Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series (arXiv:2007.15600) and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions.


Introduction
Supersymmetric quantum field theories (SQFT) can often be embedded within string theory by using either branes, geometry, or a mixture of both (distinct approaches are often related by string dualities). Over the years, string-theory techniques have led to a deeper understanding of the rich landscape of superconformal field theories (SCFT) in space-time dimension d > 2, and especially for d > 4. These SCFTs can also be analysed using SQFT methods. A typical strategy is to analyse the low-energy effective field theories obtained after breaking conformal invariance (explicitly or spontaneously). Here, we are interested in the geometric engineering approach, wherein conformal field theories correspond to certain geometric singularities, and the massive phases correspond to smoothing the singularity.
This paper is part of a series started in [1], wherein we explore the geometric engineering of 4d and 5d SCFTs at threefold isolated hypersurface singularities (IHS) We assume that X is a canonical threefold singularity, which is essentially the Calabi-Yau condition. Consequently, Type II string theory or M-theory on this geometric background preserves 8 supercharges. 1 Then, we expect the existence of two distinct maps from geometries to theories in 4d and 5d, respectively IIB : X → T 4d X , M-theory : X → T 5d X . (1.2) On the one hand, the infrared limit of Type IIB string theory on X is expected to give us a 4d N = 2 SCFT [2,3]. On the other hand, the infrared limit of M-theory on X should similarly give rise to a 5d SCFT [4,5]. The 4d SCFT T 4d X and the 5d SCFT T 5d X are closely related upon compactification to 3d, as discussed in [1,6,7].
There are two types of smoothing of X we may consider: deformations or crepant resolutions (i.e. resolutions which preserve the Calabi-Yau condition). This corresponds to turning on some complex structure moduli or some Kähler moduli, respectively. In Type IIB, families of deformation are 'classical', in that they are not subject to worldsheet nor D-brane instanton corrections. The general family of deformed singularities, denoted by X, gives us the Seiberg-Witten geometry on the extended Coulomb branch of the 4d SCFT T 4d X . Similarly, families of crepant resolutions π : X → X, denoted by X, are often 'classical' in M-theory, where they correspond to the real extended Coulomb branch of the 5d SCFT T 5d X . By contrast, deformations in M-theory and crepant resolutions in IIB are quantum-corrected, due to M2-brane instantons wrapping 3-cycles and to worldsheet, D1-and D3-brane instantons wrapping exceptional curves and divisors, respectively. These 'quantum' geometries in Mtheory and Type IIB correspond to going onto the Higgs branch of T 5d X or T 4d X , respectively. Note that this class of geometric engineering of SCFTs reverses the old QFT slogan about the Higgs branch being protected from quantum corrections (which is true for Lagrangian theories, but misleading otherwise): from the viewpoint of string theory, the Coulomb branch is 'classical' and the Higgs branch is 'quantum'.
Topologically, the crepant resolutions of canonical IHS are rather more subtle than their deformations. While the latter always have the homotopy type of a bouquet of 3-spheres, crepant resolutions of canonical IHS can have a rich topology, including compact 2-, 3-and 4-cycles. Moreover, crepant resolutions, even if they exist, are not always smooth. We might have residual terminal singularities in the resolved threefold X, or it might happen that X is smooth but the exceptional divisors themselves are not. In this paper, we will focus on isolated hypersurface singularities whose crepant resolutions are smooth. They form a small but important subset of all canonical IHS, to which we can apply a number of techniques to study the corresponding SCFT moduli spaces.
Part of the motivation for this paper is to explain some simple tools to study the key properties of deformed and resolved canonical hypersurface singularities (details which were ommited in [1]). All the necessary computations can be easily implemented on a computer.
We attached an ancillary Mathematica notebook Basic-Data-IHS.nb to the arXiv submission of this paper. The code takes as an input any canonical isolated hypersurface singularity, whose classification [3,8,9] will be streamlined and updated in section 4.1 below. (See also [10] for a related discussion.) We then provide two functions: This computational tool will hopefully be useful to the practitioner.
We will then apply these methods to explore hypersurface singularities that admit a crepant resolution that is smooth (without residual terminal singularities). It is interesting to organise the discussion in terms of the 5d rank, r, which is the number of exceptional divisors in the crepant resolution. We scanned through a large database of canonical IHS, which included a large number theories of low r. While it appears that there exists an infinite number of canonical IHS at any r (certainly, this is true at small r), we would like to put forward the following: Conjecture (1): For any r > 0, there exists a finite number of canonical IHS whose crepant resolution is smooth.
We call such a singularity with a smooth resolution a 'smoothable model'. In this paper, we list all smoothable models for r ≤ 4 -this list is complete to the best of our current knowledge.
For r ≥ 5, the scan through our dataset becomes impractical, but we can focus on a subset of 'fully smooth models', which are the smoothable models with exceptional divisors which are themselves smooth. We can scan for such models using the following: Conjecture (2): For any smoothable model with smooth exceptional divisors, the associated 4d SCFT T 4d X has an integral CB spectrum (i.e. the conformal dimensions of the CB operators satisfy ∆ ∈ Z).
This conjecture is based on observation in many examples. If true, this is a remarkable mathematical relation between the smoothness of the exceptional divisors and the spectrum of the deformed singularity. Assuming Conjecture (2), we can efficiently explore the space of all 'fully smooth' models up to r = 10, by first restricting to models with an integral 4d CB spectrum.
Further studies of more generic canonical singularities and of their 4d and 5d interpretation, as well as a more in-depth discussion of the 3d electric and magnetic quiver(ines), will be presented in part 2 of this series [7]. This paper is organised as follows. In section 2, we start with a lightning review of the geometry of isolated hypersurface singularities and of their relation to 5d and 4d SCFTs. In section 3, we discuss the topology of the link of the singularity, and its interpretation in terms of higher-form symmetries of the 5d and 4d SCFTs. In section 4, we review the classification of canonical IHS and explain how to use the ancillary Mathematica code. In section 5, the (fully) smoothable models of rank r ≤ 10 are discussed in detail, highlighting some examples with noteworthy properties, such as the models with 4d central charges satisfying a > c and a = c, and a fully resolvable model that corresponds to the AD[E 7 , E 7 ] SCFT in 4d, for which we propose a 5d IR quiver gauge theory. An appendix contains details on other smoothable models, including the infinite series of type AD[D 2n , D 2n ].

