skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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:
 [1];  [2];  [3];  [4]
  1. Univ. of Oxford (United Kingdom). Mathematical Inst.; Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
  2. 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
  3. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
  4. 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); 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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 7 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Hyperkähler metrics and supersymmetry
journal, December 1987

  • Hitchin, N. J.; Karlhede, A.; Lindström, U.
  • Communications in Mathematical Physics, Vol. 108, Issue 4
  • DOI: 10.1007/BF01214418

Three-dimensional topological field theory and symplectic algebraic geometry I
journal, August 2009


Three-dimensional topological field theory and symplectic algebraic geometry II
journal, January 2010


Ω-deformation and quantization
journal, August 2014


The geometry and topology of toric hyperkähler manifolds
journal, January 2000

  • Bielawski, Roger; Dancer, Andrew S.
  • Communications in Analysis and Geometry, Vol. 8, Issue 4
  • DOI: 10.4310/CAG.2000.v8.n4.a2

Mirror symmetry in three dimensional gauge theories
journal, October 1996


Mirror symmetry in three-dimensional gauge theories, and D-brane moduli spaces
journal, May 1997


Chern-Simons theory and S-duality
journal, May 2013


Gauge Theories Labelled by Three-Manifolds
journal, December 2013

  • Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei
  • Communications in Mathematical Physics, Vol. 325, Issue 2
  • DOI: 10.1007/s00220-013-1863-2

RG domain walls and hybrid triangulations
journal, January 2015

  • Dimofte, Tudor; Gaiotto, Davide; Veen, Roland van der
  • Advances in Theoretical and Mathematical Physics, Vol. 19, Issue 1
  • DOI: 10.4310/ATMP.2015.v19.n1.a2

Supersymmetric Boundary Conditions in $\mathcal{N}=4$ Super Yang-Mills Theory
journal, February 2009


$S$-duality of boundary conditions in ${\mathcal N}=4$ super Yang-Mills theory
journal, January 2009


Integrating over Higgs Branches
journal, January 2000

  • Moore, Gregory; Nekrasov, Nikita; Shatashvili, Samson
  • Communications in Mathematical Physics, Vol. 209, Issue 1
  • DOI: 10.1007/PL00005525

D -Particle Bound States and Generalized Instantons
journal, January 2000

  • Moore, Gregory; Nekrasov, Nikita; Shatashvili, Samson
  • Communications in Mathematical Physics, Vol. 209, Issue 1
  • DOI: 10.1007/s002200050016

Seiberg-Witten Prepotential from Instanton Counting
journal, January 2003


Quantization of Integrable Systems and four Dimensional Gauge Theories
conference, January 2012

  • Nekrasov, Nikita A.; Shatashvili, Samson L.
  • XVIth International Congress on Mathematical Physics
  • DOI: 10.1142/9789814304634_0015

Refined, Motivic, and Quantum
journal, November 2009


Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry
journal, January 2010

  • Alday, Luis F.; Gaiotto, Davide; Gukov, Sergei
  • Journal of High Energy Physics, Vol. 2010, Issue 1
  • DOI: 10.1007/JHEP01(2010)113

Gauge theory loop operators and Liouville theory
journal, February 2010

  • Drukker, Nadav; Gomis, Jaume; Okuda, Takuya
  • Journal of High Energy Physics, Vol. 2010, Issue 2
  • DOI: 10.1007/JHEP02(2010)057

The omega deformation, branes, integrability and Liouville theory
journal, September 2010


Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I
journal, January 2011


Framed BPS states
journal, January 2013

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Advances in Theoretical and Mathematical Physics, Vol. 17, Issue 2
  • DOI: 10.4310/ATMP.2013.v17.n2.a1

D-branes and topological field theories
journal, March 1996


Electric-magnetic duality and the geometric Langlands program
journal, January 2007


N = 2 dualities
journal, August 2012


Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory
journal, July 2010

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Communications in Mathematical Physics, Vol. 299, Issue 1
  • DOI: 10.1007/s00220-010-1071-2

Wall-crossing in coupled 2d-4d systems
journal, December 2012

  • Gaiotto, Davide; Moore, Gregory W.; Neitzke, Andrew
  • Journal of High Energy Physics, Vol. 2012, Issue 12
  • DOI: 10.1007/JHEP12(2012)082

Surface defects and resolvents
journal, September 2013

  • Gaiotto, Davide; Gukov, Sergei; Seiberg, Nathan
  • Journal of High Energy Physics, Vol. 2013, Issue 9
  • DOI: 10.1007/JHEP09(2013)070

Five dimensional SUSY field theories, non-trivial fixed points and string dynamics
journal, November 1996


Extremal transitions and five-dimensional supersymmetric field theories
journal, January 1997


