Abstract
We construct six-dimensional (6D) F-theory models in which discrete ℤ5, ℤ4, ℤ3, and ℤ2 gauge symmetries arise. We demonstrate that a special family of “Fano 3-folds” is a useful tool for constructing the aforementioned models. The geometry of Fano 3-folds in the constructions of models can be useful for understanding discrete gauge symmetries in 6D F-theory compactifications. We argue that the constructions of the aforementioned models are applicable to Calabi-Yau genus-one fibrations over any base space, except models with a discrete ℤ5 gauge group. We construct 6D F-theory models with a discrete ℤ5 gauge group over the del Pezzo surfaces, as well as over ℙ1 × ℙ1 and ℙ2. We also discuss some applications to four-dimensional F-theory models with discrete gauge symmetries.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. 1, Nucl. Phys.B 473 (1996) 74 [hep-th/9602114] [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].
D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, JHEP10 (2012) 128 [arXiv:1208.2695] [INSPIRE].
C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP03 (2013) 098 [arXiv:1211.6742] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, New Global F-theory GUTs with U(1) symmetries, JHEP09 (2013) 154 [arXiv:1302.1854] [INSPIRE].
J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, Elliptic fibrations for SU(5) × U(1) × U(1) F-theory vacua, Phys. Rev.D 88 (2013) 046005 [arXiv:1303.5054] [INSPIRE].
M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections, JHEP06 (2013) 067 [arXiv:1303.6970] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, Geometric Engineering in Toric F-theory and GUTs with U(1) Gauge Factors, JHEP12 (2013) 069 [arXiv:1306.0577] [INSPIRE].
M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral Four-Dimensional F-theory Compactifications With SU(5) and Multiple U(1)-Factors, JHEP04 (2014) 010 [arXiv:1306.3987] [INSPIRE].
M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Addendum, JHEP12 (2013) 056 [arXiv:1307.6425] [INSPIRE].
M. Cvetič, D. Klevers, H. Piragua and P. Song, Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1) × U(1) × U(1) gauge symmetry, JHEP03 (2014) 021 [arXiv:1310.0463] [INSPIRE].
I. Antoniadis and G.K. Leontaris, F-GUTs with Mordell-Weil U(1)’s, Phys. Lett.B 735 (2014) 226 [arXiv:1404.6720] [INSPIRE].
M. Esole, M.J. Kang and S.-T. Yau, A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations, arXiv:1410.0003 [INSPIRE].
C. Lawrie, S. Schäfer-Nameki and J.-M. Wong, F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections, JHEP09 (2015) 144 [arXiv:1504.05593] [INSPIRE].
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, JHEP11 (2015) 204 [arXiv:1507.05954] [INSPIRE].
M. Cvetič, A. Grassi, D. Klevers, M. Poretschkin and P. Song, Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality, JHEP04 (2016) 041 [arXiv:1511.08208] [INSPIRE].
D.R. Morrison and D.S. Park, Tall sections from non-minimal transformations, JHEP10 (2016) 033 [arXiv:1606.07444] [INSPIRE].
M. Bies, C. Mayrhofer and T. Weigand, Gauge Backgrounds and Zero-Mode Counting in F-theory, JHEP11 (2017) 081 [arXiv:1706.04616] [INSPIRE].
M. Cvetič and L. Lin, The Global Gauge Group Structure of F-theory Compactification with U(1)s, JHEP01 (2018) 157 [arXiv:1706.08521] [INSPIRE].
M. Bies, C. Mayrhofer and T. Weigand, Algebraic Cycles and Local Anomalies in F-theory, JHEP11 (2017) 100 [arXiv:1706.08528] [INSPIRE].
M. Esole, M.J. Kang and S.-T. Yau, Mordell-Weil Torsion, Anomalies and Phase Transitions, arXiv:1712.02337 [INSPIRE].
Y. Kimura and S. Mizoguchi, Enhancements in F-theory models on moduli spaces of K3 surfaces with ADE rank 17, PTEP2018 (2018) 043B05 [arXiv:1712.08539] [INSPIRE].
Y. Kimura, F-theory models on K3 surfaces with various Mordell-Weil ranks — constructions that use quadratic base change of rational elliptic surfaces, JHEP05 (2018) 048 [arXiv:1802.05195] [INSPIRE].
S.-J. Lee, D. Regalado and T. Weigand, 6d SCFTs and U(1) Flavour Symmetries, JHEP11 (2018) 147 [arXiv:1803.07998] [INSPIRE].
S. Mizoguchi and T. Tani, Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications, JHEP03 (2019) 121 [arXiv:1808.08001] [INSPIRE].
F.M. Cianci, D.K. Mayorga Peña and R. Valandro, High U(1) charges in type IIB models and their F-theory lift, JHEP04 (2019) 012 [arXiv:1811.11777] [INSPIRE].
