W algebras, cosets and VOAs for 4d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 SCFTs from M5 branes

We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the W algebras defined using nilpotent orbit with partition [qm, 1s]. Gauging above AD matters, we can find VOAs for more general N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 SCFTs engineered from 6d (2, 0) theories. For example, the VOA for general (AN − 1, Ak − 1) theory is found as the coset of a collection of above W algebras. Various new interesting properties of 2d VOAs such as level-rank duality, conformal embedding, collapsing levels, coset constructions for known VOAs can be derived from 4d theory.


Introduction
A remarkable correspondence between a four dimensional N = 2 superconformal field theory (SCFT) and a two dimensional vertex operator algebra (VOA) is found in [1]. which provides a promising organizing principle for the whole landscape of N = 2 theories (see  for some further developments). Once a 4d/2d pair is found, one can use the 2d theory to learn 4d theory and vice versa. For example, one can compute the Schur index JHEP04(2021)076 of 4d SCFT by calculating the vacuum character of 2d VOA which is often much easier to work out, meanwhile 4d result also motivates the study of certain 2d VOAs which received little attention before [23].
If our 4d SCFTs can be enginnered from string/M theory, it is possible to gain more insights about 4d/2d pair. In the past few years, People have found a large class of 4d N = 2 SCFTs by putting 6d (2, 0) theory on a Riemann surface with various defects [24][25][26][27][28], so we should have a map between such 6d configuration and a 2d VOA, which is schematically depicted in figure 1. The 2d VOA for the theory defined using only regular punctures was studied in [2,3,29]. The most important step is to understand the VOA for the theory defined by three full punctures (maximal flavor symmetry), since the general cases can be found from following two correspondence between operations in 4d theory and operations in 2d VOAs: • On the 4d side one can reduce the full puncture to a generic puncture labeled by a nilpotent orbit f . The 2d counterpart of such operation corresponds to the quantum Drinfeld-Sokolov (qDS) reduction of the original VOA [2], as sketched in figure 2.
• If a theory is formed by conformally gauging various matters together, the VOA is formed by performing cosets on those VOAs [1,2] as in figure 3.
Things become more interesting and complicated if we consider Argyres-Douglas (AD) theories which are engineered using one irregular singularity Φ and one regular singularity f on a sphere as in figure 1. The correspondence between 2d VOAs and certain AD theories were discussed in [5,10,11,20,28,[30][31][32][33][34][35][36][37][38][39]. More generally it was conjectured in [11,28,35] that if there is no mass parameter associated with the irregular singularity, the corresponding VOA is just the vacuum module of a W algebra denoted by W k (g, f ) which is obtained through the quantum Hamiltonian reduction from the vacuumĝ-module of level k [40], where k is determined by the data Φ. Again, the choice of the generic regular puncture f determines the qDS reduction type.
There remains the question on determining the VOA for remaining cases, and the main purpose of this paper is to partially solve this problem by using following two facts: • Irregular singularities with mass deformations often have exact marginal deformations and their weakly coupled gauge theory descriptions are found in [41][42][43]. They are described by gauging AD matters with at least two types of non-abelian flavor symmetries. 1 Therefore once we find the VOA for such AD matter, the VOA of the full theory can be found by the coset construction.
• A crucial observation for this paper is that all AD matters with two non-abelian flavor symmetries studied in [42,43] can be engineered by a different realization whose VOAs are known as certain W algebra studied in [11,28,35]. The W algebra takes the form W k (g, [q m , 1 s ]) with k depends on parameter (q, m, s), see the summary in section 3.2.5. 4d Figure 3. Four dimensional conformal gauging is interpreted as cosets of two dimensional VOAs. T 1 and T 2 are four dimensional matter with non-abelian flavor symmetry G, and one gauge them to get a new conformal field theory with exact marginal deformation. Here V 1 and V 2 are VOAs for matter T 1 and T 2 , and each of them has an affine vertex subalgebra V ki (G). g is the Lie algebra of G. Since we are taking the diagonal coset, the generators J a diag of g −2h ∨ is the sum J a diag = J a 1 + J a 2 where J a 1,2 are generators of V k1,2 (G) respectively. There are several new interesting features about VOAs of AD matters studied in this paper:

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a) The VOA has an affine VOA V k 1 (g 1 ) ⊕ V k 2 (g 2 ) as its subalgebra, where V k (g) is the affine Kac-Moody (AKM) vertex algebra 2 of Lie algebra g with level k. This affine VOA has the same central charge as the W algebra and therefore we found a large number of new possible conformal embeddings of VOAs.
b) The simple fact that a theory can be engineered in various ways can often tell us interesting properties about VOAs, see figure 4. For instance, we can derive new level-rank type duality.
c) S-duality of 4d theory implies the equivalence between different cosets constructions of a single VOA.
Once VOAs for AD matters are known, we are able to write down VOAs for more general theories engineered from M5 branes. For example, the VOA for (A N −1 , A k−1 ) theory with arbitrary N and k is found by using its weakly coupled gauge theory descriptions in section 5.1. This paper is organized as the following. Section 2 reviews known results about the mapping between AD theories engineered from M5 branes and VOAs. Section 3 studies VOAs corresponding to AD matters with two distinct non-abelian flavor symmetries. Section 4 focuses on the associated variety of the VOA for a given AD matter, which determines the Higgs branch chiral ring of the theory. Section 5 describes weakly coupled descriptions of AD theories and the coset construction of the corresponding VOA. Section 6 discusses conformal embeddings and VOAs for most general AD theories. Finally, a summary is given in section 7.

