Geodesic
An algebro-geometric proof of Witten’s conjecture
Kazarian, M.1 ; Lando, S.2
1 Steklov Institute of Mathematics, Russian Academy of Sciences, 8 Gubkina Street, Moscow, 117966 Russia, and The Poncelet Laboratory, Independent University of Moscow, 11, Bolshoy Vlasyevskiy Pereulok, Moscow, 121002 Russia
2 Institute for System Research, Russian Academy of Sciences, Nakhimovskii pr., 36 korp. 1, Moscow, 117218 Russia, and The Poncelet Laboratory, Independent University of Moscow, 11, Bolshoy Vlasyevskiy Pereulok, Moscow, 121002 Russia
Journal of the American Mathematical Society, Tome 20 (2007) no. 4, pp. 1079-1089

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-07-00566-8

Kazarian, M. 1 ; Lando, S. 2

1 Steklov Institute of Mathematics, Russian Academy of Sciences, 8 Gubkina Street, Moscow, 117966 Russia, and The Poncelet Laboratory, Independent University of Moscow, 11, Bolshoy Vlasyevskiy Pereulok, Moscow, 121002 Russia
2 Institute for System Research, Russian Academy of Sciences, Nakhimovskii pr., 36 korp. 1, Moscow, 117218 Russia, and The Poncelet Laboratory, Independent University of Moscow, 11, Bolshoy Vlasyevskiy Pereulok, Moscow, 121002 Russia 
@article{10_1090_S0894_0347_07_00566_8,
     author = {Kazarian, M. and Lando, S.},
     title = {An algebro-geometric proof of {Witten\^a€™s} conjecture},
     journal = {Journal of the American Mathematical Society},
     pages = {1079--1089},
     publisher = {mathdoc},
     volume = {20},
     number = {4},
     year = {2007},
     doi = {10.1090/S0894-0347-07-00566-8},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-07-00566-8/}
}
TY  - JOUR
AU  - Kazarian, M.
AU  - Lando, S.
TI  - An algebro-geometric proof of Witten’s conjecture
JO  - Journal of the American Mathematical Society
PY  - 2007
SP  - 1079
EP  - 1089
VL  - 20
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-07-00566-8/
DO  - 10.1090/S0894-0347-07-00566-8
ID  - 10_1090_S0894_0347_07_00566_8
ER  - 
%0 Journal Article
%A Kazarian, M.
%A Lando, S.
%T An algebro-geometric proof of Witten’s conjecture
%J Journal of the American Mathematical Society
%D 2007
%P 1079-1089
%V 20
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-07-00566-8/
%R 10.1090/S0894-0347-07-00566-8
%F 10_1090_S0894_0347_07_00566_8
Kazarian, M.; Lando, S. An algebro-geometric proof of Witten’s conjecture. Journal of the American Mathematical Society, Tome 20 (2007) no. 4, pp. 1079-1089. doi: 10.1090/S0894-0347-07-00566-8

[1] Arnol′D, V. I. Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges Funktsional. Anal. i Prilozhen. 1996

[2] Date, Etsur\B{O}, Kashiwara, Masaki, Jimbo, Michio, Miwa, Tetsuji Transformation groups for soliton equations 1983 39 119

[3] Deligne, P., Mumford, D. The irreducibility of the space of curves of given genus Inst. Hautes Études Sci. Publ. Math. 1969 75 109

[4] Ekedahl, Torsten, Lando, Sergei, Shapiro, Michael, Vainshtein, Alek On Hurwitz numbers and Hodge integrals C. R. Acad. Sci. Paris Sér. I Math. 1999 1175 1180

[5] Ekedahl, Torsten, Lando, Sergei, Shapiro, Michael, Vainshtein, Alek Hurwitz numbers and intersections on moduli spaces of curves Invent. Math. 2001 297 327

[6] Goulden, I. P., Jackson, D. M. Transitive factorisations into transpositions and holomorphic mappings on the sphere Proc. Amer. Math. Soc. 1997 51 60

[7] Goulden, I. P., Jackson, D. M., Vakil, R. The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals Proc. London Math. Soc. (3) 2001 563 581

[8] Kac, V., Schwarz, A. Geometric interpretation of the partition function of 2D gravity Phys. Lett. B 1991 329 334

[9] Kontsevich, Maxim Intersection theory on the moduli space of curves and the matrix Airy function Comm. Math. Phys. 1992 1 23

[10] Lando, Sergei K., Zvonkin, Alexander K. Graphs on surfaces and their applications 2004

[11] Mirzakhani, Maryam Weil-Petersson volumes and intersection theory on the moduli space of curves J. Amer. Math. Soc. 2007 1 23

[12] Okounkov, Andrei Toda equations for Hurwitz numbers Math. Res. Lett. 2000 447 453

[13] Pandharipande, R. The Toda equations and the Gromov-Witten theory of the Riemann sphere Lett. Math. Phys. 2000 59 74

[14] Sato, Mikio, Sato, Yasuko Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold 1983 259 271

[15] Segal, Graeme, Wilson, George Loop groups and equations of KdV type Inst. Hautes Études Sci. Publ. Math. 1985 5 65

[16] Shadrin, S. V. Geometry of meromorphic functions and intersections on moduli spaces of curves Int. Math. Res. Not. 2003 2051 2094

[17] Witten, Edward Two-dimensional gravity and intersection theory on moduli space 1991 243 310

Cité par Sources :