Chiral anomalies on a circle and their cancellation in F-theory

  • Pierre Corvilain
  • Thomas W. Grimm
  • Diego Regalado
Open Access
Regular Article - Theoretical Physics


We study in detail how four-dimensional local anomalies manifest themselves when the theory is compactified on a circle. By integrating out the Kaluza-Klein modes in a way that preserves the four-dimensional symmetries in the UV, we show that the three-dimensional theory contains field-dependent Chern-Simons terms that appear at one-loop. These vanish if and only if the four-dimensional anomaly is canceled, so the anomaly is not lost upon compactification. We extend this analysis to situations where anomalies are canceled through a Green-Schwarz mechanism. We then use these results to show automatic cancellation of local anomalies in F-theory compactifications that can be obtained as a limit of M-theory on a smooth Calabi-Yau fourfold with background flux.


Anomalies in Field and String Theories F-Theory Effective Field Theories Superstring Vacua 


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]
    A. Bilal, Lectures on anomalies, arXiv:0802.0634 [INSPIRE].
  2. [2]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    F.G. Marchesano Buznego, Intersecting D-brane models, Ph.D. thesis, Madrid, Autonoma U., 2003. hep-th/0307252 [INSPIRE].
  4. [4]
    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
  5. [5]
    E. Plauschinn, The Generalized Green-Schwarz Mechanism for Type IIB Orientifolds with D3- and D7-branes, JHEP 05 (2009) 062 [arXiv:0811.2804] [INSPIRE].ADSMathSciNetCrossRefGoogle 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]
    D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    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
  9. [9]
    M. Bies, C. Mayrhofer and T. Weigand, Algebraic cycles and local anomalies in F-theory, JHEP 11 (2017) 100 [arXiv:1706.08528] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F. Denef, Les Houches Lectures on Constructing String Vacua, in the proceedings of the Summer School in Theoretical Physics, 87th Session, July 2-27, Les Houched, France, (2008), arXiv:0803.1194 [INSPIRE].
  11. [11]
    T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    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
  13. [13]
    T.W. Grimm, A. Kapfer and D. Klevers, The arithmetic of elliptic fibrations in gauge theories on a circle, JHEP 06 (2016) 112 [arXiv:1510.04281] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    T.W. Appelquist, M.J. Bowick, D. Karabali and L.C.R. Wijewardhana, Spontaneous chiral symmetry breaking in three-dimensional QED, Phys. Rev. D 33 (1986) 3704 [INSPIRE].ADSGoogle Scholar
  18. [18]
    E. Poppitz and M. Ünsal, Index theorem for topological excitations on R 3 × S 1 and Chern-Simons theory, JHEP 03 (2009) 027 [arXiv:0812.2085] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    S. Golkar and D.T. Son, (Non)-renormalization of the chiral vortical effect coefficient, JHEP 02 (2015) 169 [arXiv:1207.5806] [INSPIRE].
  20. [20]
    L. Di Pietro and M. Honda, Cardy formula for 4d SUSY theories and localization, JHEP 04 (2017) 055 [arXiv:1611.00380] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M.J. Duff and D.J. Toms, Kaluza-Klein kounterterms, in the proceedings of the Unification of the fundamental particle interactions, Europhysics Study Conference, October 6-14, Erice, Italy (1982).Google Scholar
  22. [22]
    M.J. Duff and D.J. Toms, Divergences and anomalies in Kaluza-Klein theories, Moscow Quant. Grav. (1981) 0431.Google Scholar
  23. [23]
    L. Álvarez-Gaumé and P.H. Ginsparg, The structure of gauge and gravitational anomalies, Annals Phys. 161 (1985) 423 [Erratum ibid. 171 (1986) 233] [INSPIRE].
  24. [24]
    R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    F. Bonetti, T.W. Grimm and S. Hohenegger, One-loop Chern-Simons terms in five dimensions, JHEP 07 (2013) 043 [arXiv:1302.2918] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    P. Anastasopoulos, M. Bianchi, E. Dudas and E. Kiritsis, Anomalies, anomalous U(1)’s and generalized Chern-Simons terms, JHEP 11 (2006) 057 [hep-th/0605225] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    K. Becker and M. Becker, M theory on eight manifolds, Nucl. Phys. B 477 (1996) 155 [hep-th/9605053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    M. Haack and J. Louis, Duality in heterotic vacua with four supercharges, Nucl. Phys. B 575 (2000) 107 [hep-th/9912181] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Haack and J. Louis, M theory compactified on Calabi-Yau fourfolds with background flux, Phys. Lett. B 507 (2001) 296 [hep-th/0103068] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    D. R. Morrison and G. Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984) 15.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    A. Hanany and B. Kol, On orientifolds, discrete torsion, branes and M-theory, JHEP 06 (2000) 013 [hep-th/0003025] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    R. Donagi, S. Katz and E. Sharpe, Spectra of D-branes with Higgs vevs, Adv. Theor. Math. Phys. 8 (2004) 813 [hep-th/0309270] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-branes and monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    L.B. Anderson, J.J. Heckman and S. Katz, T-branes and geometry, JHEP 05 (2014) 080 [arXiv:1310.1931] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Collinucci and R. Savelli, F-theory on singular spaces, JHEP 09 (2015) 100 [arXiv:1410.4867] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    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
  38. [38]
    E. Cremmer, B. Julia and J. Scherk, Supergravity theory in eleven-dimensions, Phys. Lett. B 76 (1978) 409.ADSCrossRefGoogle Scholar
  39. [39]
    E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    P. Corvilain, T.W. Grimm and D. Regalado, Shift-symmetries and gauge coupling functions in orientifolds and F-theory, JHEP 05 (2017) 059 [arXiv:1607.03897] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Collinucci and R. Savelli, On flux quantization in F-theory, JHEP 02 (2012) 015 [arXiv:1011.6388] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    A. Collinucci and R. Savelli, On flux quantization in F-theory II: unitary and symplectic gauge groups, JHEP 08 (2012) 094 [arXiv:1203.4542] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    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
  44. [44]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    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
  46. [46]
    T.W. Grimm, T.G. Pugh and D. Regalado, Non-Abelian discrete gauge symmetries in F-theory, JHEP 02 (2016) 066 [arXiv:1504.06272] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Pierre Corvilain
    • 1
  • Thomas W. Grimm
    • 1
  • Diego Regalado
    • 2
  1. 1.Institute for Theoretical Physics and Center for Extreme Matter and Emergent PhenomenaUtrecht UniversityUtrechtThe Netherlands
  2. 2.Theoretical Physics DepartmentCERNGenevaSwitzerland

Personalised recommendations