Superconformal index with surface defects for class ${\cal S}_k$

Yutaka Yoshida; Yuto Ito

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Superconformal index with surface defects for class ${\cal S}_k$

Yuto Ito, Yutaka Yoshida

High Energy Physics - Theory · Submitted 6 Jun 2016

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

The authors' main claim We compute the index of the 4d-2d systems and reproduce the results obtained from the first method.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.84 · mentions code or data availability 0.02.

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We study surface defects in 4d $\mathcal{N}=1$ $SU(N)$ superconformal gauge theories of class $\mathcal{S}_k$ obtained from the 6d (1,0) theories of type $A_{N-1}$, which are worldvolume theories on $N$ M5-branes at $\mathbb{C}^2/\mathbb{Z}_k$ singularities, compactified on Riemann surfaces with punctures. First we apply a method based on Riemann surface description and obtain the superconformal index of the theories in the presence of surface defects labelled by arbitrary symmetric representations of $su(N)$. Then we propose another description for the same surface defects, which involves 4d-2d coupled systems, by identifying which 2d $\mathcal{N}=(0,2)$ theories should be coupled. We compute the index of the 4d-2d systems and reproduce the results obtained from the first method. Finally we study the 2d TQFT structure of the index for class $\mathcal{S}_{k}$ theories by obtaining several eigenfunctions and eigenvalues of the difference operators that capture the surface defects and checking their relation.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study surface defects in 4d $\mathcal{N}=1$ $SU(N)$ superconformal gauge theories of class $\mathcal{S}_k$ obtained from the 6d (1,0) theories of type $A_{N-1}$, which are worldvolume theories on $N$ M5-branes at $\mathbb{C}^2/\mathbb{Z}_k$ singularities, compactified on Riemann surfaces with punctures.	background 0.54	0.31	0.05	asserted 0.98
s1	First we apply a method based on Riemann surface description and obtain the superconformal index of the theories in the presence of surface defects labelled by arbitrary symmetric representations of $su(N)$.	method 0.63	0.95	0.12	asserted 1.00
s2	Then we propose another description for the same surface defects, which involves 4d-2d coupled systems, by identifying which 2d $\mathcal{N}=(0,2)$ theories should be coupled.	method 0.72	0.90	0.09	asserted 0.88
s3	We compute the index of the 4d-2d systems and reproduce the results obtained from the first method.	result 0.84	0.95	0.22	asserted 1.00
s4	Finally we study the 2d TQFT structure of the index for class $\mathcal{S}_{k}$ theories by obtaining several eigenfunctions and eigenvalues of the difference operators that capture the surface defects and checking their relation.	contribution 0.89	0.95	0.24	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.98.

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Original abstract

We study surface defects in 4d $\mathcal{N}=1$ $SU(N)$ superconformal gauge theories of class $\mathcal{S}_k$ obtained from the 6d (1,0) theories of type $A_{N-1}$, which are worldvolume theories on $N$ M5-branes at $\mathbb{C}^2/\mathbb{Z}_k$ singularities, compactified on Riemann surfaces with punctures. First we apply a method based on Riemann surface description and obtain the superconformal index of the theories in the presence of surface defects labelled by arbitrary symmetric representations of $su(N)$. Then we propose another description for the same surface defects, which involves 4d-2d coupled systems, by identifying which 2d $\mathcal{N}=(0,2)$ theories should be coupled. We compute the index of the 4d-2d systems and reproduce the results obtained from the first method. Finally we study the 2d TQFT structure of the index for class $\mathcal{S}_{k}$ theories by obtaining several eigenfunctions and eigenvalues of the difference operators that capture the surface defects and checking their relation.

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References

Forward citations

Cited by 3 Pith papers

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

Surface defects on lens space and transfer matrices
hep-th 2026-10 conditional novelty 7.0 of 10

On L(r,1) times S^1, all class-S surface defects are products and fusion polynomials of two commuting elementary half-holonomy-shift operators, which are traces of lens Sklyanin L-operators; class S_k defects are prop...
5d Partition Functions with A Twist
hep-th 2018-08 conditional novelty 6.0 of 10

The authors derive the exact partition function of 5d N=1 gauge theories on S^3_b x Sigma_g as a sum over Bethe-Ansatz solutions, including instantons, and use it to compute 4d superconformal indices and holographic f...
4d N=1 from 6d (1,0)
hep-th 2016-10 conditional novelty 6.0 of 10

Compactifying 6d (1,0) SCFTs on a Riemann surface yields 4d N=1 theories whose conformal manifold is enlarged by flat connections and quantized fluxes for the 6d flavor symmetry, verified for two M5 branes on a Z2 sin...

