirror theory. In this way, we can establish the constraints on the latter. To achieve this, we need to extend the Flip-Flip duality of [27] to the T [SO(2N )] theory, introduced in [28]. Hence, we propose the Flip-Flip duality for T [SO(2N )] as a by-product of this study. We also comment on the possibility that the sector of free hypermultiplets could admit a discrete gauging if the defect group of the AD theory is nontrivial.
We find that the 3d mirrors of D b p (SO(2N )) theories organize themselves in five different qualitatively distinct classes, depending on whether p is larger or smaller than b, depending on if b = 2N − 2 or b = N , and finally depending on whether GCD(b, p) is even or odd. 2 We systematically discuss each of these classes in a different section, leaving the more involved case of b = N , p < b for further study. As a consequence of the analysis of the 3d mirrors, we find that the dimensional reduction of E 6 Minahan-Nemeschansky (MN) theory [29] admits a UV completion as USp(4) with 5 flavors plus a gauging of the U(1) topological symmetry. We argue that such topological symmetry is present from the existence of monopole operators of conformal dimension 1. This provides an ADHM-like description of the moduli space of one E 6 instanton on C 2 .
The paper is organized as follows. In Section 2 we review the properties of class S theories on the sphere with one irregular and one regular puncture. In Section 3 we discuss the structure of the conformal manifold of D b p (SO(2N )) and (A n , D n ) theories. In Section 4 we collect various facts that we use in the following sections in order to derive the 3d mirrors. We discuss the Flip-Flip duality for T [SO(2N )], we review the supersymmetry enhancing MS flows for D b p (SO(2N )), and we discuss the equivalence among some 3d N = 4 Abelian quiver theories, where in one side hypermultiplets have charge one and on the other have charge two. In Sections 5 to 8 we discuss explicit examples of the 3d mirror theories, as well as non-Higgsable SCFTs. In Appendix A we provide details of the HS computation supporting the claims in the latter four sections. In Appendix B we discuss the number of marginal operators of the theories of Xie-Yau type VII. In Appendix C we provide tables listing the rank and 24(c − a) for all the non-higgsable (G, G ) theories, up to n = 100 when either G or G is A n , D n , computed using the program given in [30]. We further conjecture that such tables can be read as a prediction for the non-vanishing genus-zero Gopakumar-Vafa invariants [31][32][33] of numerous non-toric Calabi-Yau manifolds.
For the sake of the readers, we summarize the 4d theories studied in this paper together with their definitions in Table 1.

Class S theories with irregular punctures
In this section, we will review the properties of four dimensional superconformal theories obtained by compactifying 6d N = (2, 0) theories on a sphere with one irregular and one regular puncture, which are the main focus of the present work. The case of untwisted punctures of type A has already been discussed in detail in [22] and here we will mainly Hypersurface singularities of types VII and X (2.33) and (2.34) Table 1: Summary of the 4d theories studied in this paper with the reference to their definitions.
consider the D case. As we will see, in the description of the conformal manifold of these models, the superconformal theories engineered by twisted irregular punctures (both of type A and D) introduced in [20] play a key rôle, and we will therefore discuss these as well in detail.
2.1 Review of D b p (SO(2N )) theories As was discussed in [11], there are two families of irregular punctures for N = (2, 0) theories of type D N . When combined with a full regular puncture, one family leads to the D p (SO(2N )) theories introduced in [34] and the other leads to the D N p (SO(2N )) models discussed in detail in [26]. We will often refer to both families as D b p (SO(2N )), with the parameter b equal to 2N − 2 or N respectively (and p ≥ 1 in both cases). For our analysis it will be convenient to use also an alternative geometric realization of these theories as compactifications of Type IIB string theory on local Calabi-Yau threefolds. These are described by hypersurface singularities in C 3 × C * of the form for b = N . The C * variable is z and Ω denotes the holomorphic three-form. Both families enjoy a SO(2N ) global symmetry (manifest in the class S realization of the theories since it is carried by the full puncture) which is sometimes enhanced for certain values of p and b. Determining the rank of the global symmetry can be done by counting the number of deformation parameters of unit dimension. The result of this counting (see [26]) is The rank of the theory (dimension of the Coulomb branch) can be obtained using the formula where r indeed denotes the rank of the theory and rk(G F ) is the rank of the global symmetry, which we already know how to determine. The VEVs of Coulomb branch operators correspond to deformation parameters for the hypersurface singularities of dimension larger than one. Finally, we know how to compute the 't Hooft anomalies of these models. The flavor central charge of the SO(2N ) global symmetry is and the a and c central charges can be determined as follows. The combination 2a − c is given by the Shapere-Tachikawa relation [35]: where the sum runs over Coulomb branch operators and ∆ i denotes their scaling dimension. All the models discussed in this paper satisfy this relation. 3 The central charge c is given instead by the formula For p > b we can completely remove the SO(2N ) global symmetry by turning on a principal nilpotent VEV for the associated moment map, 4 an operation which is usually referred to as closure of the puncture in the class S literature and results in removing the regular puncture completely. Upon closure of the puncture we find another class of theories, associated to a sphere with an irregular puncture only, which we denote as SO (2N . When b = 2N − 2 we will always refer to these models as (A p−b−1 , D N ) theories. In addition, these models admit a description in Type IIB string theory in terms of threefold singularities very similar to (2.1) and (2.2). The main difference is that the hypersurface singularity is defined in C 4 rather than C 3 × C * and the holomorphic three-form is different. More precisely, for (A p−1 , D N ) theories (2.10)

AD theories from twisted punctures
In the study of the conformal manifold of SO(2N ) b [p] theories, we will come across AD models defined by a sphere with twisted punctures, one irregular and one regular.
As was discussed in [20] for all 6d theories of type A or D there are two families of irregular Z 2 twisted punctures, leading to two infinite sets of superconformal theories once the 6d parent theory is given. In this section we will discuss their properties, focusing on the families of models relevant for the analysis of the conformal manifolds of SO(2N ) b [p] theories: both types of irregular punctures for twisted D N and only one type for twisted A odd and twisted A even . We will first discuss the properties of theories with a full regular puncture following [20] and then move to the analysis of theories with a generic regular puncture in Section 2.2.2. This latter part is new.

Theories with a full regular puncture
We now review the properties of AD theories defined by a sphere with one irregular and one full regular punctures with a Z 2 twist, focusing on the families relevant for the analysis of the conformal manifold of SO (2N  known, the versal deformations of the ADE singularity are in one-to-one correspondence with the Casimir invariants of the underlying Lie algebra.
In the case of theories with twisted punctures, the ADE singularity is that of the parent 6d theory and the deformations associated with Casimir invariants which transform non-trivially under the action of the Z 2 outer automorphism are all proportional to half-integer powers of z. This accounts for the monodromy associated with the twist line.
Twisted D N +1 theories Let us start with the twisted D N +1 theories. As we have mentioned, there are two families of irregular punctures. One family is engineered in Type IIB by the hypersurface singularity We will call the resulting theories D 2N p (USp(2N )), 5 or simply D p (USp(2N )). The CB spectrum can be described as follows [20, 3rd row, where k is restricted to integer values such that the resulting scaling dimensions are strictly larger than one. The parameters of dimension 1 correspond to mass parameters, and we denote their number as f . The USp(2N ) flavor symmetry 7 of the full puncture has central charge 8 [20, (4.2)] Regarding the a and c central charges, the combination 2a − c is still captured by the Shapere-Tachikawa formula (2.7) and the c central charge is given by the relation [20,5 Regarding the nomenclature for the D b p (USp(2N )) theory, we choose b in such a way that the scaling dimension of z is ∆[z] = b/p. 6 According to [20,Table 14], the values of ∆[z] are needed to compute the CB spectrum. In order to reproduce correctly the results in this article, k or 2k in the expressions for ∆[z] in [20, Table 1] must be replaced by p − b t , with b t given by [20,Table 4]. 7 This USp(2N ) symmetry is denoted by C anom N in [20] due to the absence of the Witten anomaly associated with such a USp(2N ) symmetry. 8 We remark that the normalization of the flavor central charges adopted in this paper is twice of that adopted in [20]. (2.14) The second family of irregular punctures leads to a class of SCFTs which we call D N +1 p (USp(2N )) and are described in Type IIB by the hypersurface singularity where p is a half-integer. The CB spectrum is given by the formula [20, 3rd row, Table  14] with k integer. Again, the range of k is restricted by the constraint that the scaling dimensions are strictly larger than one. We denote again the number of mass parameters of dimension 1 as f . The USp(2N ) flavor central charge is (see [20, (4.2)]) and the c central charge is given by the expression [20, (4.4)] The combination 2a − c is again given by the Shapere-Tachikawa relation.
Twisted A odd theories In the case of twisted A odd theories we consider only one class of irregular punctures, whose geometric realization in Type IIB is given by the hypersurface singularity In the following we call them D p (SO(2N + 1)) where p is a positive integer. The CB spectrum is given by the formula [20, 2rd row, (2.20) 9 Minors typos in [20] have been corrected here.
where again, the range of k is constrained by the requirement that the CB operators have dimensions larger than one. The flavor central charge of the SO(2N +1) symmetry is (see [20, (4.2)]) The central charges a and c are given by the Shapere-Tachikawa relation and by the formula [20, (4.4)] where f is again the number of mass parameters of dimension 1.
Twisted A even theories Finally, for twisted A even trinions the geometric realization is given (for the class of irregular punctures we are interested in) by the hypersurface singularity and we call the resulting theories D p (USp (2N )). The value of p in this case is halfinteger. The CB spectrum is given by [20, 1st row, Table 14]

(2.24)
Again k is bounded above for consistency with the unitarity bound. The USp (2N ) flavor 10 central charge is (see [20, (4.2)]) The formula for the c central charge is, similarly to the previous cases [20, (4.4)], As usual f denotes the number of mass parameters. 10 This USp (2N ) symmetry is denoted by C anom N in [20] due to the presence of the Witten anomaly [37], as pointed out by [38].