Hypersurface Singularities and SCFTs: A Practical Summary
In this section, we review key aspects of the geometry of canonical threefold hypersurface singularities, X, and of its relation to the SCFTs T 4d X and T 5d X . By definition, a canonical singularity [28] admits a resolution π : X → X such that with a k ∈ R such that a k ≥ 0, ∀k. Here, K X denotes the canonical divisor of X, and S k denote the exceptional divisors. A canonical singularity is called terminal if a k > 0, ∀k.
In this paper, we focus on canonical singularities that admit a (complete) crepant resolution -that is, a k = 0, ∀k. 2 We also focus exclusively on quasi-homogeneous isolated hypersurface singularities (IHS) in C 4 , which are defined by a single quasi-homogeneous polynomial with a unique critical point at

Deformation and 4d Coulomb Branch Spectrum
Consider the quasi-homogeneous IHS X, with the C * -action: for some scaling weights q i ∈ Q >0 . Let us recall key properties of the deformation of such singularities [29], and its interpretation in terms of the Coulomb branch of the 4d SCFT T 4d X .
Canonical singularity condition. The quasi-homogeneous IHS X is canonical if and only if the weights q i are such that: If this inequality is not satisfied, X is worse than canonical. This condition can also be written in terms of the reduced central charge [30]: Here, c is the reduced central charge of the 2d N = (2,2) LG model with four fields x i and superpotential W = F (x).
Milnor ring and multiplicity. A deformation of an hypersurface singularity X ∼ = {F = 0} is a (smooth) hypersurface of the generic form: where {x m l } µ l=1 is a basis for the so-called Milnor ring of the singularity: The integer is the multiplicity of the singularity (also known as the Milnor number). For quasi-homogeneous singularities, M(F ) has a basis in terms of µ distinct monomials: isomorphic [31]. The multiplicity of X is related to the scaling weights according to The spectrum of X. The monomials (2.9) are assigned weights Q l ∈ Q according to Let us (partially) order the monomials by their weights, with the lowest weight being Q 1 = 0 for x m 1 = 1. Let us also introduce the shifted weights called the spectral numbers. These rational numbers are valued in the open interval (0, 2).
The so-called spectrum of the quasi-homogeneous singularity is then given by the ordered set: The ordered monomials x m l and x m µ−l+1 of M(F ) form pairs such that There can also be unpaired monomials such that l = 1. The spectral numbers define a grading of the Milnor ring There is also a simple formula for the Poincaré series of the graded ring which reduces to (2.10) in the limit q → 1. The Poincaré series gives an efficient way of computing the spectrum for quasi-homogeneous hypersurface singularities, bypassing the need for finding an explicit basis of M(F ). We denote by X a (generic) deformation of X.
Another important number that characterizes the singularity X is its modality, m(X), which is defined as the number of deformations of X which preserve its multiplicity. For quasi-homogeneous singularities, this is given by The corresponding deformation parameters t l are the 'moduli' of the singularity.
The Coulomb branch spectrum of T 4d X . In the Type IIB engineering, the family of deformations { X t } is interpreted as the Seiberg-Witten geometry of the 4d SCFT T 4d X [2,3]. The spectrum of X is equivalent to the spectrum of (extended) Coulomb branch (CB) dimensions of T 4d X , as follows. To each monomial x m l , one assigns the CB dimension We denote by r the number of such operators.
• ∆ l = 1: these geometric deformations are interpreted as the mass deformations of T 4d They correspond to conserved currents, i.e. flavor symmetries. We may equivalently view the deformation parameters as VEVs for the scalars φ l in the U (1) background vectors multiplets spanning a maximal torus of the flavour symmetry group In particular, f is the rank of the flavor symmetry group of T 4d X .
• ∆ l < 1: these geometric deformations correspond to the r monomials which are paired with the ones with ∆ µ−l > 1, such that ∆ l + ∆ µ−l = 2. They are interpreted as chiral deformations of the SCFT by the CB operators themselves In particular, if ∆ l = 0 (∆ µ−l = 2), this is a marginal deformation of the SCFTindeed, this corresponds to deforming the IHS equation by a monomial with Q l = 1, thus preserving the scaling symmetry, which is necessary for conformal invariance. Note that we can also have deformations with 0 < ∆ l < 1, corresponding to relevant deformations of the SCFT (1 < ∆ µ−l < 2), or else deformations with ∆ l < 0, corresponding to irrelevant deformations (∆ µ−l > 2). The relevant deformations only exists if there are generators of the Milnor ring with scaling dimensions in the small range Note also that the modality (2.17) is the total number of marginal and irrelevant operators of the 4d SCFT; thus, the 'moduli' of the singularity are precisely those deformations that do not trigger an RG flow away from the fixed point T 4d X .
Let us denote by B r the Coulomb branch spanned by (2.19). In particular, r is the rank of the SCFT T 4d X . We define the extended Coulomb branch M C [T 4d X ] as the total space of the Coulomb branch fibered over the mass parameters (2.21), namely The extended CB obviously has complex dimension r + f . Note also the relation The multiplicity µ gives the dimension of the charge lattice Γ of BPS particles on the CB, which arise as D3-branes wrapping the compact 3-cycles of the deformed hypersurface Let us also denote by M 3,3 the intersection form on the third homology where the 3-cycles {S 3 α } µ α=1 give some basis of H 3 ( X, Z).
Central charges. From the Coulomb branch spectrum, we can compute the conformal anomaly coefficients a and c according to [3,32] (2.28) Here, we defined with ∆ max ≡ ∆ 1 the largest CB operator dimension. Here, we also used the fact that there is no hypermultiplets on the Coulomb branch of T 4d X because H 2 ( X, Z) = 0 [3]. From a and c, we may also compute the 'effective number' of hypermultiplets and of vector multiplets of the SCFT For Lagrangian SCFTs, this gives the actual number of matter fields and gauge fields, respectively. For further discussions of the spectrum of 4d SCFTs in related contexts, see e.g. [33-36].

