Abstract

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. 1 Notation, definitions and results A central problem in Gromov-Witten theory is to determine the structure of the Gromov-Witten invariants. Of special interest is the case where the

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