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Green-Schwarz mechanism in heterotic (2,0) gauged linear sigma models: torsion and NS5 branes

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Abstract

Heterotic string compactifications can be conveniently described in the language of (2,0) gauged linear sigma models (GLSMs). Such models allow for Fayet-Iliopoulos (FI)–terms, which can be interpreted as Kähler parameters and axions on the target space geometry. We show that field dependent non-gauge invariant FI-terms lead to a Green-Schwarz-like mechanism on the worldsheet which can be used to cancel worldsheet anomalies. However, given that these FI-terms are constrained by quantization conditions due to worldsheet gauge instantons, the anomaly conditions turn out to be still rather constraining. Field dependent non-gauge invariant FI-terms result in non-Kähler, i.e. torsional, target spaces in general. When FI-terms involve logarithmic terms, the GLSM seems to describe the heterotic string in the presence of Neveu-Schwarz (NS)5 branes. In particular, the GLSM leads to a decompactified target space geometry when anti-NS5 branes are present.

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References

  1. D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 1. The Free Heterotic String, Nucl. Phys. B 256 (1985) 253 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  2. D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic String Theory. 2. The Interacting Heterotic String, Nucl. Phys. B 267 (1986) 75 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  3. L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  4. L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds. 2, Nucl. Phys. B 274 (1986) 285 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  5. 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 [SPIRES].

    Article  ADS  Google Scholar 

  6. O. Lebedev et al., Low Energy Supersymmetry from the Heterotic Landscape, Phys. Rev. Lett. 98 (2007) 181602 [hep-th/0611203] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  7. 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] [SPIRES].

    Article  ADS  Google Scholar 

  8. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  9. O. Lebedev et al., A mini-landscape of exact MSSM spectra in heterotic orbifolds, Phys. Lett. B 645 (2007) 88 [hep-th/0611095] [SPIRES].

    ADS  Google Scholar 

  10. O. Lebedev et al., The Heterotic Road to the MSSM with R parity, Phys. Rev. D 77 (2008) 046013 [arXiv:0708.2691] [SPIRES].

    ADS  Google Scholar 

  11. M.R. Douglas, B.R. Greene and D.R. Morrison, Orbifold resolution by D-branes, Nucl. Phys. B 506 (1997) 84 [hep-th/9704151] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  12. F. Denef, M.R. Douglas and B. Florea, Building a better racetrack, JHEP 06 (2004) 034 [hep-th/0404257] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  13. 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] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  14. S. Reffert, The Geometer’s Toolkit to String Compactifications, arXiv:0706.1310 [SPIRES].

  15. W. Fulton Introduction to Toric Varieties, Princeton University Press, Princeton U.S.A. (1993).

    MATH  Google Scholar 

  16. K. Hori et al., Mirror symmetry, MS Chelsea Publishing, Providence U.S.A. (2003).

    MATH  Google Scholar 

  17. G. Honecker and M. Trapletti, Merging heterotic orbifolds and K3 compactifications with line bundles, JHEP 01 (2007) 051 [hep-th/0612030] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  18. S.G. Nibbelink, T.-W. Ha and M. Trapletti, Toric Resolutions of Heterotic Orbifolds, Phys. Rev. D 77 (2008) 026002 [arXiv:0707.1597] [SPIRES].

    ADS  Google Scholar 

  19. S.G. Nibbelink, D. Klevers, F. Plöger, M. Trapletti and P.K.S. Vaudrevange, Compact heterotic orbifolds in blow-up, JHEP 04 (2008) 060 [arXiv:0802.2809] [SPIRES].

    Article  Google Scholar 

  20. S.G. 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] [SPIRES].

    Article  Google Scholar 

  21. S.G. Nibbelink, M. Trapletti and M. Walter, Resolutions of C n /Zn Orbifolds, their U(1) Bundles and Applications to String Model Building, JHEP 03 (2007) 035 [hep-th/0701227] [SPIRES].

    Article  ADS  Google Scholar 

  22. S. Groot Nibbelink, H.P. Nilles and M. Trapletti, Multiple anomalous U(1)s in heterotic blow-ups, Phys. Lett. B 652 (2007) 124 [hep-th/0703211] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  23. S.G. Nibbelink, F.P. Correia and M. Trapletti, Non-Abelian bundles on heterotic non-compact K3 orbifold blowups, JHEP 11 (2008) 044 [arXiv:0809.4430] [SPIRES].

    Article  Google Scholar 

  24. M. Blaszczyk et al., A Z2xZ2 standard model, Phys. Lett. B 683 (2010) 340 [arXiv:0911.4905] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  25. R. Donagi and K. Wendland, On orbifolds and free fermion constructions, J. Geom. Phys. 59 (2009) 942 [arXiv:0809.0330] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. M. Blaszczyk, S.G. Nibbelink, F. Ruehle, M. Trapletti and P.K.S. Vaudrevange, Heterotic MSSM on a Resolved Orbifold, JHEP 09 (2010) 065 [arXiv:1007.0203] [SPIRES].

