Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d $(2,0)$ theories

The 6d $(2,0)$ theory has codimension-one symmetry defects associated to the outer-automorphism group of the underlying ADE Lie algebra. These symmetry defects give rise to twisted sectors of codimension-two defects that are either regular or irregular corresponding to simple or higher order poles of the Higgs field. In this paper, we perform a systematic study of twisted irregular codimension-two defects generalizing our earlier work in the untwisted case. In a class S setup, such twisted defects engineer 4d ${\mathcal N}=2$ superconformal field theories of the Argyres-Douglas type whose flavor symmetries are (subgroups of) non-simply-laced Lie groups. We propose formulae for the conformal and flavor central charges of these twisted theories, accompanied by nontrivial consistency checks. We also identify the 2d chiral algebra (vertex operator algebra) of a subclass of these theories and determine their Higgs branch moduli space from the associated variety of the chiral algebra.


Introduction
The six dimensional (2, 0) superconformal theories (SCFT) are mysterious quantum field theories that arise either as low energy descriptions of five-branes in M-theory or in a decoupling limit of type IIB string probing ADE singularities [1][2][3]. They are rigid strongly coupled fixed points in six dimensions that are believed to be determined by an ADE Lie algebra and have no relevant deformations that preserve the (2, 0) supersymmetry [4][5][6]. Residing in a highly constrained structure, the richness of the (2, 0) theories lies in the collection of extended defects and their dynamics [7]. In particular, a plethora of lower dimensional supersymmetric theories have been constructed by compactifications of the (2, 0) theory on manifolds with defect insertions. The sheer existence of the 6d parent has lead to many highly nontrivial predictions for the physics of the lower dimensional theories as well as dualities between ostensibly different field-theoretic descriptions. For many cases, these predictions are verified by techniques that are accessible in the lower dimensions. In this way, even though the (2, 0) SCFT itself does not have a simple field-theoretic construction that allows direct access to its dynamics, 1 we can gain valuable insights by studying its daughter theories.
Among all defects in (2, 0) SCFTs, the half-BPS codimension-two defects are one of the central focuses of investigation in recent years. They play a crucial role in the class S construction of 4d N = 2 SCFTs by compactifying (2, 0) SCFT on a Riemann surface with punctures [10][11][12]. In this setup, the codimension-two defects that extend in the 4d spacetime directions produce the punctures. 2 They often give rise to global symmetries in the 4d theory and supply degrees of freedom that carry symmetry charges. As we review in section 2, these codimension-two defects in the (2,0) SCFT can be described by the singularities of a Lie algebra valued one-form field, the Higgs field Φ, on the Riemann surface. Furthermore, these defects come in two families: the regular (tame) defects corresponding to simple poles for Φ, and the irregular (wild) defects associated to higher order singularities. Classification of the regular defects was given in [11,13,14] and the irregular defects were studied in [15,16]. In particular the regular defects carry flavor symmetries that are subgroups of ADE Lie groups. Table 1: Outer-automorphisms of simple Lie algebras j, its invariant subalgebra g ∨ and flavor symmetry g from the Langlands dual.
The (2, 0) SCFT also has codimension-one defects that correspond to the discrete global symmetries associated to the outer-automorphism group of the ADE Lie algebra (see Table 1). 3 Thus we can consider twisted codimension-two defects that live at the ends of these symmetry defects. 4 On the Riemann surface in a class S setup, the codimension-one defect is represented by a twist line that either wraps a nontrivial cycle or connects a pair of twisted punctures (see Figure 1). Twisted regular defects carry (subgroups of) non-simply-laced BCFG flavor symmetry groups and they were studied extensively in [19,20,14,[21][22][23]. Note that the maximal Z 2 twisted regular defects in type A 2N and D N +1 (2, 0) SCFTs share the same flavor symmetry U Sp(2N ) but differ in Witten's global anomaly for the said symmetry [17,24].
One of main purposes of this paper is to classify twisted irregular defects in (2, 0) SCFTs. Before we summarize the results, let us briefly recall the classification of untwisted irregular defects performed in [15,16] since the method we pursue here is a direct generalization.
The classification of codimension-two defects in [15,16] was based on analyzing consistency conditions on the higher order singularities of the Higgs field Φ on a Riemann surface with a local holomorphic coordinate z. After the singularity is put into the convenient semisimple form by a gauge transformation, where T is a semisimple element of j, the consistency condition on Φ simply says Φ(e 2πi z) = σ g · Φ(z) (1.2) for some inner automorphism σ g of the Lie algebra j. This puts constraints on the defining data (T, k, b) of the singularity, which are then solved systematically by Kac's classification of finite order (torsion) inner automorphisms of simple Lie algebras [25]. Generic 4d N = 2 SCFTs engineered by such irregular defects are of the Argyres-Douglas (AD) type which have fractional scaling dimensions in the half-BPS Coulomb branch spectrum and are intrinsically strongly-coupled [26][27][28].
A distinguished class of solutions to (1.1) and (1.2), known as the regular-semisimple type 5 gives rise to irregular codimension-two defects that are in one-to-one correspondence with three-fold quasi-homogeneous isolated singularities of the compound Du Val (cDV) type (see Table 6). 6 This connection between the two very different types of singularities is established 5 T is a regular semisimple element of the Lie algebra j. 6 More explicitly, the moduli space of complex structure deformations (more precisely miniversal or semiuniversal deformations) of a cDV singularity is identified with the Hitchin moduli space of the Higgs bundle on a sphere with one irregular puncture of the regular-semisimple type. This identification can also be interpreted as a correspondence between non-compact "CY3" integrable systems and Hitchin integrable systems in the sense of [29] (see also [30] for a review and some recent developments). It would be interesting to make this more precise. by the observation that identical 4d N = 2 SCFTs are engineered by i) compactifying (2, 0) SCFT on P 1 with such a irregular defect inserted; ii) the decoupling limit of IIB string probing a cDV singularity. We review the general untwisted irregular defects, the resulting 4d N = 2 SCFTs as well as their physical data in section 2.
In this paper, we incorporate outer-automorphism twists into the configurations of codimensiontwo irregular defects in the (2, 0) SCFT. The consistency condition for the Higgs field singularity at z = 0 on the Riemann surface (1.2) is modified to where σ g o labels an outer-automorphism of j with o generating automorphisms of the Dynkin diagrams (see Table 1) and the parameter b in (1.1) is replaced by b t . As we explain in section 3, this constraint can be solved by invoking the classification of finite order (torsion) outer-automorphisms of simple Lie algebras, which is also given in [25]. Restricting the polar matrix T in (1.1) to be regular semisimple again gives rise to a distinguished class of twisted irregular defects, which can be put into the three-fold form in Table 4. 7 We also explicitly identify the continuous free parameters of these defects with flavor symmetry masses and exactly marginal couplings.
Moving onto section 4, we classify 4d N = 2 SCFTs that are engineered by such twisted irregular defects in class S setups which we refer to as twisted theories. The twisted theories come in infinite families (labelled by b t ) for each choice of the simple Lie algebra j (see Table 4). We spell out a simple procedure to extract physical data of such theories from our descriptions and propose formulae for the conformal and flavor central charges. Most of the theories we construct here are new but we identify in our setups several (sequences of) known constructions that only involve regular defects. They provide a nontrivial consistency check of our construction and central charge formulae.
A general 4d N = 2 SCFT is known to have a nontrivial protected sector described by a 2d chiral algebra [31]. Recent developments indicate that it captures information of both Coulomb branch and Higgs branch physics of the 4d theory [32][33][34][35][36][37][38][39][40]. In particular, the Higgs branch of the 4d theory is identified with the associated variety of the 2d chiral algebra [41]. In section 5, we identify the associated 2d chiral algebra or vertex operator algebra (VOA) for a subclass of the twisted theories and determine the Higgs branch of these theories from the associated variety of the VOA. We end in section 6. with a summary and future directions.

