Abstract
We study 4D F-theory compactifications on singular Calabi-Yau fourfolds with fluxes. The resulting \(\mathcal{N}=1\) effective theories can admit non-Abelian and U(1) gauge groups as well as charged chiral matter. In these setups we analyze anomaly cancellation and the generalized Green-Schwarz mechanism. This requires the study of 3D \(\mathcal{N}=2\) theories obtained by a circle compactification and their M-theory duals. Reducing M- theory on resolved Calabi-Yau fourfolds corresponds to considering the effective theory on the 3D Coulomb branch in which certain massive states are integrated out. Both 4D gaugings and 3D one-loop corrections of these massive states induce Chern-Simons terms. All 4D anomalies are captured by the one-loop terms. The ones corresponding to the mixed gauge-gravitational anomalies depend on the Kaluza-Klein vector and are induced by integrating out Kaluza-Klein modes of the U(1) charged matter. In M-theory all Chern-Simons terms classically arise from G 4-flux. We find that F-theory fluxes implement automatically the 4D Green-Schwarz mechanism if non-trivial geometric relations for the resolved Calabi-Yau fourfold are satisfied. We confirm these relations in various explicit examples and elucidate the general construction of U(1) symmetries in F-theory. We also compare anomaly cancellation in F-theory with its analog in Type IIB orientifold setups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Weinberg, The quantum theory of fields. Volume 2, Cambridge University Press, Cambridge U.K. (1996), chapter 22.
J. Zinn-Justin, Chiral anomalies and topology, Lect. Notes Phys. 659 (2005) 167 [hep-th/0201220] [INSPIRE].
J.A. Harvey, TASI 2003 lectures on anomalies, hep-th/0509097 [INSPIRE].
A. Bilal, Lectures on anomalies, arXiv:0802.0634 [INSPIRE].
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].
A. Sagnotti, A note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
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].
T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].
R. Donagi and M. Wijnholt, Model building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory — II: experimental predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].
H. Hayashi, R. Tatar, Y. Toda, T. Watari and M. Yamazaki, New aspects of heterotic-F theory duality, Nucl. Phys. B 806 (2009) 224 [arXiv:0805.1057] [INSPIRE].
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].
J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yau’s, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].
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].
S. Krause, C. Mayrhofer and T. Weigand, Gauge fluxes in F-theory and type IIB orientifolds, JHEP 08 (2012) 119 [arXiv:1202.3138] [INSPIRE].
M. Kuntzler and S. Schäfer-Nameki, G-flux and spectral divisors, JHEP 11 (2012) 025 [arXiv:1205.5688] [INSPIRE].
M. Cvetič, G. Shiu and A.M. Uranga, Chiral four-dimensional N = 1 supersymmetric type 2A orientifolds from intersecting D6 branes, Nucl. Phys. B 615 (2001) 3 [hep-th/0107166] [INSPIRE].
F.G. Marchesano Buznego, Intersecting D-brane models, hep-th/0307252 [INSPIRE].
R. Blumenhagen, M. Cvetič, P. Langacker and G. Shiu, Toward realistic intersecting D-brane models, Ann. Rev. Nucl. Part. Sci. 55 (2005) 71 [hep-th/0502005] [INSPIRE].
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].
E. Plauschinn, The generalized Green-Schwarz mechanism for type IIB orientifolds with D3- and D7-branes, JHEP 05 (2009) 062 [arXiv:0811.2804] [INSPIRE].
M. Cvetič and J. Halverson, TASI lectures: particle physics from perturbative and non-perturbative effects in D-braneworlds, arXiv:1101.2907 [INSPIRE].
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].
D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
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].
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].
J. Marsano, Hypercharge flux, exotics and anomaly cancellation in F-theory GUTs, Phys. Rev. Lett. 106 (2011) 081601 [arXiv:1011.2212] [INSPIRE].
E. Dudas and E. Palti, On hypercharge flux and exotics in F-theory GUTs, JHEP 09 (2010) 013 [arXiv:1007.1297] [INSPIRE].
E. Palti, A note on hypercharge flux, anomalies and U(1)s in F-theory GUTs, arXiv:1209.4421 [INSPIRE].
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].
L. Álvarez-Gaumé and E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
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].
P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].
P. Candelas, E. Perevalov and G. Rajesh, Toric geometry and enhanced gauge symmetry of F-theory/heterotic vacua, Nucl. Phys. B 507 (1997) 445 [hep-th/9704097] [INSPIRE].
