Skip to main content
SpringerLink
Log in

A global and stochastic analysis approach to bosonic strings and associated quantum fields

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

We construct a probability measure giving a mathematical realization of Polyakov's heuristic measure for bosonic strings in space-time dimensions 3≤d≤13, having as world sheet compact Riemann surfaces Λ of arbitrary genus. The measure involves the path space measures for scalar fields with exponential interaction on Λ and a measure on Teichmüller space.

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. AlvarezO.: Theory of strings with boundaries: Fluctuations, topology, and quantum geometry, Nucl. Phys. B216 (1983), 125.

    Google Scholar 

  2. Alvarez-GauméL. and NelsonP.: Riemann surfaces and string theories, in B.De-Witt and M.Grisaru (eds.), Supersymmetry, Supergravity and Superstrings, World Scientific, Singapore, 1986.

    Google Scholar 

  3. AlbeverioS., FenstadJ.E., Høegh KrohnR., and LindstrømT.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.

    Google Scholar 

  4. AlbeverioS. and Høegh-KrohnR.: The Wightman axioms and the mass gap for strong interactions of exponential type in two dimensional space-time, J. Funct. Anal. 16 (1974), 39.

    Google Scholar 

  5. AlbeverioS., Høegh-KrohnR., PaychaS., and ScarlattiS.: Path space measure for the Liouville quantum field theory and the construction of the relativistic strings, Phys. Lett. B174 (1986), 81.

    Google Scholar 

  6. Albeverio, S., Høegh-Krohn, R., Paycha, S., and Scarlatti, S.: A probability measure for random surfaces of arbitrary genus and bosonic strings in 4 dimensions, in Proc. Eugene Wigner Symposium on Space Time Symmetries, Washington 1988, Proceedings Series Nuclear Phys. B 6 (1989), 180.

  7. Bers, L.: Riemann Surfaces, Courant Institute of Mathematical Sciences, 1957–59.

  8. BrinkL., DiVecchiaP., and HoweP.: A locally supersymmetric and reparametrization invariant action for the spinning string, Phys. Lett. B65 (1971), 471.

    Google Scholar 

  9. BelavinA. and KnizhnikV.G.: Algebraic geometry and the geometry of quantum strings, Phys. Lett. B168 (1986), 201.

    Google Scholar 

  10. BeilinsonA.A. and ManinYu.I.: The Mumford form and the Polyakov measure in string theory, Comm. Math. Phys. 107 (1986), 359.

    Google Scholar 

  11. Choquet-BruhatY., DeWitt-MoretteC., and Dillard-BleickM.: Analysis, Manifolds and Physics, North Holland, Amsterdam, 1977.

    Google Scholar 

  12. Durhuus, B., Quantum Theory of Strings, Nordita lectures, 1982.

  13. DeAngelisG.F., deFalcoD., and DiGenovaG.: Random fields on Riemannian manifolds: A constructive approach, Comm. Math. Phys. 103 (1986), 297.

    Google Scholar 

  14. D'HokerE. and PhongD.H.: Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys. B269 (1986), 205.

    Google Scholar 

  15. D'HokerE. and PhongD.H.: The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988), 917.

    Google Scholar 

  16. DunfordN. and SchwartzJ.: Linear Operators, Vol. II. Spectral Theory, Interscience, New York, 1963.

    Google Scholar 

  17. DurhuusB., OlesenP., and PetersenJ.L.: Polyakov's quantized string with boundary terms, Nucl. Phys. B198 (1982), 157; B201 (1982), 176.

    Google Scholar 

  18. DurhuusB., NielsenH.B., OlesenP., and PetersonJ.L.: Dual models as saddle point approximations to Polyakov's quantized string, Nucl. Phys. B196 (1982), 498.

    Google Scholar 

  19. EellsJ.: Fibre Bundles, in Global Analysis and its Applications, Vol. I, Intern. Course Trieste, 1972, IAEA Vienna, 1974.