Closing the twisted regular puncture
In the rest of the paper, we will also need to consider the case of punctures labeled by partitions of the form a, 1 b , with b odd for twisted A odd and b even in the other cases. Since for this class of punctures the global symmetry has always embedding index one in the global symmetry of the full puncture, we can compute the flavor central charge by starting from the central charge of the theory with a full puncture, which we know, and subtracting the contribution of Goldstone multiplets, whose fermions always have charge −1 under U(1) r . These are always organized into a − 1 fundamentals of the global symmetry, which is SO(b) in the twisted A odd case and USp(b) for twisted A even and twisted D N . The contribution of Goldstone multiplets to the flavor central charge is 2a − 2 when the symmetry is SO(b) and a − 1 when the symmetry is USp(b).
We would now like to understand what sort of theories we get upon the complete closure of the regular puncture, starting from the models described above.
Twisted A 2N −1 theories Let us start by discussing the twisted A 2N −1 theories. The regular puncture is labeled by a B-partition of 2N + 1 and the fully closed (i.e. minimal) puncture corresponds to the partition [2N + 1]. We need to look at the contribution to the graded Coulomb branch of the regular puncture. In the case of the full puncture 1 2N +1 we have (see [39]) where k 2 denotes the greatest integer less than or equal to k 2 . For the minimal puncture [2N + 1] we have instead The full puncture gives us the D p (SO(2N + 1)) theory, whose CB spectrum was described before. To find the spectrum of the theory with the minimal puncture, we have to remove the operators of the highest dimension for each value of k, to account for the difference n full k − n min k . Using this prescription we conclude that upon closure we land on the theory we get by removing the regular puncture from the theory D p−N p (SO(2p − 2N )), which is described in Type IIB by the hypersurface This relation is of course valid for p ≥ N + 2. We recognize here the defining equation Let us give one example. We consider the case N = 3 and p = 8 for definiteness. For N = 3 the contribution to the CB from full and minimal twisted punctures is We can easily describe the CB spectrum of the D 8 (SO (7) We marked in red the operators which should be removed upon closing the regular puncture. It is easy to check that the resulting spectrum is the same as we get from the singularity F (u, x, y, z) = u 2 + x 4 + xy 2 + yz 3 = 0 ; Ω = dudxdydz dF .

Twisted D N theories
Let us now consider the twisted D N case, namely the D p (USp(2N −2)) and D N p (USp(2N − 2)) theories. The contribution of the full 1 2N −2 and minimal [2N − 2] punctures to the graded Coulomb branch is (see [40]) From the spectrum of the D p (USp(2N − 2)) theory described before, we therefore conclude that upon closing the full puncture the CB spectrum becomes identical to that of the theory engineered by the hypersurface singularity F (u, x, y, z) = u 2 + x p+3−2N + xy N −1 + yz 2 = 0 ; Ω = dudxdydz dF , (2.33) which corresponds to the hypersurface singularity of type VII in [21, Table 2]. We will discuss some properties of these hypersurface singularities in Appendix B. Using the same method, we can also analyze the other class of twisted D N theories. Upon closure of the twisted D N puncture we get a 4d SCFT described by the hypersurface singularity that, instead is the hypersurface singularity of type X in [21, Table 2].
Twisted A even theories Finally, let us consider twisted A even theories (or D p (USp (2N )) models). We will now see that upon higgsing D p (USp (2N )) flows in the IR to the A 2N , D2p+1−2N 2 Argyres-Douglas theories. In this case the contribution from the regular punctures to the graded Coulomb branch has not been worked out yet and therefore we cannot just quote a result from the existing literature. However, we can bypass this difficulty with a trick. The contribution to the Coulomb branch is a local property of the puncture and therefore we can determine it by considering trinions with regular punctures only. Luckily, some A even trinions have already been studied in [41,42] and we can exploit these results. We choose to compare the two trinions where the subscript t denotes the twisted puncture. Note that the difference is one twisted puncture: minimal in the first case and full in the second. The result we are looking for simply follows by comparing the CB spectra of the two theories. The first is known to describe two copies of D 2 (SU(2N + 1)) and therefore the CB spectrum is 3 2 , 3 2 , 5 2 , 5 2 , . . . , 2N +1 2 , 2N +1 2 . The spectrum of the second trinion has not been studied, but it is known that by replacing the full untwisted puncture 1 2N +1 with the puncture [N + 1, N ] we find one copy of D 2 (SU(2N + 1)). Using now the known contribution to the graded CB from untwisted punctures, we find that the spectrum of the second trinion in (2.35) contains, besides the CB operators of D 2 (SU(2N + 1)), operators of dimension 3, 4, 5, 5, 6, 6, 7, 7, 7, . . . , 2N + 1, . . . , 2N + 1 Overall, we conclude that for twisted A even punctures n full k − n min k = 0, 0, 1, 1, . . . , for k = 2, . . . , 2N +1. Furthermore, the presence of CB operators of fractional dimension in the first trinion of (2.35) suggests the presence of an a-type constraint for each odd value of k. 11 In practice this means that for k odd, besides removing k−3 2 operators, we should also trade a dimension k operator for an operator of dimension k 2 . Let us consider one example in detail. We consider the case N = 2 and p = 11 2 (remember that p is half-integer). According to (2.36) we should remove one operator of dimension 4 and one of dimension 5. Furthermore, for k = 3, 5 we should also implement the a-constraint and therefore divide by two the scaling dimension of the parameter corresponding to the leading pole. The CB spectrum of the theory with a full twisted regular puncture is k CB operators where we indicate in red the operators which should be dropped and in blue those whose dimension needs to be halved. The resulting spectrum is k CB operators which reproduces the spectrum of (A 4 , D 4 ). Repeating this analysis in general, we find a match with A 2N , D2p+1−2N 2 theories, which are described by the hypersurface singularities (2.39) 11 An a-type constraint means that one of the coefficients appearing in the deformation of the Type IIB singularity, or equivalently one of the coefficients which corresponds to a leading pole for the Hitchin field at a puncture in the class S setup, is the square of a more elementary gauge invariant parameter and we should regard the latter as a generator of the Coulomb branch. See [43] for a general discussion about such constraints.

Closure of the twisted regular puncture to a generic puncture
Let us discuss how to generalize the procedure of closing the twisted regular puncture to another puncture, but this time not the minimal one. The technique might become involved when many constraints are present. Indeed, working directly with the graded CB dimensions may fail when there are a-constraints [39] among the leading coefficients c (k) Constraints usually arise whenever it is possible to express a leading coefficient using more basic gauge-invariant coefficients. In these cases, the pole structure changes. There are two kinds of constraints, following [39]: c-constraints, for which the local contribution to n k from the pole order p k is reduced by one; and a-constraints, for which the contribution to n 2k is reduced by one, while the contribution to n k is raised by one. The presence of a-constraints modifies the graded CB dimensions by increasing the local contributions by one unit, and the previous procedure that worked directly at the level of the graded CB dimensions does not work anymore.
The correct procedure, instead, is to obtain the contribution to the graded CB dimensions of the punctures from the pole structure. Using Eq. (2.13) of [39], it is possible to compute the contribution n k for a twisted puncture. That equation already keeps into account the constraints, so it is possible to obtain the pole structure using Section 2.4 of [39]. We call p full k = n k the pole structure from the full twisted puncture, 12 and the arriving puncture will have a pole structure p fin.
k . In general p fin. k is different from n fin. k due to the constraints. We now subtract the two pole structures, i.e. p full k − p fin. k and these are the number of CB operators that must be subtracted from the original CB spectrum. Only now we can apply the constraints, reducing or increasing the spectrum according to the rules in [39].
To clarify this point, we are going to make an explicit example. Let us consider D 5 (SO(9)) and we want to close the maximal puncture [1 9 ] to [3, 1 6 ]. If we compute the contribution to the graded CB for the two punctures, we get n full k = 1, Let us now analyze the conformal manifold of D p (SO(2N )) theories. We focus on the case N > 2 since for N = 2 the theory is equivalent to two copies of D p (SU(2)) = (A 1 , D p ). Starting from (2.1), we can easily see that the allowed deformations are where P (z) is a polynomial in z. If we set we easily see that all terms of the form x N −1−nk z qk for 1 ≤ k ≤ m − 1 describe marginal deformations. We therefore conclude that the conformal manifold is at least m − 1 dimensional. There is potentially one extra marginal deformation of the form yz k . Marginality implies that k = pN 2N −2 , which is possible only if pN is a multiple of 2N − 2 and this in turn requires either p to be a multiple of 2N − 2 or N to be even and p a multiple of N − 1. In conclusion, we find that the dimension of the D p (SO(2N )) conformal manifold is Suppose we turn on a marginal deformation of the form x N z 2p with p integer. If we redefine y → z p y and u → z p u and then divide everything by z 2p . The resulting geometry is which clearly describes, as explained in [44], a SO(2N + 2) gauge theory. We can also deduce that the SO(2N +2) vector multiplet is coupled to a D p−2p (SO(2N +2)) theory at z = ∞ and another superconformal sector at z = 0. The CB operators of the sector at infinity are obtained by collecting u ij parameters with j > 2p and the terms in P (z) with degree larger than p . If instead p is half-integer, via the same steps we conclude that the gauge group is USp(2N ) and the sector at z = ∞ is a D p−2p (USp(2N )), which is engineered by twisted D N +1 punctures. The Coulomb branch sector indeed confirms this interpretation.
Notice that whenever the conformal manifold has dimension m, one of the allowed marginal deformations is xz p−q . Furthermore, p − q is necessarily even which means that the sector at infinity is D q (SO(4)) and the corresponding vectormultiplet is SO(4), which provides two marginal couplings instead of one. Conversely, whenever we have a SO(4) gauging the conformal manifold has dimension m. We therefore understand the rôle of the extra marginal deformation: it provides the second marginal coupling associated with an SO(4) gauging.

