Abstract
We prove that, for every 6D supergravity theory that has an F-theory description, the property of charge completeness for the connected component of the gauge group (meaning that all charges in the corresponding charge lattice are realized by massive or massless states in the theory) is equivalent to a standard assumption made in F-theory for how geometry encodes the global gauge theory by means of the Mordell-Weil group of the elliptic fibration. This result also holds in 4D F-theory constructions for the parts of the gauge group that come from sections and from 7-branes. We find that in many 6D F-theory models the full charge lattice of the theory is generated by massless charged states; this occurs for each gauge factor where the associated anomaly coefficient satisfies a simple positivity condition. We describe many of the cases where this massless charge sufficiency condition holds, as well as exceptions where the positivity condition fails, and analyze the related global structure of the gauge group and associated Mordell-Weil torsion in explicit F-theory models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 1. The Free Heterotic String, Nucl. Phys. B 256 (1985) 253 [INSPIRE].
D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 2. The Interacting Heterotic String, Nucl. Phys. B 267 (1986) 75 [INSPIRE].
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
A. Adams, O. DeWolfe and W. Taylor, String universality in ten dimensions, Phys. Rev. Lett. 105 (2010) 071601 [arXiv:1006.1352] [INSPIRE].
H.-C. Kim, G. Shiu and C. Vafa, Branes and the Swampland, Phys. Rev. D 100 (2019) 066006 [arXiv:1905.08261] [INSPIRE].
E. Palti, The Swampland: Introduction and Review, Fortsch. Phys. 67 (2019) 1900037 [arXiv:1903.06239] [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. 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].
L. Bhardwaj, D.R. Morrison, Y. Tachikawa and A. Tomasiello, The frozen phase of F-theory, JHEP 08 (2018) 138 [arXiv:1805.09070] [INSPIRE].
V. Kumar and W. Taylor, String Universality in Six Dimensions, Adv. Theor. Math. Phys. 15 (2011) 325 [arXiv:0906.0987] [INSPIRE].
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].
W. Taylor and A.P. Turner, An infinite swampland of U(1) charge spectra in 6D supergravity theories, JHEP 06 (2018) 010 [arXiv:1803.04447] [INSPIRE].
N. Seiberg and W. Taylor, Charge Lattices and Consistency of 6D Supergravity, JHEP 06 (2011) 001 [arXiv:1103.0019] [INSPIRE].
S. Monnier and G.W. Moore, Remarks on the Green-Schwarz Terms of Six-Dimensional Supergravity Theories, Commun. Math. Phys. 372 (2019) 963 [arXiv:1808.01334] [INSPIRE].
S.-J. Lee and T. Weigand, Swampland Bounds on the Abelian Gauge Sector, Phys. Rev. D 100 (2019) 026015 [arXiv:1905.13213] [INSPIRE].
C. Angelantonj, Q. Bonnefoy, C. Condeescu and E. Dudas, String Defects, Supersymmetry and the Swampland, JHEP 11 (2020) 125 [arXiv:2007.12722] [INSPIRE].
N. Raghuram, W. Taylor and A.P. Turner, Automatic enhancement in 6D supergravity and F-theory models, JHEP 07 (2021) 048 [arXiv:2012.01437] [INSPIRE].
H.-C. Tarazi and C. Vafa, On The Finiteness of 6d Supergravity Landscape, arXiv:2106.10839 [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B 766 (2007) 21 [hep-th/0605264] [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
J. Polchinski, Monopoles, duality, and string theory, Int. J. Mod. Phys. A 19S1 (2004) 145 [hep-th/0304042] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, Commun. Math. Phys. 383 (2021) 1669 [arXiv:1810.05338] [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].
P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds, and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].
C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory, JHEP 10 (2014) 016 [arXiv:1405.3656] [INSPIRE].
F. Apruzzi, M. Dierigl and L. Lin, The Fate of Discrete 1-Form Symmetries in 6d, arXiv:2008.09117 [INSPIRE].
M. Cvetič, M. Dierigl, L. Lin and H.Y. Zhang, String Universality and Non-Simply-Connected Gauge Groups in 8d, Phys. Rev. Lett. 125 (2020) 211602 [arXiv:2008.10605] [INSPIRE].
