×

zbMATH — the first resource for mathematics

Hodge numbers for CICYs with symmetries of order divisible by 4. (English) Zbl 1339.14023
Summary: We compute the Hodge numbers for the quotients of complete intersection Calabi-Yau three-folds by groups of orders divisible by 4. We make use of the polynomial deformation method and the counting of invariant Kähler classes. The quotients studied here have been obtained in the automated classification of V. Braun. Although the computer search found the freely acting groups, the Hodge numbers of the quotients were not calculated. The freely acting groups, \(G\), that arise in the classification are either \(\mathbb{Z}_2\) or contain \(\mathbb{Z}_4\), \(\mathbb{Z}_2 \times \mathbb{Z}_2\), \(\mathbb{Z}_3\) or \(\mathbb{Z}_5\) as a subgroup. The Hodge numbers for the quotients for which the group \(G\) contains \(\mathbb{Z}_3\) or \(\mathbb{Z}_5\) have been computed previously. This paper deals with the remaining cases, for which \(G \supseteq \mathbb{Z}_4\) or \(G \supseteq \mathbb{Z}_2 \times \mathbb{Z}_2\). We also compute the Hodge numbers for 99 of the 166 CICY’s which have \(\mathbb{Z}_2\) quotients.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q15 Kähler manifolds
32G20 Period matrices, variation of Hodge structure; degenerations
58A14 Hodge theory in global analysis
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Candelas, Vacuum Configurations for Superstrings, Nucl. Phys. B258 pp 46– (1985) · doi:10.1016/0550-3213(85)90602-9
[2] Braun, A Heterotic standard model, Phys. Lett. B618 pp 252– (2005) · Zbl 1247.81349 · doi:10.1016/j.physletb.2005.05.007
[3] Braun, A Standard model from the E(8) x E(8) heterotic superstring, JHEP 0506 pp 039– (2005) · doi:10.1088/1126-6708/2005/06/039
[4] Anderson, Exploring Positive Monad Bundles And a New Heterotic Standard Model, JHEP 1002 pp 054– (2010) · Zbl 1270.81146 · doi:10.1007/JHEP02(2010)054
[5] Bouchard, An SU(5) heterotic standard model, Phys. Lett. B633 pp 783– (2006) · Zbl 1247.81348 · doi:10.1016/j.physletb.2005.12.042
[6] Braun, The MSSM Spectrum from (0,2)-Deformations of the Heterotic Standard Embedding, JHEP 1205 pp 127– (2012) · Zbl 1348.81435 · doi:10.1007/JHEP05(2012)127
[7] Anderson, Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds, Phys. Rev. D84 pp 106005– (2011)
[8] Anderson, Heterotic Line Bundle Standard Models, JHEP 1206 pp 113– (2012) · Zbl 1397.81406 · doi:10.1007/JHEP06(2012)113
[9] Anderson, A Comprehensive Scan for Heterotic SU(5) GUT models, JHEP 01 pp 047– (2014) · Zbl 06564268 · doi:10.1007/JHEP01(2014)047
[10] Gross, Calabi-Yau threefolds and moduli of abelian surfaces I, Compositio Mathematica 127 (2) pp 169– (2001) · Zbl 1063.14051 · doi:10.1023/A:1012076503121
[11] Candelas, New Calabi-Yau Manifolds with Small Hodge Numbers, Fortsch. Phys. 58 pp 383– (2010) · Zbl 1194.14062 · doi:10.1002/prop.200900105
[12] Braun, On Free Quotients of Complete Intersection Calabi-Yau Manifolds, JHEP 1104 pp 005– (2011) · Zbl 1250.14026 · doi:10.1007/JHEP04(2011)005
[13] Candelas, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B298 pp 493– (1988) · doi:10.1016/0550-3213(88)90352-5
[14] Candelas, Completing the Web of Z3 - Quotients of Complete Intersection Calabi-Yau Manifolds, Fortsch. Phys. 60 pp 345– (2012) · Zbl 1243.81147 · doi:10.1002/prop.201200044
[15] CICY list, compiled by Andre Lukas Includes Hodge numbers and freely-acting discrete symmetries http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/cicylist/index.html
[16] Hubsch, Calabi-Yau manifolds: A Bestiary for physicists (1994)