How the literature uses it

16 arXiv papers cite this in 22 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Builds on it 1 paper

Building on [ref], a surface defect of type $(r,s)$ can be introduced via a position dependent VEV [ref] which is related to a pole at

Elliptic Quantum Curves of Class $\mathcal{S}_k$ · Higgsing to defects · 0.92

Uses its method or tool 1 paper

In [ref] , a prescription to evaluate the Schur index in the presence of a class of surface operators was proposed, which we call the “Higgsing prescription.”[See [ref] for earlier applications of this prescription.]

On the Correspondence between Surface Operators in Argyres-Douglas Theories and Modules of Chiral Algebra · Higgsing prescription · 0.91

Compares with it 1 paper

For $N > 2$, the transfer matrix is consistent with the result found in [ref] , where the action of surface defects in the symmetric powers of the vector representation was computed for restricted values of the spectral parameters.

Surface defects and elliptic quantum groups · $= 1$ supersymmetric quiver gauge theories and surface defects · 0.97

Cites it as background 3 papers

The full characteristics of the surface operator as the disorder one is less understood than those of the order operator like a Wilson line, though lots of attempts to completely classify the surface operator have been archived (e.g., see [ref] and a comprehensive review [ref] ).

Surface Operators from M-strings · Introduction and summary · 1.00
Examples of $(0,2)$ surface defects, though mostly in 4d theories, can be found in the literature [ref] , see also [ref] via breaking of $(2,2)$ defects, and [ref] for variety of other $(2,2)$ and $(4,4)$ examples, including [ref] where a $(4,4)$ surface defect is broken down to $(0,2)$ by the fugacities.

Interfaces and Quantum Algebras, II: Cigar Partition Function · Supersymmetric observables · 0.97
For example integrable systems derived using compactifications of the worldvolume theory of $k$ M5 branes probing $A_N-1$ singularity were shown to be related to the set of transfer matrices [ref] .

$C_2$ generalization of the van Diejen model from the minimal $(D_5,D_5)$ conformal matter · Discussion and outlook. · 0.88

What it cites · 29 sentences without a confident label

Builds on 1 paper

With the help of the expressions for $A_1$ case in [ref] , we expect that the normalized functions $_$ are orthogonal under the following measure $$ for $A_N-1$ case [equation] where the q-Pochhammer symbol $(x;q)$ is defined as ([ref]).

N=1 theories of class S_k · TQFT structure · 0.86

Uses the method or tool of 3 papers

In order to capture the surface defects, the authors in [ref] introduced the following procedure.

N=1 theories of class S_k · Surface defects from Riemann surface description · 0.84
The elliptic genus [ref] of this model is written as ℐ_ell(ζ^(i)_a, ζ̃^(i)_a,c,d,t;q) = 1/|W|∑_ω_* JK `-Res (𝐐(ω_*), η) Z_1`- loop .

Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups · 2d $𝒩=(0,2)$ elliptic genus and 4d-2d coupled system for surface defect · 0.95
The elliptic genus [ref] of this model is written as ℐ_ell(ζ^(i)_a, ζ̃^(i)_a,c,d,t;q) = 1/|W|∑_ω_* JK `-Res (𝐐(ω_*), η) Z_1`- loop .

Elliptic genera of 2d N=2 gauge theories · 2d $𝒩=(0,2)$ elliptic genus and 4d-2d coupled system for surface defect · 0.95

Compares with 1 paper

We evaluated the elliptic genus of these two classes of two dimensional $𝒩=(0,2)$ gauge theories and found that the 4d-2d combined index reproduces the superconformal indiex with surface defects obtained as in [ref] up to the overall fractional fugacity shift.

N=1 theories of class S_k · Conclusion · 1.00

Cites as background 8 papers

First we review class $_k$ theories, which are certain 4d $=1$ SCFTs and were associated to Riemann surfaces in [ref] .

N=1 theories of class S_k · Class $_k$ theories · 1.00
Inspired by the relation between the 4d SCFTs and the Riemann surfaces, the authors of [ref] and [ref] introduced a systematic formalism that computes superconformal index of these 4d SCFTs in terms of the Riemann surface description.

S-duality and 2d Topological QFT · Introduction · 0.86
Inspired by the relation between the 4d SCFTs and the Riemann surfaces, the authors of [ref] and [ref] introduced a systematic formalism that computes superconformal index of these 4d SCFTs in terms of the Riemann surface description.

The 4d Superconformal Index from q-deformed 2d Yang-Mills · Introduction · 0.86

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