Abstract
We perform the first systematic analysis of particle spectra obtained from heterotic string compactifications on non-Abelian toroidal orbifolds. After developing a new technique to compute the particle spectrum in the case of standard embedding based on higher dimensional supersymmetry, we compute the Hodge numbers for all recently classified 331 non-Abelian orbifold geometries which yield \( \mathcal{N}=1 \) supersymmetry for heterotic compactifications. Surprisingly, most Hodge numbers follow the empiric pattern h (1,1) − h (2,1) = 0 mod 6, which might be related to the number of three standard model generations. Furthermore, we study the fundamental groups in order to identify the possibilities for non-local gauge symmetry breaking. Three examples are discussed in detail: the simplest non-Abelian orbifold S 3 and two more elaborate examples, T 7 and Δ(27), which have only one untwisted Kähler and no untwisted complex structure modulus. Such models might be especially interesting in the context of no-scale supergravity. Finally, we briefly discuss the case of orbifolds with vanishing Euler numbers in the context of enhanced (spontaneously broken) supersymmetry.
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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].
A.E. Faraggi, A new standard-like model in the four-dimensional free fermionic string formulation, Phys. Lett. B 278 (1992) 131 [INSPIRE].
T. Dijkstra, L. Huiszoon and A. Schellekens, Supersymmetric standard model spectra from RCFT orientifolds, Nucl. Phys. B 710 (2005) 3 [hep-th/0411129] [INSPIRE].
F. Gmeiner, R. Blumenhagen, G. Honecker, D. Lüst and T. Weigand, One in a billion: MSSM-like D-brane statistics, JHEP 01 (2006) 004 [hep-th/0510170] [INSPIRE].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].
F. Gmeiner and G. Honecker, Millions of standard models on \( Z_6^{\prime } \) ?, JHEP 07 (2008) 052 [arXiv:0806.3039] [INSPIRE].
B.S. Acharya, K. Bobkov, G.L. Kane, J. Shao and P. Kumar, The G 2 -MSSM: an M-theory motivated model of particle physics, Phys. Rev. D 78 (2008) 065038 [arXiv:0801.0478] [INSPIRE].
V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two hundred heterotic standard models on smooth Calabi-Yau threefolds, Phys. Rev. D 84 (2011) 106005 [arXiv:1106.4804] [INSPIRE].
M. Blaszczyk, S. Nibbelink Groot, F. Ruehle, M. Trapletti and P.K. Vaudrevange, Heterotic MSSM on a resolved orbifold, JHEP 09 (2010) 065 [arXiv:1007.0203] [INSPIRE].
M. Blaszczyk, N.G. Cabo Bizet, H.P. Nilles and F. Ruhle, A perfect match of MSSM-like orbifold and resolution models via anomalies, JHEP 10 (2011) 117 [arXiv:1108.0667] [INSPIRE].
M. Blaszczyk, S. Groot Nibbelink and F. Ruehle, Gauged linear σ-models for toroidal orbifold resolutions, JHEP 05 (2012) 053 [arXiv:1111.5852] [INSPIRE].
W. Buchmüller, J. Louis, J. Schmidt and R. Valandro, Voisin-Borcea manifolds and heterotic orbifold models, JHEP 10 (2012) 114 [arXiv:1208.0704] [INSPIRE].
W. Buchmüller, K. Hamaguchi, O. Lebedev and M. Ratz, Supersymmetric standard model from the heterotic string, Phys. Rev. Lett. 96 (2006) 121602 [hep-ph/0511035] [INSPIRE].
W. Buchmüller, K. Hamaguchi, O. Lebedev and M. Ratz, Supersymmetric standard model from the heterotic string (II), Nucl. Phys. B 785 (2007) 149 [hep-th/0606187] [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, H.P. Nilles, S. Ramos-Sanchez, M. Ratz and P.K. Vaudrevange, Heterotic mini-landscape. (II). Completing the search for MSSM vacua in a Z 6 orbifold, Phys. Lett. B 668 (2008) 331 [arXiv:0807.4384] [INSPIRE].
O. Lebedev and S. Ramos-Sanchez, The NMSSM and string theory, Phys. Lett. B 684 (2010) 48 [arXiv:0912.0477] [INSPIRE].
I. Antoniadis, E. Gava, K. Narain and T. Taylor, Effective μ term in superstring theory, Nucl. Phys. B 432 (1994) 187 [hep-th/9405024] [INSPIRE].
