Expanding the landscape of $\mathcal{N}$=2 rank 1 SCFTs

We refine our previous proposal for systematically classifying 4d rank-1 $\mathcal N=2$ SCFTs by constructing their possible Coulomb branch geometries. Four new recently discussed rank-1 theories, including novel $\mathcal{N}=3$ SCFTs, sit beautifully in our refined classification framework. By arguing for the consistency of their RG flows we can make a strong case for the existence of at least four additional rank-1 SCFTs, nearly doubling the number of known rank-1 SCFTs. The refinement consists of relaxing the assumption that the flavor symmetries of the SCFTs have no discrete factors. This results in an enlarged (but finite) set of possible rank-1 SCFTs. Their existence can be further constrained using consistency of their central charges and RG flows.


Introduction
In a series of three papers [1][2][3] we have outlined a strategy for a systematic classification of 4d N = 2 SCFTs and carried it out for the regular rank-1 case. This is the case where the Coulomb branch (CB) is one complex-dimensional, with parameter u ∈ C. In [2] we argued the case for the existence of up to 11 rank-1 N = 2 SCFTs (see table 1 in [2]) in addition to the 11 already known in the literature [6][7][8][9][10][11].
Although this represents a dramatic enlargement of the set of possible rank-1 SCFTs, we will argue here that four of these theories have already been constructed using string theory or S-class techniques. These are: • Three new rank-1 SCFTs [4] with ∆(u) = ℓ, ℓ ∈ {3, 4, 6} and an abelian flavor symmetry f = u (1). If the single allowed mass deformation is turned off, these theories enjoy an N = 3 enlarged supersymmetry.
The flavor symmetries of these theories do not all match the ones predicted in [2]. The reason for the mismatch is the assumption made in [2] that the discrete symmetry group, Γ, of the CB geometry should be interpreted as the Weyl group of the flavor symmetry: Γ = Weyl(f). By weakening this assumption so that only a subgroup, Γ ′ ⊂ Γ, is the Weyl group of the flavor symmetry Lie algebra, Γ ′ = Weyl(f ′ ), other choices of the flavor symmetries become consistent. As we will explain in more detail below, if there is a complementary subgroup 1 Γ ′′ ⊂ Γ which acts as an outer automorphism of f ′ , then it is consistent to interpret the flavor symmetry of the CB geometry as Γ ′′ ⋉ f ′ instead of f. It turns out that the possibilities for Γ ′′ and f ′ are quite limited, so only a few additional flavor assignments are allowed for each geometry reported in [2]. Consistency of the RG flows among these theories puts additional constraints on the existence of SCFTs with these flavor symmetries. These constraints allow us to rule out certain flavor assignments as inconsistent. Conversely, RG flows from the four newly constructed theories mentioned above allow us to deduce the existence of at least four additional SCFTs and determine their flavor symmetries and central charges.
We partially summarize our results for the RG-flow consistent rank-1 theories in table 1. Theories for which new evidence for their existence is presented are shaded blue in the table. 2 We emphasize, however, that table 1 only lists a fraction (about 2/5) of the total number of possible SCFTs with internally consistent RG flows. We summarize the RG flow constraints on all the possible SCFTs in figures 1-4 in section 5. In particular there are additional theories, including one with F 4 flavor group, which fall in the I * 0 series [2] and will be discussed elsewhere [13].
The RG flow constraints can be organized in terms of three categories of flows, which we call matching, compatible, and unphysical flows. (These flows correspond to green, blue, and red arrows, respectively, in the figures in section 5.) Matching flows are ones under which precisely the subgroup of the UV flavor symmetry which is not broken by the relevant operator intiating the flow is realized as flavor symmetries of the IR SCFTs on the CB. Compatible flows are ones where the IR flavor group is accidentally enlarged, but its rank is not. All other cases are unphysical flows. These latter violate the safely irrelevant conjecture of [1], and, as discussed in [2] and in section 5 below, do not have a consistent field theory interpretation.
We use these flows to label CB geometries together with a flavor symmetry assignment as good, ugly, or bad. (These theories are shown in green, blue, and red boxes, respectively, in the figures in section 5.) Good theories are ones for which there exist matching flows for all relevant operators. All the theories listed in table 1 are good theories; however, there are additional good theories which are not shown there. Ugly theories are ones for which at 1 Γ ′ and Γ ′′ are complementary if Γ ′ Γ ′′ = Γ and Γ ′ ∩ Γ ′′ = {1}. 2 There are two theories reported in table 1 (the II * → {I1, III * } and II * → {I2, IV * Q= √ 2 } theories) which could be identified with a II * N = 3 SCFT. It is not clear whether only one or both should be identified as N = 3 theories. The Q = √ 2 subscript means that BPS states on the CB of the IV * Q= √ 2 have electric and magnetic charges which are multiples of √ 2. The reason we emphasize this is because there is a second possible frozen IV * SCFT, IV * Q=1 , whose BPS states are quantized in units of 1 (i.e., integers); see [1,2]. We should note that the interpretation -discussed at length in [2] -of the frozen I * 1 and I * 0 singularities as weakly gauged rank-0 SCFTs is not considered here. For the I * 1 singularity we focus only on the more conservative interpretation of this singularity as the lagrangian su(2) gauge theory with a single half-hypermultiplet in the spin-3/2 representation. Thus, in particular, the I * 1 series shown in table 1 -i.e., the [II * , , and [IV * , u(1)] theories -are analyzed assuming this interpretation of the theory along each flow. The I * 0 case is more interesting, and will be discussed separately [13]. The paper is organized as follows. Section 2 describes the allowed flavor symmetry identifications of CB geometries. Section 3 analyzes the CB geometries of the new rank-1 theories constructed in [4,5]. In section 4 we present evidence supporting their identification with the geometries shown in table 1 by matching their curve discriminants, RG flows, central charges, and ECB fibers. The discussion of the determination of central charges, which uses a combination of the techniques of [14] combined with constraints coming from S-dualities, RG flows, and the properties of N = 3 theories [15], is presented in more detail in [3]. The results we find for the central charges of the N = 3 theories agree with those found in [16]. Section 5 discusses more broadly the RG flow consistency constraints on all possible flavor symmetry assignments of CB geometries. We then conclude with some final remarks and open questions.