JHEP04(2021)076 2 Known results
A four dimensional 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 four dimensional 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 [44], in which short representations (where some of states in the representation are annihilated by a fraction of supercharges) were completely classified. Important half-BPS operators include primary operators of multiplets E r andB R .
The moduli space of vacua of a 4d N = 2 SCFT is extremely rich. It consists of a Coulomb branch, whose low energy effective theory involves abelian gauge theory in general. The Coulomb branch is parameterized by expectation values of primary operators of E r multiplets, and the low energy effective theory is described by a Seiberg-Witten geometry [45,46]. 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 SU(2) R × G F symmetry acts trivially.
Some theories also have a Higgs branch where the gauge group is completely broken in general should the SCFT has a gauge theory description. The Higgs branch, being a conical hyperkhaler manifold, is parameterized by expectation values of primary operators ofB R multiplets. One of important questions about the Higgs branch is to determine the affine chiral ring of the cone. The SU(2) R × G F symmetry acts non-trivially on Higgs branch, while U(1) R symmetry acts trivially.
Besides the Coulomb branch spectrum (just a set of rational numbers r i > 1) and the Higgs branch chiral ring, one would also like to determine three interesting quantities of a 4d N = 2 SCFT: central charges a 4d and c 4d which is defined using the energy-momentum tensor, and the flavor central charge k G .
There is an interesting set of short multiplets called Schur sector [47] which consists of Higgs branch operatorsB R , energy momentum tensors, and etc. Moreover, one can define a Schur index which counts those operators. It was proposed in [1] that one can get a 2d VOA from the Schur sector of a 4d N = 2 SCFT, and the basic 4d/2d correspondence used in current paper is [1]: • There is an AKM subalgebra (V k 2d (g)) in 2d VOA, where g is the Lie algebra of four dimensional 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
Finally, one can have intricated algebras built from a set of higher spin fields (W d 1 , . . . , W d i ), which are called W algebras [64]. The full algebra content and its representation theory of a W algebra is very complicated, however, if a W algebra can be derived from the qDS reduction of an AKM algebra [40], one can actually derive lots of important information of this W algebra from the representation theory of the AKM algebra.
In general it is difficult to work out the full Schur sector of a four dimensional N = 2 SCFT. However, using a surprising fact that almost all the known 2d VOA for the above 4d/2d mapping involves the W algebra derived from qDS reduction [65], one can hope that a lot can be learned about 4d theories by using existing knowledge of 2d VOAs.

AD theories correspond to W k (g, f ) algebras
One can engineer a large class of four dimensional N = 2 SCFTs by starting with a 6d (2, 0) theory of type j = ADE on a sphere with an irregular singularity and a regular singularity [24][25][26][27][28]. 4 The Coulomb branch is captured by a Hitchin system with singular boundary conditions near the singularity. The Higgs field of the Hitchin system near the irregular singularity takes the following form, Here T is determined by a positive principle grading of Lie algebra j [66], and is a regular semi-simple element of j. k > −b and is an integer. Subsequent terms are chosen such that JHEP04(2021)076  [27]. These 3d singularity is very useful in extracting the Coulomb branch spectrum, see [67].
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 puncture. Theories constructed using only above irregular singularity can also be engineered using a three dimensional singularity in type IIB string theory as summarized in table 1 [67]. 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 regular puncture with maximal flavor symmetry). A detailed discussion about these defects can be found in [68]. If there is no mass parameter encoded in the irregular singularity Φ (which means that the z −1 term in Φ is not allowed, and we also assume that singular parts of Φ are diagonalized near the irregular singularity), the VOA which corresponds to this 4d SCFT is the vacuum module of following W algebra 5 [11,35], Here h ∨ is the dual Coxeter number of j, and the W algebra is defined as the qDS reduction from the affine VOA [69].
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 [28]. Denoting the twisted Lie algebra of j as g ∨ and its Langlands dual as JHEP04(2021)076 Table 2. Outer-automorphisms of simple Lie algebras j, its invariant subalgebra g ∨ and flavor symmetry g from the Langlands dual of g ∨ .
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, Here T t is an element of Lie algebra g ∨ or other parts of the decomposition of j under outer automorphism. k > −b, and the novel thing is that k 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 of g. If there is no mass parameter in the irregular singularity, the corresponding VOA is given by following W algebra [28], 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. The Seiberg-Witten geometry of these theories are identified as the spectral curve of the Hitchin system [70].
det(x − Φ) = 0, (2.8) and one can read off the Coulomb branch spectrum from an associated Newton polygon [26][27][28], which is also reviewed in the appendix B.