    Google Scholar 

  20. EbinD.: The manifold of Riemannian metrics, Proc. Symp. Pure Math. AMS 15 (1970), 11.

    Google Scholar 

  21. EarleC.J. and EellsJ.: A fibre bundle description of Teichmüller theory. J. Diff. Geom. 3 (1969), 19.

    Google Scholar 

  22. FigariR., Høegh-KrohnR., and NappiC.: Interacting ralativistic boson fields in the De Sitter Universe with two space-time dimensions, Comm. Math. Phys. 44 (1975), 265.

    Google Scholar 

  23. FriedanD.: Introduction to Polyakov's string theory, in R.Stora and J.B.Zuber (eds.), Recent Advances in Field Theory and Statistical Mechanics, Elsevier, Amsterdam, 1984, p.839.

    Google Scholar 

  24. FarkasH.M. and KraI.: Riemann Surfaces, Springer-Verlag, Berlin, 1980.

    Google Scholar 

  25. FisherA.E. and TrombaA.J.: On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann. 267 (1984), 311.

    Google Scholar 

  26. Gilkey, P.: The Index Theorem and the Heat Equation, Publish or Perish, 1974.

  27. Gilkey, P.: Invariance Theory. The Heat Equation and the Atiyah-Singer Index Theorem, Publish or Perish, 1984.

  28. GolubitskyM. and GuilleminV.: Stable Mappings and their Singularities, Springer-Verlag, Berlin, 1973.

    Google Scholar 

  29. GilbertG.: String theory path integral: Genus two and higher, Nucl. Phys. B277 (1986), 102.

    Google Scholar 

  30. Green, M.B., Schwarz, J.H., and Witten, E.: Superstring Theory, Vols. I, II, Cambridge University Press, 1987.

  31. Høegh-KrohnR.: A general class of quantum fields without cut-off in two dimensional space-time, Comm. Math. Phys. 21 (1971), 244.

    Google Scholar 

  32. HamiltonR.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7 (1982), 65.

    Google Scholar 

  33. HabaZ.: Behavior in strong fields of Euclidean gauge theories, Phys. Rev. 26 (1982), 3506.

    Google Scholar 

  34. HabaZ.: Correlation functions of σ-fields with values in a hyperbolic space, Internat. J. Modern Phys. A4 (1989), 267.

    Google Scholar 

  35. HabaZ.: Model field theory for bag-like structure, Phys. Rev. D18 (1978), 4610.

    Google Scholar 

  36. Jackiw, R.: Liouville field theory: a two-dimensional model for gravity, in S. Christensen, and Adam Hilgen (eds.), Quantum Theory of Gravity, Bristol, 1983, p. 403.

  37. JaskólskiZ.: On the Gribov ambiguity in the Polyakov string, J. Math. Phys. 29 (1988), 1034–1043.

    Google Scholar 

  38. JaskólskiZ.: The integration of G-invariant functions and the geometry of the Faddeev-Popov procedure, Comm. Math. Phys. 111 (1987), 439.

    Google Scholar 

  39. KillingbackT.P.: Global aspects of fixing the gauge in the Polyakov string and Einstein gravity, Comm. Math. Phys. 100 (1985), 267.

    Google Scholar 

  40. KobayashiS. and NomizuK.: Foundations of Differential Geometry. Vols. I, II, Interscience, New York, 1963.

    Google Scholar 

  41. LehtoO.: Univalent Functions and Teichmüller Spaces, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  42. LangS.: Differentiable Manifolds, Addison Wesley, Reading, Mass., 1972.

    Google Scholar 

  43. MilnorJ.: Remarks on Infinite-Dimensional Lie Groups, in B.S.DeWitt and R.Stora (eds.), Relativity, Groups and Topology II, Les Houches Session XL 1983, Elsevier, Amsterdam, 1984.

