1-form symmetry, isolated N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 SCFTs, and Calabi-Yau threefolds

We systematically study 4D N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 superconformal field theories (SCFTs) that can be constructed via type IIB string theory on isolated hypersurface singularities (IHSs) embedded in ℂ4. We show that if a theory in this class has no N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2-preserving exactly marginal deformation (i.e., the theory is isolated as an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 SCFT), then it has no 1-form symmetry. This situation is somewhat reminiscent of 1-form symmetry and decomposition in 2D quantum field theory. Moreover, our result suggests that, for theories arising from IHSs, 1-form symmetries originate from gauge groups (with vanishing beta functions). One corollary of our discussion is that there is no 1-form symmetry in IHS theories that have all Coulomb branch chiral ring generators of scaling dimension less than two. In terms of the a and c central charges, this condition implies that IHS theories satisfying a<12415r+2f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ a<\frac{1}{24}\left(15r+2f\right) $$\end{document} and c<163r+f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ c<\frac{1}{6}\left(3r+f\right) $$\end{document} (where r is the complex dimension of the Coulomb branch, and f is the rank of the continuous 0-form flavor symmetry) have no 1-form symmetry. After reviewing the 1-form symmetries of other classes of theories, we are motivated to conjecture that general interacting 4D N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 SCFTs with all Coulomb branch chiral ring generators of dimension less than two have no 1-form symmetry.

The intersection of these last two points forms a starting point of our story. In particular, many 4D N = 2 conformal manifolds can be constructed by gauging a G flavor symmetry of one or more isolated 4D N = 2 SCFTs 1 (combined with unitarity, the existence of such a symmetry implies the existence of non-nilpotent moment map operators 2 [15] and, by standard lore, a Higgs branch of moduli space in each of the isolated theories we gauge).

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The resulting G gauge group, with vanishing beta functions, is a natural source for line operators (Wilson and 't Hooft lines) charged under 1-form symmetries [16].
Therefore, in order to gain an additional handle on the space of 4D N = 2 SCFTs, it is useful to understand whether these conformal gauge groups are the only sources of 1-form symmetries. To answer this question, we should study whether (1) isolated N = 2 SCFTs can have 1-form symmetries and (2) whether all exactly marginal couplings in 4D N = 2 are indeed gauge couplings.
Our focus in this paper will mostly be on the first point. 3 Moreover, we will mainly restrict our attention to the case of 4D N = 2 SCFTs arising from the low-energy limit of type IIB string theory on three-complex-dimensional isolated hypersurface singularities (IHSs) embedded in C 4 and characterized by quasi-homogeneous polynomials satisfying [21] W (x i ) : and In terms of these variables, the holomorphic 3-form is given by This vast class of SCFTs includes, among many others, the well-known (g, g ) theories of [1]. The early Argyres-Douglas (AD) SCFTs [22,23] are prominent members of this family of theories: the (A 1 , A 2 ), (A 1 , A 3 ), and (A 1 , D 4 ) theories correspond to SCFTs on the Coulomb branches of SU (2) with N f = 1, 2, 3 respectively (or alternatively, in the first case, to pure SU(3) N = 2 SYM).
For general IHS theories of this type, our main claim in this paper is the following: Claim (Main claim). Any isolated 4D N = 2 SCFT (i.e., a theory without N = 2preserving exactly marginal deformations 4 ) arising from type IIB string theory on an IHS embedded in C 4 has trivial 1-form symmetry. 5 (Unless otherwise noted, when we refer to IHS theories below, we mean theories of this type).
We believe this result extends to theories arising on IHSs embedded in other spaces like C * × C 3 , and we will discuss some of the resulting N = 2 SCFTs explicitly in section 4. We will also comment on other N = 2 SCFTs which are not realized via IHSs. 3 Regarding the second point in the previous paragraph, we are not aware of gauge coupling interpretations for most of the conformal manifolds we will come across, but we are also not aware of arguments forbidding such interpretations (similar comments apply to the vast majority of the theories discussed in [17,18]). Indeed, there are conjectures that an N = 2-preserving exactly marginal deformation always corresponds to a gauge coupling [19,20]. 4 We allow for exactly marginal deformations that only preserve N = 1. 5 Note that here we have in mind the SCFTs specified by the singularities themselves. In particular, we do not consider decoupled U(1) factors that may arise in RG flows emanating from these SCFTs as IHS theories in their own right.

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A formalism for finding the 1-form symmetries in the context of the theories covered by our main claim was developed in [24][25][26]. Using these tools, along with mathematical results in [27] and their physical interpretation in [28], we give a proof of the main claim for the subset of IHS theories corresponding to (g, g ) SCFTs in section 2.1. We relegate a complete proof of our claim for all the IHS theories to the appendix.
However, our results do not fully rely on [27]. In particular, motivated by [24,29], we allow for potentially more general IHS-defining polynomials with at least five terms. As we describe in the appendix, we are able to bypass the precise nature of these additional monomials, because we are only interested in questions regarding 1-form symmetries in isolated theories.
One intriguing aspect of the theories we study is that they are related to 2D N = (2, 2) Landau-Ginzburg (LG) models via the correspondence in [1] (indeed, the analysis of [29] mentioned in the previous paragraph proceeds from this connection). Therefore, one may wonder if there is a relation between the 1-form symmetries we study and symmetries in the corresponding LG models.
While we do not currently have a precise answer to this question, we note an intriguing parallel between our story for 4D N = 2 SCFTs and the idea of decomposition in 2D QFT (e.g., see [30] and the recent workshop [31]). In essence, we are arguing that, on their own, isolated 4D N = 2 SCFTs arising from IHSs have no 1-form symmetry. Instead, we need to combine such SCFTs into larger theories via gauging in order to have such symmetry. 6 Equivalently, if we start from a gauged collection of such isolated SCFTs and decompose the theory into its isolated SCFT constituents at zero coupling, each isolated constituent on its own will lack 1-form symmetry. Somewhat similarly, in the context of 2D QFT, the idea of decomposition states that a 2D QFT with 1-form symmetry can be rewritten as a disjoint union of QFTs without 1-form symmetry.
Given this discussion, several immediate questions arise: • Is the converse of our main claim true? In other words, do all conformal manifolds arising from IHSs have one-form symmetries? No. For example, we can have local operators in the isolated matter sectors transforming non-trivially under the centers of the exactly marginal gauge groups (thereby breaking the 1-form symmetry). More concretely, consider the realization of (A 5 , A 7 ) given in [32] as a diagonal SU(3) gauging of the D 7 (SU(3)) SCFT and the D 7 (SU(4)) SCFT. 7 The su(4) moment map of D 7 (SU(4)) decomposes into the following representations of su (3) µ A : 15 → 8 ⊕ 3 ⊕3 ⊕ 1 . (1.5)

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trivial one-form symmetry even though it has a conformal manifold and a gauge group with vanishing beta function (the result for (A 5 , A 7 ) also follows from the methods in [24][25][26]).
• Does our main claim hold for more general N = 2 theories? As we will see in section 4, the arguments in [36][37][38] suggest that many isolated (as N = 2 SCFTs) class S theories (here we mean isolated theories coming from the twisted compactification of the 6D (2, 0) theory on surfaces that do not have irregular punctures) also have no 1-form symmetry. On the other hand, recent arguments suggest that the N = 3 theory related to the G(3, 3, 3) complex reflection group [39] might have Z 3 1-form symmetry [40]. While, on general grounds [41], this theory is isolated as an N ≥ 2 theory, it may potentially be part of an N = 1 conformal manifold with a 1-form symmetry arising from an N = 1 gauging of the E 6 Minahan-Nemeschansky (MN) theory coupled to some additional matter fields. If it is true that the G(3, 3, 3) N = 3 theory has 1-form symmetry, then perhaps one can show that, for N ≥ 2 theories that are isolated as N = 1 theories and have no "conformal gauge theory origin" (i.e., do not have a formulation in terms of an N = 1 gauge group with vanishing beta function), there is no 1-form symmetry. At present, we do not have a proof of this statement. 9 • Does our result hold more generally for 4D N = 2 SCFTs related to 2D LG theories via the correspondence in [1]? For example, there are theories of this type involving more than four variables that don't typically have a presentation in terms of type IIB on an IHS. While such a result seems plausible, we do not currently have a conclusive argument one way or the other.
• Can we deduce any universal constraints from our main claim? As we will discuss in section 5, there are some tantalizing hints the answer may be yes. Indeed, one corollary that follows from our claim is that, in our class of SCFTs, any theory with all Coulomb branch generators of dimension less than two must have trivial 1-form symmetry. After reviewing other classes of theories in section 4, we are motivated to conjecture that this statement is more generally true for interacting 4D N = 2 SCFTs. Using results in [42] (see also the application in [43]) we can then argue this condition implies that for IHS theories c < 1 6 (3r + f ) , a < 15r + 2f 24 , (1.6) 9 One should be careful with such a conjecture. Indeed, if there are N = 3 theories of this type with 1-form symmetry, then the universal mass deformation δW = λµ, with µ the moment map of the U(1) N = 2 flavor symmetry all N = 3 SCFTs possess, will generate flows to theories that have (in the absence of SUSY enhancement) N = 2 SUSY. These theories may inherit the 1-form symmetry of the UV N = 3 SCFT and might be isolated (even as N = 1 theories). In this case, a more general conjecture would be that isolated theories have 1-form symmetry only if they have some (possibly UV) gauge theory origin. However, it is not clear to us how meaningful this statement is, since it may be that all 4D QFTs can be obtained from RG flows emanating from gauge theories.