Discrete parts of flavor symmetries
In [2] we classified potential rank-1 Coulomb branch geometries of SCFTs by constructing possible inequivalent regular special Kähler mass deformations of scale-invariant Kodaira singularities. The list of such deformations is given in table 1 of [2], where we also identified the flavor symmetry algebra, f, of the SCFTs associated to each geometry. We determined these flavor symmetries as follows.
A mass-deformed Coulomb branch (CB) geometry is invariant under a discrete symmetry group, Γ, which acts linearly on the r independent linear mass parameters. If we denote the complex masses by m ∈ C r , then the action of γ ∈ Γ is given by an r × r real matrix representation, can be interpreted as the Weyl group of a reductive Lie algebra, F Γ , Then the real representation ρ of Γ on C r is the action of Weyl(F Γ ) on t C ≃ C r , the complexified Cartan subalgebra of F Γ . Since the complex masses transform in the adjoint of the flavor algebra of an N = 2 SCFT, and can be rotated by a flavor transformation to lie in t C , and for generic values break the flavor symmetry as it is natural to identify the flavor symmetry with F Γ , Note that the identification (2.4) may fail to determine the flavor symmetry algebra because the simple Lie algebras of Dynkin type B r and C r share the same Weyl group, so such factors cannot be distinguished. We will therefore often call them BC r factors in what follows. 4 However, it is logically possible that only a subgroup, Γ ′ ⊂ Γ, is the flavor algebra Weyl group. Thus the connected part of the flavor algebra, f conn. , may instead be identified with The rank must stay the same, since it is the number of linearly independent mass parameters, which is fixed. So we are looking for subgroups Γ ′ ⊂ Γ which are also Weyl groups of a rank-r Lie algebra. This requirement may be satisfied, if necessary, by adding u(1) factors in F Γ ′ since Weyl(u(1)) is trivial. The elements of Γ not in Γ ′ generate a group of discrete symmetries of the CB geometry, so should be included as an additional discrete part of the flavor symmetry, f. To be a symmetry of the theory they must act by automorphisms of F Γ ′ , Upon turning on masses, this means that there must be two subgroups Γ ′ , Since Γ acts faithfully on t via ρ, this means that Γ ′′ Γ ′ = Γ (and thus Γ ′ Γ ′′ = Γ). Furthermore, Γ ′′ must a be group of outer automorphisms 4 As discussed in [2], it may happen that the poles of the SW 1-form fill out Weyl orbits in flavor weight space which lie in the root lattice of Br but not of Cr, or vice versa, in which case the 1-form distinguishes between the two Lie algebras. Also, RG flows from one theory to another with an ambiguous BCr flavor group may only be consistent for only one of Br or Cr, again determining the flavor algebra. These mechanisms account for the flavor assignments of three entries in table 1 of [2]: of the connected flavor algebra, Out(F Γ ′ ), since, by construction, its elements are non-trivial maps of the Cartan subalgebra of F Γ ′ to itself, and inner automorphisms that preserve the Cartan subalgebra as a set only act non-trivially on it as Weyl(F Γ ′ ) = Γ ′ . These conditions are quite restrictive since the set of outer automorphisms of reductive Lie algebras is small, namely and Out(g n ) = S n ⋉ Out(g) for g semi-simple, 5 and Out(u(1) n ) = O(n, R). Using some facts about the Weyl groups of simple Lie algebras [17], and the fact that if Γ ′ Γ ′′ = Γ and Γ ′ ∩ Γ ′′ = {1} then |Γ ′ | · |Γ ′′ | = |Γ|, it is not too hard to list the simple f r and reductive f ′ r of rank r and discrete group Γ ′′ ⊂ Out(f ′ r ) such that Weyl(f r ) = Γ ′′ · Weyl(f ′ r ): This gives the list of possible misidentifications of the flavor symmetry that may have been made in [2]: any theory with a flavor symmetry f in the left-most column of (2.9) can be re-interpreted as a theory with flavor symmetry Γ ′′ ⋉ f ′ instead. The first line in (2.9) is the tautological case where any rank-r Weyl group can be reinterpreted as a discrete symmetry acting on a theory with a u(1) r symmetry. The second and third lines are true for all positive r using the Lie algebra identifications In particular, the r = 1, 2, 3 cases 5 We abuse notation here by writing the n-fold direct sum, g ⊕ · · · ⊕ g, multiplicatively as g n .
of the third line in (2.9) are equivalent to Weyl(A 1 ) = Z 2 · Weyl(u(1)), We have focused so far only on the symmetry of the SW curve. But this symmetry extends to the SW 1-form as well. Since the 1-forms constructed in [2] were by design invariant under the full discrete symmetry group Γ, they are a fortiori invariant under any subgroup Γ ′ ⊂ Γ.
The rest of this paper will explore the consequences of these possible misidentifications on the classification of rank-1 SCFTs. In particular, in [2] we made just such a misidentification in the CB geometries of three of the four rank-1 SCFTs constructed in [4,5] as we will now argue.

CB geometries of new SCFTs
We start with the rank-1 SCFT found in [5].
Since ∆(u) = 6, the curve describing the mass deformations of this SCFT must be a deformation of the type II * Kodaira singularity. Among those listed in table 1 of [2], there is only one with a rank-3 flavor symmetry, namely the [II * , BC 3 ] curve: the II * → {I 3 1 , I * 1 } deformation with flavor symmetry algebra identified as BC 3 . As the third entry in (2.10) shows, this can instead be reinterpreted as having flavor symmetry where the Z 2 is the outer automorphism which acts on A 3 by conjugation. So this identification predicts that the SCFT's flavor symmetry has the discrete Z 2 factor shown in (3.1). One can identify this discrete Z 2 flavor factor from the class-S construction of the SCFT in [5]. In that construction, the manifest A 1 ⊕ A 1 ⊂ A 3 flavor symmetry comes from two Z 3twisted punctures each carrying an A 1 global symmetry and opposite Z 3 twists. The Z 2 outer automorphism of Z 3 interchanges these two punctures but leaves the fixture unchanged. This Z 2 thus interchanges the two A 1 factors in the manifest flavor algebra which is compatible with the action of the complex conjugation outer automorphism of A 3 . 6 The [II * , BC 3 ] curve reported in equations (A.25)-(A.27) of [2] is written in terms of the (m 1 , m 2 , m 3 ) linear masses associated with the flavor group BC 3 acts on the m a by independent sign flips and permutations. Abstractly, Weyl( on the masses is usually written using a basis of four masses, (µ 1 , µ 2 , µ 3 , µ 4 ), satisfying the relation i µ i = 0, corresponding to the eigenvalues of an SU (4) matrix. Then Weyl(A 3 ) acts by permutations of the µ i . An explicit relation between the BC 3 basis m a and the A 3 basis µ i is given by whose inverse is m 3 = 1 2 (µ 1 +µ 2 ) and cyclic permutations of the 123 indices. Thus we propose that the curve and one-form of the A 3 SCFT of [5] is the same as that of the BC 3 geometry described in Appendix A.1.4 of [2] but with the m a everywhere substituted with the µ i via (3.2).
