HOLOMORPHIC ANOMALIES IN TOPOLOGICAL FIELD-THEORIES

We study the stringy genus-one partition function of N = 2 SCFTs. It i s shown how to compute this using an anomaly in decoupling of BRST tri vial states from the partition function. A particular limit of this pa rtition function yields the partition function of topological theory c oupled to topological gravity. As an application we compute the number of holomorphic elliptic curves over certain Calabi-Yau manifolds incl uding the quintic threefold. This may be viewed as the first applicati on of mirror symmetry at the string quantum level.