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Journal of High Energy Physics

, 2017:147 | Cite as

6D SCFTs and phases of 5D theories

  • Michele Del Zotto
  • Jonathan J. Heckman
  • David R. Morrison
Open Access
Regular Article - Theoretical Physics

Abstract

Starting from 6D superconformal field theories (SCFTs) realized via F-theory, we show how reduction on a circle leads to a uniform perspective on the phase structure of the resulting 5D theories, and their possible conformal fixed points. Using the correspon-dence between F-theory reduced on a circle and M-theory on the corresponding elliptically fibered Calabi-Yau threefold, we show that each 6D SCFT with minimal supersymmetry directly reduces to a collection of between one and four 5D SCFTs. Additionally, we find that in most cases, reduction of the tensor branch of a 6D SCFT yields a 5D generalization of a quiver gauge theory. These two reductions of the theory often correspond to different phases in the 5D theory which are in general connected by a sequence of flop transitions in the extended Kähler cone of the Calabi-Yau threefold. We also elaborate on the structure of the resulting conformal fixed points, and emergent flavor symmetries, as realized by M-theory on a canonical singularity.

Keywords

Conformal Field Theory F-Theory Field Theories in Higher Dimensions M-Theory 

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]
    E. Witten, Some comments on string dynamics, hep-th/9507121 [INSPIRE].
  2. [2]
    A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    N. Seiberg, Nontrivial fixed points of the renormalization group in six-dimensions, Phys. Lett. B 390 (1997) 169 [hep-th/9609161] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    N. Seiberg, Notes on theories with 16 supercharges, Nucl. Phys. Proc. Suppl. 67 (1998) 158 [hep-th/9705117] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    M. Henningson, Self-dual strings in six dimensions: Anomalies, the ADE-classification and the world-sheet WZW-model, Commun. Math. Phys. 257 (2005) 291 [hep-th/0405056] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    C. Cordova, T.T. Dumitrescu and X. Yin, Higher Derivative Terms, Toroidal Compactification and Weyl Anomalies in Six-Dimensional (2,0) Theories, arXiv:1505.03850 [INSPIRE].
  7. [7]
    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].
  8. [8]
    D. Gaiotto and A. Tomasiello, Holography for (1, 0) theories in six dimensions, JHEP 12 (2014) 003 [arXiv:1404.0711] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  10. [10]
    J.J. Heckman, More on the Matter of 6D SCFTs, Phys. Lett. B 747 (2015) 73 [arXiv:1408.0006] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  11. [11]
    M. Del Zotto, J.J. Heckman, D.R. Morrison and D.S. Park, 6D SCFTs and Gravity, JHEP 06 (2015) 158 [arXiv:1412.6526] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    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
  13. [13]
    L. Bhardwaj, Classification of 6d \( \mathcal{N}=\left(1,0\right) \) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
  14. [14]
    V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, math/0005196 [INSPIRE].
  16. [16]
    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
  17. [17]
    G. Lockhart and C. Vafa, Superconformal Partition Functions and Non-perturbative Topological Strings, arXiv:1210.5909 [INSPIRE].
  18. [18]
    B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of Minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    H.-C. Kim, S. Kim and J. Park, 6d strings from new chiral gauge theories, arXiv:1608.03919 [INSPIRE].
  20. [20]
    M. Del Zotto and G. Lockhart, On Exceptional Instanton Strings, arXiv:1609.00310 [INSPIRE].
  21. [21]
    J. Gu, M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, Refined BPS invariants of 6d SCFTs from anomalies and modularity, JHEP 05 (2017) 130 [arXiv:1701.00764] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    C. Vafa, Geometric origin of Montonen-Olive duality, Adv. Theor. Math. Phys. 1 (1998) 158 [hep-th/9707131] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    P.C. Argyres and N. Seiberg, S-duality in \( \mathcal{N}=2 \) supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
  25. [25]
    D. Gaiotto, \( \mathcal{N}=2 \) dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
  26. [26]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    D. Gaiotto and S.S. Razamat, \( \mathcal{N}=1 \) theories of class \( {\mathcal{S}}_k \) , JHEP 07 (2015) 073 [arXiv:1503.05159] [INSPIRE].
  28. [28]
    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
  29. [29]
    S. Franco, H. Hayashi and A. Uranga, Charting Class \( {\mathcal{S}}_k \) Territory, Phys. Rev. D 92 (2015) 045004 [arXiv:1504.05988] [INSPIRE].
  30. [30]
    M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and \( 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
  31. [31]
    A. Hanany and K. Maruyoshi, Chiral theories of class S, JHEP 12 (2015) 080 [arXiv:1505.05053] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    M. Aganagic and N. Haouzi, ADE Little String Theory on a Riemann Surface (and Triality), arXiv:1506.04183 [INSPIRE].
  33. [33]
    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].
  34. [34]
    I. Coman, E. Pomoni, M. Taki and F. Yagi, Spectral curves of \( \mathcal{N}=1 \) theories of class \( {\mathcal{S}}_k \), JHEP 06 (2017) 136 [arXiv:1512.06079] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    D.R. Morrison and C. Vafa, F-theory and \( \mathcal{N}=1 \) SCFTs in four dimensions, JHEP 08 (2016) 070 [arXiv:1604.03560] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    J.J. Heckman, P. Jefferson, T. Rudelius and C. Vafa, Punctures for theories of class \( {\mathcal{S}}_{\varGamma } \) , JHEP 03 (2017) 171 [arXiv:1609.01281] [INSPIRE].