    Google Scholar 

  44. MooreG. and NelsonP.: Measure for moduli, Nucl. Phys. B266 (1986), 58.

    Google Scholar 

  45. MolchanovS.A.: Diffusion processes and Riemannian geometry, Russian Math. Surveys 30:1 (1975), 1–73.

    Google Scholar 

  46. NelsonP.: Lectures on strings and moduli space, Phys. Reports 149 (1987), 337.

    Google Scholar 

  47. NewlanderA. and NirenbergL.: Complex analytic coordinates in almost complex manifold, Ann. Math. 65 (1957), 391.

    Google Scholar 

  48. OmoriH.: On the group of diffeomorphisms on a compact manifold, Proc. Symp. Pure Math. AMS 15 (1970), 167.

    Google Scholar 

  49. PolyakovA.: Quantum geometry of bosonic strings, Phys. Lett. 103 B (1981), 207.

    Google Scholar 

  50. Paycha, S.: A mathematical interpretation of the Polyakov string model in non-critical dimensions, Thesis, Paris, 1990.

  51. PolchinskiJ.: Evaluation of one loop string path integral, Comm. Math. Phys. 104 (1986), 37.

    Google Scholar 

  52. Polchinski, J.: From Polyakov to moduli, in S. T. Yau (ed.), Mathematical Aspects of String Theory, Proceedings of San Diego Conference July 1986, World Scientific, Singapore, 1988.

  53. ReedM. and SimonB.: Methods of Modern Mathematical Physics Vols. I–IV, Academic Press, New York, 1975.

    Google Scholar 

  54. RellichF.: Perturbation Theory of Eigenvalue Problems, Gordon and Breach Science Publishers, New York, 1969.

    Google Scholar 

  55. Scarlatti, S.: Metodi analatici e probabilistici nello studio del modello di Polyakov per la stringa bosonica, Thesis, Rome, 1990.

  56. SchwartzA.S.: Instantons and fermions in the field of instanton, Comm. Math. Phys. 64 (1978/79), 233.

    Google Scholar 

  57. SeeleyR.: Complex powers of an elliptic operator, in Proc. Sympos. Pure Math. Vol. 10, Amer. Math. Soc., Providence, R.I., 1967, p.288.

    Google Scholar 

  58. SimonB.: The 194–1 Euclidean Quantum Field Theory, Princeton University Press, Princeton, N.J., 1974.

    Google Scholar 

  59. Simon, B.: Trace Ideals and their Applications, Cambridge University Press, 1979.

  60. SmitJ.: Algebraic geometry and strings, Comm. Math. Phys. 114 (1988), 645.

    Google Scholar 

  61. TaylorM.E.: Pseudodifferential Operators, Princeton University Press, Princeton, N.J., 1981.

    Google Scholar 

  62. TrombaA.J.: Global analysis and Teichmüller theory, in Seminar on New Results in Nonlinear Partial Differential Equations, A.J. Tromba, Public., Max Planck Inst. für Math., Vieweg-Braunschweig, Bonn, 1987.

    Google Scholar 

  63. WeinbergS.: Covariant Path Integral Approach to String Theory, in “Strings and Superstrings”, World Scient., Singapore, 1988.

    Google Scholar 

  64. WaldR.M.: On the Euclidean approach to quantum field theory in curved spacetime, Comm. Math. Phys. 70 (1979), 221.

    Google Scholar 

  65. WeidmannJ.: Linear Operators in Hilbert Spaces, Springer-Verlag, Berlin, 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematische Fakultät, Ruhr-Universität Bochum, D-4630, Bochum 1, Germany

    Sergio Albeverio

  2. Matematisk Institutt, Universitetet i Oslo, Blindern, Oslo, Norway

    Raphael Høegh-Krohn

  3. Mathematische Fakultät, Ruhr-Universität Bochum, D-4630, Bochum 1, Germany

    Sylvie Paycha

  4. Istituto di Matematica, Universitá di Roma I, Rome, Italy

    Sergio Scarlatti

Authors
  1. Sergio Albeverio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raphael Høegh-Krohn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sylvie Paycha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sergio Scarlatti
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Deceased 24 January 1988

Rights and permissions

Reprints and permissions

About this article

Cite this article

Albeverio, S., Høegh-Krohn, R., Paycha, S. et al. A global and stochastic analysis approach to bosonic strings and associated quantum fields. Acta Appl Math 26, 103–195 (1992). https://doi.org/10.1007/BF00046581

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00046581

Mathematics Subject Classifications (1991)

Key words

Search

Navigation