Curves in Calabi-Yau threefolds and topological quantum field theory

Abstract

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

We define relative invariants for local theory which give rise to a (1+1)-dimensional topological quantum field theory (TQFT) taking values in the ring $\mathbb{Q}[[t]]$. The associated Frobenius algebra over $\mathbb{Q}[[t]]$ 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 the case where either the target genus or the degree equals 1.