4d $\mathcal{N}=2$ SCFTs and lisse W-algebras

We continue our studies of the correspondence between 4d $\mathcal{N}=2$ SCFTs and 2d W-algebras. The purpose of this paper is to study the relationship between 2d lisse W-algebras and their 4d SCFT partners. The lisse W-algebra is the W-algebra whose associated Zhu's $C_2$ algebra is finite dimensional. As the associated variety of Zhu's $C_2$ algebra is identified with the Higgs branch in the 4d/2d correspondence, the lisse condition is equivalent to the absence of the Higgs branch on the 4d side. We classify 4d $\mathcal{N}=2$ SCFTs which do not admit Higgs branch, then these theories would give lisse W-algebras through the 4d/2d correspondence. In particular, we predict the existence of a large class of new non-admissible lisse W-algebras, which have not been studied before. The 4d theories corresponding to lisse W-algebra can appear in the Higgs branches of generic 4d $\mathcal{N}=2$ SCFTs, therefore they are crucial to understand the Higgs branches of $\mathcal{N}=2$ SCFTs.


Introduction
The correspondence between the Schur sector of 4d N = 2 superconformal field theories (SCFTs) and 2d vertex operator algebras (VOAs) is very useful in understanding both 4d and 2d theories [1]. It is now clear that results from one side of the correspondence can provide invaluable insights to the other side. In this paper, we continue our studies between 4d N = 2 Argyres-Douglas (AD) theories and 2d W-algebras (See [2][3][4][5] for our previous studies), and provide a systematic description of the so-called lisse W-algebras and 4d N = 2 SCFTs. Rational VOAs have attracted lots of attentions in mathematics and physics literature [6,7]. Rational VOAs arise from the study of 2d rational conformal field theories [6], and studies of such VOAs turn out to be quite useful in understanding the general structure of VOAs and conformal field theories. Rational VOAs also have interesting applications in other branches of mathematics and physics [8], i.e. three dimensional topological field theory, condensed matter physics, and etc. Given a VOA, it is in general difficult to determine whether it is rational or not. Zhu constructed a non-commutative algebra called Zhu's algebra [9] and reduced the study of rationality of VOAs to representation properties of Zhu's algebra, which is simpler. Zhu also defined a commutative algebra called Zhu's C 2 algebra, and a VOA is called C 2 co-finite if the corresponding Zhu's C 2 algebra is finite dimensional. C 2 co-finiteness is closed related to rationality 1 and is much easier to check in practice. A C 2 co-finite VOA is also called lisse [10]. Now in the context of the 4d/2d correspondence [1], Zhu's C 2 algebra also plays a crucial role: the associated variety of Zhu's C 2 algebra is identified with the Higgs branch of 4d N = 2 theories [3,11,12]. Since the associated variety of the lisse VOA is trivial, the lisse condition of a 2d VOA is just equivalent to the absence of the Higgs branch of the corresponding 4d N = 2 theory. While it is in general quite complicated to compute Zhu's C 2 algebra in the VOA context, it is a lot easier to check whether a 4d N = 2 theory has a Higgs branch by using 4d methods, i.e. the geometry of Coulomb branch.
We perform a systematic search for 4d theories which do not admit a Higgs branch. The two major classes of 4d N = 2 SCFTs are class-S theories [13,14] constructed using 6d (2, 0) SCFTs and theories constructed from three dimensional canonical singularities [15]. Class-S theories always have a Higgs branch if they are constructed using regular singularities only [13], so we are forced to consider AD theories which are constructed by using an irregular singularity and a possible regular singularity [14]. In the paper we first classify irregular singularities which do not contribute to the Higgs branch using the identification between the irregular singularity and the 3-fold singularity [16,17], then we study theories engineered using an irregular singularity and a regular singularity, and it is much harder to classify the ones without Higgs branches. To accomplish this task, we develop new methods to compute the associated variety of non-admissible affine Kac-Moody (AKM) algebra, then complete the classification of lisse W-algebras arising from 4d theories constructed using 6d (2, 0) theory. 1 It is an open question to prove that C2 co-finiteness implies rationality.
-2 -For theories constructed using three fold singularities, the Higgs branch is identified as the crepant resolution of the three dimensional canonical singularity [18]. So the classification of theory without Higgs branch is reduced to finding all 3-fold singularity with a C * action, and without the crepant resolution. Such singularity is also called a Q-factorial terminal singularity. While it might not be easy to give a classification of such singularities using algebraic geometrical method, it is much easier to check using the input of 4d physics. For example, a Gorenstein terminal singularity is Q-factorial if the corresponding 4d theory has no mass deformation, and this can be easily verified using the Coulomb branch solution.
The main purpose of this paper is to classify 4d N = 2 SCFTs which do not have a Higgs branch. According to the identification between Zhu's C 2 algebra and Higgs branch, each such 4d SCFT would give a 2d lisse VOA. The list of principal lisse W-algebras 2 are listed in table 11, 13, and 16. Some more examples are found by adding one regular singularity to the irregular singularity, see table 9, 10, 12, 15, 21 and 23. We would like to highlight two interesting aspects of our results: • Previous studies in VOA literature mainly focuses on studies of so-called admissible lisse W-algebras [19]. Our results show that there are actually a large class of nonadmissible lisse W-algebras. It is definitely interesting to study the mathematical and physical consequences of these new lisse W-algebras.
• The 4d theory corresponding to the lisse W-algebra has no Higgs branch, so they can be the IR theory in the Higgs branch deformation of a N = 2 theory. Therefore, the classification of these 4d theories are crucial to understand the Higgs branch behavior of generic 4d N = 2 SCFTs. More physical consequences of these theories on space of 4d N = 2 SCFT would be discussed somewhere else.
This paper is organized as follows: section 2 reviews basic facts of lisse VOAs and their relations to the absence of Higgs branches of 4d N = 2 SCFTs. Section 3 reviews the space of 4d N = 2 SCFTs and how their Higgs branch can be computed. Section 4 classifies 4d theories which do not have a Higgs branch. Section 5 studies 4d theories whose 2d VOAs are admissible lisse W-algebra. Section 6 studies non-admissible lisse Walgebra with classical Lie algebra. Section 7 studies non-admissible lisse W-algebra with exceptional Lie algebra. Finally a conclusion is given in section 8.

Lisse VOAs
A VOA arises as the chiral part of a 2d conformal field theory (CFT). In math literature, a VOA is a vector space V with following properties (V can be thought as the vacuum module of the chiral part of a 2d CFT) [20]: • A vacuum vector |0 . 2 The principal lisse W-algebra means that we use principal nilpotent element to define our W-algebra.
-3 - • A linear map Y : V → F(V ), a → Y (a, z) = n a n z −n−1 = a(z), (2.1) where a n ∈ End(V ). This is just the state-operator correspondence 3 . Given an operator a(z), one can recover the corresponding state |a(z) = lim z→0 a(z)|0 .
For our purpose, we need to consider the VOA with a conformal vector ω, which is nothing but the chiral part T (z) of the stress tensor. Modes in the expansion of T (z) = L n z −n−2 satisfy the Virasoro algebra (by using the standard contour integral and OPE of T (z)) [L n , L m ] = (n − m)L n+m + c(n 3 − n) 12 δ n+m,0 .
Here c is the central charge, an important quantity associated with a VOA. The normal order product of two fields a(z) and b(z) is denoted as : ab : (z), and its modes are (: ab : (z)) m = n≤−ha a n b m−n + n>−ha b m−n a n (2.3) In current convention we set h a = 1. Other properties of VOAs can be found in [20].
A VOA is called rational if • V has finite number of irreducible representations M j .
• The character ch j = tr M j (e 2πiτ (L 0 − c 24 ) ) converges to a holomorphic function on upper half plane C + 4 .
• Functions ch j span an sl 2 (Z) invariant space.
A VOA is called finitely strongly generated if there is finite number of elements a i ∈ V, i = 1, . . . , s such that the whole VOA is spanned by normal order products of the following form : ∂ k 1 a 1 . . . ∂ ks a s : . (2.4) Notice that the choice of generators may not be unique and in general there are relations among the above basis. It is an interesting problem to find a minimal generating set given a finitely strongly generated VOA. For a VOA V , there exists a Li's filtration [21] which is a decreasing filtration in which each F p is spanned by following states In physics literature, the mode expansion of a field takes the form anz −n−h with h being the scaling dimension. In VOA literature, however, the above convention of mode expansion without further shift by the scaling dimension h is used so that VOAs without the definition of scaling dimension can be considered. 4 We use ch to denote the character of a VOA and χ to denote the character of a finite Lie algebra.
-4 -then one can define a graded sum of the VOA It is obvious that F 0 = V , and F 1 is generated by {a −2 b|a ∈ V, b ∈ V }. The Zhu's C 2 algebra is defined as R V is a Poisson algebra and is finitely generated if and only if V is strongly finitely generated. Moreover the image of generators of V in R V generates R V as well. Notice that R V is in general not reduced, namely the ideal defining it would contain a nilpotent element 5 . Once we determine R V , we have now an associated scheme and an associated variety X V defined from the Zhu's C 2 algebrã A lisse VOA is defined as the VOA such that dim(X V ) = 0, which also implies that Zhu's C 2 algebra is finite dimensional. A rational VOA has to be lisse, but it is an open problem that whether a lisse VOA has to be rational. A quasi-lisse VOA is defined as the VOA whose associated variety X V has finite number of symplectic leaves. Quasi-lisse VOAs have many interesting properties [11,22]: • The VOA is strongly finitely generated.
• The Virasoro vector ω V is nilpotent in R V .
• There are finite number of ordinary modules, and they transform nicely under modular transformations. A weak V -module (M, Y M ) is called ordinary if L 0 acts semisimply on M , which means that any L 0 -eigenspace M ∆ of M of eigenvalue ∆ ∈ C is finite-dimensional, and for any ∆ ∈ C, M ∆−n = 0 for all sufficiently large n ∈ Z.
• Characters are solutions of a modular differential equation.