The conformal manifold of (A, D) Argyres-Douglas theories
As we have already discussed, the geometric engineering of (A p−1 , D N ) is given by (2.9): The allowed deformations are as in (3.1), apart from the fact that the terms proportional to z p−1 can be dropped. From this, we conclude that the conformal manifold of (A p−1 , D N ) has the same dimension as that of D p (SO(2N )). There are two exceptions to this rule: When p is a divisor of N − 1 (i.e. q = 1 in (3.2)) the dimension of the conformal manifold is reduced by 1 with respect to the D p (SO(2N )) theory. We should also subtract 1 when p = 2N −2 N −2 , since in this case the marginal deformation yz k of the D p (SO(2N )) theory is missing (because k = p − 1).
Our goal now is to exhibit a weakly coupled cusp for each exactly marginal defor- The case of qk ≥ 2N − 2 − 2nk Let us start from the case qk ≥ 2N − 2 − 2nk. The term x N −1−nk z qk then describes an exactly marginal deformation associated with (in a suitable duality frame) a SO(2N − 2nk) or USp(2N − 2nk − 2) gauge group depending on whether qk is even or odd respectively. To see this, let us consider the change of variables x = xz 2 ; y z = y, which brings the defining equation of the hypersurface (2.9) to the form Using now the same change of variables as in the previous section: y → z a+nk+1−N y and u → z a+nk+1−N u (where a is the integer part of qk/2) we find that whenever qk is even the geometry becomes which manifestly describes a SO(2N − 2nk) gauging with matter sector at z = ∞ given by Let us now discuss the other sector near z = 0. For qk even it can be obtained starting from D nk+qk/2 (SO(qk + 1)) by partially closing the regular puncture. Specifically, the full puncture 1 qk+1 is replaced by qk + 2nk + 1 − 2N, 1 2N −2nk . We can notice that the Coulomb branch is consistent with this claim and the beta function of the gauged SO(2N − 2nk) vanishes automatically with this matter content. The above discussion applies to the case qk = 2N − 2 − 2nk as well, although with one caveat: in this case the gauge group is SO(2N − 2nk − 1). For qk odd instead we conjecture that the sector at z = 0 is a higgsing of D nk+qk/2 (USp (qk − 1)) with regular puncture For qk ≤ 2N − 2nk − 2 the sectors at z = 0 and z = ∞ are interchanged with respect to the previous case, in the sense that the sector at z = ∞ has a partially closed puncture whereas the matter sector at z = 0 has a full puncture. Specifically, for qk even the sector at infinity is a descendant of D p+2N −2nk−qk−2 (SO(2N − 2nk)), with regular puncture labeled by the partition 2N − 2nk − qk − 1, 1 qk+1 . The sector at z = 0 instead is D nk+qk/2 (SO(qk + 1)) and the gauge group is SO(qk + 1). Again, the matter content ensures the vanishing of the beta function. For qk odd instead we have at z = 0 a D nk+qk/2 (USp (qk − 1)) theory. The gauge group is USp(qk − 1) and the SCFT at z = ∞ is a higgsing of D p−qk+2N −2nk−2 (USp(2N − 2nk − 2)). The regular puncture is labeled by the partition 2N − 2nk − qk − 1, 1 qk−1 .

Summary
Let us summarize the results we have found. The theories we studied above can be described schematically as follows: * where each sphere denotes a class S theory whose name is indicated below; * and * denote corresponding irregular punctures; F G denotes the full puncture whose symmetry is G; P denotes a regular puncture resulting from partially closing of the full puncture of G , where P is labelled by a partition of the form [a, 1 b ] and the associated symmetry is G; and ← G → denotes gauging of symmetry G. The details are as follows: • For qk > 2N − 2 − 2nk and qk even

Comments about the Witten Z 2 anomaly
As it was pointed out in [38], the full twisted A even puncture carries a global Z 2 anomaly analogous to the one discussed in [37] which prevents us from gauging the corresponding USp symmetry, unless we also couple to the vectormultiplet another matter sector with the same property. In this way, the anomaly cancels and the theory is consistent. This fact potentially implies that the gaugings we have discussed in this section are not consistent whenever a D p (USp (2n)) matter sector (or a descendant thereof) is involved. This happens in the second and fourth cases we have just discussed, namely, for qk odd.
We will now argue that the anomaly always cancels and the gauging is consistent for all models at hand. To do this, we exploit the fact that all theories with a USp global symmetry we have discussed are labeled by a special family of regular punctures labeled by C-partitions of the form a, 1 b , with both a and b even and we are gauging the USp(b) global symmetry. The main point is that, due to the structure of these punctures, we can also construct these theories by starting from the matter sector with a full puncture 1 a+b , gauge a USp(b) subgroup of the flavor symmetry and then activate the nilpotent VEV for the moment map which partially closes the puncture. In a sense, for this class of punctures the operations of gauging and higgsing do not interfere.
Exploiting this fact, we start from the matter sector with a full puncture and gauge the USp(b) global symmetry. If the matter sector involved is of D p (USp (a + b)) type, in order to get a consistent (not anomalous) theory we should add an odd number of half hypermultiplets in the fundamental of USp(b). If instead the matter sector is of D b p (USp(a + b)) type (and therefore belongs to the twisted D family), we can only add an even number of half hypermultiplets. If, once we have gauged, we turn on the nilpotent VEV for the moment map, in the IR we find, besides the matter sector labeled by the puncture a, 1 b , a − 1 (which is odd) Goldstone multiplets in the fundamental of USp(b) (see Section 2.2.2). Their effect clearly changes the parity of USp(b) fundamentals for any a > 0.
Since the consistency of the theory is not affected by the higgsing, we come to the following conclusion: is not. If instead we consider the descendants labeled by the puncture a, 1 b with a > 0 the opposite is true, namely we have no anomaly for twisted A even and we do have it for twisted D theories. Overall, we find that for qk > 2N − 2 − 2nk both matter sectors are not anomalous and therefore the gauging is consistent whereas for qk < 2N − 2 − 2nk both matter sectors are affected by the Z 2 anomaly and the gauging is still consistent. This confirms that all of our models are anomaly free.

Theories with
The conformal manifold of D N p (SO(2N )) theories Let us now discuss marginal deformations of D N p (SO(2N )) theories. We focus again on the case N > 2 and the allowed deformations are where P (z) is a polynomial in z. Since the defining equation (2.2) imposes the ring relation xy = z p , we can trade all terms proportional to xy for monomials of the form x m z n , possibly with n > p. This is convenient for identifying marginal deformations since the polynomial P (z) cannot provide any. We can therefore focus on the parameters we can easily see that a parameter u ij describes a marginal deformation (and has therefore dimension 0) only if the following equation holds: We should therefore find all the solutions of (3.10) with i and j both integer. We can immediately notice that, since n and q are coprime, j has to be a multiple of q. Once this constraint is satisfied, if n is even there is always exactly one value of i which solves (3.10), at least as long as j < 2mq. We therefore find 2m − 1 marginal deformations. If instead n is odd, we need to impose the constraint that j is an even multiple of q smaller than 2mq, leading to m − 1 solutions. We therefore come to the conclusion that the dimension of the D N p (SO(2N )) conformal manifold is The same counting also applies to the SO(2N ) N [p − N ] theory, since in this case closing the regular puncture does not affect at all, the dimension of the conformal manifold.
Let us now briefly discuss the weakly coupled cusps. If the marginal deformation is parametrized by u ij , we have as in the b = 2N − 2 case a SO(2i + 2) or USp(2i) gauge group depending on whether j is even or odd respectively. For j even the matter sector at z = ∞ is D i+1 p−j/2 (SO(2i + 2)). This can be seen by performing the redefinition u → uz j/2 and y → yz j/2 in (3.8) and then dividing the defining equation by z j . It is easy to see that if we keep only deformations proportional to positive powers of z in the resulting expression, we recover the spectrum of the D i+1 p−j/2 (SO(2i + 2)) theory. For j odd, we find instead a new class of twisted AD theories. The matter sector at z = ∞ in this case involves a twisted D n model with b t = 2n in the notation of [20], which is specified in Type IIB by the hypersurface singularity (2.15) with p half integer. These are indeed identified with the D n p (USp(2n − 2)) theory. Specifically, for j odd the matter sector at infinity is D i+1 p−j/2 (USp(2i)). Again, this can be derived by considering the change of variables u → uz j/2 and y → yz j/2 in (3.8) and dividing the defining equation by z j .

The conformal manifold of SO(2N ) N [p] theories
Let us analyze weakly coupled cusps in the conformal manifold of SO(2N ) N [p] theories. Consider again the marginal deformation u ij x i z j . We have to consider the two cases j ≤ 2i and j > 2i.

The case of j ≤ 2i
For j ≤ 2i the matter sector at z = ∞ can be described for j even as a higgsing of D i+1 p+i+1−j/2 (SO(2i + 2)). More precisely, the higgsing is implemented by partially closing the regular puncture from [1 2i+2 ] to [2i + 1 − j, 1 j+1 ]. The matter sector at z = 0 is instead D N +j/2−i−1 (SO(j + 1)). The two sectors are coupled via an SO(j + 1) gauging. For j odd the matter sector at z = ∞ is a higgsing of D i+1 p+i+1−j/2 (USp(2i)), with regular puncture labeled by the partition [2i − j + 1, 1 j−1 ]. The sector at z = 0 is instead D N +j/2−i−1 (USp (j − 1)) and the gauging in this case is USp(j − 1). It can be easily checked both for j even and odd that with this matter content the beta function vanishes. To see this, one should use the relation which is implied by the marginality of the term u ij x i z j .
The case of j > 2i For j > 2i the rôles of the two matter sectors are interchanged as in the b = 2N −2 case. The matter sector at z = ∞ can be described for j even as the D i+1 p+i+1−j/2 (SO(2i + 2)) theory and the matter sector at z = 0 is instead a higgsing of D N −i−1+j/2 (SO(j + 1)). The relevant twisted A j−1 puncture is labeled by the partition [j − 2i − 1, 1 2i+2 ]. The two sectors are coupled via a SO(2i + 2) gauging. For j odd instead the matter sector at z = ∞ is the D i+1 p+i+1−j/2 (USp(2i)) theory and the sector at z = 0 is a higgsing of D N +j/2−i−1 (USp (j − 1)). The corresponding twisted A j−1 puncture is labeled by the partition [j − 2i − 1, 1 2i ]. The gauging in this case is USp(2i). Again, it can be easily checked using (3.10) that, both for j even and odd, with this matter content the beta function vanishes.

Summary
The results can be represented schematically by (3.7), where the details are summarized as follows: • For j ≤ 2i and j even, (3.14) • For j ≤ 2i and j odd, • For j > 2i and j even, • For j > 2i and j odd, The cancellation of the global Z 2 anomaly of [38] works as in Section 3.2 and therefore we do not repeat the argument.

