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.

     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 = {}
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.

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