Invertible cohomological field theories and Weil-Petersson volumes
Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 519-535.

Nous montrons que la fonction génératrice des volumes de Weil-Petersson supérieurs des espaces de modules des courbes stables avec points marqués peut être obtenue à l’aide de celle de l’energie libre de Witten par un changement de variables donné par les polynômes de Schur. Comme la fonction génératrice possède un prolongement naturel à l’espace de modules des Théories Cohomologiques des Champs inversibles, ceci suggère l’existence d’un “très grand espace des phases”, dont les fonctions de corrélation incluent les intégrales de Hodge étudiées par C. Faber et R. Pandharipande. Nous dérivons de cette formule une expression asymptotique du volume de Weil-Peterson comme il est conjecturé par C. Itzykson. Nous discutons aussi d’une interprétation topologique de la formule de développement du genre de Itzykson-Zuber, ainsi que d’une bialgèbre opérant sur la cohomologie quantique qui est une version complexe du groupoïde des chemins classique.

We show that the generating function for the higher Weil–Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten’s free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the moduli space of invertible Cohomological Field Theories, this suggests the existence of a “very large phase space”, correlation functions on which include Hodge integrals studied by C. Faber and R. Pandharipande. From this formula we derive an asymptotical expression for the Weil–Petersson volume as conjectured by C. Itzykson. We also discuss a topological interpretation of the genus expansion formula of Itzykson–Zuber, as well as a related bialgebra acting upon quantum cohomology as a complex version of the classical path groupoid.

@article{AIF_2000__50_2_519_0,
     author = {Manin, Yuri I. and Zograf, Peter},
     title = {Invertible cohomological field theories and {Weil-Petersson} volumes},
     journal = {Annales de l'Institut Fourier},
     pages = {519--535},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     doi = {10.5802/aif.1764},
     mrnumber = {2001g:14046},
     zbl = {01448499},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1764/}
}
TY  - JOUR
AU  - Manin, Yuri I.
AU  - Zograf, Peter
TI  - Invertible cohomological field theories and Weil-Petersson volumes
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 519
EP  - 535
VL  - 50
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1764/
DO  - 10.5802/aif.1764
LA  - en
ID  - AIF_2000__50_2_519_0
ER  - 
%0 Journal Article
%A Manin, Yuri I.
%A Zograf, Peter
%T Invertible cohomological field theories and Weil-Petersson volumes
%J Annales de l'Institut Fourier
%D 2000
%P 519-535
%V 50
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1764/
%R 10.5802/aif.1764
%G en
%F AIF_2000__50_2_519_0
Manin, Yuri I.; Zograf, Peter. Invertible cohomological field theories and Weil-Petersson volumes. Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 519-535. doi : 10.5802/aif.1764. http://www.numdam.org/articles/10.5802/aif.1764/

[AC] E. Arbarello, M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, Journ. Alg. Geom., 5 (1996), 705-749. | MR | Zbl

[EYY] T. Eguchi, Y. Yamada, S.-K. Yang, On the genus expansion in the topological string theory, Rev. Mod. Phys., 7 (1995), 279. | MR | Zbl

[FP] C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Preprint math.AG/9810173. | Zbl

[GK] E. Getzler, M. M. Kapranov, Modular operads, Comp. Math., 110 (1998), 65-126. | MR | Zbl

[GoOrZo] V. Gorbounov, D. Orlov, P. Zograf (in preparation).

[IZu] C. Itzykson, J.-B. Zuber, Combinatorics of the modular group II: the Kont-sevich integrals, Int. J. Mod. Phys., A7 (1992), 5661. | MR | Zbl

[KabKi] A. Kabanov, T. Kimura, Intersection numbers and rank one cohomological field theories in genus one, Comm. Math. Phys., 194 (1998), 651-674. | MR | Zbl

[KaMZ] R. Kaufmann, Yu. Manin, D. Zagier, Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves, Comm. Math. Phys., 181 (1996), 763-787. | MR | Zbl

[KoM] M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525-562. | MR | Zbl

[KoMK] M. Kontsevich, Yu. Manin, (with Appendix by R. Kaufmann), Quantum cohomology of a product, Inv. Math., 124 (1996), 313-340. | MR | Zbl

[Mo] J. Morava, Schur Q-functions and a Kontsevich-Witten genus, Contemp. Math., 220 (1998), 255-266. | MR | Zbl

[Mu] D. Mumford, Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry (M. Artin and J. Tate, eds.), Part II, Birkhäuser, 1983, 271-328. | MR | Zbl

[O] F. W. J. Olver, Introduction to asymptotics and special functions, Academic Press, 1974. | MR | Zbl

[W] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom., 1 (1991), 243-310. | MR | Zbl

[Wo] S. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Diff. Geo., 31 (1990), 417-472. | MR | Zbl

[Zo] P. Zograf, Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity, Preprint math.AG/9811026.

Cité par Sources :