Classifying global symmetries of 6D SCFTs

  • Peter R. Merkx
Open Access
Regular Article - Theoretical Physics


We characterize the global symmetries for the conjecturally complete collection of all six dimensional superconformal field theories (6D SCFTs) which are realizable in F-theory and have no frozen singularities. We provide comprehensive checks of earlier 6D SCFT classification results via an alternative geometric approach yielding new restrictions which eliminate certain theories. We achieve this by directly constraining elliptically fibered Calabi-Yau (CY) threefold Weierstrass models and find this allows bypassing all anomaly cancellation machinery. This approach reduces the problem of classifying which 6D SCFT gauge and global symmetries are realizable in F-theory models before RG-flow to characterizing features of elliptic fibrations associated to these theories obtained by analysis of polynomials determining their local models. We supply an algorithm with implementation producing from a given SCFT base an explicit listing of all compatible gauge enhancements and their associated global symmetry maxima consistent with the geometric constraints we derive while making manifest the corresponding geometric ingredients for these symmetries including any possible Kodaira type realizations of each algebra summand. In mathematical terms, this amounts to determining all potentially viable non-compact CY threefold elliptic fibrations at finite distance in the moduli space with Weil-Petersson metric which meet certain requirements including the transverse pairwise intersection of singular locus components. We provide local analysis exhausting nearly all CY consistent transverse singular fiber collisions, global analysis concerning all viable gluings of these local models into larger configurations, and many novel constraints on singular locus component pair intersections and global fiber arrangements. We also investigate which transitions between 6D SCFTs can result from gauging of global symmetries and find that continuous degrees of freedom can be lost during such transitions.


Differential and Algebraic Geometry F-Theory Global Symmetries Supersymmetric Gauge Theory 


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. Some comments on string dynamics, In Future perspectives in string theory. Proceedings, Conference, Strings ‘95, Los Angeles, U.S.A., March 13-18, 1995, pp. 501-523.Google Scholar
  2. [2]
    N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
  3. [3]
    N. Seiberg, Nontrivial fixed points of the renormalization group in six-dimensions, Phys. Lett. B 390 (1997) 169 [hep-th/9609161] [INSPIRE].
  4. [4]
    W. Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
  5. [5]
    V. Kumar and W. Taylor, A bound on 6D N = 1 supergravities, JHEP 12 (2009) 050 [arXiv:0910.1586] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    V. Kumar, D.R. Morrison and W. Taylor, Mapping 6D N = 1 supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [INSPIRE].
  7. [7]
    V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].
  8. [8]
    V. Kumar and W. Taylor, String Universality in Six Dimensions, Adv. Theor. Math. Phys. 15 (2011) 325 [arXiv:0906.0987] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    T.W. Grimm and W. Taylor, Structure in 6D and 4D N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].
  11. [11]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  12. [12]
    D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
  14. [14]
    J.J. Heckman, More on the Matter of 6D SCFTs, Phys. Lett. B 747 (2015) 73 [arXiv:1408.0006] [INSPIRE].
  15. [15]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    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].ADSCrossRefMATHGoogle Scholar
  17. [17]
    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].
  18. [18]
    E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  19. [19]
    J. de Boer et al., Triples, fluxes and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Y. Tachikawa, Frozen singularities in M and F-theory, JHEP 06 (2016) 128 [arXiv:1508.06679] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    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
  22. [22]
    J.J. Heckman, T. Rudelius and A. Tomasiello, 6D RG flows and nilpotent hierarchies, JHEP 07 (2016) 082 [arXiv:1601.04078] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    D.R. Morrison and C. Vafa, F-theory and \( \mathcal{N}=1 \) SCFTs in four dimensions, JHEP 08 (2016) 070 [arXiv:1604.03560] [INSPIRE].
  24. [24]
    M. Bertolini, P.R. Merkx and D.R. Morrison, On the global symmetries of 6D superconformal field theories, JHEP 07 (2016) 005 [arXiv:1510.08056] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    D.R. Morrison and T. Rudelius, F-theory and Unpaired Tensors in 6D SCFTs and LSTs, Fortsch. Phys. 64 (2016) 645 [arXiv:1605.08045] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
  27. [27]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
  28. [28]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2., Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
  29. [29]
    T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
  30. [30]
    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].
  31. [31]
    U. Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990) 1.Google Scholar
  32. [32]
    K. Kodaira, On compact analytic surfaces: II, Annals Math. 77 (1963) 563.Google Scholar
  33. [33]
    K. Kodaira, On compact analytic surfaces, III, Annals Math. 78 (1963) 1.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    A. Néron, Modeles minimaux des variétés abéliennes sur les corps locaux et globaux, Publications mathématiques de l’IH ÉS 21 (1964) 5.Google Scholar
  35. [35]
    M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
  36. [36]
    S.-T. Yau, Calabi’s Conjecture and Some New Results in Algebraic Geometry, Proc. Natl. Acad. Sci. U.S.A. 74 (1977) 1798.ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric, in Mathematical aspects of string theory, World Scientific, (1987), pp. 629-646.Google Scholar
  39. [39]
    A.N. Todorov, The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) Calabi-Yau manifolds I, Comm. Math. Phys. 126 (1989) 325.Google Scholar
  40. [40]
    Y. Hayakawa, Degeneration of Calabi-Yau Manifold with Weil-Petersson Metric, alg-geom/9507016.
  41. [41]
    C.-L. Wang, On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds, Math. Res. Lett. 4 (1997) 157.ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik 72 (1952) 349.MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of MathematicsU.C. Santa BarbaraSanta BarbaraU.S.A.

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