Lisse W-algebras
A class of interesting VOAs are constructed from quantum Drinfeld-Sokolov (qDS) reduction of affine Kac-Moody (AKM) algebras. Namely, we start with an AKM algebra V k (g), where k is the level, and g is a simple Lie algebra, and then choose a nilpotent element f of g to perform the qDS reduction. The W-algebra obtained in this way is labeled as Now an AKM algebra is called principal admissible if the level k takes the following value

11)
5 A nilpotent element x of an ideal is an element not in I but x n ∈ I for some n.
-5 -where r ∨ is the lacety for the Lie algebra g 6 , and h ∨ is the dual Coxeter number. If we start with an admissible AKM algebra, the resulting W-algebra after qDS reduction is called admissible W-algebra. If p = h ∨ , the AKM and the corresponding W-algebra is called boundary admissible [23]. Other cases are called non-admissible. A very interesting question is to classify all possible lisse W-algebras. The admissible case has been studied in [7,24].

4d/2d correspondence
It was proposed in [1] that one can get a 2d VOA from the Schur sector of a 4d N = 2 SCFT (see also in [2-5, 11, 12, 17, 22, 25-68] for recent developments), and the basic 4d/2d dictionary used in current paper is [1]: • There is an AKM sub-algebra (V k 2d (g)) in 2d VOA, where g is the Lie algebra of 4d flavor symmetry G F .
• The 2d central charge c 2d and the level of AKM algebra k 2d are related to the 4d central charge c 4d and the flavor central charge k F as (2.12) • The (normalized) vacuum character of 2d VOA is the 4d Schur index I(q).

Lisse VOAs and the absence of Higgs branches
Given a 2d VOA corresponding to a 4d N = 2 SCFT, it is conjectured that the associated variety is identified with the Higgs branch of the 4d theory [3,11,69]. We are particularly interested in the lisse VOA whose Zhu's C 2 algebra is finite dimensional, and would like to explore what the exact meaning of the lisse property of 2d VOAs for 4d N = 2 SCFTs. A 4d N = 2 SCFT has a bosonic symmetry group SO(2, 4) × SU (2) R × U (1) r × G F , where SO(2, 4) is the 4d conformal group, SU (2) R × U (1) r is the R symmetry group which exists for every N = 2 SCFT, and G F is the flavor symmetry group which might be absent for some theories. The representation theory of 4d N = 2 superconformal algebra is studied in [70], in which short representations (where the primary states in the supermultiplet are annihilated by a fraction of supercharges) were classified. Important half-BPS operators include primary operators of multiplets E r andB R .
The moduli space of vacuum of a 4d N = 2 SCFT is extremely rich. It consists of various branches. The Coulomb branch is parameterized by expectation values of the primaries of E r multiplets, and the low energy effective theory is described by a Seiberg-Witten (SW) geometry [71,72]. The set of rational numbers [r 1 , . . . , r s ] of U (1) r charges of E r (unitarity implies that r i > 1) is an important set associated to a 4d N = 2 SCFT. The U (1) r symmetry acts non-trivially on the Coulomb branch while the SU (2) R × G F 6 r ∨ =1 for ADE Lie algebra, r ∨ = 2 for BCF Lie algebra, and r ∨ = 3 for G2 algebra. 7 Our normalization of kF is half of that of [1,25].
-6 -symmetry acts trivially. The set of Coulomb branch operators E r can be found from the SW geometry. E r operators form a ring which might be called the Coulomb branch chiral ring, and it is freely generated for almost all theories we know.
The Higgs branch is parameterized by expectation values of the primaries of half-BPS operatorsB R . These operators also form a ring which might be called the Higgs branch chiral ring. It is believed that the Higgs branch chiral ring is finitely generated, and there are in general complicated relations among generators. It is in general difficult to determine the full set ofB R operators, and relations among them. TheB 1 type operators, on the other hand, are easily determined by the SW geometry. TheB 1 multiplet consists of conserved currents of the flavor symmetry, and one can use them to deform the 4d theory as mass deformations which can be detected from the SW geometry of the Coulomb branch: they are dimension one parameters of the SW curve. Therefore if we find a mass deformation in the SW geometry, we know for sure the existence of at least oneB 1 type operator, therefore the theory has a Higgs branch. However, if there is no mass deformation in the SW geometry, one can not claim that the theory has no Higgs branch! As it is still quite possible that otherB R type operators with R > 1 exist. There are many such examples in class-S theories as we will review in later sections.
More generally, we could have a branch parameterized by expectation values of both E r andB R type operators. In general, this branch is a direct product of a component parameterized by E r type operators and a component parameterized byB R type operators. This type of branch is called the mixed branch. Now we discuss the low energy behavior in various branches. The Coulomb branch is parameterized by expectation values of E r type operators, and the low energy effective theory at generic point consists of free vector multiplets, free hypermultiplets, and interacting theories whose Coulomb branch is trivial (see [18] for examples of this type). In the literature, one mainly consider theories whose Coulomb branch consists of only free vectormultiplets. More general cases whose Coulomb branch consist of free hypermultiplets and free vector multiplets are considered in [73].
The Higgs branch is parameterized by expectation values ofB R type operators, and the low energy effective theory at generic point would also involve free hypermultiplets, free vector multiplets, and an interacting theory which does not have a Higgs branch. In literature, people mainly consider theories whose Higgs branch only consists of free hypermultiplets, and our studies in this paper shows that it is quite common that the Higgs branch of a general 4d N = 2 SCFT consists of an interacting theory and/or free vector multiplets.
Let us further clarify some definitions used in the literature: a branch of N = 2 moduli space is an irreducible component of the variety defined by full chiral ring of an N = 2 theory. A Coulomb branch is called pure Coulomb branch if the low energy effective theory on the generic point consists of only abelian gauge theory. A Higgs branch is called pure -7 -Higgs branch if the low energy effective theory on the generic point consists of only free hypermultiplets.
We now state the exact meaning of the lisse property for 4d N = 2 SCFTs. The corresponding 2d VOA counts Schur operators and in particular countB R operators, and the reduced Zhu's C 2 algebra is conjectured to be the chiral ring forB R operators. Assuming that the Higgs branch chiral ring is reduced, we have the following conjecture: • A 2d VOA corresponding to a 4d N = 2 SCFT is lisse if the moduli space of the 4d theory has no Higgs branch, or equivalently the 4d theory has noB R type operators.
3 Space of N = 2 SCFTs and their Higgs branch One can engineer a large class of 4d N = 2 SCFTs by starting with a 6d (2, 0) theory of type j = ADE on a sphere with an irregular singularity and at most one regular singularity [13,14,16,17,74] 8 . Its Coulomb branch is captured by a Hitchin system with singular boundary conditions near singularities. The Higgs field Φ of the Hitchin system near the irregular singularity takes the following form Here T is determined by a positive principal grading of Lie algebra j [75], and is a regular semi-simple element of j. k is an integer greater than −b. Subsequent terms are chosen such that they are compatible with the leading order term (essentially the grading determines the choice of these terms). We call them J (b) [k] type irregular punctures, and summarize allowed values of b in table 1 (k and b are not required to be coprime). Theories constructed using the above irregular singularity alone can also be engineered by using a three (complex) dimensional singularity in type IIB string theory which are also summarized in table 1 [15]. One can add another regular singularity which is labeled by a nilpotent orbit f of j (We use Nahm labels such that the trivial orbit corresponding to the regular puncture with maximal flavor symmetry). Detailed discussion about these defects can be found in [76]. All in all, the theory we consider could be labeled by a pair ( To get non-simply laced flavor groups, we need to consider the outer-automorphism twist of ADE Lie algebra and its Langlands dual. A systematic study of these AD theories was performed in [17]. Denoting the twisted Lie algebra of j as g ∨ and its Langlands dual as g, outer-automorphisms and twisted algebras of j are summarized in table 2. The irregular singularity of regular semi-simple type is also classified as in table 3 with the following form, 8 See the appendix A of [4] for relations between this construction and other constructions.  [16]. These 3d singularity is very useful in extracting the Coulomb branch spectrum, see [15]. Table 2: Outer-automorphisms of simple Lie algebras j, its invariant sub-algebra g ∨ and flavor symmetry g from the Langlands dual of g ∨ . Here T t is an element of Lie algebra g ∨ or other parts of the decomposition of j under the outer automorphism. k t > −b t , and the novel thing is that k t could take half-integer value or in thirds (g = G 2 ). One can also represent those irregular singularities by 3-fold singularities as in table 3. We could again add a twisted regular puncture labeled also by a nilpotent orbit f in g.
Some interesting physical properties about these theories are: • If there is no mass parameter in the irregular singularity (the constraint on k will be listed in the next section), the corresponding VOA is conjectured to be given by the following W algebra [17]: where h ∨ is the dual Coxeter number of g, n is the number listed in table 4, and k ′ is restricted to the value such that no mass parameter is in the irregular singularity. Table 3: SW geometry of twisted theories at the SCFT point. Here we also list the scaling dimension of coordinate z. All k's in this table are integer valued and the power of z coordinate in singularity is equal to k t used in equation 3.2.
k ′ is related to the value k in table 1 and 3 as follows: a) n = 1,  n is the number which appears in the level of 2d W-algebra, see 3.3.
• The SW geometry of these theories is identified with the spectral curve of the corresponding Hitchin system [77]: and one can read off the Coulomb branch spectrum from an associated Newton polygon [14,16,17], which is also reviewed in the appendix B of [4].
Remark: We could also consider theories constructed using regular singularities only, and such theories are studied extensively in [13].

