Invertible cohomological field theories and Weil-Petersson volumes

Yuri I. Manin; Peter Zograf

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 2, page 519-535
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Manin, Yuri I., and Zograf, Peter. "Invertible cohomological field theories and Weil-Petersson volumes." Annales de l'institut Fourier 50.2 (2000): 519-535. <http://eudml.org/doc/75428>.

@article{Manin2000,
abstract = {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.},
author = {Manin, Yuri I., Zograf, Peter},
journal = {Annales de l'institut Fourier},
keywords = {moduli spaces; operads; Weil-Petersson volumes; cohomological field theories; Schur polynomials; stationary phase method; Virasoro constraints; potential; quantum cohomology; KdV equation},
language = {eng},
number = {2},
pages = {519-535},
publisher = {Association des Annales de l'Institut Fourier},
title = {Invertible cohomological field theories and Weil-Petersson volumes},
url = {http://eudml.org/doc/75428},
volume = {50},
year = {2000},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 2
SP - 519
EP - 535
AB - 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.
LA - eng
KW - moduli spaces; operads; Weil-Petersson volumes; cohomological field theories; Schur polynomials; stationary phase method; Virasoro constraints; potential; quantum cohomology; KdV equation
UR - http://eudml.org/doc/75428
ER -

References

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