Theoretical and Mathematical Physics

, Volume 147, Issue 3, pp 729–754 | Cite as

Integrals over moduli spaces, ground ring, and four-point function in minimal Liouville gravity

  • A. A. Belavin
  • Al. B. Zamolodchikov


Directly evaluating the correlation functions in 2D minimal gravity requires integrating over the moduli space. For degenerate fields, the higher equations of motion of the Liouville field theory allow converting the integrand to a derivative, which reduces the integral to boundary terms and the so-called curvature contribution. The latter is directly related to the vacuum expectation value of the corresponding ground-ring element. The action of this element on the cohomology related to a generic matter primary field is evaluated directly in terms of the operator product expansions of the degenerate fields. This allows constructing the ground-ring algebra and evaluating the curvature term in the four-point function. We also analyze the operator product expansions of the Liouville “logarithmic primaries” and calculate the relevant logarithmic terms. Based on this, we obtain an explicit expression for the four-point correlation number of one degenerate and three generic matter fields. We compare this integral with the numbers obtained from the matrix models of 2D gravity and discuss some related problems and ambiguities.


Polyakov string theory Liouville gravity 


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  1. 1.
    A. Polyakov, Phys. Lett. B, 103, 207 (1981).MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    H. Dorn and H.-J. Otto, Phys. Lett. B, 291, 39 (1992); hep-th/9206053 (1992); Nucl. Phys. B, 429, 375 (1994); hep-th/9403141 (1994); A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B, 477, 577 (1996).MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    J. Teschner, Class. Q. Grav., 18, R153 (2001); hep-th/0104158 (2001).zbMATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    A. Belavin, A. Polyakov, and A. Zamolodchikov, Nucl. Phys. B, 241, 333 (1984).MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Al. Zamolodchikov, Internat. J. Mod. Phys. A, 19S2, 510 (2004); hep-th/0312279 (2003).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. Dotsenko and V. Fateev, Phys. Lett. B, 154, 291 (1985).MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Al. Zamolodchikov, Theor. Math. Phys., 142, 183 (2005); hep-th/0505063 (2005).MathSciNetGoogle Scholar
  8. 8.
    H. Poincaré, J. Math. Pure Appl., 5, No. 4, 137 (1898); A. M. Polyakov, Unpublished lectures [in Russian] (1979).zbMATHGoogle Scholar
  9. 9.
    L. Benoit and Y. Saint-Aubin, Phys. Lett. B, 215, 517 (1988).MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    C. Imbimbo, S. Mahapatra, and S. Mukhi, Nucl. Phys. B, 375, 399 (1992).MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    B. Feigin, Private communication (2005)Google Scholar
  12. 12.
    I. K. Klebanov and A. M. Polyakov, Modern Phys. Lett. A, 6, 3273 (1991); hep-th/9109032 (1991).MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    E. Witten, Nucl. Phys. B, 373, 187 (1992); hep-th/9108004 (1991).MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    I. K. Kostov and V. B. Petkova, Theor. Math. Phys., 146, 108 (2006); hep-th/0505078 (2005).CrossRefGoogle Scholar
  15. 15.
    N. Seiberg and D. Shih, JHEP, 0402, 021 (2004); hep-th/0312170 (2003); M. R. Douglas, I. R. Klebanov, D. Kutasov, J. Maldacena, E. Martinec, and N. Seiberg, “A new hat for the c=1 matrix model,” hep-th/0307195 (2003).Google Scholar
  16. 16.
    A. A. Belavin and Al. B. Zamolodchikov, JETP Letters, 82, 7 (2005).CrossRefGoogle Scholar
  17. 17.
    D. Kutasov, E. Martinec, and N. Seiberg, Phys. Lett. B, 276, 437 (1992); hep-th/9111048 (1991).MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    N. Seiberg, “Notes on quantum Liouville theory and quantum gravity,” in: Random Surfaces and Quantum Gravity (NATO ASI Ser., Ser. B, Vol. 262, O. Alvarez, E. Marinari, and P. Windey, eds.), Plenum, New York (1991), p. 363.Google Scholar
  19. 19.
    Al. Zamolodchikov, Comm. Math. Phys., 96, 419 (1984); Al. B. Zamolodchikov, Theor. Math. Phys., 73, 1088 (1987).MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    I. K. Kostov, “Thermal flow in the gravitaional O(n) model,” hep-th/0602075 (2006).Google Scholar
  21. 21.
    I. K. Kostov, Modern Phys. Lett. A, 4, 217 (1989).MathSciNetCrossRefADSGoogle Scholar
  22. 22.
    M. Gaudin and I. Kostov, Phys. Lett. B, 220, 200 (1989).MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. A. Belavin
    • 1
  • Al. B. Zamolodchikov
    • 2
    • 3
  1. 1.Landau Institute for Theoretical PhysicsRASChernogolovka, Moscow OblastRussia
  2. 2.Laboratoire de Physique Théorique et AstroparticulesUniversité Montpellier IIMontpellierFrance
  3. 3.Institute for Theoretical and Experimental PhysicsMoscowRussia

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