The local Gromov-Witten theory of curves

TL;DR

This paper provides a comprehensive solution to the local Gromov-Witten theory of curves using localization and degeneration techniques, establishing key equivalences with related theories and enabling further mathematical proofs.

Contribution

It introduces a TQFT formalism for higher genus curves and completes the calculation framework for local Gromov-Witten invariants of curves.

Findings

Exact evaluation of integrals in local Gromov-Witten theory of P^1

Definition of a TQFT formalism for higher genus curves

Establishment of equivalences with Donaldson-Thomas and quantum cohomology theories

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. A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete 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, and the orbifold quantum cohomology the symmetric product of C^2. The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.