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where r is the complex dimension of the Coulomb branch (i.e., the "rank" of the theory), and f is the rank of the continuous 0-form flavor symmetry. Note that this discussion can be phrased abstractly for any 4D N = 2 SCFT, 10 and it would be interesting to understand if these bounds imply lack of 1-form symmetry more generally.
The plan of this paper is as follows. First we briefly review the tools constructed in [24][25][26] for 1-form symmetry detection. We then give a proof of our main claim for isolated N = 2 SCFTs arising from the subset of (g, g ) SCFTs and explain how these results fit into our broader proof. After that, we derive some useful general results for our IHS theories including a lemma on the absence of 1-form symmetries in IHS theories that admit certain bilinears. Then, we give a very rough sketch of the 1-form symmetry content of more general 4D SCFTs with N ≥ 2 SUSY (including those arising from IHSs in C * × C 3 ). Before concluding and mentioning open problems, we discuss the corollary described around (1.6) and its potential implications for the space of 4D N = 2 SCFTs. We relegate a full proof of our main claim for all theories arising from IHSs in C 4 to appendix B.

Trivial 1-form symmetries in isolated SCFTs arising from IHSs
Given a realization of a 4D N = 2 SCFT, T X 6 , as the low-energy limit of type IIB string theory on an IHS 3-fold, X 6 , the authors of [24][25][26] found a prescription for computing the corresponding 1-form symmetry. The basic idea is to compute the component of the defect group, D (1) , that arises from D3 branes wrapping certain 3-cycles in X 6 (we refer the interested reader to the original papers for more details). Here we merely state the result: where the 4D 1-form symmetry group is determined by choosing maximal isotropic subgroups of the finite groups in the summand. This procedure ensures mutual locality of the spectrum (in analogy with the procedure in [44]). It turns out that these finite groups can be fixed in terms of the singularity data (1.1). In particular, taking yields [24][25][26] r i = gcd(w 1 , · · · ,ŵ i , · · · w 4 ) , w i := Dq i , D := lcm(U 1 , and The flavor symmetry rank is an unambiguous non-perturbative quantity. The rank of the Coulomb branch may be replaced by the number of generators of the N = 2 chiral ring modulo relations (here we define the chiral ring operators to be annihilated by the full anti-chiral set of N = 2 supercharges).

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Note that in (2.3) and (2.4), "x" refers to dropping the variable x from the corresponding expressions. For our purposes below, it is slightly more useful to rewrite the previous two equations in a way that does not refer directly to D. In particular, as discussed in [45], one has , (2.5) and . (2.6) To argue that a theory has no 1-form symmetry, it suffices to show that there is no i such that r i > 1 and g i > 0.
To check this absence of 1-form symmetry holds for isolated theories, we need to show that the above conditions hold for any theory that has no dimension-two Coulomb branch operator (the level-four superconformal descendant arising from the integration over all of chiral superspace becomes the dimension-four exactly marginal deformation). 11 In terms of the singularity data, this condition amounts to checking that there is no deformation in the Milnor ring of the singularity Note that the number of independent deformations in R is given by the Milnor number and that these deformations are encoded in the Poincaré polynomial where each t Q represents a non-trivial deformation of the theory with weight Q. 12 In particular, the marginal deformations correspond to the term dim H 1 t ⊂ P (t). Therefore, the condition for the SCFT to be isolated is simply dim H 1 = 0. 11 By the general discussion in [46,47], such operators are uncharged under N = 2 flavor symmetry.
Therefore, using the results of [48] (see [49] for a discussion), such deformations in the N = 2 prepotential are, in addition to being marginal, exactly marginal (these operators cannot pair up with other operators to become long multiplets). 12 Since the qi are rational, (2.10) is not technically a polynomial. On the other hand, to get a polynomial we can simply replace t → t D . Given this simple relation, we will abuse terminology and refer to (2.10) as the Poincaré polynomial.

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Equivalently, in the language of the 2D N = (2, 2) SCFT that the LG model of [1] flows to, there should be no exactly marginal deformation. 13 Indeed, we have that the scaling dimension of the coupling λ vanishes Therefore, λ can be thought of as corresponding to a coordinate on the conformal manifold.
Before making further comments, let us briefly recapitulate. Our goal is to show that when a deformation of the form (2.7) does not exist (i.e., when the theory is isolated as an N = 2 SCFT), we have no i such that r i > 1 in (2.5) and g i > 0 in (2.6).
Note that the 1-form symmetry of an IHS theory is completely fixed in terms of the weights of the coordinates (i.e., the q i ). These weights, in turn, specify the Poincaré polynomial of the singularity and, therefore, the allowed deformations. Since these geometrical deformations correspond to N = 2-preserving deformations of the SCFT and since (conformal) gauge groups often give rise to 1-form symmetry, it is natural to expect some connection between 1-form symmetry and exactly marginal deformations. On the other hand, the 1-form symmetry itself only depends on a very rough number-theoretical characterization of the weights. Therefore, very different theories (some isolated and some not) can have the same 1-form symmetry content. Moreover, given a set of weight vectors, it is not apriori obvious that there is a well-defined IHS SCFT realizing them. Therefore, our basic strategy will be to apply the above formulas, in combination with the classification of IHS theories in [27,28] (allowing for potential generalizations as in [26,29] alluded to in the introduction), to prove our main claim. In the next section, we apply this strategy to the (g, g ) subset of IHS theories and leave an exhaustive proof for all IHS theories to the appendix.

(g, g ) SCFTs and comments on type I theories
In this section we will explicitly focus on the case of the (g, g ) SCFTs [1]. Our main reasons for doing so are that these theories are well-studied (as described in the introduction, they include the original AD theories; in addition, [24][25][26] explicitly studied 1-form symmetries in these SCFTs), they illustrate some of the main techniques of this paper, and they also provide a neat entry to the classification of [27][28][29].
To that end, consider type IIB string theory on a hypersurface singularity defined via where W g and W g are given in table 1. Here W g is chosen so that w 2 + W g is the g-type Du Val singularity. Let us go through these theories in turn. First we impose absence of exactly marginal couplings as in the discussion around (2.7). Then, we check that the 1-form symmetry is indeed trivial. 13 It would be interesting to understand if theories with conformal 4D gauge couplings translate into special 2D conformal manifolds. Table 1. Useful data for the (g, g ) SCFTs.

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Let us first consider the (A p−1 , A q−1 ) SCFTs. Although the 1-form symmetry here is known to be trivial [24][25][26], these simple theories form a good starting point. A generic deformation in such a theory comes from taking a product of entries in the first row of table 1 (one for A p−1 and another for A q−1 ). The corresponding weight for this term takes the form To find a solution of Q = 1, and therefore an exactly marginal deformation, we need to solve (2.14) Clearly, k, l cannot both be zero. Moreover, when gcd(p, q) = 1, this equation has no solution (as can be seen by multiplying both sides by a factor of p or q and using the fact that p/q, q/p are never integer if p, q are co-prime 14 ). Therefore, theories with gcd(p, q) = 1 are isolated N = 2 SCFTs. Are they the only isolated (A p−1 , A q−1 ) SCFTs? To see this is indeed true, we can proceed by considering s = gcd(p, q) > 1 and defining p = sp , q = sq . Then we have Since gcd(p , q ) = 1, the only possible solution is given by However, we also need to take into account that k, l are in the restricted set of integers k = 0, · · · , p − 2 and l = 0, · · · , q − 2. As a result, when p = 2 or q = 2 or (p, q) = (3, 3), there is no solution to (2.16). Otherwise, one can always find a solution to (2.16).
To summarize, the isolated SCFTs of type (A p−1 , A q−1 ) are given by gcd(p, q) = 1 , or p = 2 , or q = 2 , or (p, q) = (3, 3) . (2.17) By plugging these results into (2.5) and (2.6) it is easy to check that either r i = 1 or g i = 0 for all i and so there is no 1-form symmetry. Alternatively one can, say, consult 14 More precisely, pl/q, qk/p ∈ Z. For these quantities to be integers, we need to have q|pl, which is equivalent to gcd(q, pl) = q. However, gcd(pl, q) = gcd(l, q) < q as l < q and p, q are coprime.  [25]. Therefore, our main claim is true in these theories (as discussed above, this statement is somewhat trivial given the fact that all (A p−1 , A q−1 ) SCFTs have no 1-form symmetry, even if they are not isolated). Next let us consider the (A p−1 , D q+1 ) SCFTs. This time, from table 1, we see that the Q's are given by Let us consider the first case (which arises from terms independent of u). The equation we want to study is This equation is almost identical to (2.14), except for the range of l. Therefore, it has no solution if and only if (note we consider q ≥ 3 here) This is a necessary condition for having an isolated SCFT. To find a sufficient condition, we must also consider the second equation in (2.18) with Q = 1, namely which arises from u-dependent deformations. For gcd(p, q) = 1 this equation obviously has no solution. Moreover, for p = 2, it also has no solution because (q + 1)/q is not an integer for q ≥ 3. Therefore we conclude that the isolated SCFTs of type (A p−1 , D q+1 ) are given by (see also the recent results in [50]) gcd(p, q) = 1, or p = 2 .
(2.22) Using (2.5) and (2.6) it is again easy to check that these isolated theories have no 1-form symmetry. Alternatively, one can again consult table 1 of [25] to verify this statement. This result is somewhat more non-trivial, since there are non-isolated (A p−1 , D q+1 ) SCFTs with 1-form symmetry (see [24][25][26]). Our discussion suggests this 1-form symmetry might arise from a conformal gauge group.
Let us now move on to the (D p+1 , D q+1 ) SCFTs. From table 1, the set of Q's are given by

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Much as before, absence of a solution to setting the first set of Q's to one is equivalent to requiring gcd(p, q) = 1 .
This is a necessary condition for having an isolated SCFT. To see it is also sufficient, note that it immediately implies that none of the remaining three sets of Q's in (2.23) can equal one. Therefore, we conclude that the isolated SCFTs of type (D p+1 , D q+1 ) are given by It is again easy to check that (2.5) and (2.6) imply that the 1-form symmetry is trivial in these cases. As before, one may also verify this statement by consulting table 1 of [25]. Note that non-isolated (D p+1 , D q+1 ) theories can, in general, have non-trivial 1-form symmetry. Let us move on to the exceptional cases. In particular, consider (A p−1 , E 6 ). In this case, table 1 implies that Setting Q = 1 is equivalent to Obviously, when 2 and 3 are coprime to p, the above equation has no solution. But we also need to take into account the range of k. It is not difficult to find that a sufficient and necessary condition for isolated (A p−1 , E 6 ) SCFTs is gcd(2, p) = gcd(3, p) = 1 , or p = 2 , or p = 3 . (2.28) Equivalently, this condition can be written as p = 2 , or p = 3 , or p = 6k ± 1 , k = 1, 2, 3, · · · . (2.29) Either by consulting An analysis similar to the one in the previous case leads to the following necessary and sufficient condition for isolated SCFTs: It is again easy to check that (2.5) and (2.6) imply that the 1-form symmetry is trivial in these cases. As before, one may also verify this statement by consulting table 1 of [25]. Note that non-isolated (A p−1 , E 7 ) theories can, in general, have non-trivial 1-form symmetry.

