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

top
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

top

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

top
  1. [AC] E. ARBARELLO, M. CORNALBA, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, Journ. Alg. Geom., 5 (1996), 705-749. Zbl0886.14007MR99c:14033
  2. [EYY] T. EGUCHI, Y. YAMADA, S.-K. YANG, On the genus expansion in the topological string theory, Rev. Mod. Phys., 7 (1995), 279. Zbl0837.58043MR96e:81169
  3. [FP] C. FABER, R. PANDHARIPANDE, Hodge integrals and Gromov-Witten theory, Preprint math.AG/9810173. Zbl0960.14031
  4. [GK] E. GETZLER, M. M. KAPRANOV, Modular operads, Comp. Math., 110 (1998), 65-126. Zbl0894.18005MR99f:18009
  5. [GoOrZo] V. GORBOUNOV, D. ORLOV, P. ZOGRAF (in preparation). 
  6. [IZu] C. ITZYKSON, J.-B. ZUBER, Combinatorics of the modular group II: the Kont-sevich integrals, Int. J. Mod. Phys., A7 (1992), 5661. Zbl0972.14500MR94m:32029
  7. [KabKi] A. KABANOV, T. KIMURA, Intersection numbers and rank one cohomological field theories in genus one, Comm. Math. Phys., 194 (1998), 651-674. Zbl0974.14018MR2000h:14046
  8. [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. Zbl0890.14011MR98i:14029
  9. [KoM] M. KONTSEVICH, Yu. MANIN, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., 164 (1994), 525-562. Zbl0853.14020MR95i:14049
  10. [KoMK] M. KONTSEVICH, Yu. MANIN, (with Appendix by R. Kaufmann), Quantum cohomology of a product, Inv. Math., 124 (1996), 313-340. Zbl0853.14021MR97e:14064
  11. [Mo] J. MORAVA, Schur Q-functions and a Kontsevich-Witten genus, Contemp. Math., 220 (1998), 255-266. Zbl0937.55003MR2000c:14034
  12. [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. Zbl0554.14008MR85j:14046
  13. [O] F. W. J. OLVER, Introduction to asymptotics and special functions, Academic Press, 1974. Zbl0308.41023MR55 #8655
  14. [W] E. WITTEN, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom., 1 (1991), 243-310. Zbl0757.53049MR93e:32028
  15. [Wo] S. WOLPERT, The hyperbolic metric and the geometry of the universal curve, J. Diff. Geo., 31 (1990), 417-472. Zbl0698.53002MR91a:32030
  16. [Zo] P. ZOGRAF, Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity, Preprint math.AG/9811026. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.