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

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.

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.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     pages = {519-535},
     doi = {10.5802/aif.1764},
     zbl = {01448499},
     mrnumber = {2001g:14046},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_2_519_0}
}
Manin, Yuri I.; Zograf, Peter. Invertible cohomological field theories and Weil-Petersson volumes. Annales de l'Institut Fourier, Volume 50 (2000) no. 2, pp. 519-535. doi : 10.5802/aif.1764. http://www.numdam.org/item/AIF_2000__50_2_519_0/

[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 99c:14033 | Zbl 0886.14007

[EYY] T. Eguchi, Y. Yamada, S.-K. Yang, On the genus expansion in the topological string theory, Rev. Mod. Phys., 7 (1995), 279. | MR 96e:81169 | Zbl 0837.58043

[FP] C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Preprint math.AG/9810173. | Zbl 0960.14031

[GK] E. Getzler, M. M. Kapranov, Modular operads, Comp. Math., 110 (1998), 65-126. | MR 99f:18009 | Zbl 0894.18005

[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 94m:32029 | Zbl 0972.14500

[KabKi] A. Kabanov, T. Kimura, Intersection numbers and rank one cohomological field theories in genus one, Comm. Math. Phys., 194 (1998), 651-674. | MR 2000h:14046 | Zbl 0974.14018

[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 98i:14029 | Zbl 0890.14011

[KoM] M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525-562. | MR 95i:14049 | Zbl 0853.14020

[KoMK] M. Kontsevich, Yu. Manin, (with Appendix by R. Kaufmann), Quantum cohomology of a product, Inv. Math., 124 (1996), 313-340. | MR 97e:14064 | Zbl 0853.14021

[Mo] J. Morava, Schur Q-functions and a Kontsevich-Witten genus, Contemp. Math., 220 (1998), 255-266. | MR 2000c:14034 | Zbl 0937.55003

[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 85j:14046 | Zbl 0554.14008

[O] F. W. J. Olver, Introduction to asymptotics and special functions, Academic Press, 1974. | MR 55 #8655 | Zbl 0308.41023

[W] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom., 1 (1991), 243-310. | MR 93e:32028 | Zbl 0808.32023

[Wo] S. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Diff. Geo., 31 (1990), 417-472. | MR 91a:32030 | Zbl 0698.53002

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