M.B. Green, J.H. Schwarz and P.C. West, Anomaly Free Chiral Theories in Six-Dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].
A. Sagnotti, A Note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].
V. Kumar and W. Taylor, A Bound on 6D N = 1 supergravities, JHEP 12 (2009) 050 [arXiv:0910.1586] [INSPIRE].
V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [INSPIRE].
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].
D.S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE].
D.S. Park, Anomaly Equations and Intersection Theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
S. Monnier, G.W. Moore and D.S. Park, Quantization of anomaly coefficients in 6D \( \mathcal{N} \) = (1, 0) supergravity, JHEP 02 (2018) 020 [arXiv:1711.04777] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum JHEP 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
D.R. Morrison, TASI lectures on compactification and duality, in proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 99): Strings, Branes, and Gravity, Boulder, CO, U.S.A., 31 May–25 June 1999, pp. 653–719 [hep-th/0411120] [INSPIRE].
F. Denef, Les Houches Lectures on Constructing String Vacua, in Les Houches 87, Elsevier B.V. (2008), pp. 483–610 [arXiv:0803.1194] [INSPIRE].
W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, arXiv:1104.2051 [INSPIRE].
T. Weigand, F-theory, PoS TASI2017 (2018) 016 [arXiv:1806.01854] [INSPIRE].
D.R. Morrison, What is F-theory?, to appear.
K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, J. Algebr. Geom. 12 (2003) 321 [math/0005196] [INSPIRE].
V. Kumar, D.R. Morrison and W. Taylor, Mapping 6D N = 1 supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [INSPIRE].
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].
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].
L. Lin and T. Weigand, Towards the Standard Model in F-theory, Fortsch. Phys. 63 (2015) 55 [arXiv:1406.6071] [INSPIRE].
M. Cvetič, D. Klevers, D.K.M. Peña, P.-K. Oehlmann and J. Reuter, Three-Family Particle Physics Models from Global F-theory Compactifications, JHEP 08 (2015) 087 [arXiv:1503.02068] [INSPIRE].
C. Lawrie, S. Schäfer-Nameki and J.-M. Wong, F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections, JHEP 09 (2015) 144 [arXiv:1504.05593] [INSPIRE].
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].
L. Lin and T. Weigand, G4-flux and standard model vacua in F-theory, Nucl. Phys. B 913 (2016) 209 [arXiv:1604.04292] [INSPIRE].
M. Cvetič and L. Lin, The Global Gauge Group Structure of F-theory Compactification with U(1)s, JHEP 01 (2018) 157 [arXiv:1706.08521] [INSPIRE].
M. Cvetič, L. Lin, M. Liu and P.-K. Oehlmann, An F-theory Realization of the Chiral MSSM with ℤ2-Parity, JHEP 09 (2018) 089 [arXiv:1807.01320] [INSPIRE].
M. Cvetič, J. Halverson, L. Lin, M. Liu and J. Tian, Quadrillion F-Theory Compactifications with the Exact Chiral Spectrum of the Standard Model, Phys. Rev. Lett. 123 (2019) 101601 [arXiv:1903.00009] [INSPIRE].
W. Taylor and A.P. Turner, Generic Construction of the Standard Model Gauge Group and Matter Representations in F-theory, Fortsch. Phys. 68 (2020) 2000009 [arXiv:1906.11092] [INSPIRE].
N. Raghuram, W. Taylor and A.P. Turner, General F-theory models with tuned (SU(3) × SU(2) × U(1))/ℤ6 symmetry, JHEP 04 (2020) 008 [arXiv:1912.10991] [INSPIRE].
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].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].
D. Klevers, D.R. Morrison, N. Raghuram and W. Taylor, Exotic matter on singular divisors in F-theory, JHEP 11 (2017) 124 [arXiv:1706.08194] [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].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, in Séminaire Nicolas Bourbak 1964/65–1965/66, W.A. Benjamin Publisher, Exp. No. 306, pp. 415–440.
T. Shioda, On elliptic modular surfaces, J. Math. Soc. Jpn. 24 (1972) 20.
R. Wazir, Arithmetic on elliptic threefolds, Compos. Math. 140 (2004) 567.
D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math. 76 (1962) 560.
