Abstract
M-theory on a Calabi-Yau threefold admitting a small resolution gives rise to an Abelian vector multiplet and a charged hypermultiplet. We introduce into this picture a procedure to construct threefolds that naturally host matter with electric charges up to six. These are built as families of Du Val ADE surfaces (or ALE spaces), and the possible charges correspond to the Dynkin labels of the adjoint of the ADE algebra. In the case of charge two, we give a new derivation of the answer originally obtained by Curto and Morrison, and explicitly relate this construction to the Morrison-Park geometry. We also give a procedure for constructing higher-charge cases, which can often be applied to F-theory models.
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S.H. Katz, D.R. Morrison and M.R. Plesser, Enhanced gauge symmetry in type-II string theory, Nucl. Phys. B 477 (1996) 105 [hep-th/9601108] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
M. Bershadsky, A. Johansen, T. Pantev, V. Sadov and C. Vafa, F theory, geometric engineering and N = 1 dualities, Nucl. Phys. B 505 (1997) 153 [hep-th/9612052] [INSPIRE].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].
S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories: 1, Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
P. Mayr, Geometric construction of N = 2 gauge theories, Fortsch. Phys. 47 (1999) 39 [hep-th/9807096] [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].
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].
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].
S.S. Gubser, TASI lectures: special holonomy in string theory and M-theory, in Strings, branes and extra dimensions. TASI 2001: proceedings, (2002), pg. 197 [hep-th/0201114] [INSPIRE].
F.M. Cianci, D.K. Mayorga 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].
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].
N. Raghuram, Abelian F-theory models with charge-3 and charge-4 matter, JHEP 05 (2018) 050 [arXiv:1711.03210] [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 D.S. Park, Tall sections from non-minimal transformations, JHEP 10 (2016) 033 [arXiv:1606.07444] [INSPIRE].
C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP 03 (2013) 098 [arXiv:1211.6742] [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].
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].
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].
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. Raghuram and W. Taylor, Large U(1) charges in F-theory, JHEP 10 (2018) 182 [arXiv:1809.01666] [INSPIRE].
T. Weigand, F-theory, PoS(TASI2017)016 (2018) [arXiv:1806.01854] [INSPIRE].
M. Cvetič and L. Lin, TASI lectures on Abelian and discrete symmetries in F-theory, PoS(TASI2017)020 (2018) [arXiv:1809.00012] [INSPIRE].
D.R. Morrison, What is F-theory?, to appear.
D. Klevers and W. Taylor, Three-index symmetric matter representations of SU(2) in F-theory from non-Tate form Weierstrass models, JHEP 06 (2016) 171 [arXiv:1604.01030] [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].
P.B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Diff. Geom. 29 (1989) 665 [INSPIRE].
E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes Congrés Intern. Math. Nice 2 (1970) 279.
P. Slodowy, Simple singularities and simple algebraic groups, Lect. Notes Math. 815, Springer, Berlin, Heidelberg, Germany (1980).
P. Slodowy, Four lectures on simple groups and singularities, in Commun. Math. Inst. 11, Mathematical Institute, Rijksuniversiteit Utrecht, Utrecht, The Netherlands (1980).
S. Katz and D.R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Alg. Geom. 1 (1992) 449 [alg-geom/9202002].
H. Clemens, J. Kollár and S. Mori, Higher dimensional complex geometry, Astérisque 166, Société Mathématique de France, Paris, France (1988).
C. Curto and D.R. Morrison, Threefold flops via matrix factorization, J. Alg. Geom. 22 (2013) 599.
J. Karmazyn, The length classification of threefold flops via noncommutative algebras, arXiv:1709.02720.
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].
A.H. Durfee, Fifteen characterizations of rational double points and simple critical points, Enseign. Math. 25 (1979) 131.
P.B. Kronheimer, A Torelli type theorem for gravitational instantons, J. Diff. Geom. 29 (1989) 685 [INSPIRE].
D.R. Morrison, Some remarks on the moduli of K3 surfaces, in Classification of algebraic and analytic manifolds, K. Ueno ed., Progr. Math. 39, Birkhäuser, Boston, MA, U.S.A., Basel, Switzerland and Stuttgart, Germany (1983), pg. 303.