Background review
In this section, we review the description of codimension-two BPS defects in 6d (2, 0) SCFTs in terms of the Higgs field and explain the class S construction of 4d N = 2 SCFTs using the Hitchin system on a Riemann surface with defect insertions. We also summarize the classification of untwisted defects in the last subsection. Readers familiar with these topics may safely skip this section.

Hitchin system and 4d N = 2 SCFTs
A large class of 4d N = 2 superconformal field theories can be engineered by the twist compactification of the 6d (2, 0) type J = ADE theories on a Riemann surface C, usually referred to as the UV curve or Gaiotto curve [10,11]. The Riemann surface C can come with punctures at isolated points {p i }, which correspond to codimensional-two BPS defects in the 6d SCFT. In M-theory, the A N type (2,0) SCFT captures the low energy dynamics of a stack of fivebranes and the codimensional-two defect is described by another stack of fivebranes which share four longitudinal directions with the former. To produce 4d theories with N = 2 supersymmetry, we have the first stack of fivebranes wrapping C, whereas the defect fivebranes extend along the cotangent fibers of C at the singularities p i . Defect fivebranes: To decode information about the 4d N = 2 SCFT from this construction where the 6d parent has no explicit description (e.g. in terms of a Lagrangian), it is useful to consider the alternate compactification of the 6d theory on a circle transverse to the Riemann surface C.
Upon twisted compactification of the 5d theory on the Riemann surface C with holomorphic coordinate z, there's a natural principal J-bundle E over C with gauge connection A = A z dz + Azdz and two of the five 5d scalars combine into a (1, 0)-form Φ = Φ z dz valued in the adjoint bundle ad(E). The supersymmetric configurations of the twisted theory are governed by the Hitchin equations where F denotes the curvature two-form of A. The pair (E, Φ) subject to (2.2) is referred to as the Higgs bundle and Φ is the Higgs field. In particular, the second line of (2.2) implies Φ is a holomorphic section of K ⊗ ad(E) where K denotes the canonical bundle on C. .

(2.3)
M H (C) is a hyper-Kähler manifold of rich structure that encodes dynamics of the 4d theory T 4 [C] as well as its 3d descendant. In one complex structure, M H (C) is equivalent to the moduli space of j C -valued flat connections A = A + iΦ + iΦ on C. For example when C has genus g with no punctures, M H (C) is parametrized by the holonomies around the 2g cycles labelled by elements U 1≤i≤g and V 1≤i≤g in J C that are subject to the constraint We can recover the usual Seiberg-Witten (SW) description [50,51] of the low energy dynamics of the 4d theory T 4 [C] as follows. The SW curve Σ governing the Coulomb branch dynamics is equivalent to the spectral curve of the Hitchin system, 10 det(xdz − Φ) = 0, (2.7) and the SW differential is identified with λ = xdz. More explicitly, the SW curves can be put in the following forms (2.8) Here φ i (z) is a degree i differential on C and they generate the ring of fundamental invariants (Casimirs) of the Lie algebra. For E N case, we only list the independent differentials. The coefficients u i,j of these differentials in the z expansion encode the Coulomb branch parameters of the theory. 11 Since the integral of the SW differential along a one-cycle gives the mass of a BPS particle, the SW differential λ = xdz has scaling dimension one and consequently

