  The local Donaldson-Thomas theory of curves is solved by localization and
degeneration methods. The results complete a triangle of equivalences relating
Gromov-Witten theory, Donaldson-Thomas theory, and the quantum cohomology of
the Hilbert scheme of points of the plane.
  The quantum differential equation of the Hilbert scheme of points of the
plane has a natural interpretation in the local Donaldson-Thomas theory of
curves. The solution determines the 1-legged equivariant vertex.
