Advertisement

Theoretical and Mathematical Physics

, Volume 147, Issue 3, pp 755–776 | Cite as

Massive Majorana fermion coupled to two-dimensional gravity and the random-lattice Ising model

  • Al. B. Zamolodchikov
  • Y. Ishimoto
Article

Abstract

We consider the partition function of the two-dimensional free massive Majorana fermion coupled to the quantized metric of the spherical topology. By adding an arbitrary conformal “spectator” matter, we gain control over the total matter central charge. This provides an interesting continuously parameterized family of critical points and also allows making a connection with the semiclassical limit. We use the Liouville field theory as the effective description of the quantized gravity. The spherical scaling function is calculated approximately, but (we believe) to a good numerical precision, in almost the whole domain of the spectator parameter. An impressive comparison with the predictions of the exactly solvable matrix model yields a more general model of random-lattice statistics, which is most probably not solvable by the matrix-model technique but reveals a more general pattern of critical behavior. We hope that numerical simulations or series extrapolation will be able to reveal our family of scaling functions.

Keywords

Liouville field theory two-dimensional gravity Ising model random lattice 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Al. Zamolodchikov, “Perturbed conformal field theory on fluctuating sphere,” hep-th/0508044 (2005).Google Scholar
  2. 2.
    V. A. Kazakov, Phys. Lett. B, 150, 282 (1985).MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    F. David, Nucl. Phys. B, 257, 543 (1985).CrossRefADSGoogle Scholar
  4. 4.
    V. Kazakov, I. Kostov, and A. Migdal, Phys. Lett. B, 157, 295 (1985).MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    D. Boulatov, V. Kazakov, I. Kostov, and A. Migdal, Nucl. Phys. B, 275, 641 (1986).MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    A. Polyakov, Phys. Lett. B, 103, 207 (1981).MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    T. Regge, Phys. Rev., 108, 558 (1961).Google Scholar
  8. 8.
    V. Knizhnik, A. Polyakov, and A. Zamolodchikov, Modern Phys. Lett. A, 3, 819 (1988).MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    F. David, Modern Phys. Lett. A, 3, 1651 (1988); J. Distler and H. Kawai, Nucl. Phys. B, 231, 509 (1989).MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    M. Goulian and M. Li, Phys. Rev. Lett., 66, 2051 (1991); V. S. Dotsenko, Modern Phys. Lett. A, 6, 3601 (1991); “Correlation functions of local operators in 2D gravity coupled to minimal matter,” in: New Symmetry Principles in Quantum Field Theory (NATO Sci. Ser. B, Vol. 295, J. Fröhlich, G. ’t Hooft, A. Jaffe, G. Mach, P. K. Mitter, and R. Stora, eds.), Plenum, New York (1992), p. 423; hep-th/9110030 (1991).CrossRefADSGoogle Scholar
  11. 11.
    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).MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    A. Zamolodchikov and Al. Zamolodchikov, Nucl. Phys. B, 477, 577 (1996).MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    J. Teschner, Phys. Lett. B, 363, 65 (1995); hep-th/9507109 (1995).CrossRefADSGoogle Scholar
  14. 14.
    V. Fateev, A. Zamolodchikov, and Al. Zamolodchikov, “Boundary Liouville field theory: I. Boundary state and boundary two-point function,” hep-th/0001012 (2000); A. Zamolodchikov and Al. Zamolodchikov, “Liouville field theory on a pseudosphere,” hep-th/0101152 (2001).Google Scholar
  15. 15.
    E. Brézin, C. Itzykson, G. Parisi, and J.-B. Zuber, Comm. Math. Phys., 59, 35 (1978).MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    V. Kazakov, JETP Letters, 44, 133 (1986).MathSciNetADSGoogle Scholar
  17. 17.
    D. Boulatov and V. Kazakov, Phys. Lett. B, 186, 379 (1987).MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    P. Di Francesco and D. Kutasov, Phys. Lett. B, 261, 385 (1991).MathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Y. Ishimoto and S. Yamaguchi, Phys. Lett. B, 607, 172 (2005); hep-th/0406262 (2004); Al. B. Zamolodchikov, Theor. Math. Phys., 142, 183 (2005); hep-th/0505063 (2005); A. A. Belavin and Al. B. Zamolodchikov, Theor. Math. Phys., 147, 729 (2006).MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Al. Zamolodchikov, JHEP, 0207, 029 (2002); hep-th/0109078 (2001).Google Scholar
  21. 21.
    F. David, Nucl. Phys. B, 257, 543 (1985).CrossRefADSGoogle Scholar
  22. 22.
    D. Boulatov, V. Kazakov, I. Kostov, and A. Migdal, Bulg. J. Phys., 13, 313 (1986).MathSciNetGoogle Scholar
  23. 23.
    P. Ginsparg and G. Moore, “Lectures on 2D gravity and 2D string theory (TASI 1992),” hep-th/9304011 (1993).Google Scholar
  24. 24.
    Z. Burda and J. Jurkiewicz, “The Ising model on a random lattice with a coordination number equal 3,” Preprint TPJU-1/89, Kraków Univ., Kraków (1989).Google Scholar
  25. 25.
    Al. Zamolodchikov, Comm. Math. Phys., 96, 419 (1984); Al. B. Zamolodchikov, Theor. Math. Phys., 73, 1088 (1987).MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Al. Zamolodchikov, “Gravitational Yang-Lee model: Four-point function,” (in preparation).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Al. B. Zamolodchikov
    • 1
    • 2
  • Y. Ishimoto
    • 3
  1. 1.Laboratoire de Physique Théorique et AstroparticulesUniversité Montpellier IIMontpellierFrance
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.Kawai Theoretical LaboratoryRIKENSaitamaJapan

Personalised recommendations