Review of untwisted (ir)regular defects
The relevant codimension-two defects in the 6d (2, 0) theory of type j can be characterized by singular boundary conditions for the Higgs field Φ on C. Supposing the defect is located at 11 Some of these coefficients are redundant (unphysical) and can be fixed by a coordinate change of (x, z) that preserves the SW differential λ up to an exact term. 12 This is meaningful since the theory T 4 [C] is assumed to be conformal. 13 There are two caveats to this prescription: (i) u i,j with integral scaling dimension ∆[u i,j ] ≥ 2 may correspond to a homogeneous polynomial (flavor Casimir) in the mass parameters; (ii) there can be constraints among u i,j . Both subtleties can be taken care of systematically from information of the puncture(s) on C [14]. z = 0 on C, it is convenient to perform a gauge transformation (that may involve fractional powers of z) to put Φ z in the following form: where each T is a semisimple element of j. In other words, by a gauge transformation, we have T ∈ h, a Cartan subalgebra of j, for all [18]. We have suppressed the non-divergent terms in (2.11). 14 To ensure the Higgs field Φ is well-defined on C, we require for some g ∈ J and ω = e The usual regular (tame) punctures correspond to b = −k in which case Φ has a simple pole (go to a branch cover if necessary) and the constraint (2.12) is trivialized by taking g = 1. The regular punctures are thus classified by the conjugacy classes of T . When j is a classical Lie algebra, these conjugacy classes are labelled by the Hitchin partitions which are related to the Nahm partitions by the Spaltenstein map. The generalization to exceptional Lie algebras involves the Bala-Carter labels [14]. The flavor symmetry associated to the puncture can be directly read off from the Nahm label and the entries of T correspond to the mass parameters. The local contributions to the conformal and flavor central charges can also be computed systematically [14].
The irregular (wild) punctures arise when k > −b where Φ has a higher order pole. 15 Now (2.12) puts nontrivial constraints on T and b which were solved by Kac [52]. 16 One can associate to g an inner automorphism of order b (torsion automorphism) σ g of j which then introduces a grading on j j = m∈Z/bZ j m . (2.14) In particular T is a semisimple element in j . Such finite-order inner automorphisms are classified by Kac (see §8 of the textbook [52]). 17 14 We emphasize here that from the perspective of the resulting 4d theory, the coefficient matrices {T } of the polar terms correspond to parameters such as chiral couplings and masses whereas the Coulomb branch moduli are encoded in the non-divergent terms of (2.11). 15 The Hitchin pole in the original holomorphic form (as opposed to (2.11)) will have higher integral pole orders but the coefficient matrices are not necessarily semisimple [18]. 16 Here we assume that the Hitchin pole is irreducible. In other words, the structure group J associated to the Higgs bundle is not reduced. 17 See [53,54] for a recent discussion relevant for 4d N = 2 SCFTs.
Since we are interested in 4d N = 2 SCFTs, the configurations of defects on the Riemann surface C must be consistent with a U (1) R symmetry. In the absence of irregular punctures on C, the U (1) R generator is identified with the rotation generator R 45 for the SO(5) R-symmetry group of the 5d MSYM that acts on the Higgs field as [R 45 , Φ] = Φ. The regular codimension-two defects are conformal 18 thus the number and positions of the regular punctures on C as well as the topology of C are unconstrained. 19 On the other hand, in the presence of an irregular defect defined by (2.11), the potential 4d U (1) R generator involves a combination of R 45 and U (1) z that acts as so that the leading polar matrix T k in (2.11) is preserved. For this to be globally defined on C, we are restricted to consider C = P 1 with either a single irregular puncture, or an irregular puncture accompanied by a regular puncture, located at the two fixed points z = 0, ∞ of U (1) z (see Figure 2). , Y ) theories in [16] respectively. Here J (b) [k] labels the irregular defect and Y labels the regular defect.
A distinguished class of irregular punctures (2.11) of the regular semisimple type in general 6d (2,0) SCFTs were classified in [16], generalizing earlier work for A-type (2,0) theories [15]. The Hitchin pole in these cases is characterized by a regular semi-simple element T k in j. The commutant of T k in j is a Cartan subalgebra h T k . Restricted to h, the inner automorphism σ g with a regular eigenvector T k corresponds to a regular element in the Weyl group W (h) in Springer's classification of regular elements for complex reflection groups [55]. The orders of these regular elements (known as regular numbers d in [55]) are listed in Table 5 and each order is associated to an unique regular element up to conjugation by W (h).
The corresponding torsion automorphisms have the same orders except for j = A 2N and 18 The more precise statement is that the boundary condition that defines a regular defect (2.13) is scaleinvariant. In 4d we expect this further implies conformal invariance.
30, 24, 20 This class of irregular punctures and the resulting 4d theories were labelled by J (b) [k] in [16] where b takes the values of the regular numbers in Table 5 including their divisors. 22 The Coulomb branch spectrum and conformal central charges were computed in [16], where it was shown that these theories are in one to one correspondence with those constructed by IIB string probing three-fold isolated quasi-homogeneous singularities of compound Du Val (cDV) type (see Table 6). The case of including a regular puncture with nontrivial flavor symmetry was also considered in [16]. In particular a class of theories denoted by (J (b) [k], F ) with nonabelian ADE flavor symmetry were constructed by including an additional maximal (full) regular puncture. A nice feature of these J (b) [k] and (J (b) [k], F ) theories is that many of them have simple 2d chiral algebras that correspond to either W-algebra minimal model or affine Kac-Moody algebra. In particular the flavor central charge of the (J (b) [k], F ) theory [56,35,53,54]: In this case, since the Weyl group W (h) acts faithfully on T ∈ h we can effectively identify b with d as opposed to the order of σ g which is twice as big. 21 Recall that T k = T 1 here since the subscript takes value mod b. 22 The associated torsion inner automorphism has order b gcd(k,b) (see around Footnote 20 for a caveat for A 2N ). We could have restricted to gcd(k, b) = 1 in the regular semisimple case and consider all possible values of b (including the divisors of the entries in Table 5) when defining the J (b) [k] punctures. However in writing down formulae (e.g. for central charges), we find it more convenient to allow gcd(k, b) = 1 while keeping in mind the identification was identified with the level of the affine Kac-Moody algebra j by k 2d = −k F . In section 5, we will see how the twisted theories realize the other types of affine Kac-Moody algebra and W-algebra associated to non-simply-laced simple Lie algebras.