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Let us move on to (A p−1 , E 8 ). From table 1 we read off the monomial weights where k = (p − 1)/2 is for the case in which x appears in the deformation. To be isolated, we have the necessary and sufficient condition To be isolated, a sufficient and necessary condition is The p = 3 option in the (A p−1 , E 8 ) case is absent here due to the larger range of k. As in the previous examples, using either (2.5) and (2.6) or table 1 of [25] shows that such theories

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have no 1-form symmetries. This is again in contrast to the general exactly marginal case where such 1-form symmetries are allowed. Finally, we come to the (E n , E m ) cases. For these 6 types, we note that In fact, since the (E n , E m ) theories more generally cannot have 1-form symmetry, this result is somewhat trivial. Therefore, we see that all isolated (as N = 2 SCFTs) (g, g ) theories have no 1-form symmetry. This establishes our main claim in this subset of the theories. In the next section, we explain how this statement fits into a similar result for type I IHS theories in the nomenclature of [28] (or the related S ×4 1,1 LG theories in [29]). We also discuss certain facts about 1-form symmetries in a subclass of type I IHS theories with exactly marginal deformations and relate these results to ones discussed in [24,26,51,52]. These conformal manifolds all have known gauge coupling interpretations, which gives us an opportunity to discuss certain global properties of the corresponding matter sectors (this type of analysis will be relevant when discussing a particular N = 3 theory in section 4). In the appendix we consider the remaining IHS theories discussed in [26,28,29] and complete the general proof of our main claim.

More general type I theories
In this section, we briefly discuss how some of the (g, g ) theories fit into the classification scheme of [26][27][28][29] and explain phenomena associated with some of the non-isolated theories that will be useful later. To that end, note that the (A p−1 , A q−1 ), (D 4 , A k−1 ), (E 6 , A k−1 ), and (E 8 , A k−1 ) SCFTs can all be written in the following form with a, b, c, and d as in table 2. 15 More generally, [28] classified so-called type I theories that come from singularities of the form in (2.42). Requiring that the singularity is at finite distance in moduli space yields the seven infinite sequences of singularities in table 2 in addition to a finite collection of sporadic solutions. We have checked that the sporadic solutions satisfy our main claim.
The three remaining classes of solutions are studied in [24,26,51,52] and are called the (3, k), (4, k) and (6, k) theories (here we use the nomenclature of [52]). These theories all have exactly marginal gauge couplings and so our main claim does not apply to them. Still JHEP12(2021)024 (a, b, c, d) other name  [27,28]. The first four entries are of the (g, g ) type indicated in the second column. The (3, k), (4, k), and (6, k) theories are studied in [24,26,51, 52] (we use the notation of [52]). There is some overlap between the different theories (e.g., (D 4 , A 2 ) (3, 2)). There are also a finite set of additional sporadic type I singularities and corresponding theories. This table is adapted from [28].
it is interesting to investigate some of these theories further for reasons that will become apparent in section 4. In particular, the (3, k) theories consist of an exactly marginal diagonal su(k) gauging of a collection of three D 3 (SU(k)) theories (these latter theories are defined in [33,34]). For gcd(3, k) = 1, the D 3 (SU(k)) matter sectors are isolated N = 2 theories. The resulting IHSs take the form From these equations, it is easy to check that the finite part of (2.1) is Z 2 k ⊂ D (1) in the case of gcd(3, k) = 1. In particular, the 1-form symmetry can be fixed by picking a maximal isotropic sub-group.
That this result is exactly the same as for su(k) N = 4 super Yang-Mills (SYM) does not seem to be a coincidence. For example, let us take k = 2. The (3, 2) theory has been studied in detail in [52]. There it was shown that a subset of the local operator content (i.e., the Schur sector) is in one-to-one correspondence (via a special map) with the Schur sector of su(2) N = 4 SYM.
Moreover, just as the adjoint hypermultiplet matter of SU(2) N = 4 SYM ensures that the Z 2 one-form symmetry is unbroken, we can argue the same is true for the D 3 (SU(2)) matter in the (3, 2) SCFT with gauge group SU(2). In particular, we claim that none of the local operators (even unprotected operators in long multiplets) of the D 3 (SU(2)) theory transform under the center of SU (2). Therefore, the flavor symmetry of this theory is SO(3) (see [53,54] for related recent discussions in other classes of theories from a different perspective). 16

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To understand this statement, let us define D 3 (SU(2)) in the way it was first constructed in [23]: as the maximally singular point on the Coulomb branch of SU(2) N = 2 SQCD with N f = 2. In this case, the UV matter hypermultiplets can be written as Q a i ⊕Q j b , where i, j = 1, 2 are color indices and a, b = 1, 2 are su(2) flavor indices. The remaining SQCD degrees of freedom (in the N = 2 vector multiplet) do not transform under the flavor symmetry. In terms of these variables, the IR flavor symmetry of the D 3 (SU(2)) theory is already manifest in the UV. Now, it is easy to show that there are no UV operators transforming in odd-spin representations of the flavor su (2). Indeed, all gauge-invariant operators transforming under flavor must have an even number of hypermultiplet fields. Since this flavor symmetry is not anomalous, we see that the IR D 3 (SU(2)) theory has no local operators tranforming in the center of SU(2) (i.e., the flavor symmetry group is SO(3)). In particular, this logic implies that if we choose an SU(2) gauge group for the (3, 2) theory, the Z 2 1-form symmetry corresponding to the center is preserved.
We will return to a similar discussion in a different non-Lagrangian theory in section 4. In any case, we see that (modulo explicit checks we have done in the case of sporadic theories and have not discussed here), our claim 1 in the introduction is true for type I IHS theories. In the appendix, we deal with the remaining cases and complete the proof of our main claim. In the next section we introduce some broader results that apply to certain theories we have just discussed and are also helpful in the proof of our main claim for more general theories appearing in the appendix.

Useful results beyond (g, g ) and type I
Recall from the introduction that our 4D N = 2 SCFTs are related to 2D LG models via the correspondence in [1]. In particular, these latter theories are described by 2D chiral superfields x, y, u, and v with superpotential given by the W describing our IHS.
From this perspective, it is intuitively clear that if W ⊃ X i X j , where X i,j ∈ {x, y, u, v} are distinct and satisfy q i + q j = 1, the theory and symmetry content should simplify (e.g., we expect a simpler Seiberg-Witten description). Indeed, from the 2D perspective we can then integrate out two of the four fields describing our theory without changing any of the scaling dimensions of operators (similar comments apply if W admits a deformation δW = λX i X j preserving the U(1) R symmetry).
It turns out this intuition is correct, in the sense that the 1-form symmetry content of such a theory is trivial: Lemma. If the sum of any two weights equals one, then the 1-form symmetry is trivial.
Proof. Without loss of generality, we can assume that This weight vector is irreducible as long as gcd(V i , U i ) = 1. In particular, the last fractional number is irreducible.
This claim is independent of any IHS SCFT classification. Of course, not all IHS theories satisfy the conditions of the lemma. 17 However, we will see that in large classes of seemingly complicated theories considered in the appendix, SCFTs that do not satisfy this lemma also have an exactly marginal deformation. This fact will lead to many simplifications.
More generally, any IHS theory either satisfies the conditions of the lemma or else admits a relevant mass deformation of the form δW = mX i X j . Indeed, suppose q X i +q X j > 1. Without loss of generality, we can assume q X i > 1/2. Then, from the argument of lemma 2.5 of [29], there must be a field X k with q X i + q X k = 1 (otherwise W will not describe an isolated singularity). On the other hand, if q X i + q X j < 1, the deformation δW = mX i X j is relevant and initiates an RG flow. The end point of this RG flow will be a W corresponding to an SCFT with trivial 1-form symmetry (although there will typically be other sources of 1-form symmetry in the IR; for example, we will generically have various decoupled U(1) factors).
Another simple consequence of the above lemma is the following claim that will be useful later and is entirely independent of any classification of IHS theories: Claim. Any IHS theory with at most two different weights has trivial 1-form symmetry or else admits a conformal manifold.
Proof. See appendix A.

