Argyres-Douglas Theories, Modularity of Minimal Models and Refined Chern-Simons

Can Koz\c{c}az; Shamil Shakirov; WenBin Yan

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Argyres-Douglas Theories, Modularity of Minimal Models and Refined Chern-Simons

Can Kozçaz, Shamil Shakirov, Wenbin Yan

High Energy Physics - Theory · Submitted 25 Jan 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.

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arXiv ID1801.08316 (version 2)

Pith IDL34RZ5TU, Pith's permanent record of this paper and its review

CitationCan Koz\c{c}az, Shamil Shakirov, WenBin Yan. "Argyres-Douglas Theories, Modularity of Minimal Models and Refined Chern-Simons." arXiv:1801.08316 (2018). https://pith.science/paper/L34RZ5TU

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The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{1},A_{2N})$ Argyres-Douglas theories on the same space. Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit.	background 1.00	0.10	0.12	asserted 1.00
s1	A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{1},A_{2N})$ Argyres-Douglas theories on the same space.	contribution 0.86	0.86	0.37	asserted 0.53
s2	Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory.	interpretation 0.55	0.20	0.08	hedged possibility 1.00
s3	The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.	result 0.53	0.87	0.70	asserted 1.00

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Original abstract

The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{1},A_{2N})$ Argyres-Douglas theories on the same space. Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity
hep-th 2026-09 conditional novelty 6.0 of 10

Closed-form Schur indices and module characters are derived for the Z3 S-fold vertex operator algebra, with a logarithmic companion completing the modular solution space.
Higher Form Symmetries of Argyres-Douglas Theories
hep-th 2020-07 conditional novelty 5.0 of 10

The defect groups of all (g,g') Argyres-Douglas theories are computed from the intersection form of the singularity and tabulated, with independent agreement against proposed N=1 Lagrangians.

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12 arXiv papers cite this in 18 sentences

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A one-parameter family of $S$ and $T$ matrices is also constructed in [ref] , which is related to our $S()$ and $T()$ by overall factors.

3d TQFTs from Argyres-Douglas theories · $(A_1, A_2)$ theory · 1.00

Cites it as background 1 paper

Various interesting properties of 4d theory using 2d VOA are studied in [ref] .

W algebra, Cosets and VOA for 4d N = 2 SCFT from M5 branes · Known results · 0.98

What it cites · 48 sentences without a confident label

Builds on 1 paper

We conjecture that Coulomb branch indices of $(A_1, A_2N)$ AD theories on lens space $L(k,1)$ times a circle can be written as (up to a proportional constant), [equation] where $_(A_1, A_2N)()$ and $_(A_1, A_2N)()$ are matrices with one parameter $$ which satisfies the following relations, [equation] Clearly $_(A_1, A_2N)()$ and $_(A_1, A_2N)()$ form a representation of $SL(2,ℤ)$ and the relation in [ref] [In fact, a slightly modified relation from [ref] is used in this paper, see equation [ref].] can be recovered by taking the limit $→exp(-2iπ/2N+3)$ under which $_(A_1, A_2N)()$ and $_(A_1, A_2N)()$ become the modular transformation matrices $S_(2N+3,2)$ and $T_(2N+3,2)$ of characters of $(2N+3,2)$ minimal models. $_(A_1, A_2N)()$ and $_(A_1, A_2N)()$ can be viewed as one parameter generalization of $S_(2N+3,2)$ and $T_(2N+3,2)$, and it will be shown in section [ref] that $_(A_1, A_2N)(

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters · AD theories, wild Hitchin systems and VOAs · 0.83

Uses data or values from 1 paper

Here we simply quote the result, [equation] where we replace the equivariant parameter $$ in [ref] by $=^1/2N+3$ for later convenience.

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters · Coulomb branch index of $(A_1, A_2N)$ AD theories · 0.93

Compares with 2 papers

In equation [ref] as in the original work of [ref] , the limit is taken to be $=e^2πi/2N+3$, whereas $=e^-2πi/2N+3$ in equation [ref] and [ref].

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters · $(A_1,A_2)$ case · 0.88
$N=4$, [equation] with [equation] and [equation] $ξ_i$'s are the diagonal elements of $_(11,2)$, [equation] The sign difference from [ref] in the above formula is originated from the sign difference in the $_(11,2)$.

Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations · S-matrix · 0.82

Cites as support 6 papers

This was verified by matching the lens space Coulomb index of AD theories and the wild Hitchin characters [ref] , I_Coulomb ([Σ, G]; L(k,1)×S^1)= _ (Σ, ^LG_ℂ; k),

Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality · AD theories, wild Hitchin systems and VOAs · 0.96
This was verified by matching the lens space Coulomb index of AD theories and the wild Hitchin characters [ref] , I_Coulomb ([Σ, G]; L(k,1)×S^1)= _ (Σ, ^LG_ℂ; k),

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters · AD theories, wild Hitchin systems and VOAs · 0.96
Notice that the characters of $(2N+3,2)$ models are identified as the Schur indices with defect insertions of $(A_1, A_2N)$ AD theories [ref] .

Line defect Schur indices, Verlinde algebras and $U(1)_r$ fixed points · Conclusions and discussions · 0.96

Cites as background 13 papers

Now recall the construction of $SU(N)$ refined CS theory [ref] .

Knot Homology from Refined Chern-Simons Theory · Relation with Refined Chern-Simons · 0.96
An AD theory $$ can be constructed by compatifying $6d$ $(2,0)$ SCFT of type $G=ADE$ on a sphere $Σ$ with one irregular singularities and possible regular singularities [ref] .

Classification of Argyres-Douglas theories from M5 branes · AD theories, wild Hitchin systems and VOAs · 0.90
An AD theory $$ can be constructed by compatifying $6d$ $(2,0)$ SCFT of type $G=ADE$ on a sphere $Σ$ with one irregular singularities and possible regular singularities [ref] .

Wild Quiver Gauge Theories · AD theories, wild Hitchin systems and VOAs · 0.90

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