Twisted irregular defects
In this section, we study the classification of general twisted codimension-two BPS defects in 6d (2, 0) SCFTs. Along the way, we make connection to known results in the literature. We then describe in detail the classification of a distinguished class of twisted irregular defects of the regular-semisimple type. The physical interpretation for the parameters of the twisted irregular defect is explained in the last subsection.

Classification of general twisted defects
When j has a nontrivial outer-automorphism group Out(j), we can decorate the puncture (defect) with a monodromy twist by o ∈ Out(j) around the singularity (see Table 1). In the IIB realization of the (2, 0) SCFT from the decoupling limit of string probing an ADE singularity, the above outer-automorphism group arises from the symmetry of the singularity [57] (see Footnote 3 and Table 2).
Locally the Higgs field behaves as and the twist amounts to modifying the requirement (2.12) to . Globally the twisted punctures must come in pairs connected by twist lines.
Compared to the untwisted case, the gauge transformation required for the Higgs field to be well-defined is now σ g o which is a twisted torsion automorphism of j of order b t that projects to a nontrivial element in Out(j). 23 It induces a grading on the Lie algebra such that T is a semisimple element of j which has eigenvalue ω under σ g o. 24 The twisted torsion automorphisms for simple Lie algebras were classified in [52].
A subclass of twisted punctures with leading simple pole are called regular twisted punctures in which case the constraint (3.2) is solved by σ g = 1 and for o of order 2 in the case of j = A n , D n , E 6 and for o of order 3 in the case of j = D 4 . Various physical information associated to these twisted punctures were studied in [14] including the local contributions to the Coulomb branch and Higgs branch, flavor symmetry and central charges. The 4d class S theories constructed from these punctures were studied extensively [23,58,22,21,59].
More generally, we have an twisted irregular puncture from (3.1). In the presence of outerautomorphism twist, to construct 4d N = 2 SCFT with one twisted irregular puncture, we must pair it with a regular twisted puncture on P 1 . 23 In particular b t is always a multiple of |o|. 24 In the superscript of j , is understood to be mod b t .

Twisted irregular defects of regular semisimple type
Similar to the J (b) [k] type of untwisted irregular punctures (defects) introduced in [16], there is a distinguished class of twisted irregular punctures of the regular semisimple type. This is achieved when T k is a regular semisimple element satisfying for ω = e 2πi b t and a twisted torsion automorphism σ g o of j.
Restricted to the commutant Cartan subalgebra h of T k , σ g o induces an element in the twisted Weyl group W t (h) ≡ W (h) Out(j). In particular, it corresponds to a twisted regular element of W t (h) in Springer's classification that generalizes the (untwisted) regular elements of W (h) which are associated to inner torsion automorphisms of j (for all simple Lie algebras) with regular eigenvectors [55]. Since we always work with semisimple elements of h in this paper, we will abuse the notation and use σ g o to denote both the twisted torsion automorphism and the corresponding twisted regular element of W t (h), similarly for σ g in the untwisted case. 25 As in the untwisted case, each twisted regular element is determined uniquely (up to conjugation) by a positive integer d t such that its regular eigenvector in h has eigenvalue e 2πi d t . The order of the twisted regular element is d t = lcm(d t , |o|). 26 In particular, the twisted regular element σ g o from (3.6) is associated with .
The twisted regular numbers d t were classified by Springer [55] and summarized in Table 7.
In particular the maximal values for d t correspond to the orders of twisted Coxeter elements and are called twisted Coxeter numbers h t of j in [55] (see the second column Table 7). In general, a twisted regular element induces a grading on the Cartan subalgebra such that h d t /dt contains a regular semisimple element. The corresponding twisted irregular Hitchin pole is specified by We group these Hitchin poles into class I and II for 2 A N , 2 D N , 3 D 4 , and class I, II and III for 2 E 6 according to the values of d t listed in the Table 7, with the understanding that their allowed divisors fall into the corresponding classes. 27 Below we give explicitly the Hitchin pole for A 2N −1 , A 2N and D N when the associated twisted torsion automorphism σ g o has the maximal order in each class.
For example, consider the case j = A 2N −1 . There are two classes of twisted irregular punctures. Denoting the standard basis of R 2N by {e 1 , e 2 , . . . , e 2N } and the Cartan subalgebra of while the Weyl group W (A 2N −1 ) acts by permutation on e i .
The required gauge transformation σ g corresponds to a permutation in the Z 2N −1 subgroup of W (A 2N −1 ) acting the lower right 2N − 1 diagonal entries of T . 14) The required σ g now corresponds to a permutation in the Z 2N subgroup of W (A 2N −1 ) .
The case j = A 2N is similar and again has two classes of twisted irregular punctures. The generator of Out(A 2N ) = Z 2 defined by 28 while the Weyl group W (A 2N +1 ) acts by permutation on e i with 1 ≤ i ≤ 2N + 1.
for gcd(k, 4N + 2) = 1, where , ω 4N +2 = 1 (3.17) 28 A special feature of the Z 2 outer-automorphism of A 2N compared to the other cases in Table 1 is that it does not fix any simple root, instead it exchanges the pair of simple roots α N = e N − e N +1 and α N +1 = e N +1 − e N +2 . This has important consequences on the 4d theories engineered by such punctures which we explain later in the paper. (3.24)

