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

## Abstract

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d $\mathcal{N}=(2,\text{}0)$ 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 ${\mathcal{W}}^{{k}_{2d}}(J,\text{}Y\text{})$, where ${k}_{2d}=-h+\frac{b}{b+k}$ 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.

- Authors:

- Univ. of California, San Diego, CA (United States). Dept. of Physics; Korea Inst. for Advanced Study, Seoul (Korea, Republic of). School of Physics
- Harvard Univ., Cambridge, MA (United States). Center of Mathematical Sciences and Applications. Jefferson Physical Lab.
- Harvard Univ., Cambridge, MA (United States). Center of Mathematical Sciences and Applications. Jefferson Physical Lab.; Tsinghua Univ., Beijing (China). Yau Mathematical Sciences Center

- Publication Date:

- Research Org.:
- Univ. of California, San Diego, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1505129

- Grant/Contract Number:
- SC0009919

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 12; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; conformal and W symmetry; supersymmetric gauge theory; supersymmetry and duality; topological field theories

### Citation Formats

```
Song, Jaewon, Xie, Dan, and Yan, Wenbin. Vertex operator algebras of Argyres-Douglas theories from M5-branes. United States: N. p., 2017.
Web. doi:10.1007/jhep12(2017)123.
```

```
Song, Jaewon, Xie, Dan, & Yan, Wenbin. Vertex operator algebras of Argyres-Douglas theories from M5-branes. United States. doi:10.1007/jhep12(2017)123.
```

```
Song, Jaewon, Xie, Dan, and Yan, Wenbin. Thu .
"Vertex operator algebras of Argyres-Douglas theories from M5-branes". United States. doi:10.1007/jhep12(2017)123. https://www.osti.gov/servlets/purl/1505129.
```

```
@article{osti_1505129,
```

title = {Vertex operator algebras of Argyres-Douglas theories from M5-branes},

author = {Song, Jaewon and Xie, Dan and Yan, Wenbin},

abstractNote = {We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N = ( 2 , 0 ) 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 ) , where k 2 d = − h + 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.},

doi = {10.1007/jhep12(2017)123},

journal = {Journal of High Energy Physics (Online)},

number = 12,

volume = 2017,

place = {United States},

year = {2017},

month = {12}

}