Abstract
Resolutions of certain toroidal orbifolds, like T6/ℤ2 × ℤ2, are far from unique, due to triangulation dependence of their resolved local singularities. This leads to an explosion of the number of topologically distinct smooth geometries associated to a single orbifold. By introducing a parameterisation to keep track of the triangulations used at all resolved singularities simultaneously, (self-)intersection numbers and integrated Chern classes can be determined for any triangulation configuration. Using this method the consistency conditions of line bundle models and the resulting chiral spectra can be worked out for any choice of triangulation. Moreover, by superimposing the Bianchi identities for all triangulation options much simpler though stronger conditions are uncovered. When these are satisfied, flop-transitions between all different triangulations are admissible. Various methods are exemplified by a number of concrete models on resolutions of the T6/ℤ2 × ℤ2 orbifold.
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References
E. Palti, The Swampland: Introduction and Review, Fortsch. Phys. 67 (2019) 1900037 [arXiv:1903.06239] [INSPIRE].
I. Antoniadis, C. P. Bachas and C. Kounnas, Four-Dimensional Superstrings, Nucl. Phys. B 289 (1987) 87 [INSPIRE].
I. Antoniadis and C. Bachas, 4-D Fermionic Superstrings with Arbitrary Twists, Nucl. Phys. B 298 (1988) 586 [INSPIRE].
H. Kawai, D. C. Lewellen and S. H. H. Tye, Construction of Fermionic String Models in Four-Dimensions, Nucl. Phys. B 288 (1987) 1 [INSPIRE].
L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [INSPIRE].
P. Athanasopoulos, A. E. Faraggi, S. Groot Nibbelink and V. M. Mehta, Heterotic free fermionic and symmetric toroidal orbifold models, JHEP 04 (2016) 038 [arXiv:1602.03082] [INSPIRE].
A. E. Faraggi, D. V. Nanopoulos and K.-j. Yuan, A Standard Like Model in the 4D Free Fermionic String Formulation, Nucl. Phys. B 335 (1990) 347 [INSPIRE].
A. E. Faraggi, A New standard-like model in the four-dimensional free fermionic string formulation, Phys. Lett. B 278 (1992) 131 [INSPIRE].
A. E. Faraggi, Construction of realistic standard-like models in the free fermionic superstring formulation, Nucl. Phys. B 387 (1992) 239 [hep-th/9208024] [INSPIRE].
G. B. Cleaver, A. E. Faraggi and D. V. Nanopoulos, String derived MSSM and M-theory unification, Phys. Lett. B 455 (1999) 135 [hep-ph/9811427] [INSPIRE].
A. E. Faraggi, E. Manno and C. Timirgaziu, Minimal Standard Heterotic String Models, Eur. Phys. J. C 50 (2007) 701 [hep-th/0610118] [INSPIRE].
A. E. Faraggi, J. Rizos and H. Sonmez, Classification of standard-like heterotic-string vacua, Nucl. Phys. B 927 (2018) 1 [arXiv:1709.08229] [INSPIRE].
O. Lebedev et al., A Mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett. B 645 (2007) 88 [hep-th/0611095] [INSPIRE].
O. Lebedev et al., The Heterotic Road to the MSSM with R parity, Phys. Rev. D 77 (2008) 046013 [arXiv:0708.2691] [INSPIRE].
O. Lebedev, H. P. Nilles, S. Ramos-Sanchez, M. Ratz and P. K. S. Vaudrevange, Heterotic mini-landscape. (II). Completing the search for MSSM vacua in a ℤ6 orbifold, Phys. Lett. B 668 (2008) 331 [arXiv:0807.4384] [INSPIRE].
M. Blaszczyk, S. Groot Nibbelink, M. Ratz, F. Ruehle, M. Trapletti and P. K. S. Vaudrevange, A Z2xZ2 standard model, Phys. Lett. B 683 (2010) 340 [arXiv:0911.4905] [INSPIRE].
P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
R. Donagi, A. Lukas, B. A. Ovrut and D. Waldram, Nonperturbative vacua and particle physics in M-theory, JHEP 05 (1999) 018 [hep-th/9811168] [INSPIRE].
R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, Standard model bundles, Adv. Theor. Math. Phys. 5 (2002) 563 [math/0008010] [INSPIRE].
V. Braun, Y.-H. He, B. A. Ovrut and T. Pantev, A Heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].
V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].
G. Honecker and M. Trapletti, Merging Heterotic Orbifolds and K3 Compactifications with Line Bundles, JHEP 01 (2007) 051 [hep-th/0612030] [INSPIRE].
R. Blumenhagen, G. Honecker and T. Weigand, Loop-corrected compactifications of the heterotic string with line bundles, JHEP 06 (2005) 020 [hep-th/0504232] [INSPIRE].
L. B. Anderson, Y.-H. He and A. Lukas, Heterotic Compactification, An Algorithmic Approach, JHEP 07 (2007) 049 [hep-th/0702210] [INSPIRE].
L. B. Anderson, Y.-H. He and A. Lukas, Monad Bundles in Heterotic String Compactifications, JHEP 07 (2008) 104 [arXiv:0805.2875] [INSPIRE].
L. B. Anderson, J. Gray, Y.-H. He and A. Lukas, Exploring Positive Monad Bundles And A New Heterotic Standard Model, JHEP 02 (2010) 054 [arXiv:0911.1569] [INSPIRE].
S. Groot Nibbelink, O. Loukas, F. Ruehle and P. K. S. Vaudrevange, Infinite number of MSSMs from heterotic line bundles?, Phys. Rev. D 92 (2015) 046002 [arXiv:1506.00879] [INSPIRE].