RG flows with supersymmetry enhancement, 3d mirrors and Abelian quivers in 3d
In this section, we collect several results and observations which are instrumental for constructing the 3d mirrors of D b p (SO(2N )) theories. We will first discuss a class of RG flows exhibiting supersymmetry enhancement and how they affect the 3d mirror theory. In passing, we will propose a new duality for T [SO(2N )]. We will then discuss our conventions for the 3d quivers and discuss several nontrivial equivalences among Abelian theories in 3d which we need to compare our findings with the results of [22].
upon closure of the regular puncture, or equivalently by giving to the SO(2N ) moment map a principal nilpotent VEV which completely breaks the global symmetry. As was pointed out in [26], we can also flow to (A p−1 , D N ) theory with a different procedure, introduced in [25, [45][46][47][48], which we refer to as MS flow: We couple to the SO(2N ) moment map a chiral multiplet (flipping field) in the adjoint of the global symmetry and we turn on a principal nilpotent VEV for this field. As we flow to the IR, N Coulomb branch operators of D p (SO(2N )) hit the unitarity bound and decouple from the theory.
We can analyze this RG flow at the level of the 3d mirror theory, as it was done for D p (SU(N )) theories in [22] (see also [49][50][51]). In the unitary case, it was argued that the introduction of the flipping field can be implemented by flipping the HB moment map of the T [SU(N )] tail and the nilpotent VEV is introduced simply by removing the tail. Furthermore, the decoupling of the N CB operators hitting the unitarity bound can be implemented by removing from the theory the Cartan components of the flipping field. Here we will follow the same procedure in the SO case. As we will see, this will provide a powerful constraint on the structure of the 3d mirror theory.

Flip-Flip duality for T [SO(2N )]
Let us now discuss a 3d duality for T [SO(2N )] which will play a key rôle in understanding the effect of the MS flow on the 3d mirror of D b p (SO(2N )). This duality was introduced for T [SU(N )] in [27] (see also [52]) and states that the theory is infrared equivalent to a variant of the model, in which both the HB and CB moment maps are flipped. We therefore introduce by hand two chiral multiplets in the adjoint of SU(N ) M C and M H and couple them via superpotential terms to the CB and HB moment maps µ C,H respectively. The duality states that the moment maps of T [SU(N )] are mapped to the corresponding flipping fields in the dual theory: We claim that this duality also applies to T [SO(2N )] and we would now like to provide a stringy argument based on the Hanany-Witten brane realization of the theory [53]. It would be interesting to also find a field theoretic derivation of this statement. We can engineer T [SO(2N )] in Type IIB on the worldvolume of N D3 branes parallel to an O3 − plane and suspended between half D5 branes on one side and half NS5 on the other. There is exactly one D3 brane ending on each 5-brane. The branes are oriented as follows: Branes 0 1 2 3 4 5 6 7 8 9 NS5 The theory is known to be self-mirror and this is reflected in the brane system being invariant under S-duality. We now exploit an observation made in [54] (see also [55]) that rotating the D5 branes into D5 extending along directions 012457 has the effect of flipping the HB moment map. If we now perform S-duality we send D5 branes to NS5 , reaching a configuration in which, starting from the T [SO(2N )] brane system, we rotate NS5 branes until they extend along directions 012389. On the other hand, we also know that S-duality implements mirror symmetry and since the brane system with rotated D5 describes a flipping of the HB moment map, we conclude that rotating the NS5 branes has the effect of flipping the CB moment map. Now, starting from the brane system in (4.2), let us rotate both D5 and NS5 branes. According to what we have said so far, this is expected to describe the Flip-Flip T [SO(2N )] theory. However, it is easy to see from (4.2) that this operation simply amounts to a rotation of the brane system (we are just interchanging the planes 4−5 and 8 − 9) and therefore we recover the brane system describing T [SO(2N )]. We therefore conclude that T [SO(2N )] is equivalent to its flipped-flipped version.
The reason this is relevant for describing the effect of the MS flow at the level of the 3d mirror is the same as in the unitary case discussed in [22]. The first step is to flip the SO(2N ) moment map of the 4d theory, which maps (at least for p > b) to the Coulomb branch moment map of the T [SO(2N )] tail in the 3d mirror theory. Since a flipping of the CB moment map is hard to describe, it is convenient to use the Flip-Flip duality to map this operation to a flipping of the HB moment map of the T [SO(2N )] tail, which is described by an ordinary superpotential interaction (we will denote the flipping field of the HB moment map in the 3d mirror by M ). This is therefore the duality frame we will focus on. The last step is to notice that the flipping field in 4d maps, after the Flip-Flip duality, to the CB moment map in the 3d mirror and therefore the effect of the nilpotent VEV is simply to remove the T [SO(2N )] tail from the quiver.
Finally, we have to take into account the fact that N CB operators in 4d hit the unitarity bound and decouple. To implement this decoupling in 3d, we should flip the corresponding operators in 3d [49]. These are HB generators in the 3d mirror. In [22] it was argued that, after the Flip-Flip duality, this operation amounts to flipping all Cartan components of M , therefore removing them from the spectrum. Here we will assume the same is true in the SO case we are interested in.
We can similarly work out the decomposition of the adjoint of SO(2N ), which is the representation in which the flipping field transforms. This will tell us how the MS flow affects the quiver. We will now describe the decomposition in detail.
• We find an adjoint of SO(2F ) and, since this symmetry is ungauged in the 3d mirror, the corresponding components of the flipping field will become F (F − 1) free hypermultiplets when we remove the tail. This counting takes into account the fact that the Cartan components of the flipping field have been removed from the spectrum. This is done to account for the CB operators in 4d which hit the unitarity bound and decouple.
• We also get bifundamentals of SO(2n i ) × SO(2F ), which become n i F hypermultiplets which transform as doublets of SO(2) i upon removal of the T [SO(2N )] tail. We shall refer to these as 2n i F hypermultiplets carrying charge 1 under • Finally, we have the adjoint of SO(2n i ) for every i. Once the T [SO(2N )] tail is removed, this provides n i (n i − 1) hypermultiplets (always taking into account the fact that the Cartan components have been removed from the spectrum). It turns out that half of them become free hypermultiplets, uncharged under the gauge group of the theory, whereas the other half provide n i (n i −1) The last statement about the SO(2n i ) adjoint requires some further explanations. Out of the SO(2n i ) global symmetry, we are gauging a SO(2) subgroup which is generated by the diagonal combination of the Cartan generators in SO(2n i ). Said differently, the SO(2) action is described by a 2n i × 2n i matrix of the form R θ ⊗ I n i , where I n i is the n i × n i identity matrix and The components of the flipping field transforming in the adjoint of SO(2n i ) can now be conveniently organized into n i (n i −1) ) and the gauged SO(2) acts on each one of these matrices as It is now convenient to rewrite each M J i as a linear combination of the Pauli matrices σ 1,2,3 and the 2 × 2 identity matrix. From (4.4) it is easy to see that the components proportional to the identity matrix and to σ 2 (the antisymmetric Pauli matrix) are invariant under SO(2) and together they provide a free hypermultiplet upon implementing the MS flow. The other two components, namely, those that are proportional to σ 1 and σ 3 , become instead a flavor of SO(2) i as we have claimed before. Since we shall refer to the above linear combinations of the latter two components as the hypermultiplets carrying charge 2 under U(1) i ∼ = SO(2) i .
Overall, we find the following result: Starting from D p (SO (2N )), upon closure of the regular SO(2N ) puncture, we get an Abelian quiver with bifundamentals and flavors plus a collection of free hypermultiplets. If instead we consider the MS flow, which amounts to increasing p by 2N − 2, i.e.
we find a similar quiver with the same number of Abelian nodes, where the number of free hypermultiplets is increased by The number of bifundamental half-hypermultiplets increases by n i n j and the number of flavors at the i-th Abelian node increases as follows: we get 2n i F hypermultiplets with charge 1 and n i (n i −1) 2 hypermultiplets with charge 2. This procedure can be easily generalized to the D N p (SO(2N )) theory simply by taking We use these facts to constrain the mirror theories discussed in the next sections.

Notations
To describe the mirror theory for D p (SO(2N )) in the subsequent sections, we adopt the following notations for the quiver diagrams.  To save space, we sometimes use the following abbreviations in the quiver diagrams: SO(2N ) = D N , USp(2N ) = C N and SO(2N + 1) = B N . We also denote by /Z 2 the diagonal Z 2 quotient of the gauge symmetry. We shall emphasize the latter again in the context.

Correspondences between certain Abelian gauge theories
In this section, we discuss some correspondences between certain Abelian gauge theories. On one side, we consider theories with SO(2) gauge groups with hypermultiplets in vector representations and possibly with those carrying charge two under U(1) ∼ = SO(2). On the other side of the correspondence, the involved theory has U(1) gauge groups with hypermultiplets carrying charge one.
The first example we would like to present is m copies of half-hypermultiplets in the representation [2; 2] of the gauge group (SO(2) × SO(2))/Z diag 2 , where Z diag 2 denotes the diagonal Z 2 gauging. As we will see in Section 6.2, this is a 3d mirror for the (A 2m−1 , D 2 ) theory. We find that this mirror theory is isomorphic to the product of two copies of the SQED with m flavors: As we will see in Section 6.2, the theory on the left is the mirror theory for (A 4m−1 , D 3 ).
Since D 3 ∼ = A 3 , the theory in question is equivalent to (A 4m−1 , A 3 ) = I 4m,4 , and so there is another description of the mirror theory in terms of a complete graph with 4 U(1) nodes where each edge has multiplicity m (see the discussion below (4.12) of [22]), which is depicted on the right-hand side. Note that the latter has an overall U(1) that decouples. Both theories have an SU(m) 6 × U(1) 3 flavor symmetry. We match the Higgs and Coulomb branch Hilbert series of the two descriptions in Appendix A.2.
The third example is the following correspondence between two descriptions of the mirror theory for (A 1 , D 2N ). We will discuss this in more detail in Section 6.3.2.
where the blue edge has multiplicity N − 1. The quiver on the right-hand side was presented in, e.g. [51, Section 2.4] and [56, Figure 36(X)]. Note that the quiver on the right-hand side has an overall U(1) that decouples. We match the Higgs and Coulomb branch Hilbert series of the theories on both sides of (4.17) in Appendix A.3. We parametrize p in the following way

D 2N
where we restrict to m such that The 3d reduction gives

Smaller non-Higgsable SCFTs
We remark that the non-Higgsable SCFT in question T , which is (A (4µ−2)m−2µ , D N ) in the current example, may 'contain' a smaller non-Higgsable SCFT T , in the sense that 1. the rank of T is smaller than that of T , 2. the Coulomb branch spectrum of T is contained in that of T , and 3. the value of 24(c − a) of T is equal to that of T .
For m = 1, we conjecture that for T = (A 2µ−2 , D N ) with constraint (5.2), i.e. 2µ−1 divides N − 1, we have 3) equal to that of T and whose rank is equal to (µ − 1) N −1 2µ−1 − 1 , less than (µ − 1)N . Below we will consider an example of T = (A 2 , D 7 ). This is a non-Higgsable SCFT as discussed above; it has 24(c − a) = 1/5 and rank 7. Nevertheless, according to the discussion below (3.4) with p = 3 and N = 7, it has a one-dimensional conformal manifold. There is a weakly coupling cusp that contains an SO(3) gauging of two theories of class S. Upon closing the puncture, we obtain the T = (A 1 , A 2 ) theory. The latter has 24(c − a) = 1/5 and rank 1. One can also show that T has a Coulomb branch operator of dimension 6/5, which is the Coulomb branch operator of T .
Currently, we do not have the full understanding of the relation between the theory T and T . Under the assumption that T has no Higgs branch, obviously one cannot obtain T from T by a Higgs branch flow. There is a possibility that such an assumption is wrong, namely, T may contain a 'Higgs branch' of which a generic point contains a collection of hypermultiplets, vector multiplets, and T , in such a way that their contributions make 24(c − a) fractional and less than 1. However, we regard this possibility as unlikely, since the mirror theory of T is simply a theory of free hypermultiplets (no vector multiplet); in other words, mirror symmetry does not give any indication of the presence of the aforementioned hypermultiplets at a generic point of the 'Higgs branch' of T . We leave the detailed study of T and its relation with T for future work.  (6)) after partially closing the SO(6) puncture is T [3,1 3 ] [SO (6)], whose description is SO(2) − USp(2) − [SO (6)]. After commonly gauging the enhanced SU(2) topological symmetries of both theory associated with each SO(2) node, we obtain [USp (2)] − [USp (6)] with 1 free hypermultiplet, i.e. 7 free hypermultiplets in total, in accordance with (5.3).

