Moduli integrals and ground ring in minimal Liouville gravity

  • A. A. Belavin
  • A. B. Zamolodchikov
Fields, Particles, and Nuclei


Straightforward evaluation of the correlation functions in 2D minimal gravity requires integration over the moduli space. For degenerate fields, the Liouville higher equations of motion allow one to turn the integrand to a derivative and, thus, to reduce it to the boundary terms plus a so-called curvature contribution. The last is directly related to the expectation value of the corresponding ground ring element. We use the operator product expansion technique to reproduce the ground ring construction explicitly in terms of the (generalized) minimal matter and Liouville degenerate fields. The action of the ground ring on the generic primary fields is evaluated explicitly. This permits us to directly construct the ground ring algebra. Detailed analysis of the ground ring mechanism is helpful in the understanding of the boundary terms and their evaluation.

PACS numbers



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Polyakov, Phys. Lett. B 103B, 207 (1981).ADSMathSciNetGoogle Scholar
  2. 2.
    H. Dorn and H.-J. Otto, Phys. Lett. B 291, 39 (1992); Nucl. Phys. B 429, 375 (1994).ADSMathSciNetGoogle Scholar
  3. 3.
    A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B 477, 577 (1996).CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    A. Belavin, A. Polyakov, and A. Zamolodchikov, Nucl. Phys. B 241, 333 (1984).CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Al. Zamolodchikov, Int. J. Mod. Phys. A 19S2, 510 (2004).MathSciNetGoogle Scholar
  6. 6.
    V. Dotsenko and V. Fateev, Phys. Lett. B 154B, 291 (1985).ADSMathSciNetGoogle Scholar
  7. 7.
    Al. Zamolodchikov, Theor. Math. Phys. 142, 183 (2005).MathSciNetGoogle Scholar
  8. 8.
    C. Imbimbo, S. Mahapatra, and S. Mukhi, Nucl. Phys. B 375, 399 (1992).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    I. Klebanov and A. Polyakov, Mod. Phys. Lett. A 6, 3273 (1991).ADSMathSciNetGoogle Scholar
  10. 10.
    E. Witten, Nucl. Phys. B 373, 187 (1992).CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    N. Seiberg and D. Shih, J. High Energy Phys. 0402, 021 (2004).Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2005

Authors and Affiliations

  • A. A. Belavin
    • 1
  • A. B. Zamolodchikov
    • 2
  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow regionRussia
  2. 2.Laboratoire de Physique Théorique et AstroparticulesUniversité Montpelier IIMontpelierFrance

Personalised recommendations