  37. [37]
    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].
  38. [38]
    S.S. Razamat, C. Vafa and G. Zafrir, 4d \( \mathcal{N}=1 \) from 6d (1, 0), JHEP 04 (2017) 064 [arXiv:1610.09178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    D.-E. Diaconescu and R. Entin, Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry, Nucl. Phys. B 538 (1999) 451 [hep-th/9807170] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS 6 duals, JHEP 07 (2012) 171 [arXiv:1206.3503] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    O. Bergman, D. Rodríguez-Gómez and G. Zafrir, 5d superconformal indices at large-N and holography, JHEP 08 (2013) 081 [arXiv:1305.6870] [INSPIRE].
  46. [46]
    F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello, AdS 6 solutions of type-II supergravity, JHEP 11 (2014) 099 [Erratum ibid. 05 (2015) 012] [arXiv:1406.0852] [INSPIRE].
  47. [47]
    S.-S. Kim, M. Taki and F. Yagi, Tao Probing the End of the World, PTEP 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].
  48. [48]
    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
  49. [49]
    O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].ADSMathSciNetGoogle Scholar
  50. [50]
    E. D’Hoker, M. Gutperle, A. Karch and C.F. Uhlemann, Warped AdS 6 × S 2 in Type IIB supergravity I: Local solutions, JHEP 08 (2016) 046 [arXiv:1606.01254] [INSPIRE].CrossRefGoogle Scholar
  51. [51]
    E. D’Hoker, M. Gutperle and C.F. Uhlemann, Holographic duals for five-dimensional superconformal quantum field theories, Phys. Rev. Lett. 118 (2017) 101601 [arXiv:1611.09411] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
  53. [53]
    O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    O. Aharony, A. Hanany and B. Kol, Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1., Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [INSPIRE].
  58. [58]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  59. [59]
    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].
  60. [60]
    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].
  61. [61]
    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].
  62. [62]
    H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    H.-C. Kim, unpublished.Google Scholar
  64. [64]
    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].
  65. [65]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  67. [67]
    N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  68. [68]
    O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E 8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  69. [69]
    R. Friedman and D.R. Morrison, The Birational geometry of degenerations, Progress in Mathematics, Birkhäuser (1983).Google Scholar
  70. [70]
    H.B. Laufer, On ℂ1 as an exceptional set, in Recent Developments in Several Complex Variables, J.E. Fornaess, ed., Princeton University Press, Ann. Math. Stud. 100 (1981) 261.Google Scholar
  71. [71]
    H. Pinkham, Factorization of birational maps in dimension 3, in Singularities, P. Orlik ed., American Mathematical Society, Proc. Symp. Pure Math. 40 (1983) 343, part 2.Google Scholar
  72. [72]
    M. Reid, Minimal models of canonical 3-folds, in Algebraic Varieties and Analytic Varieties, S. Iitaka ed., Kinokuniya, Adv. Stud. Pure Math. 1 (1983) 131.Google Scholar
  73. [73]
    S. Katz and D.R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Alg. Geom. 1 (1992) 449 [alg-geom/9202002].MathSciNetMATHGoogle Scholar
  74. [74]
    G. Baditoiu and S. Rosenberg, Feynman diagrams and Lax pair equations, math-ph/0611014 [INSPIRE].
  75. [75]
    D.R. Morrison, Beyond the Kähler cone, alg-geom/9407007.
  76. [76]
    S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4-D N = 2 gauge theories: 1., Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
  77. [77]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  78. [78]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  79. [79]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].ADSMATHGoogle Scholar
  80. [80]
    T.W. Grimm and A. Kapfer, Anomaly Cancelation in Field Theory and F-theory on a Circle, JHEP 05 (2016) 102 [arXiv:1502.05398] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  81. [81]
    Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, PTEP 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
  82. [82]
    G. Zafrir, Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories, JHEP 07 (2015) 087 [arXiv:1503.08136] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  83. [83]
    K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, JHEP 07 (2015) 167 [arXiv:1505.04743] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  84. [84]
    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].ADSMathSciNetCrossRefGoogle Scholar
  85. [85]
    P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  86. [86]
    O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP 01 (2014) 079 [arXiv:1310.2150] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    C. Papageorgakis, A. Pini and D. Rodriguez-Gomez, Nekrasov-Shatashvili limit of the 5D superconformal index, Phys. Rev. D 94 (2016) 045007 [arXiv:1602.02647] [INSPIRE].
  88. [88]
    H. Kim, N. Kim and M. Suh, Supersymmetric AdS 6 Solutions of Type IIB Supergravity, Eur. Phys. J. C 75 (2015) 484 [arXiv:1506.05480] [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Michele Del Zotto
    • 1
  • Jonathan J. Heckman
    • 2
  • David R. Morrison
    • 3
    • 4
  1. 1.Simons Center for Geometry and PhysicsStony BrookU.S.A.
  2. 2.Department of PhysicsUniversity of North CarolinaChapel HillU.S.A.
  3. 3.Department of MathematicsUniversity of California Santa BarbaraSanta BarbaraU.S.A.
  4. 4.Department of PhysicsUniversity of California Santa BarbaraSanta BarbaraU.S.A.

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