AD theories correspond to B p+1 (g) and W p+1 (g) algebras
Consider the following irregular singularity of j = ADE (2, 0) theory, Notice that there are l (the rank of j) mass parameters in this singularity. We add a trivial regular singularity (f is regular nilpotent orbit), then these theories can be engineered by following three-fold singulariities,    where h ∨ is the dual Coxeter number of j, and f ADE (x, y, z) is the famous two dimensional ADE singularity. The 4d central charge is computed using following formula [71], where α max is the maximal scaling dimension of Coulomb branch spectrum, r is the rank of Coulomb branch, and µ = 2r + f 0 with f 0 the number of mass parameters. Using the JHEP04(2021)076 result found in [35], we have, (2.12) then the central charge takes following form, When j = A N −1 , it was propsed in [20] that the corresponding VOA is the B p+1 (A N −1 ) algebra constructed in [72]. For general j = ADE, the central charge of B p+1 (j) algebra is, (2.14) One finds that c 2d (B p+1 (j)) = −12c 4d (j, p) for general j, 6 therefore we conjecture that the VOA of above 4d SCFT is given by the B p+1 (j) algebra. We could also consider twisted AD theories with the following Higgs field, Using the method proposed in [28], the 4d central charge is, (2.16) n = 1 for simply-laced cases, and values of n for non simply-laced cases are summarized in table 4. 7 This implies that it should be possible to generalize the construction in [72] to nonsimply laced Lie algebra whose central charge is given by −12c 4d with c 4d in equation (2.16). One can also add another full puncture to get a theory with G flavor symmetry whose flavor central charge is, .
The central charge for these theories are, This suggests that there should be a VOA W p+1 (g) with an affine vertex operator subalgebra V −k G (g). Some suggestions on the construction of this class of VOA are given in [20], and we will give a coset construction in later sections. The qDS reduction of W p (g) produces the B p (g) VOA. JHEP04(2021)076

VOA for AD matter with two non-abelian flavor symmetries
Theories considered in the last section usually carry only one type of non-abelian flavor symmetries, which is determined by the regular puncture. When the order of the irregular singularity considered in the last section is integral, one can consider degenerating cases and could get another type of non-abelian flavor symmetries [26]. It was realized in [42,43] that besides the theory considered in the last section, we can get new 4d N = 2 SCFT by considering more general irregular singularities (taking with T being the following diagonal matrix, Here I n×n being a diagonal matrix with eigenvalues (1, w, . . . , w n−1 ) with w the nth root of unity. To get a SCFT, coefficients of subsequent terms in the Higgs field have to take the same form as the leading one. Such irregular singularity carries a flavor symmetry U(N − n). One can also add an extra regular puncture so that there are two distinct types of non-abelian flavor symmetries. The purpose of this section is to find the VOA for the above class of theories when (k, n) = 1. The key observation is that the same theory can be realized by a different (2, 0) configuration whose VOA is already found in section 2. We first study the A N −1 case in detail in section 3.1. The same construction is then generalized to other classical Lie algebras as well.

A N −1 = sl N : SU(N ) × U(n 1 ) flavor symmetry
The regular puncture of A N −1 theory is classified by a size N Young tableaux (or the partition of N ), which also gives a nilpotent orbit of the A N −1 Lie algebra. Given a Young tableaux [h r 1 1 , h r 2 2 , . . . , h rs s ], the flavor symmetry is, We are interested in regular punctures with partitions like [m q , 1 s ], whose 4d flavor symmetry is SU(q) × SU(s) × U(1) with flavor central charges, where [z] is the scaling dimension of z coordinate in the spectral curve of A type Hitchin system. Now consider the following configuration, with k and n coprime. Here T takes the following form T = [I n 1 ×n 1 , 0, . . . , 0] with I n×n being a diagonal matrix with eigenvalues (1, w, . . . , w n−1 ) with w being the nth root of unity. This theory has a U(n 1 ) × SU(n 1 + n) flavor symmetry with 4d flavor central charges [43], 8 k SU(n 1 ) = n 1 + n n + k , k SU(n 1 +n) = n 1 + n − n n + k . (3.6) The Newton polygon for this theory is shown on the left of figure 5, from which one can read the Coulomb branch spectrum using the procedure in the appendix B.
To find its VOA, we would like to find a different realization of this theory whose VOA is known through results in the last section, which has the following data, One can show that these two realizations have the same Coulomb branch and flavor symmetries. Now we compute 4d flavor central charges of this new realization of the AD theory (3.7). Notice [z] = n 1 (n+k)+n n+k and using the formula (3.4), we have, (3.8) 8 We ignore the subscript 2d and 4d when apparenat.

New level-rank dualitys
Now consider the theory with f being a regular nilpotent orbit (hence no flavor symmetry), and the irregular singularity, Again, we take n and k to be coprime. This theory has a U(n 1 ) flavor symmetry. Using the above result (equation (3.9)) for f being a full puncture and the fact that closing of a puncture is equivalent to qDS, its VOA is Here we take k > n 1 . The same theory can be realized by the following configuration, (3.13) Using the correspondence (2.7) in section 2.1, the corresponding VOA is then Therefore, we find the following equivalence of two 2d VOAs from different realizations of the same 4d AD theories, We call this the level-rank duality as the rank and the level are sort of exchanged for these two W algebras.

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Level rank duality example. The Coulomb branch spectrum of two configurations are the same. Notice that the left hand side uses 6d A 4 (2, 0) theory while the right hand side uses 6d A 2 (2, 0) theory.