Discrete gaugings and defect groups
In some cases, the collection of free hypermultiplets indicated in (5.4) could be subject to a discrete gauging. Let us discuss this issue using an example. We consider the D 6 9 (SO(8)) theory, i.e. µ = 2, N = 4 and m = 1. Upon closing the full SO(8) puncture we obtain the (A 2 , D 4 ) theory. This 4d theory can be realized as 3 copies of the (A 1 , A 3 ) theory gauged by an SU(2) gauge algebra (see [57, Figure 1] and [58, (4.30)]). As can be seen from the latter reference, there are two choices for the corresponding gauge symmetry, namely SU(2) or SU(2)/Z 2 . 14 Upon reducing the (A 2 , D 4 ) theory to 3d, we have a star-shaped quiver with the U(1) gauge node at each of its three legs and central node being SU(2) or SU(2)/Z 2 (see also [58, (4.31)]). If the central node is SU(2)/Z 2 , this is precisely the mirror of the theory of four free hypermultiplets (the T 2 theory), in agreement with H free in (5.4). On the other hand, if the central node is SU(2), this is the mirror of the Z 2 discrete gauging of the T 2 theory, i.e. the mirror of the O(1) − [USp(8)] theory. As explained below [58, (4.30)], the two aforementioned choices originate from the fact that the defect group of the (A 2 , D 4 ) theory is Z 2 2 (see also [60, Table 1]), and so there are two versions of the (A 2 , D 4 ) theory, namely that with a Z 2 electric one-form symmetry and the other with a Z 2 magnetic one-form symmetry. In four dimensions, the former corresponds to the choice of the SU(2) gauge group and the latter corresponds to the choice of the SU(2)/Z 2 gauge group.
In general, the defect group is expected to indicate the presence of the one-form global symmetry and can be used to determine whether it is possible to apply a discrete gauging to the set of free hypermultiplets.
As a consequence, when the defect group is empty, such as in the case of the (A N −1 , A k−1 ) theory with GCD(N, k) = 1, the (A 2 , D 3 ) ∼ = (A 2 , A 3 ) theory, and the (A 2 , D 5 ) theory [60, Table 1], it is expected that the one-form global symmetry is absent in such non-Higgsable SCFTs. Upon reduction to 3d and applying mirror symmetry, one obtains a collection of hypermultiplets, and we do not expect any discrete gauging for the latter. 15 6 D 2N −2 p (SO(2N )) with p ≥ 2N − 2 and GCD(2N − 2, p) even In this section, we discuss 3d mirror theories for D 2N −2 p (SO(2N )) with p ≥ 2N − 2 and GCD(2N − 2, p) even. We first discuss the general results and then provide several explicit examples.

General results
Let us write GCD(2N − 2, p) = 2µ . (6.1) There are two possibilities to consider, depending whether 2N − 2 divided by 2µ is an even number or an odd number. There is one mass parameter in addition to those associated with the SO(2N ) flavor symmetry. Let us parametrize p as whose quiver description is 16 Note that the total number of gauge groups is 2x + 1 + 2(2µN − 1) = p − 1.

GCD(2N − 2, p) is even and
There are µ+1 mass parameters in addition to those associated with the SO(2N ) flavor symmetry. We parametrize p as follows: The mirror theory involves H free = µ(m − 1)(N − 1) (6.10) free hypermultiplets, together with the following quiver gauge theory: the T [SO(2N )] tail It can be checked that the expression of H free given by (6.4) is consistent with the fact that the Milnor number is equal to twice the rank r of the CB of the 4d theory plus the rank rk(G F ) of the global symmetry [8]. Suppose that we focus on the 4d (A m , D n ) theory (or the SO(2N ) b [p] theory in general). The Milnor number is mn (or can be computed from [26, (2.9)] respectively). By mirror symmetry, r is equal to the dimension of the HB of the 3d mirror theory plus H free , and rk(G F ) is equal to the number of SO(2) nodes in the 3d mirror theory.
connected to a complete graph of µ + 1 SO(2) nodes in the following way: 18 1. Among the µ + 1 SO(2) nodes in the complete graph, µ of them are connected to the C N −1 node in the tail by edges with multiplicity 13) and the remaining SO(2) node is connected to the C N −1 node in the tail by an edge with multiplicity 1. Note that the total number of flavors of the C N −1 node is indeed µR + 1 = 2µN − µ + 1, as required. For convenience, we refer to the edges with multiplicity R as A 1 , A 2 , . . . , A µ and the edge with multiplicity one as B. (2)  3. Each edge in the complete graph that connects any two SO(2) gauge nodes attached to the edges A i and A j has multiplicity

Each SO
and each edge that connect any two SO(2) gauge nodes attached to the edges B and A i multiplicity m.

