# Argyres-Douglas theories, the Macdonald index, and an RG inequality

## Abstract

Here we conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A _{1},A _{2n₋3}) and (A _{1},D _{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2) _{R} currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S ^{1} reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general $N = 2$ superconformal field theories.

- Authors:

- Rutgers Univ., Piscataway, NJ (United States). NHETC and Dept. of Physics and Astronomy; Univ. of Chicago, IL (United States). Enrico Fermi Inst.
- Rutgers Univ., Piscataway, NJ (United States). NHETC and Dept. of Physics and Astronomy; Kyoto Univ. (Japan). Yukawa Inst. for Theoretical Physics

- Publication Date:

- Research Org.:
- Rutgers Univ., Piscataway, NJ (United States); Univ. of Chicago, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)

- OSTI Identifier:
- 1327220

- Grant/Contract Number:
- SC0010008; SC0003883; SC0007897; SC0009924; PHY-1066293

- 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: 2016; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Supersymmetric gauge theory; Extended Supersymmetry; Conformal and W Symmetry; Renormalization Group

### Citation Formats

```
Buican, Matthew, and Nishinaka, Takahiro. Argyres-Douglas theories, the Macdonald index, and an RG inequality. United States: N. p., 2016.
Web. doi:10.1007/JHEP02(2016)159.
```

```
Buican, Matthew, & Nishinaka, Takahiro. Argyres-Douglas theories, the Macdonald index, and an RG inequality. United States. doi:10.1007/JHEP02(2016)159.
```

```
Buican, Matthew, and Nishinaka, Takahiro. Wed .
"Argyres-Douglas theories, the Macdonald index, and an RG inequality". United States. doi:10.1007/JHEP02(2016)159. https://www.osti.gov/servlets/purl/1327220.
```

```
@article{osti_1327220,
```

title = {Argyres-Douglas theories, the Macdonald index, and an RG inequality},

author = {Buican, Matthew and Nishinaka, Takahiro},

abstractNote = {Here we conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A1,A2n₋3) and (A1,D2n) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general $N = 2$ superconformal field theories.},

doi = {10.1007/JHEP02(2016)159},

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

number = 2,

volume = 2016,

place = {United States},

year = {2016},

month = {2}

}

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Works referencing / citing this record:

##
Bootstrapping the (A1, A2) Argyres-Douglas theory

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- Cornagliotto, Martina; Lemos, Madalena; Liendo, Pedro
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