[17] He, On the Number of Complete Intersection Calabi-Yau Manifolds, Commun. Math. Phys. 135 pp 193– (1990) · Zbl 0722.53061 · doi:10.1007/BF02097661
[18] S.-T. Yau Compact Three-Dimensional Kähler Manifolds with Zero Ricci Curvature In *Argonne/chicago 1985, Proceedings, Anomalies, Geometry, Topology*, 395-406
[19] Green, All Hodge Numbers of All Complete Intersection Calabi-Yau Manifolds, Class. Quant. Grav. 6 pp 105– (1989) · Zbl 0657.53063 · doi:10.1088/0264-9381/6/2/006
[20] Rodland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian G(2, 7), Compositio Mathematica 122 (02) pp 135– (2000) · Zbl 0974.14026 · doi:10.1023/A:1001847914402
[21] Kreuzer, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 pp 1209– (2002) · Zbl 1017.52007 · doi:10.4310/ATMP.2000.v4.n6.a2
[22] Kreuzer, Toric complete intersections and weighted projective space, J. Geom. Phys. 46 pp 159– (2003) · Zbl 1061.14037 · doi:10.1016/S0393-0440(02)00124-9
[23] Klemm, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 pp 023– (2005) · doi:10.1088/1126-6708/2005/05/023
[24] Tonoli, Construction of Calabi-Yau 3-folds in P6, J. Algebraic Geometry 13 pp 209– (2004) · Zbl 1060.14060 · doi:10.1090/S1056-3911-03-00371-0
[25] Kreuzer, Classification of toric Fano 5-folds, Adv. Geom. 9 pp 85– (2009) · Zbl 1193.14067 · doi:10.1515/ADVGEOM.2009.005
[26] Candelas, Triadophilia: A Special Corner in the Landscape, Adv. Theor. Math. Phys. 12 pp 429– (2008) · Zbl 1144.81499 · doi:10.4310/ATMP.2008.v12.n2.a6
[27] Hua, Classification of free actions on complete intersections of four quadrics, Advances in Theoretical and Mathematical Physics 15 (4) pp 973– (2011) · Zbl 1252.14025 · doi:10.4310/ATMP.2011.v15.n4.a2
[28] Kapustka, Primitive contractions of Calabi-Yau threefolds II, Journal of the London Mathematical Society 79 (1) pp 259– (2009) · Zbl 1170.14025 · doi:10.1112/jlms/jdn069
[29] Bouchard, On a class of non-simply connected Calabi-Yau threefolds, Commun. Num. Theor. Phys. 2 pp 1– (2008) · Zbl 1165.14032 · doi:10.4310/CNTP.2008.v2.n1.a1
[30] Batyrev, Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions, Adv. Theor. Math. Phys. 14 pp 879– (2010) · Zbl 1242.14037 · doi:10.4310/ATMP.2010.v14.n3.a3
[31] Braun, A Three-Generation Calabi-Yau Manifold with Small Hodge Numbers, Fortsch. Phys. 58 pp 467– (2010) · Zbl 1194.14061 · doi:10.1002/prop.200900106
[32] Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds, Advances in Geometry 15 (2) pp 143– (2015) · Zbl 1326.14083 · doi:10.1515/advgeom-2015-0002
[33] A. Garbagnati New families of Calabi-Yau 3-folds without maximal unipotent monodromy 1005.0094
[34] A. Stapledon New Mirror Pairs of Calabi-Yau Orbifolds 1011.5006
[35] Davies, The Expanding Zoo of Calabi-Yau Threefolds, Adv. High Energy Phys. 2011 pp 901898– (2011) · Zbl 1234.81110 · doi:10.1155/2011/901898
[36] Davies, Hyperconifold Transitions, Mirror Symmetry, and String Theory, Nucl. Phys. B850 pp 214– (2011) · Zbl 1215.81081 · doi:10.1016/j.nuclphysb.2011.04.010
[37] Braun, The 24-Cell and Calabi-Yau Threefolds with Hodge Numbers (1,1), JHEP 1205 pp 101– (2012) · Zbl 1348.14104 · doi:10.1007/JHEP05(2012)101
[38] E. Freitag R. Salvati Manni On Siegel threefolds with a projective Calabi-Yau model 1103.2040
[39] Filippini, A Rigid Calabi-Yau 3-fold, Adv. Theor. Math. Phys. 15 pp 1745– (2011) · Zbl 1435.14036 · doi:10.4310/ATMP.2011.v15.n6.a4
[40] L. A. Borisov H. J. Nuer On (2,4) complete intersection threefolds that contain an Enriques surface 1210.1903
[41] Bini, Groups acting freely on Calabi-Yau threefolds embedded in a product of del Pezzo surfaces, Adv. Theor. Math. Phys. 16 pp 887– (2012) · Zbl 1271.14054 · doi:10.4310/ATMP.2012.v16.n3.a4
