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Journal of High Energy Physics

, 2014:103 | Cite as

Coulomb branch and the moduli space of instantons

  • Stefano Cremonesi
  • Giulia Ferlito
  • Amihay Hanany
  • Noppadol MekareeyaEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

The moduli space of instantons on ℂ2 for any simple gauge group is studied using the Coulomb branch of \( \mathcal{N}=4 \) gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescription for computing the Coulomb branch Hilbert series of such diagrams is very simple. For instanton numbers one and two, the results are in agreement with previous works. New results and general features for the moduli spaces of three and higher instanton numbers are reported and discussed in detail.

Keywords

Supersymmetric gauge theory Solitons Monopoles and Instantons Brane Dynamics in Gauge Theories Supersymmetry and Duality 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y. Tyupkin, Pseudoparticle Solutions of the Yang-Mills Equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    G. ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14 (1976) 3432.Google Scholar
  3. [3]
    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    M.R. Douglas, Branes within branes, hep-th/9512077 [INSPIRE].
  6. [6]
    P.C. Argyres, M.R. Plesser and N. Seiberg, The moduli space of vacua of N = 2 SUSY QCD and duality in N = 1 SUSY QCD, Nucl. Phys. B 471 (1996) 159 [hep-th/9603042] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    D. Gaiotto and S.S. Razamat, Exceptional indices, JHEP 05 (2012) 145 [arXiv:1203.5517] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, math/0505553 [INSPIRE].
  15. [15]
    C.A. Keller and J. Song, Counting exceptional instantons, JHEP 07 (2012) 085 [arXiv:1205.4722] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert Series of the One Instanton Moduli Space, JHEP 06 (2010) 100 [arXiv:1005.3026] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of Instantons and W-algebras, JHEP 03 (2012) 045 [arXiv:1111.5624] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    A. Hanany, N. Mekareeya and S.S. Razamat, Hilbert Series for Moduli Spaces of Two Instantons, JHEP 01 (2013) 070 [arXiv:1205.4741] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  21. [21]
    N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    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].ADSCrossRefGoogle Scholar
  23. [23]
    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].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Hall-Littlewood polynomials, JHEP 09 (2014) 178 [arXiv:1403.0585] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    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].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    V. Borokhov, A. Kapustin and X.-k. Wu, Topological disorder operators in three-dimensional conformal field theory, JHEP 11 (2002) 049 [hep-th/0206054] [INSPIRE].
  27. [27]
    P.B. Kronheimer, The Construction of ALE spaces as hyperKähler quotients, J. Diff. Geom. 29 (1989) 665 [INSPIRE].zbMATHMathSciNetGoogle Scholar
  28. [28]
    P. Kronheimer and H. Nakajima, Yang-mills instantons on ale gravitational instantons, Math. Ann. 288 (1990) 263.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    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].ADSCrossRefGoogle Scholar
  30. [30]
    M. Porrati and A. Zaffaroni, M theory origin of mirror symmetry in three-dimensional gauge theories, Nucl. Phys. B 490 (1997) 107 [hep-th/9611201] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    S.S. Razamat and B. Willett, Down the rabbit hole with theories of class S, JHEP 1410 (2014) 99 [arXiv:1403.6107] [INSPIRE].CrossRefGoogle Scholar
  32. [32]
    S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
  33. [33]
    Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    C. Krattenthaler, V.P. Spiridonov and G.S. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, JHEP 06 (2011) 008 [arXiv:1103.4075] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional Field Theories, arXiv:1106.2484 [INSPIRE].
  36. [36]
    J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, \( \mathrm{\mathbb{Z}} \)) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    A. Kapustin, D(n) quivers from branes, JHEP 12 (1998) 015 [hep-th/9806238] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    A. Hanany and A. Zaffaroni, Issues on orientifolds: On the brane construction of gauge theories with SO(2N) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    A. Hanany and J. Troost, Orientifold planes, affine algebras and magnetic monopoles, JHEP 08 (2001) 021 [hep-th/0107153] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    B. Julia, Kac-Moody symmetry of gravitation and supergravity theories, talk at AMS summer seminar on Appication of Group Theory in Physics and Mathematical Physics, Chicago U.S.A. (1982).Google Scholar
  41. [41]
    A. Sen, Stable nonBPS bound states of BPS D-branes, JHEP 08 (1998) 010 [hep-th/9805019] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
  43. [43]
    F. Englert and P. Windey, Quantization Condition fort Hooft Monopoles in Compact Simple Lie Groups, Phys. Rev. D 14 (1976) 2728 [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    A. Kapustin, Wilson-t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].ADSMathSciNetGoogle Scholar
  46. [46]
    V. Borokhov, A. Kapustin and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].
  47. [47]
    V. Borokhov, Monopole operators in three-dimensional N = 4 SYM and mirror symmetry, JHEP 03 (2004) 008 [hep-th/0310254] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    M.K. Benna, I.R. Klebanov and T. Klose, Charges of Monopole Operators in Chern-Simons Yang-Mills Theory, JHEP 01 (2010) 110 [arXiv:0906.3008] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    D. Bashkirov and A. Kapustin, Supersymmetry enhancement by monopole operators, JHEP 05 (2011) 015 [arXiv:1007.4861] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  50. [50]
    P.B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Diff. Geom. 32 (1990) 473.zbMATHMathSciNetGoogle Scholar
  51. [51]
    R. Brylinski, Instantons and Kähler geometry of nilpotent orbits, in NATO Sci. Ser. C. Vol. 514: Representation theories and algebraic geometry [math/9811032] [INSPIRE].
  52. [52]
    P. Kobak and A. Swann, The hyperkähler geometry associated to Wolf spaces, Boll. Unione Mat. Ital. B 4 (2001) 587 [math/0001025].
  53. [53]
    E.B. Vinberg and V.L. Popov, On a class of quasihomogeneous affine varieties,” Math. USSR Izv. 6 (1972) 743.Google Scholar
  54. [54]
    D. Garfinkle, A new construction of the Joseph ideal (1982), http://hdl.handle.net/1721.1/15620.
  55. [55]
    A. Joseph, The minimal orbit in a simple lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup. 9 (1976) 1.zbMATHGoogle Scholar
  56. [56]
    D. Gaiotto, A. Neitzke and Y. Tachikawa, Argyres-Seiberg duality and the Higgs branch, Commun. Math. Phys. 294 (2010) 389 [arXiv:0810.4541] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  57. [57]
    D. Bashkirov, Examples of global symmetry enhancement by monopole operators, arXiv:1009.3477 [INSPIRE].
  58. [58]
    A. Dey, A. Hanany, P. Koroteev and N. Mekareeya, Mirror Symmetry in Three Dimensions via Gauged Linear Quivers, JHEP 06 (2014) 059 [arXiv:1402.0016] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Stefano Cremonesi
    • 1
  • Giulia Ferlito
    • 1
  • Amihay Hanany
    • 1
  • Noppadol Mekareeya
    • 2
    Email author
  1. 1.Theoretical Physics GroupImperial College LondonLondonU.K.
  2. 2.Theory Division, Physics DepartmentCERNGeneva 23Switzerland

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