Physical parameters from the punctures
The defining data of the punctures can be identified with the parameters of the resulting 4d theories. In particular, for SCFTs, we are interested in the masses for flavor symmetries and exactly marginal couplings. Of course one can enumerate such parameters in the SW (spectral) curve. Here we describe how these data can be extracted directly from the punctures.
The grading (3.3) induces a natural conjugation action of a reductive Lie group J 0 associated to j 0 on j . Therefore j are J 0 -modules of finite ranks. Furthermore, the U (1) R symmetry associated with the singularity acts by (3.25) such that T k ∈ j d t /dt has zero U (1) R charge. In general T has U (1) R charge Hence T −bt ∈ j 0 has U (1) R charge q R = 1. From N = 2 superconformal symmetry, we deduce that T −bt contains the mass parameters of the theory whereas T k is associated to exactly marginal couplings.
The maximal number of exactly marginal couplings is determined by the rank of j d t /dt as a J 0 -module to be Note that we have fixed the redundancy due to conjugation by J 0 as well as rescaling of the z coordinate.
The maximal number of mass parameters is captured by the dimension of the intersection between the centralizer of the semisimple part of j d t /dt and j 0 (3.28) Both n marg and n mass can be extracted from the grading (3.3) of h described in [55]. We summarize the results in Table 8

Twisted theories and central charges
Given the classification of the twisted irregular defects in the previous section, we now use them to construct 4d N = 2 SCFTs and study properties of the resulting theories. In particular, we determine their flavor symmetry and Coulomb branch spectrum, and propose conjectured formulae for the flavor and conformal central charges. We later offer nontrivial checks for these conjectures.

Classification of theories from twisted irregular defects
We are interested in 4d N = 2 SCFTs engineered by compactifiying six dimensional (2, 0) SCFT of ADE type on twice-punctured P 1 with outer-automorphism twist. The twist line connects one twisted irregular singularity and one twisted regular singularity on P 1 . We will refer to such 4d theories as twisted theories (see Figure 3). One common feature of the twisted theories is their non-simply-laced flavor groups G (see Table 1) coming from the twisted regular punctures. Figure 3: The class S setup for twisted theories: one twisted irregular defect (star) and one twisted regular defect (dot) on a sphere.
As we have reviewed in the last section, the classification of twisted irregular punctures is reduced to that of torsion outer-automorphisms of the Lie algebra j and the associated gradings (3.3). In this section, we focus on the regular semisimple type (see Section 3.2). The torsion outer-automorphism in this case corresponds to a regular element of the twisted Weyl group W t (h) and induces a grading (3.8) on the Cartan subalgebra h, labelled by d t in Table 7.
The corresponding twisted irregular singularity of the Higgs field takes the following forms Here T is regular semisimple, b t takes the values of d t as in (3.9) and k is an arbitrary integer such that the leading pole order is larger than one. Recall that h t denotes the twisted Coxeter number.
For simplicity we take the regular twisted puncture to be maximal. Some physical properties of the resulting twisted theories can be extracted as follows: • The SW curve at the conformal point takes the forms listed in Table 4. Note that now the z variable admits fractional powers.
• The regular twisted puncture gives rise to non-Abelian flavor symmetry G (see Table 1). There are two cases with C N flavor symmetry from Z 2 twisted defects in A 2N and D N +1 (2, 0) theories. We label them by C anom N and C anom N respectively. 29 They differ by Witten's global anomaly for U Sp(2N ) [24]: the former carries a nontrivial anomaly whereas the latter is non-anomalous.
• The number of mass parameters for U (1) flavor symmetries associated to the twisted 29 We thank Yuji Tachikawa for suggesting these names. j • The dimension of the conformal manifold can be extracted from ∆ = 0 parameters for the Hitchin poles, as summarized in Table 8-12.

Degrees of the Casimirs
• The full Coulomb branch spectrum can be found by expanding the degree d i differentials φ d i (z) in z, where d i labels the degrees of the Casimirs of the parent ADE theory, see Table 13 for these numbers. The novelty here is that some of the differentials are no longer holomorphic and they often have a Laurent expansion with half integral (or plus-minus one third) powers of z, according to their transformation rule under the outer-automorphism in Table 13. The spectrum of half-BPS Coulomb branch chiral primaries is then summarized in Table 14.
Flavor group G Coulomb branch spectrum ∆ C anom Table 14: Coulomb branch spectrum of the twisted theories from a twisted irregular puncture of the regular-semisimple type and a maximal twisted regular puncture. The Coulomb branch chiral primaries are constrained to have ∆ > 1 by unitarity.  Table 15: Some useful Lie algebra data. h is the Coxeter number and h ∨ is the dual Coxeter number. r ∨ is the lacety of the Lie algebra and n is equal to r ∨ except for C anom N .
• We present conjectures for the flavor and conformal central charges with nontrivial evidences in the next section.

