Advertisement

Journal of High Energy Physics

, 2017:144 | Cite as

E 8 instantons on type-A ALE spaces and supersymmetric field theories

  • Noppadol Mekareeya
  • Kantaro Ohmori
  • Yuji Tachikawa
  • Gabi Zafrir
Open Access
Regular Article - Theoretical Physics

Abstract

We consider the 6d superconformal field theory realized on M5-branes probing the E 8 end-of-the-world brane on the deformed and resolved 2/ k singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T 2 compactification, and the star-shaped quiver obtained as the mirror of the T 3 compactification.

Keywords

Supersymmetry and Duality Field Theories in Higher Dimensions Field Theories in Lower Dimensions 

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]
    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu. I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    P.B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990) 263.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Bianchi, F. Fucito, G. Rossi and M. Martellini, Explicit construction of Yang-Mills instantons on ALE spaces, Nucl. Phys. B 473 (1996) 367 [hep-th/9601162] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
  6. [6]
    P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D. Gaiotto, \( \mathcal{N}=2 \) dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
  8. [8]
    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].
  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. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
  10. [10]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    G. Zafrir, Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,0\right) \) SCFT’s, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
  13. [13]
    K. Ohmori and H. Shimizu, S 1 /T 2 compactifications of 6d \( \mathcal{N}=\left(1,0\right) \) theories and brane webs, JHEP 03 (2016) 024 [arXiv:1509.03195] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams, arXiv:1509.03300 [INSPIRE].
  15. [15]
    K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,0\right) \) theories on T 2 and class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    P.B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Diff. Geom. 32 (1990) 473 [INSPIRE].
  17. [17]
    Y. Tachikawa, Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d N = 2 supersymmetric theories, JHEP 06 (2014) 056 [arXiv:1402.4200] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    H. Nakajima, Questions on provisional Coulomb branches of 3-dimensional \( \mathcal{N}=4 \) gauge theories, arXiv:1510.03908 [INSPIRE].
  21. [21]
    S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Coulomb Branch and The Moduli Space of Instantons, JHEP 12 (2014) 103 [arXiv:1408.6835] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    N. Mekareeya, The moduli space of instantons on an ALE space from 3d N = 4 field theories, JHEP 12 (2015) 174 [arXiv:1508.06813] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    V.G. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge U.K. (1994).Google Scholar
  24. [24]
    H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102 (1990) 267.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
  26. [26]
    K. Intriligator, 6d, \( \mathcal{N}=\left(1,0\right) \) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    K. Ohmori, H. Shimizu and Y. Tachikawa, Anomaly polynomial of E-string theories, JHEP 08 (2014) 002 [arXiv:1404.3887] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS 6 duals, JHEP 07 (2012) 171 [arXiv:1206.3503] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, More on 5d descriptions of 6d SCFTs, JHEP 10 (2016) 126 [arXiv:1512.08239] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    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
  35. [35]
    O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Y. Tachikawa, A review of the T N theory and its cousins, PTEP 2015 (2015) 11B102 [arXiv:1504.01481] [INSPIRE].
  37. [37]
    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
  38. [38]
    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].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,0\right) \) theories on S 1 /T 2 and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].ADSGoogle Scholar
  40. [40]
    D. Gaiotto and S.S. Razamat, Exceptional Indices, JHEP 05 (2012) 145 [arXiv:1203.5517] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    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
  42. [42]
    I. Bah, A. Passias and A. Tomasiello, AdS5 compactifications with punctures in massive IIA supergravity, arXiv:1704.07389 [INSPIRE].
  43. [43]
    Y. Tachikawa, \( \mathcal{N}=2 \) Supersymmetric Dynamics for Pedestrians, Lect. Notes Phys. 890 (2013) 2014 [arXiv:1312.2684] [INSPIRE].Google Scholar
  44. [44]
    J.J. Heckman, T. Rudelius and A. Tomasiello, 6D RG Flows and Nilpotent Hierarchies, JHEP 07 (2016) 082 [arXiv:1601.04078] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    N. Mekareeya, T. Rudelius and A. Tomasiello, T-branes, Anomalies and Moduli Spaces in 6D SCFTs, arXiv:1612.06399 [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Noppadol Mekareeya
    • 1
  • Kantaro Ohmori
    • 2
  • Yuji Tachikawa
    • 3
  • Gabi Zafrir
    • 3
  1. 1.Dipartimento di FisicaUniversità di Milano-Bicocca, and INFN, sezione di Milano-BicoccaMilanoItaly
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  3. 3.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

Personalised recommendations