Abstract
We study to what extent Wilson lines in heterotic Calabi-Yau compactifications lead to non-trivial H-flux via Chern-Simons terms. Wilson lines are basic ingredients for Standard Model constructions but their induced H-flux may affect the consistency of the leading order background geometry and of the two-dimensional worldsheet theory. Moreover H-flux in heterotic compactifications would play an important role for moduli stabilization and could strongly constrain the supersymmetry breaking scale. We show how to compute H-flux and the corresponding superpotential, given an explicit complete intersection Calabi-Yau compactification and choice of Wilson lines. We do so by identifying large classes of special Lagrangian submanifolds in the Calabi-Yau, understanding how the Wilson lines project onto these submanifolds, and computing their Chern-Simons invariants. We illustrate our procedure with the quintic hypersurface as well as the split-bicubic, which can provide a potentially realistic three generation model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The Exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [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, 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, P. Candelas, R. Davies and R. Donagi, The MSSM Spectrum from (0,2)-Deformations of the Heterotic Standard Embedding, JHEP 05 (2012) 127 [arXiv:1112.1097] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic Line Bundle Standard Models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].
V. Braun, Y.-H. He and B.A. Ovrut, Supersymmetric Hidden Sectors for Heterotic Standard Models, JHEP 09 (2013) 008 [arXiv:1301.6767] [INSPIRE].
E. Witten, New Issues in Manifolds of SU(3) Holonomy, Nucl. Phys. B 268 (1986) 79 [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua, JHEP 02 (2011) 088 [arXiv:1010.0255] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua, Phys. Rev. D 83 (2011) 106011 [arXiv:1102.0011] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, The Atiyah Class and Complex Structure Stabilization in Heterotic Calabi-Yau Compactifications, JHEP 10 (2011) 032 [arXiv:1107.5076] [INSPIRE].
L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories, JHEP 07 (2013) 017 [arXiv:1304.2704] [INSPIRE].
S. Gukov, S. Kachru, X. Liu and L. McAllister, Heterotic moduli stabilization with fractional Chern-Simons invariants, Phys. Rev. D 69 (2004) 086008 [hep-th/0310159] [INSPIRE].
M. Cicoli, S. de Alwis and A. Westphal, Heterotic Moduli Stabilisation, JHEP 10 (2013) 199 [arXiv:1304.1809] [INSPIRE].
A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].
E. Witten, Global Anomalies in String Theory, Symposium on Anomalies, Geometry, Topology, edited by W.A. Bardeen and A.R. White, World Scientific, Singapore (1985), pg. 61–99.
H. Partouche and B. Pioline, Rolling among G 2 vacua, JHEP 03 (2001) 005 [hep-th/0011130] [INSPIRE].
M. Dine and N. Seiberg, Nonrenormalization Theorems in Superstring Theory, Phys. Rev. Lett. 57 (1986) 2625 [INSPIRE].
G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, Heterotic string theory on nonKähler manifolds with H flux and gaugino condensate, Fortsch. Phys. 52 (2004) 483 [hep-th/0310021] [INSPIRE].
A.R. Frey and M. Lippert, AdS strings with torsion: Non-complex heterotic compactifications, Phys. Rev. D 72 (2005) 126001 [hep-th/0507202] [INSPIRE].
L. Witten and E. Witten, Large Radius Expansion of Superstring Compactifications, Nucl. Phys. B 281 (1987) 109 [INSPIRE].
M.B. Green, J. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Monographs on Mathematical Physics (1988).
E.A. Bergshoeff and M. de Roo, The Quartic Effective Action of the Heterotic String and Supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].
G. Curio, A. Krause and D. Lüst, Moduli stabilization in the heterotic/ IIB discretuum, Fortsch. Phys. 54 (2006) 225 [hep-th/0502168] [INSPIRE].
P. Manousselis, N. Prezas and G. Zoupanos, Supersymmetric compactifications of heterotic strings with fluxes and condensates, Nucl. Phys. B 739 (2006) 85 [hep-th/0511122] [INSPIRE].
O. Lechtenfeld, C. Nolle and A.D. Popov, Heterotic compactifications on nearly Kähler manifolds, JHEP 09 (2010) 074 [arXiv:1007.0236] [INSPIRE].
A. Chatzistavrakidis, O. Lechtenfeld and A.D. Popov, Nearly Kähler heterotic compactifications with fermion condensates, JHEP 04 (2012) 114 [arXiv:1202.1278] [INSPIRE].
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
M. Klaput, A. Lukas and E.E. Svanes, Heterotic Calabi-Yau Compactifications with Flux, JHEP 09 (2013) 034 [arXiv:1305.0594] [INSPIRE].
M. Dine, R. Rohm, N. Seiberg and E. Witten, Gluino Condensation in Superstring Models, Phys. Lett. B 156 (1985) 55 [INSPIRE].
J.P. Derendinger, L.E. Ibáñez and H.P. Nilles, On the Low-Energy D = 4, N = 1 Supergravity Theory Extracted from the D = 10, N = 1 Superstring, Phys. Lett. B 155 (1985) 65 [INSPIRE].
