Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Witten E.: Two dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243 (1991)
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.235
Kontsevich M.: Intersection Theory on the moduli Space of Curves and Matrix Airy Function. Commun. Math. Phys. 147, 1 (1992)
Brézin E., Hikami S.: Vertices from replica in a random matrix theory. J. Phys. A 40, 13545 (2007)
Brézin E., Hikami S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697 (1996)
Brézin E., Hikami S.: Extension of level-spacing universality. Phys. Rev. E56, 264 (1997)
Brézin E., Hikami S.: Spectral form factor in a random matrix theory. Phys. Rev. E55, 4067 (1997)
Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers, and Matrix models, I. http://arxiv.org/list/math.AG/0101147, 2001
Okounkov A.: Generating functions for the intersection numbers on moduli spaces of curves. Intern. Math. Research. Notices 18, 933 (2002)
Okounkov A.: Random matrices and random permutations. Intern. Math. Research. Notices 20, 1043 (2000)
Harish-Chandra.: Proc. Nat. Acad. Sci. 42, 252 (1956)
Itzykson C., Zuber J.-B.: The planar approximation II. J. Math. Phys. 21, 411 (1980)
Brézin E., Hikami S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111–135 (2000)
Hashimoto, A., Min-xin Huang, Klemm, A., Shih, D.: Open/closed string duality for topological gravity with matter. JHEP 05, 007 (2005)
Kazakov V.A.: External matrix field problem and new multicriticalities in (−2)-dimensional random surfaces. Nucl. Phys. B 354, 614 (1991)
Itzykson C., Zuber J.-B.: Combinatorics of the Modular Group II. The Kontsevich integrals. Int. J. Mod. Phys. A7, 5661 (1992)
Liu K., Xu H.: New properties of the intersection numbers on moduli spaces of curves. Math. Res. Lett. 14, 1041–1054 (2007)
Brézin E., Hikami S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E57, 4140 (1998)
Brézin E., Hikami S.: Level spacing of random matrices in an external source. Phys. Rev. E58, 7176 (1998)
Jarvis T.J., Kimura T., Vaintrob A.: Moduli Spaces of Higher Spin Curves and Integrable Hierarchies. Comp. Math. 126, 157–212 (2001)
Adler M., van Moerbeke P.: A matrix integral solution to two-dimensional Wp-gravity. Commun. Math. Phys. 147, 25–56 (1992)
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. http://arxiv.org/list/math/0612510, 2006
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.R. Douglas
Unité Mixte de Recherche 8549 du Centre National de la Recherche Scientifique et de l’École Normale Supérieure
Rights and permissions
About this article
Cite this article
Brézin, E., Hikami, S. Intersection Theory from Duality and Replica. Commun. Math. Phys. 283, 507–521 (2008). https://doi.org/10.1007/s00220-008-0519-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0519-0