
ABSTRACT
Four dimensional 𝒩=2 superconformal field theories are extremely rich in dynamics, and yet, constrained heavily by superconformal symmetry algebra, leading to numerous fascinating dualities and applications. Among them are the stronglycoupled ArgyresDouglas theories without any Lagrangian description. They were previously believed to be engineered by the class𝒮 construction involving irregular punctures, until very recently. Beem and Peelaers [1] proposed a novel class𝒮 construction for some ArgyresDouglas theories using only regular ones. Here we briefly review this important progress and mention a few possible open problems.
4d 𝓝=2 SCFTs and class 𝓢 construction
4d 𝒩=2 superconformal field theories (SCFTs) strike a fascinating balance between dazzling richness and mathematical rigidity (for pedagogical reviews, see e.g. [2, 3, 4, 5]). On the one hand, unlike the unique 4d 𝒩=4 superYangMills, there are infinitely many 4d 𝒩=2 SCFTs one can define by a range of different approaches, thus generating a vast and varying landscape for physicists to explore. On the other hand, they harbor a variety of rigid structures and exactly computable quantities that are highly constrained by the 4d 𝒩=2 superconformal symmetry. Countless dualities originate from such richness and rigidity, such as the renowned DonaldsonWitten [6] and SeibergWitten invariants [7], the famous AldayGaiottoTachikawa duality (AGT duality) [8] between S^{4}partition functions and correlation functions of Liouville/Toda theories on Riemann surfaces, and the SCFT/VOA (vertex operator algebra) correspondence [9]. These structures further relate to physics in other dimensions. For example, one can map [10, 11, 12] the correlation functions in the VOA [13] to the topological correlators [14] appearing in the 3d 𝒩=4 SCFTs and their mirror dual [15, 16].
Fig. 1: A three punctured Riemann sphere. If the underlying 𝔧 = 𝖘𝖚(2) and all three punctures are the nontwisted regular punctures associated with trivial embedding, the resulting 4d effective theory is simply that of 4 free hypermultiplets. If 𝔧 = 𝖘𝖚 (N) and all three punctures are associated with trivial embedding of Λ : 𝖘𝖚 (2) → 𝖘𝖚 (N >2), the corresponding 4d 𝒩=2 SCFT is the wellknown nonLagrangian T_{N} theory.
Upon the wonderland of 4d 𝒩=2 SCFTs, there are infinitely many pointlike isolated kingdoms which are strongly coupled, and do not admit Lagrangian descriptions. Among them are the particularly interesting, yet simple, T_{N} > 2 theories [17, 18, 19] and ArgyresDouglas theories [20, 21]. Both of these theories admit "class𝒮" construction [22, 23], i.e., compactifying some particular 6d theories on Riemann surfaces to engineer 4d effective theories. The T_{N} theories are constructed using only "regular punctures", while the latter always involve "irregular punctures", until very recently. A paper [1] proposes a particular construction of the ArgyresDouglas theories which only involve regular punctures of a special type. To further explain detail of their construction, let us first review the class S construction.
Classification of regular/irregular punctures
One way to construct 4d 𝒩=2 SCFTs is to start from the famous and yet mysterious 6d (0,2) superconformal theories classified by a ADE algebra 𝔧, say, 𝔧 = a_{n}, with the associated simply connected Lie group J. One can then put such a theory on a product manifold ℝ^{4}×C_{g,n}where C_{g,n}denotes a genusg Riemann surface with n marked points. To consistently send the size of C_{g,n}to zero and therefore engineer a 4d effective theory on ℝ^{4} with the 4d 𝒩=2 superconformal symmetry, one should first solve a system of partial differential equations of the 6d fields on the Riemann surface C_{g,n}, referred to as the Hitchin system. The 1form Hitchin field in the equation is required to have prescribed singularities at those marked points: if the leading singularity is a simple pole, then the marked point is called a regular puncture, otherwise an irregular puncture. These punctures are further determined by additional data as follows.
• A nontwisted regular puncture is simplest and is further specified by its residue, which itself is defined by an embedding Λ : 𝖘𝖚(2) → 𝔧. Recall that such an embedding is equivalent to a certain Young diagram.
• A less simple regular puncture, refereed to as a "twisted puncture", is defined by an outerautomorphism σ of 𝔧 (which then further specifies the invariant subalgebra 𝔧_{0} of 𝔧), and an embedding Λ : 𝖘𝖚(2) → _{}, the Langlands dual of the 𝔧_{0}. Table 1 lists all the allowed embedding for the case, 𝔧 = a_{2}, corresponding to untwisted and twisted regular punctures.
• An irregular puncture J^{b} [κ] is specified by two integers b, κ which control the order of the pole.
The number and detail of these punctures fix all the properties of the effective 4d SCFT. For example, they determine the (manifest) flavor symmetries and the associated flavor central charges, a,c central charges , the Coulomb branch spectrum, etc. In Figure^{1}, we display a threepunctured sphere. The detail of punctures then fixes the effective 4d theory to be 4 free hypermultiplets, or T_{N}theories, or other more complicated ones.
ArgyresDouglas theories
ArgyresDouglas theories are traditionally defined by C = S^{2} with one irregular puncture, and optionally an additional regular puncture. They are strongly coupled, and have no deformation that preserves the 4d 𝒩=2 superconformal symmetry. In particular, they have no Lagrangian description. The latter statement is manifested in the fact that they contain Coulomb branch chiral operators having fractional dimensions, which no 4d 𝒩=2 Lagrangian theory possesses.
