Gaiotto and Witten found that one can construct 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Chern-Simons matter theories by using   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SCFT whose momentum map of global symmetries satisfy special condition. Usually, one uses free hypermultiplet and twisted hypermultiplet, and more recently it was found that strongly coupled theory such as 3d version of T  N  theory and Argyres-Douglas matter can also be used. In this paper, we compute the superconformal index of these   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 theories and derive the Coulomb/Higgs limit. Our results determine the moduli space of vacua, which is used to check various interesting mirror symmetry involving CSM theory and usual   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 gauge theory.

China

Bohan Li

Dan Xie

Wenbin Yan

Superconformal indices of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Chern-Simons matter theories

There are two major ways of constructing 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories (SCFTs): the first one is putting a 6d (2, 0) theory on a punctured Riemann surface (class-S theory), and the second one is putting type IIB string theory on a 3d canonical singularity. As there are interests on low rank theories, we search all the possibilities from above two constructions. Most of those theories are engineered by class-S theory with irregular singularities, and we find a universal formula for the rank of theory so that a complete search is possible. We then compute various physical quantities of those theories, such as the central charges, flavor symmetry, associated vertex operator algebra and Higgs branch, etc. One of interesting consequence of our results are the prediction of many new isomorphism of 2d vertex operator algebra.

On low rank 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs

One can derive a large class of new   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFTs by turning on   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 preserving deformations for   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Argyres-Dougals theories. In this work, we use   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal indices to get indices of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFTs, then use these indices to derive chiral rings of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFTs. For a large class of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 theories, we find that the IR theory contains only free chirals if we deform the parent   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 theory using the Coulomb branch operator with smallest scaling dimension. Our results provide interesting lessons on studies of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 theories, such as a-maximization, accidental symmetries, chiral ring, etc.

A study of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFT derived from   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFT: index and chiral ring

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

4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs and lisse W-algebras

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 [q  m  , 1 s ]. Gauging above AD matters, we can find VOAs for more general   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs engineered from 6d (2, 0) theories. For example, the VOA for general (A  N − 1 , A  k − 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.

W algebras, cosets and VOAs for 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs from M5 branes

We study chiral algebras associated with Argyres-Douglas theories engineered from M5 branes. For the theory engineered using 6D (2,0) type <math><mi>J</mi></math> theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of <math><mi>J</mi></math> type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of <math><mi>J</mi></math> type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.

United States

Shing-Tung Yau

Chiral algebra of the Argyres-Douglas theory from M5 branes

The 6D (2,0) theory has codimension-one symmetry defects associated to the outer-automorphism group of the underlying the simply-laced Lie algebra of ADE type. 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 <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories of the Argyres-Douglas type whose flavor symmetries are (subgroups of) nonsimply laced Lie groups. We propose formulas 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.

Yifan Wang

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

We study S-duality of Argyres-Douglas theories obtained by compactification of 6d (2,0) theories of ADE type on a sphere with irregular punctures. The weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points, among which three punctured sphere represents isolated superconformal theories. We also discuss twisted irregular punctures and their S-duality.

Ke Ye

Argyres-Douglas matter and S-duality. Part II

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N = 2 , 0 $$ \mathcal{N}=\left(2,\ 0\right) $$ theory of type J on a punctured sphere. We denote the AD theories as ( J b [ k ], Y ), where J b [ k ] and Y represent an irregular and a regular singularity respectively. We restrict to the ‘minimal’ case where J b [ k ] has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra W k 2 d J , Y $$ {\mathcal{W}}^{k_{2d}}\left(J,\ Y\ \right) $$ , where k 2 d = − h + b b + k $$ {k}_{2d}=-h+\frac{b}{b+k} $$ with h being the dual Coxeter number of J . We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for b = h . We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.

South Korea

Jaewon Song

Vertex operator algebras of Argyres-Douglas theories from M5-branes

We conjecture that every three dimensional canonical singularity defines a five dimensional N $$ \mathcal{N} $$ = 1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a) a chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b) Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.