Crepant Resolution and 5d Coulomb Branch
For any canonical threefold singularity, there exists a crepant resolution π : X → X, such that the resolved singularity X has at worst terminal singularities [37,38]. In M-theory, the resolution X corresponds to probing the extended Coulomb branch of the 5d SCFT T 5d X [39]. (See also e.g. [40][41][42][43][44] for some further discussions.) Topology of the crepant resolution X. We will call 'a crepant resolution' any (partial) resolution π : X → X that introduces r exceptional divisors with a k = 0 in (2.1). This is excepted to preserves 5d N = 1 (or 4d N = 2) supersymmetry. A smooth (or 'complete') crepant resolution, corresponding to a k = 0, ∀k, in (2.1), is a crepant resolution that is smooth.
In general, there may be many distinct crepant resolutions of X, but some of their key topological properties are invariant. The set of residual terminal singularities is such an invariant, and so are the Betti numbers Here, r is the number of exceptional divisors in the crepant resolution, and f = ρ(X) is the rank of the divisor class group of the singularity X. We can also understand f as the number of 2-cycles of X that are dual to non-compact divisors. They correspond to U (1) flavor symmetries of the 5d SCFT T 5d X . Note also that the crepant resolution X generally has three-cycles, as we will explain momentarily.
There is a remarkable relation between f as defined here and the integer f defined in (2.20) from the spectrum of the singularity. It is a non-trivial mathematical fact that the two quantities agree [45] f = ρ(X) = #{ l | l = 1} . (2.32) Physically, this can be interpreted in terms of generalised geometric transitions, wherein f 2-cycles in the resolution X become f 3-cycles in the deformation X [1]. 3 A third way to compute f is from the five-dimensional link of the singularity, as we will review in section 3.
Newton polytope and weightings. For any quasi-homogeneous IHS, the Betti numbers (2.31) can be computed in the following way [45]. which may be reducible. 3 The proper mathematical understanding of this relation involves discussing the mixed Hodge structure of the singularity, which we will review in more detail in [7].
Computation of r and b 3 . The topological numbers r and b 3 are independent of the choice of resolution, and they can computed as follows [45,46] (analogously to the Batyrev formula [47]). Consider the dual polytope Γ of W (F ), defined as The dual face Γ α ⊂ Γ of α ∈ W (F ), defined as can have different dimensions, denoted by dim(Γ α ). The explicit formula for r is then where l (Γ α ) is number of interior lattice points in Γ α . When dim(Γ α ) = 1 and l (Γ α ) > 0, the compact divisor corresponding to α has l (Γ α ) + 1 irreducible components.
To compute b 3 , the number of 3-cycles in the crepant resolution, we first note that each compact divisor associated to the weighting α ∈ W + (F ) has the structure of P 1 fibration over a complex curve with genus g C (α). The numbers g C (α) are computed as .

(2.41)
The formula for b 3 is then In other words, there is a one-to-one correspondence between the 3-cycles and the 1-cycles at the base of the ruled exceptional divisors. Note that the resolved threefold X is itself simply-connected.
2. (y 1 , y 2 , y 3 ; δ): blow up the locus y 1 = y 2 = y 3 = 0. The exceptional divisor δ = 0 in the ambient space is associated to the weighting In the hypersurface equation, one replaces and then divide the equation by δ 2 . There is a new SR ideal generator y 1 y 2 y 3 after the resolution.
After the resolution sequence, one can also compute the topological quantities associated to each exceptional divisor. In particular, we can compute the intersection numbers of the exceptional divisors S i inside X, according to This determines the prepotential on the 5d CB of T 5d X [39].

Link Topology and Higher-form Symmetries
The five-manifold L 5 at the boundary of a non-compact Calabi-Yau three-fold encodes global symmetries of the field theories, both in 4d and 5d. In particular, the torsion part of H * (L 5 , Z) encodes the higher-form symmetries [50] of both the 5d SCFT T 5d X and the 4d SCFT T 4d X [1,[51][52][53][54]. For isolated hypersurface singularities, L 5 is simply connected, and the 5d field theory T 5d X does not have any 1-form symmetries. On the other hand, we often have non-zero torsion in H 2 (L 5 , Z), which can be computed from the weights q i of the quasi-homogeneous polynomial F (x) [55,56]. In M-theory, this translates into 3-form symmetries of T 5d X that act on M5-branes wrapping relative 3-cycles (and/or its dual 0-form symmetry acting on wrapped M2-branes). In Type IIB string theory, this torsion group is related to the 1-form symmetry of the 4d SCFT T 4d X , with the charged line operators arising from wrapped D3-branes. In the rest of this section, we first review an explicit formula for the link homology, and we then briefly discuss its 5d and 4d interpretation.

Topology of the Link
The link of the singularity X is the 5-dimensional compact manifold L 5 (X) obtained by intersecting the hypersurface with a seven-sphere S 7 ε centered at the origin of C 4 for a small radius ε > 0. For quasi-homogeneous singularities, we may pick any value of ε due to the scaling symmetry. Therefore, by sending ε → ∞, we see that the link is also the boundary manifold of X and of any local desingularization of the singularity; in particular, for any smooth deformation X of X, we have ∂ X ∼ = L 5 (X). The five-manifold L 5 is connected and simply-connected, and its homology groups are where denotes the torsion part of the second homology. This leads to our third definition of the 'flavor rank' f , as the Betti number of the link This gives us an intuitive way of understanding the 'generalised geometric transition' mentioned below (2.32), generalising the conifold transition [57]: as we resolve or deform the singularity X, we have f exceptional 2-cycles that survive 'at the tip' inside X, or f 3-cycles that survive inside X, respectively.
The homology of L 5 (X): explicit formulas. The non-trivial quantities f and t 2 in (3.2) can be computed from the scaling weights q i in (2.3) as follows. Let us define the quantities Let us consider the set I = {1, 2, 3, 4}, and let I s denote any particular subset I s ⊂ I of s elements. Then, we have the explicit formula [55,56] .
The sum includes all subsets I s of I, including the trivial subset I 0 = {}, which contributes +1. More generally, we have the following formula for the torsion part, t 2 [56]. For every and (3.8) We then define a set of integers, C(I s ), inductively, starting with and . (3.10) Then, for the integers j > 1, we define We then have: The torsion group t 2 from the deformation X. Using a long-exact sequence for the (relative) homology groups of the deformed threefold X, as reviewed in the next subsection, one can show that the torsion group t 2 is also isomorphic to the cokernel of the map M 3,3 : Z µ → Z µ given by the intersection form on H 3 ( X, Z), namely We computed this cokernel in many examples, allowing us to check the general formula (3.12) for t 2 , which appeared as a conjecture in [56], on a case-by-case basis. 4