W. Taylor and A.P. Turner, Generic matter representations in 6D supergravity theories, JHEP05 (2019) 081 [arXiv:1901.02012] [INSPIRE].
Y. Kimura, F-theory models with 3 to 8 U(1) factors on K3 surfaces, arXiv:1903.03608 [INSPIRE].
V. Braun and D.R. Morrison, F-theory on Genus-One Fibrations, JHEP08 (2014) 132 [arXiv:1401.7844] [INSPIRE].
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys.B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models, Nucl. Phys.B 820 (2009) 317 [arXiv:0904.2631] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev.D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
S. Hellerman and E. Sharpe, Sums over topological sectors and quantization of Fayet-Iliopoulos parameters, Adv. Theor. Math. Phys.15 (2011) 1141 [arXiv:1012.5999] [INSPIRE].
P.G. Camara, L.E. Ibáñez and F. Marchesano, RR photons, JHEP09 (2011) 110 [arXiv:1106.0060] [INSPIRE].
M. Berasaluce-Gonzalez, L.E. Ibáñez, P. Soler and A.M. Uranga, Discrete gauge symmetries in D-brane models, JHEP12 (2011) 113 [arXiv:1106.4169] [INSPIRE].
L.E. Ibáñez, A.N. Schellekens and A.M. Uranga, Discrete Gauge Symmetries in Discrete MSSM-like Orientifolds, Nucl. Phys.B 865 (2012) 509 [arXiv:1205.5364] [INSPIRE].
M. Berasaluce-Gonzalez, P.G. Camara, F. Marchesano, D. Regalado and A.M. Uranga, Non-Abelian discrete gauge symmetries in 4d string models, JHEP09 (2012) 059 [arXiv:1206.2383] [INSPIRE].
M. Berasaluce-Gonzalez, P.G. Camara, F. Marchesano and A.M. Uranga, Zp charged branes in flux compactifications, JHEP04 (2013) 138 [arXiv:1211.5317] [INSPIRE].
F. Marchesano, D. Regalado and L. Vazquez-Mercado, Discrete flavor symmetries in D-brane models, JHEP09 (2013) 028 [arXiv:1306.1284] [INSPIRE].
G. Honecker and W. Staessens, To Tilt or Not To Tilt: Discrete Gauge Symmetries in Global Intersecting D-brane Models, JHEP10 (2013) 146 [arXiv:1303.4415] [INSPIRE].
M. Berasaluce-González, G. Ramírez and A.M. Uranga, Antisymmetric tensor Z pgauge symmetries in field theory and string theory, JHEP01 (2014) 059 [arXiv:1310.5582] [INSPIRE].
A. Karozas, S.F. King, G.K. Leontaris and A. Meadowcroft, Discrete Family Symmetry from F-theory GUTs, JHEP09 (2014) 107 [arXiv:1406.6290] [INSPIRE].
G. Honecker and W. Staessens, Discrete Abelian gauge symmetries and axions, J. Phys. Conf. Ser.631 (2015) 012080 [arXiv:1502.00985] [INSPIRE].
T.W. Grimm, T.G. Pugh and D. Regalado, Non-Abelian discrete gauge symmetries in F-theory, JHEP02 (2016) 066 [arXiv:1504.06272] [INSPIRE].
L.B. Anderson, I. García-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP12 (2014) 156 [arXiv:1406.5180] [INSPIRE].
D. Klevers, D.K. Mayorga Pena, P.-K. Oehlmann, H. Piragua and J. Reuter, F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches, JHEP01 (2015) 142 [arXiv:1408.4808] [INSPIRE].
I. García-Etxebarria, T.W. Grimm and J. Keitel, Yukawas and discrete symmetries in F-theory compactifications without section, JHEP11 (2014) 125 [arXiv:1408.6448] [INSPIRE].
C. Mayrhofer, E. Palti, O. Till and T. Weigand, Discrete Gauge Symmetries by Higgsing in four-dimensional F-theory Compactifications, JHEP12 (2014) 068 [arXiv:1408.6831] [INSPIRE].
C. Mayrhofer, E. Palti, O. Till and T. Weigand, On Discrete Symmetries and Torsion Homology in F-theory, JHEP06 (2015) 029 [arXiv:1410.7814] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, Complete Intersection Fibers in F-theory, JHEP03 (2015) 125 [arXiv:1411.2615] [INSPIRE].
M. Cvetič, R. Donagi, D. Klevers, H. Piragua and M. Poretschkin, F-theory vacua with ℤ3gauge symmetry, Nucl. Phys.B 898 (2015) 736 [arXiv:1502.06953] [INSPIRE].
L. Lin, C. Mayrhofer, O. Till and T. Weigand, Fluxes in F-theory Compactifications on Genus-One Fibrations, JHEP01 (2016) 098 [arXiv:1508.00162] [INSPIRE].
Y. Kimura, Gauge Groups and Matter Fields on Some Models of F-theory without section, JHEP03 (2016) 042 [arXiv:1511.06912] [INSPIRE].
