Abstract
We establish an orientifold Calabi-Yau threefold database for h1,1(X) ≤ 6 by considering non-trivial ℤ2 divisor exchange involutions, using a toric Calabi-Yau database (www.rossealtman.com/tcy). We first determine the topology for each individual divisor (Hodge diamond), then identify and classify the proper involutions which are globally consistent across all disjoint phases of the Kähler cone for each unique geometry. Each of the proper involutions will result in an orientifold Calabi-Yau manifold. Then we clarify all possible fixed loci under the proper involution, thereby determining the locations of different types of O-planes. It is shown that under the proper involutions, one typically ends up with a system of O3/O7-planes, and most of these will further admit naive Type IIB string vacua. The geometries with freely acting involutions are also determined. We further determine the splitting of the Hodge numbers into odd/even parity in the orbifold limit. The final result is a class of orientifold Calabi-Yau threefolds with non-trivial odd class cohomology (\( {h}_{-}^{1,1} \)(X/σ*) ≠ 0).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
X. Gao and P. Shukla, On Classifying the Divisor Involutions in Calabi-Yau Threefolds, JHEP 11 (2013) 170 [arXiv:1307.1139] [INSPIRE].
R. Altman, J. Gray, Y.-H. He, V. Jejjala and B.D. Nelson, A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List, JHEP 02 (2015) 158 [arXiv:1411.1418] [INSPIRE].
M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2000) 1209 [hep-th/0002240] [INSPIRE].
M. Graña, Flux compactifications in string theory: A Comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].
H. Jockers, The Effective Action of D-branes in Calabi-Yau Orientifold Compactifications, Fortsch. Phys. 53 (2005) 1087 [hep-th/0507042] [INSPIRE].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].
D. Lüst, S. Reffert, E. Scheidegger, W. Schulgin and S. Stieberger, Moduli Stabilization in Type IIB Orientifolds (II), Nucl. Phys. B 766 (2007) 178 [hep-th/0609013] [INSPIRE].
F. Denef, Les Houches Lectures on Constructing String Vacua, Les Houches 87 (2008) 483 [arXiv:0803.1194] [INSPIRE].
R. Blumenhagen, V. Braun, T.W. Grimm and T. Weigand, GUTs in Type IIB Orientifold Compactifications, Nucl. Phys. B 815 (2009) 1 [arXiv:0811.2936] [INSPIRE].
L.E. Ibanez and A.M. Uranga, String theory and particle physics: An introduction to string phenomenology, Cambridge University Press, Cambridge U.K. (2012).
A. Hebecker, Lectures on Naturalness, String Landscape and Multiverse, arXiv:2008.10625 [INSPIRE].
B.S. Acharya, M. Aganagic, K. Hori and C. Vafa, Orientifolds, mirror symmetry and superpotentials, hep-th/0202208 [INSPIRE].
I. Brunner and K. Hori, Orientifolds and mirror symmetry, JHEP 11 (2004) 005 [hep-th/0303135] [INSPIRE].
H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys. B 718 (2005) 203 [hep-th/0502059] [INSPIRE].
X. Gao and P. Shukla, F-term Stabilization of Odd Axions in LARGE Volume Scenario, Nucl. Phys. B 878 (2014) 269 [arXiv:1307.1141] [INSPIRE].
M. Cicoli, A. Schachner and P. Shukla, Systematics of type IIB moduli stabilisation with odd axions, arXiv:2109.14624 [INSPIRE].
S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, de Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].
V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactifications, JHEP 03 (2005) 007 [hep-th/0502058] [INSPIRE].
T.W. Grimm, Axion inflation in type-II string theory, Phys. Rev. D 77 (2008) 126007 [arXiv:0710.3883] [INSPIRE].
L. McAllister, E. Silverstein and A. Westphal, Gravity Waves and Linear Inflation from Axion Monodromy, Phys. Rev. D 82 (2010) 046003 [arXiv:0808.0706] [INSPIRE].
R. Flauger, L. McAllister, E. Pajer, A. Westphal and G. Xu, Oscillations in the CMB from Axion Monodromy Inflation, JCAP 06 (2010) 009 [arXiv:0907.2916] [INSPIRE].
A. Hebecker, S.C. Kraus, D. Lüst, S. Steinfurt and T. Weigand, Fluxbrane Inflation, Nucl. Phys. B 854 (2012) 509 [arXiv:1104.5016] [INSPIRE].
M. Arends et al., D7-Brane Moduli Space in Axion Monodromy and Fluxbrane Inflation, Fortsch. Phys. 62 (2014) 647 [arXiv:1405.0283] [INSPIRE].
