Journal of High Energy Physics

, 2019:68 | Cite as

Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states

  • Fabio Apruzzi
  • Craig LawrieEmail author
  • Ling Lin
  • Sakura Schäfer-Nameki
  • Yi-Nan Wang
Open Access
Regular Article - Theoretical Physics


We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d \( \mathcal{N} \) = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations. The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, how- ever, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.


Conformal Field Models in String Theory F-Theory M-Theory Supersymmetry and Duality 


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 the proceedings of Future perspectives in string theory (Strings’95), March 13–18, Los Angeles, U.S.A. (1995), hep-th/9507121 [INSPIRE].
  2. [2]
    J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP05 (2014) 028 [Erratum ibid.06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
  3. [3]
    L. Bhardwaj, Classification of 6d \( \mathcal{N} \) = (1, 0) gauge theories, JHEP11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
  4. [4]
    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].
  5. [5]
    K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys.B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
  6. [6]
    P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
  7. [7]
    P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part I. Physical constraints on relevant deformations, JHEP02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
  8. [8]
    F. Apruzzi et al., 5d superconformal field theories and graphs, arXiv:1906.11820 [INSPIRE].
  9. [9]
    D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys.B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
  10. [10]
    D. Xie and S.-T. Yau, Three dimensional canonical singularity and five dimensional \( \mathcal{N} \) = 1 SCFT, JHEP06 (2017) 134 [arXiv:1704.00799] [INSPIRE].
  11. [11]
    M. Del Zotto, J.J. Heckman and D.R. Morrison, 6D SCFTs and phases of 5D theories, JHEP09 (2017) 147 [arXiv:1703.02981] [INSPIRE].
  12. [12]
    P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: single gauge node, arXiv:1705.05836 [INSPIRE].
  13. [13]
    F. Apruzzi, L. Lin and C. Mayrhofer, Phases of 5d SCFTs from M-/F-theory on non-flat fibrations, JHEP05 (2019) 187 [arXiv:1811.12400] [INSPIRE].
  14. [14]
    C. Closset, M. Del Zotto and V. Saxena, Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective, SciPost Phys.6 (2019) 052 [arXiv:1812.10451] [INSPIRE].
  15. [15]
    O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys.B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
  16. [16]
    O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP01 (1998) 002 [hep-th/9710116] [INSPIRE].
  17. [17]
    O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional E(n) field theories, JHEP03 (1999) 006 [hep-th/9902179] [INSPIRE].
  18. [18]
    O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
  19. [19]
    G. Zafrir, Brane webs, 5d gauge theories and 6d \( \mathcal{N} \) = (1, 0) SCFT’s, JHEP12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
  20. [20]
    G. Zafrir, Brane webs and O5-planes, JHEP03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
  21. [21]
    K. Ohmori and H. Shimizu, S 1/T 2compactifications of 6d \( \mathcal{N} \) = (1, 0) theories and brane webs, JHEP03 (2016) 024 [arXiv:1509.03195] [INSPIRE].
  22. [22]
    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Discrete theta angle from an O5-plane, JHEP11 (2017) 041 [arXiv:1707.07181] [INSPIRE].
  23. [23]
    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 5-brane webs for 5d \( \mathcal{N} \) = 1 G 2gauge theories, JHEP03 (2018) 125 [arXiv:1801.03916] [INSPIRE].
  24. [24]
    H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Dualities and 5-brane webs for 5d rank 2 SCFTs, JHEP12 (2018) 016 [arXiv:1806.10569] [INSPIRE].
  25. [25]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett.B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
  26. [26]
    A. Brandhuber and Y. Oz, The D4-D8 brane system and five-dimensional fixed points, Phys. Lett.B 460 (1999) 307 [hep-th/9905148] [INSPIRE].
  27. [27]
    O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS 6duals, JHEP07 (2012) 171 [arXiv:1206.3503] [INSPIRE].
  28. [28]
    H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box graphs and singular fibers, JHEP05 (2014) 048 [arXiv:1402.2653] [INSPIRE].
  29. [29]
    J. Tian and Y.-N. Wang, E-string spectrum and typical F-theory geometry, arXiv:1811.02837 [INSPIRE].
  30. [30]
    F. Apruzzi et al., Fibers add flavor. Part III. Higher rank,Google Scholar
  31. [31]
    F. Apruzzi et al., Fibers add flavor. Part II. 5d SCFTs, gauge theories and dualities, arXiv:1909.09128 [INSPIRE].
  32. [32]