For later use, it will be useful to identify the action of the Z 2 outer automorphism of A 3 in terms of both the A 3 and the BC 3 mass bases. A standard basis of simple roots of A 3 is one in which the (α 1 )−(α 2 )−(α 3 ) Dynkin diagram has with the e i a basis of (R 4 ) * ⊃ t * dual to a basis of an R 4 in which t is embedded as the subspace annihilated by i e i . Thus denoting the A 3 masses as µ ∈ C 4 ⊃ t C , we have The Z 2 outer automorphism of A 3 is generated by an element, o, which exchanges α 1 ↔ α 3 and leaves α 2 invariant. This therefore acts on the A 3 masses as In terms of the BC 3 masses, this is the Z 2 generated by as follows from (3.2).
3.2 The f = u(1) N = 3 theories with ∆(u) = 6, 4, 3 In [4], García-Etxebarria and Regalado propose a novel F-theory construction of 4d field theories preserving N = 3 supersymmetry. These are necessarily isolated superconformal field theories with U (3) R-symmetry group, Coulomb branch vevs of dimension ∆(u) = ℓ, ℓ ∈ {3, 4, 6}, and central charges a = c = (2ℓ − 1)/4 [15,16]. Furthermore, they also have no N = 3 relevant deformations, and so, in particular, no continuous flavor symmetry. The authors of [4] propose 3 series of such N = 3 theories, labelled by ℓ ∈ {3, 4, 6} and L ∈ Z + . These theories have moduli spaces M ℓ,L = (C 3 /Z ℓ ) L /S L , describing vevs of operators with scaling dimensions nℓ, n ∈ {1, . . . , L}. In fact, [4] discusses evidence for multiple inequivalent versions of each of these theories, with the same low energy description, but presumably differing by some non-perturbative analog of discrete gauge factors [16]. Here the "rank", L, of these theories (the number of D3 brane probes of the F-theory geometry) controls the dimension of the moduli space. We will concentrate on the L = 1 case, since those will have rank-1 Coulomb branches when reinterpreted as N = 2 theories. We will call these the ℓ = 3, 4, 6 rank-1 N = 3 theories. Now let us reinterpret these theories as Higgs branch over the origin, and H 1 is a 1-quaternionic-dimensional hyperkähler fiber of a mixed branch over the generic points of the Coulomb branch. Thus, as N = 2 theories, the N = 3 ℓ = 3, 4, 6 theories have 1-dimensional Coulomb branches with parameter u of dimension ∆(u) = ℓ, and all have a U (1) F flavor symmetry, and thus a single mass deformation. (Since ∆(u) > 2 there are no relevant or marginal chiral deformations.) In the table 1 of [2] there is only one deformation of the III * , and IV * singularities with a single mass parameter (i.e., rank-1 flavor algebra) while there are two possibilities in the II * case. The compatible identifications are: The ℓ = 3 geometry's flavor symmetry matches the u(1) predicted for the N = 3 theory, As the first entry in (2.10) shows, the ℓ = 6 and ℓ = 4 geometries can instead be reinterpreted as having flavor symmetry where the Z 2 is the outer automorphism which acts on u(1) by reversing the sign of charges, and so as m → −m on the complex mass. So, if this identification with the N = 3 SCFTs is correct, we predict that the flavor groups of the ℓ = 6 and 4 theories have the extra discrete Z 2 factor shown in (3.9). This holds true for both identifications of the ℓ = 6 N = 3 theory presented in 3.7 which also have the same values for the a and c central charges (see 1). The only way to distinguish these two inequivalent identifications from their low energy data is through their RG flows (as explained in more detail in section 4.2). This discrete Z 2 flavor factor can in fact be seen in the F-theory construction of the SCFTs in [4]. In that construction, the moduli space of the rank-1 theories is (3.10) (z 1 , z 2 , z 3 ) transform as a triplet under the N = 3 U (3) R-symmetry. Take the z 2 = z 3 = 0 subspace to be the N = 2 CB, so that u := z ℓ 1 is its Z ℓ -invariant coordinate. Then (z 2 , z 3 ) transform as a doublet under the N = 2 SU (2) R symmetry, and the Higgs branch is C 2 /Z ℓ with C 2 the z 2,3 -plane. In terms of Z ℓ -invariant holomorphic coordinates it is given by Similarly, the fiber of the mixed branch over a generic point (e.g., z ℓ 1 = 1) of the CB is simply a copy of the z 2,3 -plane ≃ C 2 . Since the Higgs branch operators are neutral under U (1) R , it follows that the U (1) R is the subgroup of the U (3) which leaves z 2 and z 3 invariant, so acts by phase rotations of z 1 only.
For ℓ even, since p interchanges V and W which have opposite U (1) F charges, it is plausible that the Z 2 acts as the outer automorphism on U (1) F flipping the sign of the linear mass parameter m → −m. The latter action also realizes the sign flip on the X as it can be seen from the form of the mass term ∼ mX which appears in the lagrangian. 8 The situation is different for ℓ odd. The discrete isometry p in (3.13) cannot be interpreted as a flavor symmetry since p 2 = −I on (V, W ) which is the action of the center of SU (2) R . So the p action on X must instead be interpreted as a composition of an SU (2) R transformation, r : X → −X, with CPT conjugation, c : X → X. This realizes X → −X as a non-flavor symmetry action. As far as X goes the two actions are indistinguishable, yet they have different actions on (W, V ). In particular: (3.14) The transformations above follow since V and W transform as the highest-and lowest-weight components of a spin-ℓ/2 representation of SU (2) R , while X is the highest-weight component of a spin-1 representation (i.e., the moment map for the Higgs branch). The composition of r and c reproduces (3.13). Thus for ℓ odd we are led to the following identification: Since r is in SU (2) R it does not act on the mass parameters. It follows that p acts on the mass parameters as m → m from the charge conjugation, and thus should not be identified as the U (1) F outer automorphism m → −m. This non-holomorphic action on the masses is not visible in the CB geometry, and so for ℓ = 3 we expect the flavor symmetry visible in the deformed CB geometry is simply f 3 = u(1), as in (3.8).

