Advertisement

Journal of High Energy Physics

, 2015:54 | Cite as

6d Conformal matter

  • Michele Del Zotto
  • Jonathan J. HeckmanEmail author
  • Alessandro Tomasiello
  • Cumrun Vafa
Open Access
Regular Article - Theoretical Physics
First Online:

Abstract

A single M5-brane probing G, an ADE-type singularity, leads to a system which has G × G global symmetry and can be viewed as “bifundamental” (G, G) matter. For the A N series, this leads to the usual notion of bifundamental matter. For the other cases it corresponds to a strongly interacting (1, 0) superconformal system in six dimensions. Similarly, an ADE singularity intersecting the Hořava-Witten wall leads to a superconformal matter system with E 8 × G global symmetry. Using the F-theory realization of these theories, we elucidate the Coulomb/tensor branch of (G, G′) conformal matter. This leads to the notion of fractionalization of an M5-brane on an ADE singularity as well as fractionalization of the intersection point of the ADE singularity with the Hořava-Witten wall. Partial Higgsing of these theories leads to new 6d SCFTs in the infrared, which we also characterize. This generalizes the class of (1, 0) theories which can be perturbatively realized by suspended branes in IIA string theory. By reducing on a circle, we arrive at novel duals for 5d affine quiver theories. Introducing many M5-branes leads to large N gravity duals.

Keywords

F-Theory AdS-CFT Correspondence Field Theories in Higher Dimensions 
Download to read the full article text

