The local Gromov-Witten theory of curves

Bryan, Jim; Pandharipande, Rahul

doi:10.1090/S0894-0347-06-00545-5

The local Gromov-Witten theory of curves

Authors: Jim Bryan and Rahul Pandharipande; \break with an appendix by Jim Bryan; C. Faber; A. Okounkov; Rahul Pandharipande
Journal: J. Amer. Math. Soc. 21 (2008), 101-136
MSC (2000): Primary 14N35
DOI: https://doi.org/10.1090/S0894-0347-06-00545-5
Published electronically: December 6, 2006
MathSciNet review: 2350052
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Abstract: The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evaluation of basic integrals in the local Gromov-Witten theory of $\mathbb{P}^1$ . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a complete and effective solution.

The local Gromov-Witten theory of curves is equivalent to the local Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of $\mathbb{C}^2$ , and the orbifold quantum cohomology of the symmetric product of $\mathbb{C}^2$ . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

1. Mina Aganagic, Hirosi Ooguri, Natalia Saulina, and Cumrun Vafa, Black holes, 𝑞-deformed 2d Yang-Mills, and non-perturbative topological strings, Nuclear Phys. B 715 (2005), no. 1-2, 304–348. MR 2135642, https://doi.org/10.1016/j.nuclphysb.2005.02.035
2. J. Bryan and T. Graber, The crepant resolution conjecture, math.AG/0610129.
3. J. Bryan and R. Pandharipande, On the rigidity of stable maps to Calabi-Yau threefolds, Proceedings of the BIRS workshop on the interaction of finite type and Gromov-Witten invariants, Geometry and Topology Monographs, Vol. 8 (2006), pp. 97-104.
4. Jim Bryan and Rahul Pandharipande, BPS states of curves in Calabi-Yau 3-folds, Geom. Topol. 5 (2001), 287–318. MR 1825668, https://doi.org/10.2140/gt.2001.5.287
5. Jim Bryan and Rahul Pandharipande, Curves in Calabi-Yau threefolds and topological quantum field theory, Duke Math. J. 126 (2005), no. 2, 369–396. MR 2115262, https://doi.org/10.1215/S0012-7094-04-12626-0
6. Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699
7. Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, https://doi.org/10.1007/978-3-0346-0425-3_4
8. C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, https://doi.org/10.1007/s002229900028
9. C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215–252. With an appendix by Don Zagier; Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488, https://doi.org/10.1307/mmj/1030132716
10. Daniel S. Freed and Frank Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no. 3, 435–472. MR 1240583
11. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, https://doi.org/10.1007/s002220050293
12. Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, https://doi.org/10.4007/annals.2003.157.45
13. Eleny-Nicoleta Ionel and Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935–1025. MR 2113018, https://doi.org/10.4007/annals.2004.159.935
14. Joachim Kock, Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, vol. 59, Cambridge University Press, Cambridge, 2004. MR 2037238
15. An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218. MR 1839289, https://doi.org/10.1007/s002220100146
16. Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667
17. Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
18. Eduard Looijenga, On the tautological ring of ℳ_{ℊ}, Invent. Math. 121 (1995), no. 2, 411–419. MR 1346214, https://doi.org/10.1007/BF01884306
19. I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
20. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory I (to appear in Comp. Math.), math.AG/0312059.
21. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory II (to appear in Comp. Math.), math.AG/0406092.
22. A. Okounkov and R. Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004), 675–699. MR 2057777, https://doi.org/10.2140/gt.2004.8.675
23. A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR 2199225, https://doi.org/10.4007/annals.2006.163.517
24. A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, math.AG/0411120.
25. A. Okounkov and R. Pandharipande, Local Donaldson-Thomas theory of curves, math.AG/0512573.
26. R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999), no. 2, 489–506. MR 1729095, https://doi.org/10.1007/s002200050766
27. R. Pandharipande, The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett. Math. Phys. 53 (2000), no. 1, 59–74. MR 1799843, https://doi.org/10.1023/A:1026571018707
28. R. Pandharipande, Three questions in Gromov-Witten theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 503–512. MR 1957060
29. R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on 𝐾3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR 1818182
30. C. Vafa, Two dimensional Yang-Mills, black holes and topological strings, hep-th/0406058.