Some checks
There is further evidence supporting the identifications discussed above. This evidence comes from: comparing the discriminant locus of the curve in [5] with ours; revisiting the RG flow consistency conditions (for details see [1,2]) in light of the new flavor groups; and central charge computations using the technique of [14] as described in [3].

Curve discriminants
Let us start by comparing the curve in [5] with the one we constructed in [2] but with the newly identified flavor group. In particular we can explicitly check that once we turn on the same mass deformations the locations of the singularities for the two curves coincide.
When the A 1 ⊕ A 1 ⊂ A 3 mass deformations are turned on, the curve for the [II * , A 3 ⋊ Z 2 ] CB is given in [5] by Here z is a coordinate on the Riemann sphere, x is a coordinate on the cotangent space to the sphere, {z 1 , z 2 , z 3 } are the (arbitrary) locations of the three punctures,  .1) is singular for values of u solving F = ∂F/∂x = ∂F/∂z = 0. Locating the punctures at z 1 = 1, z 2 = −1, and z 3 = 0 for convenience, we find that the curve has three singular loci in the u plane located at the zeros of the polynomial 9 We now repeat this analysis for the curve found in [2]. This is a straightforward procedure once we take care of two subtleties. First, the curve reported in the appendix of [2] is written in terms of the linear masses (m 1 , m 2 , m 3 ) associated with the wrong (BC 3 ) flavor group. The A 3 form of the curve is given by using (3.2) to rewrite the curve in terms of A 3 linear masses (µ 1 , µ 2 , µ 3 , µ 4 ) satisfying i µ i = 0.
The second subtlety is to identify the directions in the (µ 1 , µ 2 , µ 3 , µ 4 ) mass deformation space corresponding to turning on only the M ± mass parameters in (4.1). The manifest A 1 ⊕ A 1 ⊂ A 3 flavor symmetry of the curve (4.1) has two quadratic mass Casimirs, M 2 ± , while the full A 3 symmetry has three independent Casimirs which we can take to be N a = 4 i=1 µ a i , for a = 2, 3, 4. So if only A 1 ⊕ A 1 masses are turned on, only N 2 and N 4 can be non-zero, and we must have Take the solution so that, with respect to the basis of simple roots of A 3 in (3.4), the A 3 outer automorphism (3.5) acts non-trivially on the chosen A 1 ⊕ A 1 subgroup. 10 Then, writing the SW curve of appendix A.1.4 of [2] in terms of the µ i using (3.2) and substituting for µ 1 using (4.5), the resulting curve becomes singular at the zeros of the discriminant The discriminants, D and D ′ , of the two curves clearly agree after identifying their linear mass parameters as M + = ( 6 √ 2/ √ 3) µ 2 and M − = ( 6 √ 2/ √ 3) µ 3 . As mentioned above the Z 2 factor of the flavor symmetry should be identified with the interchange of the two A 1 factors. From the explicit expression of M ± in terms of the µ 2,3 we see that this action is in fact compatible with the action of the outer automorphism of the full A 3 as identified in (3.5).
We cannot perform a similar discriminant check for the ℓ = 3, 4, 6 N = 3 theories because it is not clear how to modify the string construction in [4] to turn on the N = 2 u(1) mass deformation.

RG flows
In [2] we claimed that the [II * , BC 3 ] theory did not pass the RG flow condition if the frozen I * 1 singularity was interpreted as a lagrangian field theory. The RG flow test depends on the identification of the global flavor group. Thus we should redo the analysis of minimal adjoint flavor breaking RG flows for the [II * , A 3 ⋊Z 2 ] theory. A 3 has two inequivalent minimal adjoint breakings, one from turning on a vev for either node at the end of the Dynkin diagram, and one for turning on a vev for the middle node. Keeping track of the discrete Z 2 factor as well, it is easy to see that these give rise to the following flavor breakings, For each flavor breaking we have also recorded deformation pattern of the parent II * singularity which results from putting in the specific breaking masses in the [II * , A 3 ⋊ Z 2 ] SW curve described in section 3.1.
In the first line of (4.7) the I * 1 is frozen while the I 3 must be interpreted as a u(1) theory with three charge one hypermultiplets providing a u(3) ≡ A 2 ⊕ u(1) flavor symmetry. In the second line, the I 1 provides a u(1) flavor factor while the I * 3 should be interpreted as an su(2) w/ 4 · 2 + 1 · 4 lagrangian theory with so(4) ≃ A 1 ⊕ A 1 flavor symmetry and charge normalization a = 1. (For details on these identifications see [1,2].) Since these IR singularities precisely reproduce the expected unbroken flavor symmetries, we conclude that the [II * , A 3 ⋊ Z 2 ] theory passes the RG flow consistency condition.
Once the existence of the [II * , A 3 ⋊ Z 2 ] theory is accepted, any other SCFTs it flows to must also be consistent. We will now check that that is the case.
In [2] we found that one of the [II * , BC 3 ] minimal adjoint breakings generates the deformation II * → {III * , I 1 }. This direction is no longer a minimal adjoint breaking in the A 3 ⋊ Z 2 interpretation of the theory but instead corresponds to setting µ 1 = µ 2 = 0 and µ 3 = −µ 4 in the A 3 linear masses defined in (3.2). Along this direction we expect an unbroken A 1 ⊕ u(1) 2 flavor group. Because the I 1 only contributes a u(1) factor, the remaining part should be identified as the flavor group of the CFT at the III * singularity.
In table 1 of [2] the only deformation of a III * singularity with a rank 2 flavor group is the [III * , A 1 ⊕ A 1 ] curve (which also failed the RG flow test for a lagrangian interpretation of the I * 1 singularity). But our table (2.9) of possible flavor misidentifications allows for this curve to be interpreted instead as the curve of a [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory. Notice that the Z 2 factor of the initial II * singularity is broken along this RG flow direction, thus the Z 2 factor of the III * is a new one.
