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Title: 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:
ORCiD logo [1];  [2];  [3]
  1. Inst. for Advanced Study, Princeton, NJ (United States); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
  2. Inst. for Advanced Study, Princeton, NJ (United States)
  3. Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
Publication Date:
Research Org.:
Institute 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:
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 = {2017},
month = {6}
}

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Cited by: 17 works
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Works referenced in this record:

Instantons, three dimensional gauge theories, and monopole moduli spaces
journal, August 1998


Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence
journal, February 2015

  • Luo, Yuan; Tan, Meng-Chwan; Yagi, Junya
  • Journal of High Energy Physics, Vol. 2015, Issue 2
  • DOI: 10.1007/JHEP02(2015)047

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

Aspects of N = 2 supersymmetric gauge theories in three dimensions
journal, August 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

Parameter counting for singular monopoles on ℝ3
journal, October 2014

  • Moore, Gregory W.; Royston, Andrew B.; Van den Bleeken, Dieter
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)142

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

Noncommutative geometry and quiver algebras
journal, February 2007

  • Crawley-Boevey, William; Etingof, Pavel; Ginzburg, Victor
  • Advances in Mathematics, Vol. 209, Issue 1
  • DOI: 10.1016/j.aim.2006.05.004

Seiberg-Witten Prepotential from Instanton Counting
journal, January 2003


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


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

Singular Monopoles and Gravitational Instantons
journal, June 1999

  • Cherkis, Sergey A.; Kapustin, Anton
  • Communications in Mathematical Physics, Vol. 203, Issue 3
  • DOI: 10.1007/s002200050632

The moduli space metric for well-separated BPS monopoles
journal, August 1995


Geometry of type ii Superstrings and the Moduli of Superconformal Field Theories
journal, June 1989

  • Cecotti, S.; Ferrara, S.; Girardello, L.
  • International Journal of Modern Physics A, Vol. 04, Issue 10
  • DOI: 10.1142/S0217751X89000972

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


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


Hypermultiplets and topological strings
journal, February 2006


Mirror symmetry in three-dimensional gauge theories, quivers and D-branes
journal, May 1997


Quantum Riemann surfaces in Chern-Simons theory
journal, January 2013


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


Aspects of 3d $ \mathcal{N}=2 $ Chern-Simons-Matter theories
journal, July 2013

  • Intriligator, Kenneth; Seiberg, Nathan
  • Journal of High Energy Physics, Vol. 2013, Issue 7
  • DOI: 10.1007/JHEP07(2013)079

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

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory
journal, October 2016

  • Bullimore, Mathew; Dimofte, Tudor; Gaiotto, Davide
  • Journal of High Energy Physics, Vol. 2016, Issue 10
  • DOI: 10.1007/JHEP10(2016)108

Ω-deformation and quantization
journal, August 2014


Singular monopoles and supersymmetric gauge theories in three dimensions
journal, August 1998


Complete intersection moduli spaces in $ \mathcal{N} = 4 $ gauge theories in three dimensions
journal, January 2012

  • Hanany, Amihay; Mekareeya, Noppadol
  • Journal of High Energy Physics, Vol. 2012, Issue 1
  • DOI: 10.1007/JHEP01(2012)079

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


Brane bending and monopole moduli
journal, October 2014

  • Moore, Gregory W.; Royston, Andrew B.; Van den Bleeken, Dieter
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)157

Exact results for ’t Hooft loops in Gauge theories on S 4
journal, May 2012

  • Gomis, Jaume; Okuda, Takuya; Pestun, Vasily
  • Journal of High Energy Physics, Vol. 2012, Issue 5
  • DOI: 10.1007/JHEP05(2012)141

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


Line operators on $ {S^{{1}}} \times {\mathbb{R}^{{3}}} $ and quantization of the Hitchin moduli space
journal, April 2012

  • Ito, Yuto; Okuda, Takuya; Taki, Masato
  • Journal of High Energy Physics, Vol. 2012, Issue 4
  • DOI: 10.1007/JHEP04(2012)010

Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
journal, September 1994


$D_k$ gravitational instantons and Nahm equations
journal, January 1998

  • Cherkis, Sergey A.; Kapustin, Anton
  • Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 6
  • DOI: 10.4310/ATMP.1998.v2.n6.a3

