Abstract
We study the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold. Focusing on the simplest example of a \( {\mathbb{Z}}_2 \) symmetry, we show that there are two physically distinct ways that such a discrete gauge symmetry can arise. First, compactifications of M-Theory on Calabi-Yau threefolds which support a genus-one fibration with a bi-section are known to be dual to six-dimensional F-theory vacua with a \( {\mathbb{Z}}_2 \) gauge symmetry. We show that the resulting five-dimensional theories do not have a \( {\mathbb{Z}}_2 \) symmetry but that the latter emerges only in the F-theory decompactification limit. Accordingly the genus-one fibred Calabi-Yau manifolds do not exhibit torsion in homology. Associated to the bi-section fibration is a Jacobian fibration which does support a section. Compactifying on these related but distinct varieties does lead to a \( {\mathbb{Z}}_2 \) symmetry in five dimensions and, accordingly, we find explicitly an associated torsion cycle. We identify the expected particle and membrane system of the discrete symmetry in terms of wrapped M2 and M5 branes and present a field-theory description of the physics for both cases in terms of circle reductions of six-dimensional theories. Our results and methods generalise straightforwardly to larger discrete symmetries and to four-dimensional compactifications.
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References
T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
P.G. Camára, L.E. Ibáñez and F. Marchesano, RR photons, JHEP 09 (2011) 110 [arXiv:1106.0060] [INSPIRE].
M. Berasaluce-González, L.E. Ibáñez, P. Soler and A.M. Uranga, Discrete gauge symmetries in D-brane models, JHEP 12 (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-González, P.G. Camára, F. Marchesano, D. Regalado and A.M. Uranga, Non-Abelian discrete gauge symmetries in 4d string models, JHEP 09 (2012) 059 [arXiv:1206.2383] [INSPIRE].
P. Anastasopoulos, M. Cvetič, R. Richter and P.K.S. Vaudrevange, String constraints on discrete symmetries in MSSM type II quivers, JHEP 03 (2013) 011 [arXiv:1211.1017] [INSPIRE].
M. Berasaluce-González, P.G. Cámara, F. Marchesano and A.M. Uranga, Zp charged branes in flux compactifications, JHEP 04 (2013) 138 [arXiv:1211.5317] [INSPIRE].
G. Honecker and W. Staessens, To tilt or not to tilt: discrete gauge symmetries in global intersecting D-brane models, JHEP 10 (2013) 146 [arXiv:1303.4415] [INSPIRE].
M. Berasaluce-González, M. Montero, A. Retolaza and A.M. Uranga, Discrete gauge symmetries from (closed string) tachyon condensation, JHEP 11 (2013) 144 [arXiv:1305.6788] [INSPIRE].
M. Berasaluce-González, G. Ramírez and A.M. Uranga, Antisymmetric tensor Zp gauge symmetries in field theory and string theory, JHEP 01 (2014) 059 [arXiv:1310.5582] [INSPIRE].
V. Braun and D.R. Morrison, F-theory on genus-one fibrations, JHEP 08 (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].
L.B. Anderson, I. García-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP 12 (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, JHEP 01 (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, JHEP 11 (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, JHEP 12 (2014) 068 [arXiv:1408.6831] [INSPIRE].
A. Karozas, S.F. King, G.K. Leontaris and A. Meadowcroft, Discrete family symmetry from F-theory GUTs, JHEP 09 (2014) 107 [arXiv:1406.6290] [INSPIRE].
E. Witten, Nonperturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Non-Abelian gauge symmetry and the Higgs mechanism in F-theory, Commun. Math. Phys. 336 (2015) 1231 [arXiv:1402.5962] [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].
V. Batyrev and M. Kreuzer, Intergral cohomology and miror symmetry for Calabi-Yau 3-folds, math.AG/0505432.
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].
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].
S. Krause, C. Mayrhofer and T. Weigand, Gauge fluxes in F-theory and type IIB orientifolds, JHEP 08 (2012) 119 [arXiv:1202.3138] [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, Adv. Theor. Math. Phys. 17 (2013) 601 [arXiv:1203.6662] [INSPIRE].
A. Strominger, Massless black holes and conifolds in string theory, Nucl. Phys. B 451 (1995) 96 [hep-th/9504090] [INSPIRE].
B.R. Greene, D.R. Morrison and A. Strominger, Black hole condensation and the unification of string vacua, Nucl. Phys. B 451 (1995) 109 [hep-th/9504145] [INSPIRE].
B.R. Greene, D.R. Morrison and C. Vafa, A Geometric realization of confinement, Nucl. Phys. B 481 (1996) 513 [hep-th/9608039] [INSPIRE].
M. Dolgachev, I. Gross, Elliptic three-folds I: Ogg-Shafarevich theory, J. Algebraic Geom. 3 (1994) 38, math.AG/9210009.
P.S. Aspinwall, D.R. Morrison and M. Gross, Stable singularities in string theory, Commun. Math. Phys. 178 (1996) 115 [hep-th/9503208] [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].
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].
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].
W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3-1-6 — A computer algebra system for polynomial computations, (2012) http://www.singular.uni-kl.de.
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Mayrhofer, C., Palti, E., Till, O. et al. On discrete symmetries and torsion homology in F-theory. J. High Energ. Phys. 2015, 29 (2015). https://doi.org/10.1007/JHEP06(2015)029
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DOI: https://doi.org/10.1007/JHEP06(2015)029