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, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    M. Bershadsky and A. Johansen, Colliding singularities in F-theory and phase transitions, Nucl. Phys. B 489 (1997) 122 [hep-th/9610111] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    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].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    J.D. Blum and K.A. Intriligator, Consistency conditions for branes at orbifold singularities, Nucl. Phys. B 506 (1997) 223 [hep-th/9705030] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    K.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [INSPIRE].MathSciNetGoogle Scholar
  11. [11]
    A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [arXiv:1312.5746] [INSPIRE].CrossRefADSGoogle Scholar
  13. [13]
    D. Gaiotto and A. Tomasiello, Holography for (1, 0) theories in six dimensions, JHEP 12 (2014) 003 [arXiv:1404.0711] [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    P. Hořava and E. Witten, Heterotic and type-I string dynamics from eleven-dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209] [INSPIRE].ADSGoogle Scholar
  15. [15]
    P. Hořava and E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl. Phys. B 475 (1996) 94 [hep-th/9603142] [INSPIRE].ADSGoogle Scholar
  16. [16]
    J. de Boer et al., Triples, fluxes and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].Google Scholar
  17. [17]
    P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-branes and monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    C.-C. Chiou, A.E. Faraggi, R. Tatar and W. Walters, T-branes and Yukawa couplings, JHEP 05 (2011) 023 [arXiv:1101.2455] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    R. Donagi and M. Wijnholt, Gluing branes I, JHEP 05 (2013) 068 [arXiv:1104.2610] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  21. [21]
    R. Donagi and M. Wijnholt, Gluing branes II: flavour physics and string duality, JHEP 05 (2013) 092 [arXiv:1112.4854] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  22. [22]
    L.B. Anderson, J.J. Heckman and S. Katz, T-branes and geometry, JHEP 05 (2014) 080 [arXiv:1310.1931] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  23. [23]
    P.S. Aspinwall and D.R. Morrison, Nonsimply connected gauge groups and rational points on elliptic curves, JHEP 07 (1998) 012 [hep-th/9805206] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    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].CrossRefADSMathSciNetGoogle Scholar
  25. [25]
    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].zbMATHMathSciNetGoogle Scholar
  27. [27]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  29. [29]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    J.J. Heckman, More on the matter of 6D SCFTs, arXiv:1408.0006 [INSPIRE].
  31. [31]
    D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  32. [32]
    A. Sen, A note on enhanced gauge symmetries in M and string theory, JHEP 09 (1997) 001 [hep-th/9707123] [INSPIRE].CrossRefADSGoogle Scholar
  33. [33]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
  34. [34]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  35. [35]
    K. Ohmori, H. Shimizu and Y. Tachikawa, Anomaly polynomial of E-string theories, JHEP 08 (2014) 002 [arXiv:1404.3887] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  36. [36]
    K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, Prog. Theor. Exp. Phys. 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].CrossRefGoogle Scholar
  37. [37]
    E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].ADSGoogle Scholar
  38. [38]
    B. Haghighat, A. Iqbal, C. Kozcaz, G. Lockhart and C. Vafa, M-strings, arXiv:1305.6322 [INSPIRE].
  39. [39]
    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
  40. [40]
    S. Hohenegger and A. Iqbal, M-strings, elliptic genera and N = 4 string amplitudes, Fortsch. Phys. 62 (2014) 155 [arXiv:1310.1325] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  41. [41]
    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].MathSciNetGoogle Scholar
  42. [42]
    E. Perevalov and G. Rajesh, Mirror symmetry via deformation of bundles on K3 surfaces, Phys. Rev. Lett. 79 (1997) 2931 [hep-th/9706005] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. [43]
    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].CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    M. Porrati and A. Zaffaroni, M theory origin of mirror symmetry in three-dimensional gauge theories, Nucl. Phys. B 490 (1997) 107 [hep-th/9611201] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  45. [45]
    A. Dey and J. Distler, Three dimensional mirror symmetry and partition function on S 3, JHEP 10 (2013) 086 [arXiv:1301.1731] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  46. [46]
    S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
  47. [47]
    D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  48. [48]
    C. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990) 713.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [49]
    N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    R. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley, New York U.S.A. (1985).zbMATHGoogle Scholar
  51. [51]
    O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6D N = (2, 0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  52. [52]
    O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E 6 theory, arXiv:1403.4604 [INSPIRE].
  53. [53]
    J.J. Heckman and C. Vafa, An exceptional sector for F-theory GUTs, Phys. Rev. D 83 (2011) 026006 [arXiv:1006.5459] [INSPIRE].ADSGoogle Scholar
  54. [54]
    J.J. Heckman, Y. Tachikawa, C. Vafa and B. Wecht, N = 1 SCFTs from brane monodromy, JHEP 11 (2010) 132 [arXiv:1009.0017] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  55. [55]
    J.J. Heckman, C. Vafa and B. Wecht, The conformal sector of F-theory GUTs, JHEP 07 (2011) 075 [arXiv:1103.3287] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  56. [56]
    J.J. Heckman and S.-J. Rey, Baryon and dark matter genesis from strongly coupled strings, JHEP 06 (2011) 120 [arXiv:1102.5346] [INSPIRE].CrossRefADSGoogle Scholar
  57. [57]
    J.J. Heckman, P. Kumar and B. Wecht, Oblique electroweak parameters S and T for superconformal field theories, Phys. Rev. D 88 (2013) 065016 [arXiv:1212.2979] [INSPIRE].ADSGoogle Scholar
  58. [58]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  59. [59]
    A. Hanany and A. Zaffaroni, Issues on orientifolds: on the brane construction of gauge theories with SO(2N ) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  60. [60]
    S. Ferrara, A. Kehagias, H. Partouche and A. Zaffaroni, Membranes and five-branes with lower supersymmetry and their AdS supergravity duals, Phys. Lett. B 431 (1998) 42 [hep-th/9803109] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  61. [61]
    C.-H. Ahn, K. Oh and R. Tatar, Orbifolds of AdS 7 × S 4 and six-dimensional (0, 1) SCFT, Phys. Lett. B 442 (1998) 109 [hep-th/9804093] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  62. [62]
    M. Berkooz, A supergravity dual of a (1, 0) field theory in six-dimensions, Phys. Lett. B 437 (1998) 315 [hep-th/9802195] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  63. [63]
    F. Apruzzi, M. Fazzi, D. Rosa and A. Tomasiello, All AdS 7 solutions of type-II supergravity, JHEP 04 (2014) 064 [arXiv:1309.2949] [INSPIRE].CrossRefADSGoogle Scholar
  64. [64]
    J. Polchinski and E. Witten, Evidence for heterotic type-I string duality, Nucl. Phys. B 460 (1996) 525 [hep-th/9510169] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  65. [65]
    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].CrossRefzbMATHMathSciNetGoogle Scholar
  66. [66]
    P. Slodowy, Simple singularities and simple algebraic groups, Springer Verlag, Germany (1980).zbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Michele Del Zotto
    • 1
  • Jonathan J. Heckman
    • 1
    • 2
    Email author
  • Alessandro Tomasiello
    • 3
  • Cumrun Vafa
    • 1
  1. 1.Jefferson Physical LaboratoryHarvard UniversityCambridgeU.S.A.
  2. 2.Department of PhysicsUniversity of North CarolinaChapel HillU.S.A.
  3. 3.Dipartimento di FisicaUniversità di Milano BicoccaMilanItaly

Personalised recommendations