Coulomb branches of star-shaped quivers

  • Tudor DimofteEmail author
  • Niklas Garner
Open Access
Regular Article - Theoretical Physics


We study the Coulomb branches of 3d \( \mathcal{N}=4 \) “star-shaped” quiver gauge theories and their deformation quantizations, by applying algebraic techniques that have been developed in the mathematics and physics literature over the last few years. The algebraic techniques supply an abelianization map, which embeds the Coulomb-branch chiral ring into a vastly simpler abelian algebra \( \mathcal{A} \). Relations among chiral-ring operators, and their deformation quantization, are canonically induced from the embedding into \( \mathcal{A} \). In the case of star-shaped quivers — whose Coulomb branches are related to Higgs branches of 4d \( \mathcal{N}=2 \) theories of Class \( \mathcal{S} \) — this allows us to systematically verify known relations, to generalize them, and to quantize them. In the quantized setting, we find several new families of relations.


Supersymmetric Gauge Theory Topological Field Theories Differential and Algebraic Geometry 


Open Access

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  1. [1]
    N. Seiberg, IR dynamics on branes and space-time geometry, Phys. Lett. B 384 (1996) 81 [hep-th/9606017] [INSPIRE].
  2. [2]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson. Proceedings, Conference, Saclay, France, June 5-7, 1996, pp. 333-366, 1996, hep-th/9607163 [INSPIRE].
  3. [3]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
  4. [4]
    J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
  5. [5]
    J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, ℤ) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].
  6. [6]
    G. Chalmers and A. Hanany, Three-dimensional gauge theories and monopoles, Nucl. Phys. B 489 (1997) 223 [hep-th/9608105] [INSPIRE].
  7. [7]
    A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
  8. [8]
    N. Dorey, V.V. Khoze, M.P. Mattis, D. Tong and S. Vandoren, Instantons, three-dimensional gauge theory and the Atiyah-Hitchin manifold, Nucl. Phys. B 502 (1997) 59 [hep-th/9703228] [INSPIRE].
  9. [9]
    C. Fraser and D. Tong, Instantons, three-dimensional gauge theories and monopole moduli spaces, Phys. Rev. D 58 (1998) 085001 [hep-th/9710098] [INSPIRE].
  10. [10]
    S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N}=4 \) gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
  11. [11]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb Branch of 3d \( \mathcal{N}=4 \) Theories, Commun. Math. Phys. 354 (2017) 671 [arXiv:1503.04817] [INSPIRE].
  13. [13]
    M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn and H.-C. Kim, Vortices and Vermas, Adv. Theor. Math. Phys. 22 (2018) 803 [arXiv:1609.04406] [INSPIRE].CrossRefGoogle Scholar
  14. [14]
    C. Teleman, Gauge theory and mirror symmetry, arXiv:1404.6305 [INSPIRE].
  15. [15]
    H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N}=4 \) gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 [arXiv:1503.03676] [INSPIRE].
  16. [16]
    A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N}=4 \) gauge theories, II, arXiv:1601.03586 [INSPIRE].
  17. [17]
    A. Braverman, M. Finkelberg and H. Nakajima, Coulomb branches of 3d \( \mathcal{N}=4 \) quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster and Alex Weekes), arXiv:1604.03625 [INSPIRE].
  18. [18]
    B. Webster, Koszul duality between Higgs and Coulomb categories \( \mathcal{O} \), arXiv:1611.06541.
  19. [19]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
  20. [20]
    G.W. Moore and Y. Tachikawa, On 2d TQFTs whose values are holomorphic symplectic varieties, Proc. Symp. Pure Math. 85 (2012) 191 [arXiv:1106.5698] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  21. [21]
    F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].
  22. [22]
    A. Gadde, K. Maruyoshi, Y. Tachikawa and W. Yan, New N = 1 Dualities, JHEP 06 (2013) 056 [arXiv:1303.0836] [INSPIRE].
  23. [23]
    K. Yonekura, Supersymmetric gauge theory, (2, 0) theory and twisted 5d Super-Yang-Mills, JHEP 01 (2014) 142 [arXiv:1310.7943] [INSPIRE].
  24. [24]
    K. Maruyoshi, Y. Tachikawa, W. Yan and K. Yonekura, N = 1 dynamics with T N theory, JHEP 10 (2013) 010 [arXiv:1305.5250] [INSPIRE].
  25. [25]
    H. Hayashi, Y. Tachikawa and K. Yonekura, Mass-deformed T N as a linear quiver, JHEP 02 (2015) 089 [arXiv:1410.6868] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Lemos and W. Peelaers, Chiral Algebras for Trinion Theories, JHEP 02 (2015) 113 [arXiv:1411.3252] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Y. Tachikawa, A review of the T N theory and its cousins, PTEP 2015 (2015) 11B102 [arXiv:1504.01481] [INSPIRE].
  28. [28]
    V. Ginzburg and D. Kazhdan, Construction of symplectic varieties arising in ‘sicilian theories’, unpublished.Google Scholar
  29. [29]
    A. Braverman, M. Finkelberg and H. Nakajima, Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan), arXiv:1706.02112 [INSPIRE].
  30. [30]
    A. Hanany and D. Miketa, Nilpotent orbit Coulomb branches of types AD, arXiv:1807.11491 [INSPIRE].