Comments on 1-form symmetries in more general isolated N ≥ 2 SCFTs
In this section, we briefly discuss 1-form symmetries in more general isolated 4D N ≥ 2 SCFTs. We leave a more detailed accounting for later work. Let us start with the maximal amount of SUSY: N = 4 SCFTs. These theories may or may not have 1-form symmetry (e.g., g 2 N = 4 SYM does not have 1-form symmetry, but su(N ) does). However, if we assume locality, then N = 4 SCFTs are never isolated, since the energy-momentum tensor is in the same multiplet with an exactly marginal deformation. As a result, our claim has nothing to say about local N = 4 theories. 18 17 Examples of theories satisfying this lemma include the (Ap−1, Aq−1) SCFTs. In particular, the lemma gives a simple proof that these theories have trivial 1-form symmetry (even when there is an exactly marginal deformation). 18 We expect non-local N = 4 theories to be quite constrained in their structure. For example, it seems unlikely that we can engineer them as boundaries of 5D QFTs, since Nahm's classification [59] forbids bulk SCFTs with sufficiently many supercharges. Similar comments apply if we try to engineer non-local N = 4 theories on defects in D ≥ 5 dimensional QFTs.
In fact, from the N = 1 Lagrangian conformal manifold construction in [40], one can argue that a c = 5/4 rank-one N = 3 theory (related to the G(3, 1, 1) complex reflection group in the language of [39]) has trivial one-form symmetry.
On the other hand, the proposed N = 1 conformal manifold for the N = 3 rank-three theory related to the G(3, 3, 3) complex reflection group has Z 3 one-form symmetry [40]. This statement follows from the fact that the N = 1 theory involves gauging a diagonal SU(3) 3 ⊂ E 6 flavor subgroup of the E 6 MN theory along with some additional chiral superfields that are manifestly invariant under the diagonal Z 3 center. To check whether the strongly interacting E 6 MN sector is invariant under this diagonal center is not too difficult. Indeed, since the fundamental of E 6 decomposes as 27 = (3,3, 1) ⊕ (3, 1, 3) ⊕ (1, 3,3), no local operator in the E 6 MN theory is charged under the diagonal Z 3 center (this agrees also with discussions in [36,53]). 19 This conclusion on the local operator spectrum is reminiscent of our discussion for the D 3 (SU(2)) matter sector of the (3, 2) SCFT in the previous section (although the argument here is purely group theoretical while our argument in the previous section involved invoking an RG flow).
As a result, if the G(3, 3, 3) SCFT indeed lies on this N = 1 conformal manifold, it would likely be an example of an N = 3 theory with a non-trivial one-form symmetry. Clearly, it is worth answering this question definitively (perhaps by a computation in the associated chiral algebra).
Finally, it might be that some of the N = 3 theories that can be engineered by gauging certain discrete su(N ) N = 4 SYM 0-form symmetries may potentially have non-trivial 1-form symmetry (whether such 1-form symmetry is inherited or not would clearly be interesting to check). If these theories possess 1-form symmetry, then one might be able to argue that isolated N ≥ 2 theories have 1-form symmetry only if they have a gauge theory origin (i.e., via flowing from a gauge theory and/or discrete gauging of global symmetries of a gauge theory). Of course, this seems like a very expansive class of theories.
Next let us focus on other constructions of N = 2 theories. In particular, let us briefly discuss how some of the class S results of [36,38] fit in with our discussion. Of the theories we have checked in these references, all have trivial one-form symmetry when they are isolated. For example, consider (4.6) of [38], which describes the abelian group of line operators, L, (including mutually non-local lines) of the (2, 0) theory compactified on a genus g surface with n regular twisted punctures Since L is trivial for g = 0, we require g > 0 in order to have 1-form symmetry. Having non-trivial genus leads in turn to a theory with an exactly marginal gauge coupling. Somewhat more generally, consider (4.21) in [38] L

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This expression corresponds to L for the case of the (2, 0) theory compactified on a Riemann surface of genus g with n regular twisted punctures and 2k Z 2 -twisted regular punctures.
In order for L to be non-trivial, we require that g > 1 or k > 1. If g > 1, then the theory is not an isolated N = 2 SCFT. On the other hand, if g = 0 and k > 1, then the theory again has an exactly marginal gauge coupling. Therefore, these theories seem to be compatible with a generalization of our claim on isolated IHS N = 2 SCFTs. Finally, we come to a set of theories that flow to the (A k−1 , G) theories we have studied in section 2.1. In particular, there are well-known RG flows of the following type [60][61][62][63] [33,34], h ∨ is the dual Coxeter number, and are alternate names for the UV and IR theories that will connect with further generalizations we will discuss.
As emphasized in [63,64], the UV D k (G) theory has a type IIB string theory realization in terms of the same IHS singularity as the IR In terms of these variables, we have It is natural to ask if the UV theories in (4.3) obey our main claim as well. We will shortly see that the answer is yes, provided we can identify the UV and IR 1-form symmetries. 21 The reason is that the UV and IR Milnor rings are closely related: In particular, we see that the UV Milnor ring is larger than the IR Milnor ring since, in essence, we take ∂ y W → t∂ t W . Said differently, monomials involving t k−1 are non-trivial in the UV, while monomials involving y k−1 are trivial in the IR. Hence, we can canonically identify the IR ring with the UV subring involving monomials with t β and β < k − 1. 22 20 Here we drop decoupled matter fields. 21 Note that under the RG flow (4.3), the UV and IR ranks are the same, so there is no possibility of decoupled free U(1)'s and accidental continuous IR 1-form symmetry. This statement means that (up to caveats we omit) one can reasonably guess that the UV and IR 1-form symmetries match. This guess is confirmed by the BPS quiver computations in [35] for all the many cases that were checked. 22 As a concrete example, consider the Maruyoshi-Song flow (see also [65]) D 4 2 (SU(4)) → A 4 3 (2) (where the UV is N = 2 SU(2) SQCD with N f = 4, and the IR is the (A1, A3) SCFT). The UV theory admits six deformations (four related to mass parameters, one related to the dimension two Coulomb branch vev, and

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While the scaling dimensions of the UV coordinates differ from those of the IR (this fact follows from demanding that the different Ω in (4.5) have scaling dimension one), the weight vectors, q, are the same. Therefore, we see that if the UV theory lacks a marginal deformation (i.e., a parameter, λ, with weight zero), then so too does the IR theory. 23 Assuming we can equate the UV and IR 1-form symmetry (see footnote 21 for evidence in favor of this hypothesis for general D k (G) and conclusive proof in certain cases), we see that our UV theories can have 1-form symmetry only if they have conformal manifolds. Therefore, we strongly suspect that the D k (G) theories satisfy our main claim as well, even though they are related to singularities embedded in a different ambient space. 24 More generally, we have flows of the form [63] where we can have b = h ∨ , and G b (k) is in our class of SCFTs. We may use similar logic to that used for b = h ∨ to conclude that, if we can equate the UV and IR 1-form symmetries (as assumed in [35] and as hinted at by the agreement of the UV and IR ranks), the D b k (G) theories have 1-form symmetry only if they are part of a conformal manifold.
Furthermore, if one can identify 1-form symmetry in certain more general flows between theories related to IHSs embedded in C * × C 3 and those related to IHSs embedded in C 4 , then we expect the above comments to generalize.

Some rigorous bounds and a (less rigorous) conjecture
In this section, we first establish a simple but useful claim showing that theories with i q i > 3/2 are isolated SCFTs. From this result and claim 1, it follows that theories with i q i > 3/2 have no 1-form symmetry. Using these results we suggest a few conjectures for more general 4D N = 2 SCFTs.

Claim. IHS theories with
Proof. We give two proofs of this statement. The first follows from unitarity bounds in the (possibly trivial!) (2, 2) 2D SCFT that the LG model with superpotential W flows to. 26 The second proof follows from an analysis of the Poincaré polynomial.
one related to the gauge coupling) while the IR theory admits three deformations (one related to the SU(2) mass parameter, one related to the dimension 4/3 vev, and one related to the corresponding dimension 2/3 coupling). The UV deformations λ0, λ1v, and λ2v 2 can be identified with the three IR deformations (though they are related to UV mass parameters!), whileλ0t,λ1vt, andλ2v 2 t are set to zero in the IR Milnor ring (after substituting t → y). 23 Note that the converse does not hold. Indeed, see the example in footnote 22. The issue is that UV deformations may trivialize in the IR. 24 Note that some Dp(G) theories can be realized via (different) IHSs embedded in either ambient space.
For example, D3(SU (2)) also has a realization as (A1, A3) A 4 3 (2). This subset of Dp(G) theories is, of course, directly subject to our main claim. 25 Again, we allow for the possibility that they are special points on an N = 1 conformal manifold. 26 We thank Z. Komargodski for bringing these bounds to our attention in an unrelated context.

