A genus-4 topological recursion relation for Gromov-Witten invariants

  • Xin WangEmail author


In this paper, we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton's relations on the moduli space of curves. As an application, we prove that Pixton's relations imply a known topological recursion relation on \(\bar{\mathcal{M}}_{g,1}\) for genus g ≤ 4.


topological recursion relation moduli space of curves Gromov-Witten invariants 


14N35 53D45 


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This work was supported by National Natural Science Foundation of China (Grant No. 11601279) and the Fundamental Research Funds of Shandong University. The author thanks Professor Xiaobo Liu for helpful discussion. The author also thanks Felix Janda for lots of helpful comments on the paper.


  1. 1.
    Belorousski P, Pandharipande R. A descendent relation in genus 2. Ann Sc Norm Super Pisa Cl Sci (5), 2000, 29: 171–191MathSciNetzbMATHGoogle Scholar
  2. 2.
    Getzler E. Intersection theory on M1;4 and elliptic Gromov-Witten invariants. J Amer Math Soc, 1997, 10: 973–998MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Getzler E. Topological recursion relations in genus 2. In: Integrable Systems and Algebraic Geometry. Singapore: World Scientific, 1997, 73–106zbMATHGoogle Scholar
  4. 4.
    Kimura T, Liu X. A genus-3 topological recursion relation. Comm Math Phys, 2006, 262: 645–661MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kimura T, Liu X. Topological recursion relations on M3;2. Sci China Math, 2015, 58: 1909–1922MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li J, Tian G. Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds. In: Topics in Symplectic 4-Manifolds. Cambridge: Internatinal Press, 1998, 47–83zbMATHGoogle Scholar
  7. 7.
    Lin Y J, Zhou J. Topological recursion relations from Pixton relations. Acta Math Sin Engl Ser, 2017, 33: 470–494MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu X. Quantum product on the big phase space and the Virasoro conjecture. Adv Math, 2002, 169: 313–375MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu X. Gromov-Witten invariants and moduli spaces of curves. In: Proceedings of the International Congress of Mathematicians. Zürich: Eur Math Soc, 2006, 791–812zbMATHGoogle Scholar
  10. 10.
    Liu X, Pandharipande R. New topological recursion relations. J Algebraic Geom, 2011, 20: 479–494MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pandharipande R, Pixton A, Zvonkine D. Relations on Mg;n via 3-spin structures. J Amer Math Soc, 2015, 28: 279–309MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pixton A. Conjectural relations in the tautological ring of Mg;n. ArXiv:1207.1918, 2012Google Scholar
  13. 13.
    Ruan Y, Tian G. Higher genus symplectic invariants and sigma models coupled with gravity. Invent Math, 1997, 130: 455–516MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Witten E. Two dimensional gravity and intersection theory on moduli space. Surv Differ Geom, 1991, 1: 243–310MathSciNetCrossRefzbMATHGoogle Scholar

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© Science in China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShandong UniversityJinanChina

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