    H.-C. Kim, S.-S. Kim and K. Lee, 5-dim superconformal index with enhanced en global symmetry, JHEP10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
  33. [33]
    O. Bergman, D. Rodríguez-Gómez and G. Zafrir, 5-brane webs, symmetry enhancement and duality in 5d supersymmetric gauge theory, JHEP03 (2014) 112 [arXiv:1311.4199] [INSPIRE].
  34. [34]
    G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP12 (2014) 116 [arXiv:1408.4040] [INSPIRE].
  35. [35]
    V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-base duality and global symmetry enhancement, JHEP04 (2015) 052 [arXiv:1411.2450] [INSPIRE].
  36. [36]
    C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP07 (2015) 063 [arXiv:1406.6793] [INSPIRE].
  37. [37]
    D. Gaiotto and H.-C. Kim, Duality walls and defects in 5d \( \mathcal{N} \) = 1 theories, JHEP01 (2017) 019 [arXiv:1506.03871] [INSPIRE].
  38. [38]
    Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, PTEP2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
  39. [39]
    K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, JHEP07 (2015) 167 [arXiv:1505.04743] [INSPIRE].
  40. [40]
    G. Zafrir, Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories, JHEP07 (2015) 087 [arXiv:1503.08136] [INSPIRE].
  41. [41]
    O. Bergman and D. Rodriguez-Gomez, A note on instanton operators, instanton particles and supersymmetry, JHEP05 (2016) 068 [arXiv:1601.00752] [INSPIRE].
  42. [42]
    H. Hayashi et al., A new 5d description of 6d D-type minimal conformal matter, JHEP08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
  43. [43]
    G. Ferlito, A. Hanany, N. Mekareeya and G. Zafrir, 3d Coulomb branch and 5d Higgs branch at infinite coupling, JHEP07 (2018) 061 [arXiv:1712.06604] [INSPIRE].
  44. [44]
    S. Cabrera, A. Hanany and F. Yagi, Tropical geometry and five dimensional Higgs branches at infinite coupling, JHEP01 (2019) 068 [arXiv:1810.01379] [INSPIRE].
  45. [45]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
  46. [46]
    C. Lawrie and S. Schäfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP04 (2013) 061 [arXiv:1212.2949] [INSPIRE].
  47. [47]
    P. Candelas et al., Codimension three bundle singularities in F-theory, JHEP06 (2002) 014 [hep-th/0009228] [INSPIRE].
  48. [48]
    V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP01 (2013) 016 [arXiv:1110.4883] [INSPIRE].
  49. [49]
    V. Braun, T.W. Grimm and J. Keitel, Geometric engineering in toric F-theory and GUTs with U(1) gauge factors, JHEP12 (2013) 069 [arXiv:1306.0577] [INSPIRE].
  50. [50]
    M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral four-dimensional f-theory compactifications with SU(5) and multiple U(1)-factors, JHEP04 (2014) 010 [arXiv:1306.3987] [INSPIRE].
  51. [51]
    F. Baume, E. Palti and S. Schwieger, On E8and F-theory GUTs, JHEP06 (2015) 039 [arXiv:1502.03878] [INSPIRE].
  52. [52]
    L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Tools for CICYs in F-theory, JHEP11 (2016) 004 [arXiv:1608.07554] [INSPIRE].
  53. [53]
    W. Buchmüller, M. Dierigl, P.-K. Oehlmann and F. Ruehle, The toric SO(10) F-theory landscape, JHEP12 (2017) 035 [arXiv:1709.06609] [INSPIRE].
  54. [54]
    L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY threefolds, JHEP10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
  55. [55]
    Y.-C. Huang and W. Taylor, Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers, JHEP02 (2019) 087 [arXiv:1805.05907] [INSPIRE].
  56. [56]
    M. Dierigl, P.-K. Oehlmann and F. Ruehle, Global tensor-matter transitions in F-theory, Fortsch. Phys.66 (2018) 1800037 [arXiv:1804.07386] [INSPIRE].
  57. [57]
    I. Achmed-Zade, I. García-Etxebarria and C. Mayrhofer, A note on non-flat points in the SU(5) × U(1)PQF-theory model, JHEP05 (2019) 013 [arXiv:1806.05612] [INSPIRE].
  58. [58]
    R. Miranda, Smooth models for elliptic threefolds, in The birational geometry of degenerations, R. Friedman ed., Progress in Mathematics volume 29, Birkhäuser, Boston U.S.A. (1983).Google Scholar
  59. [59]
    M. Esole, M.J. Kang and S.-T. Yau, Mordell-Weil torsion, anomalies and phase transitions, arXiv:1712.02337 [INSPIRE].
  60. [60]
    M. Esole, R. Jagadeesan and M.J. Kang, 48 Crepant paths to SU(2) × SU(3), arXiv:1905.05174 [INSPIRE].
  61. [61]
    M. Esole and M.J. Kang, Flopping and slicing: SO(4) and Spin(4)-models, arXiv:1802.04802 [INSPIRE].
  62. [62]
    M. Esole and M.J. Kang, The geometry of the SU(2) × G 2-model, JHEP02 (2019) 091 [arXiv:1805.03214] [INSPIRE].
  63. [63]
    M. Bertolini, P.R. Merkx and D.R. Morrison, On the global symmetries of 6D superconformal field theories, JHEP07 (2016) 005 [arXiv:1510.08056] [INSPIRE].
  64. [64]
    P.R. Merkx, Classifying global symmetries of 6D SCFTs, JHEP03 (2018) 163 [arXiv:1711.05155] [INSPIRE].
  65. [65]
