Learning non-Higgsable gauge groups in 4D F-theory

  • Yi-Nan WangEmail author
  • Zhibai Zhang
Open Access
Regular Article - Theoretical Physics


We apply machine learning techniques to solve a specific classification problem in 4D F-theory. For a divisor D on a given complex threefold base, we want to read out the non-Higgsable gauge group on it using local geometric information near D. The input features are the triple intersection numbers among divisors near D and the output label is the non-Higgsable gauge group. We use decision tree to solve this problem and achieved 85%-98% out-of-sample accuracies for different classes of divisors, where the data sets are generated from toric threefold bases without (4,6) curves. We have explicitly generated a large number of analytic rules directly from the decision tree and proved a small number of them. As a crosscheck, we applied these decision trees on bases with (4,6) curves as well and achieved high accuracies. Additionally, we have trained a decision tree to distinguish toric (4,6) curves as well. Finally, we present an application of these analytic rules to construct local base configurations with interesting gauge groups such as SU(3).


Differential and Algebraic Geometry F-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]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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].
  3. [3]
    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].
  4. [4]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
  5. [5]
    W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    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].
  7. [7]
    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].
  8. [8]
    J. Halverson and W. Taylor, ℙ1 -bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models, JHEP 09 (2015) 086 [arXiv:1506.03204] [INSPIRE].
  9. [9]
    W. Taylor and Y.-N. Wang, A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua, JHEP 01 (2016) 137 [arXiv:1510.04978] [INSPIRE].
  10. [10]
    J. Halverson, C. Long and B. Sung, Algorithmic universality in F-theory compactifications, Phys. Rev. D 96 (2017) 126006 [arXiv:1706.02299] [INSPIRE].
  11. [11]
    W. Taylor and Y.-N. Wang, Scanning the skeleton of the 4D F-theory landscape, JHEP 01 (2018) 111 [arXiv:1710.11235] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
  13. [13]
    D.R. Morrison and W. Taylor, Non-Higgsable clusters for 4D F-theory models, JHEP 05 (2015) 080 [arXiv:1412.6112] [INSPIRE].
  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].
  15. [15]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
  16. [16]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    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].
  18. [18]
    D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S.B. Johnson and W. Taylor, Enhanced gauge symmetry in 6D F-theory models and tuned elliptic Calabi-Yau threefolds, Fortsch. Phys. 64 (2016) 581 [arXiv:1605.08052] [INSPIRE].
  20. [20]
    M. Graña, Flux compactifications in string theory: a comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    F. Denef, Les Houches lectures on constructing string vacua, Les Houches 87 (2008) 483 [arXiv:0803.1194] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Type IIA moduli stabilization, JHEP 07 (2005) 066 [hep-th/0505160] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    B.S. Acharya and M.R. Douglas, A finite landscape?, hep-th/0606212 [INSPIRE].
  25. [25]
    A.P. Braun and T. Watari, Distribution of the number of generations in flux compactifications, Phys. Rev. D 90 (2014) 121901 [arXiv:1408.6156] [INSPIRE].
  26. [26]
    A.P. Braun and T. Watari, The vertical, the horizontal and the rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications, JHEP 01 (2015) 047 [arXiv:1408.6167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    T. Watari, Statistics of F-theory flux vacua for particle physics, JHEP 11 (2015) 065 [arXiv:1506.08433] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    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].
  29. [29]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
  30. [30]
    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].
  31. [31]
    Y.-H. He, Deep-learning the landscape, arXiv:1706.02714 [INSPIRE].
  32. [32]
    D. Krefl and R.-K. Seong, Machine learning of Calabi-Yau volumes, Phys. Rev. D 96 (2017) 066014 [arXiv:1706.03346] [INSPIRE].
  33. [33]
    F. Ruehle, Evolving neural networks with genetic algorithms to study the string landscape, JHEP 08 (2017) 038 [arXiv:1706.07024] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    J. Carifio, J. Halverson, D. Krioukov and B.D. Nelson, Machine learning in the string landscape, JHEP 09 (2017) 157 [arXiv:1707.00655] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Y.-H. He, Machine-learning the string landscape, Phys. Lett. B 774 (2017) 564 [INSPIRE].
  36. [36]
    K. Hashimoto, S. Sugishita, A. Tanaka and A. Tomiya, Deep learning and AdS/CFT, arXiv:1802.08313 [INSPIRE].
  37. [37]
    W. Fulton, Introduction to toric varieties, Ann. Math. 131, Princeton University Press, Princeton, U.S.A., (1993).Google Scholar
  38. [38]
    V.I. Danilov, The geometry of toric varieties, Russ. Math. Surv. 33 (1978) 97.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    D.R. Morrison, D.S. Park and W. Taylor, Non-Higgsable Abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces, arXiv:1610.06929 [INSPIRE].
  41. [41]
    Y.-N. Wang, Tuned and non-Higgsable U(1)s in F-theory, JHEP 03 (2017) 140 [arXiv:1611.08665] [INSPIRE].
  42. [42]
    F. Apruzzi, J.J. Heckman, D.R. Morrison and L. Tizzano, 4D gauge theories with conformal matter, arXiv:1803.00582 [INSPIRE].
  43. [43]
    P. Candelas, D.-E. Diaconescu, B. Florea, D.R. Morrison and G. Rajesh, Codimension three bundle singularities in F-theory, JHEP 06 (2002) 014 [hep-th/0009228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    F. Baume, E. Palti and S. Schwieger, On E 8 and F-theory GUTs, JHEP 06 (2015) 039 [arXiv:1502.03878] [INSPIRE].
  45. [45]
    P. Arras, A. Grassi and T. Weigand, Terminal singularities, Milnor numbers and matter in F-theory, J. Geom. Phys. 123 (2018) 71 [arXiv:1612.05646] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    P. Mehta et al., A high-bias, low-variance introduction to machine learning for physicists, arXiv:1803.08823 [INSPIRE].
  47. [47]
    F. Pedregosa et al., Scikit-learn: machine learning in Python, J. Mach. Learn. Res. 12 (2011) 2825.MathSciNetzbMATHGoogle Scholar
  48. [48]
    A. Grassi, J. Halverson, J. Shaneson and W. Taylor, Non-Higgsable QCD and the Standard Model spectrum in F-theory, JHEP 01 (2015) 086 [arXiv:1409.8295] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Center for Theoretical Physics, Department of PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.
  2. 2.Department of Finance and Risk Engineering, Tandon School of EngineeringNew York UniversityBrooklynU.S.A.

Personalised recommendations