Communications in Mathematical Physics

, Volume 283, Issue 2, pp 507–521 | Cite as

Intersection Theory from Duality and Replica

  • E. Brézin
  • S. Hikami


Kontsevich’s work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In this article we show that a duality between k-point functions on N × N matrices and N-point functions of k × k matrices, plus the replica method, familiar in the theory of disordered systems, allows one to recover Kontsevich’s results on the intersection numbers, and to generalize them to other models. This provides an alternative and simple way to compute intersection numbers with one marked point, and leads also to some new results.


Modulus Space Marked Point Intersection Number Intersection Theory Airy Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Witten E.: Two dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243 (1991)MathSciNetGoogle Scholar
  2. 2.
    Witten, E.: Algebraic geometry associated with matrix models of two dimensional gravity. In: Topological methods in modern mathematics (Stony Brook, NY, 1991), Houston, TX: Publish or Perish, 1993, p.235Google Scholar
  3. 3.
    Kontsevich M.: Intersection Theory on the moduli Space of Curves and Matrix Airy Function. Commun. Math. Phys. 147, 1 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Brézin E., Hikami S.: Vertices from replica in a random matrix theory. J. Phys. A 40, 13545 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Brézin E., Hikami S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697 (1996)zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Brézin E., Hikami S.: Extension of level-spacing universality. Phys. Rev. E56, 264 (1997)ADSGoogle Scholar
  7. 7.
    Brézin E., Hikami S.: Spectral form factor in a random matrix theory. Phys. Rev. E55, 4067 (1997)ADSGoogle Scholar
  8. 8.
    Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers, and Matrix models, I., 2001
  9. 9.
    Okounkov A.: Generating functions for the intersection numbers on moduli spaces of curves. Intern. Math. Research. Notices 18, 933 (2002)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Okounkov A.: Random matrices and random permutations. Intern. Math. Research. Notices 20, 1043 (2000)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Harish-Chandra.: Proc. Nat. Acad. Sci. 42, 252 (1956)zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Itzykson C., Zuber J.-B.: The planar approximation II. J. Math. Phys. 21, 411 (1980)zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Brézin E., Hikami S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111–135 (2000)zbMATHCrossRefADSGoogle Scholar
  14. 14.
    Hashimoto, A., Min-xin Huang, Klemm, A., Shih, D.: Open/closed string duality for topological gravity with matter. JHEP 05, 007 (2005)Google Scholar
  15. 15.
    Kazakov V.A.: External matrix field problem and new multicriticalities in (−2)-dimensional random surfaces. Nucl. Phys. B 354, 614 (1991)CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Itzykson C., Zuber J.-B.: Combinatorics of the Modular Group II. The Kontsevich integrals. Int. J. Mod. Phys. A7, 5661 (1992)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Liu K., Xu H.: New properties of the intersection numbers on moduli spaces of curves. Math. Res. Lett. 14, 1041–1054 (2007)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Brézin E., Hikami S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E57, 4140 (1998)ADSGoogle Scholar
  19. 19.
    Brézin E., Hikami S.: Level spacing of random matrices in an external source. Phys. Rev. E58, 7176 (1998)ADSGoogle Scholar
  20. 20.
    Jarvis T.J., Kimura T., Vaintrob A.: Moduli Spaces of Higher Spin Curves and Integrable Hierarchies. Comp. Math. 126, 157–212 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Adler M., van Moerbeke P.: A matrix integral solution to two-dimensional Wp-gravity. Commun. Math. Phys. 147, 25–56 (1992)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    Shadrin, S.: Geometriy of meromorphic functions and intersections on moduli spaces of curves. Int. Math. Res. Not. 38, 2051 (2003); Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the r-spin Witten conjecture., 2006

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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique ThéoriqueEcole Normale SupérieureParis Cedex 05France
  2. 2.Department of Basic SciencesUniversity of TokyoTokyoJapan

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