Flavor and conformal central charges
Before we state the conjectural formulae for the flavor and conformal central charges, let us provide some motivations for how they come about. The important observation is, in all previously known class S constructions that involve maximal regular punctures, it appears that the flavor central charge is determined by certain maximal scaling dimension among the CB spectrum contributed by the maximal puncture: 30 Here ∆ max denotes the maximal scaling dimension contributed by the maximal regular puncture which determines the flavor central charge in the untwisted theories. In twisted theories, empirical evidence suggests that one should instead take the maximal scaling dimension ∆ max from the twisted differentials (i.e. a differential that transforms non-trivially under the twist, see Table 13) at the maximal puncture. The Z 2 twisted maximal punctures of A 2N type re- 30 Note that our normalization of the flavor central charges is related to the one in [14,22] by k ours quires special attention. 31 For example, in a Z 2 twisted A 2N type class S construction without irregular punctures, the maximal scaling dimension contributed by the twisted differentials at the maximal twisted regular singularity is 2N + 1 from φ 2N +1 , and the U Sp(2N ) flavor central charge computed in [22] is equal to (in our normalization) h ∨ (U Sp(2N )) = N + 1 in agreement with (4.2). The introduction of irregular punctures into the setup modifies the U (1) R symmetry of the 4d theory, but we expect (4.2) continues to hold. The formula for 2a − c is given in [62] while our new formula for c has a close relation to the 2d chiral algebra which we will explain in Section 5. 32 Conjecture 1 (Flavor central charge k G ). The central charge of the flavor symmetry G of a twisted theory defined by a twisted irregular defect of the regular semisimple type in (4.1) and a maximal twisted regular defect takes the following form: The Lie algebra data h ∨ and n are listed in Table 15.
Conjecture 2 (Conformal central charges a and c). The conformal central charges of a general twisted theory are determined by Here f is the number of mass parameters contributed by the irregular singularity, k G denotes the flavor symmetry central charge listed in (4.3), and ∆[u i ] are the Coulomb branch scaling dimensions listed in Table 14.
Let us provide some evidences for the above conjectures by considering twisted theories defined with integral order Hitchin poles, in which case the irregular singularity takes the form Φ z = T z 2+k + . . .
(4.5) 31 Note that the 1 2 shift in the third equation of (4.2) is exactly the contribution from a half-hyper multiplet in the fundamental representation of U Sp(2N ) which would also saturate Witten's global anomaly for these punctures [24]. Thus it's tempting to say that this 1 2 contribution comes from the minimal boundary modes of Z 2 symmetry defect. The other 1 2 factor in (4.2) should be related to the fact that the Langlands dual SO(2N + 1) of the U Sp(2N ) has an index of embedding equal to one in D N +1 but two in A 2N . A better understanding of the anomaly inflow from 6d along the lines of [61] should give a rigorous argument for (4.2). 32 The essential statement here is that the 2d chiral algebra contains the affine Kac-Moody algebras associated to both the simple and U (1) factors of the flavor symmetry, and the 2d stress-tensor is given by the Sugawara construction.
where T is regular semisimple and the associated grading of h corresponds to d t = 1. We will use following formula to compute their central charges. The second equation above is known to hold for the untwisted irregular puncture defined by integral order Hitchin poles [15] and we assume that it is also valid for the current situation. 33 Here we take the regular twisted puncture to be maximal in which case the Higgs branch dimension is [14,16] dim H Higgs = 1 2 (dim(G) − rank (G)) + rank (G). (4.7) The last term above comes from the twisted irregular puncture.
On the other hand, since the irregular singularity contributes f = rank (G) mass parameters (see top rows of Table 8- Here the conjectured flavor central charge (4.3) takes the form We have verified that that the two formulae (4.8) and (4.6) give the same answers. Below we give some details for two instances of such checks for illustration.
Example 1. Let's consider the C anom N −1 theory which is constructed by Z 2 twist of the D N (2, 0) theory. The Hitchin pole (4.5) fixes the U (1) R charge (hence scaling dimension) of z, (4.10) and the Coulomb branch spectrum can be enumerated using Table 14 Along with the Higgs branch dimension from (4.7) dim H Higgs = (N − 1)N (4.13) we obtain from (4.6) which is in agreement with (4.8) where the flavor central charge is determined by (4.9) to be

Twisted theories with Lagrangians
It turns out that many sub-families of the theories engineered from twisted irregular defects actually have Lagrangian descriptions. 34 Since the Coulomb branch spectrum for Lagrangian theories have integral scaling dimensions, a necessary condition is ∆[z] ≡ 0 (mod 2) for A n , D n , E 6 theories with Z 2 twist, and ∆[z] ≡ 0 (mod 3) for D 4 with Z 3 twist. This can be achieved by choosing k appropriately with respect to b t in the Hitchin pole (4.1).
Since such theories have a weakly coupled frame, we can use the formulae to compute the conformal central charges. Here n v and n h count the number of vector and hypermultiplets in the quiver gauge theory description. Similarly the central charges associated to the G flavor flavor symmetry of hypermultiplets can be obtained straightforwardly from identifying the embedding G flavor × G gauge ⊂ U Sp(2n h ). This allows us to verify the conjectured formulae (4.3) and (4.4).
As we will see, often times the Lagrangian description only makes manifest a subgroup of the full flavor symmetry which is realized in our description by the single regular puncture. so that we have the scaling dimension ∆[z] = 2. The Coulomb branch spectrum for this theory (using Table 4 and 14) is The central charges from (4.4) and (4.3) are (4.25) 34 In this paper, we implicitly assume that the SW geometry together with all of its N = 2 deformations fixes the N = 2 SCFT uniquely. To our best knowledge there is no counter-example but it would be interesting to prove this rigorously.
Note that from Table 10, the N even case has N marginal couplings and no mass parameters from the irregular puncture whereas the N odd case has N − 1 marginal couplings and an extra mass parameter. This results in the different expressions for the conformal central charges above.
The boxed nodes label the flavor symmetry of hypermultiplets. Here the symmetry is as provided by 2N hypermultiplets in the fundamental representation of U Sp(2N ). We can also count the number of hyper and vector multiplets from the quiver.  so that ∆[z] = 2. As before, the Coulomb branch spectrum (from Table 14) and central charges (from (4.3) and (4.4)) are (4.30) In particular, the irregular singularity contributes no additional mass parameters and the theory has 2N −1 marginal couplings (see Table 8). The theory has a Lagrangian description by so we can compute the central charges using the field content (4.35) From Table 9 the irregular singularity contributes no additional mass parameters and 2N −2 marginal couplings.
The theory has a Lagrangian description as so we can compute the central charges using the field content Moreover the flavor central charge is supplied by 2N − 2 half-hypers in the fundamental representation of SO(2n + 1) thus Again we see they are in agreement with (4.35),

Example 6. Half-hypermultiplet
A free half-hyper multiplet in the fundamental representation of U Sp(2N ) flavor symmetry can be constructed using A 2N (2, 0) theory with Z 2 twist. The irregular puncture is specified by It is easy to see from our general formulae that the Coulomb branch is empty in this case and the central charges are as expected for a half-hyper in the fundamental representation of U Sp(2N ) (or N free halfhypers).
We emphasize here that this is the only twisted theory within our construction that has an empty Coulomb branch yet nonvanishing central charge.
Example 7. Let's take G = C anom N −1 which is derived from the Z 2 twist of D N (2, 0) theory. We take N = 3n with n ∈ Z + and the irregular puncture is specified by From Table 10 the irregular singularity contributes no additional mass parameters and 2 marginal couplings.
The theory has a Lagrangian description by We can then check the central charge by counting the multiplets Furthermore the flavor central charge for U SP (6N − 2) is from the quiver, in perfect agreement with (4.42).
The simplest example in this sequence of theories is when n = 1 which can be equivalently described by a cyclic quiver with two SU (2) nodes,

