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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
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.


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Additional Information

Jim Bryan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
Email: jbryan@math.ubc.ca

Rahul Pandharipande
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
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

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00545-5
PII: S 0894-0347(06)00545-5
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.