Example.
Taking n 1 = 0, we have the following equivalence of VOAs: This is the familiar level-rank duality discovered in [73]. See figure 6 for an illustration from the four dimensional theory point of view. The above level-rank duality can be further generalized as follows. Consider two configurations, Choose n+n 1 −n 2 > 1, k+n 2 −n 1 > 1 and T 1 = diag(I n 1 ×n 1 , 0 n×n ), T 2 = diag(I n 2 ×n 2 , 0 k×k ). These two configurations give the same Coulomb branch spectrum, therefore we conjecture that they give the same 4d theory. The manifest flavor symmetry is U(n 1 ) × U(n 2 ) with following flavor central charges, The two seemingly different VOAs should be the same, so we have following (conjectured) equivalence, An example is illustrated in figure 7.

Conformal embedding in W algebra
Conformal embedding is defined as the following [74]. Let V be a vertex algebra with a Virasoro (= conformal) vector w V and let W be a vertex subalgebra of V endowed with a Virasoro vector w W . The embedding W ⊂ V is called conformal if w W = w V . A necessary condition for conformal embedding is that c V = c W . For our W algebra VOA A defined in equation (3.9), there is an affine vertex operator subalgebra u(1) ⊕ V −(n 1 + n n+k ) (sl n 1 ) ⊕ V −(n+n 1 )+ n n+k (sl n+n 1 ). It is interesting to note that the central charge of VOA A is equal to the central charge of this affine vertex operator subalgebra. So the necessary condition for conformal embedding is achieved, and it is interesting to check whether the following embedding, is indeed the conformal embedding (see [52] for physical discussions).

Collapsing levels and decoupling of flavor symmetry
We have found VOAs of some AD matters with two distinct non-abelian flavor symmetries by finding an alternative (2, 0) construction. 9 Notice that the above construction does not give a new realization for theories with just a SU(n) flavor symmetry arising only from regular singularity. We would like to find a different realization for those theories too. Such realization is not very good for 4d theory as there appears to be more flavor symmetries which will decouple in the IR, but they do have interesting implications for VOAs. The basic idea is the following. Start with a theory engineered by the following configuration, We take (k, n) = 1, and T being the principle type (so no mass term is allowed in the irregular singularity) so that the VOA is the affine vertex operator algebra Now we would like to find another realization whose assosiated VOA is known by the result of section 2 too, so it should take the following form, Naively, this configuration has flavor symmetry SU(n) × U(m). The flavor central charge for SU(n) flavor group is n + m − N n+k . The necessary condition for the equivalence is that the flavor central charge for SU(n) group should be the same, therefore we find N = m(n + k) + k, so q = (n + k). The flavor central charge of SU(m) group is Physically, we interpret that this result implies that the U(m) flavor symmetry is decoupled in the IR 4d SCFT. These two configuration defines the same 4d SCFT in the IR (One can check that they give the same Coulomb branch spectrum), and we have the following equivalence of VOAs, Mathematically, this means that the W algebra collapses to its affine subalgebra [74,75]. We could generalize the above collapsing story as follows. Consider more general theory engineered by following data,

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Again T = diag(I n×n , 0 n 1 ). This theory has flavor symmetry U(n 1 )×SU(n+n 1 ). We found that its VOA is a W algebra given by the nilpotent orbit [(n + k − 1) n 1 , 1 n+n 1 ]. However, the theory can be engineered by the following configuration as well, (3.29) We have N = (m + n 1 )(n + k) + n and N + m = n + k. The VOA for above configuration is This configuration has the naive flavor symmetry U(m) × U(n 1 ) × SU(n + n 1 ). The flavor central charge for the U(m) flavor group is zero, and therefore is decoupled in the IR. The above W algebra (3.30) is therefore collapsed to VOA A defined in (3.9).

Classical Lie algebra
We now discuss how to generaize the above construction to other Lie algebras. The idea is similar: we consider AD matters engineered from one 6d realization whose VOA is not known, then we find an equivalent realization whose VOA can be read from results in section 2.