    A.P. Braun and S. Schäfer-Nameki, Box graphs and resolutions I, Nucl. Phys.B 905 (2016) 447 [arXiv:1407.3520] [INSPIRE].
  66. [66]
    A.P. Braun and S. Schäfer-Nameki, Box graphs and resolutions II: from Coulomb phases to fiber faces, Nucl. Phys.B 905 (2016) 480 [arXiv:1511.01801] [INSPIRE].
  67. [67]
    C. Lawrie, S. Schäfer-Nameki and J.-M. Wong, F-theory and all things rational: surveying U(1) symmetries with rational sections, JHEP09 (2015) 144 [arXiv:1504.05593] [INSPIRE].
  68. [68]
    O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E8theory, Nucl. Phys.B 487 (1997) 93 [hep-th/9610251] [INSPIRE].
  69. [69]
    A.C. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett.B 357 (1995) 76 [hep-th/9506144] [INSPIRE].
  70. [70]
    S. Ferrara, R.R. Khuri and R. Minasian, M theory on a Calabi-Yau manifold, Phys. Lett.B 375 (1996) 81 [hep-th/9602102] [INSPIRE].
  71. [71]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys.B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
  72. [72]
    S. Ferrara, R. Minasian and A. Sagnotti, Low-energy analysis of M and F theories on Calabi-Yau threefolds, Nucl. Phys.B 474 (1996) 323 [hep-th/9604097] [INSPIRE].
  73. [73]
    L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: rank one, JHEP07 (2019) 178 [arXiv:1809.01650] [INSPIRE].
  74. [74]
    L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: arbitrary rank, arXiv:1811.10616 [INSPIRE].
  75. [75]
    C. Vafa, Evidence for F-theory, Nucl. Phys.B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
  76. [76]
    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].
  77. [77]
    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].
  78. [78]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys.B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
  79. [79]
    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys.B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
  80. [80]
    S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP08 (2011) 094 [arXiv:1106.3854] [INSPIRE].
  81. [81]
    D.S. Park, Anomaly equations and intersection theory, JHEP01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
  82. [82]
    M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math. Phys.17 (2013) 1195 [arXiv:1107.0733] [INSPIRE].
  83. [83]
    J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yau’s, JHEP11 (2011) 098 [arXiv:1108.1794] [INSPIRE].
  84. [84]
    S. Krause, C. Mayrhofer and T. Weigand, G 4flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys.B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].
  85. [85]
    M. Esole, P. Jefferson and M.J. Kang, Euler characteristics of crepant resolutions of Weierstrass models, Commun. Math. Phys.371 (2019) 99 [arXiv:1703.00905] [INSPIRE].
  86. [86]
    N. Mekareeya, K. Ohmori, Y. Tachikawa and G. Zafrir, E8instantons on type-A ALE spaces and supersymmetric field theories, JHEP09 (2017) 144 [arXiv:1707.04370] [INSPIRE].
  87. [87]
    Y. Tachikawa, On S-duality of 5d super Yang-Mills on S 1 , JHEP11 (2011) 123 [arXiv:1110.0531] [INSPIRE].
  88. [88]
    L. Bhardwaj, D.R. Morrison, Y. Tachikawa and A. Tomasiello, The frozen phase of F-theory, JHEP08 (2018) 138 [arXiv:1805.09070] [INSPIRE].
  89. [89]
    F. Apruzzi et al., Supplementary material: CFD-trees,
  90. [90]
    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
  91. [91]
    S. Kachru and M. Zimet, A comment on 4d and 5d BPS states, arXiv:1808.01529 [INSPIRE].
  92. [92]
    M. Taki, Seiberg duality, 5D SCFTs and Nekrasov partition functions, arXiv:1401.7200 [INSPIRE].
  93. [93]
    N. Yamatsu, Finite-dimensional Lie algebras and their representations for unified model building, arXiv:1511.08771 [INSPIRE].
  94. [94]
    M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
  95. [95]
    S. Cecotti, D. Gaiotto and C. Vafa, tt geometry in 3 and 4 dimensions, JHEP05 (2014) 055 [arXiv:1312.1008] [INSPIRE].
  96. [96]
    S. Banerjee, P. Longhi and M. Romo, Exploring 5d BPS spectra with exponential networks, Annales Henri Poincaré20 (2019) 4055 [arXiv:1811.02875] [INSPIRE].
  97. [97]
    U. Derenthal, Geometry of universal torsors, Ph.D. thesis, Universit¨at G¨ottingen, G¨ottingen, Germany (2006).Google Scholar
  98. [98]
    U. Derenthal, Singular del pezzo surfaces whose universal torsors are hypersurfaces, Proc. London Math. Soc.108 (2014) 638.Google Scholar
  99. [99]
    W. Taylor and Y.-N. Wang, Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua, Adv. Theor. Math. Phys.21 (2017) 1063 [arXiv:1504.07689] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Fabio Apruzzi
    • 1
  • Craig Lawrie
    • 2
    Email author
  • Ling Lin
    • 2
  • Sakura Schäfer-Nameki
    • 1
  • Yi-Nan Wang
    • 1
  1. 1.Mathematical InstituteUniversity of OxfordOxfordU.K.
  2. 2.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaU.S.A.

Personalised recommendations