Defects and Higher-Form Symmetries in 5d from M-theory
Consider the canonical singularity X, any desingularisation X 6 of X (which will be either the deformation X or the crepant resolution X), and the boundary five-manifold L 5 ∼ = ∂X 6 .
M-theory on X 6 provides us with a natural set of defect operators of the 5d low-energy theory, from membranes wrapping non-compact k-cycles. The wrapped M2-and M5-branes give rise to defect operators of dimension q = k − 3 and q = 6 − k, respectively, which can be charged under a non-trivial q-form symmetry. The sets of such charged 'electric' and 'magnetic' defect operators of dimension q are denoted by respectively. They are computed as where f k is the map of f k : H k (X 6 , Z) → H k (X 6 , L 5 , Z) that sends compact k-cycles to relative k-cycles. It sits inside a long exact sequence for the relative homology (with integral coefficients) The objects counted by h (k) are the defects whose charge cannot be screened by the dynamical particles that arise from membranes wrapping compact k-cycles of X 6 . For q ≤ 1, for definiteness, let us define the 'defect groups' of the 5d theory, in its massive phase determined The defect operators in Γ can have a non-trivial Dirac pairing given by the intersection of the relative (3 − k)-and (3 + k)-cycles inside X 6 . The higher-form symmetry of the 5d theory is determined after one chooses a polarisation of the defect group (3.17), by fixing a maximal set of mutually commuting defects [51,53,54,[59][60][61][62][63]. The higher-form symmetry group itself is given by the Pontryagin dual group Note that it is generally a mixture of q-and (3 − q)-form symmetries. We may also choose, for simplicity, the 'purely electric' or the 'purely magnetic' polarisation, consisting in picking one of the two summands in (3.17).
Given a general local Calabi-Yau threefold in M-theory, there are two defects groups of corresponding to the '0-form symmetry' and the '1-form symmetry', respectively, allowing for a slight abuse of language. Using the exact sequence (3.16), we can write the groups h (k) in terms of the homology of the boundary: The charged defect operators wrap relative k-cycles Σ k with non-trivial boundary on L 5 , . For L 5 any connected, orientable closed manifold, there are two independent torsion groups in its homology Let us now see how these torsion groups are related to higher-form symmetries in 5d.

1-form symmetry and Coulomb-branch defects.
For the sake of generality, let us first discuss the possibility of a non-trivial 1-form symmetry in 5d [51,53,63]. We will assume that the smooth threefold X 6 is simply-connected and that it has no torsion in homology, which is the case for all examples we have studied (such as toric threefolds or hypersurface singularities). It directly follows from (3.16) and (3.21) that the 'electric' part of the defect group is isomorphic to the the torsion part of the first homology group of the boundary.
Alternatively, we can use the definition (3.15) of h (2) as the cokernel of the map f 2 , which gives where b j denote the Betti numbers of X 6 . Similarly, we have The 1-form (and/or 2-form) symmetry is then determined by the intersection pairing between the compact 2-cycles and 4-cycles of X 6 . The charged defect operators naturally live on the Coulomb branch of the 5d SCFT, in which case X 6 = X, b 2 = r + f and b 4 = f . For the 5d SCFTs T 5d X of interest in this paper, which arise at hypersurface singularities, the link manifold L 5 is simply connected, and therefore we find that h (2) = h (4) = 0. 0-and 3-form symmetry and Higgs-branch defects. The 0-form/3-form symmetries of T 5d X act on defect operators that are more naturally defined on the Higgs branch of T 5d X . Let us then take X 6 = X a deformed singularity, and assume that H 2 ( X) = 0 for simplicity, which is the case, in particular, for the generic deformation of an hypersurface singularity.
We then find that with b 3 = µ = 2 r+f for an IHS. It would be interesting to have a more detailed understanding of these charged HB defect operators from the 5d field-theory point of view. This is left as a challenge for future work.

Defects and Higher-Form Symmetries in 4d from Type IIB
In the Type IIB compactification on X 6 , the natural set of charged defect operators of the 4d field theory arise from wrapped D-branes. We may define the defect group corresponding to D1-, D5-and D3-branes wrapping 2-and 4-cycles, as well as the defect from D3-brane wrapping relative 3-cycles inside X 6 . Since our focus is on hypersurface singularities, we will not consider further the interesting possibilities suggested by (3.27). In other words, our 4d theories T 4d X will have no defects charged under non-trivial 0-form or 2-form symmetries. On the other hand, the defect group (3.28) is generally non-trivial, and it encodes all the ('electric' and 'magnetic') line operators of T 4d X [1, 53] -see also [10,[62][63][64] for related discussions. It is interesting to note that, for 4d SCFTs T 4d X engineered at IHS, non-trivial 1-form symmetries can only arise if the SCFT is non-isolated, as shown in [10]. 5 The actual one-form symmetry Λ For our hypersurface singularities, the torsion in the second homology of the link always takes the form: so that there exists a 'purely electric' choice of polarisation for which the one-form symmetry

Classification of Canonical IHS and Mathematica Code
In this section, we review the complete classification of canonical isolated singularities, which is equivalent to the Kreuzer-Skarke classification of 2d N = (2, 2) Landau-Ginzburg (LG) superpotentials [65]. This 2d perspective clarifies some aspects of the discussion in [3,8].
We then present an ancillary Mathematica code [66] (available with the arXiv version of this paper) which implements the computations explained in the previous sections for any such singularity. The code computes the spectrum of X, the basic topological data of its crepant resolution X, as well as the homology of the link L 5 . This directly gives all the 'basic data' associated to the 5d SCFT T 5d X and the 4d SCFT T 4d X . The code also implements the crepant resolution procedure explained in section 2.2.

