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Journal of High Energy Physics

, 2017:100 | Cite as

Algebraic cycles and local anomalies in F-theory

  • Martin Bies
  • Christoph Mayrhofer
  • Timo Weigand
Open Access
Regular Article - Theoretical Physics

Abstract

We introduce a set of identities in the cohomology ring of elliptic fibrations which are equivalent to the cancellation of gauge and mixed gauge-gravitational anomalies in F-theory compactifications to four and six dimensions. The identities consist in (co)homological relations between complex codimension-two cycles. The same set of relations, once evaluated on elliptic Calabi-Yau three-folds and four-folds, is shown to universally govern the structure of anomalies and their Green-Schwarz cancellation in six- and four-dimensional F-theory vacua, respectively. We furthermore conjecture that these relations hold not only within the cohomology ring, but even at the level of the Chow ring, i.e. as relations among codimension-two cycles modulo rational equivalence. We verify this conjecture in non-trivial examples with Abelian and non-Abelian gauge groups factors. Apart from governing the structure of local anomalies, the identities in the Chow ring relate different types of gauge backgrounds on elliptically fibred Calabi-Yau four-folds.

Keywords

Anomalies in Field and String Theories F-Theory Differential and Algebraic Geometry Flux compactifications 

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]
    M.B. Green and J.H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    M.B. Green and J.H. Schwarz, Infinity Cancellations in SO(32) Superstring Theory, Phys. Lett. B 151 (1985) 21 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    M.B. Green and J.H. Schwarz, The Hexagon Gauge Anomaly in Type I Superstring Theory, Nucl. Phys. B 255 (1985) 93 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Polchinski and Y. Cai, Consistency of Open Superstring Theories, Nucl. Phys. B 296 (1988) 91 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Sagnotti, Some properties of open string theories, in proceedings of the International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY 95), Palaiseau, France, 15-19 May 1995, pp. 473-484 [hep-th/9509080] [INSPIRE].
  6. [6]
    R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    L.E. Ibanez and A.M. Uranga, String theory and particle physics: An introduction to string phenomenology, Cambridge University Press (2012).Google Scholar
  8. [8]
    V. Kumar and W. Taylor, String Universality in Six Dimensions, Adv. Theor. Math. Phys. 15 (2011) 325 [arXiv:0906.0987] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    V. Kumar, D.R. Morrison and W. Taylor, Mapping 6D N = 1 supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    N. Seiberg and W. Taylor, Charge Lattices and Consistency of 6D Supergravity, JHEP 06 (2011) 001 [arXiv:1103.0019] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    T.W. Grimm and W. Taylor, Structure in 6D and 4D N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    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
  15. [15]
    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
  16. [16]
    V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, math.AG/0005196 [INSPIRE].
  18. [18]
    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
  19. [19]
    D.S. Park, Anomaly Equations and Intersection Theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Bonetti and T.W. Grimm, Six-dimensional (1, 0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    M. Cvetič, T.W. Grimm and D. Klevers, Anomaly Cancellation And Abelian Gauge Symmetries In F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Esole and S.-H. Shao, M-theory on Elliptic Calabi-Yau Threefolds and 6d Anomalies, arXiv:1504.01387 [INSPIRE].
  25. [25]
    L. Lin and T. Weigand, G 4 -flux and standard model vacua in F-theory, Nucl. Phys. B 913 (2016) 209 [arXiv:1604.04292] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Bies, C. Mayrhofer, C. Pehle and T. Weigand, Chow groups, Deligne cohomology and massless matter in F-theory, arXiv:1402.5144 [INSPIRE].
  27. [27]
    M. Bies, C. Mayrhofer and T. Weigand, Gauge Backgrounds and Zero-Mode Counting in F-theory, arXiv:1706.04616 [INSPIRE].
  28. [28]
    S. Krause, C. Mayrhofer and T. Weigand, Gauge Fluxes in F-theory and Type IIB Orientifolds, JHEP 08 (2012) 119 [arXiv:1202.3138] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    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
  30. [30]
    T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972) 20.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    J.T. Tate, Algebraic cycles and poles of zeta functions, in Arithmetical Algebraic Geometry (Proceedings of a Conference held at Purdue University, Dec. 1963), Harper & Row, New York U.S.A. (1965), pp. 93-110.Google Scholar
  32. [32]
    J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki 9 (1964-1966) 415.Google Scholar
  33. [33]
    R. Wazir, Arithmetic on Elliptic Threefolds, math.NT/0112259.
  34. [34]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    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
  36. [36]
    M. Esole, S.-H. Shao and S.-T. Yau, Singularities and Gauge Theory Phases, Adv. Theor. Math. Phys. 19 (2015) 1183 [arXiv:1402.6331] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys. 173 (1995) 559 [hep-th/9402119] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    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
  39. [39]
    A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl. Phys. B 856 (2012) 129 [arXiv:1107.5337] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  40. [40]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A.P. Braun, A. Collinucci and R. Valandro, Hypercharge flux in F-theory and the stable Sen limit, JHEP 07 (2014) 121 [arXiv:1402.4096] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) Tops with Multiple U (1)s in F-theory, Nucl. Phys. B 882 (2014) 1 [arXiv:1307.2902] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    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
  44. [44]
    S. Krause, C. Mayrhofer and T. Weigand, G 4 -flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    W. Fulton, Intersection Theory, Princeton University Press (1993).Google Scholar
  48. [48]
    D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate Studies in Mathematics, American Mathematical Society (2011).Google Scholar
  49. [49]
    L. Lin, C. Mayrhofer, O. Till and T. Weigand, Fluxes in F-theory Compactifications on Genus-One Fibrations, JHEP 01 (2016) 098 [arXiv:1508.00162] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    S. Schäfer-Nameki and T. Weigand, F-theory and 2d (0, 2) theories, JHEP 05 (2016) 059 [arXiv:1601.02015] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, UV Completions for Non-Critical Strings, JHEP 07 (2016) 045 [arXiv:1602.04221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    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].ADSGoogle Scholar
  53. [53]
    C. Lawrie, S. Schäfer-Nameki and T. Weigand, The gravitational sector of 2d (0, 2) F-theory vacua, JHEP 05 (2017) 103 [arXiv:1612.06393] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP 04 (2017) 111 [arXiv:1612.05640] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    A.M. Uranga, D-brane probes, RR tadpole cancellation and k-theory charge, Nucl. Phys. B 598 (2001) 225 [hep-th/0011048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    I. Garcia-Etxebarria and A.M. Uranga, From F/M-theory to k-theory and back, JHEP 02 (2006) 008 [hep-th/0510073] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikRuprecht-Karls-UniversitätHeidelbergGermany
  2. 2.Arnold Sommerfeld Center for Theoretical PhysicsMünchenGermany
  3. 3.CERN, Theory DivisionGeneva 23Switzerland

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