Matter in transition

  • Lara B. AndersonEmail author
  • James Gray
  • Nikhil Raghuram
  • Washington Taylor
Open Access
Regular Article - Theoretical Physics


We explore a novel type of transition in certain 6D and 4D quantum field theories, in which the matter content of the theory changes while the gauge group and other parts of the spectrum remain invariant. Such transitions can occur, for example, for SU(6) and SU(7) gauge groups, where matter fields in a three-index antisymmetric representation and the fundamental representation are exchanged in the transition for matter in the two-index antisymmetric representation. These matter transitions are realized by passing through superconformal theories at the transition point. We explore these transitions in dual F-theory and heterotic descriptions, where a number of novel features arise. For example, in the heterotic description the relevant 6D SU(7) theories are described by bundles on K3 surfaces where the geometry of the K3 is constrained in addition to the bundle structure. On the F-theory side, non-standard representations such as the three-index antisymmetric representation of SU(N) require Weierstrass models that cannot be realized from the standard SU(N) Tate form. We also briefly describe some other situations, with groups such as Sp(3), SO(12), and SU(3), where analogous matter transitions can occur between different representations. For SU(3), in particular, we find a matter transition between adjoint matter and matter in the symmetric representation, giving an explicit Weierstrass model for the F-theory description of the symmetric representation that complements another recent analogous construction.


F-Theory Superstring Vacua Superstrings and Heterotic 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]
    N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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].
  3. [3]
    S. Kachru and E. Silverstein, Chirality changing phase transitions in 4 − D string vacua, Nucl. Phys. B 504 (1997) 272 [hep-th/9704185] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    B.A. Ovrut, T. Pantev and J. Park, Small instanton transitions in heterotic M-theory, JHEP 05 (2000) 045 [hep-th/0001133] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M.B. Green, J.H. Schwarz and P.C. West, Anomaly Free Chiral Theories in Six-Dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Sagnotti, A Note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D \( \mathcal{N}=1 \) supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Donagi, B.A. Ovrut and D. Waldram, Moduli spaces of five-branes on elliptic Calabi-Yau threefolds, JHEP 11 (1999) 030 [hep-th/9904054] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. Buchbinder, R. Donagi and B.A. Ovrut, Vector bundle moduli and small instanton transitions, JHEP 06 (2002) 054 [hep-th/0202084] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D.R. Morrison, TASI lectures on compactification and duality, hep-th/0411120 [INSPIRE].
  16. [16]
    F. Denef, Les Houches Lectures on Constructing String Vacua, arXiv:0803.1194 [INSPIRE].
  17. [17]
    W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, arXiv:1104.2051 [INSPIRE].
  18. [18]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    D.R. Morrison and W. Taylor, Non-Higgsable clusters for 4D F-theory models, JHEP 05 (2015) 080 [arXiv:1412.6112] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.CrossRefzbMATHGoogle Scholar
  23. [23]
    K. Kodaira, K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    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].ADSCrossRefzbMATHGoogle Scholar
  25. [25]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].ADSzbMATHGoogle Scholar
  26. [26]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    C. Lawrie and S. Schäfer-Nameki, The Tate Form on Steroids: Resolution and Higher Codimension Fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    H. Hayashi, C. Lawrie and S. Schäfer-Nameki, Phases, Flops and F-theory: SU(5) Gauge Theories, JHEP 10 (2013) 046 [arXiv:1304.1678] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box Graphs and Singular Fibers, JHEP 05 (2014) 048 [arXiv:1402.2653] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    M. Esole, S.-H. Shao and S.-T. Yau, Singularities and Gauge Theory Phases, arXiv:1402.6331 [INSPIRE].