Classification of Canonical Isolated Hypersurface Singularities in C 4
Consider a quasi-homogeneous IHS in C 4 , defined by the polynomial with the scaling weights q i as in (2.3). Such quasi-homogeneous polynomials F (x) were classified by Kreuzer and Skarke (KS) many years ago [65]. A classification of isolated hypersurface threefold singularities was also given by Yau and Yu (YY) [8], which was used by Xie and 5 This claim was checked systematically in [10], and it is easily checked with the 'experimental' methods of this paper. In particular, in our finite database of models discussed in section 5, 56% of the models are isolated (no marginal deformations) and all those isolated models have t2 = 0. Only 15% of all models (amounting to 35% of the models with marginal deformations) have a non-trivial one-form symmetry.
It is very useful to think of the polynomial (4.1) as defining a 2d Landau-Ginzburg (LG) model with four chiral superfields x i and a superpotential W = F (x) [2]. Any quasihomogeneous IHS then defines a 2d N = (2, 2) SCFT as well as the 4d SCFT T 4d X , with the two perspectives related by a 4d/2d correspondence [67], provided that the reduced central charge of the 2d theory satisfies [2,68,69]: In fact, the IHS is canonical if and only if (4.2) holds [28,70]. Such LG models were recently classified by Davenport and Melnikov [9]. 6 Let us review their result, while keeping the YY notation for ease of comparison. One finds that the YY classification needs to be refined, by introducing 'subtypes' for some of the 19 'types'. Since this classification is essentially equivalent the KS classification of quasi-homogeneous polynomials, we will call it the KS-YY classification.
The full KS-YY classification. The YY classification distinguishes between 19 types of polynomials F (x), denoted by the roman numerals I to XIX, and we denote the subtypes necessary for the full classification by subscripts. 7 Then, for a given type and subtype, the positive integers (a, b, c, d) fully determine the singularity (up to a choice of solution to some linear Diophantine equation, in some cases, which we discuss momentarily). For instance, the singularity IX 2 (2, 3, 4, 2) is given by The complete classification is given in tables 1, 2 and 3.
It is very important to note that this classification includes many redundacies: several singularities of distinct types can be physically equivalent -two quasi-homogeneous singularities are physically equivalent if and only they have same scaling weights, up to ordering. From the 2d LG model perspective, these scaling dimensions (q i ) are the fundamental quantities, and any two superpotentials W = F 1 (x) and W = F 2 (x) that realises the same scaling dimensions must flow to the same 2d SCFT. From the 4d SCFT point of view, two singularities with the same scaling weights obviously give rise to the same 4d CB spectrum, as per (2.16).
In order to explain the KS-YY classification, let us introduce some useful terminology and notation [9,65]. First of all, an (LG) field x i that appears in F (x) is called a 'root' if it 6 Note that [9] also considered LG models with five fields such that c < 2. We only consider four fields. 7 We also use the notation {type, subtype} with type ∈ {1, 2, · · · , 19}, as in the Mathematica notebook.
Skeleton and links Skeleton Table 1. The KS-YY classification of isolated hypersurface singularities in C 4 : Types I to XIII. Note that types I-VI exhaust the possibilities using skeleton with up to n = 2 fields, while the cases I-X exhaust the possibilities using skeletons with up to n = 3 fields.

Type
F (x 1 , x 2 , x 3 , x 4 ) Skeleton and links Skeleton appears as x m i i , and it is called a 'pointer' if it appears as x m i i x j , for some positive integer m i . One can show that every x i in F (x) must be either a root or a pointer. We say that a pointer x i 'points at' the field x j . One can then introduce a convenient graphical notation for the polynomial. Each x i , root or pointer, is denoted by a dot accompanied with the integer m i , and for each pointer we draw an arrow from x i to x j . For instance, the polynomial in With four fields x 1 , · · · , x 4 , there are 19 distinct possible 'skeletons', corresponding to the 19 classes in the YY classification. As we see from the tables, some of the skeletons are reducible. The basic building blocks are the skeletons denoted by S n,α in [9], where n denotes Type F (x 1 , x 2 , x 3 , x 4 ) Skeleton and links Skeleton Whenever a skeleton contains a pair of pointers x i and x j that point to the same node , the corresponding polynomial is not an isolated singularity. One should introduce a 'link' or a 'pointing link', which is an additional monomial involving x i and x j such that the singularity is isolated. The link is denoted by the dashed line connecting the two nodes The link, and therefore the singularity, exists only if there exists positive integers p and q such that the polynomial is quasi-homogeneous. We must then solve a Diophantine equation determined by the scaling weight q i of the four fields. For instance, in (4.4), if q k is the scaling weight of x j , we must have p 1−q k m i + q 1−q k m j = 1. A pointing link is a link that is itself a pointer to another field y, of the form x p i x q j y. A link associated to a field x k can only point to a field y that is not x k nor any of the fields that are pointed at by the fields x i and x j involved in the link. For instance, we can have We denote the pointed link as shown here. As for simple links, pointed links can only exists if they preserve the quasi-homogeneity of the polynomial.

Computing the Basic Data of Any Canonical IHS: Mathematica Code
Given any F (x) defining a canonical threefold singularity X, we can easily compute all the mathematical and physical quantities introduced in the previous sections, which we call 'the basic data' associated to X. We implemented these computations on a computer using Mathematica [66], and we provided the Mathematica notebook Basic-Data-IHS.nb as an ancillary file to the arXiv version of this paper. Here, we describe the main routines in that notebook.
Basic Characteristics of IHSs. The code implements many computations which are collated through the main routine: The code also generate any allowed polynomial F (x) given a type, subtype and the positive integers (a, b, c, d): with {t, s} = {type, subtype}. The 'basic data' routine can be called directly as: The output is given as a list with the following information: • F (x) itself; • the scaling weights (q i ); • r: the rank of the 5d SCFT T 5d X (that is, the number of exceptional divisors in X); • f : the flavor rank preserved on the Coulomb branch of T 4d X and T 5d X ; • d H = r + f : the quaternionic Higgs branch dimension of T 5d X ; • r: the rank of the 4d SCFT T 4d X ; • d H = r + f : the quaternionic Higgs branch dimension of T 4d X ; • µ = 2 r + f : the multiplicity (Milnor number) of X; • modality: the modality m(X) of X; • n h : the effective number of hypermultiplets of T 4d X ; • n v : the effective number of vector multiplets of T 4d X ; • b 3 : the number of 3-cycles in the resolved threefold X; • ∆A: the difference between the 'virtual dimension' n h − n v of the HB of T 4d X and the actual HB dimension, ∆A = n h − n v − d H ; • g C : the list {g 1 , 2 , · · · } of genera for the higher-genus (g > 0) complex curves at the base of the exceptional divisors inside X; • specL: the spectrum { l } of the Milnor ring M(F ); • CB spectrum: the Coulomb branch spectrum of T 4d X ; • a and c: the conformal anomaly coefficients of T 4d X ; • marginal Defs: the number of marginal deformations of T 4d X ; • TorH 2 : the torsional part of the second homology of the link L 5 , Tor H 2 (L 5 , Z) = t 2 .
Smoothing Operator. The crepant resolution procedure described in secton 2.2 is encoded in an additional routine, which constructs the (partially) resolved Calabi-Yau threefold X of any IHS {F (x 1 , x 2 , x 3 , x 4 ) = 0}. Using the function: Note that this code also works in the generic case with residual terminal singularities. In these cases, the resolution sequence is a partial crepant resolution that gives rise to X with at most terminal singularities.
As an example, take The patches are then given by