4d N = 2 SCFTs from 3-fold singularities
One can also engineer a large class of 4d N = 2 SCFTs by putting type IIB string theory on a 3-fold canonical singularity with a C * action [15]. We first review the hypersurface case: starting with an isolated singularity which is defined by a polynomial f : (C 4 , 0) → (C, 0), which is required to have a C * action satisfying following condition The SW geometry of the theory defined by f is identified with the mini-versal deformation of f , i.e.
with φ α being a monomial basis of the Jacobi algebra and λ α is identified with the Coulomb branch parameters with scaling dimension Here Q α is the charge of φ α under the C * action. In particular, if [λ α ] = 1, it is a mass parameter. Theories constructed using complete intersection singularity are discussed in [78,79]. Theories which do not have Coulomb branch are discussed in [18]. Some further physical aspects of those theories are discussed in [80,81].

The Higgs branch
As reviewed in the last subsection, one can find the SW geometry for the Coulomb branch once the geometric data specifying the theory is given. The Higgs branch, on the other hand, is more complicated and less studied. However, the basic ideas for studying the Higgs branch were actually already given in [18,82], and we review those methods first.

4d SCFTs from (2, 0) theories
Higgs branches of 4d SCFTs from 6d (2, 0) theories have been studied in [82]. Taking theories constructed from 6d A N (2, 0) theory as examples, the low energy effective theory of N M5 branes is a 6d (2, 0) theory with transverse deformations described by 5 real scalars. Then put the (2, 0) theory on a Riemann surface Σ with certain partial topological twist on Σ so that the 4d theory has N = 2 supersymmetry. After the twist, the transverse deformations are described by two complex scalars Φ 1 and Φ 2 defined on Σ. Φ 1 is a section of the canonical bundle on Σ, while Φ 2 is a section of the trivial bundle. The low energy effective theory is described by the following spectral curve where φ i is a section of K i Σ , and c i are constants. Above equations are independent. However, the two Higgs fields Φ 1 and Φ 2 are actually commuting [83], which give a holomorphic factorization condition on the above curve. Different branches are given by following curves: where n i = N and c i = 0. Assuming n 1 ≥ n 2 ≥ . . . ≥ n s , the branch is then labeled by a Young Tableaux [n 1 , . . . , n s ], although not all Young Tableaux can appear. Here are some further remarks: • It is always possible to take the Young Tableaux Y = [N ] (A single column of N boxes), and all cs are equal to zero, so we have a Coulomb branch.
• The Higgs branch appears if Y = [1 N ] (A single row of N boxes), and v N = 0. Notice that this is not always possible.
• If our theory is constructed using regular singularities only, it is always possible to have a Higgs branch or a mixed branch, as we can always turn off all the deformations in v direction, i.e. v N = 0, and it has at least a N − 1 dimensional Higgs branch.
So to find a theory without Higgs branch, we have to look at the theory constructed using irregular singularities. Furthermore, the necessary condition is that there is no holomorphic factorization of the Coulomb branch spectral curve! This can be easily checked for the A N −1 type theory, as the SW curve at the SCFT point takes the form x N + z k = 0, and it is easy to see that no holomorphic factorization implies that N and k are coprime. For other Lie algebras, this condition is not easy to check, but luckily it can be translated into a much simpler criteria which will be discussed in the next section. Things become more complicated if there is a regular singularity, as one could have local contribution to Higgs branch. The detailed discussion on regular singularities will also be given in next section.
Example 1: Consider a theory engineered by putting 6d A 1 (2, 0) theory on a sphere with the following irregular singularity We have only one branch which is the usual Coulomb branch. This theory has no Higgs branch. Example 2: Consider a theory engineered by putting 6d A 1 (2, 0) theory on a sphere with the following irregular singularity and there are two branches Branch I is a pure Coulomb branch, while branch II is a mixed branch.

4d SCFTs from 3-fold singularities
For a 4d theory engineered from a 3-fold singularity, its Coulomb branch is described by a mini-versal deformation of the singularity [15]. The Higgs branch, on the other hand, is described by its crepant resolution [18]. A crucial theorem by Kawamata [84] is that every 3-fold canonical singularity has a partial crepant resolution f : Y → X, where Y is a Q-factorial terminal singularity. Gorenstein terminal singularity takes the following form, 14) Here f ADE (x, y, z) is the familiar 2d ADE singularity. g(x, y, z, t) is a polynomial such that the singularity is isolated. Other terminal singularities are constructed from the quotient of Gorenstein terminal singularities. The crepant resolution of a three-fold singularity is not unique, but the number of crepant divisor C(X) is constant. Given a partial resolution, the number of free hypermultiplets is given by b 2 (Y ) which is given by following formula [85]: Here ρ(X) is the rank of local class group of the singularity. In the hypersurface case, ρ(x) is identified with the number of mass parameter in the SW geometry. We are looking for the singularity X with b 2 (Y ) = 0, therefore • C(X) has to be zero, which implies that X has to be a terminal singularity.
• ρ(X) is zero, which further implies that X is a Q-factorial terminal singularity.
The conclusion is then theories without Higgs branch corresponds to Q-factorial terminal singularity! In general b 3 (Y ) is nonzero, and this means that the low energy theory in the Higgs branch contains free vector multiplets. Moreover, if Y is not smooth, one can have an interacting theory in the IR which is described by the singularity of Y .
Example: Consider the singularity X: x 3 + y 3 + z 3 + t n = 0, and it is possible to do a weighted blow up so that the resolution is a crepant resolution, the new variety has singularity of type x 3 + y 3 + z 3 + t n−3 = 0. So we have a chain of blow ups The ending variety X s is either a non-singular variety if n ≡ 0 or 1 mod 3, or a Q-factorial terminal singularity if n ≡ 2 mod 3. The topology of X s is then computed as follows. We have b 2 (X s ) = C(X) + ρ(X) with and The third Betti number b 3 (X s ) can be computed as All in all we have following interesting situations:

Absence of mass deformation: Classification
Now we would like to classify 4d SCFTs which do not have Higgs branch. A necessary condition is that the theory does not have a mass deformation, as the existence of mass deformation implies the existence ofB 1 operators. We first consider 4d theories constructed from 6d (2, 0) theories, and classify irregular singularities which do not have mass parameters. We then list those regular singularities which do not have a flavor symmetry and therefore do not have mass deformation. For theories constructed from three fold singularities, the Higgs branch deformation is identified with the crepant resolution of the singularity, and a very powerful theorem by Kawamata reduces the classification of theory without Higgs branch to the classification of Q-factorial terminal singularity. The classification of quasi-homogeneous Gorenstein Q-factorial terminal singularities actually coincides with the classification of special irregular singularity which do not have mass parameter in class S construction.