We should now redo the RG flow analysis for the newly identified flavor group: In the first case the I * 1 is frozen while the I 2 provides the non-abelian, A 1 , component of the flavor group. The second case now also passes the RG flow test since the I 1 provides one u(1) factor while the I * 2 must be interpreted as an su(2) w/ 2 · 2 + 1 · 4 gauge theory with so(2) ≃ u(1) flavor group with charge normalization a = 1. We thus conclude that the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] passes the RG flow condition as well.
Next, we can study the flow from [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] to the [IV * , u(1)] theory, which, as we argued earlier, can be identified with the ℓ = 3 N = 3 theory. The right RG flow direction was already identified in [2] when we studied the non-adjoint breaking We can analyze in a similar manner the RG flows involving the other two (ℓ = 4, 6) N = 3 theories. As discussed above, there are two theories which are compatible with the properties of II * N = 3 theories (3.7). Both are [II * , u(1) ⋊ Z 2 ] theories with a single mass parameter. Turning such mass deformation on splits the II * singularity as II * → {I 1 , III * } and II * → {I 2 , IV * √ 2 } respectively. It was argued in [1,2] that both the III * and the IV * theory. In section 5, however, we will see by examining the full space of RG flows that the [II * , A 2 ⋊ Z 2 ] is a "good" theory while the [II * , u(1) 2 ⋊ (Z 2 ⋉ S 3 )] theory is "ugly" (it requires accidental flavor symmetry enhancements in the IR). For this reason we show only the [II * , A 2 ⋊Z 2 ] theory in table 1, and will discuss only its central charges and Higgs branches below.
We emphasize that the [II * , u(1) 2 ⋊ (Z 2 ⋉ S 3 )] theory is not logically excluded: we have excluded it only to keep our discussion relatively short. We will discuss consistency of these flows as well as of flows among rank-1 SCFTs with all possible flavor symmetry assignments in section 5.

Central charges
We will summarize here how the a, c, and k central charges of rank-1 N = 2 SCFTs can be computed from a generalization of the argument developed by Shapere and Tachikawa in [14]. The a and c central charges of the 4d conformal algebra are certain coefficients in OPEs of energy-momentum tensors, and the k central charges appear in the OPEs of flavor currents. We use the standard normalizations of the central charges where for n v free vector multiplets and n h free hypermultiplets transforming under a nonabelian global symmetry f, 24a = 5n v + n h , 12c = 2n v + n h , and k = T (2n h ). Here 2n h is the representation of f under which the half-hypermultiplets transform. The quadratic index is defined as T (r) := [rank(f)] −1 λ∈r (λ, λ), where the weights are normalized so that the long roots of f have length-squared 2. In this normalization T (n) = 1 for su(n).
We obtain the a, c, and k central charges of each entry of table 1 as a function of a few parameters involving mostly data from the deformation pattern singularities. These can be used both as checks for the correctness of the identifications made in sections 3.1 and 3.2, and also to deduce more information about the various SCFTs in table 1 and their RG flows.
The a and c central charges The following formulas for a and c are derived in [3]: Here ∆ = ∆(u) is the scaling dimension of the CB parameter, and h 1 is the quaternionic dimension of the Higgs fiber of the "enhanced Coulomb branch" (ECB) of the SCFT. Z and N i refer to properties of the generic mass deformation of the SCFT. In particular, Z counts the number of undeformable Kodaira singularities the initial singularity of the SCFT splits into upon turning on a generic relevant deformation, and N i is the central charge contribution of the conformal or IR-free theory corrsponding to the ith such singularity. It is given by [3] where c i , h i and ∆ i are respectively the c central charge, the quaternionic ECB dimension, and the CB scaling dimension of the SCFT or IR free field theory corresponding to the ith Kodaira singularity in the deformation pattern. When these undeformable singularities have a lagrangian interpretation, N i is easily computable. For undeformable I n singularities N In = 1 while for a frozen I * 1 singularity, N I * 1 = 3; see [3] for the details. Since ECBs might not be familiar, we pause to summarize their main properties; the structure of ECBs is discussed in more detail in [3]. "Enhanced Coulomb branch" is our name for a mixed Higgs-Coulomb branch that occurs over the whole CB; thus the CB proper is a sub-variety of the ECB, and the ECB is in effect an enlarged Coulomb branch. The ECB locally has a direct product geometry U i × H 1 where {U i } is an open covering of the regular points of the CB, and H 1 is a hyperkähler space. h 1 is the quaternionic dimension of H 1 , so the total complex dimension of the ECB is 2h 1 + 1 (since we are discussing here only theories with rank-1 CBs). Over a generic point on the CB, the 2h 1 complex scalars whose vevs parameterize the ECB fiber are neutral under the low energy electromagnetic u(1) gauge group, so the ECB fiber over a regular CB point is a flat hyperkähler space, H 1 = H h 1 . The moduli spaces of N = 4 theories as well as of the N = 3 SCFTs described in section 3.2 are examples of ECBs. But ECBs commonly occur in N = 2 field theories as well. Even when there is an ECB, there can be additional mixed and Higgs branches. In the case of a SCFT with rank-1 CB, the only possibility for an additional branch is a Higgs branch, H 0 , which is a hyperkähler cone with tip touching the CB at its singular point (the "origin"). It is a logical possibility that H 0 might have multiple components and that the intersection of H 0 with the H 1 fiber of the ECB over the origin might be any hyperkähler cone from the empty one (the origin istelf) to all of H 1 .
The a and c central charges for this theory were computed in [5] to be 24a = 75 and 12c = 42.
Plugging into (4.10) using its deformation pattern II * → {I 1 3 , I * 1 } and that ∆(u) = 6, one finds that h 1 = 4. Thus we make a prediction that the [II * , su(4) ⋊ Z 2 ] has a 4 quaternionic dimensional ECB fiber. It is worth noting that the fact that h 1 comes out as an integer is a non-trivial check of the corectness of our identification. A sharper check will be found when we compute the flavor central charge, k, below. It would also be interesting to determine the value for h 1 independently from the superconformal index of this theory, or by embedding this theory in a web of S-dualities.