The gauge symmetry of the theory is in fact
where the factors C 1 × D 1 × C 2 × D 2 × · · · × C N −1 come from the tail, the factor D µ+1 1 comes the complete graph and Z diag 2 denotes the quotient of the diagonal Z 2 symmetry (see a detailed discussion in [61]). As pointed out in [61], the Z diag 2 quotient affects the magnetic fluxes of each gauge factor in such a way that the half-integral values must be taken into account. We will denote this Z diag 2 quotient by /Z 2 in the subsequent part of the paper. 18 We find that the total number of hypermultiplets in the complete graph part (not including the tail and connections) In general, we also conjecture that the non-Higgsable SCFTs are giving rise to H free free hypermultiplets, as required. The mirror theory for (A 2µm−1 , D 2µN−µ+1 ) is therefore the complete graph as described above, together with H free free hypermultiplets.
Upon closing the full puncture, the D 2N −2 (2N −2)r (SO(2N )) theory flows upon higgsing to the AD theory A (2N −2)m−1 , D N , with m = r − 1. The 3d mirror for the latter is a complete graph with N SO(2) nodes with edge multiplicity m. The gauge symmetry is SO(2) N /Z 2 . There are two interesting special cases to consider: • For the special case of N = 2, the theory in question is (A 2m−1 , D 2 ), with m = r−1.
We have discussed this theory and its correspondence with two copies of SQED with m flavors in (4.15). 19 The total number of the hypermultiplets in the complete graph is 2 × 1 2 N (N − 1) × (r − 1) = N (N − 1)(r − 1) in agreement with (6.12).
• For the special case of N = 3, the mirror theory for (A 4m−1 , D 3 ) was presented in (4.16). Its correspondence with the mirror theory for (A 4m−1 , A 3 ), namely the complete graph with 4 U(1) nodes where each edge has multiplicity m was also discussed there.
There are two cases for D 2N −2 p (SO(2N )), with GCD(2N − 2, p) = 2 and N ≥ 2, to be considered: those are odd after divided by 2, and those are even after divided by 2.
Note that although this parametrization includes all p such that GCD(4N, p) = 2, it also include those with GCD(4N, p) = 2, in which case we should exclude those values of p from the following analysis. In other words, we consider m such that GCD(2N, 2m − 1) = 1 .
whose quiver description is The [D 1 ] flavor symmetry corresponds to the fact that all of the 4d theories in this class has one mass parameter in addition to those associated with the SO(4N + 2) flavor symmetry. The mirror theory consists of H free free hypermultiplets (6.24) and the T ρ σ [SO(2n)] theory whose quiver description is Upon decoupling the tail, we obtain the mirror theory for (A 4m−3 , D 2N+1 ) such that GCD(2N, 2m − 1) = 1, namely with H free free hypermultiplets. In particular, setting m = 1, we propose that the mirror for the (A 1 , D 2N+1 ) theory is the SQED with 2 flavors with N − 1 free hypermultiplets. For N = 1 (i.e. the case of SO(6) whose algebra is isomorphic to SU(4)), the non-Higgsable SCFTs are (A 1 , A 2m−2 ) ⊗2 [22, (3.27)]; these give rise to 2(m − 1) twisted hypermultiplets, as expected.
We tabulate some non-Higgsable SCFTs, including some cases for m ≥ 2 and N ≥ 2, below; 8 + rank(X) (6.27) Let us comment on some of the above cases.
• Next, we comment on the D 12 22 (SO (14)) theory, i.e. (b, p) = (12,22) and (N, m) = (3, 3). It is convenient to close the full SO (14) puncture and analyze the (A 9 , D 7 ) theory, which admits the description (6)) (6.29) where this was described in Section 3.2, with m = 2, q = 5, n = 3, k = 1. Upon closing the USp(4) puncture, we obtain the (A 4 , D 4 ) theory from D 11/2 (USp (4)) according to (2.39), whereas from the D 11 (USp(6)) theory we obtain a theory that is described by the Type IIB hypersurface singularity given by (2.33) with N = 4, p = 11: according to (2.33). To the best of our knowledge, the latter theory has not been studied anywhere in the literature. We denote the by X the non-Higgsable SCFT(s) for such a theory. Since we know that the values of 24(c − a) of the D 12 22 (SO (14)) theory and the (A 4 , D 4 ) theory are 14/11 and 8/11 respectively, the value of 24(c − a) of X should be 6/11. Moreover, we expect that a generic point of the Higgs branch of theory (6.30) should contain X and a collection of h hypermultiplets, satisfying 8 + rank(X) + h = H free = 22 (6.31) and so rank(X) + h = 14. We leave the detailed study of (6.30) as well as X for future work. We propose that the mirror theory of D 4N−2 4N−2+2m (SO(4N)) with the condition (6.33) is 20 (6.35) The notation in the above quiver is as described in section 4.2.
The non-Higgsable SCFTs for this class of theory are Decoupling the tail, we see that the mirror for the (A 2m−1 , D 2N ) theory, with the condition (6.33), is together with H free free hypermultiplets. Let us test this proposal in two special cases as follows: • For m = 1, we expect that this reduces the theory depicted on the left-hand side of (4.17), with no free hypermultiplets. In fact, the 3d mirror for the (A 1 , D 2N ) theory admits another description in terms of unitary gauge groups as depicted on the right-hand side of (4.17). We have also discussed the correspondence between the two descriptions.
• The special case of N = 1 corresponds to the (A 2m−1 , D 2 ) theory. We have discussed the mirror for this theory and its factorization in (4.15).
There are two cases for 2N − 2, with N ≥ 2, to be considered: those are even after divided by 4, and those are odd after divided by 4; • For 2N − 2 = 4(2N), i.e. 2N = 8N + 2, there is only one mass parameter in addition to those associated with the SO(2N ) flavor symmetry. The 3d reduction of this theory gives twisted hypermultiplets, together with the T σ ρ [SO(2n)] theory whose quiver description is the quiver The mirror theory for this theory is described by together with H free free hypermultiplets.
Upon decoupling the tail, we obtain the mirror theory for (A 8m−5 , D 4N+1 ) theory. It is described by with H free free hypermultiplets.
We parametrize p as follows: with the restriction We propose the following mirror theory: where each red line has multiplicity 4N − 2, each gray line has multiplicity m and the blue line has multiplicity 21 Decoupling the tail, we obtain the mirror theory for (A 4m−1 , D 4N−1 ) with GCD(2N− 1, m) = 1: The total number of hypermultiplets in the complete graph part (not included the tail and the connections) is 6mN.
For N = 1, we have M = m and so we recover the mirror for the (A 4m−1 , D 3 ) theory, as discussed around (4.16), as expected.
For m = 1, there is no free hypermultiplet. For m = 2, the non-Higgsable SCFTs are (A 1 , A 2N−2 ) ⊗2 , giving rise to 2N − 2 free hypermultiplets. We conjecture that, in general, the non-Higgsable SCFTs are (6.50) There are two cases for 2N − 2, with N ≥ 2, to be considered: those are even after divided by 6, and those are odd after divided by 6.
The first case will be discussed as a special case of the general result in Section 6 with µ = 3.
For the second case, namely D 12N−6 p (SO (12N − 4)) with GCD(p, 12N − 6) = 6, we write We propose that the mirror theory of D 12N−6 p (SO (12N − 4)), is free hypermultiplets, together with 22 The total number of hypermultiplets in the complete graph part (not including the tail and not including the connections) is 3m (5N − 1).
where M = m (2N − 1). We also claim that the non-Higgsable SCFTs are Decoupling the tail, we obtain the following mirror theory for (A 6m−1 , D 6N−2 ): In this section, we study the D N p (SO(2N )) theory with p ≥ N and the corresponding mirror theory. There are two subclasses to consider: 1. The subclass containing the theories of which N/GCD(N, p) is odd; 2. The subclass containing the theories of which N/GCD(N, p) is even.
In Subclass 1, each theory has no additional mass parameters to those associated with the SO(2N ) flavor symmetry. Furthermore, as discussed in (3.11), the conformal manifold of the theory in this subclass is GCD(N, p) − 1 dimensional. In Subclass 2, each theory has GCD(N, p) mass deformations in addition to those associated with the SO(2N ) flavor symmetry. The conformal manifold of the theory in this subclass is 2GCD(N, p) − 1.  N (N − 1). This is in accordance with (7.2), where δH free = 2m(2mn − 1)δk = 2mn(2mn − 1) = N (N − 1).
Upon closing the full SO(2N ) puncture, we obtain the non-Higgsable SCFT whereby the reduction to 3d and applying mirror symmetry giving rise to H free free hypermultiplets, given by (7.2). For k = 1, we claim that the non-Higgsable SCFT T = SO(4mn) 2mn [m] contains the smaller non-Higgsable SCFTs T = (A 1 , A 2(n−1) ) ⊗m (7.4) as discussed towards the end of Section 5.1. We will soon provide an example of SO(32) 16 [2] to illustrate this point. Note that (7.4) has 24(c − a) = m n−1 2n+1 and gives rise to m (n − 1) twisted hypermultiplets upon reduction to 3d.
Moreover, for k = 1 and m = 1, we have the following identification: This relation can be seen from the curve. That of the theory of the left-hand side reads u 2 + x 2n−1 + xy 2 + yz = 0. Since y and z are massive and can be integrated out, we obtain u 2 + x 2n−1 = 0, which corresponds to the curve of the (A 1 , A 2n−2 ) theory.
The descriptions of the theory at other weakly coupled cusps can also be analyzed, but they are more complicated. For example, that associated with the marginal deformation x 7 y 2 is SO(32) 16 [2] = D 9/2 (USp(6)) ←− USp(2) −→ D 27/2 (USp (2)) (7.7) The properties of the D 9/2 (USp(6) theory are little known, and so we are not analyzing this theory further. Nevertheless, as described at the end of Section 2.2.2, upon closing the USp(2) puncture, the D 27/2 (USp (2)) becomes the (A 2 , D 13 ) theory. We will next show that the latter, in fact, contains the (A 1 , A 6 ) theory. Although the (A 2 , D 13 ) theory is a non-Higgsable SCFT, it has a one dimensional conformal manifold, associated with the deformation x 4 y 2 . The theory can be described by This theory has m mass deformations. In general, we propose that the mirror theory contains H free = m(n − 1)(k − 1) , (7.10) free hypermultiplets, together with a quiver gauge theory that can be constructed as follows: 23 1. Construct a complete graph with m SO(2) nodes such that every edge has multiplicity M = kN/m = (2n − 1)k . (7.11) We shall use the same notation as in (4.13) and (4.14). 23 The total number of hypermultiplets in the complete graph part (not including the tail and the connections) is mk[m(2n − 1) − n].

Construct the T [SO(2N )] tail:
4. Connect the C N −1 gauge node in the tail to each SO(2) gauge group in the complete graph with the edges, such that each edge has multiplicity is We shall use the same notation as in (4.9) and (4.10).
5. The gauge symmetry of the theory is where we shall denote the diagonal Z 2 quotient Z diag 2 (see [61] for a detailed discussion) by a shorthand notation /Z 2 in the subsequent part of the paper.
We propose that the non-Higgsable SCFTs are giving rise to H free free hypermultiplets, as expected.

The case in which N divides p
Let us consider the case in which m = N , n = 1 , p = N (k + 1) , (7.16) i.e., the D N N (k+1) (SO(2N )) theory. From the prescription above we have The mirror theory consists of a complete graph of N SO(2) gauge nodes such that every edge of the graph has multiplicity kN/m = k and each SO(2) node connects to the C N −1 gauge node connects to the T [SO(2N )] tail with an edge with multiplicity N/m = 1. There is an overall Z 2 quotient in the gauge symmetry as indicated in (7.13).
Note that there is no flavor of hypermultiplets with charge 2 under any U(1) ∼ = SO(2) gauge group. Recall from Section 6.2 that this is the same mirror theory as for the 2N )). We thus claim the following identification: This is analogous to [22, (6.18)] for the SU(N ) case; in particular, for N = 3, this is in agreement with [22, (6.18)] with N = 4. Decoupling the tail, we obtain the mirror theory for SO(2N ) N [N k], whose description is the same as that for the SO(2N ) 2N −2 [(2N − 2)k] = (A (2N −2)k−1 , D N ) theory, namely a complete graph of N SO(2) gauge nodes such that every edge of the graph has multiplicity k, and the gauge symmetry is SO(2) N /Z 2 . We also propose the identification

The case of m = 1
The mirror theory consists of (n − 1)(k − 1) free hypermultiplets, together with the following theory where the red edge has multiplicity N/m = 2n − 1 and /Z 2 indicates that the gauge symmetry is as indicated in (7.13).
Decoupling the tail, we obtain the mirror for the SO(4n − 2) (2n−1) [k] theory, whose description is as follows: (n − 1)(k − 1) free hypermultiplets, together with i.e., the U(1)/Z 2 gauge theory with (n − 1)k flavors of hypermultiplets with charge 2. This is equivalent to the U(1) gauge theory with (n − 1)k flavors of hypermultiplets with charge 1. Hence, the mirror theory for SO(2N ) (2n−1) [k] can be rewritten as We emphasize the importance of the Z 2 quotient, discussed in (7.13), in reaching this conclusion. Let us provide a test for this proposal. We consider the case of N = 3 and so the only possibility that is relevant to our restriction is to have m = 1 and n = 2, namely D 3 3+k (SO(6)) theory. The above proposal for the mirror theory is Since D 3 3+k (SO(6)) = D 3 3+k (SU(4)), there is an alternative description of the mirror theory given by [22, (6.13)] (with N = 4 and p = 3 + k): 3+k (SU(4)) 3d mirr : Indeed, the Higgs branch symmetry SU(k) × SU(3) × U(1) is manifest in both descriptions. We match the Higgs and Coulomb branch Hilbert series of the two quiver descriptions in Appendix A.4. Upon decoupling the tail in either description, we indeed obtain the mirror theory which is the SQED with k flavors, together with k − 1 free hypermultiplets, as expected.

The case of m = 2
The mirror theory consists of 2(n − 1)(k − 1) free hypermultiplets, together with the following theory where each red line has multiplicity N/m = 2n − 1 and each blue line has multiplicity kN/m. Decoupling the tail, we obtain the mirror for the SO(8n − 4) (4n−2) [2k] theory, whose description is as follows: Let us provide a test for this proposal. We consider the case of N = 2 and so the only possibility that is relevant to our restriction is to have m = 2 and n = 1, namely D 2 2(k+1) (SO(4)) theory. Since SO(4) = SU(2)×SU(2), we expect that this theory factorizes into (D 2 2(k+1) (SU(2))) ⊗2 . Indeed, this can be checked using the Higgs and Coulomb branch Hilbert series. Upon closing the full SU(2) puncture in each factor, we obtain the (A 2k−1 , D 2 ) theory, whose mirror theory was discussed in (4.15), with m being k. This indeed in agreement with (7.26).

