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
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by Jim Bryan and Rahul Pandharipande; \break with an appendix by Jim Bryan; Rahul Pandharipande

J. Amer. Math. Soc. 21 (2008), 101-136

DOI: https://doi.org/10.1090/S0894-0347-06-00545-5

Published electronically: December 6, 2006

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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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