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 P<sup>1</sup>. A TQFT
formalism is defined via degeneration to capture higher genus
curves. Together, the results provide a compete and effective
solution.

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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 in the plane, and the orbifold quantum
cohomology the symmetric product of the plane.  The results of the
paper provide the local Gromov-Witten calculations required for the
proofs of these equivalences.