R. Blumenhagen and E. Plauschinn, Towards Universal Axion Inflation and Reheating in String Theory, Phys. Lett. B 736 (2014) 482 [arXiv:1404.3542] [INSPIRE].
F. Marchesano, G. Shiu and A.M. Uranga, F-term Axion Monodromy Inflation, JHEP 09 (2014) 184 [arXiv:1404.3040] [INSPIRE].
A. Hebecker, S.C. Kraus and L.T. Witkowski, D7-Brane Chaotic Inflation, Phys. Lett. B 737 (2014) 16 [arXiv:1404.3711] [INSPIRE].
I. Ben-Dayan, F.G. Pedro and A. Westphal, Hierarchical Axion Inflation, Phys. Rev. Lett. 113 (2014) 261301 [arXiv:1404.7773] [INSPIRE].
C. Long, L. McAllister and P. McGuirk, Aligned Natural Inflation in String Theory, Phys. Rev. D 90 (2014) 023501 [arXiv:1404.7852] [INSPIRE].
X. Gao, T. Li and P. Shukla, Combining Universal and Odd RR Axions for Aligned Natural Inflation, JCAP 10 (2014) 048 [arXiv:1406.0341] [INSPIRE].
I. Ben-Dayan, F.G. Pedro and A. Westphal, Towards Natural Inflation in String Theory, Phys. Rev. D 92 (2015) 023515 [arXiv:1407.2562] [INSPIRE].
G. Shiu, W. Staessens and F. Ye, Large Field Inflation from Axion Mixing, JHEP 06 (2015) 026 [arXiv:1503.02965] [INSPIRE].
D. Escobar, A. Landete, F. Marchesano and D. Regalado, D6-branes and axion monodromy inflation, JHEP 03 (2016) 113 [arXiv:1511.08820] [INSPIRE].
R. Blumenhagen, D. Herschmann and F. Wolf, String Moduli Stabilization at the Conifold, JHEP 08 (2016) 110 [arXiv:1605.06299] [INSPIRE].
A. Landete, F. Marchesano, G. Shiu and G. Zoccarato, Flux Flattening in Axion Monodromy Inflation, JHEP 06 (2017) 071 [arXiv:1703.09729] [INSPIRE].
R. Blumenhagen, I. Valenzuela and F. Wolf, The Swampland Conjecture and F-term Axion Monodromy Inflation, JHEP 07 (2017) 145 [arXiv:1703.05776] [INSPIRE].
A. Hebecker, S. Leonhardt, J. Moritz and A. Westphal, Thraxions: Ultralight Throat Axions, JHEP 04 (2019) 158 [arXiv:1812.03999] [INSPIRE].
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
T.D. Brennan, F. Carta and C. Vafa, The String Landscape, the Swampland, and the Missing Corner, PoS TASI2017 (2017) 015 [arXiv:1711.00864] [INSPIRE].
G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, de Sitter Space and the Swampland, arXiv:1806.08362 [INSPIRE].
E. Palti, The Swampland: Introduction and Review, Fortsch. Phys. 67 (2019) 1900037 [arXiv:1903.06239] [INSPIRE].
R. Blumenhagen, S. Moster and E. Plauschinn, Moduli Stabilisation versus Chirality for MSSM like Type IIB Orientifolds, JHEP 01 (2008) 058 [arXiv:0711.3389] [INSPIRE].
A. Collinucci, M. Kreuzer, C. Mayrhofer and N.-O. Walliser, Four-modulus ’Swiss Cheese’ chiral models, JHEP 07 (2009) 074 [arXiv:0811.4599] [INSPIRE].
T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, On Fluxed Instantons and Moduli Stabilisation in IIB Orientifolds and F-theory, Phys. Rev. D 84 (2011) 066001 [arXiv:1105.3193] [INSPIRE].
M. Kerstan and T. Weigand, Fluxed M5-instantons in F-theory, Nucl. Phys. B 864 (2012) 597 [arXiv:1205.4720] [INSPIRE].
M. Cicoli, C. Mayrhofer and R. Valandro, Moduli Stabilisation for Chiral Global Models, JHEP 02 (2012) 062 [arXiv:1110.3333] [INSPIRE].
V. Balasubramanian, P. Berglund, V. Braun and I. Garcia-Etxebarria, Global embeddings for branes at toric singularities, JHEP 10 (2012) 132 [arXiv:1201.5379] [INSPIRE].
M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, D-Branes at del Pezzo Singularities: Global Embedding and Moduli Stabilisation, JHEP 09 (2012) 019 [arXiv:1206.5237] [INSPIRE].