SU (2)
SU (2) and comes from type A 1 (2, 0) theory on T 2 with two punctures. This theory has U Sp(4) enhanced flavor symmetry which is manifest in our description from type D 3 (2, 0) theory on P 1 with twisted irregular punctures. 35 Example 8. Let's take G = C anom N which is engineered from the Z 2 twist of D N +1 (2, 0) theory. We take N = 3n and the irregular singularity specified by b t = 6n, k = 3 − 6n From Table 10 the irregular singularity contributes one mass parameter and 2 marginal couplings.
Example 9. Consider the F 4 theory constructed from E 6 (2, 0) theory with Z 2 twist and the irregular puncture is specified by b t = 12, k = −9.  They are consistent with (4.52) since the index of embedding I so 9 →f 4 = 1 and our description with irregular puncture makes manifest the enhanced F 4 flavor symmetry of the theory. 36

Twisted theories and non-Lagrangian conformal matter
In addition to the Lagrangian examples discussed in the last section, our twisted theories also generalize many non-Lagrangian conformal matter theories constructed in class S with regular (tame) punctures. Below we provide various examples and their reincarnation in our construction with twisted irregular punctures. Various physical data of these conformal matter theories have been extracted from the SW geometry, superconformal index, and decoupling limits of certain Lagrangian theories. We will view these information as support for our construction of the much larger class of theories and nontrivial evidence for our conjectured formulae for the central charges (4.3) and (4.4). As we will see in the examples, our construction often makes manifest the enhanced global symmetry which is obscure in the ordinary (regular punctures) class S setup.
Example 10. R 2,2N conformal matter The R 2,2N non-Lagrangian theory was constructed in [22] from the Z 2 twist of type A 2N (2, 0) theory with three regular punctures on a sphere: one minimal untwisted puncture, and two maximal twisted punctures. 37 This theory also arises in the decoupling limit of N = 2 SU (2N + 1) SYM coupled to one symmetric and one antisymmetric rank two tensor hypermultiplets in a S-dual frame [22].
The Coulomb branch spectrum of R 2,2N SCFT to be ∆ = {3, 5, 7, . . . , 2N + 1} (4.56) and the conformal central charges are (4.57) 36 Recall from [64] that the Dynkin index of embedding for G ⊂ J is computed by where r denotes a representation of J which decomposes into ⊕ i r i under G, and T (·) computes the quadratic index of the representation (which can be found for example in [65]). 37 The theory was also constructed by the circle compactification of a 5d N = 1 SCFT with Z 2 twist [66]. Alternatively, the R 2,2N theories can be constructed from type A 4N (2, 0) theory with Z 2 twist and the twisted irregular puncture is specified by such that ∆[z] = 4N . One can immediately read off the Coulomb branch spectrum from Table 14 and the result coincides with (4.56). The manifest U Sp(4N ) flavor symmetry comes from the regular twisted puncture and its flavor central charge is determined by (4.3) to be (4.58). Our description also makes obvious Witten's global anomaly for U Sp(4N ) [24]. From Table 8 we see that the irregular puncture provides the additional mass parameter responsible for the U (1) factor in the flavor symmetry. It is also easy to check that the central charges computed from (4.4) is consistent with the result (4.57) from [22]. (4.61) From Table 8 we see the irregular puncture contributes one marginal coupling but no mass parameters.
For N = 1, we get the familiar N = 4 SU (2) SYM with n h = n v = 3 ! The SU (2) flavor symmetry with central charge k SU (2) = 3 2 is now realized manifestly by the twisted regular puncture in A 2 (2, 0) theory. It also carries Witten's global anomaly for SU (2).
and the conformal central charges (4.64) From Table 9, in the N even case the irregular puncture contributes no marginal couplings but one additional mass parameter, whereas the N odd case has one marginal coupling and no extra mass parameter.
Upon closer inspection, it turns out that this Z 2 twisted class A 2N −1 setup does not make manifest the full flavor symmetry. For N odd, the theory is identical to the U Sp(N − 1) SYM conformally coupled to N + 1 fundamental flavors which has SO(2N + 2) symmetry. For N even the theory is identified with the R 2,N −1 theories in [67]. Our formulae above again give the correct conformal and current central charges as computed previously with standard methods.
Example 13. G 2 conformal matter Let's consider the D 4 (2, 0) theory with Z 3 twist and choose the irregular puncture to be given by b t = 12, k = −8. (4.66) From Table 12 we see the irregular puncture contributes one marginal coupling but no mass parameters.
This is identified with the E 6 Minahan-Nemeschansky (MN) theory [68] with an SU (3) subgroup of E 6 flavor symmetry gauged.
This theory is identified with the SO(9) 5 × U (1) SCFT in [21]. Note that our description predicts the enhancement of flavor symmetry from SO(9) 5 to (F 4 ) 5 .

Example 15. F 4 conformal matter
Let's consider another irregular defect in the E 6 theory with Z 2 twist specified by b t = 12, k = −10. From Table 11, the irregular puncture contributes two marginal coupling but no mass parameters.