  32. [32]
    A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions I, Nucl. Phys. B 905 (2016) 447 [arXiv:1407.3520] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Grassi, J. Halverson and J.L. Shaneson, Matter From Geometry Without Resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, JHEP 06 (2015) 061 [arXiv:1404.6300] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Bershadsky and A. Johansen, Colliding singularities in F-theory and phase transitions, Nucl. Phys. B 489 (1997) 122 [hep-th/9610111] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    N. Seiberg, Nontrivial fixed points of the renormalization group in six-dimensions, Phys. Lett. B 390 (1997) 169 [hep-th/9609161] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    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. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
  40. [40]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  41. [41]
    B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of Minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    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
  43. [43]
    J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Geometry of 6D RG Flows, JHEP 09 (2015) 052 [arXiv:1505.00009] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  44. [44]
    A. Grassi, J. Halverson and J.L. Shaneson, Non-Abelian Gauge Symmetry and the Higgs Mechanism in F-theory, Commun. Math. Phys. 336 (2015) 1231 [arXiv:1402.5962] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology, Cambridge Monographs On Mathematical Physics, Cambridge University Press, Cambridge U.K. (1987).Google Scholar
  46. [46]
    F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, Heidelberg Germany (1995).zbMATHGoogle Scholar
  47. [47]
    M. Dine, N. Seiberg and E. Witten, Fayet-Iliopoulos Terms in String Theory, Nucl. Phys. B 289 (1987) 589 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    E.R. Sharpe, Kähler cone substructure, Adv. Theor. Math. Phys. 2 (1999) 1441 [hep-th/9810064] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    A. Lukas and K.S. Stelle, Heterotic anomaly cancellation in five-dimensions, JHEP 01 (2000) 010 [hep-th/9911156] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    R. Blumenhagen, G. Honecker and T. Weigand, Loop-corrected compactifications of the heterotic string with line bundles, JHEP 06 (2005) 020 [hep-th/0504232] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, The Edge Of Supersymmetry: Stability Walls in Heterotic Theory, Phys. Lett. B 677 (2009) 190 [arXiv:0903.5088] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stability Walls in Heterotic Theories, JHEP 09 (2009) 026 [arXiv:0905.1748] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    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].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    J. Li and S.-T. Yau, The Existence of supersymmetric string theory with torsion, J. Diff. Geom. 70 (2005) 143 [hep-th/0411136] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  55. [55]
    R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    G. Curio and R.Y. Donagi, Moduli in N = 1 heterotic/F theory duality, Nucl. Phys. B 518 (1998) 603 [hep-th/9801057] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    P.S. Aspinwall, Aspects of the hypermultiplet moduli space in string duality, JHEP 04 (1998) 019 [hep-th/9802194] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    R. Donagi, S. Katz and M. Wijnholt, Weak Coupling, Degeneration and Log Calabi-Yau Spaces, arXiv:1212.0553 [INSPIRE].
  59. [59]
    R. Friedman and J. Morgan, Smooth Four-Manifolds and Complex Surfaces, Springer, Heidelberg Germany (1994).CrossRefzbMATHGoogle Scholar
  60. [60]
    R. Friedman, J.W. Morgan and E. Witten, Vector bundles over elliptic fibrations, alg-geom/9709029 [INSPIRE].
  61. [61]
    R.Y. Donagi, Principal bundles on elliptic fibrations, Asian J. Math. 1 (1997) 214 [alg-geom/9702002] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    R. Donagi, Spectral Covers, in Mathematical Sciences Research Institute Publications. Book 28: Current topics in complex algebraic geometry, Cambridge University Press, Cambridge U.K. (1992), pg. 65 [alg-geom/9505009].
  63. [63]
    R. Donagi, Heterotic/F theory duality: ICMP lecture, hep-th/9802093 [INSPIRE].
  64. [64]
    B.A. Ovrut, T. Pantev and J. Park, Small instanton transitions in heterotic M-theory, JHEP 05 (2000) 045 [hep-th/0001133] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    G. Curio, Chiral matter and transitions in heterotic string models, Phys. Lett. B 435 (1998) 39 [hep-th/9803224] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    C. Lazaroiu, On degree zero semistable bundles over an elliptic curve, physics/9712054.