A set of extremely useful diagnostic tools are the superconformal index and its various limits. The most relevant ones in the paper we are describing are the Macdonald index and Schur index. Both of them are easy to compute via suitable topological quantum field theory on C_{g}_{,n}. These quantities reveals the detailed structure of the operator algebra, including the representations in which operators transform, their quantum relations, and even their correlation functions.
^{1} The a,c central captures the conformal anomaly in four dimensions, given schematically by T_{} ~ c(Weyl)^{2 }+ a(Euler). A flavor central charge associated to a flavor symmetry captures the leading singularity in the currentcurrent OPE j_{}(x)j_{}(y).
Class 𝓢 construction of AD theories with regular punctures
In Ref. [1], the authors consider a Riemann surface S^{2} with three regular punctures, one untwisted, and a pair of twisted punctures. In particular, for the 𝔧 = a_{2} case, there are five possibilities, corresponding to different arrangements of the embeddings, as shown in Figure2.
Combining with ShapereTachikawa's central charge formula, they deduce the a,c anomaly coeffcients and the dimensions of the Coulomb branch chiral operators for the twisted a_{2}, and later twisted a_{2n} theories. By analyzing their Macdonald index and subsequently the Schur and Hilbert series limit, they also gain insights into the operator contents and some quantum relations in the Higgs branch chiral ring. The associated VOAs are also constructed by working out the operator product expansions of the strong generators.
Table 1. Relevant embedding for 𝔧 = a_{2}. Note that the second row correspond to twisted punctures, while the first row untwisted punctures. Twisted punctures must appear in pair. Here we use blue and green colors to denote embeddings for untwisted and twisted punctures respectively.
Λ : 𝖘𝖚(2) → a_{2} 
[1,1,1], [2,1] 
Λ : 𝖘𝖚(2) → c_{1}

[1,1], [2] 
Fig. 2: Five twisted a_{2} theories associated to different combinations of punctures. The third and the fifth theories will be most interesting, since they are related to ArgyresDouglas theories as discussed later in the main text. Again we use blue and green colors to denote untwisted and twisted punctures respectively.
Among the five possible twisted a_{2} theories, two actually contain Coulomb branch operators of fractional dimensions! In particular, their Macdonald indices reveal that one of them (the last one in Figure 2) is actually the product of the wellstudied (A_{1}, D_{4}) theory and a doublet of free hypermultiplets, while the other (the third one) is the product of two (A_{1}, D_{4}) theories. The former theory has SU (3) × SU (2) flavor symmetry where the SU (2) acts on the free hypermultiplets, and the latter theory has SU (3)^{2} symmetry which acts on the two tensor product ingredients individually. In both cases, the full flavor symmetry is enhanced from the manifest flavor symmetry visible from the punctures on the Riemann sphere. Furthermore, the five twisted a_{2} theories are related by partial Higgsing.
Finally, the authors elaborate on some interesting Sdualities involving (A_{1}, D_{4}) theories. For example, take the product of two (A_{1}, D_{4}) theories. The diagonal SU (3) ⊂ SU (3)^{2} of the product theory is gauged by an SU (3) gauge group, and further coupled to a usual untwisted a_{2} theory, aka 3^{2} free hypermultiplets. Sduality then shows the equivalence of the two quiver theories shown in figure3, by performing pants decomposition of the punctured Riemann sphere in two different ways. On the right, the boxed 1/2 denotes a fundamental halfhypermultiplet, where the SU (2) ⊂ SU (3) flavor of 𝒯_{a}_{2}^{(1)}, and the SU (2) ⊂ SU (3) × SU (2) of the 𝒯_{a}_{2}^{(2)} are gauged by the central SU (2).
Fig. 3: Figures from Ref. [1] showing two Sdual theories made out of the twisted ArgyresDouglas theories 𝒯_{a}_{2}^{(i)}.
There are open questions remaining to be explored. Detailed study of these twisted a_{2n} theories with n >1 are required to further understand the Coulomb branch spectrum. Also, 4d 𝒩=2 theories admit a variety of nonlocal operators, while in the VOA language, surface operators are associated with nonvacuum modules [24]. Therefore, it would be interesting to study the module structure of these twisted theories to better understand the intricate dynamics brought about by nonlocal operators. Moreover, it is also natural to explore the 3d mirror dual of these theories and explicitly uncover the relation between the Higgs branch in the twisted ArgyresDouglas theories and the Coulomb branch correlation functions in the 3d mirror dual [10,11]. Finally, to further complete the AGT dictionary [8], it maybe important and plausible to find the corresponding vertex operators in the Liouville/Toda theory, and in turn define the "instanton partition function" for these twisted ArgyresDouglas theories, following the idea of an unpublished work by Nishinaka and collaborators.
Acknowledgments: This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11875327 and 11905301, the Natural Science Foundation of Guangdong Province under Grant No. 2016A030313313, the Fundamental Research Funds for the Central Universities, and the Sun YatSen University Science Foundation.
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Yiwen Pan is an associate professor of physics at the School of Physics, Sun Yatsen University in Guangzhou, China. He received his PhD degree in 2015 from YITP at SUNY Stony Brook University. He continued his study as a postdoc at Uppsala University in Sweden, and later became a faculty member of Sun Yatsen University in Guangzhou. His research focuses on supersymmetric field theories and various dualities. 

HongHao Zhang is a professor of physics at the School of Physics, Sun Yatsen University in Guangzhou, China. He received his PhD degree in 2007 from Physics Department, Tsinghua University in Beijing. Thereafter he worked as a faculty member at the Sun Yatsen University in Guangzhou. He was promoted to a full professor in January 2017. His research focuses on quantum field theory, particle physics and cosmology. 