Vertex operator algebra of twisted theories
It was shown in [31] that for any 4d N = 2 SCFT, one can associate a 2d chiral algebra or vertex operator algebra (VOA). The basic correspondence is as follows: • The 2d Virasoro central charge c 2d is given in terms of the conformal anomaly c 4d of the 4d theory as c 2d = −12c 4d . (5.1) • The global symmetry algebra g becomes an affine Kac-Moody algebraĝ k 2d and the level of affine Kac-Moody algebra k 2d is related to the 4d the flavor central charge k F by • The (normalized) vacuum character of the chiral algebra/VOA is identical to the Schur index of the 4d N = 2 theory: We focus on the twisted theory where there is no mass parameter from the irregular singularity, and the regular singularity is labeled by a nilpotent orbit Y . 39 Our proposal for the corresponding VOA is following Conjecture 3. The VOA for the twisted theory is the W-algebra W k 2d (G, Y ). Here G is the flavor symmetry corresponding to the maximal twisted regular puncture, and Y labels the corresponding nilpotent orbit. This W algebra is derived as the Drinfeld-Sokolov reduction of the Kac-Moody algebra g k 2d associated to the simple Lie algebra g at level k 2d .
The ADE cases of these W-algebras is considered in [35,38], and here we discover the correspondence for non-simply-laced simple Lie groups. We have verified the relation between the 4d central charges and the 2d central charges.

Admissible levels of the 2d current algebra
A 2d current algebra level is called admissible if it can be written in one of the following forms: Recall the 2d levels in our case are given by following formula: where b t takes the values of d t in Table 7 and n is as listed in Table 15. We observe that the 2d current algebra levels of the class I twisted theories are admissible (for generic k ∈ Z): Note that for these cases the relevant torsion outer-automorphism that defines the twisted irregular defect is always generated by the twisted Coxeter element.
The Schur index of the twisted theory is then identified with the vacuum character of vertex operator algebra, which is particularly simple for the boundary levels [69]: Comparing with the 2d levels in our list, we see that the boundary levels appear for the B N , C anom N , G 2 , F 4 cases.

Associated variety and Higgs branch
The Higgs branch of the 4d N = 2 SCFT is identified with the associated variety X V of the VOA [38,39] M Higgs = X V . (5.8) For affine Kac-Moody algebra with an admissible level (see (5.6) which corresponds to the case Y = F and we have a maximal regular puncture), the associated variety X V = X M is found to be the closure of certain nilpotent orbits in [41] (5.9) We summarize the result of [41] for the relevant orbits O q and L O q here in Table 16-19.
For q ≥ h(g), O q is the same as the principal (maximal) nilpotent orbit O prin with quaternionic dimension dim H O prin = 1 2 (dim g − rank g). For other values of q, in Table 16 where [s i ] is the transpose partition to [n i ], and r j counts the number of appearances of the part j in [n i ]. For exceptional Lie algebras, we use the Bala-Carter labels [70] for the nilpotent orbits in Table 17-16. We also include the quaternionic dimensions for the reader's convenience.
Near the lower end of the list of nilpotent orbits, we have the minimal nilpotent orbit of the smallest nonzero dimension. It corresponds to the centered one-instanton moduli space of g. Here the minimal nilpotent orbit of C N labelled by partition [2, 1 2N −2 ] shows up in Table 16 at q = 1, in which case the twisted theory is nothing but N free hypermultiplets (see Example 6). On the other hand, the minimal nilpotent orbits of B N , G 2 and F 4 do not appear to be Higgs branches of our twisted theories of the regular semisimple type. These are consistent with the results of anomaly matching on the Higgs branch in [63].
Example: For illustration we consider a class I F 4 theory in (5.6). We take k = −7, so the 2d current algebra level k 2d = −9 + 9 5 and we have q = 5. Looking at Table 19, we conclude that the corresponding Higgs branch is the closure of nilpotent orbit with label F 4 (a 3 ) which has quaternionic dimension 20. The Coulomb branch spectrum of this theory is ∆ = 6 5 , 12 5 , 16 5 , 18 5 , 22 5 , 24 5 , 36 5 , 42 5 and the central charges are a = 247 15 , c = 52 3 . Let's consider the more general twisted theories where the twisted irregular puncture is as before in (5.6) but the twisted regular puncture is now labeled by a general nilpotent orbit Y of the group G. The associated variety X V is then given by (5.12) which describes the Higgs branch for these theories. Here S Y is the Slodowy slice defined by the nilpotent orbit Y [38,39].

Conclusion
We systematically studied irregular codimension-two defects twisted by outer-automorphism symmetries in 6d (2, 0) theories. They engineer 4d N = 2 Argyres-Douglas (AD) SCFTs that admit non-simply-laced flavor groups. We completed the classification of twisted irregular defects of the regular semisimple type, and the result was summarized in Table 4. Together with the classification of the ADE cases in [16], we have a large class of of Argyres-Douglas theories with arbitrary simple flavor groups. We outlined a simple procedure to extract their Coulomb branch spectrum, central charges and in some cases, the 2d chiral algebra and Higgs branch. One can also consider the degenerations of irregular singularities and regular singularities as in [16,53,54] which will give rise to many new AD theories.
The theories we constructed here should be thought of building blocks towards a better understanding of the full space of 4d N = 2 SCFTs. On one hand, by conformally gauging the flavor symmetries of these AD theories we can form new 4d N = 2 SCFTs. On the other hand, some of our theories admit exact marginal deformations, and it is interesting to study S-duality and weakly coupled gauge theory descriptions (which may involve non-Lagrangian matters) of these theories using the method in [53,54].
In this large space of theories, we saw various relations between the physical data, such as those between the central charges and Coulomb branch spectrum, that call for explanations.
For example it would be nice to develop field-theoretic proofs for our conjectured formulae for the central charges (4.4) and (4.3) perhaps along the lines of [62]. It would also be interesting to verify that the vacuum character of our proposed VOA matches with the Schur index of the 4d theory using the 2d TQFT [71][72][73][74].
Lastly, we identified the VOA for the twisted theories defined using irregular defects which do not carry any flavor symmetry. It would be interesting to identify the VOA for the remaining theories. 40 We hope to address some of these directions in future work.