Untwisted theory and Q factorial Gorenstein terminal singularity
Let's consider class S theory constructed using 6d (2, 0) theory of type g = ADE, and f to be a regular nilpotent orbit of g. The same theory can be then engineered from the three fold singularity listed in table 1. They can be written in the form with f ADE (x, y, z) being the famous 2d ADE singularity. The Higgs branch of these theories is conjectured to be described by crepant resolutions of these singularities. So we would like to classify above singularities which do not admit crepant resolutions. The singularity of above type is called Gorenstein terminal singularity, and those which do not admit crepant resolution is called Q-factorial terminal singularity [84], which is actually equivalent to the fact that the corresponding 4d N = 2 SCFT does not admit a mass parameter. Such singularities have already been classified in [2], and the complete list is shown in table 5. Table 5: Constraint on k so that irregular singularity denoted by J b [k] has no mass deformation.

Twisted theory
Now we would like to consider the classification of twisted irregular singularity which does not admit a Higgs branch. We do not know the geometric criteria for them yet, however, we would still like to conjecture that the absence mass parameters in the Coulomb branch implies that there is no higgs branch! The complete list of these irregular singularities is shown in table 6.
-15 -j with twist SW geometry at SCFT point no mass and k even Table 6: Constraint on twisted irregular singularity which does not have mass deformation.

Classification using the positive grading
As described in [16], the irregular singularity described in section 2 is classified by the positive grading of regular semi-simple type 9 , i.e.
The coefficient of the z −1 term in Higgs field takes value in g 0 . The coefficient T of the leading order term is chosen as a regular semi-simple element in g 1 , and the number of mass parameter is determined by the dimension of {g T 0 } which are all semi-simple elements in g 0 which also commute with T .
Some positive gradings of regular semi-simple type are determined by a nilpotent element e, and the number of mass parameter has a simple description. Each nilpotent element gives a sl 2 triple φ, and g T 0 is given by g φ (this is the sub-algebra which commutes with sl 2 triple). In particular, if b = h ∨ and (k, h ∨ ) = 1, the nilpotent element is given by the regular nilpotent element whose flavor symmetry is trivial, so the corresponding irregular singularity does not have any flavor symmetries. This agrees with results shown in table. 5, which is derived using the explicit computation of Coulomb branch spectrum.

Regular singularity: absence of mass deformation
A necessary condition for the absence of Higgs branch is that there is no flavor symmetry in the regular singularity. Such regular singularities can be found from the table listed in [87][88][89][90][91][92][93][94]. Here we reproduce them in table 7. 9 We take g to be simply-laced here. g

No flavor symmetry
[n 1 , . . . , n s ],n i = n j and n i are all odd C N [n 1 , . . . , n s ],n i = n j and n i are all even D N [n 1 , . . . , n s ],n i = n j and n i are all odd Table 7: Nilpotent orbits which do not have flavor symmetry. A nilpotent orbit is labeled by a partition for classical groups, and by a Bala-Carter label for exceptional groups.

Admissible lisse W-algebras
Considering 4d N = 2 SCFTs constructed from 6d (2, 0) theory on a sphere with an irregular singularity and a regular singularity discussed in section 2, the necessary condition for a theory in this class to have no Higgs branch is that the irregular singularity does not admit mass deformation, which provides a constraint on the irregular singularity (such singularities are summarized in table 5 and 6). The choice of the regular singularity is subtle on the other hand. Naively, we would think that we have to choose regular singularity which do not have a flavor symmetry, however this is not that simple for following reasons. First of all, there is still possible contribution to Higgs branch from regular singularity even if there is no mass deformation in it, and we have to choose very special regular singularity (whose choice depends on the choice of irregular singularity) to make sure there is no Higgs branch. The main tool to select such special regular singularities is the associated variety of the AKM algebra which corresponds to the theory defined by the same irregular singularity and a regular singularity with trivial nilpotent orbit. Second of all, even if a regular puncture carries a flavor symmetry, it is possible that the 4d theory actually does not have such flavor symmetry which is related to the so-called collapsing levels of 2d W-algebra [4].
In this section, we focus on the irregular singularity such that the level of the corresponding W-algebra is admissible, namely, the level takes the following form Here (k ′ , h ∨ ) = 1 10 , see equation 3.3 for the convention. The associated variety of the where S f is the Slowdoy slice associated with the nilpotent orbit f , O u is the nilpotent orbit depending on u, see tables in [19], and O u is the closure of O u . O u is actually the Higgs branch of the theory defined by an irregular singularity determined by the pair (h ∨ , u) and a regular puncture labeled by trivial nilpotent orbit (the corresponding VOA is the AKM with level 5.1). So the regular singularity f should be taken as an and the corresponding 4d theory would have no Higgs branch! Since O u is determined by the data of irregular singularities, the above condition further restrict the choice of f given the choice of irregular singularity. We first consider all f 's which do not have flavor symmetries which are summarized in 8. More data of those 4d theories defined using exceptional Lie algebra are listed in table 9. Other cases where the regular singularity f has non-trivial flavor symmetry but the whole theory has no Higgs branch are summarized in table 10 and 12. g Exceptional Pair (f, u), and gcd(u, Table 8: f is a nilpotent element labelling a regular singularity. u specifies the irregular singularity. The level k of the corresponding 2d VOA is k = −h ∨ + h ∨ u . These pairs give rise to 4d N = 2 SCFTs with no Higgs branch. Here we list only those regular singularities without mass deformation.
For theories whose VOA is admissible lisse W-algebra, we can compute following things: • Coulomb branch spectrum: One can compute the Coulomb branch spectrum of these theories using the spectral curve of the corresponding Hitchin system. The irregular singularity part can be found from the Newton polygon, and one also need to use the pole structure of regular singularities listed in [87][88][89][90][91][92][93][94].
(g, f, u) Coulomb branch Schur index 4d Isomorphism W-algebra Vir 2,7 Table 10: Theories without the Higgs branch but whose regular singularity has a flavor symmetry (mass deformation). f denotes the nilpotent orbit of the regular singularity, and u specifies the irregular singularity. The corresponding W algebra is of SU (2) group. In our case u = q + k ′ , and (u, h ∨ ) = 1, (u, r) = 1. Here h ∨ is the dual Coxeter number, and r is the lacety of lie algebra.
• Zhu's C 2 algebra: Zhu's C 2 algebra can be found from Jacobi algebra of an isolated hypersurface singularity [5].
Here are some further remarks about those 4d theories whose VOA is admissible lisse -19 -W-algebras: 1 If u > h ∨ , the nilpotent orbit O u is just the regular nilpotent orbit, so f has to be taken to regular nilpotent orbit. The corresponding W-algebra is just the minimal model of W-algebra listed in table 11. Some of the physical information such as the Coulomb branch, the Schur index, central charge, etc, can be found in [5]. These might be called admissible principal lisse W-algebra. These theories (for g = ADE ) can be engineered by following singularity and it is also labeled as (G, A k−1 ) (or (A k−1 , G) theory) theory using the notation of [4]. Here G is the ADE type appearing in above singularity.
2 If u < h ∨ and for classical lie algebra, we can actually see from the index that most of them are isomorphic to an admissible principal lisse W-algebra. For example, consider D type case with the regular singularity of the form [u, s] (here u iand s are odd integers), and the corresponding W-algebra is Using the index formula in [5], we have We now recognize that this is actually the vacuum character of W −(s−2)+ s−2 u (so s , f prin )! In fact, one can check that 4d Coulomb branch spectrum agrees with each other. 3 The data for exceptional case is listed in table 9 and 10. Almost all of them is actually isomorphic to principal W-algebras (All of them are admissible principal W-algebra except E 8 theory with u = 29.). Table 11: 4d SCFT whose VOA is admissible lisse W-algebra. Here k is constrained so that 4d theory has no mass deformation and the level of W-algebra is admissible. Namely, (k, h ∨ ) = 1 for ADE case, (2k + 1, h ∨ ) = 1 for BCF case, and finally (3k ± 1, h ∨ ) = 1 for G 2 case! The corresponding field theory has no exact marginal deformation.
g Nilpotent orbit f Vacuum character VOA Isomorphism , where g is classical lie algebra. u is an odd integer and (u, h ∨ ) = 1.
6 Non-admissible lisse W-algebra: Classical Lie algebra Now consider 4d N = 2 SCFT whose associated VOA is non-admissible lisse W-algebra with classical Lie algebra. The 4d theories are constructed from 6d (2, 0) theory of A and D type (We also consider outer automorphism twist so that VOAs also include W-algebra with non-simply-laced classical Lie algebra). The irregular singularity is chosen so that there is no mass parameter in it, as in table 5 and 6, and if the regular singularity is chosen to be the principal one, the corresponding W-algebra is conjectured to be lisse. Such theories are summarized in table 13.
-21 -j with twist SW geometry at SCFT point n even and g = 2N n even n u Table 13: 4d N = 2 SCFT whose 2d VOA is non-admissible lisse W-algebra defined using classical Lie algebra! u is coprime with n, and u is always odd.