  68. [68]
    K.-S. Choi and H. Hayashi, U(n) Spectral Covers from Decomposition, JHEP 06 (2012) 009 [arXiv:1203.3812] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    L.B. Anderson, J.J. Heckman and S. Katz, T-Branes and Geometry, JHEP 05 (2014) 080 [arXiv:1310.1931] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    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
  71. [71]
    H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys. B 718 (2005) 203 [hep-th/0502059] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    M. Buican, D. Malyshev, D.R. Morrison, H. Verlinde and M. Wijnholt, D-branes at Singularities, Compactification and Hypercharge, JHEP 01 (2007) 107 [hep-th/0610007] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    T.W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [INSPIRE].ADSGoogle Scholar
  74. [74]
    T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive Abelian Gauge Symmetries and Fluxes in F-theory, JHEP 12 (2011) 004 [arXiv:1107.3842] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    A.P. Braun, A. Collinucci and R. Valandro, The fate of U(1)’s at strong coupling in F-theory, JHEP 07 (2014) 028 [arXiv:1402.4054] [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    L.B. Anderson, I. García-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    I. García-Etxebarria, T.W. Grimm and J. Keitel, Yukawas and discrete symmetries in F-theory compactifications without section, JHEP 11 (2014) 125 [arXiv:1408.6448] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    M. Cvetič, A. Grassi, D. Klevers, M. Poretschkin and P. Song, Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality, JHEP 04 (2016) 041 [arXiv:1511.08208] [INSPIRE].CrossRefGoogle Scholar
  79. [79]
    J. Gray, Y.-H. He, A. Ilderton and A. Lukas, STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology, Comput. Phys. Commun. 180 (2009) 107 [arXiv:0801.1508] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    M. Cvetič, D. Klevers, H. Piragua and W. Taylor, General U(1) × U(1) F-theory compactifications and beyond: geometry of unHiggsings and novel matter structure, JHEP 11 (2015) 204 [arXiv:1507.05954] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    M.R. Douglas and C.-g. Zhou, Chirality change in string theory, JHEP 06 (2004) 014 [hep-th/0403018] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  82. [82]
    M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, The Web of D-branes at Singularities in Compact Calabi-Yau Manifolds, JHEP 05 (2013) 114 [arXiv:1304.2771] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  83. [83]
    P.S. Aspinwall and M. Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B 387 (1996) 735 [hep-th/9605131] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  84. [84]
    V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  85. [85]
    L.B. Anderson, A. Constantin, S.-J. Lee and A. Lukas, Hypercharge Flux in Heterotic Compactifications, Phys. Rev. D 91 (2015) 046008 [arXiv:1411.0034] [INSPIRE].ADSGoogle Scholar
  86. [86]
    L.B. Anderson, A. Constantin, J. Gray, A. Lukas and E. Palti, A Comprehensive Scan for Heterotic SU(5) GUT models, JHEP 01 (2014) 047 [arXiv:1307.4787] [INSPIRE].ADSCrossRefGoogle Scholar
  87. [87]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic Line Bundle Standard Models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  88. [88]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds, Phys. Rev. D 84 (2011) 106005 [arXiv:1106.4804] [INSPIRE].ADSGoogle Scholar
  89. [89]
    V. Kumar and W. Taylor, String Universality in Six Dimensions, Adv. Theor. Math. Phys. 15 (2011) 325 [arXiv:0906.0987] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    R. Blumenhagen, S. Moster and T. Weigand, Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys. B 751 (2006) 186 [hep-th/0603015] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    M. Kuriyama, H. Nakajima and T. Watari, Theoretical Framework for R-parity Violation, Phys. Rev. D 79 (2009) 075002 [arXiv:0802.2584] [INSPIRE].ADSGoogle Scholar
  92. [92]
    L.B. Anderson, J. Gray and B. Ovrut, Yukawa Textures From Heterotic Stability Walls, JHEP 05 (2010) 086 [arXiv:1001.2317] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  93. [93]
    L.B. Anderson, J. Gray and B.A. Ovrut, Transitions in the Web of Heterotic Vacua, Fortsch. Phys. 59 (2011) 327 [arXiv:1012.3179] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    R. Hartshorne, Algebraic Geometry, Springer-Verlag, Heidelberg Germany (1977).CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Lara B. Anderson
    • 1
    Email author
  • James Gray
    • 1
  • Nikhil Raghuram
    • 2
  • Washington Taylor
    • 2
  1. 1.Department of PhysicsVirginia TechBlacksburgU.S.A.
  2. 2.Center for Theoretical Physics, Department of PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.

Personalised recommendations