R. Kappl et al., Large hierarchies from approximate R symmetries, Phys. Rev. Lett. 102 (2009) 121602 [arXiv:0812.2120] [INSPIRE].
T. Kobayashi, H.P. Nilles, F. Plöger, S. Raby and M. Ratz, Stringy origin of non-abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
H.M. Lee et al., A unique \( Z_4^R \) symmetry for the MSSM, Phys. Lett. B 694 (2011) 491 [arXiv:1009.0905] [INSPIRE].
S. Förste, H.P. Nilles, S. Ramos-Sánchez and P.K. Vaudrevange, Proton hexality in local grand unification, Phys. Lett. B 693 (2010) 386 [arXiv:1007.3915] [INSPIRE].
T. Kobayashi and N. Ohtsubo, Geometrical aspects of Z N orbifold phenomenology, Int. J. Mod. Phys. A 9 (1994) 87 [INSPIRE].
D. Bailin and A. Love, Orbifold compactifications of string theory, Phys. Rept. 315 (1999) 285 [INSPIRE].
R. Blumenhagen and E. Plauschinn, Intersecting D-branes on shift Z 2 × Z 2 orientifolds, JHEP 08 (2006) 031 [hep-th/0604033] [INSPIRE].
R. Donagi and K. Wendland, On orbifolds and free fermion constructions, J. Geom. Phys. 59 (2009) 942 [arXiv:0809.0330] [INSPIRE].
R. Donagi and A.E. Faraggi, On the number of chiral generations in Z 2 × Z 2 orbifolds, Nucl. Phys. B 694 (2004) 187 [hep-th/0403272] [INSPIRE].
S. Förste, T. Kobayashi, H. Ohki and K.-j. Takahashi, Non-factorisable Z 2 × Z 2 heterotic orbifold models and Yukawa couplings, JHEP 03 (2007) 011 [hep-th/0612044] [INSPIRE].
F. Beye, T. Kobayashi and S. Kuwakino, Gauge symmetries in heterotic asymmetric orbifolds, arXiv:1304.5621 [INSPIRE].
M. Fischer, M. Ratz, J. Torrado and P.K. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 01 (2013) 084 [arXiv:1209.3906] [INSPIRE].
Z. Kakushadze, G. Shiu and S.H. Tye, Asymmetric non-abelian orbifolds and model building, Phys. Rev. D 54 (1996) 7545 [hep-th/9607137] [INSPIRE].
S.J. Konopka, Non abelian orbifold compactifications of the heterotic string, arXiv:1210.5040 [INSPIRE].
G. Ross, Wilson line breaking and gauge coupling unification, hep-ph/0411057 [INSPIRE].
A. Hebecker and M. Trapletti, Gauge unification in highly anisotropic string compactifications, Nucl. Phys. B 713 (2005) 173 [hep-th/0411131] [INSPIRE].
A. Anandakrishnan and S. Raby, SU(6) GUT breaking on a projective plane, Nucl. Phys. B 868 (2013) 627 [arXiv:1205.1228] [INSPIRE].
T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4d string models with a Pati-Salam symmetry: orbifold grand unified theories from heterotic string compactification on a Z 6 orbifold, Nucl. Phys. B 704 (2005) 3 [hep-ph/0409098] [INSPIRE].
J.E. Kim, J.-H. Kim and B. Kyae, Superstring standard model from Z 12−I orbifold compactification with and without exotics and effective R-parity, JHEP 06 (2007) 034 [hep-ph/0702278] [INSPIRE].
M. Blaszczyk et al., A Z 2 × Z 2 standard model, Phys. Lett. B 683 (2010) 340 [arXiv:0911.4905] [INSPIRE].
D.K.M. Pena, H.P. Nilles and P.-K. Oehlmann, A zip-code for quarks, leptons and Higgs bosons, JHEP 12 (2012) 024 [arXiv:1209.6041] [INSPIRE].
L.E. Ibáñez, J. Mas, H.-P. Nilles and F. Quevedo, Heterotic strings in symmetric and asymmetric orbifold backgrounds, Nucl. Phys. B 301 (1988) 157 [INSPIRE].
J. Casas, A. de la Macorra, M. Mondragón and C. Muñoz, Z 7 phenomenology, Phys. Lett. B 247 (1990) 50 [INSPIRE].
H. Brown et al., Crystallographic groups of four-dimensional space, J. Wiley & Sons, U.S.A. (1978).
J. Opgenorth, W. Plesken, and T. Schulz, Cristallographic algorithms and tables, Acta Cryst. Sect. A 54 (1998).
The GAP Group, GAP — Groups, Algorithms, and Programming, version 4.6.3, (2013).