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Let us consider the proof by unitarity bounds first. To that end, take an IHS 4D N = 2 SCFT described by W (x, y, u, v) = 0. The corresponding LG model with chiral superfields x, y, u, v and superpotential W will flow to some (possibly trivial) (2, 2) SCFT in the IR.
In this IR SCFT, unitarity bounds require that (2, 2) chiral operators have left scaling dimension (e.g., see [66]) 27 h ≤ c 2d 6 , where c 2d is the 2D central charge. Using the fact that c 2d = 3ĉ = 6(2 − i q i ) via the correspondence in [1] and the fact that the superconformal R-charge for a chiral operator satisfies Q = 2h, we see that In particular, if i q i > 3/2, we are done since all chiral operators have Q < 1 and therefore cannot give rise to exactly marginal deformations.
Alternate proof. We can arrive at the same result by considering the Poincaré polynomial introduced in (2.10). In particular, the weights appearing in (2.10) are bounded [28] 0 Combining this result with the proof of our main claim (see the appendix), we have that: Corollary. IHS theories with i q i > 3/2 have no 1-form symmetry. Equivalently, IHS theories with all N = 2 chiral ring generators (i.e., Coulomb branch chiral ring generators) having scaling dimension less than two have trivial 1-form symmetry.
This corollary follows from our main claim, but we can also prove it directly: Direct proof. First, we should require all q i ≤ 1/2 in order to have a hope of finding nontrivial 1-form symmetry. 28 Then, if there are two or more weights equal to 1/2, the 1-form symmetry is still trivial because 1/2 + 1/2 = 1.
Assuming the discussion in [42] applies to our theories of interest we can rephrase the above corollary in terms of bounds on a and c. To understand this statement, we substitute i q i > 3/2 into (4.5) and (4.6) of [43] to obtain where r is the complex Coulomb branch dimension of our SCFT (i.e., the rank of the theory), and f is the rank of its continuous 0-form flavor symmetry. One may wonder how robust (5.4) is. For example, recall that for theories arising from gauging a discrete 0-form symmetry in a parent SCFT, the formalism of [42] need not apply [39,67]. However, we do not expect discrete gauging to increase r or f . Moreover, the central charges are unaffected. Now, if we start from a parent theory that has 1form symmetry and therefore does not obey (5.4), the daughter theory will not obey (5.4) either. As a result, we believe that for theories satisfying (5.4), the one-form symmetry is indeed trivial. 30 How general can we make the above statements? One reasonable conjecture is as follows: Conjecture. Interacting 4D N = 2 SCFTs with all N = 2 chiral ring generators (i.e., Coulomb branch chiral ring generators) of scaling dimension less than two have trivial 1-form symmetry.
We have assembled some evidence for this conjecture. Indeed, it is satisfied by theories related to IHSs embedded in C 4 (we have also seen evidence that this statement extends to more general ambient spaces). 31 Moreover, typical class S theories (with regular punctures) involve at least some N = 2 chiral ring generators with scaling dimensions ∆ ≥ 3 and therefore satisfy the conjecture trivially. Finally, N = 3 SCFTs (and, in particular, the potentially troublesome G(3, 3, 3) theory discussed in section 4) satisfy this conjecture 30 Since our theories are strongly coupled, it is natural to give a more universal criterion for when the analysis in [42] applies. We are agnostic on this point, but one interesting possibility, following the discussion in [39], is that the correct criterion is the absence of 2-form symmetry (and that gauging this 2-form symmetry takes us back to the parent theory where [42] applies). 31 More explicitly, here we have

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trivially as well. Indeed, all Coulomb branch operators in these theories have ∆ ≥ 3. 32 The fact that N = 3 theories -and particularly those related to N = 4 SCFTs via discrete gauging -satisfy this constraint is important. Indeed, this statement implies that the fate of the conjecture does not depend on what happens to any N = 4 1-form symmetries in the discrete 0-form symmetry gauging process (i.e., on whether there are mixed 0-form / 1-form 't Hooft anomalies).
In the case of IHS theories (and any discrete gaugings thereof), we have seen that the central charge bounds (5.4) also imply trivial 1-form symmetry. It would be interesting to understand if this statement is more universally true in the space of interacting 4D N = 2 SCFTs.
Finally, let us conclude with a discussion of other possible bounds and their implications for 1-form symmetry. For example, it may well be true that, for sufficiently small a or c (independently of the rank and flavor symmetry of the theory), interacting 4D N = 2 SCFTs do not have 1-form symmetry. Indeed, we know from [69] that for interacting theories c ≥ 11/30 and that this value of the central charge is saturated by the (A 1 , A 2 ) SCFT. We know this AD theory has trivial 1-form symmetry. Therefore, if one can show that (A 1 , A 2 ) is the unique lowest c theory, we have the somewhat trivial result that for sufficiently small c (i.e., c = 11/30), interacting SCFTs have no 1-form symmetry.
On the other hand, we do not expect an absolute a or c bound on 1-form symmetry to be much larger than 11/30. Indeed, su(2) N = 4 SYM has a Z 2 1-form symmetry and has a = c = 3/4. Still, it would be interesting to use a version of the bootstrap including extended operators and prove from first principles that interacting theories with a, c < 3/4 have no 1-form symmetry (or else to find a counterexample).

Conclusions
We have argued that 4D N = 2 SCFTs arising from type IIB string theory on IHSs have 1-form symmetry only if they also have an exactly marginal deformation (see the appendix for the bulk of the proof). We saw some evidence that this behavior extends to certain other classes of N ≥ 2 theories, but there are some potential obstructions for the G(3, 3, 3) N = 3 theory. It would be interesting to understand whether this N = 3 theory has 1-form symmetry or not. If one could indeed prove that it has 1-form symmetry, then one possibility might be that our main claim applies to N = 2 theories subject to the 4D/2D correspondence in [1] and that the N = 3 theory in question does not fall into this class (one consequence of our results here is that this theory would not have a realization as an IHS theory). 33 In section 5 we saw that by focusing on theories with all N = 2 chiral ring generators of dimension less than two we could formulate a conjecture for interacting SCFTs having 32 This statement follows from the fact that Coulomb branch operators in N = 3 theories have integer dimension [68]. Since these operators cannot have dimension one (this would correspond to a free decoupled vector multiplet) or two (N = 3 theories are isolated as N = 2 SCFTs), we have ∆ ≥ 3. 33 Perhaps ideas related to those in [30] will be relevant for demonstrating this claim. In addition, perhaps one can make contact with the discussion in [70].

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no 1-form symmetry that is compatible with all the data we are aware of (including being trivially satisfied by theories for which we are not able to explicitly compute the 1-form symmetry). Moreover, we argued that for IHS theories, this statement could be reformulated in terms of bounds on a and c (it would be interesting to understand if this is true more generally).
In addition to trying to prove (or disprove) the conjecture, there is much work to be done. For example, it would be interesting to explore what happens with less (or no) supersymmetry or to understand if constraints discussed in [20] and [71] are relevant here.

A Theories with at most two different weights
In this section we consider SCFTs with at most two distinct q i and prove claim 2 in section 3. While these theories are also covered by the proof we give in section 2.1 and appendix B, our result here is entirely independent of any IHS classification. Therefore, we consider it worthwhile to exhibit this proof independently. Moreover, we will see that this result has its uses in appendix B.
To that end, without loss of generality, we may take q x ≥ q y ≥ q u ≥ q v . Let us first discuss the case where all q i = q. Then all U i = U and so, from (2.5), we have that r i = 1 for all i and the 1-form symmetry is trivial.
Next let us consider the situation in which there are two different weights. This scenario breaks up into various cases: First, suppose that N ∈ Z. It then follows that U = pU x for some integer p ≥ 1 and so gcd(U, U x ) = U x . As a result, applying (2.5) yields and so the 1-form symmetry is trivial. Let us now consider N ∈ Z. 34 An upper bound on the number of independent constraints on marginal (i.e., Q = 1) terms arises from considering 35 34 In this case, we can have gcd(U, Ux) = 1. Then we would find ry = ru = rv = 1 and rx = Ux > 1. 35 In principle, some of these constraints could be redundant.

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This amounts to at most ten constraints. How many marginal terms can we generate? Let us first recall that any variable appearing in W must be a root or a pointer [29,72]. 36 Suppose first that x is a root so that x Nx ⊂ W . Let us also suppose that x N x y a is marginal for some value of the exponents. Then As a result, a > 1 since otherwise we contradict the statement that q < q x . In fact In all this amounts to at least eleven marginal monomials and at most ten constraints. So, the theory has at least one exactly marginal deformation.
Let us now suppose that there is no a such that the above holds. In this case, we still need marginal terms built out of y, u, and v. Since N x ≥ 2 (otherwise the singularity is not isolated; from the 2D perspective, x would be a supersymmetry-breaking Polonyi field), we must have terms of the form y e u f v g with e + f + g ≥ 3 (since q x > q). This again amounts to at least eleven marginal monomials and at most ten constraints. Therefore, we have a conformal manifold.
Finally, let us consider the case that x is a pointer. Without loss of generality, we may assume x N x y ⊂ W . Then clearly x N x u and x N x v are also marginal. Therefore, we need at least one more marginal term and it cannot depend on x (otherwise, setting x = 0 implies that ∂ u W = ∂ v W = ∂ y W = 0, and the singularity will not be isolated). As a result, we need an x-independent term. Since q x > q, we need again at least ten marginal terms of the form y e u f v g with e + f + g ≥ 3. This gives at least thirteen marginal terms and at most ten constraints. Therefore the theory has a conformal manifold. and so the 1-form symmetry is trivial.
It then follows that U v = pU for some integer p ≥ 1 and so gcd(U , U v ) = U v . As a result, applying (2.5) yields and so the 1-form symmetry is trivial (if N ∈ Z, this statement need not hold). 36 Recall that a variable is a root if it appears as W ⊃ X Nx and is a pointer if W ⊃ X N x Y . In this latter case, we say that X points to Y .

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Let us now consider N ∈ Z. An upper bound on the number of independent constraints on marginal (i.e., Q = 1) terms arises from considering This amounts to at most ten constraints. How many marginal terms can we generate? As in case 1, x must be a root or a pointer. First suppose that x is a root so that x Nx ⊂ W (with N x ≥ 2). If N x > 2, we are done: we get at least ten marginal terms of the form x e y f u g with e + f + g ≥ 3. In addition, we require at least one more term involving v (otherwise the singularity is not isolated). On the other hand, if N x = 2 we are also done: q = 1/2 and lemma 1 guarantees the 1-form symmetry is trivial.
Next, suppose that x is a pointer. If we have x N x y ⊂ W we are again done if N x > 2 by the same logic as above. Similarly, if N x = 1, we are done by lemma 1 (again q = 1/2), since the 1-form symmetry is trivial. The same argument applies if x points to u instead.
The final case to check is that where N x > 1 (otherwise lemma 1 again guarantees trivial 1-form symmetry). We need a v-independent term to add to W since otherwise the singularity is not isolated (setting v = 0 solves ∂ x W = ∂ y W = ∂ u W = 0 without any further constraints). Such a term cannot be consistent with q v < q and marginality.