S. Groot Nibbelink, O. Loukas and F. Ruehle, (MS)SM-like models on smooth Calabi-Yau manifolds from all three heterotic string theories, Fortsch. Phys. 63 (2015) 609 [arXiv:1507.07559] [INSPIRE].
S. Reffert, Toroidal Orbifolds: Resolutions, Orientifolds and Applications in String Phenomenology, Ph.D. Thesis, Ludwig-Maximilians-Universität (LMU) München (2006) [hep-th/0609040] [INSPIRE].
S. Groot Nibbelink, M. Trapletti and M. Walter, Resolutions of Cn/Zn Orbifolds, their U(1) Bundles, and Applications to String Model Building, JHEP 03 (2007) 035 [hep-th/0701227] [INSPIRE].
S. Groot Nibbelink, T.-W. Ha and M. Trapletti, Toric Resolutions of Heterotic Orbifolds, Phys. Rev. D 77 (2008) 026002 [arXiv:0707.1597] [INSPIRE].
S. Groot Nibbelink, Blowups of Heterotic Orbifolds using Toric Geometry, in 15th International Conference on Supersymmetry and the Unification of Fundamental Interactions (SUSY07), pp. 590–593 (2007) [arXiv:0708.1875] [INSPIRE].
S. Groot Nibbelink, D. Klevers, F. Ploger, M. Trapletti and P. K. S. Vaudrevange, Compact heterotic orbifolds in blow-up, JHEP 04 (2008) 060 [arXiv:0802.2809] [INSPIRE].
D. Lüst, S. Reffert, E. Scheidegger and S. Stieberger, Resolved Toroidal Orbifolds and their Orientifolds, Adv. Theor. Math. Phys. 12 (2008) 67 [hep-th/0609014] [INSPIRE].
S. Groot Nibbelink, J. Held, F. Ruehle, M. Trapletti and P. K. S. Vaudrevange, Heterotic Z6−II MSSM Orbifolds in Blowup, JHEP 03 (2009) 005 [arXiv:0901.3059] [INSPIRE].
M. Blaszczyk, S. Groot Nibbelink, F. Ruehle, M. Trapletti and P. K. S. Vaudrevange, Heterotic MSSM on a Resolved Orbifold, JHEP 09 (2010) 065 [arXiv:1007.0203] [INSPIRE].
R. Donagi and A. E. Faraggi, On the number of chiral generations in Z2 × Z2 orbifolds, Nucl. Phys. B 694 (2004) 187 [hep-th/0403272] [INSPIRE].
R. Donagi and K. Wendland, On orbifolds and free fermion constructions, J. Geom. Phys. 59 (2009) 942 [arXiv:0809.0330] [INSPIRE].
M. Fischer, M. Ratz, J. Torrado and P. K. S. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 01 (2013) 084 [arXiv:1209.3906] [INSPIRE].
G. Honecker, Massive U(1)s and heterotic five-branes on K3, Nucl. Phys. B 748 (2006) 126 [hep-th/0602101] [INSPIRE].
S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1 [INSPIRE].
K. Uhlenbeck and S. Yau On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 19 (1986) 257.
M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2000) 1209 [hep-th/0002240] [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].
A. E. Faraggi, C. Kounnas and J. Rizos, Spinor-Vector Duality in fermionic Z2 × Z2 heterotic orbifold models, Nucl. Phys. B 774 (2007) 208 [hep-th/0611251] [INSPIRE].
A. E. Faraggi, C. Kounnas and J. Rizos, Spinor-vector duality in N = 2 heterotic string vacua, Nucl. Phys. B 799 (2008) 19 [arXiv:0712.0747] [INSPIRE].
T. Catelin-Jullien, A. E. Faraggi, C. Kounnas and J. Rizos, Spinor-Vector Duality in Heterotic SUSY Vacua, Nucl. Phys. B 812 (2009) 103 [arXiv:0807.4084] [INSPIRE].
C. Angelantonj, A. E. Faraggi and M. Tsulaia, Spinor-Vector Duality in Heterotic String Orbifolds, JHEP 07 (2010) 004 [arXiv:1003.5801] [INSPIRE].
A. E. Faraggi, I. Florakis, T. Mohaupt and M. Tsulaia, Conformal Aspects of Spinor-Vector Duality, Nucl. Phys. B 848 (2011) 332 [arXiv:1101.4194] [INSPIRE].
A. E. Faraggi, S. Groot Nibbelink and M. Hurtado-Heredia, Uncovering a spinor-vector duality on a resolved orbifold, Nucl. Phys. B 969 (2021) 115473 [arXiv:2103.13442] [INSPIRE].
A. E. Faraggi, S. Groot Nibbelink and M. Hurtado Heredia, Constraint on spinor-vector dualities in six dimensions, Phys. Rev. D 103 (2021) 126016 [arXiv:2103.14684] [INSPIRE].
P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [INSPIRE].
B. R. Greene and M. R. Plesser, Duality in Calabi-Yau Moduli Space, Nucl. Phys. B 338 (1990) 15 [INSPIRE].
C. Vafa and E. Witten, On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995) 189 [hep-th/9409188] [INSPIRE].
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Faraggi, A.E., Nibbelink, S.G. & Heredia, M.H. Taming triangulation dependence of T6/ℤ2 × ℤ2 resolutions. J. High Energ. Phys. 2022, 169 (2022). https://doi.org/10.1007/JHEP01(2022)169
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DOI: https://doi.org/10.1007/JHEP01(2022)169