Built on

T. Shioda, Mordell-Weil lattices and Galois representation. I , Proc. Japan Acad. Ser. A Math. Sci. 65
1989
Earlier work this paper cites.
T. Shioda, On the Mordell-Weil lattices , Comment. Math. Univ. St. Paul. 39
1990
Earlier work this paper cites.
Springer-Verlag, New York, 1994, 10.1007/978-1-4612-0851-8
J. H. Silverman, Advanced topics in the arithmetic of elliptic curves , vol. 151 of Graduate Texts in Mathematics · 1994
Earlier work this paper cites.
I. Dolgachev and M. Gross, Elliptic threefolds. I. Ogg-Shafarevich theory , J. Algebraic Geom. 3
1994
Earlier work this paper cites.
Frontiers in Physics. Avalon Publishing, 1995
M. Peskin and D. Schroeder, An Introduction To Quantum Field Theory · 1995
Earlier work this paper cites.
C. Vafa, Evidence for F theory , Nucl. Phys. B469
1996
Earlier work this paper cites.
D. R. Morrison and C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 1 , Nucl. Phys. B473
1996
Earlier work this paper cites.
D. R. Morrison and C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 2. , Nucl. Phys. B476
1996
Earlier work this paper cites.
E. Witten, Nonperturbative superpotentials in string theory , Nucl. Phys. B474
1996
Earlier work this paper cites.
P. Candelas and A. Font, Duality between the webs of heterotic and type II vacua , Nucl. Phys. B511
1998
Earlier work this paper cites.
V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification of tops , Adv. Theor. Math. Phys. 7
2003
Earlier work this paper cites.
R. Wazir, Arithmetic on elliptic threefolds , Compos. Math. 140
2004
Earlier work this paper cites.
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland , Nucl. Phys. B766
2007
Earlier work this paper cites.
0803.1194
Original
F. Denef, Les Houches Lectures on Constructing String Vacua , in String theory and the real world: From particle physics to astrophysics. Proceedings, Summer School in Theoretical Physics, 87th Session, Les Houches, France, July 2-27, 2007 , pp. 483–610, 2008 · 2008
Earlier work this paper cites.
J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory Compactifications for Supersymmetric GUTs , JHEP 08
Original
2009
Earlier work this paper cites.
J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs , JHEP 08
Original
2009
Earlier work this paper cites.
Springer, Dordrecht, second ed., 2009, 10.1007/978-0-387-09494-6
J. H. Silverman, The arithmetic of elliptic curves , vol. 106 of Graduate Texts in Mathematics · 2009
Earlier work this paper cites.
R. Blumenhagen, T. W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs , Nucl. Phys. B829
Original
2010
Earlier work this paper cites.
J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1) (PQ) , JHEP 04
Original
2010
Earlier work this paper cites.
T. W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs , Phys. Rev. D82
Original
2010
Earlier work this paper cites.
T. Weigand, Lectures on F-theory compactifications and model building , Class. Quant. Grav. 27
Original
2010
Earlier work this paper cites.
V. Kumar, D. R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities , JHEP 11
Original
2010
Earlier work this paper cites.
M. Cvetič, I. Garcia-Etxebarria and J. Halverson, Global F-theory Models: Instantons and Gauge Dynamics , JHEP 01
Original
2011
Earlier work this paper cites.
M. J. Dolan, J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory GUTs with U(1) Symmetries: Generalities and Survey , Phys. Rev. D84
Original
2011
Earlier work this paper cites.
M. J. Dolan, J. Marsano and S. Schäfer-Nameki, Unification and Phenomenology of F-Theory GUTs with U ( 1 ) P Q U(1)_{PQ} , JHEP 12
Original
2011
Earlier work this paper cites.
T. W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive Abelian Gauge Symmetries and Fluxes in F-theory , JHEP 12
Original
2011
Earlier work this paper cites.
S. Krause, C. Mayrhofer and T. Weigand, G 4 G_{4} flux, chiral matter and singularity resolution in F-theory compactifications , Nucl. Phys. B858
Original
2012
Earlier work this paper cites.
D. S. Park, Anomaly Equations and Intersection Theory , JHEP 01
Original
2012
Earlier work this paper cites.
D. R. Morrison and D. S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds , JHEP 10
Original
2012
Earlier work this paper cites.
D. S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry , JHEP 01
Original
2012
Earlier work this paper cites.
A. P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles , Nucl. Phys. B856
Original
2012
Earlier work this paper cites.

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2018

TASI Lectures on Abelian and Discrete Symmetries in F-theory

Cvetic, Mirjam, Lin, Ling

Understand

In F-theory compactifications, the abelian gauge sector is encoded in global structures of the internal geometry.

