The local Gromov-Witten theory of curves

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 P 1 \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 C 2 \mathbb C^2 , and the orbifold quantum cohomology of the symmetric product of C 2 \mathbb C^2 . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.