R. Rohm and E. Witten, The Antisymmetric Tensor Field in Superstring Theory, Annals Phys. 170 (1986) 454 [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477-478] [hep-th/9906070] [INSPIRE].
M. Becker and D. Constantin, A Note on flux induced superpotentials in string theory, JHEP 08 (2003) 015 [hep-th/0210131] [INSPIRE].
M.A. Shifman and A.I. Vainshtein, On Gluino Condensation in Supersymmetric Gauge Theories. SU(N) and O(N) Groups, Nucl. Phys. B 296 (1988) 445 [INSPIRE].
G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 10 (2003) 004 [hep-th/0306088] [INSPIRE].
E. Witten, Symmetry Breaking Patterns in Superstring Models, Nucl. Phys. B 258 (1985) 75 [INSPIRE].
P.A. Kirk and E.P. Klassen, Chern-simons invariants of 3-manifolds and representation spaces of knot groups, Math. Ann. 287 (1990) 343.
L. Rozansky, A Large k asymptotics of Witten’s invariant of Seifert manifolds, Commun. Math. Phys. 171 (1995) 279 [hep-th/9303099] [INSPIRE].
H. Nishi, Su(n)-chern-simons invariants of seifert fibered 3-manifolds, Int.J.Math. 09 (1998) 295.
P. Green and T. Hübsch, Calabi-yau Manifolds as Complete Intersections in Products of Complex Projective Spaces, Commun. Math. Phys. 109 (1987) 99.
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].
T. Hübsch, Calabi-Yau manifolds: A Bestiary for physicists, World Scientific Publishing, Singapore (1992).
D. Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, math/0108088 [INSPIRE].
N.J. Hitchin, Lectures on special Lagrangian submanifolds, math/9907034 [INSPIRE].
V. Braun, On Free Quotients of Complete Intersection Calabi-Yau Manifolds, JHEP 04 (2011) 005 [arXiv:1003.3235] [INSPIRE].
L.B. Anderson, A. Constantin, J. Gray, A. Lukas and E. Palti, A Comprehensive Scan for Heterotic SU(5) GUT models, JHEP 01 (2014) 047 [arXiv:1307.4787] [INSPIRE].
A. Hatcher, Notes on Basic 3-Manifold Topology, http://www.math.cornell.edu/~hatcher/3M/3M.pdf (2007).
D.R. Auckly, Topological methods to compute chern-simons invariants, Math. Proc. Camb. Phil. Soc. 115 (1994) 229.
M. Jankins and W.D. Neumann, Lectures on Seifert Manifolds, Brandeis Lecture Notes 2, Brandais University, Waltham MA, U.S.A. (1983).
M.G. Brin, Sifert fibered spaces, arXiv:0711.1346.
J. Montesinos, Classical tessellations and three-manifolds, Universitext, Springer-Verlag, Germany (1979).
P.A. Kirk and E.P. Klassen, Chern-simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of t2, Comm. Math. Phys. 153 (1993) 521.
I. Brunner, M.R. Douglas, A.E. Lawrence and C. Romelsberger, D-branes on the quintic, JHEP 08 (2000) 015 [hep-th/9906200] [INSPIRE].
B.A. Ovrut, T. Pantev and R. Reinbacher, Torus fibered Calabi-Yau threefolds with nontrivial fundamental group, JHEP 05 (2003) 040 [hep-th/0212221] [INSPIRE].
V. Braun, B.A. Ovrut, T. Pantev and R. Reinbacher, Elliptic Calabi-Yau threefolds with Z 3 × Z 3 Wilson lines, JHEP 12 (2004) 062 [hep-th/0410055] [INSPIRE].
D. McDuff and S. Dusa, Introduction to Symplectic Topology, Oxford University Press, Oxford U.K. (1999).
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].
V. Braun, T. Brelidze, M.R. Douglas and B.A. Ovrut, Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 05 (2008) 080 [arXiv:0712.3563] [INSPIRE].
C.G.A. Harnack, Über Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876) 189.
R. Blumenhagen, S. Moster and T. Weigand, Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys. B 751 (2006) 186 [hep-th/0603015] [INSPIRE].
L. Anguelova, C. Quigley and S. Sethi, The Leading Quantum Corrections to Stringy Kähler Potentials, JHEP 10 (2010) 065 [arXiv:1007.4793] [INSPIRE].
L. Anguelova and C. Quigley, Quantum Corrections to Heterotic Moduli Potentials, JHEP 02 (2011) 113 [arXiv:1007.5047] [INSPIRE].
E. Palti, Model building with intersecting D6-branes on smooth Calabi-Yau manifolds, JHEP 04 (2009) 099 [arXiv:0902.3546] [INSPIRE].
F. Denef, (Dis)assembling special Lagrangians, hep-th/0107152 [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1410.2603
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Apruzzi, F., Gautason, F.F., Parameswaran, S. et al. Wilson lines and Chern-Simons flux in explicit heterotic Calabi-Yau compactifications. J. High Energ. Phys. 2015, 183 (2015). https://doi.org/10.1007/JHEP02(2015)183
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)183