  31. [31]
    F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  33. [33]
    J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
  34. [34]
    D. Gaiotto, A. Neitzke and Y. Tachikawa, Argyres-Seiberg duality and the Higgs branch, Commun. Math. Phys. 294 (2010) 389 [arXiv:0810.4541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Three Dimensional Sicilian Theories, JHEP 09 (2014) 185 [arXiv:1403.2384] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Blau and G. Thompson, Aspects of N(T)two topological gauge theories and D-branes, Nucl. Phys. B 492 (1997) 545 [hep-th/9612143] [INSPIRE].
  37. [37]
    L. Rozansky and E. Witten, HyperKähler geometry and invariants of three manifolds, Selecta Math. 3 (1997) 401 [hep-th/9612216] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    C. Beem, D. Ben-Zvi, M. Bullimore, T. Dimofte and A. Neitzke, Secondary products in supersymmetric field theory, arXiv:1809.00009 [INSPIRE].
  39. [39]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    J. Yagi, Ω-deformation and quantization, JHEP 08 (2014) 112 [arXiv:1405.6714] [INSPIRE].
  41. [41]
    C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys. 354 (2017) 345 [arXiv:1601.05378] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    M. Dedushenko, S.S. Pufu and R. Yacoby, A one-dimensional theory for Higgs branch operators, JHEP 03 (2018) 138 [arXiv:1610.00740] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
  44. [44]
    T. Dimofte and S. Gukov, Refined, Motivic and Quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09): Prague, Czech Republic, August 3-8, 2009, pp. 265-289, arXiv:0908.4052 [INSPIRE].
  46. [46]
    D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    T. Hikita, An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane, arXiv:1501.02430.
  49. [49]
    J. Kamnitzer, P. Tingley, B. Webster, A. Weekes and O. Yacobi, Highest weights for truncated shifted Yangians and product monomial crystals, arXiv:1511.09131.
  50. [50]
    T. Braden, A. Licata, N. Proudfoot and B. Webster, Quantizations of conical symplectic resolutions II: category \( \mathcal{O} \) and symplectic duality, arXiv:1407.0964 [INSPIRE].
  51. [51]
    M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, Mirror Symmetry and Symplectic Duality in 3d \( \mathcal{N}=4 \) Gauge Theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].
  52. [52]
    P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
  53. [53]
    Y. Tachikawa, Six-dimensional D N theory and four-dimensional SO-USp quivers, JHEP 07 (2009) 067 [arXiv:0905.4074] [INSPIRE].
  54. [54]
    O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    O. Chacaltana and J. Distler, Tinkertoys for the D N series, JHEP 02 (2013) 110 [arXiv:1106.5410] [INSPIRE].
  56. [56]
    C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  59. [59]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  60. [60]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
  61. [61]
    R.Y. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010 [INSPIRE].
  62. [62]
    V. Borokhov, A. Kapustin and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  63. [63]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    J. Gomis, T. Okuda and V. Pestun, Exact Results for ’t Hooft Loops in Gauge Theories on S4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
  65. [65]
    Y. Ito, T. Okuda and M. Taki, Line operators on S1 × ℝ3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [Erratum ibid. 03 (2016) 085] [arXiv:1111.4221] [INSPIRE].
  66. [66]
    T. Dimofte, N. Garner, M. Geracie and J. Hilburn, work in progress.Google Scholar
  67. [67]
    R. Bezrukavnikov, M. Finkelberg and I. Mirković, Equivariant homology and K-theory of affine Grassmannians and Toda lattices, Compos. Math. 141 (2005) 746.Google Scholar
  68. [68]
    V. Ginzburg, Nil Hecke algebras and Whittaker D-modules, arXiv:1706.06751.
  69. [69]
    J. Kamnitzer, B. Webster, A. Weekes and O. Yacobi, Yangians and quantizations of slices in the affine Grassmannian, Alg. Numb. Theor. 8 (2014) 857 [arXiv:1209.0349].MathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
  71. [71]
    D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
  72. [72]
    A. Hanany and A. Zajac, Discrete Gauging in Coulomb branches of Three Dimensional \( \mathcal{N}=4 \) Supersymmetric Gauge Theories, JHEP 08 (2018) 158 [arXiv:1807.03221] [INSPIRE].
  73. [73]
    A. Hanany and N. Mekareeya, Tri-vertices and SU(2)’s, JHEP 02 (2011) 069 [arXiv:1012.2119] [INSPIRE].
  74. [74]
    A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    A. Kapustin and N. Saulina, The algebra of Wilson-’t Hooft operators, Nucl. Phys. B 814 (2009) 327 [arXiv:0710.2097] [INSPIRE].
  76. [76]
    E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    M. Atiyah and N. Hitchin, The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, U.S.A., (1988), [].

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Quantum Mathematics and Physics (QMAP)UC DavisDavisU.S.A.
  2. 2.Department of Physics and QMAPUC DavisDavisU.S.A.

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