Magnetic quivers, Higgs branches and 6d \( \mathcal{N} = \left(1,\kern0.5em 0\right) \) theories

  • Santiago Cabrera
  • Amihay Hanany
  • Marcus SperlingEmail author
Open Access
Regular Article - Theoretical Physics


The physics of M5 branes placed near an M9 plane on an A-type ALE singularity exhibits a variety of phenomena that introduce additional massless degrees of freedom. There are tensionless strings whenever two M5 branes coincide or whenever an M5 brane approaches the M9 plane. These systems do not admit a low-energy Lagrangian description so new techniques are desirable to shed light on the physics of these phenomena. The 6-dimensional \( \mathcal{N} = \left(1,\kern0.5em 0\right) \) world-volume theory on the M5 branes is composed of massless vector, tensor, and hyper multiplets, and has two branches of the vacuum moduli space where either the scalar fields in the tensor or hyper multiplets receive vacuum expectation values. Focusing on the Higgs branch of the low-energy theory, previous works suggest the conjecture that a new Higgs branch arises whenever a BPS-string becomes tensionless. Consequently, a single theory admits a multitude of Higgs branches depending on the types of tensionless strings in the spectrum. The two main phenomena discrete gauging and small E8instanton transition can be treated in a concise and effective manner by means of Coulomb branches of 3-dimensional \( \mathcal{N} = 4 \) gauge theories. In this paper, a formalism is introduced that allows to derive a novel object from a brane configuration, called the magnetic quiver. The main features are as follows: (i) the 3d Coulomb branch of the magnetic quiver yields the Higgs branch of the 6d system, (ii) all discrete gauging and E8 instanton transitions have an explicit brane realisation, and (iii) exceptional symmetries arise directly from brane configurations. The formalism facilitates the description of Higgs branches at finite and infinite gauge coupling as spaces of dressed monopole operators.


Brane Dynamics in Gauge Theories D-branes Extended Supersymmetry Supersymmetric Gauge Theory 


Open Access

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  1. [1]
    E. Witten, Some comments on string dynamics, in Future perspectives in string theory. Proceedings, Conference, Strings ′95, Los Angeles, CA, U.S.A., 13-18 March 1995, pg. 501 [hep-th/9507121] [INSPIRE].
  2. [2]
    A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Sagnotti, A note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    U.H. Danielsson, G. Ferretti, J. Kalkkinen and P. Stjernberg, Notes on supersymmetric gauge theories in five-dimensions and six-dimensions, Phys. Lett. B 405 (1997) 265 [hep-th/9703098] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    M. Bershadsky and C. Vafa, Global anomalies and geometric engineering of critical theories in six-dimensions, hep-th/9703167 [INSPIRE].
  6. [6]
    A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    I. Brunner and A. Karch, Branes and six-dimensional fixed points, Phys. Lett. B 409 (1997) 109 [hep-th/9705022] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Hanany and A. Zaffaroni, Chiral symmetry from type IIA branes, Nucl. Phys. B 509 (1998) 145 [hep-th/9706047] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
  10. [10]
    J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M.B. Green, J.H. Schwarz and P.C. West, Anomaly free chiral theories in six-dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    S. Randjbar-Daemi, A. Salam, E. Sezgin and J.A. Strathdee, An anomaly free model in six-dimensions, Phys. Lett. B 151 (1985) 351 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. Dabholkar and J. Park, An orientifold of type IIB theory on K3, Nucl. Phys. B 472 (1996) 207 [hep-th/9602030] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Del Zotto and A. Hanany, Complete graphs, Hilbert series and the Higgs branch of the 4d N = 2 (A n ,A m) SCFTs,Nucl. Phys. B 894(2015) 439 [arXiv:1403.6523] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Instanton operators and the Higgs branch at infinite coupling, JHEP 04 (2017) 042 [arXiv:1505.06302] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    G. Ferlito, A. Hanany, N. Mekareeya and G. Zafrir, 3d Coulomb branch and 5d Higgs branch at infinite coupling, JHEP 07 (2018) 061 [arXiv:1712.06604] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. Cabrera, A. Hanany and F. Yagi, Tropical geometry and five dimensional Higgs branches at infinite coupling, JHEP 01 (2019) 068 [arXiv:1810.01379] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    N. Mekareeya, K. Ohmori, Y. Tachikawa and G. Zafrir, E 8 instantons on type-A ALE spaces and supersymmetric field theories, JHEP 09 (2017) 144 [arXiv:1707.04370] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Hanany and N. Mekareeya, The small E 8 instanton and the Kraft Procesi transition, JHEP 07 (2018) 098 [arXiv:1801.01129] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    A. Hanany and G. Zafrir, Discrete gauging in six dimensions, JHEP 07 (2018) 168 [arXiv:1804.08857] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    N. Mekareeya, K. Ohmori, H. Shimizu and A. Tomasiello, Small instanton transitions for M5 fractions, JHEP 10 (2017) 055 [arXiv:1707.05785] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  25. [25]
    N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    K.A. Intriligator, RG fixed points in six-dimensions via branes at orbifold singularities, Nucl. Phys. B 496 (1997) 177 [hep-th/9702038] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    J.D. Blum and K.A. Intriligator, New phases of string theory and 6D RG fixed points via branes at orbifold singularities, Nucl. Phys. B 506 (1997) 199 [hep-th/9705044] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    G. Zafrir, Brane webs, 5d gauge theories and 6d N = (1, 0) SCFTs, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].ADSzbMATHGoogle Scholar
  29. [29]
    K. Ohmori and H. Shimizu, S 1 /T 2 compactifications of 6d N = (1, 0) theories and brane webs, JHEP 03 (2016) 024 [arXiv:1509.03195] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d dualities and tao web diagrams, JHEP 05 (2019) 203 [arXiv:1509.03300] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    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
  32. [32]
    S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition, JHEP 11 (2016) 175 [arXiv:1609.07798] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition: classical case, JHEP 04 (2018) 127 [arXiv:1711.02378] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    S. Cabrera and A. Hanany, Quiver subtractions, JHEP 09 (2018) 008 [arXiv:1803.11205] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Hanany and M. Sperling, Discrete quotients of 3-dimensional N = 4 Coulomb branches via the cycle index, JHEP 08 (2018) 157 [arXiv:1807.02784] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Hanany and A. Zajac, Discrete gauging in Coulomb branches of three dimensional N = 4 supersymmetric gauge theories, JHEP 08 (2018) 158 [arXiv:1807.03221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    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
  39. [39]
    A. Beauville, Symplectic singularities, Invent. Math. 139 (2000) 541 [math.AG/9903070].
  40. [40]
    A. Dancer, F. Kirwan and A. Swann, Implosion for hyper-Kähler manifolds, arXiv:1209.1578 [INSPIRE].
  41. [41]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    G. Ferlito and A. Hanany, A tale of two cones: the Higgs branch of Sp(n) theories with 2n flavours, arXiv:1609.06724 [INSPIRE].
  43. [43]
    V. Kac, Infinite-dimensional Lie algebras, Progr. Math., Cambridge University Press, Cambridge, U.K. (1994).Google Scholar
  44. [44]
    D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, del Pezzo surfaces and type-I theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, S. Theisen and S. Yankielowicz, On heterotic orbifolds, M-theory and type-Ibrane engineering, JHEP 05 (2002) 015 [hep-th/0108135] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    R. Bott, The stable homotopy of the classical groups, Ann. Math. 70 (1959) 313.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Theoretical Physics GroupImperial College LondonLondonU.K.
  2. 2.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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