joint with Rahul Pandharipande


We continue our study of the local Gromov-Witten invariants of curves in
Calabi-Yau 3-folds.

We define relative invariants for the local theory which give rise to a
1+1-dimensional TQFT taking values in the ring Q[[t]]. The associated
Frobenius algebra over Q[[t]] is semisimple. Consequently, we obtain a
structure result for the local invariants. As an easy consequence of our
structure formula, we recover the closed formulas for the local invariants
in case either the target genus or the degree equals 1.

We prove there exist degree 2 rigid curves of any genus. Hence, our degree
2 theory agrees with the double cover contributions to the Gromov-Witten
invariants of the ambient 3-folds.