5d E n Seiberg-Witten curve via toric-like diagram

  • Sung-Soo KimEmail author
  • Futoshi Yagi
Open Access
Regular Article - Theoretical Physics


We consider 5d Sp(1) gauge theory with \( {E}_N{{}_{{}_f}}_{{}_{+1}} \) global symmetries based on toric(-like) diagram constructed from (p, q)-web with 7-branes. We propose a systematic procedure to compute the Seiberg-Witten curve for generic toric-like diagram. For N f = 6, 7 flavors, we explicitly compute the Seiberg-Witten curves for 5d Sp(1) gauge theory, and show that these Seiberg-Witten curves agree with already known E 7,8 results. We also discuss a generalization of the Seiberg-Witten curve to rank-N cases.


Supersymmetric gauge theory Brane Dynamics in Gauge Theories Global Symmetries 


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]
    E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Brandhuber, N. Itzhaki, J. Sonnenschein, S. Theisen and S. Yankielowicz, On the M-theory approach to (compactified) 5D field theories, Phys. Lett. B 415 (1997) 127 [hep-th/9709010] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    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
  4. [4]
    I. Brunner and A. Karch, Branes and six-dimensional fixed points, Phys. Lett. B 409 (1997) 109 [hep-th/9705022] [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]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    H.-C. Kim, S.-S. Kim and K. Lee, 5-dim superconformal index with enhanced E n global symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, arXiv:1406.6793 [INSPIRE].
  9. [9]
    V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-base duality and global symmetry enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories: 1, Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].CrossRefzbMATHGoogle Scholar
  11. [11]
    L. Bao, E. Pomoni, M. Taki and F. Yagi, M 5-branes, toric diagrams and gauge theory duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L. Bao, E. Pomoni, M. Taki and F. Yagi, M 5-branes, toric diagrams and gauge theory duality, Int. J. Mod. Phys. Conf. Ser. 21 (2013) 136 [INSPIRE].CrossRefzbMATHGoogle Scholar
  13. [13]
    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].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E n global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    J.A. Minahan, D. Nemeschansky and N.P. Warner, Investigating the BPS spectrum of noncritical E n strings, Nucl. Phys. B 508 (1997) 64 [hep-th/9705237] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    T. Eguchi and K. Sakai, Seiberg-Witten curve for the E string theory, JHEP 05 (2002) 058 [hep-th/0203025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    T. Eguchi and K. Sakai, Seiberg-Witten curve for E string theory revisited, Adv. Theor. Math. Phys. 7 (2004) 419 [hep-th/0211213] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Eguchi and H. Kanno, Five-dimensional gauge theories and local mirror symmetry, Nucl. Phys. B 586 (2000) 331 [hep-th/0005008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional E n field theories, JHEP 03 (1999) 006 [hep-th/9902179] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Taki, Seiberg duality, 5d SCFTs and Nekrasov partition functions, arXiv:1401.7200 [INSPIRE].
  24. [24]
    L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-Lagrangian theories from brane junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d T N partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    H. Hayashi and G. Zoccarato, Exact partition functions of Higgsed 5d T N theories, JHEP 01 (2015) 093 [arXiv:1409.0571] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141 [arXiv:1410.2806] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M - and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    R. Feger and T.W. Kephart, LieARTa Mathematica application for Lie algebras and representation theory, Comput. Phys. Commun. 192 (2015) 166 [arXiv:1206.6379] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
  31. [31]
  32. [32]
    R.M. Fonseca, Calculating the renormalisation group equations of a SUSY model with Susyno, Comput. Phys. Commun. 183 (2012) 2298 [arXiv:1106.5016] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A. Iqbal and C. Vafa, BPS degeneracies and superconformal index in diverse dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE].ADSGoogle Scholar
  34. [34]
    H. Hayashi, Y. Tachikawa and K. Yonekura, Mass-deformed T N as a linear quiver, JHEP 02 (2015) 089 [arXiv:1410.6868] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Isachenkov, V. Mitev and E. Pomoni, Toda 3-point functions from topological strings II, arXiv:1412.3395 [INSPIRE].
  36. [36]
    P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.School of PhysicsKorea Institute for Advanced StudySeoulSouth Korea

Personalised recommendations