R. Kobayashi and A.N. Todorov, Polarized period map for generalized K3 surfaces and the moduli of Einstein metrics, Tôhoku Math. J. 39 (1987) 341.
S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation, I, Commun. Pure Appl. Math. 31 (1978) 339.
M.T. Anderson, The L2 structure of moduli spaces of Einstein metrics on 4-manifolds, Geom. Funct. Anal. 2 (1992) 29.
S.W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977) 81 [INSPIRE].
H.-J. Hein, Gravitational instantons from rational elliptic surfaces, J. Amer. Math. Soc. 25 (2012) 355.
G. Chen and X. Chen, Gravitational instantons with faster than quadratic curvature decay (I), arXiv:1505.01790 [INSPIRE].
G. Chen and X. Chen, Gravitational instantons with faster than quadratic curvature decay (II), arXiv:1508.07908 [INSPIRE].
G. Chen and X. Chen, Gravitational instantons with faster than quadratic curvature decay (III), arXiv:1603.08465 [INSPIRE].
H.-J. Hein, ALG and ALH spaces, seminar given at Metric and analytic aspects of moduli spaces, Isaac Newton Institute, University of Cambridge, Cambridge, U.K. (2015).
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds, Nucl. Phys. B 298 (1988) 493 [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].
H. Clemens, Double solids, Adv. Math. 47 (1983) 107 [INSPIRE].
R. Friedman, Simultaneous resolution of threefold double points, Math. Ann. 274 (1986) 671.
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1, Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [INSPIRE].
M.F. Atiyah, On analytic surfaces with double points, Proc. Roy. Soc. London Ser. A 247 (1958) 237.
M. Reid, Minimal models of canonical 3-folds, in Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, The Netherlands (1983), pg. 131.
F. Cachazo, S. Katz and C. Vafa, Geometric transitions and N = 1 quiver theories, hep-th/0108120 [INSPIRE].
A. Collinucci, M. Fazzi and R. Valandro, Geometric engineering on flops of length two, JHEP 04 (2018) 090 [arXiv:1802.00813] [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].
A.P. Braun, A. Collinucci and R. Valandro, The fate of U(1)’s at strong coupling in F-theory, JHEP 07 (2014) 028 [arXiv:1402.4054] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Quivers from matrix factorizations, Commun. Math. Phys. 313 (2012) 607 [arXiv:1005.1042] [INSPIRE].
R. Vakil, The rising sea: foundations of algebraic geometry, http://math.stanford.edu/∼vakil/216blog/index.html.
H.B. Laufer, On CP1 as an exceptional set, in Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, NJ, 1979), Ann. Math. Stud. 100, Princeton Univ. Press, Princeton, NJ, U.S.A. (1981), pg. 261.
H.C. Pinkham, Factorization of birational maps in dimension 3, in Singularities, part 2 (Arcata, CA, 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, RI, U.S.A. (1983), pg. 343.
D.R. Morrison, The birational geometry of surfaces with rational double points, Math. Ann. 271 (1985) 415.
M. Rossi, Geometric transitions, J. Geom. Phys. 56 (2006) 1940 [math.AG/0412514] [INSPIRE].
B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].
P. Candelas, P.S. Green and T. Hubsch, Rolling among Calabi-Yau vacua, Nucl. Phys. B 330 (1990) 49 [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].
W. Bosma, J. Cannon and C. Playoust, The magma algebra system. I. The user language, J. Symbol. Comput. 24 (1997) 235.
C. Curto, Matrix model superpotentials and ADE singularities, Adv. Theor. Math. Phys. 12 (2008) 353 [hep-th/0612172] [INSPIRE].
C. Curto, Matrix model superpotentials and Calabi-Yau spaces: an A-D-E classification, Ph.D. thesis, Duke University, ProQuest LLC, Ann Arbor, MI, U.S.A. (2005).
T. Ando, Some examples of simple small singularities, Commun. Alg. 41 (2013) 2193.
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Collinucci, A., Fazzi, M., Morrison, D.R. et al. High electric charges in M-theory from quiver varieties. J. High Energ. Phys. 2019, 111 (2019). https://doi.org/10.1007/JHEP11(2019)111
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DOI: https://doi.org/10.1007/JHEP11(2019)111