Abstract
A large number of examples of compact G 2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes. In particular, this enables us to prove combinatorial formulas for the Hodge numbers and other relevant topological data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R.L. Bryant and S.M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989) 829.
G.W. Gibbons, D.N. Page and C.N. Pope, Einstein Metrics on S 3 R 3 and R 4 Bundles, Commun. Math. Phys. 127 (1990) 529 [INSPIRE].
M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, Cohomogeneity one manifolds of spin(7) and G 2 holonomy, Phys. Rev. D 65 (2002) 106004 [hep-th/0108245] [INSPIRE].
M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, M theory conifolds, Phys. Rev. Lett. 88 (2002) 121602 [hep-th/0112098] [INSPIRE].
D. Joyce, Compact Manifolds with Special Holonomy, Oxford mathematical monographs, Oxford University Press, (2000).
A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003) 125 [INSPIRE].
A. Corti, M. Haskins, J. Nordström and T. Pacini, Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds, Geom. Topology 17 (2013) 1955 [arXiv:1206.2277].
A. Corti, M. Haskins, J. Nordström and T. Pacini, G2 -manifolds and associative submanifolds via semi-Fano 3-folds, Duke Math. J. 164 (2015) 1971 [arXiv:1207.4470] [INSPIRE].
A. Kovalev and N.-H. Lee, K3 surfaces with non-symplectic involution and compact irreducible g 2 -manifolds, Math. Proc. Cambridge Philos. Soc. 151 (2011) 193.
V.V. Nikulin, On factor groups of the automorphism group of hyperbolic forms modulo subgroups generated by 2-reflections, Soviet. Math. Dokl. 20 (1979) 1156.
V.V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. algebro-geometric applications, J. Soviet Math. 22 (1983) 1401.
V.V. Nikulin, Discrete reflection groups in lobachevsky spaces and algebraic surfaces, in Proceedings of the International Congress of Mathematicians at Berkeley, 1986, vol. 1, 2, Amer. Math. Soc., (1987).
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].
A. Klemm, W. Lerche and P. Mayr, K3 Fibrations and heterotic type-II string duality, Phys. Lett. B 357 (1995) 313 [hep-th/9506112] [INSPIRE].
S. Hosono, B.H. Lian and S.-T. Yau, GKZ generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Commun. Math. Phys. 182 (1996) 535 [alg-geom/9511001] [INSPIRE].
A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, Searching for K3 fibrations, Nucl. Phys. B 494 (1997) 567 [hep-th/9610154] [INSPIRE].
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
M. Kreuzer and H. Skarke, Reflexive polyhedra, weights and toric Calabi-Yau fibrations, Rev. Math. Phys. 14 (2002) 343 [math/0001106] [INSPIRE].
P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].
M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].
P. Candelas, A. Constantin and H. Skarke, An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts, Commun. Math. Phys. 324 (2013) 937 [arXiv:1207.4792] [INSPIRE].
V.I. Danilov and A.G. Khovanskii, Newton Polyhedra and an Algorithm for Computing Hodge-Deligne Numbers, Math. USSR Izvestija 29 (1987) 279.
J. Halverson and D.R. Morrison, The landscape of M-theory compactifications on seven-manifolds with G 2 holonomy, JHEP 04 (2015) 047 [arXiv:1412.4123] [INSPIRE].
D. Crowley, S. Goette and J. Nordström, An analytic invariant of G 2 manifolds, arXiv:1505.02734.
M. Kreuzer, Toric geometry and Calabi-Yau compactifications, Ukr. J. Phys. 55 (2010) 613 [hep-th/0612307] [INSPIRE].
W. Fulton, Introduction to toric varieties, Princeton University Press, Princeton, U.S.A. (1993).
V.I. Danilov, The geometry of toric varieties, Russ. Math. Surv. 33 (1978) 97.
V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification of tops, Adv. Theor. Math. Phys. 7 (2003) 205 [hep-th/0303218] [INSPIRE].
J. De Loera, J. Rambau and F. Santos, Triangulations, Springer, (2010).
R. Davis et al., Short tops and semistable degenerations, Exper. Math. 23 (2014) 351 [arXiv:1307.6514].
P. Candelas, E. Perevalov and G. Rajesh, Comments on A, B, C chains of heterotic and type-II vacua, Nucl. Phys. B 502 (1997) 594 [hep-th/9703148] [INSPIRE].
E. Perevalov and H. Skarke, Enhanced gauged symmetry in type-II and F theory compactifications: Dynkin diagrams from polyhedra, Nucl. Phys. B 505 (1997) 679 [hep-th/9704129] [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].
J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) Tops with Multiple U(1)s in F-theory, Nucl. Phys. B 882 (2014) 1 [arXiv:1307.2902] [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions II: From Coulomb Phases to Fiber Faces, Nucl. Phys. B 905 (2016) 480 [arXiv:1511.01801] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].
A. Clingher, R. Donagi and M. Wijnholt, The Sen Limit, Adv. Theor. Math. Phys. 18 (2014) 613 [arXiv:1212.4505] [INSPIRE].
V.V. Batyrev and M. Kreuzer, Integral Cohomology and Mirror Symmetry for Calabi-Yau threefolds, math/0505432.
Y.-H. He, S.-J. Lee, A. Lukas and C. Sun, Heterotic Model Building: 16 Special Manifolds, JHEP 06 (2014) 077 [arXiv:1309.0223] [INSPIRE].
W. Stein et al., Sage Mathematics Software (Version 6.7), The Sage Development Team, (2015), http://www.sagemath.org.
J. Halverson and D.R. Morrison, On gauge enhancement and singular limits in G 2 compactifications of M-theory, JHEP 04 (2016) 100 [arXiv:1507.05965] [INSPIRE].
V. Kulikov, Degenerations of k3 surfaces and enriques surfaces, Math. USSR Izvestija 11 (1977) 957.
R. Friedman and D.R. Morrison, The birational geometry of degenerations: an overview, in The birational geometry of degenerations, Cambridge U.S.A. (1981), vol. 29 of Progr. Math., pp. 1–32, Birkhäuser, Boston, U.S.A. (1983).
V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izvestija 14 (1980) 103.
H. Pinkham, Singularités exceptionnelles, la dualité érange d’Arnold et les surfaces K3, C.R. Acad. Sci. Paris. Ser. A-B 284 (1977) 615.
I. Dolgachev and V. Nikulin, Exceptional singularities of V.I. Arnold and K3 surfaces, Proceedings of the USSR Topological Conference in Minsk, (1977).
I. Dolgachev, Integral quadratic forms:applications to algebraic geometry, n° 611, Asterisque, Soc. Math. France 105/106 (1982/83) 251.
P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, hep-th/9404151 [INSPIRE].
I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, alg-geom/9502005 [INSPIRE].
U. Bruzzo, A. Grassi, Picard group of hypersurfaces in toric 3-folds, Int. J. Math. 23 (2012) 1250028 [arXiv:1011.1003].
F. Rohsiepe, Lattice polarized toric K3 surfaces, hep-th/0409290 [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].
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: 1602.03521
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Braun, A.P. Tops as building blocks for G 2 manifolds. J. High Energ. Phys. 2017, 83 (2017). https://doi.org/10.1007/JHEP10(2017)083
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)083