M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, D3/D7 Branes at Singularities: Constraints from Global Embedding and Moduli Stabilisation, JHEP 07 (2013) 150 [arXiv:1304.0022] [INSPIRE].
M. Cicoli, F. Muia and P. Shukla, Global Embedding of Fibre Inflation Models, JHEP 11 (2016) 182 [arXiv:1611.04612] [INSPIRE].
M. Cicoli, I. Garcìa-Etxebarria, C. Mayrhofer, F. Quevedo, P. Shukla and R. Valandro, Global Orientifolded Quivers with Inflation, JHEP 11 (2017) 134 [arXiv:1706.06128] [INSPIRE].
M. Cicoli, D. Ciupke, V.A. Diaz, V. Guidetti, F. Muia and P. Shukla, Chiral Global Embedding of Fibre Inflation Models, JHEP 11 (2017) 207 [arXiv:1709.01518] [INSPIRE].
M. Cicoli, I.G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla and R. Valandro, The Standard Model quiver in de Sitter string compactifications, JHEP 08 (2021) 109 [arXiv:2106.11964] [INSPIRE].
J. Gray, Y.-H. He, V. Jejjala, B. Jurke, B.D. Nelson and J. Simon, Calabi-Yau Manifolds with Large Volume Vacua, Phys. Rev. D 86 (2012) 101901 [arXiv:1207.5801] [INSPIRE].
C. Long, L. McAllister and P. McGuirk, Heavy Tails in Calabi-Yau Moduli Spaces, JHEP 10 (2014) 187 [arXiv:1407.0709] [INSPIRE].
Y.-H. He, V. Jejjala and L. Pontiggia, Patterns in Calabi-Yau Distributions, Commun. Math. Phys. 354 (2017) 477 [arXiv:1512.01579] [INSPIRE].
R. Galvez, Kähler Moduli Inflation in Type IIB Compactifications: A random tumble through the Calabi-Yau landscape, Phys. Rev. D 94 (2016) 103521 [arXiv:1603.06631] [INSPIRE].
C. Long, L. McAllister and J. Stout, Systematics of Axion Inflation in Calabi-Yau Hypersurfaces, JHEP 02 (2017) 014 [arXiv:1603.01259] [INSPIRE].
R. Altman, Y.-H. He, V. Jejjala and B.D. Nelson, New large volume Calabi-Yau threefolds, Phys. Rev. D 97 (2018) 046003 [arXiv:1706.09070] [INSPIRE].
M. Demirtas, C. Long, L. McAllister and M. Stillman, The Kreuzer-Skarke Axiverse, JHEP 04 (2020) 138 [arXiv:1808.01282] [INSPIRE].
J. Halverson, C. Long, B. Nelson and G. Salinas, Towards string theory expectations for photon couplings to axionlike particles, Phys. Rev. D 100 (2019) 106010 [arXiv:1909.05257] [INSPIRE].
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].
F. Carta, J. Moritz and A. Westphal, A landscape of orientifold vacua, JHEP 05 (2020) 107 [arXiv:2003.04902] [INSPIRE].
F. Carta, A. Mininno, N. Righi and A. Westphal, Thraxions: towards full string models, JHEP 01 (2022) 082 [arXiv:2110.02963] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY Threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
V. Braun, On Free Quotients of Complete Intersection Calabi-Yau Manifolds, JHEP 04 (2011) 005 [arXiv:1003.3235] [INSPIRE].
J. Gray, A.S. Haupt and A. Lukas, All Complete Intersection Calabi-Yau Four-Folds, JHEP 07 (2013) 070 [arXiv:1303.1832] [INSPIRE].
P. Candelas, A. Constantin and C. Mishra, Hodge Numbers for CICYs with Symmetries of Order Divisible by 4, Fortsch. Phys. 64 (2016) 463 [arXiv:1511.01103] [INSPIRE].
A. Constantin, J. Gray and A. Lukas, Hodge Numbers for All CICY Quotients, JHEP 01 (2017) 001 [arXiv:1607.01830] [INSPIRE].
A. Braun, A. Lukas and C. Sun, Discrete Symmetries of Calabi–Yau Hypersurfaces in Toric Four-Folds, Commun. Math. Phys. 360 (2018) 935 [arXiv:1704.07812] [INSPIRE].
Y.-H. He, Calabi-Yau Spaces in the String Landscape, arXiv:2006.16623 [INSPIRE].
R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, A Note on Poly-Instanton Effects in Type IIB Orientifolds on Calabi-Yau Threefolds, JHEP 06 (2012) 162 [arXiv:1205.2485] [INSPIRE].
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].