The case of m = 3
We propose the following mirror theory: where each red line has multiplicity N/m = 2n − 1 and each blue line has multiplicity kN/m = (2n − 1)k. Decoupling the tail, we obtain the following mirror theory for (14)) theory Let us analyze this theory along the line of Section 3.1. We have m = 2, n = 3 and q = 2. From (3.3), the D 12 4 (SO (14)) theory has a one dimensional conformal manifold, and the corresponding marginal deformation is x 3 z 2 , which gives k = 1, N = 3 and p = 1. We thus see that the sector at z = ∞ is an SO(8) gauge theory coupled to D 6 2 (SO (8)). We propose the following description: Reduction to 3d and the mirror theory As pointed out in [34, (7.19)], the D 6 2 (SO(8)) theory, which coincides with the E 6 Minahan-Nemeschansky theory, can be realized as an IR fixed point of the USp(4) gauge theory with 5 flavors: Upon reduction to 3d, (8.3) gives rise to the T [5,5] [SO (10)] theory, which has two quaternionic dimensional Coulomb branch. However, the reduction of the E 6 MN theory to 3d yields an SCFT with one quaternionic dimensional Coulomb branch. In Appendix A.5, we demonstrate that upon gauging the SO(2) Coulomb branch symmetry of the T [5,5] [SO (10)] theory (unfortunately, this symmetry is not manifest in the UV description (8.3) upon reduction), we obtain the 3d reduction of the E 6 MN theory: where the notation /SO(2) C denotes the 3d N = 4 gauging of the SO(2) Coulomb branch symmetry. Note that this SO (2) is the global symmetry associated with the D-partition [5,5] of SO (10).
Since the Higgs branch of the E 6 MN theory is the reduced moduli space of one E 6 instanton on C 2 [62], the leftmost theory in (8.4), namely the 3d N = 4 USp(4) gauge theory with 5 flavors with the SO(2) topological symmetry being gauged, provides the ADHM construction of such an instanton moduli space.
Let us now consider the mirror theory of (8.8  (20)], whose quiver description is where we use the shorthand notations C n = USp(2n) and D m = SO(2m). Under mirror symmetry, the SO(2) C 1 ×SO(2) C 2 in the denominator of (8.8) becomes the SO(2)×SO(2) subgroup of the flavor symmetry [D 2 ] = [SO(4)] in the above mirror theory. After gauging, we obtain a quiver description of the mirror theory of (D 12 4 (SO (14))) 3d as follows: where, as before, /Z 2 denotes the diagonal Z 2 quotient. This is a star-shaped quiver which is the mirror theory of the 3d reduction of the following theory of class S: described in (8.11) can be computed from the information given by [43,  ⇒ (a, c) = (167/8, 45/2) , (8.13) in agreement with that of the D 12 4 (SO (14)) theory. It can also be checked that the Coulomb branch spectra of the two theories match perfectly. (SO(8M +6)) theory has a one dimensional conformal manifold, and the corresponding marginal deformation is x 2M +1 z 2 , which gives k = 1, N = 2M + 1 and p = 1. We thus see that the sector at z = ∞ is an SO(4M + 4) gauge theory coupled to D 4M +2 2 (SO(4M + 4)). We propose the following description: where the D 12M +6 2 (SO(12M + 8)) theory can be determined from the (a, c) central charges and Coulomb branch spectra of the above theories.

Reduction to 3d and mirror theory
The D 4M +2 2 (SO(4M +4)) theory can be realized as an IR fixed point of the USp(2M +2) gauge theory with 2M + 3 flavors: Upon compactifying to 3d, (8.15) flows to the T [2M +3,2M +3] [SO(4M + 6)] theory, with the U(1) C Coulomb branch symmetry. Upon gauging this U(1) C symmetry, we obtain the 3d reduction of the D 4M +2 2 (SO(4M + 4)) theory. The similar argument applies also for D 12M +6 2 (SO(12M + 8)), whose corresponding quiver is This again flows to an SCFT with the U(1) Coulomb branch symmetry, where, upon gauging this symmetry, we obtain the 3d reduction of the D 12M +6 2 (SO(12M +8)) theory. To study the 3d reduction of the D 8M +4 4 (SO(8M + 6)) theory, we first consider the following quiver theory (SO(8M + 6)) to 3d: Let us now consider the mirror theory. The mirror theory of T , whose quiver description is (SO(8M + 6)) 3d mirr : This is a star-shaped quiver which is the mirror theory of the 3d reduction of the following theory of class S: The special case of M = 0 Due to the isomorphism between the Lie algebras of SU(4) and SO (6), the case of M = 0 in the above discussion gives an alternative description of the D 4 4 (SU(4)) theory, which is a Lagrangian theory described by [34] (see also [22, (4.12)]): In particular, it is instructive to compare this to (8.14) with M = 0, in which case D 2 2 (SO(4)) = D 2 2 (SU(2)) × D 2 2 (SU(2)) is simply two copies of hypermultiplets, and the D 6 2 (SO(8)) theory is the E 6 MN theory [34]. Theories (8.14) for M = 0 and (8.22) are related to each other by the Argyres-Seiberg duality [63], where we dualize the SU(3) node, which has six fundamental flavors transforming under it, to the E 6 MN theory with an SU(2) subgroup of E 6 being gauged and coupled to one flavor of the hypermultiplet. Moreover, (8.20) with M = 0, namely 4 (SO(6)) 3d mirr : gives an alternative description of the mirror theory of the reduction of D 4 4 (SU(4)) to 3d: (D 4 4 (SU(4))) 3d mirr : It can be checked similarly to Appendix A.4 that the Coulomb and Higgs branch Hilbert series of (8.23) and (8.24) are in agreement with one another (see also Section 2 of [64]).

The D
2p(2M +1) 2p (SO(4pM + 2p + 2)) theory Similarly to the previous discussion, we propose that the D 2p(2M +1) 2p (SO(4pM + 2p + 2)) theory admits the following class S description According to (3.3), this theory has a (p − 1)-dimensional conformal manifold. [SO(4pM + 6p)] theory is gauged. Note that the quiver description for the latter is [SO(4pM + 6p)] : where there are 2p − 1 gauge groups in total. The mirror of this theory is where we define The mirror theory of the 3d reduction of D 2p(2M +1) 2p (SO(4pM + 2p + 2)) can then be obtained by gauging the Cartan subalgebra of the flavor symmetry D 2p in the above quiver. As a result, we obtain D 2p(2M +1) 2p (SO(4pM + 2p + 2)) 3d mirr For the special case of M = 0, the above mirror theory reduces to D 2p 2p (SO(2p + 2)) 3d mirr :  2M p+2)) is in fact a Lagrangian theory, whose quiver description is In these 4d N = 2 theories, each C and D gauge group has zero beta-function, 24 i.e. all gauge groups are conformal. However, upon reduction to 3d, if we assume that we obtain the same quiver gauge theory with 3d N = 4 supersymmetry, then each conformal Cgauge group is overbalanced, and each conformal D-gauge group is underbalanced. 25 The presence of the latter renders the quiver gauge theory in question a "bad theory" in the sense of [28]. For example, the D 8M 4 (SO(8M + 2)) theory has the following Lagrangian description: As a 3d N = 4 gauge theory, this quiver is a bad theory, due to the presence of the underbalanced SO(4M + 2) node. Nevertheless, it can be identified with the T σ ρ [SO(8M + 8)] theory, where σ = 3 2 , 1 8M +2 and ρ = [2M + 3, (2M + 2) 2 , 2M + 1] The mirror theory of the latter, namely T ρ σ [SO(8M + 8)], admits the following Lagrangian description (8.34) We propose that this is a mirror for the D 8M 4 (SO(8M + 2)) theory. Another example is the D 10M 5 (SO(10M + 2)) theory, whose quiver description is The beta-functions of the C N gauge group with 2N + 2 flavors of fundamental hypermultiplets and the D N gauge group with 2N − 2 flavors of vector hypermultiplets are zero. 25 The conditions for a C N gauge group with F C N flavors of fundamental hypermultiplets and a D N gauge group with F D N flavors of vector hypermultiplets to be balanced are, respectively, F C N = 2N +1 and F D N = 2N − 1. In each case, if the number of flavors are fewer (resp. greater) than the said F C N or F D N , then the corresponding gauge group is said to be underbalanced (resp. overbalanced) [28]. theory. The mirror theory T [2M+1,(2M) 4 ,1] [SO(10M + 2)] admits the following quiver description: We propose that this is a mirror for the D 10M

5
(SO(10M + 2)) theory. The special case of D 8 2 (SO(10)) is also worth discussing. This 4d theory admits the Lagrangian description in terms of the USp(4) gauge theory with 6 hypermultiplets in the fundamental representation [34]. In terms of a 3d N = 4 gauge theory, this flows to the T [7,5] [SO (12)] theory. The mirror theory, namely, T [7,5] [SO (12)], admits the following quiver description This is precisely the 3d mirror theory [65] for the class S theory of the twisted D 3 type associated with a sphere with two untwisted punctures [1 6 ], [1 6 ] and two twisted punctures [4] t , [4] t . Such a theory of class S indeed describes the 4d N = 2 USp(4) gauge theory with 6 fundamental flavors 26 [40]. We thus conclude that (8.37) is a mirror theory for the D 8 2 (SO(10)) theory. Moreover, due to the isomorphism between D 3 and A 3 , the aforementioned class S theory can also be described as that of the twisted A 3 type [39] associated with a sphere with two untwisted punctures [1 4 ], [1 4 ] and two twisted punctures [5] t , [5] t . We thus propose another description of the mirror theory for the D 8 2 (SO(10)) theory as follows:

A Hilbert series
In this appendix, we compute the Hilbert series of various theories discussed in the main text. In particular, we match the Higgs and Coulomb branch Hilbert series for the dual theories that admit different quiver descriptions, as well as discuss certain properties of the moduli space. Regarding the Coulomb branch Hilbert series computations, the magnetic lattices for orthosymplectic quivers were spelled out explicitly in [61].