R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].
T.W. Grimm, S. Krause and T. Weigand, F-theory GUT vacua on compact Calabi-Yau fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [INSPIRE].
M. Cvetič, I. Garcia-Etxebarria and J. Halverson, Global F-theory models: instantons and gauge dynamics, JHEP 01 (2011) 073 [arXiv:1003.5337] [INSPIRE].
C.-M. Chen, J. Knapp, M. Kreuzer and C. Mayrhofer, Global SO(10) F-theory GUTs, JHEP 10 (2010) 057 [arXiv:1005.5735] [INSPIRE].
J. Knapp, M. Kreuzer, C. Mayrhofer and N.-O. Walliser, Toric construction of global F-theory GUTs, JHEP 03 (2011) 138 [arXiv:1101.4908] [INSPIRE].
V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP 01 (2013) 016 [arXiv:1110.4883] [INSPIRE].
M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, arXiv:1107.0733 [INSPIRE].
T.W. Grimm and R. Savelli, Gravitational instantons and fluxes from M/F-theory on Calabi-Yau fourfolds, Phys. Rev. D 85 (2012) 026003 [arXiv:1109.3191] [INSPIRE].
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].
T. Shioda, Mordell-Weil lattices and Galois representation. I, P. Jpn. Acad. 65A (1989) 268.
T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul 39 (1990) 211.
T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].
E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].
A. Collinucci and R. Savelli, On flux quantization in F-theory, JHEP 02 (2012) 015 [arXiv:1011.6388] [INSPIRE].
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].
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].
P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four folds, Nucl. Phys. B 494 (1997) 489 [hep-th/9610162] [INSPIRE].
A. Klemm, B. Lian, S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys. B 518 (1998) 515 [hep-th/9701023] [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477-478] [hep-th/9906070] [INSPIRE].
D. Lüst, P. Mayr, S. Reffert and S. Stieberger, F-theory flux, destabilization of orientifolds and soft terms on D7-branes, Nucl. Phys. B 732 (2006) 243 [hep-th/0501139] [INSPIRE].
T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing brane and flux superpotentials in F-theory compactifications, JHEP 04 (2010) 015 [arXiv:0909.2025] [INSPIRE].
T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Five-brane superpotentials and heterotic/F-theory duality, Nucl. Phys. B 838 (2010) 458 [arXiv:0912.3250] [INSPIRE].
H. Jockers, P. Mayr and J. Walcher, On N = 1 4D effective couplings for F-theory and Heterotic Vacua, Adv. Theor. Math. Phys. 14 (2010) 1433 [arXiv:0912.3265] [INSPIRE].
M. Alim et al., Type II/F-theory superpotentials with several deformations and N = 1 mirror symmetry, JHEP 06 (2011) 103 [arXiv:1010.0977] [INSPIRE].
D. Klevers, Holomorphic couplings in non-perturbative string compactifications, Fortsch. Phys. 60 (2012) 3 [arXiv:1106.6259] [INSPIRE].
K. Intriligator, H. Jockers, P. Mayr, D.R. Morrison and M.R. Plesser, Conifold transitions in M-theory on Calabi-Yau fourfolds with background fluxes, arXiv:1203.6662 [INSPIRE].
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].
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].
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
H.-C. Kao, K.-M. Lee and T. Lee, The Chern-Simons coefficient in supersymmetric Yang-Mills Chern-Simons theories, Phys. Lett. B 373 (1996) 94 [hep-th/9506170] [INSPIRE].
A.J. Niemi and G.W. Semenoff, Axial anomaly induced fermion fractionization and effective gauge theory actions in odd dimensional space-times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].
A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
F. Bonetti, T.W. Grimm and S. Hohenegger, A Kaluza-Klein inspired action for chiral p-forms and their anomalies, arXiv:1206.1600 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [INSPIRE].
H. Jockers and J. Louis, The effective action of D7-branes in N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 705 (2005) 167 [hep-th/0409098] [INSPIRE].
H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys. B 718 (2005) 203 [hep-th/0502059] [INSPIRE].
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].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1210.6034
Rights and permissions
About this article
Cite this article
Cvetič, M., Grimm, T.W. & Klevers, D. Anomaly cancellation and abelian gauge symmetries in F-theory. J. High Energ. Phys. 2013, 101 (2013). https://doi.org/10.1007/JHEP02(2013)101
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2013)101