D N = so 2N : SO(2N ) × Sp(n 1 ) flavor symmetry
We start with AD theories engineered from 6d D N (2, 0) theory. The regular puncture of D N theory is classified by nilpotent orbits of so 2N Lie algebra and is labeled by a size 2N Young tableaux whose even parts has even multiplicities. Given a Young tableau [r h 1 1 , r h 2 2 , . . .], the flavor symmetry is Given the puncture [m q , 1 s ] with (q, m, s) all even, the flavor symmetry is Sp(q) × SO(s) whose central charges are Here [z] is the scaling dimension of z coordinate in the spectral curve of D N type. Now consider following configuration, with (k, n) = 1, and T takes a specific form, see [43]. We choose n to be even and consider the irregular singularity with which no flavor symmetry is associated. This also implies that n + k is odd. See figure 8 for its corresponding Newton polygon. The flavor symmetry is Sp(n 1 ) × SO(n + n 1 + 2). The flavor central charges of them are k SO(2N ) = n + n 1 − n n + k , k Sp(n 1 ) = n 1 + 2 2 + n 2(n + k) . (3.34)
In previous discussions, we require n to be even. We can also consider the case where n is odd. The AD matter with two non-abelian flavor symmetries are given by following configuration, Here n 1 is even and n is odd, and T = diag(I 2n×2n , 0 n 1 ), see [43] for the specific form of diagonal matrix I 2n×2n . We also require (n, k) = 1. The flavor symmetry is SO(n 1 ) × SO(2n + n 1 ) × SO (2). Unlike the previous case, there is an extra SO(2) flavor symmetry besides two simple flavor groups. The same theory can be described by following configuration, We have N = n 1 (n + k − 1) + n + n 1 and N + k = n + k, hence Here N is odd, and there is a SO(2) flavor symmetry in irregular singularity. Unfortunately, we do not know the VOA for this configuration yet. The regular puncture of twisted D N theory is classified by nilpotent orbits of sp 2N −2 Lie algebra and is labeled by a 2N − 2 size Young tableaux whose odd parts has even multiplicities. Given a Young tableau [r h 1 1 , r h 2 2 , . . .], the flavor symmetry is Given the partition [m q , 1 s ] with (q, m, s) even, the flavor groups are SO(q) × Sp(s) with following central charge: . (3.43) here [z] is the scaling dimension of the coordinate z in spectral curve of twisted D N theory. Next consider the twisted theory with the following data, We take (n, k) = 1 and n is even so that there is no flavor symmetry associated with the irregular part (this implies that n + k is odd), and figure 8 illustrates its Newton polygon. The flavor symmetry is SO(n 1 ) × Sp(n + n 1 − 2), and flavor central charges are , k SO(n 1 ) = n 1 − 2 + n n + k . (3.45) To find its VOA, we realize that there is another equivalent description: k Sp(n+n 1 −2) = n + 2n 1 2 − n 1 (n + k) + n 2(n + k) = n + n 1 − n 2(n + k) , k SO(n 1 ) = n 1 (n + k) + n − 2 − (n + k − 1) n 1 (n + k) + n n + k = n 1 − 2 + n n + k . (3.47) which are the same as results from other description, see (3.45). One can also check that the two configurations have the same Coulomb branch spectrum. Therefore the corresponding VOA is the W algebra (3.48)

Twisted sl 2N theories: SO(2N + 1) × Sp(n 1 ) flavor symmetry
Let's now consider twisted sl 2N theory from which we can get SO 2N +1 flavor symmetry. The regular puncture is labeled by a Young tableaux with size 2N + 1, and the constraint is that even parts has even multiplicities. Given a Young tableaux [r h 1 1 , r h 2 2 , . . .], and the flavor symmetry is We are interested in punctures like [m q , 1 s ], with m even, and (q, s) odd. The flavor symmetry is SO(s) × Sp(q) with following central charge: Here [z] is the scaling dimension of z coordinate in spectral curve of twisted sl 2N type. The defining data for our theory is, Here n is odd and n 1 is even (cf. figure 9). The flavor symmetry is SO(n + n 1 + 2) × Sp(n 1 ) with following flavor central charges, k SO(n+n 1 +2) = n + n 1 − n 2n + 2k + 1 , k Sp(n 1 ) = n 1 + 2 2 + n 2(2n + 2k + 1) .

(3.68)
Recalling the central charge of AKM algebra g k is

Exceptional Lie algebra
Now consider AD theories constructed using exceptional 6d (2, 0) theory. In general, we have a theory with flavor symmetry E n ×G, where G is some subgroup of E n . Unfortunately, we do not know how to realize the above theory using the configuration presented in section 2. Instead we analyze its Coulomb branch when a generic regular singularity is present, and check whether one can find the same Coulomb branch spectrum using constructions presented in section 2. We do not attempt to do a general analysis, and only give some examples here: • Start with e 8 theory and look at the regular puncture whose Nahm label is [A 1 ] [68], then the flavor symmetry is E 7 . The flavor central charge is, We would like to realize this theory by using a 6d (2, 0) e 7 theory with a full puncture of E 7 type. The flavor central charge in e 7 construction is, We find a solution with 30 + k = 2 = 18 + k = 2. Both theory has the Coulomb branch spectrum [9,5,3] (these numbers can be derived using Newton polygon of E 8 type theory [43]  Here [A 1 ] denotes minimal nilpotent orbit of e 8 Lie algebra. The interpretation is that the e 8 description misses the U(1) flavor symmetry.
• Now look at e 7 theory with regular puncture whose Nahm label is [A 1 ]. The flavor symmetry is SO (12), and the flavor central charge is, On the other hand, starting with the so 12 realization, the flavor central charge is, Again, we find solution 18 + k = 18 + k = 2. For this value, E 7 configuration has an extra U(1) flavor symmetry. The so 12 descrition also has an extra U(1) flavor symmetry. The Coulomb branch spectrum is [5,3], which is read from Newton polygon. We can not find its VOA because of this extra U(1) flavor symmetry.

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• Finally we look at an e 6 theory with a regular puncture whose Nahm label is [A 1 ]. The flavor symmetry is SU (6), and the flavor central charge is, k SO(10) = 9 − 12 12 + k . (3.75) In an su 6 realization, the flavor central charge is, The matching of flavor central charges k SO(10) = k SU(6) requires 12 + k = 6 + k = 2. Now in the SU (6)  Now we move on to more interesting examples with exceptional flavor symmetries. Some rank one theories with following data were found in [77], The (a, c) central charges of B 3 and G 2 theory are the same as the N = 2 SU(2) SQCD with four flavors. The F 4 theory also has the same (a, c) central charge as the E 6 Minahan-Nemeschansky theory [78]. There are interesting relations between corresponding VOAs of these 4d theories. In fact, following conformal embeddings were proven in [79], It would be interesting to study further the relation between these VOAs and what they imply for 4d theories.