K. Oguiso and T. Peternell, Calabi-Yau threefolds with positive second Chern class,” Commun. Anal. Geom. 6 (1998) 153.
V. Lazić, K. Oguiso and T. Peternell, The Morrison-Kawamata Cone Conjecture and Abundance on Ricci flat manifolds, arXiv:1611.00556.
C. Wall, Classification problems in differential topology. V, Invent. Math. 1 (1966) 355.
R. Blumenhagen, B. Jurke, T. Rahn and H. Roschy, Cohomology of Line Bundles: A Computational Algorithm, J. Math. Phys. 51 (2010) 103525 [arXiv:1003.5217] [INSPIRE].
cohomCalg package, Download link: http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg/ (2010).
A. Collinucci, New F-theory lifts. II. Permutation orientifolds and enhanced singularities, JHEP 04 (2010) 076 [arXiv:0906.0003] [INSPIRE].
P. Shanahan, The Atiyah-Singer Index Theorem: An Introduction, Springer-Verlag, Heidelberg Germany (1978).
R. Blumenhagen, A. Collinucci and B. Jurke, On Instanton Effects in F-theory, JHEP 08 (2010) 079 [arXiv:1002.1894] [INSPIRE].
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-6 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de.
W. Stein et al., Sage Mathematics Software (Version 9.1), The Sage Development Team, http://www.sagemath.org (2020).
R. Altman, J. Carifio, X. Gao and B. D. Nelson, Type II Orientifold Vacua in Kreuzer-Skarke Database, work in progress.
M. Demirtas, L. McAllister and A. Rios-Tascon, Bounding the Kreuzer-Skarke Landscape, Fortsch. Phys. 68 (2020) 2000086 [arXiv:2008.01730] [INSPIRE].
R. Altman, J. Carifio, J. Halverson and B.D. Nelson, Estimating Calabi-Yau Hypersurface and Triangulation Counts with Equation Learners, JHEP 03 (2019) 186 [arXiv:1811.06490] [INSPIRE].
X. Gao and H. Zou, Applying machine learning to the Calabi-Yau orientifolds with string vacua, Phys. Rev. D 105 (2022) 046017 [arXiv:2112.04950] [INSPIRE].
P. Candelas and X.C. de la Ossa, Comments on Conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].
P. Candelas, P.S. Green and T. Hubsch, Rolling Among Calabi-Yau Vacua, Nucl. Phys. B 330 (1990) 49 [INSPIRE].
L.B. Anderson, F. Apruzzi, X. Gao, J. Gray and S.-J. Lee, A new construction of Calabi–Yau manifolds: Generalized CICYs, Nucl. Phys. B 906 (2016) 441 [arXiv:1507.03235] [INSPIRE].
P. Berglund and T. Hübsch, On Calabi–Yau generalized complete intersections from Hirzebruch varieties and novel K 3-fibrations, Adv. Theor. Math. Phys. 22 (2018) 261 [arXiv:1606.07420] [INSPIRE].
P. Berglund and T. Hubsch, A Generalized Construction of Calabi-Yau Models and Mirror Symmetry, SciPost Phys. 4 (2018) 009 [arXiv:1611.10300] [INSPIRE].
P. Candelas, A. Constantin and C. Mishra, Calabi-Yau Threefolds with Small Hodge Numbers, Fortsch. Phys. 66 (2018) 1800029 [arXiv:1602.06303] [INSPIRE].
A. Garbagnati and B. van Geemen, A remark on generalized complete intersections, Nucl. Phys. B 925 (2017) 135 [arXiv:1708.00517] [INSPIRE].
Q. Jia and H. Lin, Calabi-Yau generalized complete intersections and aspects of cohomology of sheaves, J. Math. Phys. 61 (2020) 052301 [arXiv:1809.04714] [INSPIRE].
L.B. Anderson, F. Apruzzi, X. Gao, J. Gray and S.-J. Lee, Instanton superpotentials, Calabi-Yau geometry, and fibrations, Phys. Rev. D 93 (2016) 086001 [arXiv:1511.05188] [INSPIRE].
M. Larfors, D. Passaro and R. Schneider, Heterotic Line Bundle Models on Generalized Complete Intersection Calabi Yau Manifolds, JHEP 05 (2021) 105 [arXiv:2010.09763] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2111.03078
Rights and permissions
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.
About this article
Cite this article
Altman, R., Carifio, J., Gao, X. et al. Orientifold Calabi-Yau threefolds with divisor involutions and string landscape. J. High Energ. Phys. 2022, 87 (2022). https://doi.org/10.1007/JHEP03(2022)087
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2022)087