# The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories

## Abstract

We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory with N=4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
- Inst. for Advanced Study, Princeton, NJ (United States)
- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)

- Publication Date:

- Research Org.:
- Inst. for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1423961

- Grant/Contract Number:
- sc0009988

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Communications in Mathematical Physics

- Additional Journal Information:
- Journal Volume: 354; Journal Issue: 2; Journal ID: ISSN 0010-3616

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Bullimore, Mathew, Dimofte, Tudor, and Gaiotto, Davide.
```*The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories*. United States: N. p., 2017.
Web. doi:10.1007/s00220-017-2903-0.

```
Bullimore, Mathew, Dimofte, Tudor, & Gaiotto, Davide.
```*The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories*. United States. doi:10.1007/s00220-017-2903-0.

```
Bullimore, Mathew, Dimofte, Tudor, and Gaiotto, Davide. Sat .
"The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories". United States.
doi:10.1007/s00220-017-2903-0. https://www.osti.gov/servlets/purl/1423961.
```

```
@article{osti_1423961,
```

title = {The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories},

author = {Bullimore, Mathew and Dimofte, Tudor and Gaiotto, Davide},

abstractNote = {We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory with N=4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.},

doi = {10.1007/s00220-017-2903-0},

journal = {Communications in Mathematical Physics},

number = 2,

volume = 354,

place = {United States},

year = {Sat Jun 03 00:00:00 EDT 2017},

month = {Sat Jun 03 00:00:00 EDT 2017}

}

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