    Article  ADS  Google Scholar 

  27. P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  28. S. Donalson Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Londan Math. Soc. 50 (1985) 1.

    Article  Google Scholar 

  29. K. Uhlenbeck and S. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure and Appl. Math. 19 (1986) 257.

    Article  MathSciNet  Google Scholar 

  30. M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [SPIRES].

    MathSciNet  Google Scholar 

  31. T.L. Gomez, S. Lukic and I. Sols, Constraining the Kähler moduli in the heterotic standard model, Commun. Math. Phys. 276 (2007) 1 [hep-th/0512205] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. R. Donagi, B.A. Ovrut, T. Pantev and D. Waldram, Standard models from heterotic M-theory, Adv. Theor. Math. Phys. 5 (2002) 93 [hep-th/9912208] [SPIRES].

    MathSciNet  Google Scholar 

  33. R. Donagi, B.A. Ovrut, T. Pantev and D. Waldram, Standard-model bundles, Adv. Theor. Math. Phys. 5 (2002) 563 [math/0008010]. = MATH/0008010;

    MathSciNet  Google Scholar 

  34. R. Donagi, B.A. Ovrut, T. Pantev and D. Waldram, Spectral involutions on rational elliptic surfaces, Adv. Theor. Math. Phys. 5 (2002) 499 [math/0008011].

    MathSciNet  Google Scholar 

  35. V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  36. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  37. V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  38. V. Bouchard and R. Donagi, On heterotic model constraints, JHEP 08 (2008) 060 [arXiv:0804.2096] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  39. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  40. L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds, arXiv:1106.4804 [SPIRES].

  41. S. Kachru, M.B. Schulz and S. Trivedi, Moduli stabilization from fluxes in a simple IIB orientifold, JHEP 10 (2003) 007 [hep-th/0201028] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  42. 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] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  43. M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. F. Denef, M.R. Douglas and S. Kachru, Physics of string flux compactifications, Ann. Rev. Nucl. Part. Sci. 57 (2007) 119 [hep-th/0701050] [SPIRES].

    Article  ADS  Google Scholar 

  45. A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  46. K. Becker, M. Becker, K. Dasgupta and P.S. Green, Compactifications of heterotic theory on non-Kähler complex manifolds. I, JHEP 04 (2003) 007 [hep-th/0301161] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  47. K. Becker, M. Becker, P.S. Green, K. Dasgupta and E. Sharpe, Compactifications of heterotic strings on non-Kähler complex manifolds. II, Nucl. Phys. B 678 (2004) 19 [hep-th/0310058] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  48. K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and G-flux, JHEP 08 (1999) 023 [hep-th/9908088] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  49. K. Becker and K. Dasgupta, Heterotic strings with torsion, JHEP 11 (2002) 006 [hep-th/0209077] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  50. K. Becker and S. Sethi, Torsional Heterotic Geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  51. J.-X. Fu and S.-T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampere equation, J. Diff. Geom. 78 (2009) 369 [hep-th/0604063] [SPIRES].

    MathSciNet  Google Scholar 

  52. K. Becker, M. Becker, J.-X. Fu, L.-S. Tseng and S.-T. Yau, Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory, Nucl. Phys. B 751 (2006) 108 [hep-th/0604137] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  53. J.-X. Fu, L.-S. Tseng and S.-T. Yau, Local Heterotic Torsional Models, Commun. Math. Phys. 289 (2009) 1151 [arXiv:0806.2392] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  54. B. Andreas and M. Garcia-Fernandez, Solutions of the Strominger System via Stable Bundles on Calabi-Yau Threefolds, arXiv:1008.1018 [SPIRES].

  55. B. Andreas and M. Garcia-Fernandez, Heterotic Non-Kähler Geometries via Polystable Bundles on Calabi-Yau Threefolds, arXiv:1011.6246 [SPIRES].

  56. A. Adams, M. Ernebjerg and J.M. Lapan, Linear models for flux vacua, Adv. Theor. Math. Phys. 12 (2008) 817 [hep-th/0611084] [SPIRES].

    MATH  MathSciNet  Google Scholar 

  57. A. Adams and D. Guarrera, Heterotic Flux Vacua from Hybrid Linear Models, arXiv:0902.4440 [SPIRES].

  58. A. Adams and J.M. Lapan, Computing the Spectrum of a Heterotic Flux Vacuum, JHEP 03 (2011) 045 [arXiv:0908.4294] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  59. K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  60. G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, New branches of string compactifications and their F-theory duals, Nucl. Phys. B 492 (1997) 119 [hep-th/9607121] [SPIRES].