B Completing the proof of the main claim
The authors of [28] argued for a classification of 4D N = 2 SCFTs realized via type IIB string theory based on the classification of singularities in [27]. In parts B.1 and B.2 of the appendix, we follow this classification and prove our main claim in this set of theories.
In appendix B.3, we give arguments that do not depend on the classification of [27]. In particular, we allow for potentially more general terms in the IHS polynomials than those considered in B.2 (compatible with discussion in [26,29]).
Up to these subtleties which we will address in B.3, there are 19 types of theories we need to consider (see table 3). To define an SCFT, one needs to impose the constraint i q i > 1. For each type of singularity, the solution falls into two classes: some infinite sequences and a finite number of sporadic cases. To check the conjecture, one can consider the sporadic cases one by one since there are only a finite number of them. Indeed, we have checked the conjecture is true for these sporadic cases. Therefore, in the following, we will only explicitly discuss the infinite sequences. 37 The 19 types of singularities can be divided into two broad classes. In the first class, there are only four monomials in the defining polynomials (e.g., as in the case of type I discussed in (2.42)). We discuss these theories in section B.1. In the second class, there are five or more monomials in the defining polynomials (e.g., as in type VIII). We will discuss these theories in section B.2. We consider generalizations of these latter theories in section B.3.

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Throughout this appendix, we will make use of the notion of reduction or of reducing one theory to an other. By this we mean that the weights of the singularities corresponding to the theories in question coincide. As a result, the Poicaré polynomials and deformation spectra coincide as well.

B.1 Singularities with four monomials
In this section we prove our main claim for IHS theories with at most four monomials in W . In the body of the paper we already showed that theories of type I satisfy our claim. Therefore, after briefly recapitulating this case, we move onto the remaining types of theories with four monomials.
Before discussing each singularity one by one, we observe that the type I-VI singularities in table 3 can be regarded as the composition of two subpolynomials in terms of x, y and u, v. More precisely, we can think of the singularities of types I, II, · · · , VI as types XX, XY, XZ, YY, ZY, ZZ where the X, Y, Z data is given in table 4.
With this decomposition, one has W (x, y, u, v) = W (x, y) + W (u, v) where W , W ∈ W X,Y,Z , and Q = Q + Q where Q , Q ∈ Q X,Y,Z (recall that Q is the scaling weight of a monomial deformation built from sub-monomials of scaling weights Q and Q in x, y and u, v respectively). To find the isolated set of SCFTs, we need to find the set where Q = 1 has no solution in the Milnor ring. In particular, this means Q = 1 or Q = 1 also have no solutions in their own Milnor rings.
For type X, the condition for the theory being isolated is just the condition for For type Y, the equation Q = 1 is equivalent to (a − 1)l = b(a − k), which can be further rewritten as Then the resulting condition for the theory to be isolated can be found to be which can be further rewritten as It is then easy to find the condition for the theory to be isolated  [28]. Note that for those types of singularities with five or more monomials, the specific monomials are chosen to make the singularity isolated following [28]. But there are also other choices of these additional monomials, as mentioned in [26,29]. Our proof in B.3 does not rely on the structure of these additional terms. Type I: the type I singularity is given by

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where a, b, c, d ≥ 2 so that there is an isolated singularity at the origin. The weights and the Milnor number are given by The condition i q i > 1 imposes constraints on the possible values of a, b, c, d. The solution was found in [28]. It includes several infinite sequences listed in table 2 and a finite number of sporadic cases. Since the sporadic cases are finite in number, one can check the conjecture explicitly case-by-case. Henceforth, we will not list the solution corresponding to the sporadic cases.
As discussed in the main text, from table 2, we learn that the infinite sequences of Type I singularities are just certain (g, g ) theories and the (p, k) SCFTs considered before. We have already shown how they satisfy our main claim. Therefore, in addition to the checks of the sporadic cases we have performed, the conjecture holds for SCFTs realized via type I singularities.
Type II: the type II singularity is given by where we require a, b, c ≥ 2, d ≥ 1 in order to have an isolated singularity at the origin. The weights and the Milnor numbers are given by A monomial basis of the Milnor ring can be obtained from that of X a,b and Y c,d in table 4. Note that there are actually some overlaps with Type I singularities. If there exits an integer n such that n(c−1) dc = 1, then v n is an exactly marginal deformation and the singularity can be put in the equivalent formW = x a + y b + u c + v n . In this case, the type JHEP12(2021) 024   (a, b, c, 1) (2, 2, c, d) (2, b, 2, d) (3, b, 2, 2)    II singularity reduces to the type I singularity. The reducibility condition is 38 Imposing the condition q i > 1 as well as a ≤ b and c ≤ d to remove redundancy, one gets the infinite sequences in table 5 and a finite number of sporadic cases which will not be listed explicitly. In particular, we have marked all the reducible cases satisfying (B.10) in gray. Note that exchanging (c, d) leaves i q i invariant, but this is not a symmetry of the SCFTs. Therefore, in table 5 we also list another sub-table where the entries of (c, d) are exchanged. In this way, we give a complete list of SCFTs of type II (for the infinite sequences).
The finite sporadic cases can be checked explicitly one by one and indeed our claim holds. So we next check the infinite sequences in table 5. In particular, we only need to consider the irreducible ones there because reducible entries in gray are just the type I we considered already.
We will first need to find the isolated set and then check the 1-form symmetry. In practice, our procedure is as follows: by considering specific types of monomials in the Milnor ring, we can find some necessary condition for the theory being isolated; then for an irreducible weight (namely a weight satisfying (2.2)) we apply our formulas (2.5) and (2.6) to compute the 1-form symmetry. As discussed in the main text, to show the absence of 1-form symmetry, we only need to show that g i = 0 or r i = 1 for each i = 1, 2, 3, 4.
(2, 6, 3, d) -It is easy to see y 4 u has Q = 1, so this theory is never isolated.
(2, b, c, 2) -Consider y k u l . The condition Q = 1 leads to b(c − l) = kc, namely So a necessary condition for being isolated is gcd(b, c) = 1 or b = 2.
If c is odd, c = 2p + 1, the weight vector
Then g i = 0, and the 1-form symmetry is trivial.
The reducibility condition is Imposing the constraint i q i > 1 as well as b ≥ a and d ≥ c to remove redundancy, one gets the infinite sequences in table 6 as well as sporadic cases we do not list but have checked are consistent with our claim. In this case, the monomial basis of the Milnor ring can be obtained from that in X a,b and Z c,d in table 4. A necessary condition for being isolated is given by equations of the form (B.1) and (B.5).
In this case, the weight vector is irreducible and r i = 1, which implies trivial 1-form symmetry.

Type IV: in this case
We require a, c ≥ 2 and b, d ≥ 1 in order to get an isolated singularity at the origin. The weights and Milnor number are given by The reducibility condition is
(a, 1, c, 1) -In this case, the weight vector is irreducible. Then we find r i = 1, so there is no 1-form symmetry.
If a > 2, consider x k u with k = 0, · · · , a − 1 in the Milnor ring. Q = 1 leads to 2a = 3k. It is then easy to figure out a necessary condition for being isolated is gcd(a, 3) = 1.
If a is even, the weight vector is irreducible. Then one finds r 1 = r 3 = 1, g 2 = g 4 = 0 due to gcd(3, a) = 1. So the 1-form symmetry is trivial.
If a is odd, a = 2p + 1, the weight vector becomes and is irreducible. Then we find r 1 = r 2 = r 3 = 1, g 4 = 0. Again, the 1-form symmetry is trivial. Type V: here the singularity is given by the polynomial