Small instantons, del Pezzo surfaces and type I' theory
journal, July 1997


Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces
journal, July 1997


Branes, superpotentials and superconformal fixed points
journal, October 1997


Hyper-Kähler geometry and invariants of three-manifolds
journal, September 1997


Aspects of NT ⩾ 2 topological gauge theories and D-branes
journal, May 1997


Walls, lines, and spectral dualities in 3d gauge theories
journal, May 2014

  • Gadde, Abhijit; Gukov, Sergei; Putrov, Pavel
  • Journal of High Energy Physics, Vol. 2014, Issue 5
  • DOI: 10.1007/JHEP05(2014)047

Mirror symmetry and loop operators
journal, November 2015


Holomorphic blocks in three dimensions
journal, December 2014

  • Beem, Christopher; Dimofte, Tudor; Pasquetti, Sara
  • Journal of High Energy Physics, Vol. 2014, Issue 12
  • DOI: 10.1007/JHEP12(2014)177

Remarks on A-branes, mirror symmetry, and the Fukaya category
journal, October 2003


Branes and quantization
journal, January 2009


A link invariant from the symplectic geometry of nilpotent slices
journal, September 2006


Knot homology via derived categories of coherent sheaves, I: The $\mathfrak{sl}(2)$ -case
journal, April 2008


Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case
journal, July 2008


Knot invariants from four-dimensional gauge theory
journal, January 2012


AGT on the S-duality wall
journal, December 2010

  • Hosomichi, Kazuo; Lee, Sungjay; Park, Jaemo
  • Journal of High Energy Physics, Vol. 2010, Issue 12
  • DOI: 10.1007/JHEP12(2010)079

3-Manifolds and 3d indices
journal, January 2013

  • Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei
  • Advances in Theoretical and Mathematical Physics, Vol. 17, Issue 5
  • DOI: 10.4310/ATMP.2013.v17.n5.a3

Exact results in supersymmetric field theories on manifolds with boundaries
journal, November 2013

  • Sugishita, Sotaro; Terashima, Seiji
  • Journal of High Energy Physics, Vol. 2013, Issue 11
  • DOI: 10.1007/JHEP11(2013)021

Exact results for boundaries and domain walls in 2d supersymmetric theories
journal, September 2015


Wave-equation scattering in even-dimensional spaces
journal, January 1976

  • Adamyan, V. M.
  • Functional Analysis and Its Applications, Vol. 10, Issue 1
  • DOI: 10.1007/BF01075764

𝒩 = 4 Superconformal Chern-Simons theories with hyper and twisted hyper multiplets
journal, July 2008


Phases of N = 2 theories in two dimensions
journal, August 1993


Morse Theory and Tilting Sheaves
journal, January 2006


Quadratic duals, Koszul dual functors, and applications
journal, October 2008

  • Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina
  • Transactions of the American Mathematical Society, Vol. 361, Issue 03
  • DOI: 10.1090/S0002-9947-08-04539-X

Gale duality and Koszul duality
journal, November 2010

  • Braden, Tom; Licata, Anthony; Proudfoot, Nicholas
  • Advances in Mathematics, Vol. 225, Issue 4
  • DOI: 10.1016/j.aim.2010.04.011

Hypertoric category <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi mathvariant="script">O</mml:mi></mml:math>
journal, October 2012

  • Braden, Tom; Licata, Anthony; Proudfoot, Nicholas
  • Advances in Mathematics, Vol. 231, Issue 3-4
  • DOI: 10.1016/j.aim.2012.06.019

Isomorphisms of quantizations via quantization of resolutions
journal, October 2012


Supersymmetric Vacua and Bethe Ansatz
journal, July 2009


On three dimensional quiver gauge theories and integrability
journal, May 2013


Fivebranes and knots
journal, January 2012


Knot invariants and topological strings
journal, June 2000


Khovanov-Rozansky Homology and Topological Strings
journal, October 2005

  • Gukov, Sergei; Schwarz, Albert; Vafa, Cumrun
  • Letters in Mathematical Physics, Vol. 74, Issue 1
  • DOI: 10.1007/s11005-005-0008-8

Sequencing BPS spectra
journal, March 2016

  • Gukov, Sergei; Nawata, Satoshi; Saberi, Ingmar
  • Journal of High Energy Physics, Vol. 2016, Issue 3
  • DOI: 10.1007/JHEP03(2016)004

Matrix factorizations and link homology
journal, January 2008

  • Khovanov, Mikhail; Rozansky, Lev
  • Fundamenta Mathematicae, Vol. 199, Issue 1
  • DOI: 10.4064/fm199-1-1

Matrix factorizations and link homology II
journal, January 2008


An E 7 surprise
journal, October 2012


On mirror symmetry in three dimensional Abelian gauge theories
journal, April 1999


Derived equivalence for quantum symplectic resolutions
journal, October 2013


Supersymmetry and Morse theory
journal, January 1982


Duality of the fermionic 2d black hole and N = 2 Liouville theory as mirror symmetry
journal, August 2001


Monopole Operators and Mirror Symmetry in Three Dimensions
journal, December 2002


Monopole operators and Hilbert series of Coulomb branches of 3d $ \mathcal{N} $ = 4 gauge theories
journal, January 2014