IR dynamics on branes and space-time geometry
journal, September 1996


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

Mirror symmetry by O3-planes
journal, November 2000


Geometric Langlands duality and representations of algebraic groups over commutative rings
journal, January 2007


Kac-Moody algebras
journal, November 1994


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

Three-dimensional gauge theories and monopoles
journal, March 1997


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


Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD
journal, December 1994


Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces
journal, March 1990


Supersymmetric σ-models, twistors, and the Atiyah-Hitchin metric
journal, December 1996

  • Ivanov, Ivan T.; Roček, M.
  • Communications in Mathematical Physics, Vol. 182, Issue 2
  • DOI: 10.1007/BF02517891

Mirror symmetry in three dimensional gauge theories
journal, October 1996


Instantons, three-dimensional gauge theory, and the Atiyah-Hitchin manifold
journal, September 1997


Branes and quantization
journal, January 2009


Issues in topological gauge theory
journal, November 1998


Three-dimensional gauge theories and ADE monopoles
journal, February 1999


On F-term contribution to effective action
journal, August 2007


On a Class of Representations of the Yangian and Moduli Space of Monopoles
journal, August 2005

  • Gerasimov, A.; Kharchev, S.; Lebedev, D.
  • Communications in Mathematical Physics, Vol. 260, Issue 3
  • DOI: 10.1007/s00220-005-1417-3

Quiver varieties of type A
journal, January 2005

  • Maffei, Andrea
  • Commentarii Mathematici Helvetici
  • DOI: 10.4171/CMH/1

Gauge theory, ramification, and the geometric Langlands program
journal, January 2006


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


Linear Perturbations of Quaternionic Metrics
journal, February 2010

  • Alexandrov, Sergei; Pioline, Boris; Saueressig, Frank
  • Communications in Mathematical Physics, Vol. 296, Issue 2
  • DOI: 10.1007/s00220-010-1022-y

On the instantons and the hypermultiplet mass of $ \mathcal{N} = {{2}^{ * }} $ super Yang-Mills on S 4
journal, March 2012


Monopoles and rational maps: A note on a theorem of Donaldson
journal, June 1985

  • Hurtubise, Jacques
  • Communications in Mathematical Physics, Vol. 100, Issue 2
  • DOI: 10.1007/BF01212447

Issues on orientifolds: on the brane construction of gauge theories with SO(2 n ) global symmetry
journal, July 1999


Isomorphisms of quantizations via quantization of resolutions
journal, October 2012


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

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

Shifted Yangians and finite W -algebras
journal, February 2006


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


Equivariant homology and K-theory of affine Grassmannians and Toda lattices
journal, April 2005

  • Bezrukavnikov, Roman; Finkelberg, Michael; Mirkovic, Ivan
  • Compositio Mathematica, Vol. 141, Issue 03
  • DOI: 10.1112/S0010437X04001228

Linear Perturbations of Hyperkähler Metrics
journal, February 2009

  • Alexandrov, Sergei; Pioline, Boris; Saueressig, Frank
  • Letters in Mathematical Physics, Vol. 87, Issue 3
  • DOI: 10.1007/s11005-009-0305-8

Monopole operators in three-dimensional N =4 SYM and mirror symmetry
journal, March 2004


New hyperkähler metrics and new supermultiplets
journal, March 1988

  • Lindström, U.; Roček, M.
  • Communications in Mathematical Physics, Vol. 115, Issue 1
  • DOI: 10.1007/BF01238851

Nahm's equations and the classification of monopoles
journal, September 1984

  • Donaldson, S. K.
  • Communications in Mathematical Physics, Vol. 96, Issue 3
  • DOI: 10.1007/BF01214583

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


Quantization and representation theory of finiteW algebras
journal, December 1993

  • de Boer, Jan; Tjin, Tjark
  • Communications in Mathematical Physics, Vol. 158, Issue 3
  • DOI: 10.1007/BF02096800

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


Fedosov Quantization in Algebraic Context
journal, January 2004


Representations of shifted Yangians and finite 𝑊-algebras
journal, January 2008

  • Brundan, Jonathan; Kleshchev, Alexander
  • Memoirs of the American Mathematical Society, Vol. 196, Issue 918
  • DOI: 10.1090/memo/0918

    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