Skip to main content
SpringerLink
Log in

Super Weyl invariance: BPS equations from heterotic worldsheets

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

It is well-known that the beta functions on a string worldsheet correspond to the target space equations of motion, e.g. the Einstein equations. We show that the BPS equations, i.e. the conditions of vanishing supersymmetry variations of the space-time fermions, can be directly derived from the worldsheet. To this end we consider the RNSformulation of the heterotic string with (2,0) supersymmetry, which describes a complex torsion target space that supports a holomorphic vector bundle. After a detailed account of its quantization and renormalization, we establish that the cancellation of the Weyl anomaly combined with (2,0) finiteness implies the heterotic BPS conditions: At the one loop level the geometry is required to be conformally balanced and the gauge background has to satisfy the Hermitean Yang-Mills equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, The Heterotic String, Phys. Rev. Lett. 54 (1985) 502 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. 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 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  4. M.B. Green, J.H. Schwarz and E. Witten, Superstring theory vol. 2: Loop amplitudes, anomalies and phenomenology, Cambridge University Press, Cambridge, U.K. (1987) [INSPIRE].

    Google Scholar 

  5. S. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1 [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  8. C. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  11. J. Gillard, G. Papadopoulos and D. Tsimpis, Anomaly, fluxes and (2,0) heterotic string compactifications, JHEP 06 (2003) 035 [hep-th/0304126] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. 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] [INSPIRE].

    MathSciNet  Google Scholar 

  13. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

  15. E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Ten-Dimensional Maxwell-Einstein Supergravity, Its Currents and the Issue of Its Auxiliary Fields, Nucl. Phys. B 195 (1982) 97 [INSPIRE].

    Article  ADS  Google Scholar 

  16. E. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].

    Article  ADS  Google Scholar 

  17. W. Chemissany, M. de Roo and S. Panda, α′ -Corrections to Heterotic Superstring Effective Action Revisited, JHEP 08 (2007) 037 DOI:dx.doi.org [arXiv:0706.3636] [INSPIRE].

    Article  ADS  Google Scholar 

  18. R. Metsaev and A.A. Tseytlin, On loop corrections to string theory effective actions, Nucl. Phys. B 298 (1988) 109 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. R. Metsaev and A.A. Tseytlin, Order α′ (Two Loop) Equivalence of the String Equations of Motion and the σ-model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor, Nucl. Phys. B 293 (1987) 385 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. D. Friedan, Nonlinear Models in Two Epsilon Dimensions, Phys. Rev. Lett. 45 (1980) 1057 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. D.H. Friedan, Nonlinear Models in Two + Epsilon Dimensions, Annals Phys. 163 (1985) 318 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. L. Álvarez-Gaumé and D.Z. Freedman, Kähler Geometry and the Renormalization of Supersymmetric σ-models, Phys. Rev. D 22 (1980) 846 [INSPIRE].

    ADS  Google Scholar 

  23. C.G. Callan Jr., E. Martinec, M. Perry and D. Friedan, Strings in Background Fields, Nucl. Phys. B 262 (1985) 593 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. C. Hull and P. Townsend, String effective actions from σ-model conformal anomalies, Nucl. Phys. B 301 (1988) 197 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  26. M. Dine and N. Seiberg, (2,0) superspace, Phys. Lett. B 180 (1986) 364 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. M. Evans and B.A. Ovrut, A two-dimensional superfield formulation of the heterotic string, Phys. Lett. B 175 (1986) 145 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. R. Brooks, F. Muhammad and S. Gates, Unidexterous D = 2 Supersymmetry in Superspace, Nucl. Phys. B 268 (1986) 599 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. X. Cui and M. Shifman, Perturbative Aspects of Heterotically Deformed CP(N-1) σ-model. I, Phys. Rev. D 82 (2010) 105022 [arXiv:1009.4421] [INSPIRE].

    ADS  Google Scholar 

  30. X. Cui and M. Shifman, N = (0, 2) Supersymmetry and a Nonrenormalization Theorem, Phys. Rev. D 84 (2011) 105016 [arXiv:1105.5107] [INSPIRE].

    ADS  Google Scholar 

  31. C. Hull, σ-model β-functions and string compactifications, Nucl. Phys. B 267 (1986) 266 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. C. Hull and P. Townsend, finiteness and conformal invariance in nonlinear σ-models, Nucl. Phys. B 274 (1986) 349 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. M. Evans and B.A. Ovrut, Super-Weyl transformations in heterotic superstring geometry, Phys. Lett. B 184 (1987) 153 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. R. Brooks, F. Muhammad and S.J. Gates Jr., Extended D = 2 supergravity theories and their lower superspace realizations, Class. Quant. Grav. 5 (1988) 785 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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

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

    Article  MathSciNet  ADS  Google Scholar 

  38. S. Groot Nibbelink, Heterotic orbifold resolutions as (2,0) gauged linear σ-models, Fortsch. Phys. 59 (2011) 454 [arXiv:1012.3350] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. G. Lopes Cardoso et al., NonKähler string backgrounds and their five torsion classes, Nucl. Phys. B 652 (2003) 5 [hep-th/0211118] [INSPIRE].

