An algebro-geometric proof of Witten’s conjecture

Kazarian, M.; Lando, S.

doi:10.1090/S0894-0347-07-00566-8

An algebro-geometric proof of Witten’s conjecture
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by M. E. Kazarian and S. K. Lando;

J. Amer. Math. Soc. 20 (2007), 1079-1089

DOI: https://doi.org/10.1090/S0894-0347-07-00566-8

Published electronically: March 23, 2007

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Abstract:

We present a new proof of Witten’s conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the $2$-sphere.

References

V. I. Arnol′d, Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges, Funktsional. Anal. i Prilozhen. 30 (1996), no. 1, 1–17, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 30 (1996), no. 1, 1–14. MR 1387484, DOI 10.1007/BF02383392
Etsur\B{o} Date, Masaki Kashiwara, Michio Jimbo, and Tetsuji Miwa, Transformation groups for soliton equations, Nonlinear integrable systems—classical theory and quantum theory (Kyoto, 1981) World Sci. Publishing, Singapore, 1983, pp. 39–119. MR 725700
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240, DOI 10.1007/BF02684599
Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein, On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1175–1180 (English, with English and French summaries). MR 1701381, DOI 10.1016/S0764-4442(99)80435-2
Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), no. 2, 297–327. MR 1864018, DOI 10.1007/s002220100164
I. P. Goulden and D. M. Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997), no. 1, 51–60. MR 1396978, DOI 10.1090/S0002-9939-97-03880-X
I. P. Goulden, D. M. Jackson, and R. Vakil, The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc. (3) 83 (2001), no. 3, 563–581. MR 1851082, DOI 10.1112/plms/83.3.563
V. Kac and A. Schwarz, Geometric interpretation of the partition function of $2$D gravity, Phys. Lett. B 257 (1991), no. 3-4, 329–334. MR 1100639, DOI 10.1016/0370-2693(91)91901-7
Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23. MR 1171758, DOI 10.1007/BF02099526
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004. With an appendix by Don B. Zagier; Low-Dimensional Topology, II. MR 2036721, DOI 10.1007/978-3-540-38361-1
Maryam Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), no. 1, 1–23. MR 2257394, DOI 10.1090/S0894-0347-06-00526-1
Andrei Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), no. 4, 447–453. MR 1783622, DOI 10.4310/MRL.2000.v7.n4.a10

Gromov–Witten theory, Hurwitz numbers, and matrix models I

R. Pandharipande, The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett. Math. Phys. 53 (2000), no. 1, 59–74. MR 1799843, DOI 10.1023/A:1026571018707
Mikio Sato and Yasuko Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, Nonlinear partial differential equations in applied science (Tokyo, 1982) North-Holland Math. Stud., vol. 81, North-Holland, Amsterdam, 1983, pp. 259–271. MR 730247
Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65. MR 783348, DOI 10.1007/BF02698802
S. V. Shadrin, Geometry of meromorphic functions and intersections on moduli spaces of curves, Int. Math. Res. Not. 38 (2003), 2051–2094. MR 1994776, DOI 10.1155/S1073792803212101
Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR 1144529

Enumeration of ramified coverings of the sphere and $2$-dimensional gravity