Smoothable Models
In the previous section, we spelled out a general approach to determine the basic properties of the SCFTs T 5d X and T 4d X for any IHS X. We also explained how to explicitly construct the crepant resolution π : X → X. This allows us to determine the smoothness properties of the resolved model X (whether it contains residual singularities or not), as well as possible 5d infrared gauge theory descriptions. In earlier works [1,6], we studied interesting models with terminal singularities (whether X was a terminal singularity with r = 0, or a singularity with r > 0 that contains remnant terminal singularities after a crepant resolution). In this section, we would like to focus on smoothable models -that is, singularities X whose crepant resolution X is a smooth CY threefold (below, we also call such a CY threefold 'a smooth model').

Tabulating the Smoothable Models
We found it useful to organise the smoothable singularities X into three categories: It is also useful to organise the models by their 5d rank, r, i.e. by the number of exceptional divisors. For low rank, we gathered (conjecturally) all such models by brute force, by generating a large dataset of 15,142 physically distinct models with r > 0. 8 The number of models for r ≤ 10 in that dataset is given in Table 4.
For r = 1, 2, 3, we resolved all the models explicitly and found all the smoothable models in our dataset. For larger r, this becomes impractical, but we can reduce strongly the number of cases to consider by realising that a necessary condition for having a smoothable model is that ∆A ∈ Z, because ∆A / ∈ Z can only arise from residual terminal singularities [1]. We are also particularly interested in the smoothable models with smooth exceptional divisors (the 'fully smooth models', for short). We then make the following conjecture, based on explicit analysis of our dataset: Note that the converse is not true. On the other hand, we propose the following: Conjecture 5.2 Any model X with an integral CB spectrum is smoothable (not necessarily with smooth divisors).
In summary, we have the following inclusions of sets, at any fixed rank r > 0: where the first and second inclusions are conjectures. The number of models at fixed rank appears to be infinite, but we conjecture that the number of smoothable models of a fixed rank is finite. For small r, we also believe that all these models are included in our dataset. 9 The numbers of models with various properties (in our dataset) are shown in Table 4. For r ≤ 10, we considered all the models with integral CB spectrum and extracted all the 'fully smooth' models from that set. They are discussed in detail in the following, and in appendix.
Models with b 3 = 0. We first discuss the smoothable models with smooth exceptional divisors with b 3 (X) = 0. We studied all such models with rank r ≤ 10 in our dataset in tables 8 and 9. 9 We do not have a proof that our lists of 'smoothable' and 'fully smooth' models up to r = 10 is complete within the infinite class of models defined by canonical IHSs. This is largely due to the fact that the correlation between rank r and the hypersurface equation types is not one we understand systematically at this point. It would be very interesting to determine more direct relations between the (a, b, c, d)'s defining the equation F (x) = 0 and the rank r, if possible.
This class of models have the simplest 5d interpretation, as they correspond to 5d SCFTs T 5d X whose Coulomb-branch low-energy physics is described entirely by r 5d U (1) vector multiples, with a prepotential determined by the intersection numbers (2.49) in the smooth resolved threefold. Similarly, the 4d SCFT T 4d X is such that the Higgs-branch low-energy physics consists of only d H = r + f hypermultiplets [1].
Note that, in all these tables, we color the labels {t, s}{a, b, c, d} in (i) black for the cases with a Lagrangian T 4d X ; (ii) blue for the cases with a D b p (G) trinion description studied in [6]; (iii) red for the cases with a generalised D b p (G) description of T 4d X (to be discussed below); (iv) orange for the cases without a known description of T 4d X yet. For the case (i), the Lagrangian quiver description of T 4d X is written down in the table. For the other cases, we simply list the CB spectrum. Many of theses models were already discussed in [1,6].
The physical interpretation of the 3-cycles in X was already discussed in [1]. The 5d SCFT T 5d X has an 'enhanced Coulomb branch' with 1 2 b 3 free hypermultiplets at any generic point on the CB. The 4d SCFT T 4d X has an Higgs branch whose low-energy physics includes To precisely characterise the physics of enhanced Coulomb branches in 5d is an interesting question, which is left for future work (for a discussion of 4d enhanced Coulomb branches, see [72]).
Smoothable models with singular divisors. Finally, one can consider the smoothable models whose exceptional divisors are not all smooth. The interpretation of such singularities in the divisors (while the threefold itself is smooth) appears to be more subtle, and we hope to come back to this in future work. Here, we simply list all the smoothable models up to rank r = 4 in the tables 15, 16, 17, 18, 19, 20 and 21.

