## Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory

## Abstract

We introduce several families of N = (2, 2) UV boundary conditions in 3d N=4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality $-$ an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

- Authors:

- Univ. of Oxford (United Kingdom). Mathematical Inst.; Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences; Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada); Univ. of California, Davis, CA (United States). Center for Quantum Mathematics and Physics, Dept. of Mathematics
- Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
- Univ. of Oregon, Eugene, OR (United States). Dept. of Mathematics

- Publication Date:

- Research Org.:
- Univ. of Oxford (United Kingdom)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); European Research Council (ERC)

- OSTI Identifier:
- 1358546

- Grant/Contract Number:
- SC0009988; 335739; 306260

- 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: 10; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Supersymmetric Gauge Theory; Supersymmetry and Duality; Topological Field Theories; Solitons Monopoles and Instantons

### Citation Formats

```
Bullimore, Mathew, Dimofte, Tudor, Gaiotto, Davide, and Hilburn, Justin. Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory. United States: N. p., 2016.
Web. doi:10.1007/JHEP10(2016)108.
```

```
Bullimore, Mathew, Dimofte, Tudor, Gaiotto, Davide, & Hilburn, Justin. Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory. United States. doi:10.1007/JHEP10(2016)108.
```

```
Bullimore, Mathew, Dimofte, Tudor, Gaiotto, Davide, and Hilburn, Justin. Thu .
"Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory". United States. doi:10.1007/JHEP10(2016)108. https://www.osti.gov/servlets/purl/1358546.
```

```
@article{osti_1358546,
```

title = {Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory},

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

abstractNote = {We introduce several families of N = (2, 2) UV boundary conditions in 3d N=4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality $-$ an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.},

doi = {10.1007/JHEP10(2016)108},

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

number = 10,

volume = 2016,

place = {United States},

year = {2016},

month = {10}

}

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