A physical derivation of associated variety
Now we consider adding a regular singularity of other type instead of the principal one, and try to find the criteria on f such that the resulting theory has no Higgs branch. As discussed in last section, the crucial input is the associated variety of W-algebra with trivial nilpotent orbit, and the corresponding W-algebra is actually the AKM In our case the growth function of VOA (6.1) can be summarized by the following formula with d g (n) summarized in table 14. We found this formula by explicitly computing the 4d central charges.
The lisse condition on 4d theory is equivalent to the following choices of irregular and regular singularity: • The irregular singularity is restricted to those listed in table 13.
• The regular singularity is chosen such that f ∈ O g,n,u , therefore S f ∩ O g,n,u is trivial.
So the choice of regular singularity is determined once we find the associated variety O g,n,u .
Unlike the admissible case, associated varieties of VOA (6.1) are not known in the mathematics literature. Here we will use a simple physical method to compute it. The idea is as the following: our 4d theory is engineered by a 6d (2, 0) theory on a sphere with -22 -j with twist g Constraint on n d g (n) A 2N /Z 2 sp 2N n odd and g = 2N +1 n odd g 2 n 2 − 3n 2 + g sp 2N n odd and g = 2N n even g 2 n 2 + g A 2N −1 /Z 2 so 2N +1 n odd and g = 2N −1 n odd g 2 n 2 + 3n 2 + g so 2N +1 n odd and g = 2N n even g 2 n 2 + g D N /Z 2 sp 2N −2 n even and g = 2N n odd g 2 n 2 − 3n 2 + g sp 2N −2 n even and g = 2N −2 n even g 2 n 2 + g D N so 2N n even and g = 2N −2 n odd g 2 n 2 + 3n 2 + g so 2N n even and g = 2N n even g 2 n 2 + g Table 14: Data required to compute the growth function G of VOA V −h ∨ + n ru (g). G = dim(g) − dg(n) u .
an irregular and a regular singularity labeled by a trivial nilpotent orbit. Going to the Higgs branch of this theory is equivalent to closing the regular puncture, in other words, changing the type of regular puncture (equivalent to choice a different nilpotent orbit) as sketched in figure 1. The IR theory consists of free hypers and an interacting theory which is described by 6d (2, 0) theory on a sphere with one irregular and one regular puncture labeled by the nilpotent element f . We would like to choose the right f such that O f gives the Higgs branch of original theory (the associated variety of the non-admissible AKM).
To accomplish that, f has to be chosen as the following: 1. The number of Coulomb branch operators in the leading order differential of the corresponding SW curve defined by f should be non-negative. Since otherwise, the right configuration in figure 1 consists of free hypermultiplets, and we can not simply count the dimension of O f as the dimension of free hypermultiplets. We also imposing the constraint that the Coulomb branch parameters in the leading order differential is as small as possible.
2. Once we use above condition to constrain possible f , we choose the maximal among them, namely, there is an order among nilpotent orbit, and we choose the maximal one among the nilpotent orbits satisfying condition 1.
Once such a f is found, we conjecture that the Higgs branch (associated variety) of the left theory (non-admissible AKM) of figure 1 is the closure of O f . Example 1: First we illustrate the idea for the simple case u > n, and in this case, we can completely close off the regular puncture, i.e. we can take the regular puncture to be labeled by the principal nilpotent orbit (which is the maximal regular singularity we can choose), and the remaining theory is defined by an irregular singularity which has no Higgs branch. So we claim that the associated variety in this case is given by the closure of principal nilpotent orbit.
Example 2: Now consider 4d theory whose VOA is V −N + N u (sl N ), here u is coprime with N so that there is no mass parameter in irregular singularity of corresponding (2, 0) -23 -  construction. We consider the case u < N . The SW curve takes the form Given a general regular puncture labeled by a Young Tableaux [h r 1 1 , h r 2 2 , . . . , h rs s ], the maximal scaling dimension of Coulomb branch operator in differential φ N is where s N is the height of N th box of the Young Tableaux (we label the Young Tableaux row by row starting with the bottom-left corner), which is equal to h 1 . The maximal one we can choose so that it satisfies condition 1 is h 1 = u, and the maximal scaling dimension in φ N is then 0. Using the partial ordering of nilpotent orbit of sl N Lie algebra described in [95], we find that the regular puncture f which satisfies condition 2 is [u, . . . , u, s], s ≤ u, which agrees with results found by Arakawa [19]. Using the condition 1 and 2, we are able to find associated varieties for non-admissible VOAs listed in table 15.
Alternatively, we notice that we can find the associated variety using the isomorphism found in [4]: our theory can be denoted as ((g, n, u), f ), and if f can be chosen so that the above theory is isomorphic to another configuration ((g ′ , n ′ , u ′ ), f prin ) which is conjectured to have no Higgs branch, then the associated variety of original theory is equal to the closure of f . This method can be used to confirm the result from using previous result, and furthermore we can find the resulting lisse W algebra using this method.
Example: Let's consider following example: consider a four dimensional theory whose 2d VOA is (here n is even, and g = 2N −2 n is odd)   The associated variety of non-admissible lisse W-algebra defined using classical Lie algebra. u is coprime with n, and u is always odd. The associated variety is the closure of the nilpotent orbit of Lie algebra used to define W-algebra.

Some details of D type theory
For the untwisted D N type theory, the irregular singularity takes following form with (n, k) = 1, and n is chosen such that there is no mass parameter in the irregular singularity: either a) n is an even divisor of 2N − 2 and 2N −2 n is odd, or b) n is an even -25 -divisor of N . T is chosen to be a regular semi-simple element of D type. We first consider adding a full regular singularity of D N type besides the irregular singularity. These theories have a weakly coupled gauge theory descriptions [96] with its quiver shown in figure 2. The corresponding 2d VOA is AKM The Schur index can be computed from the index of each individual matter system. We illustrate this by using the quiver listed in top row of figure 2. Define u = n + k. Each AD matter T u i carries flavor symmetry C i and D i (Here C means the symplectic group, and D means orthogonal group). The character of the AD matter is given by the following formula [4] ch(T u i ) = PE . (6.10) Here χ adj C i is the character of the adjoint representation of C i Lie algebra, and χ f C i is the character of the fundamental representation of C i , and etc. It is interesting to note that when u = 1, the above index reduces to the index of the free bi-fundamental hypermultiplets in C i and D i group. So our matter T u i can be thought of as an infinite sequence of matter generalizing the usual bi-fundamental matter. Now the vector multiplet (the gauge group is g) would contribute to the index by Combine the contribution from matters and vector multiplets, we have the following index formula Notice that the integral does not involve the flavor group SO(2N ) and the integral actually does depend on fugacities of SO(2N ) flavor group. This formula can be generalized to the bottom quiver in figure 2. The rule is quite simple: u = 1 is just the quiver gauge theory with free bi-fundamental hypermultiplets as matter, and we write down the index using the free field realization, and to get the index for general case one simply replaces q by q u in the full index (a prefactor which is zero for u = 1 is also needed).
The growth function can be computed using the following two formulas l odd, n even l even, n even Here we use the fact that the 2d VOA of above theories are AKM with level k 2d = −h ∨ + n u , and the relation c 4d = − 1 12 c 2d . Since we know the Coulomb branch spectrum, we can compute the central charge a 4d and therefore compute the growth function of the character G = −48(a 4d − c 4d ). We find that the growth function takes the following form with the function d so 2N (n) given by: Now we discuss the associated variety (the Higgs branch) of out theory. Consider u = 1 case where the theory is just a linear quiver with Lagrangian description and the Higgs branch is just the closure of a nilpotent orbit, which can be read from the quiver shown in figure 2.
First we recall how to associate a quiver tail to a D type nilpotent orbit [97]. For a nilpotent orbit with partition Y = [h 1 , h 2 , . . . , h J ] (where even h i appears even times), we attach a quiver tail (See appendix for brane construction): Here J is always even (The end USp gauge group might be rank zero.). The sizes are To match the UV anomaly −48(a 4d − c 4d ) (growth function), the gauge group is not completely Higgsed, and the there are l−1 2 free vector multiplet for l odd, and l 2 free vector multiplet for l even.
We can compute the Schur index or the vacuum character of non-admissible W-algebra using the gauge theory description too. We illustrate for the theory with only irregular singularity, which is in the form of 6.8 with the following constraint: We take u > n for simplicity. Now define an integer number a = [ 2N k ] and b = l − a. If a = 0, the quiver takes the same form as in figure 2, and the only difference is that T u l+1 has no SO(2N ) flavor symmetry (The matter systems and the gauge groups are different though). The index takes following form Here d i are Casmiers of SO(2N ) group. If a = 0, the quiver takes a more complicated form: if a is even, then G 0 (bn) = Sp(bn − 2), G 1 (ak) = SO(ak + 1); and if a is odd, then G 0 (bn) = SO(bn), G 1 (ak) = Sp(ak − 1). Notice that there is half-hypermultiplet coupled with Sp(k − 1) gauge group. The Schur index takes the following form: Isomorphism: if n = N with N even, then l = 1, the theory is actually defined by following singularity: x 2 + y N −1 + yz 2 + zw = 0, which is actually equivalent to (A 1 , A 2N ) theory.