The AD[E 7 , E 7 ] model
We now illustrate one example of 'fully smooth' model with b 3 = 0, which turns out to have r = 5. We consider the geometry that is associated to the Argyres-Douglas-type theory . This is the Type{4, 1}{3, 3, 3, 3} singularity, with basic data: Deformations and 4d SCFT. In this case, the CB spectrum of T 4d X contains operators with ∆ = 7 and ∆ = 9, but there is no operator with ∆ = 8. Although the CB spectrum is fully integral, it cannot match any set of Casimir operators of Lie groups. Hence there is no Lagrangian description of T 4d X . There is, however, a generalised quiver using D b p (G) building blocks. The conjectured 4d generalised D p (G) description is: 10  (8)) is the E 6 MN theory [73,74].
Resolutions and 5d SCFT. This is a fully resolvable IHS, and a resolution sequence is given by: After the resolution, the equation is smooth. The exceptional divisor δ 1 = 0 has three components: The exceptional divisors δ 2 = 0 and δ 3 = 0 are irreducible, thus resulting in total in a rank r = 5 theory. The intersection numbers of the compact surfaces are shown on the LHS of figure 1. The three components of δ 1 = 0 are S 2 , S 3 and S 4 , which are all P 2 s. S 1 corresponds to δ 3 = 0 and S 5 corresponds to δ 2 = 0.
The conventions for the intersection diagrams are as follows (similarly to the ones in [75,76]): 10 We thank Simone Giacomelli for discussions of this model and for suggesting this particular quiver. (1) Each oval denotes an irreducible compact divisor S i . The number in the bracket after S i is the self-triple intersection number S 3 i .
(2) The line segment between ovals S i and S j corresponds to the complete intersection curve S i · S j . The numbers in the boxes at the end of the line segment describe the self-intersection numbers of the curves inside their respective divisors. The number on S i corresponds to S i · S 2 j while the number of S j corresponds to S j · S 2 i .
(3) For each triangle among three ovals S i , S j and S k , we use a boxed number to denote the triple intersection number S i · S j · S k .
To see the structure of the curves more clearly, we may do four flops by blowing up S 2 , S 3 , S 4 , S 5 once and blowing down S 1 four times, which gives the local model shown on the right-hand-side of figure 1. Here, S 1 is a gdP 6 of type 4A 1 , which is also known as the (resolution of) Cayley's cubic surface [77]. We plot the Mori cone generators of S 1 and their representations under the Picard group generators h, e i (h 2 = 1 , h · e i = 0 , e i · e j = −δ ij ), showing the four (−2)-curves S 1 · S 2 , S 1 · S 3 , S 1 · S 4 and S 1 · S 5 : We also plot the curves on S 2 , S 3 , S 4 and S 5 : X by applying the procedure in [80]. The CFDs are -in briefa collection of (usually rational) curves, which encode the flavor symmetries (in terms of the subgraph of green-colored nodes) and the (−1)-self-intersection curves that can be flopped and the associated hypermultiplet decoupled.
The CFD is constructed by connected to the same S i · S j are combined, with the proper multiplicities. For example, on S 1 the six (−1)-curves between the four (−2)-curves have multiplicities two, which correspond to (−2) nodes in the CFD. Hence the CFD can be read off as: The UV flavor symmetry algebra is therefore g F = su(2) 6 ⊕ u(1).
On the geometry, we can also identify a ruling structure by taking S 1 · S 2 and S 1 · S 5 to be the section curves, S 1 · S 3 and S 1 · S 4 to be the fiber. From the spectrum of massless hypermultiplets in the IR gauge theory limit, we can identify the following 5d gauge theory description: If we decouple the two SU (2) gauge factors by sending the volume of S 3 and S 4 to infinity, the surfaces S 1 , S 2 , S 5 give rise to IR gauge theory SU (4) − 1 2 + 1AS + 5F. It would be interesting to see whether this IR gauge theory description can inform the computation of the magnetic quiver for this theory, and to then match this with the conjectured 4d quiver (5.4).

4d SCFTs: D b p (G)-Trinions
We now turn our attention 'fully smooth' models that have b 3 > 0. The 5d interpretation becomes more difficult in general, due to existence of an enhanced Coulomb branch. As before, the 4d SCFTs T 5d X may not have a Lagrangian description, as can be seen from tables 10, 11, 12, 13 and 14. Some of these models have an interpretation in terms of the D b p (G)-trinions introduced in [6]. Using the notation for the trinions introduced therein, namely: the models with 4d trinion description are summarized in table 5.

4d SCFTs with c − a < 0
Another interesting class of models are those for which the conformal anomalies of T 4d X satisfy the inequality c < a. Such 4d SCFTs are slightly peculiar, and appear to be more 'rare'; from the point of view of the superconformal index, c − a < 0 is analogous to having c < 0 for a 2d CFT -see e.g. [81][82][83][84]. It turns out that there are only 8 canonical IHS for which c < a in our database, and they are all smoothable models with b 3 > 0. Their basic data is shown in table 6.
The 4d Coulomb branch spectrum of all these singularities except for the type {1,1}{2,3,8,8} is integer-valued, ∆ α ∈ Z. (And these 7 models are actually 'fully smooth'.) Note that we  have:  in agreement with the physical interpretation of the 3-cycles as vector multiplets in IIB. We will now provide the corresponding 4d SCFTs T 4d X (whose description is simply related to the electric and magnetic quiverines in 3d, since f = 0 [1,6]). We also study the resolution of these singularities and identify the 5d gauge theory phase if it exists.
Let us also discuss the resolution of this model. Starting with the hypersurface equation The resolved equation is The set δ 1 = 0 is empty, so there are only two exceptional divisors S 1 : δ 2 = 0 and S 2 : δ 3 = 0.
The presence of the 3-cycles in X makes the M-theory interpretation more subtle, as mentioned above. We can obtain a distinct and more 'conventional' model by performing a geometric transition, as described in [71], to obtain a new Calabi-Yau threefold with distinct asymptotics and b 3 = 0. In the present example, one performs a complex structure deformation of the geometry, such that the intersection curve S 1 · S 2 has six double point singularities.
Then we perform flop operations six times by blowing up six double points on S 2 . After these flops, the triple intersection numbers of the new surfaces S 1 and S 2 are Hence S 1 is a F 3 and S 2 is a dP 6 . These triple intersection numbers exactly give rise to the 5d SCFT with IR description SU (3) 9/2 + 5F , Sp(2) + 3F + 2AS or G 2 + 5F , and UV flavor symmetry G 5d F = Sp (6). Before the flops, the geometry gives f = 0. After the flops, there are six new 2-cycles in the flopped geometry, which give rise to the flavor rank f = 6 that matches G 5d F .

Rank 2: {1,1}{2,3,8,8}
The 4d CB spectrum of this model is non-integral. In [6], a non-Lagrangian description of [74,85,86], here with G = E 6 for the central gauge group: The singularity The resolved equation is completely smooth: The triple intersection numbers of S 1 : δ 1 = 0 and S 2 : δ 2 = 0 are computed to be 4 S 1 (9) -6 S 2 (-2) However, this model has flavor rank f = 0, which leads to a contradiction. The reason is that the equation of S 2 is From the genus-3 curve on S 2 , we have although S 1 · S 2 has genus 0. The 5d physical interpretation of this geometry is unclear.

