# Coulomb branch operators and mirror symmetry in three dimensions

## Abstract

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional N = 4 abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the (n ≤ 3)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere *HS* ^{3} with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the *HS* ^{3} wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on *S* ^{3} with an arbitrary number of operator insertions. Lastly, we show that our results can be recovered by dimensionally reducing the Schur index of 4D N=2 theories decorated by BPS ’t Hooft-Wilson loops.

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics
- Princeton Univ., Princeton, NJ (United States). Dept. of Physics
- Weizmann Inst. of Science, Rehovot (Israel). Dept. of Particle Physics and Astrophysics

- Publication Date:

- Research Org.:
- California Inst. of Technology (CalTech), Pasadena, CA (United States)

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

- OSTI Identifier:
- 1499694

- Grant/Contract Number:
- SC0011632

- Resource Type:
- Journal Article: Accepted Manuscript

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

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 4; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Extended Supersymmetry; Supersymmetric Gauge Theory; Supersymmetry and Duality; Conformal Field Theory

### Citation Formats

```
Dedushenko, Mykola, Fan, Yale, Pufu, Silviu S., and Yacoby, Ran.
```*Coulomb branch operators and mirror symmetry in three dimensions*. United States: N. p., 2018.
Web. doi:10.1007/jhep04(2018)037.

```
Dedushenko, Mykola, Fan, Yale, Pufu, Silviu S., & Yacoby, Ran.
```*Coulomb branch operators and mirror symmetry in three dimensions*. United States. doi:10.1007/jhep04(2018)037.

```
Dedushenko, Mykola, Fan, Yale, Pufu, Silviu S., and Yacoby, Ran. Mon .
"Coulomb branch operators and mirror symmetry in three dimensions". United States. doi:10.1007/jhep04(2018)037. https://www.osti.gov/servlets/purl/1499694.
```

```
@article{osti_1499694,
```

title = {Coulomb branch operators and mirror symmetry in three dimensions},

author = {Dedushenko, Mykola and Fan, Yale and Pufu, Silviu S. and Yacoby, Ran},

abstractNote = {We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional N = 4 abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the (n ≤ 3)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere HS3 with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the HS3 wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on S3 with an arbitrary number of operator insertions. Lastly, we show that our results can be recovered by dimensionally reducing the Schur index of 4D N=2 theories decorated by BPS ’t Hooft-Wilson loops.},

doi = {10.1007/jhep04(2018)037},

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

issn = {1029-8479},

number = 4,

volume = 2018,

place = {United States},

year = {2018},

month = {4}

}