We proposed that these curves are identified with the ℓ = 6 N = 3 theory. As explained in [15], N = 3 supersymmetry requires a = c. This, together with ∆(u) = 6 and (4.10) determine a = c = 11/4. Furthermore, as reviewed in section 3.2, it also implies that this theory has a one-quaternionic-dimensional ECB fiber, thus h 1 = 1. This theory is identified with the ℓ = 4 N = 3 theory. As in the previous two cases we get from N = 3 supersymmetry the central charges and ECB fiber dimension shown above. Unlike the previous two cases, however, this theory's deformation pattern, III * → {I 1 , I * 1 }, is to IR free lagrangian theories, and so one can independently compute c from (4.10) to obtain the same answer. This is a strong indication that this theory should be identified as the ℓ = 3 N = 3 theory constructed in [4].
Finally, we note that the a = c central charges of the N = 3 theories found here agree with those found in [16], who also find further evidence in support of those values coming from the structure of the chiral algebras associated to the Schur operators of those theories [18].

Flavor central charges
As explained in [14], the flavor central charges for u(1) factors of flavor groups are difficult to determine because of the possibility of them mixing under RG flows with the low energy global electric and magnetic u(1)'s on the CB. So we restrict ourselves to computing the flavor central charges, k, for nonabelian factors of the flavor symmetry. Also, we can no longer use the strategy of turning on a generic mass deformation to compute k since under such a deformation the low energy flavor group is entirely broken to u(1) factors. Thus we must instead use special (e.g., minimal adjoint breaking) mass deformations which leave some nonabelian subgroup of the SCFT flavor symmetry unbroken.
Let's say that under one such special mass deformation, m, our [K, f] SCFT (with K the Kodaira type and f the flavor symmetry) deforms to Y singularities as Consequently the flavor symmetry breaks to f Ignoring any u(1) factors in this breaking, put the (topologically twisted) theory in a background of n i instantons for each (nonabelian) f i . This corresponds to a total n-instanton background for the original f flavor symmetry where n = Y i=1 n i d i , and the d i are the Dynkin indices of embedding f i ֒→ f. Then, as long as one knows the flavor central charges, k i , for the [K i , f i ] theories (e.g., if they are lagrangian theories) one deduces from the arguments of [14] that [3] for all i such that f i is nonabelian. We now apply (4.13) to the [II * , A 3 ⋊ Z 2 ] theory. The minimal adjoint breaking A 3 → A 2 ⊕ u(1) mass deformation, m 1 , deforms the singularity as (4.14) Since the non-abelian flavor factor appearing in the second (I 3 ) singularity is A 2 ⊂ A 3 with index of embedding 1, we set d 2 = 1 in (4.13). The I 3 singularity is an IR free u(1) gauge theory with 3 massless charge-1 hypermultiplets transforming in the 3 of the A 2 flavor symmetry. They thus contribute k 2 = T (3 ⊕ 3) = 2 to the A 2 flavor central charge of the I 3 theory. Also, the CB parameter of an IR free u(1) gauge theory gives ∆ 2 = 1. Thus (4.13) gives us that k = 6[2 − T (2h 1 )] + T (2h 1 ). Now, we have seen from matching to the a and c central charges from [5] that h 1 = 4, corresponding to 2h 1 = 8 complex scalars (the "half-hypermultiplets"). 8 free halfhypermultiplets can only transform in the 2h 1 = 8 · 1 (giving T (2h 1 ) = 0) or 2h 1 = 4 ⊕ 4 (giving T (2h 1 ) = 2) representations of an A 3 flavor group [19]. In the first case, since the ECB fibers are flavor singlets, they are not lifted under the flavor breaking, but, as singlets, they do not contribute to the index. In the second case, under the adjoint flavor breaking A 3 → A 2 ⊕u(1) all the ECB fibers are lifted, so h (2) 1 = 0. So in either case we find T (2h (2) 1 ) = 0 and thus we find from (4.13) that either k = 12 or k = 14. The second is the value found in [5] from the S class construction, and we learn that the ECB fiber transforms in the 4 ⊕ 4 of the flavor symmetry.
These conclusions also follow from turning on other adjoint breakings of the flavor symmetry. For example, the minimal adjoint breaking A 3 → A 1 2 ⊕ u(1) mass deformation, m 2 , deforms the singularity as Since the non-abelian factor appearing in the first ( index of embedding 1, we set d 1 = 1 in (4.13). The I * 3 singularity is the IR free gauge theory su(2) w/ 4 · 2 ⊕ 1 · 4 massless half-hypermultiplets. The four doublet half-hypermultiplets transform in the 4 of the D 2 flavor symmetry. They thus contribute k 1 = 2 · T (4) = 4 to the D 2 flavor central charge of the I * 3 theory. The ECB fibers are either lifted or are flavor singlets. Thus (4.13) again gives us that k = 12 + T (2h 1 ) (as it must).
The ℓ = 6, 4, 3 N = 3 SCFTs all have abelian flavor symmetries, so their central charges cannot be computed by (4.13). Note that, since the u(1) flavor symmetry of these theories is part of the N = 3 U (3) R-symmetry, its central charge is proportional to the a = c central charge. (The coefficient of proportionality depends on an arbitrary normalization of the u(1) flavor generator.)

ECB fibers
The central charge matching performed above showed that the ECB fiber of the [II * , A 3 ⋊ Z 2 ] SCFT has complex dimension 2h 1 = 8 which transform as 4⊕4 under the A 3 flavor symmetry. Also, we saw that the ℓ = 6, 4, 3 N = 3 theories of each have ECB fiber of complex dimension 2 transforming as (+1) ⊕ (−1) under the u(1) flavor symmetry. As we now explain, through RG flows we can compute the ECB fiber dimensions of the remaining 4 blue-shaded theories in table 1.
Consider the [II * , A 3 ⋊ Z 2 ] theory. In section 4.2 we found that the A 3 mass deformation µ 1 = µ 2 = 0 and µ 3 = −µ 4 is the one which flows to the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory. Since the half-hypers of the ECB fiber of the UV theory transform as 4 ⊕ 4 under the A 3 flavor symmetry, upon turning on this adjoint A 3 mass two of the four hypermultiplet directions are lifted, leaving unlifted half-hypers in the 2 +q ⊕ 2 −q of the IR A 1 ⊕ (u(1) ⋊ Z 2 ) symmetry. The non-adjoint breaking in (4.9) flows to the [IV * , u(1)] theory which, since it is an N = 3 theory, has h 1 = 1. Under this breaking the 4 half-hypermultiplets of the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory will receive masses ±m 1 ± qm 2 ∝ ±(i ± q √ 3)m 2 , where q is the u(1) flavor charge of the half-hypermultiplets. Thus, in order for an ECB fiber hypermultiplet of the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory not to be lifted by this breaking, we must have that q = i/ √ 3 relative to the (arbitrary) normalization of the u(1) flavor factor chosen by the normalization of the m 1 and m 2 masses appearing in the SW curve constructed in [2]. (The phase, i, in the charge is also arbitrary, since the masses are in the complexified Cartan subalgebra of the flavor symmetry.) In any case, we learn that the u(1) flavor charges of the ECB hypermultiplets of the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory are non-vanishing.
