ADE string chains and mirror symmetry

Open Access
Regular Article - Theoretical Physics


6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by −2 curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the A case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the D and E cases as well.


F-Theory Field Theories in Higher Dimensions Supersymmetric Gauge Theory Topological Strings 


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]
    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].
  2. [2]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    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].CrossRefMATHADSGoogle Scholar
  4. [4]
    J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Geometry of 6D RG flows, JHEP 09 (2015) 052 [arXiv:1505.00009] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the classification of little strings, Phys. Rev. D 93 (2016) 086002 [arXiv:1511.05565] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and 6d (1,0) → 4d (N = 2), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
  7. [7]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  8. [8]
    A. Gadde, B. Haghighat, J. Kim, S. Kim, G. Lockhart and C. Vafa, 6d string chains, arXiv:1504.04614 [INSPIRE].
  9. [9]
    B. Haghighat and W. Yan, M-strings in thermodynamic limit: Seiberg-Witten geometry, arXiv:1607.07873 [INSPIRE].
  10. [10]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
  12. [12]
    A. Kanazawa and S.-C. Lau, Local Calabi-Yau manifolds of affine type A and open Yau-Zaslow formula via SYZ mirror symmetry, arXiv:1605.00342 [INSPIRE].
  13. [13]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
  14. [14]
    K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d N = (1, 0) theories on S 1 /T 2 and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].MATHGoogle Scholar
  15. [15]
    M.B. Green, J.H. Schwarz and P.C. West, Anomaly free chiral theories in six-dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  16. [16]
    V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  17. [17]
    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].MathSciNetCrossRefMATHADSGoogle Scholar
  18. [18]
    F. Riccioni and A. Sagnotti, Consistent and covariant anomalies in six-dimensional supergravity, Phys. Lett. B 436 (1998) 298 [hep-th/9806129] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  19. [19]
    A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Exact Gell-Mann-Low function of supersymmetric Yang-Mills theories from instanton calculus, Nucl. Phys. B 229 (1983) 381 [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].CrossRefMATHADSGoogle Scholar
  22. [22]
    E. Witten, Geometric Langlands from six dimensions, arXiv:0905.2720 [INSPIRE].
  23. [23]
    M. Del Zotto, J.J. Heckman, D.S. Park and T. Rudelius, On the defect group of a 6D SCFT, Lett. Math. Phys. 106 (2016) 765 [arXiv:1503.04806] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  24. [24]
    H. Shimizu and Y. Tachikawa, Anomaly of strings of 6d N = (1, 0) theories, JHEP 11 (2016) 165 [arXiv:1608.05894] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  25. [25]
    F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, From 6D SCFTs to dynamic GLSMs, Phys. Rev. D 96 (2017) 066015 [arXiv:1610.00718] [INSPIRE].ADSGoogle Scholar
  26. [26]
    M. Del Zotto and G. Lockhart, On exceptional instanton strings, JHEP 09 (2017) 081 [arXiv:1609.00310] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].ADSGoogle Scholar
  28. [28]
    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].CrossRefMATHGoogle Scholar
  29. [29]
    J. Kim, S. Kim and K. Lee, Little strings and T-duality, JHEP 02 (2016) 170 [arXiv:1503.07277] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  30. [30]
    M. Aganagic and N. Haouzi, ADE little string theory on a Riemann surface (and triality), arXiv:1506.04183 [INSPIRE].
  31. [31]
    J. Kim and K. Lee, Little strings on D n orbifolds, JHEP 10 (2017) 045 [arXiv:1702.03116] [INSPIRE].CrossRefGoogle Scholar
  32. [32]
    B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  33. [33]
    A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004) 247 [math.QA/0306164].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Center of Mathematical Sciences and ApplicationsHarvard UniversityCambridgeU.S.A.
  3. 3.Department of MathematicsHarvard UniversityCambridgeU.S.A.

Personalised recommendations