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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- 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, DOI https://doi.org/10.1016/j.nuclphysb.2005.02.035 Bryan-Graber J. Bryan and T. Graber, The crepant resolution conjecture, math.AG/0610129. Br-Pa-rigidity 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.
- Jim Bryan and Rahul Pandharipande, BPS states of curves in Calabi-Yau 3-folds, Geom. Topol. 5 (2001), 287–318. MR 1825668, DOI https://doi.org/10.2140/gt.2001.5.287
- 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, DOI https://doi.org/10.1215/S0012-7094-04-12626-0
- Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699
- 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, DOI https://doi.org/10.1007/978-3-0346-0425-3_4
- C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, DOI https://doi.org/10.1007/s002229900028
- 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, DOI https://doi.org/10.1307/mmj/1030132716
- Daniel S. Freed and Frank Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no. 3, 435–472. MR 1240583
- T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, DOI https://doi.org/10.1007/s002220050293
- Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI https://doi.org/10.4007/annals.2003.157.45
- 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, DOI https://doi.org/10.4007/annals.2004.159.935
- Joachim Kock, Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, vol. 59, Cambridge University Press, Cambridge, 2004. MR 2037238
- 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, DOI https://doi.org/10.1007/s002220100146
- Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667
- Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
- Eduard Looijenga, On the tautological ring of ${\scr M}_g$, Invent. Math. 121 (1995), no. 2, 411–419. MR 1346214, DOI https://doi.org/10.1007/BF01884306
- 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 MNOP1 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. MNOP2 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.
- A. Okounkov and R. Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004), 675–699. MR 2057777, DOI https://doi.org/10.2140/gt.2004.8.675
- 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, DOI https://doi.org/10.4007/annals.2006.163.517 Ok-Pan-Hilb A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, math.AG/0411120. dtlc A. Okounkov and R. Pandharipande, Local Donaldson-Thomas theory of curves, math.AG/0512573.
- R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999), no. 2, 489–506. MR 1729095, DOI https://doi.org/10.1007/s002200050766
- 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, DOI https://doi.org/10.1023/A%3A1026571018707
- 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
- 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 Vafa-04-2dYang-Mills C. Vafa, Two dimensional Yang-Mills, black holes and topological strings, hep-th/0406058.
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14N35
Retrieve articles in all journals with MSC (2000): 14N35
Additional Information
Jim Bryan
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
ORCID:
0000-0003-2541-5678
Email:
jbryan@math.ubc.ca
Rahul Pandharipande
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID:
357813
Email:
rahulp@math.princeton.edu
C. Faber
Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Email:
faber@math.jhu.edu
A. Okounkov
Affiliation:
Department of Mathematics, Princeton University, Washington Road Fine Hall, Princeton, NJ 08544
Email:
okounkov@math.princeton.edu
Received by editor(s):
December 5, 2005
Published electronically:
December 6, 2006
Additional Notes:
The first author was partially supported by the NSERC, the Clay Institute, and the Aspen Institute.
The second author was partially supported by the Packard foundation and the NSF
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.