Comments on collapsing levels
We now make some remarks on collapsing levels of a W algebra into its affine piece, namely finding the proper nilpotent orbit f of the Lie algebra g and level k such that the following equivalence between two VOAs holds,

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We determine f and k by matching the Coulomb branch spectrum and other data of corresponding 4d theories. In practice, important insights can be gained if one first requires that flavor central charges should be equal for two descriptions. First consider some examples in detail. The (2, 0) configuration is the following, where k is integer-valued for g = ADE, and half-integral valued for twisted theories. k is restricted such that there is no flavor symmetry associated with the irregular singularity.
• Consider g = so 2N , and the 4d flavor To find a W algebra whose 4d partner has the same Coulomb branch spectrum, we consider so qm+2N theory with the regular singularity f = [q m , 1 2N ]. Naively, this puncture has the flavor symmetry SO(2N ) × SO(m). The 4d flavor central charge −k of SO(2N ) flavor group in this description is which is odd, so there might be the following equality of VOAs, , (so 2N ).  (3.85) which implies that the SO(m) flavor symmetry is decoupled in the IR. One can check that above two configurations give the same Coulomb branch spectrum.
• Given g = sp 2N −2 , one has the following collapsing levels and nilpotent orbits, • Taking g = so 2N +1 , one finds the following collapsing levels and nilpotent orbits,

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In above cases, we can choose more general puncture on AKM side, and we then have collapsing of one W algebra into another W algebra.
We interpret our results as follows. Taking g = sl N (other cases are similar) and the irregular singularity, the Seiberg-Witten curve takes the following form If there is no Coulomb branch operators in φ N , it is possible to find a (2, 0) realization with a lower rank Lie algebra. Assume we have a generic regular puncture, and denote h N (f ) the height of the N th box in Young tableaux f with h N ≤ N . Denote u as the Coulomb branch operator in φ N , then its scaling dimension is To have a reduced theory (no Coulomb branch operators in φ N (z)), [u] should have scaling dimension less or equal to one, so φ N is zero and the above is factorized as, Then it is possible to find a description with lower rank (2, 0) theory. The constraint on h N is then, Since h N ≤ N , the above equation has solution if k < 0. So if we have following situation there is a collapsing of sl N type W algebra into a sl N type W algebra with N < N .

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We already know the flavor symmetry on the Higgs branch of 4d theories we studied in this paper, in this section we will use the associated variety of their corresponding VOAs to learn the Higgs branch chiral ring of these theories.

The Higgs branch as the associated variety of the VOA
The W algebras appear in section 3 take following form, W k (g, f ) with f = [m q , 1 s ]. The associated variety of the above W algebra is given by following formula [80], Here S f is the Slowdoy slice associated with the nilpotent orbit f , and X M is the associated variety of the affine vertex operator algebra g with the level k . If the level k is admissible, the associated variety of AKM is found in [80]. We list the result below (we only show result for k > 2 here, interested readers can work out the general case):  Here X M is the nilpotent orbit specified by the listed partition. In the following, we will describe these Higgs branches in some detail.

The Higgs branch as a quiver variety
Let us compactify our 4d theory on a circle and flow to deep IR to get a 3d N = 4 SCFT. The Higgs branch of the 3d theory is the same as the 4d theory, so it is described by the associated variety of the corresponding VOA. Meanwhile the 3d theory has a Coulomb branch which is also a hyperkahler manifold. The 3d N = 4 theory has an interesting mirror symmetry: there is a mirror theory B whose Higgs branch is the Coulomb branch of theory A, and vice versa. If the mirror theory has a Lagrangian description, its Higgs branch is described by the classical hyperkahler quotient. Let's look at our class A theory whose VOA is: For the special case k = 1, we conjecture that the mirror theory is given by the quiver in figure 11. The simple counting of dimensions of Coulomb and Higgs branch of mirror quiver is: The dimension n H of 3d mirror is equal to the Coulomb branch dimension of original 4d theory, see (4.1).
In the following, we use the methods proposed in [81] to describe the Higgs branch as a quiver variety. Considering a nilpotent orbit f = [(n + k − 1) n 1 , 1 n+n 1 ] and X M with JHEP04(2021)076 Figure 11. 3d mirror for 4d theory whose associated VOA is (3.9) with k = 1. From this quiver, one can actually read another Hitchin system description which uses type III irregular singularity as discussed in [26], and we check the Coulomb branch spectrum which is the same as the one found using the construction in section 3. which is actually independent of k.

VOA for theories with exact marginal deformations
If a four dimensional N = 2 SCFT has exact marginal deformations, then it is possible to write down a weakly coupled gauge theory description which typically looks like figure 15.
If we know the VOA V i for each matter (which should have an affine vertex subalgebra V k i (G i )), the VOA for the parent theory is given by the following coset, We use the fact that the conformal gauging condition implies that the sum of levels from the matter gauged by gauge group G i is −2h ∨ i . If there are more than one weakly coupled gauge theory descriptions, then we have found equivalence between non-trivial coset constructions.