These structures lie at the intersection of algebraic and arithmetic description of elliptic fibrations: While the Mordell--Weil lattice is related to the continuous abelian sector, the Tate--Shafarevich group is conjectured to encode discrete abelian symmetries in F-theory.
In these notes we review both subjects with a focus on recent findings such as the global gauge group and gauge enhancements.
We then highlight the application to F-theory model building.

Built on

T. Shioda, Mordell-Weil lattices and Galois representation. I , Proc. Japan Acad. Ser. A Math. Sci. 65
1989
Earlier work this paper cites.
T. Shioda, On the Mordell-Weil lattices , Comment. Math. Univ. St. Paul. 39
1990
Earlier work this paper cites.
Springer-Verlag, New York, 1994, 10.1007/978-1-4612-0851-8
J. H. Silverman, Advanced topics in the arithmetic of elliptic curves , vol. 151 of Graduate Texts in Mathematics · 1994
Earlier work this paper cites.
I. Dolgachev and M. Gross, Elliptic threefolds. I. Ogg-Shafarevich theory , J. Algebraic Geom. 3
1994
Earlier work this paper cites.
Frontiers in Physics. Avalon Publishing, 1995
M. Peskin and D. Schroeder, An Introduction To Quantum Field Theory · 1995
Earlier work this paper cites.
C. Vafa, Evidence for F theory , Nucl. Phys. B469
1996
Earlier work this paper cites.
D. R. Morrison and C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 1 , Nucl. Phys. B473
1996
Earlier work this paper cites.
D. R. Morrison and C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 2. , Nucl. Phys. B476
1996
Earlier work this paper cites.
E. Witten, Nonperturbative superpotentials in string theory , Nucl. Phys. B474
1996
Earlier work this paper cites.
P. Candelas and A. Font, Duality between the webs of heterotic and type II vacua , Nucl. Phys. B511
1998
Earlier work this paper cites.
V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification of tops , Adv. Theor. Math. Phys. 7
2003
Earlier work this paper cites.
R. Wazir, Arithmetic on elliptic threefolds , Compos. Math. 140
2004
Earlier work this paper cites.
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland , Nucl. Phys. B766
2007
Earlier work this paper cites.
0803.1194
Original
F. Denef, Les Houches Lectures on Constructing String Vacua , in String theory and the real world: From particle physics to astrophysics. Proceedings, Summer School in Theoretical Physics, 87th Session, Les Houches, France, July 2-27, 2007 , pp. 483–610, 2008 · 2008
Earlier work this paper cites.
J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory Compactifications for Supersymmetric GUTs , JHEP 08
Original
2009
Earlier work this paper cites.
J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs , JHEP 08
Original
2009
Earlier work this paper cites.
Springer, Dordrecht, second ed., 2009, 10.1007/978-0-387-09494-6
J. H. Silverman, The arithmetic of elliptic curves , vol. 106 of Graduate Texts in Mathematics · 2009
Earlier work this paper cites.
R. Blumenhagen, T. W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs , Nucl. Phys. B829
Original
2010
Earlier work this paper cites.
J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1) (PQ) , JHEP 04
Original
2010
Earlier work this paper cites.
T. W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs , Phys. Rev. D82
Original
2010
Earlier work this paper cites.
T. Weigand, Lectures on F-theory compactifications and model building , Class. Quant. Grav. 27
Original
2010
Earlier work this paper cites.
V. Kumar, D. R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities , JHEP 11
Original
2010
Earlier work this paper cites.
M. Cvetič, I. Garcia-Etxebarria and J. Halverson, Global F-theory Models: Instantons and Gauge Dynamics , JHEP 01
Original
2011
Earlier work this paper cites.
M. J. Dolan, J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory GUTs with U(1) Symmetries: Generalities and Survey , Phys. Rev. D84
Original
2011
Earlier work this paper cites.
M. J. Dolan, J. Marsano and S. Schäfer-Nameki, Unification and Phenomenology of F-Theory GUTs with U ( 1 ) P Q U(1)_{PQ} , JHEP 12
Original
2011
Earlier work this paper cites.
T. W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive Abelian Gauge Symmetries and Fluxes in F-theory , JHEP 12
Original
2011
Earlier work this paper cites.
S. Krause, C. Mayrhofer and T. Weigand, G 4 G_{4} flux, chiral matter and singularity resolution in F-theory compactifications , Nucl. Phys. B858
Original
2012
Earlier work this paper cites.
D. S. Park, Anomaly Equations and Intersection Theory , JHEP 01
Original
2012
Earlier work this paper cites.
D. R. Morrison and D. S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds , JHEP 10
Original
2012
Earlier work this paper cites.
D. S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry , JHEP 01
Original
2012
Earlier work this paper cites.
A. P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles , Nucl. Phys. B856
Original
2012
Earlier work this paper cites.