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Observables in the Kontsevich Model

  • P. Di Francesco
Part of the NATO ASI Series book series (NSSB, volume 315)

Abstract

Kontsevich introduced a hermitian random matrix model to compute the generating function of intersection numbers of the moduli space of (punctured) Riemann surfaces. He showed that this generating function is also a τ-function for the Korteveg-de Vries (KdV) hierarchy of differential equations. This model is fundamentally different from the usual double scaling limit of random matrix models known to yield analogous τ-functions. Our aim is to clarify the notion of “observables” in both pictures, as related to KdV time evolutions. As a result we prove two conjectures by Kontsevich and Witten about the form of these observables, which involve polynomial matrix averages.

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References

  1. [1]
    E. Brézin and V. Kazakov, Phys. Lett. B236 (1990) 144.ADSGoogle Scholar
  2. M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 127.MathSciNetGoogle Scholar
  3. D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127.MathSciNetADSMATHCrossRefGoogle Scholar
  4. [2]
    E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber, Comm. Math. Phys. 69 (1979) 147.MathSciNetADSCrossRefGoogle Scholar
  5. [3]
    M. Douglas, Phys. Lett. B238 (1990) 176.ADSGoogle Scholar
  6. [4]
    P. Di Francesco and D. Kutasov, Nucl. Phys. B342 (1990) 589.ADSCrossRefGoogle Scholar
  7. [5]
    V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819.MathSciNetADSGoogle Scholar
  8. F. David, Mod. Phys. Lett. A3 (1988) 1651.ADSGoogle Scholar
  9. J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509.MathSciNetADSCrossRefGoogle Scholar
  10. [6]
    P. Di Francesco and D. Kutasov, Phys. Lett. B261 (1991) 385.ADSGoogle Scholar
  11. P. Di Francesco and D. Kutasov, and Nucl. Phys. B375 (1992) 119.ADSCrossRefGoogle Scholar
  12. [7]
    T. Banks, M. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B238 (1990) 279.MathSciNetADSGoogle Scholar
  13. R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B348 (1991) 435.MathSciNetADSCrossRefGoogle Scholar
  14. [8]
    E. Witten, Comm. Math. Phys. 117 (1988) 353.MathSciNetADSMATHCrossRefGoogle Scholar
  15. E. Witten, Phys. Lett. B206 (1988) 601.MathSciNetADSGoogle Scholar
  16. [9]
    E. Witten, Nucl. Phys. B240 (1990) 281.MathSciNetADSCrossRefGoogle Scholar
  17. [10]
    E. Witten, Surv. DifF. Geom. 1 (1991) 243.MathSciNetGoogle Scholar
  18. [11]
    M. Kontsevich, Intersection theory on the moduli space of curves, Funk. Anal. & Prilozh., 25 (1991) 50.-57MathSciNetCrossRefGoogle Scholar
  19. Intersection theory on the moduli space of curves and the matrix Airy function, lecture at the Arbeitstagung, Bonn, June 1991 and Comm. Math. Phys. 147 (1992) 1.Google Scholar
  20. [12]
    E. Witten, On the Kontsevich model and other models of two dimensional gravity, preprint IASSNS-HEP-91/24Google Scholar
  21. [13]
    C. Itzykson and J.-B. Zuber, Int. J. Mod. Phys. A7 (1992) 5661.MathSciNetADSGoogle Scholar
  22. [14]
    S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Phys. Lett. B275 (1992) 311.MathSciNetADSGoogle Scholar
  23. [15]
    M. Adler and P. van Moerbeke, The W p-gravity version of the Witten-Kontsevich model, Brandeis preprint, September 1991.Google Scholar
  24. [16]
    E. Witten, Algebraic Geometry associated with matrix models of two dimensional gravity, preprint IASSNS-HEP-91/74.Google Scholar
  25. [17]
    P. Di Francesco, C. Itzykson and J.-B. Zuber, Comm. Math. Phys. 151 (1993) 193.MathSciNetADSMATHCrossRefGoogle Scholar
  26. [18]
    Harish-Chandra, Amer. J. Math. 79 (1957) 87.-120MathSciNetMATHCrossRefGoogle Scholar
  27. C. Itzykson and J.-B. Zuber, J. Math. Phys. 21 (1980) 411–421.MathSciNetADSMATHCrossRefGoogle Scholar
  28. [19]
    J. Duistermaat and G. Heckman, Invent. Math. 69 (1982) 259.MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • P. Di Francesco
    • 1
  1. 1.Service de Physique Théorique de SaclayGif sur Yvette CedexFrance

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