Sp(6)
Spin (20) Sp (8) Spin (16) Sp (6) Spin (12) Sp (4) Spin (8) Sp (2) Spin (4) Sp(4) It is a simple exercise to check that this quiver has a 1-form symmetry f = Z 7 2 , in agreement with the geometric determination. The resolution of the singularity x 2 1 + x 3 2 + x 9 3 + x 9 4 = 0 is achieved by The resolved equation is The intersection curve S 2 · S 3 is a genus-7 curve, and the surface S 2 is ruled over that curve. The model does not appear to have a 5d gauge-theory description.
Finally, we find six 'sporadic' models with r > 0: We will now discuss the 'fully smooth' model in more detail. The theory does not have known quivernines EQ (4) and MQ (5) . The resolution sequence of the singularity x 4 The resolved equation is The UV flavor symmetry is G F = SO(4n + 2) × U (1). Let us first summarize the properties of these singularities: Now we check these properties from both deformation and resolution of X.

A.1.1 Deformation and Magnetic Quiver
From the spectrum of the singularity, we can check that it is a Lagrangian theory with so that we find the virtual dimension of the 4d HB, n h − n v = 3n + 1. This equals the actual dimension as computed from the geometry, d H = r + f = 3n + 1. The 4d Lagrangian reads [6]: where the nodes are special unitary gauge groups. The 5d magnetic quiver is obtained by gauging all the special-unitary and flavor nodes in (A.5), and then modding out a common U (1): The Hasse diagram is shown in Figure 2 (it was already computed in [11]). Note that the AD[D 4 , D 4 ] theory is also equivalent to the rank-1 Seiberg E 6 theory. The 3d mirror MQ (4) to (A.5) was recently discussed in [88], together with the magnetic quiver for any of the

A.1.2 Resolution and 5d SCFT Interpretation
The singularity (A.1) allows for a complete resolution, with the following sequence: The resulting equation is completely smooth, and all the exceptional divisors S i : δ i = 0 are smooth as well. We can compute the non-vanishing triple intersection numbers among the different divisors, which are represented as The intersection numbers are consistent with the 5d IR gauge theory description (A.2) as well.

A.2 Rank 1
There are only three rank r = 1 5d SCFTs in this cateory: the Seiberg rank 1 E 6 , E 7 and E 8 SCFTs, and we discussed them in [1]. They are described by the following type {1, 1} Table 7. All smooth rank 1 models with smooth divisors and b 3 = 0.
isolated hypersurface singularities: (A.10) The resolution of these singularities consists of a single weighted blow-up in the ambient space: 3 , x The exceptional divisors S 1 : δ 1 = 0 are del Pezzo surfaces dP 6 , dP 7 and dP 8 respectively.
We also list the full data of these theories in table 7. The EQ (4) for the rank 1 E k theory are SU (n) quivers in the shape of affine E k Dynkin diagram, while the MQ (5) are U (n) quivers in the same shape. The Hasse diagram is simply a single e k , which corresponds to the minimal nilpotent orbit of E k .

A.3.1 Deformations and Magnetic Quiver
From the CB spectrum we can infer the EQ (4) and MQ (5) . The magnetic quiver for the 5d theory is From the Hasse diagram, we can read off the flavor symmetry G F = E 7 × U (1).

A.3.2 Resolution and 5d SCFT
The resolution sequence that completely resolves this model is 2 , x 4 , δ with the intersection numbers This theory does not have a 5d IR gauge theory description. Nonetheless, the theory is a descendant of 6d (E 7 , SO (7)) conformal matter [79] with the following CFD transitions The CFD of this 5d SCFT with G F = E 7 × U (1) is at the bottom-right corner.

A.4.1 Deformations and Magnetic Quiver
The magnetic quiver again follows from the CB spectrum of T 4d X : From the Hasse diagram, we can read off the flavor symmetry is which is different from the subgraph of the balanced nodes in the magnetic quiver.

A.4.2 Resolution and 5d SCFT
The resolution sequence is 3 , x The non-vanishing triple intersection numbers are This theory does not have a 5d IR gauge theory description. Nonetheless, the theory is a descendant of 6d (E 7 , SO (7)) conformal matter as well, with the following CFD transitions The theory on the bottom-right has the correct r F and G F . There is no valid BG-CFD embedding.
The CB spectrum cannot match any set of Casimir operators of Lie groups, and there is no Lagrangian description of T 4d X . Again there is a conjectural generalised quiver that matches all the data of this model (flavor rank and CB spectrum) Similarly, we gauge an SO(12) × SU (9) subgroup of the SO(30) × U (1) flavor symmetry of D 2 (SO(28)).

A.5.2 Resolution and 5d SCFT
Resolution sequence that fully resolves this model is 3 , δ The intersection relations among the compact surfaces S i : δ i = 0 are Note that S 1 · S 2 · S 6 = S 1 · S 3 · S 6 = S 2 · S 3 · S 6 = 1, the curve S 1 · S 2 = S 2 · S 3 = S 1 · S 3 coincides, S 1 · S 2 · S 3 = −1. Since the curve S 3 · S 1 is a section curve on S 3 and a ruling curve on S 1 , the geometry does not have a consistent ruling structure. The model has the following basic characteristics

A.6.1 Deformations and Magnetic Quiver
From the CB spectrum we can identify the magnetic quiver as MQ (5) = A.6.2 Resolution and 5d SCFT The following sequence of resolutions smoothes this model 3 , x 3 , x The triple intersection numbers of S i : δ i = 0 are given in the following diagram: S 2 (7) S 4 (5) S 3 (5) For the non-Abelian part of flavor symmetry G F , it comes from the extra (−2)-curves on S 6 . The topology of S 6 can be chosen as gdP 8 of type E 7 + A 1 [89], and the (−2)-curves arranged in an E 7 Dynkin diagram exactly gives rise to G F,nA = E 7 .
A.7 Rank 9: {4,1}{4,2,4,4} We consider the type {4, 1}{4, 2, 4, 4} singularity, with the data 36) In this case the CB spectrum of T 4d X contains operators with ∆ = 13 and ∆ = 16, but there is no operator with ∆ = 14 or 15. The CB spectrum cannot match any set of Casimir operators of Lie groups, and there is no Lagrangian description of T 4d X . It would be interesting to identify similarly to the other models a generalised quiver.
For the flavor symmetry G F , note that the topology of S 11 is exactly the same as the topology of S 6 in the rank-6 model (A.24). Following the same argument based on extra (−2)-curves on S 11 , we get Magnetic Quiver (MQ (5) ).