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and we need a, b, c ≥ 2, d ≥ 1 to ensure an isolated singularity at the origin. The weights and Milnor number are given by The reducibility condition is  (a, b, c, 1) -From (B.5), we learn that a necessary condition to have an isolated theory is gcd(a − 1, b − 1) = 1. The weight vector is then irreducible, and r i = 1. Therefore the theory has no 1-form symmetry.
Type VI: here and a, b, c, d ≥ 2 in order to have an isolated singularity at the origin. The weights and Milnor number are given by
The weight vector can be written in an irreducible form Then r 1 = r 2 = r 3 = 0, g 4 = 0, implying trivial 1-form symmetry.
In all cases, the weight is irreducible, and we have r 2 = r 3 = 1, g 1 = g 4 = 0, which gives trivial 1-form symmetry.
( 4, 2, c, 4) -In this case x 2 y has Q = 1, and we see this theory is not isolated.
( 3, 3, c, 3) -In this case xy 2 has Q = 1, and we again see this theory is not isolated.
(4, b, 2, 2) -As in the previous case, we still have even b which is a necessary condition for being isolated. But then x 2 y b/2 has Q = 1. So this theory is not isolated.
( 2, 6, c, 3) -A necessary condition for being isolated is gcd(5, c) = 1. The weight is irreducible and r 1 = r 2 = r 3 = 1, g 4 = 0, implying no 1-form symmetry. Therefore, in all cases the 1-form symmetry is trivial. 3, c, 2) -A necessary condition for being isolated is gcd(2, c) = 1. Then the weight is irreducible and r 2 = r 3 = 1, g 1 = 0, g 4 = 0. This gives no 1-form symmetry. 3, c, 2) -The first condition for being isolated is gcd(2, c) = 1. Then the weight is irreducible. Furthermore, consider x 2 yu k . Q = 1 leads to 2c = 5k. So another condition for being isolated is gcd(5, c) = 1. This gives r 2 = r 3 = 1, g 1 = g 4 = 0 and therefore no 1-form symmetry.   Type X: the singularity is given by and we require a ≥ 2 and b, c, d ≥ 2 to get an isolated singularity at the origin. The weights and Milnor number are given by The singularity is reducible to a previous type if Imposing i q i > 1 and b ≤ c ≤ d leads to the infinite sequences in table 11 and the sporadic solutions we have checked separately. Note that (B.40) is cyclically symmetric in b, c, d. In general, there is another inequivalent SCFTs obtained by exchanging any two in b, c, d.
Consider yuv k . Q = 1 leads to d + 1 = 3k. So a necessary condition for being isolated is gcd(3, d + 1) = 1. Then one can show that the weight above is irreducible. As a result g 1 = 0, r 2 = r 3 = r 4 = 1, implying that the 1-form symmetry is trivial.
(4, 2, 2, d) -As in the above (3, 2, 2, d) case, a necessary condition for being isolated is gcd(3, d + 1) = 1. Then one can show that the weight above is irreducible. As a result g i = 0, implying that the 1-form symmetry is trivial.
If d = 2 and ac is even, then the weight above is irreducible, and we have r 1 = r 2 = r 3 = 1, g 4 = 0, which implies trivial 1-form symmetry.
If d = 2 and ac is odd, ac = 2p + 1, then the weight vector becomes and is irreducible. One can compute r i = 1, implying trivial 1-form symmetry.
( a, b, 1, d) - For b = d = 1, the weight vector is irreducible and r i = 1, so there is no 1-form symmetry. For b = 1, d = 2 or b = 2, d = 1, one can also check that the 1-form symmetry is trivial.
We can rewrite the weight vector as (B.64) Then the above weight vector is irreducible. One can show that r i = 1, which implies no 1-form symmetry.
(a, 2, c, 2) - If c = 1, we can check that the 1-form symmetry is always trivial. This can be seen by observing that q y + q u = 1 and using the lemma in section 3.
If c is even, then the weight vector above is irreducible. One finds r 1 = r 2 = r 3 = 1, g 4 = 0, giving no 1-form symmetry.
The x, y part in the form of (B.3) gives a necessary condition for being isolated: gcd(2, b) = 1. Then the weight vector is irreducible and we have r 1 = r 2 = r 3 = 1, g 4 = 0, which implies trivial 1-form symmetry.
( 3, b, 3, 2) - The x, y part in the form of (B.3) gives a necessary condition for being isolated: gcd(2, b) = 1. Then the weight vector is irreducible, and we have r i = 1, meaning trivial 1-form symmetry.
Type XIX: the singularity is defined by The weight vector is (B.73) and the Milnor number is µ = abcd. There are some sporadic cases that we have checked separately, and we include the infinite sequences in table 13.
A monomial basis in the Milnor ring is 1, b, c, d) -The weight vector becomes The Milnor ring is The monomial basis can be chosen as The equation Q = 1 is equivalent to Then it is not difficult to show that the condition for an isolated theory is This guarantees that the weight vector (B.75) is irreducible, and one can check r i = 1, so the 1-form symmetry is trivial.
( 2, 2, c, 3) - In this case, the weight is always irreducible. Moreover, we can compute r i = 1, showing the absence of 1-form symmetry.

B.2 Singularities with more than four monomials
In this section, we consider theories with more than four monomials in W , following the classification of [27]. We will extend our discussion to theories arising in a refinement of this classification in B.3 (although, as we will see, our goal does not require us to make this classification explicit). By analyzing the weights of each type of singularity, we find that if any of a, b, c, d are allowed to equal one, then there are two weights whose sum equals one. Then using the lemma in section 3, we learn that there is no 1-form symmetry. In particular, the case i q i > 3/2, which corresponds to isolated SCFTs according to Claim 3 in section 5, only arises when a specific a, b, c, d equals to 1. As a consequence, Corollary 1 in section 5 holds in these situations.
As a result, we only need to consider cases where all a, b, c, d ≥ 2. For these cases, we have numerically checked that, for all 2 ≤ a, b, c, d ≤ 13 giving an isolated singularity, 39 the corresponding Milnor ring always has a monomial with weight 1. This numerical result motivates us to make the following conjecture: W and a, b, c, d ≥ 2, the corresponding Milnor ring always has a marginal monomial with weight one.

Conjecture. For the isolated singularities in table 3 defined with 5 or more monomials in
In the rest of this appendix, we will prove a slightly weaker statement. Namely, we will show that the conjecture holds in most situations, except for some special values of a, b, c, d. For those exceptional case we show that the singularities reduce to previous types or that the SCFTs have no 1-form symmetry. This statement is enough for us to establish a JHEP12(2021)024 complete proof of our main claim for SCFTs related to all these singularities. We complete our proof for the most general theories in B.3.
We begin with a simple observation: Claim. Consider those singularities involving 5 or more monomials in W . The weights are fixed by a, b, c, d in table 3 and are independent of the remaining parameters. If no entry in the weight vector can be written as a non-negative linear combination of the rest of the entries, namely, ∀i, q i = j =i M ij q j , M ij ∈ N, then the theory has a conformal manifold.
Proof. We need to consider elements of the Milnor ring (i.e., monomials which are nontrivial even when dW = 0 is imposed). We are particularly interested in weight-one constraints since these will affect the exactly marginal elements of the Milnor ring. Such constraints take the following form Since we assume that j =i n j q j /q i = 1, these are the only constraints. 40 Therefore, at weight one there are at most four constraints. However, for these types of singularities, by construction, we have at least five monomials with weight one appearing in W . As a result, there is at least one non-trivial element with weight one in the Milnor ring. This is an exactly marginal deformation, so the theory is not isolated.
The more difficult situation to deal with then is one in which there are integral relations among weights: We denote the number of such distinct relations for each i as N i ≥ 0. Each such relation enables us to generate an additional weight-one constraint on the Milnor ring: 41 As a result, we generate (at most) N x + N y + N u + N v additional constraints at weight one on top of the four constraints, X i ∂ X i W . Therefore, to show that a theory of the type we are considering has a marginal deformation, it is enough to show that we can generate one more distinct weight one monomial for each relation in (B.85). In such a case, we are always left with at least one non-trivial element with weight one in the Milnor ring, which thus gives rise to a marginal deformation.
Type VIII: let us start with type VIII (B.87) 40 If, say, qi/qj = k ∈ N, then we can have a further constraint, q k j ∂x i W , which is trivial in the Milnor ring and has weight 1. 41 Some of these constraints may be redundant.

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We want to show that for a, b, c, d ≥ 2, the theory has a marginal deformation (c, d ≥ 2 or else the lemma of section 3 guarantees the 1-form symmetry is trivial; the singularity cannot be isolated unless a, b ≥ 2). Instead of directly analyzing the Milnor ring, R, of W , we consider the subring R sub = R/ x . Our strategy will be to show that there is a weight one monomial in this subring. Therefore, the Milnor ring R itself will have at least one weight one monomial. Practically speaking, going to the subring can be done by setting x → 0. As a result, (B.87) reduces to To simplify the discussion, we introduce the notion of a monomial vector, which represents each monomial in terms of a vector of entries corresponding to the power of a given variable in the monomial. For example, we can represent the monomials in (B.88) as There are also replacement vectors, which represent each weight relation (B.85) in terms of a vector where m, n, e, f, k, l are non-negative integers subject to q·S = 0. The meaning of (−1, m, n) is that we can replace y → u m v n . Therefore, we can use the V + S combinations to generate new monomials. Each replacement vector enables us to generate one more monomial. To show that there are marginal deformations, it is enough to show that the new monomial vectors generated in this way are all different from each other (and from the five terms appearing in W ).
It turns out we can consider the following special combinations: where * , , # ∈ N, and # > 0, = 1. Comparing all the V,Ṽ above, it is not difficult to see that they are distinct except for the possibilities that

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We assume p, q > 0 (if either p = 0 or q = 0, this theory reduces to type II, which we have dealt with in section B.1). So m, n ≥ 1 and we can replace For b = 2, we have y 2 + yu c + yv d + u p v q . In this case, the weight vector is given by This reduces to a type I theory, and it also has an exactly marginal deformation since the u, v part has gcd(2c, 2d) > 1.
For the caseṼ c =Ṽ d , we have It is easy to find that this equation implies Since q y = q u = q v , claim 2 (proven in appendix A) implies that the one-form symmetry is trivial (it is also easy to check there is an exactly marginal deformation).
Type IX: in this case, we can again consider the subring gotten by setting x → 0. For convenience, we reshuffle the variables by exchanging y ↔ v, b ↔ d, p ↔ q. Then the problem boils down to analyzing the subring corresponding to This is almost identical to (B.88) in the type VIII case except for the first monomial, vy b . The monomial vectors are and the replacement vectors arẽ where * , , # ∈ N, and # > 0, = 1. All the vectors above are distinct except possibly In the case thatṼ b = V pq , we have and this reduces to the previous type VII. Thus we just need to consider b > 2. For b > 2, we can replacẽ It may happen that this still coincides withṼ c ,Ṽ d .
Solving the equation for p, q, one further finds that d = 2s+1 must be odd. This case is reducible to the previous type VII singularity as then we should have d = 2, k = 1, and we can replacẽ This vector is different from all the rest.
In the case thatṼ c =Ṽ d , we are led to c = d = 2. Solving the equation for p, q we find the only solution b = 2, p + q = 3. So we have vy 2 + yu 2 + yv 2 + uv 2 . One can explicitly check that this theory has a weight 1 marginal monomial in the Milnor ring. Finally Therefore this case reduces to the type VII considered before [28].
Type XII: we can consider the subring R/ v and therefore set v → 0. The resulting W is identical to (B.88) in the type VIII case after reshuffling the variables.
Type XIII: the monomial vectors are given by where , #, * ∈ N and = 1, # ≥ 1, * ≥ 0. All the vectors V,Ṽ above are distinct except Obviously this new vector is distinct from all the rest of the vectors above.