J.R. Ellis, A. Lahanas, D.V. Nanopoulos and K. Tamvakis, No-scale supersymmetric standard model, Phys. Lett. B 134 (1984) 429 [INSPIRE].
M. Cvetič, J. Louis and B.A. Ovrut, A string calculation of the Kähler potentials for moduli of Z N orbifolds, Phys. Lett. B 206 (1988) 227 [INSPIRE].
A. Brignole, L.E. Ibáñez and C. Muñoz, Soft supersymmetry breaking terms from supergravity and superstring models, hep-ph/9707209 [INSPIRE].
L. Covi et al., De Sitter vacua in no-scale supergravities and Calabi-Yau string models, JHEP 06 (2008) 057 [arXiv:0804.1073] [INSPIRE].
L.E. Ibáñez and D. Lüst, Duality anomaly cancellation, minimal string unification and the effective low-energy Lagrangian of 4D strings, Nucl. Phys. B 382 (1992) 305 [hep-th/9202046] [INSPIRE].
M. Dine, N. Seiberg, X. Wen and E. Witten, Nonperturbative effects on the string world sheet, Nucl. Phys. B 278 (1986) 769 [INSPIRE].
L.J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2, 2) vacua of the heterotic string, Nucl. Phys. B 329 (1990) 27 [INSPIRE].
S. Groot Nibbelink and P.K.S. Vaudrevange, Schoen manifold with line bundles as resolved magnetized orbifolds, JHEP 03 (2013) 142 [arXiv:1212.4033] [INSPIRE].
H.P. Nilles, S. Ramos-Sanchez, P.K.S. Vaudrevange and A. Wingerter, The orbifolder: a tool to study the low energy effective theory of heterotic orbifolds, Comput. Phys. Commun. 183 (2012) 1363 [arXiv:1110.5229] [INSPIRE].
L.E. Ibáñez, H.P. Nilles and F. Quevedo, Orbifolds and Wilson lines, Phys. Lett. B 187 (1987) 25 [INSPIRE].
R. Brown and P.J. Higgins, The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action, math/0212271.
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].
V. Braun, On free quotients of complete intersection Calabi-Yau manifolds, JHEP 04 (2011) 005 [arXiv:1003.3235] [INSPIRE].
E. Witten, Strong coupling expansion of Calabi-Yau compactification, Nucl. Phys. B 471 (1996) 135 [hep-th/9602070] [INSPIRE].
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Magnetic flux, Wilson line and orbifold, Phys. Rev. D 80 (2009) 126006 [arXiv:0907.5274] [INSPIRE].
A.-K. Kashani-Poor, R. Minasian and H. Triendl, Enhanced supersymmetry from vanishing Euler number, JHEP 04 (2013) 058 [arXiv:1301.5031] [INSPIRE].
H. Ishimori et al., An introduction to non-abelian discrete symmetries for particle physicists, Lect. Notes Phys. 858 (2012) 1.
C. Luhn, S. Nasri and P. Ramond, Tri-bimaximal neutrino mixing and the family symmetry semidirect product of Z 7 and Z 3, Phys. Lett. B 652 (2007) 27 [arXiv:0706.2341] [INSPIRE].
C. Luhn, K.M. Parattu and A. Wingerter, A minimal model of neutrino flavor, JHEP 12 (2012) 096 [arXiv:1210.1197] [INSPIRE].
R. Blumenhagen, L. Görlich and B. Körs, Supersymmetric 4D orientifolds of type IIA with D6-branes at angles, JHEP 01 (2000) 040 [hep-th/9912204] [INSPIRE].
S. Förste, G. Honecker and R. Schreyer, Supersymmetric Z N × Z M orientifolds in 4D with D-branes at angles, Nucl. Phys. B 593 (2001) 127 [hep-th/0008250] [INSPIRE].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].
G. Honecker and M. Trapletti, Merging heterotic orbifolds and K3 compactifications with line bundles, JHEP 01 (2007) 051 [hep-th/0612030] [INSPIRE].
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Fischer, M., Ramos-Sánchez, S. & Vaudrevange, P.K. Heterotic non-abelian orbifolds. J. High Energ. Phys. 2013, 80 (2013). https://doi.org/10.1007/JHEP07(2013)080
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DOI: https://doi.org/10.1007/JHEP07(2013)080