  • Cremonesi, Stefano; Hanany, Amihay; Zaffaroni, Alberto
  • Journal of High Energy Physics, Vol. 2014, Issue 1
  • DOI: 10.1007/JHEP01(2014)005

Supersymmetric Yang-Mills theory and integrable systems
journal, February 1996


Quantal Phase Factors Accompanying Adiabatic Changes
journal, March 1984

  • Berry, M. V.
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 392, Issue 1802
  • DOI: 10.1098/rspa.1984.0023

A geometric Jacquet functor
journal, November 2004


On primitive ideals
journal, September 2003


Special Transverse Slices and Their Enveloping Algebras
journal, September 2002


A Remark on Rational Cherednik Algebras and Differential Operators on the Cyclic Quiver
journal, February 2006


Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products
journal, June 2007

  • Etingof, Pavel; Gan, Wee Liang; Ginzburg, Victor
  • Publications mathématiques de l'IHÉS, Vol. 105, Issue 1
  • DOI: 10.1007/s10240-007-0005-9

On classification ofN=2 supersymmetric theories
journal, December 1993

  • Cecotti, Sergio; Vafa, Cumrun
  • Communications in Mathematical Physics, Vol. 158, Issue 3
  • DOI: 10.1007/BF02096804

Topological—anti-topological fusion
journal, December 1991


A dilatonic deformation of AdS 5 and its field theory dual
journal, May 2003


Janus configurations, Chern-Simons couplings, and The θ-Angle in $ \mathcal{N} = 4 $ super Yang-Mills theory
journal, June 2010


Quantum Cohomology of Flag Manifolds G/B and Quantum Toda Lattices
journal, January 1999

  • Kim, Bumsig
  • The Annals of Mathematics, Vol. 149, Issue 1
  • DOI: 10.2307/121021

Quantum cohomology of partial flag manifolds $$F_{n_1 } \ldots _{n_k }$$
journal, June 1995

  • Astashkevich, Alexander; Sadov, Vladimir
  • Communications in Mathematical Physics, Vol. 170, Issue 3
  • DOI: 10.1007/BF02099147

Quantum cohomology of flag manifolds and Toda lattices
journal, April 1995

  • Givental, Alexander; Kim, Bumsig
  • Communications in Mathematical Physics, Vol. 168, Issue 3
  • DOI: 10.1007/BF02101846

Yangians and quantizations of slices in the affine Grassmannian
journal, January 2014

  • Kamnitzer, Joel; Webster, Ben; Weekes, Alex
  • Algebra & Number Theory, Vol. 8, Issue 4
  • DOI: 10.2140/ant.2014.8.857

Constructible sheaves and the Fukaya category
journal, September 2008


Categorifications and cyclotomic rational double affine Hecke algebras
journal, September 2015

  • Rouquier, Raphaël; Shan, Peng; Varagnolo, Michela
  • Inventiones mathematicae, Vol. 204, Issue 3
  • DOI: 10.1007/s00222-015-0623-7

Koszul duality of affine Kac–Moody algebras and cyclotomic rational double affine Hecke algebras
journal, September 2014


Gauge theory and knot homologies
journal, May 2007


Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics
journal, May 1997


Topological quantum field theory
journal, September 1988

  • Witten, Edward
  • Communications in Mathematical Physics, Vol. 117, Issue 3
  • DOI: 10.1007/BF01223371

Topological sigma models
journal, September 1988

  • Witten, Edward
  • Communications in Mathematical Physics, Vol. 118, Issue 3
  • DOI: 10.1007/BF01466725

    Works referencing / citing this record:

    Twisted indices of 3d N $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps
    journal, July 2019

    • Bullimore, Mathew; Ferrari, Andrea; Kim, Heeyeon
    • Journal of High Energy Physics, Vol. 2019, Issue 7
    • DOI: 10.1007/jhep07(2019)014

    Twisted indices of 3d N $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps
    journal, July 2019

    • Bullimore, Mathew; Ferrari, Andrea; Kim, Heeyeon
    • Journal of High Energy Physics, Vol. 2019, Issue 7
    • DOI: 10.1007/jhep07(2019)014

    Vertex Operator Algebras and 3d N $$ \mathcal{N} $$ = 4 gauge theories
    journal, May 2019


    Coulomb branch quantization and abelianized monopole bubbling
    journal, October 2019

    • Dedushenko, Mykola; Fan, Yale; Pufu, Silviu S.
    • Journal of High Energy Physics, Vol. 2019, Issue 10
    • DOI: 10.1007/jhep10(2019)179

    Deformation Quantization and Superconformal Symmetry in Three Dimensions
    journal, February 2017

    • Beem, Christopher; Peelaers, Wolfger; Rastelli, Leonardo
    • Communications in Mathematical Physics, Vol. 354, Issue 1
    • DOI: 10.1007/s00220-017-2845-6

    The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories
    journal, June 2017

    • Bullimore, Mathew; Dimofte, Tudor; Gaiotto, Davide
    • Communications in Mathematical Physics, Vol. 354, Issue 2
    • DOI: 10.1007/s00220-017-2903-0