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Communications in Mathematical Physics

, Volume 327, Issue 3, pp 779–814 | Cite as

The Effective Theory of Strings

  • Dorothea Bahns
  • Katarzyna Rejzner
  • Jochen ZahnEmail author
Article

Abstract

We show that the Nambu–Goto string, and its higher dimensional generalizations, can be quantized, in the sense of an effective theory, in any dimension of the target space. The crucial point is to consider expansions around classical string configurations. We are using tools from perturbative algebraic quantum field theory, quantum field theory on curved spacetimes, and the Batalin–Vilkovisky formalism. Our model has some similarities with the Lüscher–Weisz string, but we allow for arbitrary classical background string configurations and keep the diffeomorphism invariance.

Keywords

Target Space Formal Power Series Covariant Functor Ghost Number Quantum Master Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Green, M.B., Schwarz, J.H. Witten, E.: Superstring theory, Vol. 1, Cambridge: Cambridge University Press, 1987Google Scholar
  2. 2.
    Rebbi C.: Dual models and relativistic quantum strings. Phys. Rept. 12, 1–73 (1974)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Pohlmeyer K.: A group theoretical approach to the quantization of the free relativistic closed string. Phys. Lett. B119, 100 (1982)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Thiemann T.: The LQG string: Loop quantum gravity quantization of string theory I: Flat target space. Class. Quant. Grav. 23, 1923–1970 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Meusburger C., Rehren K.-H.: Algebraic quantization of the closed bosonic string. Commun. Math. Phys. 237, 69–85 (2003)ADSzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brandt F., Troost W., van Proeyen A.: The BRST-antibracket cohomology of 2-d gravity conformally coupled to scalar matter. Nucl. Phys. B 464, 353–408 (1996)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Gomis J., Paris J., Samuel S.: Antibracket, antifields and gauge theory quantization. Phys. Rept. 259, 1–145 (1995)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grundling H., Hurst C.A.: The operator quantization of the open bosonic string: field algebra. Commun. Math. Phys. 156, 473–525 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dimock J.: Locality in free string field theory-II. Annales Henri Poincaré 3, 613–634 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Brink, L., Henneaux, M.: Principles of String Theory. New York: Plenum Press, 1988Google Scholar
  11. 11.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: A New paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003)ADSzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 231, 309–345 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Barnich G., Brandt F., Henneaux M.: Local BRST cohomology in gauge theories. Phys. Rept. 338, 439–569 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fredenhagen K., Rejzner K.: Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697–725 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lüscher, M., Weisz, P.: Quark confinement and the bosonicstring. JHEP 0207, 049 (2002)Google Scholar
  16. 16.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton: Princeton University Press, 1992Google Scholar
  17. 17.
    Aharony O., Dodelson M.: Effective string theory and nonlinear Lorentz invariance. JHEP 1202, 008 (2012)ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dubovsky S., Flauger R., Gorbenko V.: Effective string theory revisited. JHEP 1209, 044 (2012)ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    Polchinski J., Strominger A.: Effective string theory. Phys. Rev. Lett. 67, 1681–1684 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Mund J., Schroer B., Yngvason J.: String localized quantum fields from Wigner representations. Phys. Lett. B596, 156–162 (2004)ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    Brunetti, R., Fredenhagen, K.: Towards a background independent formulation of perturbative quantum gravity. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Quantum Gravity: Mathematical Models and Experimental Bounds, Boston: Birkhäuser, 2007, p. 151Google Scholar
  22. 22.
    Hollands S., Wald R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227–312 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Wald, R.M. General Relativity, Chicago: University of Chicago Press, 1984Google Scholar
  24. 24.
    Eggers J., Hoppe J.: Singularity formation for time-like extremal hypersurfaces. Phys. Lett. B680, 274–278 (2009)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Müller O.: The Cauchy problem of Lorentzian minimal surfaces in globally hyperbolic manifolds. Ann. Global Anal. Geom. 32(1), 67–85 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Allen P., Andersson L., Isenberg J.: Timelike minimal submanifolds of general co-dimension in Minkowski space time. J. Hyperbolic Differ. Equ. 3(4), 691–700 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Fredenhagen K., Rejzner K.: Batalin-Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314, 93–127 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Brunetti R., Dütsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Providence: American Mathematical Society, 1997Google Scholar
  31. 31.
    Lang, S.: Differential and Riemannian Manifolds, Berlin: Springer, 1995Google Scholar
  32. 32.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization, Billingsley: European Mathematical Society, 2007Google Scholar
  33. 33.
    Kleinert H.: The membrane properties of condensing strings. Phys. Lett. B 174, 335–338 (1986)ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Bogoliubov, N., Shirkov, D.: Introduction to the Theory of Quantized Fields. New York: Interscience Publishers, Inc., 1959Google Scholar
  35. 35.
    Hollands S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033–1172 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Rejzner, K.: Batalin-Vilkovisky formalism in locally covariant field theory Ph.D. thesis, Hamburg University, 2011Google Scholar
  37. 37.
    Kontsevich M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 223, 289–326 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Fulling S., Narcowich F., Wald R.M.: Singularity structure of the two point function in quantum field theory in curved space-time II.. Annals Phys. 136, 243–272 (1981)ADSCrossRefMathSciNetGoogle Scholar
  40. 40.
    Dütsch M., Fredenhagen K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16, 1291–1348 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Brennecke F., Dütsch M.: Removal of violations of the Master Ward Identity in perturbative QFT. Rev. Math. Phys. 20, 119–172 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Guichardet, A.: Cohomologie des groupes topologiques et des algèbres de Lie, Vol. 2 of Textes Mathématiques [Mathematical Texts], CEDIC, Paris (1980)Google Scholar
  43. 43.
    Piguet, O., Sorella, S.P.:Algebraic Renormalization. In: Lecture Notes in Physics, Vol. 28, Berlin: Springer, 1995Google Scholar
  44. 44.
    Brandt F., Dragon N., Kreuzer M.: All consistent Yang-Mills anomalies. Phys. Lett. B 231, 263–270 (1989)ADSCrossRefMathSciNetGoogle Scholar
  45. 45.
    Polyakov A.M.: Fine structure of strings. Nucl. Phys. B 268, 406–412 (1986)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Dorothea Bahns
    • 1
  • Katarzyna Rejzner
    • 2
  • Jochen Zahn
    • 1
    Email author
  1. 1.Courant Research Centre “Higher Order Structures”University of GöttingenGöttingenGermany
  2. 2.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany

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