3d \( \mathcal{N} \) = 2 mirror symmetry, pq-webs and monopole superpotentials

  • Sergio Benvenuti
  • Sara PasquettiEmail author
Open Access
Regular Article - Theoretical Physics


D3 branes stretching between webs of (p,q) 5branes provide an interesting class of 3d \( \mathcal{N} \) = 2 theories. For generic pq-webs however the low energy field theory is not known. We use 3d mirror symmetry and Type IIB S-duality to construct Abelian gauge theories corresponding to D3 branes ending on both sides of a pq-web made of many coincident N S5’s intersecting one D5. These theories contain chiral monopole operators in the superpotential and enjoy a non trivial pattern of global symmetry enhancements. In the special case of the pq-web with one D5 and one N S5, the 3d low energy SCFT admits three dual formulations. This triality can be applied locally inside bigger quiver gauge theories. We prove our statements using partial mirror symmetry à la Kapustin-Strassler, showing the equality of the S b 3 partition functions and studying the quantum chiral rings.


D-branes Supersymmetry and Duality 


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.


  1. [1]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    I. Brunner, A. Hanany, A. Karch and D. Lüst, Brane dynamics and chiral nonchiral transitions, Nucl. Phys. B 528 (1998) 197 [hep-th/9801017] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    O. Aharony, A. Hanany and B. Kol, Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    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
  10. [10]
    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].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    D. Bashkirov and A. Kapustin, Supersymmetry enhancement by monopole operators, JHEP 05 (2011) 015 [arXiv:1007.4861] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    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
  17. [17]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    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
  19. [19]
    A.M. Polyakov, Quark Confinement and Topology of Gauge Groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    I. Affleck, J.A. Harvey and E. Witten, Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions, Nucl. Phys. B 206 (1982) 413 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  22. [22]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    A. Collinucci, S. Giacomelli, R. Savelli and R. Valandro, T-branes through 3d mirror symmetry, JHEP 07 (2016) 093 [arXiv:1603.00062] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    T. Dimofte and D. Gaiotto, An E7 Surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2N c) and U(N c) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A. Yu. Volkov, Noncommutative hypergeometry, Commun. Math. Phys. 258 (2005) 257 [math/0312084] [INSPIRE].
  32. [32]
    R. Kashaev, F. Luo and G. Vartanov, A TQFT of Turaev-Viro type on shaped triangulations, Annales Henri Poincaré 17 (2016) 1109 [arXiv:1210.8393] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    V.P. Spiridonov, On the elliptic beta function, Uspekhi Mat. Nauk 56 (2001) 181 [Russian Math. Surveys 56 (2001) 185].Google Scholar
  34. [34]
    I. Gahramanov and H. Rosengren, A new pentagon identity for the tetrahedron index, JHEP 11 (2013) 128 [arXiv:1309.2195] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    F. Benini, S. Benvenuti and S. Pasquetti, to appear.Google Scholar
  36. [36]
    D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, arXiv:1412.2781 [INSPIRE].
  37. [37]
    S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A. Hanany, C. Hwang, H. Kim, J. Park and R.-K. Seong, Hilbert Series for Theories with Aharony Duals, JHEP 11 (2015) 132 [arXiv:1505.02160] Addendum ibid. 04 (2016) 064 [INSPIRE].
  39. [39]
    S. Cremonesi, The Hilbert series of 3d \( \mathcal{N} \) = 2 Yang-Mills theories with vectorlike matter, J. Phys. A 48 (2015) 455401 [arXiv:1505.02409] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    S. Cremonesi, Type IIB construction of flavoured ABJ(M) and fractional M2 branes, JHEP 01 (2011) 076 [arXiv:1007.4562] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Braverman, B. Feigin, M. Finkelberg and L. Rybnikov, A Finite analog of the AGT relation I: F inite W -algebras and quasimapsspaces, Commun. Math. Phys. 308 (2011) 457 [arXiv:1008.3655] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    H. Nakajima, Handsaw quiver varieties and finite W-algebras, arXiv:1107.5073 [INSPIRE].
  43. [43]
    M. Aganagic, N. Haouzi and S. Shakirov, A n -Triality, arXiv:1403.3657 [INSPIRE].

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.International School of Advanced Studies (SISSA)TriesteItaly
  2. 2.INFN, Sezione di TriesteTriesteItaly
  3. 3.Dipartimento di FisicaUniversità di Milano-BicoccaMilanoItaly

Personalised recommendations