Other cases
Twisted D N type theory For the twisted D N type theory, the irregular singularity takes the following form with (n, k) = 1. Here n is chosen such that there is no mass parameter in the irregular singularity: either a) n is an even divisor of 2N − 2 and 2N −2 n is even, or b) n is a even divisor of 2N , and 2N n is odd. T t is chosen to be a regular semi-simple element of C type Lie algebra. These theories also have a weakly coupled gauge theory description, see figure  3 for the theory defined by the above irregular singularity plus a regular singularity labeled by trivial nilpotent orbit.
The Schur index for the quiver gauge theory shown in figure 3 can be computed using the index for each matter content, and the result is (6.21) here ch n,u=1 (q) is the Schur index for the theory defined with u = 1, which is a Lagrangian theory. For other theories defined by the same irregular singularity and different regular singularity, we could also write down its Schur index using the weakly coupled gauge theory description, and we leave it for the interested reader.
Let us now discuss a little bit of the Higgs branch of the theory shown in figure 3. If u = 1, the theory is just a Lagrangian theory. The idea is: We start from a B N −1 partition Y B and find its associated quiver which has C N −1 type flavor symmetry (See appendix for the explanation), we get the higgs branch of the associated quiver by finding its dual partition Y d C in its Langlands dual C N −1 algebra. For the quiver shown in figure 3 with u = 1, we have:  Comparing with the UV growth, we find that in the Higgs branch there are l 2 free vector multiplets for l even, and l+1 2 free vector multiplets for l odd. Twisted sl 2N type theory: For the twisted sl 2N theory, the Higgs field takes following form Here n can be chosen as follows: either a) n is a odd divisor of 2N , or b) n is a divisor of 2N − 1. The corresponding quiver description is shown in figure 4. T t is a regular semi-simple element of C type Lie algebra.
To associate a quiver tail whose Higgs branch is B N type nilpotent orbit (so the Higgs branch has B N type flavor symmetry), we actually need to start with a nilpotent orbit of C N type with partition Y = [h 1 , h 2 , . . . , h J ] (where odd h i appears even times), then attach the following quiver tail Here J is always even (The end USp gauge group might be rank zero). The sizes are The dimension is To match the UV growth, the gauge group is not completely Higgsed, and the there are l+1 2 free vector multiplet for l odd, and l 2 free vector multiplet for l even. Twisted sl 2N +1 type theory: For the twisted sl 2N +1 theory, the Higgs field takes following form Here n can be chosen as follows: either a) n is an odd divisor of 2N , or b) n is a divisor of 2N + 1. The corresponding quiver is shown in figure 5. We can not find the Higgs branch l even, n even l odd, n even    for u = 1 theory using previous methods as the quiver can not be realized as the quiver tail of a B type partition, as the quiver constructed from B type partition would have D type and C type quiver nodes, but our quiver shown in figure. 5 has B type and C type quiver nodes.
In this section, we are going to classify non-admissible W-algebra whose corresponding four dimensional N = 2 SCFT is engineered from six dimensional (2, 0) theory of exceptional type. The classification of principal lisse W-algebra is equivalent to the classification of E type irregular singularity which do not admit any mass deformation (the regular singularity is given by the principal nilpotent orbit), see table 5 and 6 for the classification.
The classification of lisse W-algebra from 4d theory engineered using non-trivial regular singularities is more difficult, as little is known in the literature about the associated variety of non-admissible W-algebra with exceptional Lie algebras. However, motivated by considerations of four dimensional physics, we develop a simple method to determine the associated variety of non-admissible AKM, and complete the full classification of those 4d theories whose 2d VOAs are lisse W-algebras.