A.1 Relation (4.15)
The Higgs branch Hilbert series The Higgs branch Hilbert series of the theory on the left-hand side of (4.15) can be written as follows, following the convention (4.14): where z 1 and z 2 are gauge fugacities for each U(1) gauge factor, and x 1 , x 2 , q are the fugacities for each factor of the SU(m)×SU(m)×U(1) flavor symmetry respectively. We will see that the fugacity q can be absorbed into the gauge fugacities by a redefinition. Let us rewrite the gauge fugacities as follows: Then, the above Hilbert series can be rewritten as [r,0,··· ,0,r] (x j )t 2r . (A. 3) The right-hand side of the second equality is indeed the product of two HB Hilbert series of the SQED with m hypermultiplets with charge 1. This matches the Higgs branches of the theories on both sides of (4.15). The right-hand side of the third equality is the product of two Hilbert series of the reduced moduli space of one SU(m) instanton on C 2 [62], which is the closure of the minimal nilpotent orbit of SU(m).

The Coulomb branch Hilbert series
On the other hand, the Coulomb branch (CB) Hilbert series of the theory on the lefthand side of (4.15) can be written as [61,66] 27 (see also [67,68]): where ∆(r, s) is the dimension of the monopole operator with the Abelian gauge flux (r, s) whose expression is given by x 1 and x 2 are fugacities for the topological symmetries for each SO(2) gauge group; ω is the discrete fugacity for the topological symmetry for the Z 2 quotient in (SO(2) × SO(2))/Z 2 which satisfies ω 2 = 1 ; (A.6) 27 Note that the fugacity t in [66] corresponds to t 2 in this paper.
and (1 − t 2 ) −1 is the dressing factor associated with each SO(2) gauge group. We can rewrite and obtain H[CB(4.15)](t; ω, y 1 y 2 , y −1 The right-hand side of the second equality is indeed the CB Hilbert series of two copies of the SQED with m hypermultiplets with charge 1. This matches the Coulomb branches of the theories on both sides of (4.15). The right-hand side of the third equality is the Hilbert series for the product of two copies of C 2 /Z m .

A.2 Relation (4.16)
The Higgs branch Hilbert series Following the convention (4.14), we see that the HB Hilbert series of the theory on the left-hand side of (4.16) is given by H[HB (4.16)](t; x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , q 1 , q 2 , q 3 ) where each x 1,2,3 and y 1,2,3 denotes the fugacities of each SU(m) factor of the SU(m) 6 flavor symmetry and z 1,2,3 are the fugacities for each U(1) gauge symmetry. Here we identify z 4 ≡ z 1 . Similarly to the previous discussion, we define There are, however, the relations: We can therefore rewrite the above Hilbert series as H[HB (4.16)](t; x 1 , x 2 , x 3 , y 1 , y 2 , y 3 , q 1 , q 2 , q 3 ) where we identify u 4 ≡ u 1 , u 0 ≡ u 3 and define w i = q −1 i u i . This is the HB Hilbert series for the following quiver gauge theory This is actually equivalent to the theory on the right-hand side of (4.16), namely a complete graph of 4 U(1) gauge nodes where each edge has multiplicity m, upon decoupling an overall U(1) in the latter theory. We indeed have an SU(m) 6 × U(1) 3 flavor symmetry, as can also be seen from the t 2 terms in the power expansion of the Hilbert [1,0,··· ,0,1] (y i ) . (A.14) The Coulomb branch Hilbert series On the other hand, the Coulomb branch (CB) Hilbert series of the theory on the lefthand side of (4.16) can be written as where ω is the discrete fugacity for the topological symmetry associated with the Z 2 quotient satisfying ω 2 = 1 and with r 4 ≡ r 1 . We define, for i = 1, 2, 3, There are the following relations: The dimension of monopole operators can be rewritten as 28 For reference, we provide a partially unrefined Hilbert series for m = 3, with all elements of x 1,2,3 and y 1,2,3 set to 1, as follows: where we remark that 51 is the dimension of SU(3) 6 × U(1) 3 , which is the flavor symmetry of the theory as claimed.
The factor 3 i=1 x r i i in (A.15) can be written as We can rewrite the above CB Hilbert series as We thus obtained the CB Hilbert series of (A.13) such that the topological symmetry for the i-th node is ω y 4 i (y 1 y 2 y 3 ) −1 .

A.3 Relation (4.17)
The Higgs branch Hilbert series For the sake of conciseness, let us write The Higgs branch Hilbert series of the theory of the left-hand side of (4.17) is where x denotes the fugacities of the SU(P ) flavor symmetry and q denotes the fugacity for U(1) flavour symmetry arising from the edge between two SO(2) gauge groups. We change the integration variables as in (A.2) and obtain This is indeed the HB Hilbert series for the theory on the right-hand side of (4.17), upon decoupling an overall U(1), where u and v are the gauge fugacities for the U(1) nodes connected by the blue line. The flavor symmetry is indeed SU(P ) × U(1), as can be seen from order t 2 in the power series of the above Hilbert series.

The Coulomb branch Hilbert series
The CB Hilbert series of the theory on the left-hand side of (4.17) is given by where ω is the discrete fugacity for the topological symmetry associated with the Z 2 quotient, satisfying ω 2 = 1, and ∆(r 1 , r 2 ) = 1 2 We change the variables as in (A.7) and obtain H[CB(4.17)](t; ω, y 1 y 2 , y −1 This is indeed the CB Hilbert series for the theory on the right-hand side of (4.17), upon decoupling an overall U(1), where (u 1 , u 2 ) and (ωy 1 , ωy 2 ) are, respectively, the gauge fluxes and the fugacities for the topological symmetries for the U(1) nodes connected by the blue line. In order to compute the HB Hilbert series with that of (7.24), we proceed similarly, noting that there is an overall U(1) that decouples. Specifically, this can be realized as follows. Suppose we denote by z 1 and z 2 the gauge fugacities of the two U(1) nodes in the triangle. The hypermultiplets associated with the blue line transform as z 1 z −1 2 and z −1 1 z 2 , whereas there is no matter field that transforms as z 1 z 2 or (z 1 z 2 ) −1 . The latter correspond to the combination of an overall U(1) that decouples, and so we may set z 1 z 2 = 1, i.e. z 1 = z −1 2 ≡ z. Hence, the hypermultiplets associated with the blue line transform as z 2 and z −2 , corresponding to the terms indicated in blue in the above expression.

The Coulomb branch Hilbert series
The CB Hilbert series of (7.23) can be written as H[CB(7.23)](t; ω, x 1 , x 2 ) where we use the small case letters to denote the gauge fugacities of the corresponding groups; the dimension of the monopole operator is with |x±y| = |x+y|+|x−y|; x 1 and x 2 are the fugacities for the topological symmetries associated with the left D 1 and right D 1 respectively; ω is the discrete fugacity for the topological symmetry associated with the Z 2 quotient, satisfying ω 2 = 1; (1 − t 2 ) −1 is the dressing factor for each D 1 gauge group, whereas P G denotes the dressing factor for the group G given by [66, Appendix A] (with t in that reference being t 2 in this article). In particular, the explicit expression for P C 2 (t; c 2 ) was provided in [66, (A.18)].
It should be noted that the Coulomb branch symmetry SU(4) × U(1) is not manifest in the Hilbert series (A.31); this is analogous to the discussion in [70].
On the other hand, the CB Hilbert series of (7.24) can be written as H[CB(7.24)](t; y 1 , y 2 , y 3 , r) where (y 1 , y 2 , y 3 ) are fugacities for the SU(4) Coulomb branch symmetry and r is that for the U(1) Coulomb branch symmetry. The dimension of the monopole operators is (A.34) The dressing factor P U (2) and P U (3) are given by [66, (A.4)] (with t in that reference being t 2 in this article). The Coulomb branch symmetry SU(4) × U(1) is manifest in this expression.
(A. 39) This is precisely the Hilbert series of C 2 /E 6 , which is the Coulomb branch of the E 6 MN theory.
B Results regarding Eqs. (2.33) and (2.34) The geometries in Eqs. (2.33) and (2.34) have been identified, respectively, as the hypersurface singularities of type VII and X, following the notation of [21]. In particular, we can compute their Milnor numbers to be µ (2.33) We can also discuss the number of marginal operators for theory (2.33). Such theory has a number of marginal operators whose patterns depend on if N is even or odd.
• If N is odd, for any i, j, k ≥ 0, there are 1. 1 marginal operators when p + 3 − 2N is odd.
4. 2j + 2 marginal operators when N = (4j + 6)(k + 1) + 1 and p = (3 + 2j)(5 + 2i + 4k). Whenever different values of i, j, k gives the same value of N and p, the number of marginal operators is the one that have the largest j.
5. 2j + 3 marginal operators when N = (2j + 4)k + 4j + 9 and p = 2(5 + 2i + 2k)(2 + j). Whenever different values of i, j, k gives the same value of N and p, the number of marginal operators is the one that have the largest j.

C Examples of non-Higgsable SCFTs
Here we list (G, G ) theories that have 0 ≤ 24(c − a) < 1. They have been computed using the program provided in [30]. We are dividing the theories in three main tables: 1. (A n , A m ) with 1 ≤ n, m ≤ 100 in Table 2. 2. (A n , D m ) with 1 ≤ n, m ≤ 100 in Table 3.
All the theories that are not in the tables have 24(c − a) ≥ 1. For instance, there are no (D n , D m ) or (E, E) theories. All the non-Higgsable SCFTs are expected not to have mass parameters since they have no Higgs branch. Indeed, theories in Tables 2 to 4 have rank equal to rank (G n , G m ) = nm 2 , for G = A, D, E. (C.1) The rank in Eq. (C.1) is exactly equal to half the Milnor number, computed in [21], of the associated hypersurface singularity, as expected by theories without mass parameters.
It is natural to conjecture that the rank of such theories equals the sum of all the non-vanishing genus-zero Gopakumar-Vafa (GV) invariants [31][32][33] of the corresponding geometry. The logic for this is the following. The magnetic quiver of a 4d theory engineered by IIB on a given CY X is related to the electric quiver of the 5d theory engineered by A-theory on the same CY X by an operation consisting in gauging the topological symmetry (see Figure 1 in [23]). However, the mirrors we consider here are just free hypers, so there is no topological symmetry that can be gauged. Therefore, they directly correspond to the electric quivers (i.e., the 3d dimensional reduction) of the 5d theory obtained by compactifying M-theory on X. GV invariants at genus zero count hypers of the 5d theory, which in this case are free hypers, and are clearly in oneto-one correspondence with those of the considered 3d mirror. Their number equals the rank of the original 4d theory. Some of those invariants were computed independently in [73,74]. We checked that our prediction matches the results of [73,74] in all cases in which the same CY was considered in both papers.