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Invariance of tautological equations II: Gromov-Witten theory

Author: Y.-P. Lee; with Appendix A by Y. Iwao; Y.-P. Lee
Journal: J. Amer. Math. Soc. 22 (2009), 331-352
MSC (2000): Primary 14N35, 14H10
Published electronically: September 24, 2008
MathSciNet review: 2476776
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Abstract: The aim of Part II is to explore the technique of invariance of tautological equations in the realm of Gromov-Witten theory. The relationship between Gromov-Witten theory and the tautological rings of the moduli of curves is studied from Givental's point of view via deformation theory of semisimple axiomatic Gromov-Witten theory.

References [Enhancements On Off] (What's this?)

  • 1. D. Arcara and Y.-P. Lee, Tautological equations in genus 2 via invariance constraints, Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007), no. 1, 1–27. MR 2294106
  • 2. D. Arcara, Y.-P. Lee, Tautological equation in $ \overline{M}_{3,1}$ via invariance constraints, math.AG/0503184, to appear in Canadian Mathematical Bulletin.
  • 3. D. Arcara, Y.-P. Lee, On independence of generators of the tautological rings, math.AG/0605488, to appear in Compositio Math.
  • 4. Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53. MR 2276766, 10.4007/annals.2007.165.15
  • 5. Robbert Dijkgraaf and Edward Witten, Mean field theory, topological field theory, and multi-matrix models, Nuclear Phys. B 342 (1990), no. 3, 486–522. MR 1072731, 10.1016/0550-3213(90)90324-7
  • 6. C. Faber, S. Shadrin, D. Zvonkine, Tautological relations and the $ r$-spin Witten conjecture, arXiv:math/0612510.
  • 7. C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, 10.1007/s002229900028
  • 8. C. Faber and R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13–49. MR 2120989, 10.4171/JEMS/20
  • 9. E. Getzler, Topological recursion relations in genus 2, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 73–106. MR 1672112
  • 10. Ezra Getzler, The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants, Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 45–89. MR 2115766
  • 11. Alexander B. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), no. 4, 551–568, 645 (English, with English and Russian summaries). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. MR 1901075
  • 12. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. MR 2115767
  • 13. A. Givental, Y.-P. Lee, unpublished.
  • 14. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
  • 15. M. Kontsevich and Yu. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998), no. 2, 385–398. MR 1645019, 10.1007/s002200050426
  • 16. Y.-P. Lee, Invariance of tautological equations. I. Conjectures and applications, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 399–413. MR 2390329, 10.4171/JEMS/115
  • 17. Y.-P. Lee, Witten's conjecture, Virasoro conjecture, and invariance of tautological equations, math.AG/0311100.
  • 18. Y.-P. Lee, Witten’s conjecture and the Virasoro conjecture for genus up to two, Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 31–42. MR 2234883, 10.1090/conm/403/07594
  • 19. Y.-P. Lee, R. Pandharipande, Frobenius manifolds, Gromov-Witten theory, and Virasoro constraints, in preparation. Materials needed for this paper are available from˜rahulp/
  • 20. C. Teleman, The structure of 2D semi-simple field theories, arXiv:0712.0160.
  • 21. R. Vakil, The moduli space of curves and Gromov-Witten theory, math.AG/0602347.

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Additional Information

Y.-P. Lee
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090

Y. Iwao
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090

Keywords: Gromov--Witten theory, moduli of curves
Received by editor(s): May 30, 2006
Published electronically: September 24, 2008
Additional Notes: This research was partially supported by NSF and an AMS Centennial Fellowship
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.