Principal lisse W-algebra
First we consider theories engineered using the irregular singularity only. The lisse condition is equivalent to the absence of mass parameter of the theory, and the constraint on irregular singularity is listed in 5 and 6. The corresponding lisse W algebra is listed in table. 16. Some of the physical information of these 4d theories are: 1. The W-algebra takes the following form with h ∨ being the dual Coxeter number. The allowed value of n is listed in table 16.
2. The 4d theory has no Higgs branch, and the Coulomb branch spectrum can be easily found from the mini-versal deformation of the corresponding 3-fold singularity.
3. We do not know the character of these W-algebras yet, but the growth of the character is given as G = −48(a 4d − c 4d ), which can be calculated using Coulomb branch data, and it takes the following form: Since there is very little study on non-admissible lisse W-algebras, it is definitely good to know some isomorphism between non-admissible and known lisse W-algebras. We can use the isomorphism between 3-fold singularities to infer isomorphisms between W-algebra. The idea is that a single 3d singularity which defines our 4d theory can be written in different ways, which could have different 6d (2, 0) realization. One can read different W-algebra from these different 6d realizations, and therefore we can find nontrivial isomorphism between W-algebras. Some examples are shown in table 18.
• Now for the W-algebra W −h ∨ + n u (g, f ), its associated variety is so if we choose f ∈ O g,n,u , the above associated variety would be trivial, and the corresponding W-algebra is lisse.
So the crucial part of finding f is to determine the associated variety of non-admissible AKM, which has not been studied much yet. It might be possible to directly compute it using the method developed in [19], here we use a method which is motivated from four dimensional physics. To find the associated variety of a non-admissible AKM from using the data of four dimensional N = 2 SCFT, we need to use following facts: • First we need to use anomaly matching. The difference of central charges a 4d − c 4d can be computed from the Coulomb branch data [99,100], and is related to the growth function of W-algebra vacuum character as follows a 4d and c 4d can be explicitly calculated using the method developed in [16,17], and it takes the following simple form Here the data d g (n) is given in Here G IR is the growth function of the IR theory T IR . Since the Higgs branch is assumed to be the closure of a nilpotent orbit O g,n,u , 2n h has to equal to the dimension of O g,n,u , so we have dim(O g,n,u ) = G U V − G IR + 2n v . (7.9) One simple consequence of formula (7.9) is that if G U V is fractional, then T IR can not be trivial, as the contribution of free hypermultiplet and free vector multiplet has to be an integer. The theory T IR is the four dimensional theory whose Higgs branch is trivial and therefore whose 2d associated W-algebra is lisse.
To determine the dimension of associated variety, we need to have some information of n v and G IR in the IR. Furthermore, even if we determine the dimension of O g,n,u , it is still possible that there are more than one nilpotent orbit with same dimension. We need to use more four dimensional input.
• The T IR in the Higgs branch can actually be identified as follows: it is just the theory defined by adding a regular singularity f IR because of the fact that going to the Higgs branch of a class S theory is equivalent to closing the puncture. We determine f IR using the Coulomb branch data: 1. If the UV growth G U V is fractional, then the IR interacting theory T IR has to be non-trivial and has Coulomb branch operator(s). We choose maximal f IR (using the order of nilpotent orbit) such that there are non-trivial Coulomb branch operators. To ensure that there are no free hypermultiplets for the theory defined using f IR , we need to ensure that the leading order differential of the SW curve of corresponding 6d configuration has non-negative Coulomb branch operators.
2. If the UV growth G U V is integral, we can still find the maximal f IR such that there are non-trivial Coulomb branch operators. However, there is a further possibility that the interacting theory is trivial, and we actually need to consider larger f IR , and there is no simple way to determine f due to the possibility of existence of free vector multiplets. We will then need to use Coulomb branch data to further constrain f IR .
In practice, we can first try f IR whose dimension is just below the growth function, and gradually increase f IR . Our assumption is that the orbit of f IR which gives T IR is O g,n,u O f IR = O g,n,u . (7.10) • We cannot completely determine f IR and n v using a single equation 7.9, so we need another equation. There is one further constraint from the Coulomb branch: The dimension of Hitchin moduli space d f IR should be equal to the number of free vector multiplets and the rank of T IR On the other hand, the dimension of the Hitchin moduli space defined using the regular singularity f is (dim(g) − rank(g)), (7.12) where r U V is the Coulomb branch dimension of U V theory, and f d IR is the dual Hitchin label for the puncture. Hence we have the constraint ). (7.13) In practice, using equation 7.9 and 7.13, one can determine the associated variety of AKM with exceptional Lie algebra as in table 19, 20, and 22. Once we determine the associated variety O g,n,u of AKM V −h ∨ + n u (g), we can find f ∈ O g,n,u such that W −h ∨ + n u (g, f ) is lisse. Full results are listed in table 21 and 23.
Example 1: Consider a theory engineered by an irregular singularity of the type g = e 6 , n = 9, u = 8, and a regular singularity labeled by trivial nilpotent orbit. The growth function is G U V = 78 − 56/8 = 71, we can try f IR = E 6 (a 1 ) whose dimension is 70 which is just below the growth function, and the theory has Coulomb branch spectrum 15 8 , 9 8 , 5 4 , 3 2 , which can actually be engineered by the singularity F IR : x 2 + y 2 + z 3 + w 5 = 0. Theory with f IR has no Higgs branch, so we conclude that the associated variety of the V −12+9/8 (e 6 ) is just E 6 (a 1 ). The growth function of IR interacting theory is G IR = 1, so using formula, 7.9, we conclude that n v = 0, and there is no free vector multiplets in the generic point of Higgs branch. One can also check that formula 7.13 is also satisfied: we have r U V = 29, and r IR = 4, and the dual of E 6 (a 1 ) is A 1 orbit, whose dimension is 22.
Example 2: Consider a theory engineered by an irregular singularity of the type g = e 6 , n = 3, u = 2, and a regular singularity labeled by trivial nilpotent orbit. The growth function is G = 78 − 24/2 = 66. Now change the regular singularity to f 1 = E 6 (a 3 ) whose dimension is equal to 66. The Coulomb branch spectrum of the IR theory is {3, 2, ( 3 2 ) 4 }. This theory has a gauge theory description which is just SU (3) coupled with four copies of D 2 (SU (3)) theory. This theory can also be engineered by a complete intersection singularity defined with two polynomials. The complete intersection singularity is not terminal, and has a crepant resolution, so it has a Higgs branch. In fact, the Higgs branch of the theory F IR has n h = 1, n v = 1, and no further interacting theory. This implies that dim(O e 6 ,3,2 ) = G U V − G IR + 2n v = 68, (7.14) so the associated variety O e 6 ,3,2 is just closure of nilpotent orbit with the label D 5 , as this is the unique E 6 nilpotent orbit whose dimension is 68. The Higgs branch of the original theory therefore has 34 free hypermultipelts, and one free vector multiplet. Example 3: Now consider a more complicated example. Consider a theory engineered by an irregular singularity of the type g = e 8 , n = 5, q = 3, we also have the regular singularity with label f trivial. and the growth of this theory is G = 248 − 48 3 = 232, which is an integer and we are not sure whether there would be an interacting piece in the Higgs branch.
To find the effective theory on the Higgs branch, we first choose f 1 = E 8 (a 4 ) whose dimension is 232, and the theory has Coulomb branch spectrum {5, 4, ( 10 3 ) 3 , 3, ( 7 3 ) 3 , 2, ( 5 3 ) 4 , ( 4 3 ) 3 }. It is interesting that this theory has the same Coulomb branch spectrum as the theory engineered using three-fold singularity x 3 + y 3 + z 3 + w 5 = 0, which actually has a Higgs branch whose IR theory consists of n h = 1, n v = 1, together with an interacting theory engineered by the singularity F IR : x 3 + y 3 + z 3 + w 2 = 0. The theory engineered by F IR has no Higgs branch, so there is no further Higgs branch deformation. Using the formula 7.9, we see that (We use the fact G IR = 0): (7.15) and there is a unique nilpotent orbit E 8 (a 3 ) whose dimension is equal to 234, so the associated variety O e 8 ,5,3 is the closure of E 8 (a 3 ) orbit. The IR theory on the Higgs branch consists of free hypermultipelts, free vector multiplet, and an interacting theory.
Example 4: For e 6 , n = 3, u = 1, we have G U V = 54, so we first take f = 2A 2 + A 1 whose dimension is 54, the Coulomb branch spectrum of the theory defined with f is {2, 3}. To match the a − c anomaly, the theory should be N = 4 SU (3) SYM. We can further Higgs △ SYM, and in the IR, there are n v = 2 and n h = 2, and no interacting theory. So the associated variety of original theory should have dimension 58, and there is a unique E 6 nilpotent orbit with this dimension D 4 (a 1 ), and we conclude that this is the associated variety of the original theory.
Example 5: Now we give a more detailed explanation for the case g = E 8 , n = 3, u = 1, and G U V = 216. We take f = E 6 (a 1 ), and the Coulomb branch spectrum is {2 4 , 3 7 }. This theory is actually isomorphic to A 2 (2, 0) theory on a genus two Riemann surface with one regular puncture. It is known that one can further higgs this theory, and the Higgs branch is not a pure Higgs branch, and we have n v = 4 with no interacting part left. Using formula 7.9, we can see that the dimension of the associated variety of original theory should be 224, and there are two choices for it: E 8 (a 6 ) and E 7 (a 2 ). Using formula 7.13, we can conclude the associated variety should be E 8 (a 6 ). Table 19: Associated variety for G 2 and F 4 type theory. The rank of Coulomb branch of G 2 type theory: r = 4 n u − 1. The rank of Coulomb branch of F 4 type theory: r = 18u n − 2. The one with # implies that the IR theory has free vector multiplets in the Higgs branch, and n v is the number of free vector multiplets. The one with * means that the theory has pure Higgs branch, i.e. the IR theory on the Higgs branch consists of free hypermultiplets only. Table 20: Associated variety for E 6 and E 7 type AKM V −h ∨ + n u (g). The one with star implies that that theory has a pure Higgs branch! The one with # implies that the IR theory has free vector multiplets in the Higgs branch, and n v is the number of free vector multiplets. The one with * means that the theory has pure Higgs branch, i.e. the IR theory on the Higgs branch consists of free hypermultiplets only. The IR theory in the Higgs branch of other theories has an interacting piece, which is listed in table 21. The rank of Coulomb branch of E 6 type theory: r U V = 36 n u − 3. The rank of Coulomb branch of E 7 type theory: .
-38 -(f, n, u) Coulomb branch Isomorphism W-algebra (G 2 (a 1 ), 2, 3) (2, ( 4 3 ) 3 ) x 3 + y 3 + z 3 + 2 2 = 0 W −6+ 2 3 (so 8 , f prin ) (E 6 (a 1 ), 9,8)  (E 7 (a 5 ), 14, 5) 5   Table 21: The physical data for 4d theory whose 2d VOA is W −h ∨ + n u (g, f ). Table 22: Associated variety for E 8 type theory. The rank of the Coulomb branch is: r = 120 n u − 4. The one with star implies that the theory has a pure Higgs branch! The one with # implies that the IR theory in the Higgs branch has free vector multiplets, and n v is the number. For n = 24, u = 5, there are actually two possible choices of nilpotent orbits (D 6 (a 1 ) and A 6 ) which satisfy our formula 7.9 and 7.13 (with n v = 0); To pick D 6 (a 1 ), we use the following closure relation: D 5 + A 1 orbit is in the closure of D 6 (a 1 ) but is not in closure of A 6 , now we can compute the Coulomb branch operator of the theory defined using D 5 + A 1 which is non-empty, and we can further Higgs this particular theory. This observation then implies that we have to choose D 6 (a 1 ) as the Higgs branch of theory defined using regular puncture with trivial nilpotent orbit! -40 - Table 23: Physical data for 4d theories whose VOA is W −h ∨ + n u (e 8 , f ). Here f denotes regular singularity.

