Abstract
We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in ref. [1]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there are at least 36,779 topologically distinct manifolds in our list. We also study the fibration structure of these manifolds and find that 99.95 percent can be described as elliptic fibrations. In total, we find 50,114,908 elliptic fibrations, demonstrating the multitude of ways in which many manifolds are fibered. A sub-class of 26,088,498 fibrations satisfy necessary conditions for admitting sections. The complete data set can be downloaded here.
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References
J. Gray, A.S. Haupt and A. Lukas, All Complete Intersection Calabi-Yau Four-Folds, JHEP 07 (2013) 070 [arXiv:1303.1832] [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].
M. Kreuzer and H. Skarke, PALP: A Package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun. 157 (2004) 87 [math/0204356] [INSPIRE].
T. Hübsch, Calabi-Yau Manifolds: Motivations and Constructions, Commun. Math. Phys. 108 (1987) 291 [INSPIRE].
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].
P. Green and T. Hübsch, Calabi-Yau Manifolds as Complete Intersections in Products of Complex Projective Spaces, Commun. Math. Phys. 109 (1987) 99 [INSPIRE].
P. Candelas, C.A. Lütken and R. Schimmrigk, Complete intersection calabi-yau manifolds. 2. three generation manifolds, Nucl. Phys. B 306 (1988) 113 [INSPIRE].
I. Brunner, M. Lynker and R. Schimmrigk, Unification of M-theory and F theory Calabi-Yau fourfold vacua, Nucl. Phys. B 498 (1997) 156 [hep-th/9610195] [INSPIRE].
A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys. B 518 (1998) 515 [hep-th/9701023] [INSPIRE].
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
M. Lynker, R. Schimmrigk and A. Wisskirchen, Landau-Ginzburg vacua of string, M-theory and F-theory at c = 12, Nucl. Phys. B 550 (1999) 123 [hep-th/9812195] [INSPIRE].
L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].
R. Hartshorne, Algebraic Geometry, Springer, New York (1977).
P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York (1978).
J. Distler and B.R. Greene, Aspects of (2,0) String Compactifications, Nucl. Phys. B 304 (1988) 1 [INSPIRE].
T. Hübsch, Calabi-Yau manifolds: A Bestiary for physicists, World Scientific, Singapore (1992).
L.B. Anderson, Heterotic and M-theory Compactifications for String Phenomenology, arXiv:0808.3621 [INSPIRE].
H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge (1986).
P.S. Green and T. Hübsch, (1, 1)3 Couplings in Calabi-Yau Threefolds, Class. Quant. Grav. 6 (1989) 311 [INSPIRE].
J. Kollar, Deformations of elliptic Calabi-Yau manifolds, arXiv:1206.5721 [INSPIRE].
The full CICY four-fold data set (configuration matrices, Euler characteristics, Hodge data, fibration structure) can be downloaded at http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html.
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].
P. Candelas, X. de la Ossa, Y.-H. He and B. Szendroi, Triadophilia: A Special Corner in the Landscape, Adv. Theor. Math. Phys. 12 (2008) 429 [arXiv:0706.3134] [INSPIRE].
N.C. Bizet, A. Klemm and D.V. Lopes, Landscaping with fluxes and the E8 Yukawa Point in F-theory, arXiv:1404.7645 [INSPIRE].
W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].
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Gray, J., Haupt, A.S. & Lukas, A. Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds. J. High Energ. Phys. 2014, 93 (2014). https://doi.org/10.1007/JHEP09(2014)093
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DOI: https://doi.org/10.1007/JHEP09(2014)093