    Article  ADS  Google Scholar 

  40. M. Ademollo et al., Dual String Models with Nonabelian Color and Flavor Symmetries, Nucl. Phys. B 114 (1976) 297 [INSPIRE].

    Article  ADS  Google Scholar 

  41. N. Marcus, A Tour through N = 2 strings, hep-th/9211059 [INSPIRE].

  42. M. Evans and B.A. Ovrut, The world sheet supergravity of the heterotic string, Phys. Lett. B 171 (1986) 177 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. S. Govindarajan and B.A. Ovrut, The Anomaly structure of (2,0) heterotic world sheet supergravity with gauged R invariance, Nucl. Phys. B 385 (1992) 251 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. E. Fradkin and A.A. Tseytlin, Effective Action Approach to Superstring Theory, Phys. Lett. B 160 (1985) 69 [INSPIRE].

    Article  ADS  Google Scholar 

  45. P.S. Howe, G. Papadopoulos and K. Stelle, The background field method and the nonlinear σ-model, Nucl. Phys. B 296 (1988) 26 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. C.M. Hull, Lectures on nonlinear sigma models and strings, lectures given at Super Field Theories Workshop, Vancouver, Canada, Jul 25 - Aug 6, 1986.

  47. K. Higashijima and M. Nitta, Kähler normal coordinate expansion in supersymmetric theories, Prog. Theor. Phys. 105 (2001) 243 [hep-th/0006027] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. K. Higashijima, E. Itou and M. Nitta, Normal coordinates in Kähler manifolds and the background field method, Prog. Theor. Phys. 108 (2002) 185 [hep-th/0203081] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. R. Brooks and S.J. Gates Jr., Unidexterous D = 2 supersymmetry in superspace. 2. Quantization, Phys. Lett. B 184 (1987) 217 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. S. Groot Nibbelink and T.S. Nyawelo, Two Loop effective Kähler potential of (non-)renormalizable supersymmetric models, JHEP 01 (2006) 034 [hep-th/0511004] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  51. S. Deguchi, Proving the Absence of the Perturbative Corrections to the N = 2 U(1) Kähler Potential Using the N = 1 Supergraph Techniques, arXiv:1111.3723 [INSPIRE].

  52. G. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. J.C. Collins, Renormalization. An introduction to renormalization, the renormalization group, and the operator product expansion, Cambridge University Press, Cambridge, U.K. (1986).

    Google Scholar 

  54. M. Sher, Electroweak Higgs Potentials and Vacuum Stability, Phys. Rept. 179 (1989) 273 [INSPIRE].

    Article  ADS  Google Scholar 

  55. G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. L. Álvarez-Gaumé, D.Z. Freedman and S. Mukhi, The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear σ-model, Annals Phys. 134 (1981) 85 [INSPIRE].

    Article  ADS  Google Scholar 

  57. C. Hull, U. Lindström, M. Roček, R. von Unge and M. Zabzine, Generalized Káhler Geometry in (2,1) superspace, JHEP 06 (2012) 013 [arXiv:1202.5624] [INSPIRE].

    Article  ADS  Google Scholar 

  58. K. Becker, M. Becker and J. Schwarz String theory and M-theory: A modern introduction, Cambridge University Press, Cambridge, U.K. (2007).

    MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  60. X. Cui and M. Shifman, N = (0, 2) Deformation of CP(1) Model: Two-dimensional Analog of N = 1 Yang-Mills Theory in Four Dimensions, Phys. Rev. D 85 (2012) 045004 [arXiv:1111.6350] [INSPIRE].

    ADS  Google Scholar 

  61. J. Bagger and J. Wess, Supersymmetry and supergravity, Princeton University Press, Princeton, U.S.A. (1992).

    Google Scholar 

  62. S. Gates, M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].

    Google Scholar 

  63. C. Hull and P. Townsend, The two loop β-function for σ-models with torsion, Phys. Lett. B 191 (1987) 115 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333, München, Germany

    Stefan Groot Nibbelink & Leonhard Horstmeyer

Authors
  1. Leonhard Horstmeyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Groot Nibbelink.

Additional information

ArXiv ePrint: 1203.6827

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nibbelink, S.G., Horstmeyer, L. Super Weyl invariance: BPS equations from heterotic worldsheets. J. High Energ. Phys. 2012, 54 (2012). https://doi.org/10.1007/JHEP07(2012)054

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2012)054

Keywords

Search

Navigation