A proposal for the character
Now we discuss the character of lisse W-algebra with exceptional Lie algebra. We first consider the case when the regular puncture has full flavor symmetry group (the nilpotent orbit f is trivial). The corresponding VOA is the affine Kac-Moody algebra V k (g), with with η [2 2 ] (c) = (q; q)(qc ± ; q)(q 2 ; q)(q 2 c ± , q) = w∈adj su (2) (qc w ; q)(q 2 c w ; q) (7.21) andc = (q 1 2 c, q, q 1 2 c). (7.22) We use the shorthand notation (qa ± ; q) = (qa; q)(qa −1 ; q). Again the sum is over all highest weight representations of SU (4) and the result has an E 7 symmetry.    Now the sum is over all highest weight representations of SU (6). Again, there is an enhancement to E 8 symmetry.

Conclusion
We continue our studies of the correspondence between 2d W-algebras and 4d N = 2 SCFTs. Lisse W-algebras are defined as those W-algebras whose corresponding Zhu's C 2 algebra are finite dimensional. As the associated variety of Zhu's C 2 algebra is identified with the Higgs branch in the 4d/2d correspondence, the lisse condition is equivalent to the absence of Higgs branch of the 4d theory. We classify 4d N = 2 SCFTs which do not admit a Higgs branch, and the corresponding 2d W-algebra of these theories should be lisse. In particular, we predict the existence of a large class of new non-admissible lisse W-algebra which has not been found before (see few examples considered in [102]). These 4d theories can appear in the IR theory of the Higgs branch of a general 4d N = 2 SCFT, and therefore they are crucial to understand the behavior of Higgs branch of general 4d N = 2 SCFTs.
In our study of 4d N = 2 theory with no Higgs branch, we encounter the following generic situation. Consider 4d N = 2 SCFT constructed from 6d (2, 0) theory on a sphere -43 -with one irregular singularity and one regular singularity labeled by trivial nilpotent orbit. We call this theory U V theory, and the corresponding VOA is just the AKM algebra. It is easy to study the IR theory of its Higgs branch: one simply change the type of regular puncture to the one labeled by the nilpotent element f , whose orbit leads to the associated variety of the AKM. The IR theory generically consists of free hypermultiplets, free vectormultiplets, and an interacting theory which has no Higgs branch. This suggests that the UV AKM might be related to the VOA of free hypermultiplets and vectormultiplets, plus a lisse W-algebra. Indeed such realizations are discussed in [66,[103][104][105][106]. Our study reveals that such constructions are much more general and we can identify all the IR pieces in the Higgs branch using (2, 0) construction. Moreover, our results suggest that it is also possible to have free vectormultiplets in the IR. It would be interesting to further study the relations between these W-algebras.
We found new non-admissible lisse W-algebras, which deserve further studies from physical and mathematical point of view. We give a formula for vacuum characters for these W-algebras in the classical Lie algebra case, and it would be interesting to verify them using 2d methods. The module structure is also very interesting to study, and some results will appear in [107].
The lisse W-algebra in our context has interesting connection to three dimensional Q-factorial terminal singularity. In our paper, we mainly study the Gorenstein Q-terminal singularity, which gives the results in table 5. The non-Gorenstein ones needs further study, and might be related to the lisse W-algebra with non-simply laced Lie algebra, and it is interesting to further study them as well. quiver N U (r s−1 ) U (r s−2 ) U (r 1 ) Figure 6: Upper row: We construct a D5 − D3 brane system for a A type partition. Middle row: We perform S duality on the brane system in upper row, which turns a D5 brane to a NS5 brane, and a D3 brane to a D3 brane. Bottom row: We read a quiver gauge theory from the middle row. The rule is following, we associate a U (r) gauge group for r D3 branes suspending between NS5 brane, and each NS5 brane contributes a hypermultiplet in bi-fundamental representation.
First consider g = A N −1 , and its Langlands dual is just itself. A nilpotent orbit in A N −1 Lie algebra is specified by a partition [h 1 , . . . , h s ], and to find its associated quiver, we first construct a D3 − D5 brane system as in figure 6. The number of D3 branes suspended between D5 branes are r 1 = N − h 1 , and r 2 = N − h 1 − h 2 , and etc. The quiver gauge theory is found by doing S duality on above brane configuration: D3 branes stay as D3 branes, while D5 branes become N S5 branes. The quiver gauge theory can be easily read from the N S5 − D3 brane system. The process was summarized in figure 6.
For other classical Lie algebras, a nilpotent orbit is also labeled by a partition [h 1 , . . . , h s ], but with following constraints: for (B N , D N ), even parts appear even time; for C N , odd parts appear even times. To engineer gauge theories with orthogonal and symplectic gauge groups, we need to use O3 planes. There are a total of four kinds of O3 planes, and we call them B D, and C, C ′ type. The name of such O3 planes are clear: if we put D3 branes on certain type of O3 planes, we would get corresponding gauge theory types (the number -45 -of D3 branes on corresponding O3 plane is then constrained by the type, i.e. there are 2r for D type, 2r + 1 for B type, and 2r for C type). S duality exchange those O3 planes as follows: Now we construct a quiver from a D type or B type partition, where the even parts appear even times. We first start with a D3 − O3 − D5 brane system, and there are two new facts: • The type of O3 planes are alternating between D3 branes. Here we have D type and B type alternating.
• There are only an odd number of D3 branes ending on a half D5 brane.
For the B type or D type partition where parts are all odd, we can construct a D3−O3−D5 brane system using the same rule as A type partition. The quiver gauge theory can be found from S dual brane configuration, see figure 7. To deal with a partition with even parts, we need to do following modification: • We add one more D3 brane to O3 brane for the segment which violates the corresponding rule for O3 plane. For example: if there are 2r D3 brane on a B type O3 plane, we add one more D3 brane, or if there are 2r + 1 D3 brane on D type O3 plane, we add one more D3 brane.
After this modification, it is straightforward to find the quiver gauge theory by doing S duality on D3 − O3 − D5 brane system. Notice that the corresponding quiver gauge theory involves only D type and C type gauge groups (the flavor group is D type or C type). Now we consider how to construct a quiver from a C type partition, where the even parts appear even times. We first start with a D3 − O3 − D5 brane system, and there are two new facts • The type of O3 planes are alternating between D3 branes. Here we have C type and C ′ type alternating.
• There are only an even number of D3 branes ending on a half D5 brane.
For the C type partition where the parts are all even, we can construct a D3 − O3 − D5 brane system using the same rule as A type partition. The quiver gauge theory can be found from S dual brane configuration, see figure 8. To deal with a partition with even parts, we need to do following modification: • We add one more D3 brane to O3 brane for the segment which violates the corresponding rule for O3 plane. For example: if there are 2r + 1 D3 brane on a C type or C ′ O3 plane, we add one more D3 brane.
After this modification, it is straightforward to find the quiver gauge theory by doing S duality on D3 − O3 − D5 brane system. Notice that the corresponding quiver gauge theory involves only B type and C type gauge groups (flavor group is of C type). Notice  We construct a D5 − D3 brane system for a D type partition and B type partition respectively. There should be even number of D3 branes for a D type O3 plane, and odd number of D3 branes for a B type O3 plane. For a general D type or B type partition, the brane configuration would violate the rule for D type or B type O3 planes, and we add one more D3 brane so that the number is consistent with the corresponding O3 plane. Middle: We perform S duality on brane system in upper row, which turn a D5 brane to a NS5 brane, and D3 brane to a D3 brane. Bottom row: We read a quiver gauge theory from brane configuration of middle row. that if we fixe the flavor group to be C type, we do not get a quiver with alternating D type and C type gauge groups by using this method.
Given a nilpotent orbit e in classical algebra g, and one can find its dual in g ∨ using the Spaltenstein map. The rule is • For A type: the partition of e d is given by the transpose of partition of e.
• For D type: the partition of e d is given as follows: first find the transpose of partition e, and then do D collapse to get a D partition.
• For B type: the partition of e d is given as follows: first subtract one box from last row of e, and then do the transpose, and then do C collapse to get a C partition.
• For C type: the partition of e d is given as follows: first add a box to first row of e and find its transpose, and then do B collapse to get a B partition.
-47 -  We construct a D5−D3 brane system for a C type partition. There should be even number of D3 branes for C type or C ′ type O3 planes. For a general C type partition, the brane configuration would violate the rule for O3 planes, and we add one more D3 brane so that the number is consistent with the O plane type. Middle: We perform S duality on brane system in upper row, which turn a D5 brane to a NS5 brane, and D3 brane to a D3 brane. Bottom row: We read a quiver gauge theory from brane configuration of middle row.