    ADS  Google Scholar 

  61. R. Donagi, A. Lukas, B.A. Ovrut and D. Waldram, Non-perturbative vacua and particle physics in M-theory, JHEP 05 (1999) 018 [hep-th/9811168] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  62. R. Blumenhagen, G. Honecker and T. Weigand, Non-abelian brane worlds: The heterotic string story, JHEP 10 (2005) 086 [hep-th/0510049] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  63. G. Honecker, Massive U(1)s and heterotic five-branes on K3, Nucl. Phys. B 748 (2006) 126 [hep-th/0602101] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  64. L.B. Anderson, Y.-H. He and A. Lukas, Monad Bundles in Heterotic String Compactifications, JHEP 07 (2008) 104 [arXiv:0805.2875] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  65. E. Witten, Small Instantons in String Theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  66. A. Strominger, Heterotic solitons, Nucl. Phys. B 343 (1990) 167 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  67. L. Carlevaro and D. Israel, Heterotic Resolved Conifolds with Torsion, from Supergravity to CFT , JHEP 01 (2010) 083 [arXiv:0910.3190] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  68. E. Witten, Phases of N = 2 theories in two dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  69. J. Distler and B.R. Greene, Aspects of (2, 0) String Compactifications, Nucl. Phys. B 304 (1988) 1 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  70. J. Distler, Notes on N = 2 σ-models, hep-th/9212062 [SPIRES].

  71. J. Distler, Notes on (0, 2) superconformal field theories, hep-th/9502012 [SPIRES].

  72. E. Silverstein and E. Witten, Global U(1) R symmetry and conformal invariance of (0, 2) models, Phys. Lett. B 328 (1994) 307 [hep-th/9403054] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  73. E. Silverstein and E. Witten, Criteria for conformal invariance of (0, 2) models, Nucl. Phys. B 444 (1995) 161 [hep-th/9503212] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  74. A. Basu and S. Sethi, World-sheet stability of (0, 2) linear σ-models, Phys. Rev. D 68 (2003) 025003 [hep-th/0303066] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  75. C. Beasley and E. Witten, Residues and world-sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  76. S.G. Nibbelink, Heterotic orbifold resolutions as (2, 0) gauged linear σ-models, arXiv:1012.3350 [SPIRES].

  77. C. Quigley and S. Sethi, Linear σ-models with Torsion, arXiv:1107.0714 [SPIRES].

  78. R. Blumenhagen and T. Rahn, Landscape Study of Target Space Duality of (0, 2) Heterotic String Models, arXiv:1106.4998 [SPIRES].

  79. B. Zumino, Y.-S. Wu and A. Zee, Chiral Anomalies, Higher Dimensions and Differential Geometry, Nucl. Phys. B 239 (1984) 477 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  80. O. Alvarez, I.M. Singer and B. Zumino, Gravitational anomalies and the family’s index theorem, Commun. Math. Phys. 96 (1984) 409 [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  81. B. Zumino, Cohomology of Gauge Groups: Cocycles and Schwinger Terms, Nucl. Phys. B 253 (1985) 477 [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  82. A. Adams, Orbifold Phases of Heterotic Flux Vacua, arXiv:0908.2994 [SPIRES].

  83. A. Adams, A. Basu and S. Sethi, (0, 2) duality, Adv. Theor. Math. Phys. 7 (2004) 865 [hep-th/0309226] [SPIRES].

    MathSciNet  Google Scholar 

  84. M. Becker, L.-S. Tseng and S.-T. Yau, New Heterotic Non-Kähler Geometries, arXiv:0807.0827 [SPIRES].

  85. C.M. Hull and E. Witten, Supersymmetric σ-models and the Heterotic String, Phys. Lett. B 160 (1985) 398 [SPIRES].

    ADS  MathSciNet  Google Scholar 

  86. C.M. Hull and P.K. Townsend, World sheet supersymmetry and anomaly cancellation in the heterotic string, Phys. Lett. B 178 (1986) 187 [SPIRES].

    ADS  MathSciNet  Google Scholar 

  87. D. Tong, NS5-branes, T-duality and worldsheet instantons, JHEP 07 (2002) 013 [hep-th/0204186] [SPIRES].

    Article  ADS  Google Scholar 

  88. L.B. Anderson, Y.-H. He and A. Lukas, Heterotic compactification, an algorithmic approach, JHEP 07 (2007) 049 [hep-th/0702210] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

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  1. Bethe Center for Theoretical Physics and Physikalisches Institut der Universität Bonn, Nussallee 12, 53115, Bonn, Germany

    Michael Blaszczyk & Fabian Ruehle

  2. Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, 80333, München, Germany

    Stefan Groot Nibbelink

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  1. Michael Blaszczyk
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Blaszczyk, M., Nibbelink, S.G. & Ruehle, F. Green-Schwarz mechanism in heterotic (2,0) gauged linear sigma models: torsion and NS5 branes. J. High Energ. Phys. 2011, 83 (2011). https://doi.org/10.1007/JHEP08(2011)083

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