T. Bauer, M. Caibăr and G. Kennedy, Zariski decomposition: a new (old) chapter of linear algebra, Am. Math. Mon. 119 (2012) 25 [arXiv:0911.4500].
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].
Y.-N. Wang, Tuned and Non-Higgsable U(1)s in F-theory, JHEP 03 (2017) 140 [arXiv:1611.08665] [INSPIRE].
D.R. Morrison, D.S. Park and W. Taylor, Non-Higgsable Abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces, Adv. Theor. Math. Phys. 22 (2018) 177 [arXiv:1610.06929] [INSPIRE].
W. Taylor and A.P. Turner, Generic matter representations in 6D supergravity theories, JHEP 05 (2019) 081 [arXiv:1901.02012] [INSPIRE].
J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer (1992).
J.H. Silverman, The Arithmetic of Elliptic Curves, Springer (1986).
D. Kubert, Universal bounds on the torsion of elliptic curves, Proc. Lond. Math. Soc. 33 (1976) 193.
N. Hajouji and P.-K. Oehlmann, Modular Curves and Mordell-Weil Torsion in F-theory, JHEP 04 (2020) 103 [arXiv:1910.04095] [INSPIRE].
B. Mazur, Modular curves and Eisenstein ideal, Publ. Math. IHÉS 47 (1977) 33.
B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978) 129.
N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris France (1968), chapters IV, V, VI.
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].
L.B. Anderson, J. Gray, N. Raghuram and W. Taylor, Matter in transition, JHEP 04 (2016) 080 [arXiv:1512.05791] [INSPIRE].
Y. Kimura, Unbroken E7 × E7 nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals, arXiv:1902.00944 [INSPIRE].
M. Cvetič, J.J. Heckman and L. Lin, Towards Exotic Matter and Discrete Non-Abelian Symmetries in F-theory, JHEP 11 (2018) 001 [arXiv:1806.10594] [INSPIRE].
Y.-C. Huang and W. Taylor, Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers, JHEP 02 (2019) 087 [arXiv:1805.05907] [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].
D.R. Morrison and W. Taylor, Sections, multisections, and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
D. Klevers, D.K.M. Peña, P.-K. Oehlmann, H. Piragua and J. Reuter, F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE].
N. Raghuram, Abelian F-theory Models with Charge-3 and Charge-4 Matter, JHEP 05 (2018) 050 [arXiv:1711.03210] [INSPIRE].
F.M. Cianci, D.K.M. Peña and R. Valandro, High U(1) charges in type IIB models and their F-theory lift, JHEP 04 (2019) 012 [arXiv:1811.11777] [INSPIRE].
A. Collinucci, M. Fazzi, D.R. Morrison and R. Valandro, High electric charges in M-theory from quiver varieties, JHEP 11 (2019) 111 [arXiv:1906.02202] [INSPIRE].
N. Raghuram and W. Taylor, Large U(1) charges in F-theory, JHEP 10 (2018) 182 [arXiv:1809.01666] [INSPIRE].
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY Threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
U. Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990) 1.
R. Miranda, Persson’s list of singular fibers for a rational elliptic surface, Math. Z. 205 (1990) 191.
B.C. Hall, Lie groups, Lie algebras, and representations: An elementary introduction, second edition, in Graduate Texts in Mathematics 222, Springer-Verlag, New York NY U.S.A. (2015).
T. Bröcker and T. tom Dieck, Representations of compact Lie groups, in Graduate Texts in Mathematics 98, Springer-Verlag, New York NY U.S.A. (1995).
J.C. Baez and J. Huerta, The Algebra of Grand Unified Theories, Bull. Am. Math. Soc. 47 (2010) 483 [arXiv:0904.1556] [INSPIRE].
H. Georgi and S.L. Glashow, Unity of All Elementary Particle Forces, Phys. Rev. Lett. 32 (1974) 438 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2108.02309
Rights and permissions
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.
About this article
Cite this article
Morrison, D.R., Taylor, W. Charge completeness and the massless charge lattice in F-theory models of supergravity. J. High Energ. Phys. 2021, 40 (2021). https://doi.org/10.1007/JHEP12(2021)040
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)040