Y. Kimura, Gauge symmetries and matter fields in F-theory models without section — compactifications on double cover and Fermat quartic K3 constructions times K3, Adv. Theor. Math. Phys.21 (2017) 2087 [arXiv:1603.03212] [INSPIRE].
P.-K. Oehlmann, J. Reuter and T. Schimannek, Mordell-Weil Torsion in the Mirror of Multi-Sections, JHEP12 (2016) 031 [arXiv:1604.00011] [INSPIRE].
Y. Kimura, Gauge groups and matter spectra in F-theory compactifications on genus-one fibered Calabi-Yau 4-folds without section: Hypersurface and double cover constructions, Adv. Theor. Math. Phys.22 (2018) 1489 [arXiv:1607.02978] [INSPIRE].
M. Cvetič, A. Grassi and M. Poretschkin, Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry, JHEP06 (2017) 156 [arXiv:1607.03176] [INSPIRE].
Y. Kimura, Discrete Gauge Groups in F-theory Models on Genus-One Fibered Calabi-Yau 4-folds without section, JHEP04 (2017) 168 [arXiv:1608.07219] [INSPIRE].
Y. Kimura, K3 surfaces without section as double covers of Halphen surfaces and F-theory compactifications, PTEP2018 (2018) 043B06 [arXiv:1801.06525] [INSPIRE].
L.B. Anderson, A. Grassi, J. Gray and P.-K. Oehlmann, F-theory on Quotient Threefolds with (2,0) Discrete Superconformal Matter, JHEP06 (2018) 098 [arXiv:1801.08658] [INSPIRE].
Y. Kimura, SU(N) × ℤ2in F-theory on K3 surfaces without section as double covers of Halphen surfaces, arXiv:1806.01727 [INSPIRE].
T. Weigand, F-theory, PoSTASI2017 (2018) 016 [arXiv:1806.01854] [INSPIRE].
M. Cvetič, L. Lin, M. Liu and P.-K. Oehlmann, An F-theory Realization of the Chiral MSSM with ℤ2-Parity, JHEP09 (2018) 089 [arXiv:1807.01320] [INSPIRE].
M. Cvetič and L. Lin, TASI Lectures on Abelian and Discrete Symmetries in F-theory, PoS(TASI2017)020 (2018) [arXiv:1809.00012] [INSPIRE].
Y.-C. Huang and W. Taylor, On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds, JHEP03 (2019) 014 [arXiv:1809.05160] [INSPIRE].
Y. Kimura, Nongeometric heterotic strings and dual F-theory with enhanced gauge groups, JHEP02 (2019) 036 [arXiv:1810.07657] [INSPIRE].
Y. Kimura, Unbroken E 7 × E 7nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals, arXiv:1902.00944 [INSPIRE].
P. Berglund, J.R. Ellis, A.E. Faraggi, D.V. Nanopoulos and Z. Qiu, Elevating the free fermion Z(2) x Z(2) orbifold model to a compactification of F-theory, Int. J. Mod. Phys.A 15 (2000) 1345 [hep-th/9812141] [INSPIRE].
J. de Boer et al., Triples, fluxes and strings, Adv. Theor. Math. Phys.4 (2002) 995 [hep-th/0103170] [INSPIRE].
J.W.S. Cassels, Lectures on Elliptic Curves, London Math. Society Student Texts, vol. 24, Cambridge University Press (1991).
N. Nakayama, On Weierstrass Models, in Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Academic Press (1988).
I. Dolgachev and M. Gross, Elliptic Three-folds I: Ogg-Shafarevich Theory, J. Alg. Geom.3 (1994) 39.
M. Gross, Elliptic Three-folds II: Multiple Fibres, Trans. Am. Math. Soc.349 (1997) 3409.
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, JHEP01 (2009) 058 [arXiv:0802.3391] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — II: Experimental Predictions, JHEP01 (2009) 059 [arXiv:0806.0102] [INSPIRE].
R. Donagi and M. Wijnholt, Breaking GUT Groups in F-theory, Adv. Theor. Math. Phys.15 (2011) 1523 [arXiv:0808.2223] [INSPIRE].
V.A. Iskovskih, Fano 3-folds. I, Math. USSR Izv. 11 (1977) 485.
T. Fujita, On the structure of polarized manifolds with total deficiency one, I, J. Math. Soc. Jap.32 (1980) 709.
T. Fujita, On the structure of polarized manifolds with total deficiency one, II, J. Math. Soc. Jap.33 (1981) 415.
T. Fujita, On the structure of certain types of polarized varieties, II, Proc. Japan Acad.50 (1974) 411.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.03775
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kimura, Y. Discrete gauge groups in certain F-theory models in six dimensions. J. High Energ. Phys. 2019, 27 (2019). https://doi.org/10.1007/JHEP07(2019)027
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2019)027