The minimal adjoint breaking in the top line of (4.8) leaves the nonabelian A 1 flavor factor unbroken while flowing to lagrangian IR theories. This breaking can thus be used as in the flavor central charge discussion of section 4.3 to compute the A 1 flavor factor central charge, k, of the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory. Since this breaking only turns on a mass for the u(1) flavor factor, and since we have just learned that the ECB hypermultiplets are charged under this u(1), it follows that they will be lifted by this flow. This means that h (i) 1 = 0 in (4.13), giving k = 10.
We have, in this way, determined the central charges and ECB fiber dimensions of the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory shown in table 1. Perhaps the information on the (pure) Higgs branch structure of the [II * , A 3 ⋊ Z 2 ] theory computed in [5] together with its flow to the [III * , A 1 ⊕ (u(1) ⋊ Z 2 )] theory can be used to also determine the latter's Higgs branch, and, in particular, its quaternionic dimension h 0 .
Next consider the [II * , A 2 ⋊Z 2 ] theory, which the analysis of section 4.2 showed might flow to the [III * , u(1) ⋊ Z 2 ] theory. If we describe the mass deformations of this theory in terms of A 2 masses, {m 1 , m 2 , m 3 } with i m i = 0, then the flow to the [III * , u(1) ⋊ Z 2 ] theory is in the m 1 = 0 direction. Thus if the [II * , A 2 ⋊ Z 2 ] has an ECB fiber transforming in the 3 ⊕ 3 of A 2 , one of its hypermultiplets will not be lifted in this breaking, implying (correctly) that the [III * , u(1) ⋊ Z 2 ] has h 1 = 1 ECB hypermultiplet transforming as (+1) ⊕ (−1) under the unbroken u(1). It is not hard to see that no other assignment of A 2 transformation properties of the [II * , A 2 ⋊ Z 2 ] ECB fiber gives the correct result.
We can use this flow to determine the a and c central charges of the [II * , A 2 ⋊ Z 2 ] theory. Since the singularity splits as II * → {I 1 , III * } which contribute N (I 1 ) = 1 and N (III * ) = 9/2 (since the III * singularity must be identified with the [III * , u(1) ⋊ Z 2 ] theory), (4.10) gives If we instead consider the flow where we turn on A 2 masses m 1 = m 2 , breaking A 2 → A 1 ⊕ u(1), the [II * , A 2 ⋊ Z 2 ] singularity splits as II * → {I 2 , IV * Q=1 }. This adjoint breaking lifts the ECB fiber so contributes h (i) 1 = 0 and d i = 1 on the right side of (4.13). The I 2 is the IR free u(1) gauge theory with two charge-1 hypermultiplets, so is the one carrying the unbroken non-abelian A 1 flavor factor, and contributes ∆ i = 1 and k i = 2 to the right side of (4.13). We thus learn that the flavor central charge of the [II * , A 2 ⋊ Z 2 ] is k = 14.
Finally, consider the flows from the three possible ℓ = 4, 6 N = 3 theories to the frozen [III * , ∅], [IV * q= √ 2 , ∅], and [IV * q=1 , ∅] SCFTs. In each case the u(1) mass lifts the ECB fiber and Higgs branch of the N = 3 theory, and so we conclude that h 0 = h 1 = 0 for the frozen SCFTs. (Note that the mechanism described in [20] where Higgs branches of SCFTs at the IR end of RG flows are lifted all along the flow does not apply here: because the dimension of the CBs of both the UV and IR SCFTs is the same, there can be no irrelevant gauging of flavor symmetries.)

RG flow constraints for all flavor assignments
In section 2 we pointed out that a given SW curve with discrete symmetry group Γ is compatible with multiple choices of the flavor group; see (2.9) and (2.10). In this section we systematically analyze each of these possibilities and discuss which alternative interpretations of the flavor symmetry algebras are allowed. The main constraint comes from a careful analysis of RG flows and the pattern of factorizations of the curve discriminant.
Turning on masses breaks the flavor symmetry f of the original SCFT. Since the masses appear as vevs of vector multiplets upon weakly gauging f, they can be thought of as linear coordinates on t C , the complexified Cartan subalgebra of f. Thus the subalgebra of f which leaves invariant a given mass deformation is a symmetry of the IR theory. We call this the expected IR flavor symmetry.
The IR flavor symmetry also manifests itself in the flavor symmetries of the massless degrees of freedom associated to the singularities on the CB. Mass deformations which leave non-abelian factors of the flavor symmetry unbroken do not fully split the initial singularity. This is reflected in the occurence of higher-order zeros of the discriminant and correspond to non-frozen conformal or IR-free theories which themselves have unbroken flavor symmetries. All of these factors will be part of the flavor symmetry in the IR. We call this the curve flavor symmetry which need not be the same as the expected IR flavor symmetry.
We can then distinguish three types of RG flows: matching flows are those for which the curve and expected symmetries match; compatible flows are those where the expected flavor symmetry is a subalgebra of the curve flavor symmetry of the same rank; and unphysical flows are the remainder, i.e., flows for which the curve flavor symmetry either does not contain or is of larger rank than the expected symmetry. As was argued in [1,2], unphysical flows are indeed unphysical; we will give examples below. Compatible flows require an accidental enlargement of the flavor symmetry in the IR, while matching ones do not.
This classification of flows gives rise to a classification of the (possible) SCFTs corresponding to the original (UV) singularity from which the flows originate: good theories are ones for which all flows are matching; ugly theories are ones for which at least one flow is compatible and none are unphysical; and bad theories have at least one unphysical flow. We have no rational reason to exclude ugly theories, only prejudice.