[42] L. B. Anderson F. Apruzzi X. Gao J. Gray S.-J. Lee A New Construction of Calabi-Yau Manifolds: Generalized CICYs 1507.03235 · Zbl 1334.14023
[43] Witten, Symmetry Breaking Patterns in Superstring Models, Nucl. Phys. B258 pp 75– (1985) · doi:10.1016/0550-3213(85)90603-0
[44] Green, Polynomial Deformations and Cohomology of Calabi-yau Manifolds, Commun. Math. Phys. 113 pp 505– (1987) · Zbl 0633.53089 · doi:10.1007/BF01221257
[45] Anderson, Vacuum varieties, holomorphic bundles and complex structure stabilization in heterotic theories, Journal of High Energy Physics 2013 (7) pp 1– (2013) · Zbl 1342.81391 · doi:10.1007/JHEP07(2013)017
[46] Dolgachev, Classical algebraic geometry. A modern view (2012) · Zbl 1252.14001 · doi:10.1017/CBO9781139084437
[47] M. A. Reid The complete intersection of two or more quadrics PhD thesis University of Cambridge 1972
[48] Buchbinder, The Moduli Space of Heterotic Line Bundle Models: a Case Study for the Tetra-Quadric, JHEP 1403 pp 025– (2014) · Zbl 06564603 · doi:10.1007/JHEP03(2014)025
[49] Buchbinder, A heterotic standard model with B-L symmetry and a stable proton, JHEP 1406 pp 100– (2014) · Zbl 1390.81572 · doi:10.1007/JHEP06(2014)100
[50] Buchbinder, Non-generic Couplings in Supersymmetric Standard Models, Phys. Lett. B748 pp 251– (2015) · Zbl 1345.81138 · doi:10.1016/j.physletb.2015.07.012
[51] A. Constantin A. Lukas C. Mishra The Family Problem: Hints from Heterotic Line Bundle Models 1509.02729 · Zbl 1388.81512
[52] Buchbinder, Heterotic QCD axion, Phys. Rev. D91 (4) pp 046010– (2015)
[53] Greene, A Three Generation Superstring Model. 1. Compactification and Discrete Symmetries, Nucl. Phys. B278 pp 667– (1986) · doi:10.1016/0550-3213(86)90057-X
[54] Greene, A Three Generation Superstring Model. 2. Symmetry Breaking and the Low-Energy Theory, Nucl. Phys. B292 pp 606– (1987) · doi:10.1016/0550-3213(87)90662-6
[55] Donagi, Standard model bundles on nonsimply connected Calabi-Yau threefolds, JHEP 0108 pp 053– (2001) · doi:10.1088/1126-6708/2001/08/053
[56] Donagi, Standard model bundles, Adv. Theor. Math. Phys. 5 pp 563– (2002) · Zbl 1027.14005 · doi:10.4310/ATMP.2001.v5.n3.a5
[57] Donagi, The Spectra of heterotic standard model vacua, JHEP 0506 pp 070– (2005) · doi:10.1088/1126-6708/2005/06/070
[58] Braun, The Exact MSSM spectrum from string theory, JHEP 0605 pp 043– (2006) · doi:10.1088/1126-6708/2006/05/043
[59] Davies, Quotients of the conifold in compact Calabi-Yau threefolds, and new topological transitions, Adv. Theor. Math. Phys. 14 pp 965– (2010) · Zbl 1242.14038 · doi:10.4310/ATMP.2010.v14.n3.a6
[60] R. Davies Classification and Properties of Hyperconifold Singularities and Transitions 1309.6778
[61] Strominger, New Manifolds for Superstring Compactification, Commun. Math. Phys. 101 pp 341– (1985) · doi:10.1007/BF01216094
[62] Anderson, Heterotic Compactification, An Algorithmic Approach, JHEP 0707 pp 049– (2007) · doi:10.1088/1126-6708/2007/07/049
[63] L. B. Anderson A. Constantin S.-J. Lee A. Lukas Hypercharge Flux in Heterotic Compactifications 1411.0034
[64] He, Heterotic Models from Vector Bundles on Toric Calabi-Yau Manifolds, JHEP 1005 pp 071– (2010) · Zbl 1287.81094 · doi:10.1007/JHEP05(2010)071
[65] He, Heterotic Bundles on Calabi-Yau Manifolds with Small Picard Number, JHEP 1112 pp 039– (2011) · Zbl 1306.81246 · doi:10.1007/JHEP12(2011)039
[66] Y.-H. He S.-J. Lee A. Lukas C. Sun Heterotic Model Building: 16 Special Manifolds 1309.0223
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.