Weakly coupled gauge theory descriptions for AD theories
Consider AD theory engineered using following data, where (k, n) is coprime, and T is taken to be the most general matrix allowed by the grading. The corresponding AD theory often has exact marginal deformations, and its weakly coupled gauge theory description is found in [42,43]. Here we summarize the basic ideas: JHEP04(2021)076  Figure 15. A typical quiver for the weakly coupled gauge theory description of a 4d N=2 SCFT. T i 's are matter systems with non-abelian flavor symmetries, and G i 's are gauge groups.
• We first represent the above theory by an auxiliary punctured sphere Σ with n a black marked points (n a is equal to the number of exact marginal deformation plus one), one blue marked point representing the irregular singularity with flavor symmetries, and one red point representing the regular singularity.
• The weakly coupled description is found by finding a pair-of-pants decomposition of Σ with the rules, a) In degenerating a tube, one create a pair of blue marked point and red marked point.
b) Each three punctured sphere in the pants decomposition has one black, one red and one blue puncture.
Now the crucial thing is to determine the puncture type created in the degeneration limit and the matter which is identified with the three punctured sphere. It turns out that the matter appearing in the above degeneration is exactly the matter studied in section 3.
Since we have already figured out the VOA for the matter part, 10 we can now describe the VOA for the full theory as a coset. Now we discuss in more detail about VOA of general A type theory. A general AD theory of A type is represented by following configuration, Here (k, n) = 1. T = diag(I nq×nq , 0 n 1 ). This theory has flavor symmetries U(n 1 ) × 11 where G F is the flavor symmetry associated to the regular puncture, Weakly coupled gauge theory descriptions are given in [43]. The idea is to represent our theory by an auxilliary punctured sphere Σ with q black marked points, one red marked 10 In fact, we only have the full VOA information for following cases: n arbitrary of A type theory; n even of DN and DN twisted type theory; and n odd for twisted sl2N and sl2N+1 theory. 11 In general, we can choose a partition of size n1 such that one can have more general flavor symmetries other than U(n1). Figure 16. Puncture Riemann surface and its degeneration: each three punctured sphere represents a AD matter, and each tube represents a gauge group.

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point reprenting the data of regular singularity, and one blue marked point represent U(n 1 ) flavor symmetry for the irregular singularity. 12 In particular, the theory considered in last section is represented by a three puncture sphere, whose VOA is identified as a W algebra. The weakly coupled gauge theory description is given by finding a pair-of-pants decomposition of Σ such that each tube is connected by a blue and red punctrue. Moreover, each three punctured sphere has to have one blue, one red and one black puncture as shown in figure 16. Now since we have already identified the VOA for each three punctured sphere, the VOA for the original theory is constructed from cosets of the VOA of the matter system.

Example.
Taking n 1 = 0 and f = [nq] in (5.2), this theory is also called the (A nq−1 , A kq−1 ) theory. To find the weakly coupled gauge theory description, we represent our theory by a sphere Σ with q black points, one trivial blue point and one trivial red point. The weakly coupled description is described by taking a pants decomposition of Σ such that each pant has a black puncture, a red punctrue, and a blue puncture. We assume k ≥ n without losing any generality. The weakly coupled gauge theory description is shown in figure 17, where a = qk n + k , b = qn n + k . (5.5) The square bracket means the integral part of the number inside. The flavor symmetry of the matter content is, T i is the theory studied in section 3, which is engineered by following configuration, with T = diag(I n×n , 0 (i−1)n ). The matter system L i is engineered by following configuration, There is another descrition of the same theory: which is just the level-rank dual of the original 4d theory. The weakly coupled gauge theory description is found in similar way, but the VOA of each matter is described differently (which is just the level rank dual discussed in section 3.1.1). So the level rank duality of above theory is the consequence of the level-rank duality of each AD matter.

Theory with Lagrangian description
Some theories have Lagrangian descriptions, therefore one can find the VOA by cosets of symplectic bosons. Here are some examples, This theory has a Lagrangian description given by the quiver in figure 18. Now each matter is just bifundamental hypers and its VOA is a set of symplectic bosons, and the VOA of the full theory is just the coset of symplectic bosons.

More on the conformal embedding
VOAs for AD matters considered in this paper have the interesting property that their AKM subalgebras have the same central charges as full VOAs, so they define possible conformal embeddings. In this section, we will show that such possible conformal embeddings are much more general in our theory space. Consider following configuration, where T is given by a principle grading, and the classification of b is summarized in section 2.
The flavor symmetry group is U(1) f 0 × G F , where f 0 is the number of mass parameters JHEP04(2021)076 encoded in irregular singularity and G F is the flavor symmetry from the regular singularity. It was noticed in [35] that the 4d central charge has the following form: Here k G is the flavor central charge of flavor group G F . So the corresponding central charge of 2d VOA is Here we use the correspondence k 2d = −k G . We know that the 2d VOA has a AKM subalgebra V k (G) × U(1) f 0 , and the central charge of AKM sector is exactly the central charge of full VOA. Therefore, potentially we have a conformal embedding of AKM subalgebra V k (G) × U(1) f 0 into the VOA. If we can indeed prove the above conformal embedding, we could define the full VOA as a reducible module of AKM subalgebra, which would provide us a definition of the full VOA. The conformal embedding of the AKM algebra into the full VOA is generally not true if we change the regular singularity to a generic one. The conformal embedding is possible only for very special choice of level and nilpotent orbit f , see examples in section 3. Now consider a theory which has exact marginal deformations, and one can find a weakly coupled gauge theory description. Assuming that each matter has a conformal embedding of AKM, we now prove that the full theory also has a possible conformal embedding (at least the total central charges of AKM pieces and that of the full VOA are the same). Assume that the weakly coupled gauge theory description has the form, and also assume that T L part has flavor symmetry G L ×G, and T R part has flavor symmetry G R × G, here G is a simple factor of flavor symmetry group of two matter systems for simplicity. The full theory then has the flavor symmetry G L × G R . The central charge of 4d theory is, If each individual piece has a conformal embedding, we have where c(G L ) and c(G R ) is the AKM central charge from the flavor symmetry G L and G R respectively, and k LG and k RG are flavor central charges. Conformal gauging requires, Substiting (6.6) and (6.7) into (6.5), we find that