It is a daunting task to algebraically locate all flows which do not fully split a singularity, and classify them as matching, compatible, or unphysical depending on the possible flavor symmetry assignments of the UV and IR singularities. We are not able to fully perform this task, but instead have examined all minimal adjoint breaking flows and all flows for theories with just two relevant deformations as in [2]. The results are most easily summarized graphically as a web of RG flows among the possible SCFTs which connect different interpretations of the various SW curves. These are shown in figures 1, 2, 3 and 4, where green, blue, and red arrows denote matching, compatible, and unphysical flows, respectively. Similarly, theories with a green, blue, or red background are good, ugly, or bad, respectively. Some compatible flows are not shown in the figures (because they would make them too hard to read) but are explained in the figure captions.
We now illustrate these considerations with a few examples. First, consider the [II * , C 5 ] theory in the I 4 series, shown in figure 1. The arrows from the [II * , C 5 ] theory represent the minimal adjoint breaking where the adjoint C 5 mass breaks C 5 → C 3 ⊕ u(1) ⊕ A 1 , so C 3 ⊕ A 1 is the expected (non-abelian) IR flavor symmetry. From the curve, one finds that this mass splits the singularity as II * → {III * , I 1 }. The I 1 singularity only has the interpretation as the IR free u(1) gauge theory with a massless (charge-1) hypermultiplet. The III * singularity is some SCFT with a 4-parameter family of relevant deformations. The seven different possible flavor symmetry interpretations (from the discussion in section 2) of this III * curve are shown in the III * row in figure 1. The [III * , C 3 ⊕ A 1 ] curve has a flavor symmetry which matches the expected IR symmetry, and so this flow is a matching flow. In contrast, the [III * , (A 3 ⋊ Z 2 ) ⊕ A 1 ] theory has a smaller than expected flavor symmetry, so the flow to it from the [II * , C 5 ] theory is unphysical. The same is true for the remaining flows from the [II * , C 5 ] theory to five other theories in the III * row: this is indicated in figure 1 by the red III * : flow. The singularity of the curve splits as above, and, as above, there is a single flavor interpretation of the III * singularity which gives a matching flavor symmetry. Two of the remaining six flavor interpretations of the III * singularity have smaller-than-expected flavor symmetries, so flows to them are unphysical, while the other four have larger flavor symmetry algebras but with the same expected rank. Flows to these four are compatible flows (shown as blue arrows in the figures): the accidental IR enlargement of the flavor symmetry is physically allowed, and does not contradict the safely irrelevant conjecture of [1] (which states that there Green, blue and red arrows label matching, compatible and unphysical RG flows, while green, blue and red backgrounds indicate "good", "ugly" and "bad" theories, respectively.  one arranges the theories in each row from largest to smallest flavor algebra, then the flows to the left of the matching flow are all compatible, while flows to its right are all unphysical. Note that for each theory in figure 1 there is a path of matching flows. Nevertheless, this does not mean that all the theories have intepretations as "good" theories. The reason is that even for interpretations of the minimal adjoint flows shown as matching for these theories, there are other non-minimal flows for which the flavor symmetry does not match. These flows are not shown in the figure, but are described in the caption. In the above examples, because the singularity splits to {III * , I 1 }, it was easy to figure out the possible IR flavor symmetry assignments since the I 1 singularity has a unique interpretation as an IR free theory. But when the the singularity splits into I n>1 or I * n>0 singularities there can be multiple IR-free interpretations of these singularities. 11 Often in these cases the number of possibilities can be greatly reduced by looking for consistent interpretations for whole sets of flows. For example, an I * 1 can be interpreted as an IR free theory as the su(2) w/ 10 · 2 theory with f = D 5 , or as su(2) w/ 2 · 2 ⊕ 2 · 3 with f = u(1) ⊕ A 1 , or as su(2) w/ 4 · 3 with f = C 2 (with charge normalization a = 1/2), or as su(2) w/ 1 · 4 with f = ∅. But if there were another mass which further split I * 1 → {I 1 7 }, then only the first, f = D 5 , assignment would be consistent. But usually we need to account for a large web of possibilities and seek a pattern of matching or compatible RG flows. Three final notes on the figures. First, we have introduced a compact notation Γ X for the Weyl group of the Lie algebra with Dynkin name X. Second, it is important to note that the "bad" [II * , BC 3 ] and [III * , A 1 2 ] theories in the I * 1 series become "good" theories if the frozen I * 1 singularity is interpreted as a non-lagrangian field theory as discussed in [2].
Finally, we have not analyzed the RG flows for the I * 0 series here since it will be the subject of [13]

Conclusion
Our main result is to provide evidence for the existence of at least an extra 8 rank 1 4d N = 2 SCFTs in addition to the 11 already known. Four of them were recently discussed in [4,5].
Here we not only point out that they fit into our classification of rank 1 N = 2 SCFTs, but also that their existence implies the existence of additional rank 1 theories through RG flow consistency arguments. Furthermore, using the techniques developed in [1][2][3] we are also able to further characterize the central charges, ECB fibers, and RG flows of the recently proposed theories.
Technically, we lift an implicit assumption made in [2] that flavor symmetries of N = 2 SCFTs have no discrete factors. Lifting this assumption effectively allows multiple different flavor symmetry interpretations of each CB geometry found in [2]. We have characterized here precisely what the freedom in flavor interpretations is, and have presented a discussion of all the allowed possibilities, summarized in table 1 and especially figures 1-4.
Most notably, the new interpretation of the flavor symmetries of some of the theories in the I * 1 series has "rehabilitated" the lagrangian interpretation of the frozen I * 1 . That is, the undeformable I * 1 which appears in the deformation patterns of these theories can simply be interpreted as an su(2) w/ 1 · 4 lagrangian theory [1] and not as a non-lagrangian, weakly gauged rank-0 theory, X 1 , as proposed in [2]. A similar but more subtle story holds for the I * 0 series and will be the subject of [13]. We believe that being able to systematically discuss the set of possible N = 2 SCFTs which could appear at rank 1 is a remarkable result. Our findings show that despite decades of continuous advances in our understanding of N = 2 SCFTs, the landscape even of rank 1 theories is not well understood. Other systematic explorations of the landscape of low-rank N = 2 SCFTs using techniques such as the bootstrap [21][22][23][24], S-class constructions [5,[25][26][27][28][29][30][31][32], geometric engineering [33][34][35], BPS quivers [36][37][38][39][40][41][42], and clarifying and generalizing the F-theory construction of [4] will undoubtably help sharpen our understanding.