Skip to main content
SpringerLink
Log in

Virasoro constraints for target curves

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are completely and explicitly determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. Sel. Math., New Ser. 5, 423–466 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Eguchi, T., Hori, K., Xiong, C.-S.: Quantum cohomology and Virasoro algebra. Phys. Lett. B 402, 71–80 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. GAFA 2000 (Tel Aviv 1999). Geom. Funct. Anal. 2000, Special Volume, Part II, pp. 560–673

  4. Faber, C., Pandharipande, R.: Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS) 7, 13–49 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Givental, A.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Dedicated to the memory of I.G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1, 551–568 (2001)

  6. Getzler, E.: The Virasoro conjecture for Gromov-Witten invariants. Algebraic geometry: Hirzebruch 70 (Warsaw 1998), pp. 147–176. Contemp. Math., vol. 241. Providence, RI: Am. Math. Soc. 1999

  7. Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135, 487–518 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. math.AG/0309227

  9. Ionel, E., Parker, T.: Relative Gromov-Witten invariants. Ann. Math. 157, 45–96 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kac, V.: Infinite dimensional Lie algebras. Cambridge: Cambridge University Press 1990

  11. Li, A.-M., Ruan, Y.: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I. Invent. Math. 145, 151–218 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60, 199–293 (2002)

    MATH  Google Scholar 

  13. Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers, and matrix models, I. math.AG/0101147

  14. Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory, and completed cycles. To appear in Ann. Math.

  15. Okounkov, A., Pandharipande, R.: The equivariant Gromov-Witten theory of P 1. To appear in Ann. Math.

  16. Pandharipande, R.: Hodge integrals and degenerate contributions. Commun. Math. Phys. 208, 489–506 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    A. Okounkov & R. Pandharipande

Authors
  1. A. Okounkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Pandharipande
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Okounkov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Okounkov, A., Pandharipande, R. Virasoro constraints for target curves. Invent. math. 163, 47–108 (2006). https://doi.org/10.1007/s00222-005-0455-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-005-0455-y

Keywords

Search

Navigation