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This vector is also different from all the rest except, potentially, for (B.128) which happens when a = b = 2. IfṼ b =Ṽ d andṼ b =Ṽ c , we then haveṼ c =Ṽ d with c = d = 2 which we have already considered.
Therefore, we only need to study the case a = b = 2, which has weight vector The weights coincide with those of the type VIII theory.  where #, * ∈ N and # ≥ 1, * ≥ 0.
One can easily check that for a, b, c, d ≥ 2, all the monomial vectors above are distinct unless a, b, c, d equals to 2. More precisely, the two vectors above coincide if: Therefore for all a, b, c, d ≥ 3, the corresponding SCFTs always have exactly marginal deformations, and our claim on isolated SCFTs and 1-form symmetry trivially holds. We need only check the case when any of a, b, c, d equals to 2.
In the case of a = b = c = d = 2, we have

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The singularity is isolated and one can explicitly check that all the monomials in W are in the Milnor ring.
On the other hand, we expect the Milnor ring has the property that it increases with increasing a, b, c, d (where we define "increasing" to be in the sense of (B.145) below). 42 This is because whenever one increase the value of a, b, c, d, the constraints dW = 0 become weaker due to the higher power of the variables. The only concern is that p, q may not increase with a, b, c, d. However, the Milnor ring should be independent of the specific value of p, q as long as the singularity is isolated. This statement is consistent with the fact that the Milnor number µ is a monotonically increasing function of a, b, c, d. As a consequence, we have We can then immediately show that as long as at least one of a, b, c, d equals to two, there is a weight one monomial in the Milnor ring. For example, when a = 2, we have ux 2 ∈ R (2,2,2,2) ⊂ R (2,b,c,d) , and ux 2 has weight one. Therefore, we justify that even when one of the a, b, c, d equal two, there is still at least a marginal monomial with weight one in the Milnor ring. The corresponding SCFT thus has a conformal manifold. Alternatively, we can also justify our claim without relying on the property (B.145). Indeed, if three of a, b, c, d equal to two, we have explicitly checked the conjecture holds. Then, if two of a, b, c, d equal to 2, by analysing the cases whereṼ 's coincide, one can show that the possibilities are (a, b, 2, 2),(a, 2, c, 2), and (a, 2, 2, d) . For (a, b, 2, 2), one can show the 1-form symmetry of the corresponding SCFT is always trivial. 43 The (a, 2, c, 2) and (a, 2, 2, d) cases are more complicated to analyze directly, but can be discussed by imposing the further requirement that q i > 1 for SCFTs (which we did not impose explicitly before). 44 Finally, we can discuss the case when only one of a, b, c, d equals two. If a = 2, we haveṼ a =Ṽ b = (1, 1, * , * ) = (2, 0, 0, 0) + (−1, 1, * , * ). We can then replaceṼ a → (2, 0, 0, 0) + 2(−1, 1, * , * ) = (0, 2, * , * ). This new vector never coincides with any other vectors. In particular, it is different fromṼ c ,Ṽ d due to our assumption that c, d > 2. If b = 2, andṼ b =Ṽ d , we can replaceṼ b → ( * , 0, * , 2). If b = 2 andṼ b =Ṽ c , we can replacẽ V b → ( * , 0, 2, * ). If c = 2, namelyṼ a =Ṽ c , we can replaceṼ c → (2, * , 0, * ). If d = 2, namelỹ V a =Ṽ d , we can replaceṼ d → (2, * , * , 0). 42 One simple explicit example illustrating this property is when a = b = c = 2. The singularity is isolated and defined by W = uv d + yv d + u 2 y + ux 2 + xy 2 . One can check that the Milnor ring has monomial basis x k y l u m v n with k, l = 0, 1, m = 0, 1, 2, n = 0, · · · , d − 2 or k, l = 0, 1, m = 0, n = d − 1. The Milnor ring and the Milnor number µ = 12d − 8 increase with d. 43 This is because the denominators of all the weights are the same. 44 The requirement i qi > 1 then leads to two infinite sequences, a = 2 or c = 2 (or d = 2), as well as a finite number of cases with a, c ≥ 3 (or a, d ≥ 3). If either of a, c, d equals two, we have at least three twos in (a, b, c, d) which was already considered. For the finite set, we also need to have a solution for p, q in order to get an isolated singularity (within the scheme of [28]). For (a, 2, c, 2) with a, c > 3, we get the unique case (4, 2, 3, 2) which indeed has weight one monomial in the Milnor ring. While for (a, 2, 2, d) wtih a, d > 3, there is no solution. We can also use the extra condition, i qi > 1 for SCFTs, to discuss the case of (a, b, 2, 2) as well as other types of singularities.

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To conclude, we have shown that for a, b, c, d ≥ 2, the theory has marginal deformations and our claim holds. This completes the proof of our main claim for the theories discussed in [28].

B.3 1-form symmetry and more general singularities with five or more monomials
In this appendix, we wish to study the theories covered in B.2 having five or more monomials in W while allowing for more general possibilities regarding the extra monomials rendering the singularities isolated (e.g., see the recent discussion in [26,29]). Therefore, we study these singularities directly from their weight vectors without relying on particular choices for the extra monomials. Since the weights are determined by a, b, c, d, they are independent of the extra monomials. As a result, we neither need nor attempt to perform a classification of the extra terms.
To have a well-defined SCFT, the condition i q i > 1 should be imposed as well, and the resulting solutions were classified in [28] (here we only allow for more general additional monomials rendering the singularity isolated than those considered in [28]). We will use the results there and discuss each set of solutions one-by-one. Note that not all the solutions in [28] give well-defined SCFTs. In particular, we will impose the condition that the corresponding Milnor number is integral and that the corresponding Poincare polynomial indeed truncates to a polynomial (instead of an infinite series). These are all necessary conditions for a well-defined SCFT.
Type VIII: the singularity is defined by the polynomial 45 where the dots represent extra weight 1 monomials to make the singularity isolated. It is easy to show that, as discussed in [26,29], these extra terms must include monomials of the form u p v q or xu p v q (we have explicitly carried out an analysis akin to that conducted in B.2 for u p v q → xu p v q ). 46 Besides similar cursory comments in the type IX case, we will not attempt to systematically classify the possible additional terms. The weight vector and Milnor number are Note that i q i > 3/2 only holds when c = 1 or d = 1 (therefore, the corresponding SCFTs are isolated by our claim 3). 47 If c = 1, we have q y + q u = 1. If d = 1, we have q y + q v = 1. 45 Note that we can impose c ≥ d, because exchanging c ↔ d is a symmetry. 46 To understand this statement, let us denote the ellipses in (B.146) as f (x, y, u, v). Suppose that y, u, v) and that there is no x-dependent and y-independent term in f (x, y, u, v) either, i.e., y, u, v). Then, setting x = y = 0, we solve ∂uW = ∂vW = ∂xW = 0. This means that the final constraint yields 0 = ∂yW |x=y=0 = g(u, v), and the singularity is not isolated. Similar comments apply if all x α u p v q terms appear with α > 1. 47 Note that this is a necessary but not sufficient condition. There may still be no p, q leading to an isolated singularity. On the other hand, i qi > 3/2 does not lead to an empty set of well-defined SCFTs. One example is W = uv 3 + uy + v 6 y + x 2 + y 3 which has isolated singularity and satisfies i qi > 3/2.
In addition to the finite number of sporadic cases which we have checked separately, the infinite sequences solving the condition i q i > 1 are given in table 14.
( 2, b, 2, d) -The Milnor number is If d/(b − 1) ∈ Z then it reduces to previous types. So we just need to consider d/(b − 1) ∈ Z but 2d/(b − 1) ∈ Z in order to have an integral Milnor number. Thus  a, b, 2, 2) -If b = 2, 3, the theory reduces to type I. Since q u = (b − 1)/(2b) = q v , we have 1 3 < q u,v < 1 2 for b > 3. According to lemma 2.8 of [29], we should have another q i = 1 − 2q u = 1/b, q j = 1 − 2q v = 1/b. Therefore a = b, and there is no 1-form symmetry by the discussion in appendix A.     In the first four cases, there is a weight-one term in the Milnor ring. For the last two, the singularity is not well defined.
(3, b, 2, 4) -One finds b = 2, 3, 5, 9, 17. The first four cases have a weight one term, while the last one is not well defined.  49 There is a simpler way to show this by noticing that u 2 y ∈ R (2,3,2,3) ⊂ R (2,b,2,d) . Since u 2 y always has weight 1 for (2, b, 2, d), it is thus an exactly marginal deformation. 50 From now on, we will use the notation ¡ x to indicate that the corresponding singularity is not welldefined due to a lack of truncation in the Poincare polynomial. For the rest of the cases, there is always a weight-one term, but we will not repeat this statement any more.
Our singularity reduces to previous types if
( 2, b, 3, 4) -To be isolated, we must have an extra y p u q term (this statement follows from an argument analogous to the one in footnotes 46 and 48). Using an argument similar to one in the previous subsection after eq. (B.105), we conclude this theory has an exactly marginal deformation.
Type XV: the singularity is defined by the polynomial W = yx a + xy b + xu c + uv d + · · · , a, b ≥ 2 , c, d ≥ 1 . (B.175) The weight vector and Milnor number are  We have that i q i > 3/2 holds only when c = 1 or d = 1. We then have q x + q u = 1 and q u + q v = 1 respectively. If a = 2, the singularity reduces to type XI(2b − 1, 2, c, d). If b = 2, it reduces to XII(2a − 1, c, 2, d).

(B.177)
We can impose c ≥ d, as this is a symmetry. The weight vector and Milnor number are We find that i q i > 3/2 holds only when c = 1 or d = 1. Then, q x + q u = 1 or q x + q v = 1 respectively. If a = 2 the singularity reduces to type XIII. On the other hand, if b = 2 it reduces to type VIV.

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