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So the full theory also has a possible conformal embedding. We have concluded that the gauged system has the conformal embedding if each matter piece has the conformal embedding. On the other hand, if we assume that one piece of matter and the gauged system has the conformal embedding, the other piece of matter would also have the conformal embedding.
The general AD matter has three non-abelian flavor symmetries [41,42], and the addition of a third non-abelian flavor symmetries would not change the flavor central charge of the other two non-abelian flavor symmetries. We have shown in section 3 that conformal embedding of two non-abelian AKMs into a W algebra W k (g, f ) is possible, one might wonder whether it is possible to have a conformal embedding of general AD theory into a W algebra. However the analysis of section 3.1.3 shows that the conformal embedding of three AKM into a W algebra is not possible (the third non-abelian flavor symmetry would have flavor central charge zero and is decoupled).

Conclusion
We have identified VOAs of a class of AD matters with two distinct non-abelian flavor symmetries as W algebras. Using weakly coupled gauge theory descriptions formed by gauging above types of AD matters, we found the VOA for more general AD theories engineered from 6d (2, 0) SCFTs, i.e. VOA for general (A N −1 , A k−1 ) theory is found.
One usually learns many interesting properties of 4d theory by using properties of 2d VOA, since 4d theory is strongly coupled and little is known about their spectrum while many aspects of 2d VOA are much more well understood. Therefore it is pleasant that 4d theory can actually predict many interesting features about 2d VOAs. In this paper, we show that the simple fact that a single 4d SCFT can be engineered by different 6d configurations can often teach us very interesting lessons about VOAs. For example, we find new level-rank duality, coset descriptions, possible conformal embeddings, and etc. Although VOA is mainly about the Schur sector which includes the Higgs branch information, the usage of Coulomb branch data is often very useful in telling whether two configurations are the same or not, which in turn teaches us interesting lessons of VOA.
One of interesting lesson we learned in this paper is that the flavor symmetries of SCFTs defined by a 6d (2, 0) construction is a subtle issue. There are situations where the naive flavor symmetry is actually decoupled in the IR, which corresponds to collapsing levels of 2d VOA. There are also situations where there is extra flavor symmetries which are not manifest in certain 6d descriptions. Therefore it is interesting to understand the emergency of symmetry from the VOA point of view.
The general AD matter has three non-abelian flavor symmetries, and the remaining task is to identify VOAs for them. Once we find VOAs for these AD matters, we can find the VOA for all SCFTs constructed from 6d (2, 0) construction. VOAs defined using junctions of N = 4 boundary conditions are studied in [82,83]. It appears that they have similar structures involving two or three Lie algebras, and it would be interesting to figure out whether these VOAs have anything to do with VOAs studied in this paper.

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We mainly identify the 4d/2d pair in this paper, and a detailed study of characters and its physical implication will be given in a follow-up paper.

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it is called type I theory in [26], and for b = N − 1, it is called type II theory in [26].
We have the following mapping between the third class of theories and the first two class of theories: Here h ∨ is the dual Coxeter number.

B Coulomb branch spectrum from the Newton polygon
Let us now briefly review how to find the Coulomb branch spectrum from the Newton polygon: • The SW curve at SCFT point is

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Now if there is a generic puncture with Young tableaux of size n+n 1 , we label the boxes of Young tableaux from one to n + n 1 row by row, and record the height of the lth box as h(l), 14 then the Coulomb branch spectrum has the following description, l − jn n + k , l = 2, . . . , n, j = h(l), . . . , (l − 1)(n + k) n , l − jn n + k , l = n + 1, . . . , n + n 1 , j = h(l), . . . , n + k − 1. The set of positive roots is {e i − e j | i < j}. The (e i − e j )-root space is spanned by the elementary matrix E i,j with its ij-entry 1 and zeros otherwise. Nilpotent orbits in A n−1 are labelled by partitions of n (or Young tableaux of n boxes). Following the notation and recipe in [88], for the partition Y = [d 1 , · · · , d i , · · · , d l ], choose a block of consecutive indices {N i + 1, · · · , N i + d i } in such way that disjoint blocks are attached to different d i 's. Then define the set of simple roots for each d i , with C + empty whenever d i = 1. One choice of the standard triple {H, X, Y } for Y is, and
(C.15) For other Lie algebra g, partitions will be specified to particular cases used in the main context. The general recipe for sl 2 -triple can be found in [88]. In the case of g = so 2n = D n and the partition Y = [q m , 1 2n−qm ] with q and m even, one can work out x 0 ,

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.